Constructiveempiricismclassnote xMcMullinTheinferencethatmakesscience MaddyIndispensabilityandpractice VanFraassenMcMullinonRetroduction LivioWhyMathWorks 3-01_Maxwell
1. Explain (a) or (b) using between 200 and 300 words. Give as much detail as you can, including the relevant background knowledge. a. The Original Indispensability Argument and its problems b. The Enhanced Indispensability Argument and its problems 2. Explain (a) using between 200 and 350 words. Give as much detail as you can, including the relevant background knowledge. a. Reintroduction 3. Explain (c) using between 200 and 300 words. Give as much detail as you can, including the relevant background knowledge. c. Constructive empiricism 4. 1. Answer (a) or (b) using between 200 and 300 words. Give as much detail as you can, including the relevant background knowledge. a. How do mathematical models represent the world? b. Why is the applicability of mathematics in science important to the mathematical realism debate?
3. Van Fraassen’s constructive empiricism
Van Fraassen’s constructive empiricism is one of the most discussed current alternatives to scientific realism. Contrary to positivists, Van Fraassen does believe that scientific theories must be taken literally. In that sense, he rejects the positivist reinterpretations of scientific statements, according to which talk of unobservablesis only a convenient abbreviation of complicated talk about observables. For the positivists, in principle,everything scientists say about unobservable reality can be expressed, without any kind of loss whatsoever, into statements referring only to observables. But Van Fraassen disagrees. He thinks that scientific statements regarding unobservables are meaningful. If a theory says that ‘electrons are not planets’ then the theory is asserting the existence of both electrons and planets. And if the theory happens to be true, both electrons and planets would exist. But here is the twist in Van Fraassen’s story. According to him, “there is no need to believe good theories to be true, nor to believe ipso facto that the entities they postulate are real” (1065). He defines Constructive Empiricism like this:
“Science aims to give us theories which are empirically adequate; and acceptance of a theory involves a beliefonly that it isempirically adequate” (1065)
A theory is empirically adequate “if what it says about the observable things and eventsin the world is true –exactly if it ‘saves the phenomena’” (1065). Contrast this with what he takes to be the correct definition of Scientific Realism:
“Science aims to give us, in its theories, a literally true story of what the world is like; and acceptance of a scientific theory involves the belief that it is true” (1062)
The first thing to note is that both definitions focus on the aims of science, and not on its actual accomplishments. According to the constructive empiricist picture, science does posit unobservables, but its aim is only to be correct about its observational claims. Against this, the scientific realist picture says that the aim of science is to be actually correct in their claims about unobservables. By focusing merely on the aims of science, Van Fraassen in a way lowers the stakes of this debate. It is no longer about whether or not current scientific theories are correct regarding the unobservable part of the world, but about what is supposed to be the main goal of these theories. Now, whether or not this is a good way of framing the debate is debatable. Most realistsbelieve, I think, that science gets at least part of the unobservable world right. (Van Fraassen’s reply, as we saw in week 6, is that there is no noncircular way to verify the truth of claims about unobservables. But as we saw, the same happens with claims regarding observables!)
Now, something can be observable and yet unobserved. ‘Stones are hard’ aims to be true about both observed stonesand those that haven’t been. That statement, however, is not about ‘unobservables’, in the sense of objects that cannot be observed in principle by any human. ‘Stones are hard’ refers to the observable object stone and the observable property of being hard. It’s not about observations made at one point in time. Accepting a theory means accepting what it says about the observable part of the world, past, present, and future. Observable, on Van Fraassen’s view, is what we get in unaided acts of perception (1067).
“The human organism” says Van Fraassen, “is, from the point of view of physics, a certain kind of measuring apparatus. As such, it has certain inherent limitations… It is these limitations to which the ‘able’ in ‘observable’ refers—our limitations, qua human beings” (1070).
But with respect to the unobservable reality, an accepted theory may as well be false. It really doesn’t matter for the understanding of science. In that sense, constructive empiricism advocates agnosticism regarding the theory’s claims about the unobservable reality.
In terms of our discussion regarding scientific inference, we can say that Van Fraassen accepts ‘horizontal inferences’, or if you want ‘inferences that remain at the surface’, that is, inductive inferences from observable cases to observable cases. But he doesn’t accept ‘vertical inferences’, ‘inferences that go deeper than the surface’, that is, abductive inferences which conclusions refer to the unobservable causes of the phenomena.
Many realists have contested this distinctionas arbitrary. Why believingin induction, despite Hume, but not believing in abduction? There are many things that are unobserved but observable in Van Fraassen’s lights, like the core of the Earth, but it seems that our understanding of it is purely theoretical and also based on abductive inferences. There’s no reason to reject these inferences in those cases, but then if that’s true there wouldn’t be any reason to rejecting these inferences in other cases as well. The laws of nature go well beyond evidence, just like abduction does. Why would we prefer one over the other is not clear, despite Van Fraassen’s argument.
One important pragmatic aspect of Van Fraassen’s view, is that although accepting a theory only means accepting that the theory is empirically adequate, one can still use the whole machinery the theory puts at one’s disposal, which includes the unobservable structures and processes. Van Fraassen is not advocating for a change in the practice of science. His point is rather that if the theory works and accommodates past, present and future empirical evidence, then we don’t need to worry about whether the theory is true. ‘Why is the theory so successful?’ is not an interesting question. Perhaps this theory is successful because all other unsuccessful theories died out and were abandoned (this is a kind of ‘survival of the fittest’ argument). The reason our theories are successful is that we reject those that aren’t. A meta-scientific explanation regarding why they are successful is out of order, on Van Fraassen’s view. This has been contested by many philosophers who think that explaining the success of scienceis one of the most interesting tasks of philosophy of science, and there’s no reason to abandon it.
A final remark with respect to Van Fraassen’s criticism of the overlap argument. We saw that the overlap argument proposes some sort of instrument calibration. For example, we can verify the magnification powers of a microscope by applying it to observable objects first. Once we do that, there is no reason not to trust it when applied to things that would be unobservable to the naked eye, like proteins. Van Fraassen’s reliance on human capacities of observation seems arbitrary, because if we accept what we see through our spectacles, there’s no in-principle reason not to accept what we see through a microscope. Against this, Van Fraassen says that that is debatable, because the term observable is just vague, but there are clear cases of unobservableentities, like so-called unobservables in principle, and the overlap argument does not extend to them. As I mentioned before, it doesn’t seem that the overlap argument itself can deal with this objection, which doesn’t mean that antirealism wins, because as we will see, there are many more arguments defending this view.
McMullin’s Inference: A Case for Realism?
with Bas C. van Fraassen, “Scientific Realism and the Empiricist Challenge: An
Introduction to Ernan McMullin’s Aquinas Lecture”; and Ernan McMullin, “The
Inference that Makes Science”
THE INFERENCE THAT MAKES SCIENCE
by Ernan McMullin
Abstract. In his Aquinas Lecture 1992 at Marquette University,
Ernan McMullin discusses whether there is a pattern of inference that
particularly characterizes the sciences of nature. He pursues this theme
both on a historical and a systematic level. There is a continuity of
concern across the ages that separate the Greek inquiry into nature
from our own vastly more complex scientific enterprise. But there is
also discontinuity, the abandonment of earlier ideals as unworkable.
The natural sciences involve many types of inference; three of these
interlock in a special way to produce “retroductive inference,” the
kind of complex inference that supports causal theory.
Keywords: abduction; Thomas Aquinas; Aristotle; causality;
demonstration; Galileo Galilei; inference; realism; science; theory
Is there a pattern of inference that particularly characterizes the sciences of
nature? Theorists of science, from Aristotle’s day to our own, have on the
whole tended to answer in the affirmative, though views have changed as to
what that pattern is. It has usually been linked, in one way or another, with
explanation. To demonstrate in proper scientific form, Aristotle noted, is
also to explain. The credibility of a theoretical inference, it might be said
today, is proportionate to its explanatory success.
My aim in this essay is to pursue this theme, the nature of the inference
that constitutes a claim as “science,” both on a historical and a systematic
level. As historians, we shall find a continuity of concern, a link across the
ages that separate the Greek inquiry into nature from our own vastly more
complex scientific enterprise. But we shall also discover discontinuity, the
Ernan McMullin (1924–2011) held the John Cardinal O’Hara Chair of Philosophy,
and was director of the program in history and philosophy of science at the University
of Notre Dame, South Bend, IN, USA. The text is reproduced from Ernan McMullin,
The Inference that Makes Science (Milwaukee: Marquette University Press, 1992, 1–112).
Beginnings of pages in the original are indicated with the page number between / and /.
C© Marquette University Press, 1992. Reprinted with permission.
[Zygon, vol. 48, no. 1 (March 2013)]
C© 2013 by the Joint Publication Board of Zygon ISSN 0591-2385 www.zygonjournal.org
143
144 Zygon
abandonment of earlier ideals as unworkable. Indeed, it is arguable that
the failure (in its own terms) of Aristotelian natural philosophy may to
some degree have been linked with its emphasis on demonstration, on a
science of nature that would rest on causal claims held to be evident in their
own right. We shall find /2/ a worry among medieval Aristotelians that
a demonstrative science of nature might be very difficult of achievement
or might even be out of reach. The deep shift that we have come to call
the “Scientific Revolution” can be regarded as in large part an attempt to
construct an alternative to demonstration, a “New Organon,” as its most
influential protagonist dubbed it. We shall discover that the New Organon
was fundamentally ambiguous, that it involved two quite different patterns
of inference. It took more than two centuries before this was finally
recognized. And even in our own century, it was implicitly denied, first by
the logical positivists, and more recently by those who, for whatever reason,
reject scientific realism. Since our canvas is such a large one, we shall have
to be content with broad strokes. Besides Aristotle, there will be a host of
other characters: Grosseteste, Zabarella, Bacon, Whewell, Peirce . . . . And
Aquinas, needless to say, will not be forgotten.
We shall come to see that the natural sciences involve many types of
inference-pattern; three of these interlock in a special way to produce what
we shall call retroductive inference, the kind of complex inference that
supports causal theory. Since theories are primarily designed to explain,
explanatory power obviously plays a major part in their warranting. But
there is a good deal /3/ of disagreement about how this warranting role
may best be understood. We shall, for example, challenge the thesis often
associated with the hypothetico-deductive (H-D) account of scientific
knowledge which would limit the warrant of a theory to the sum of the
verified consequences deductively derivable from it.
It may be worth noting from the beginning that the attempt to define
“the inference that makes science” is not intended to furnish a criterion of
demarcation between science and nonscience. The issue of demarcation has
been actively debated ever since Popper made it central to his philosophy
of science. We shall not address it here. Suffice it to say that retroductive
inference makes use of ingredients that are commonplace in human
reasoning generally. One finds them in any inquiry into causes, in the
work of a detective or a newspaper reporter, for example. What is distinctive
about the way in which explanatory theories are constructed and tested
in natural science is the precision, as well as the explicitness, with which
retroductive inference is deployed. But this alone is not enough to enable
a sharp boundary line to be drawn. There will be large areas where a
clear cut decision will not be possible, where, for example, the questions:
“good science or bad science?” and “science or non-science?” will inevitably
overlap. /4/
Ernan McMullin 145
The sciences of human behavior pose a further, and equally debatable,
question. Is the pattern of inference that constitutes these as “science” the
same as (or, at least, very similar to) that employed by the natural sciences?
Do they explain in more or less the sense in which, say, chemistry explains?
Do they work back from observed effects to underlying structural causes as
chemical theories do? Once again, we shall have to set aside an important
issue in order to focus on the already-large one at hand. Our concern here
is with the natural sciences, and with a single question: in what kind of
complex inference do they (ideally) culminate? Of course, this limitation
would have been foreign to the intentions of Aristotle, from whose seminal
work on the theory of science, the Posterior Analytics, our inquiry takes
its start. His aim was to discover what the ideal of knowledge (episteme)
should be, while warning against seeking a greater degree of precision in
any domain than the nature of the inquiry admits.
PART ONE: DEMONSTRATION ALONE
Aristotle on Demonstration. What makes knowledge “scientific”
(epistemonikos) according to Aristotle is that it should constitute strict
demonstration (apodeixis). And by demonstration /5/ he means an
inference from premisses which are true, primary, immediate, more
knowable (gnorimos) than, and prior to the conclusion, and further that
the premisses furnish an explanation of the conclusion.1 It is not enough
that the inference be a deductively valid syllogism; logical validity does not
suffice to render a piece of reasoning scientific. It is not even enough that
the inference be a valid one from true premisses. The premisses must be of
a quite definite kind, and they must specify in a unique way the cause of
the effect or property of which scientific knowledge is desired.
How are the premisses of the requisite sort to be obtained? Not by
further demonstration, for that would lead to regress. The premisses must
be primary and immediate; that is, they must carry conviction in their own
right once they are properly understood. (The English term “self-evident,”
with its overtone of “obvious,” can be misleading in this context.) But how
is such an understanding to be attained? Aristotle knew perfectly well that
on an answer to this question his entire account of science would stand or
fall. But he is famously laconic in his response.2
Experience (empeiria) is, it appears, crucial to the discovery of the
necessary premisses, the starting points of demonstration in natural science:
/6/
It pertains to experience to provide the principles of any subject. In astronomy, for
example, astronomical experience supplied the principles of the science; it was only
when the phenomena were adequately grasped that the demonstrations proper to
astronomy were discovered. Similarly with any other art or science.3
146 Zygon
Through perception we register particulars, but these particulars
themselves are not objects of scientific knowledge, which is directed
to universals.4 The process leading from the perception of particular
things to the grasp of universals Aristotle calls epagoge, which is often
translated as “induction.” Is there, then, a second sort of inference, a
form of systematic generalization, that provides the starting point for
demonstration? Ought we say that Aristotle proposes not one but two
forms of inference, demonstration and induction, together leading to
science (episteme)? Epagoge is, indeed, sometimes described as though it
proceeded by enumeration, or depended on a systematic comparison of
instances.5
But any resemblance to what Bacon will later call induction is
misleading. In Aristotle’s view, it seems, rather, to be a process of recognizing
the universal in a few particulars, of grasping the phenomena as instances
of a specific universal. It does not depend on sample size.6 There is first the
/7/ ability to perceive (which humans share with animals); the perceptions
persist and constitute memory. And “out of frequently repeated memories
of the same thing develops experience.”7 In this way the universal is, as it
were, “stabilized” in the soul, bringing about a state of mind called nous
(insight, intuition, comprehension). Nous is a direct grasp of the universals
already implicit in perception, and is brought about by epagoge.8 It is
more basic than demonstration; it is, Aristotle assures us, the originative
source of science since it anchors the premisses from which demonstration
begins.9
Underlying this analysis, of course, is Aristotle’s doctrine of the mind’s
ability to receive the form of an object. “The thinking part of the soul
must therefore be, while impassible, capable of receiving the form of
an object; that is, it must be potentially identical in character with its
object without being the object.”10 So that “mind is what it is in virtue of
becoming all things.”11 The veridical character of episteme depends on this
ability of mind to grasp form, as presented in perceived appearance. The
form conveys the essential nature of the thing perceived, and so the basic
premisses of demonstration can be required not only to be true but to be
necessarily true, displaying causal relationships that are “more knowable”
in themselves than the fact to be demonstrated. /8/
Here in brief and familiar outline is how Aristotle proposes that science
should be acquired. There are obviously many difficulties and obscurities
in the account. How, for example, is one to deal with the obvious problem
of sense-error? Aristotle himself points it out: “We must maintain that not
everything which appears is true; firstly, because even if sensation . . . is not
false, still appearance is not the same as sensation.”12 Only “reliable” (aei
kurios) phenomena can serve as a basis for natural science, he reminds his
reader.13 But how is one to know, in an absolutely assured way, which of
the phenomena can be counted as reliable?
Ernan McMullin 147
More fundamentally, what justifies us in supposing that the forms given
us in perception really do convey the essence of the thing perceived? Aristotle
recognizes in passing that there may be a “failure” in perception when we
are unable to perceive the inner structures of a substance on which a
property, like the ability of a burning-glass to set objects on fire, may
depend.14 If we were to be able to see pores in the glass and the light
passing through these pores, then “the reason of the kindling would be
clear to us.” But as it is, such microstructures lie permanently outside the
range of our senses. “Light shines through a lantern because that which
consists of relatively small particles necessarily passes through pores /9/
larger than those particles.”15 Aristotle is clearly aware of the challenge
this sort of explanation poses for his phenomenalist account of the natural
sciences, but he nowhere deals with this directly.16 Instead, he restricts
himself to observed correlations in the examples on which he relies (as
in his celebrated explanation of the lack of incisors in the upper jaws of
horned animals in terms of the nutriment needed for their horns17), or to
simple causal analyses, as in his frequent references to eclipses.
In a significant passage, he draws a distinction between knowledge “of the
fact” (oti, quia) and knowledge “of the reasoned fact” (dioti, propter quid ).
Since he is trying in this passage to explain how demonstration works, the
examples he chooses are of special interest. They are drawn from astronomy,
an odd choice it might seem. Our perceptual knowledge of the heavenly
bodies is obviously very limited; they are, he notes elsewhere:
excellent beyond compare and divine, but less accessible to knowledge. The
evidence that might throw light on them, and on the problems we long to
solve respecting them, is furnished but scantily by sensation. Whereas respecting
perishable plants and animals we have abundant information, living as we do in
their midst. Both domains, /10/ however, have their special charm. The scanty
conceptions to which we can attain of celestial things give us, from their excellence,
more pleasure then all our knowledge of the world in which we live . . . . On the
other hand, in certitude and completeness our knowledge of terrestrial things
has the advantage. Moreover, their greater nearness and affinity to us balances
somewhat the loftier interest of the heavenly things . . . .18
Where the presumptive pores in glass that allow light to pass are
imperceptible to us because of their minute size, the difficulty with the
heavenly bodies is one both of distance and of nature. Not only does
their great distance prevent us from observing their properties in any other
than a perfunctory way, but (in Aristotle’s view, at least) we have reason
to believe that these bodies are fundamentally different in nature to the
bodies of earth by means of which our perceptual expectations have been
molded. So our explorations of the skies must be regarded as conjectural.
Why, then, choose examples drawn from astronomy to illustrate a thesis
about strict demonstration in natural science? Was it just because of his
general fondness for astronomical illustrations (“a half-glimpse of persons
148 Zygon
that we love is more delightful than a more leisurely look at others”19), or
was it /11/ because these examples were in some special way apposite?
The Nontwinkling Planets. The distinction he draws between two
grades of knowledge was intended in part to help overcome the difficulty
of discovering a unique causal explanation when one has to work backward
from perceived effect to less familiar cause. To see this will require a detailed
analysis of the key passage in the Posterior Analytics (I, 13). He gives two
examples of the sort of problem that, despite appearances, lends itself to
demonstration. The most striking property of the planets (other than the
“wandering” motion that gave them their original Greek name) is that they
do not twinkle. Alone among the heavenly bodies they shine with a steady
light. How are we to explain this? How are we to “demonstrate” the property
of nontwinkling they possess? Only by finding the more basic property of
planets responsible for the fact that they do not twinkle. Aristotle proposes
nearness as a plausible candidate. But are the planets nearer than the
other heavenly bodies? A confident assertion follows: “That which does
not twinkle is near: we must take this truth as having been reached by
induction or sense-perception.”20 /12/
This gives him an apparent proof of nearness:
S1 A That which does not twinkle is near
B The planets do not twinkle
Therefore the planets are near
This he calls a demonstration of the fact (oti). It is an improper
demonstration because it is not causally explanatory: nontwinkling does
not explain the nearness. The major premiss is merely an observed
correlation between two properties of shining bodies: if they do not twinkle,
then they are observed to be near. This is sufficient, however, to prove the
truth of the conclusion. And now this conclusion can become the minor
premiss of a new syllogism:
S2 A Nearby (shining) objects do not twinkle
B Planets are near
Therefore planets do not twinkle
This is (Aristotle says) a demonstration of the reasoned fact, a proper
demonstration, because it gives the cause of (or reason for) the fact. The
middle term joining the extremes functions to explain the link between
them: nearness is the reason why planets do not twinkle. What gives
this demonstration force as demonstration for Aristotle is not merely its
syllogistic validity but its explanatory force. /13/
Ernan McMullin 149
But is S2 a proper demonstration? It would appear not, and for two
separate reasons. Neither premiss seems to qualify as the sort of necessary
truth that a demonstration requires as starting point. How would one
establish the necessity of S2A, the claim that nearby shining objects do
not twinkle? It is not enough that it just happens to be true (if indeed it
is true). “True in every instance,” Aristotle himself reminds us, does not
suffice; the attribute (nontwinkling, in this case) must be “commensurately
universal,” that is, it must belong to every instance (of nearby shining
object) essentially.21 It must be shown to “inhere necessarily in the subject.”
Induction-as-generalization will not do; at best, all it can show is factual
correlation of attribute and subject. Epagoge cannot (as Aristotle knows)
reduce to induction, in the sense of generalization.
It is worth noting, indeed emphasizing, that exactly the same issue arose
for the logical positivists when they tried to define the notion of “law” that
was so basic to their account of explanation. It is not enough for an inductive
generalization to be factually true (“everyone in this room is over five feet
tall”) for it to serve as the starting point of a scientific explanation. An
“accidental” universal will not sustain the sort of counterfactual conditional
(“if x had been in this room . . . ”) that is taken to be diagnostic of “genu-
/14/ ine” (what Aristotle would call “essential”) lawlikeness. We shall return
to this later. Suffice for the moment to say that any account of science that
rests (as Aristotle’s does) on attributes given in perception is bound to have
trouble in separating “essential” from accidental linkages, in construing
causality as anything more than invariable correlation.
How is epagoge supposed to lead us to the insight that nearness is
the cause of nontwinkling in the planets? Is some kind of immediate
grasp of the universals, nearness and nontwinkling (in the case of planets),
sufficient? It is clearly not enough for epagoge to bring us to recognize the
two universals in their particular instances; they have also to be seen as
causally (necessarily) related. In On the Heavens, Aristotle does give a hint
as to what the causal relationship might be. Noting that the sun appears to
twinkle at sunrise and sunset, he goes on:
This appearance is due not to the sun itself but to the distance from which we
observe it. The visual ray being excessively prolonged becomes weak and wavering.
The same reason probably accounts for the apparent twinkling of the fixed stars
and the absence of twinkling in the planets. The planets are near, so that the
visual ray reaches them in full vigor, but when it /15/ comes to the fixed stars
it is quivering because of the distance and its excessive extension; and its tremor
produces an appearance of motion in the star.22
Here is a theoretical account of why twinkling occurs, and how it may be
due to distance. It relies on the notion of a “visual ray” that goes out from
the eye, and is attenuated by distance. This is obviously not something
that could be derived directly by epagoge from perception of particulars. It
is a tentative conjecture about an underlying process that might account
150 Zygon
for the twinkling of the distant stars. Its explanatory force comes from its
ampliative character: it does not just associate twinkling with great distance,
but suggests why this association might betoken a causal connection. The
necessity is of a weak hypothetical sort: if there are visual rays and if
visual rays tend to attenuate with distance (more theory needed here), then
the stars will (necessarily) twinkle. What allows one to transcend mere
factual correlation in this case is not nous as direct insight into essence, into
causal relations themselves not given in perception, but plausible theoretical
reconstruction in terms of postulated underlying structures.
In his “official” account of the nature of demonstration in natural science
in the Posterior Analytics, Aristotle nowhere explicitly admits the /16/
mediating role played by theory in the establishing of causal connections.
He leaves the reader to believe that there is a power of mind which can
somehow, subsequent to perception, attain to the essence of natural things
immediately. It is not hard to see why he does this. It is crucial, in his
mind, that the premisses from which science begins be “primary,” that
is, not themselves in need of further evidential support. They must be
definitively true. Unless this be granted, there is no hope of attaining
the “eternal and necessary knowledge” that he holds out as the aim of his
inquiry into nature. But once one admits that either premiss is “theoretical”
in the sense sketched above, one has implicitly given up on this aim. For
theory (e.g., about visual rays) is clearly not primary; it is in need of further
corroboration, of systematic testing. Nor is it definitive; Aristotle himself
allows that his suggestion that visual rays attenuate in vigor the further
they travel is at best only probable.
His attempt to supplement the phenomenalism of his starting point with
an optimistic rationalist account of what epagoge can accomplish, brings
out the main weakness in his account of demonstration. This can be seen
in another way if we shift attention to the minor premiss, S2B. How are
we to know that this premiss is true? Perception alone does not allow us to
claim that the /17/ planets are near. Their nearness is not perceived; it has
to be inferred. How, then, can the minor premiss be regarded as primary?
Aristotle introduces a distinction between something “more knowable in
itself’ and something “more knowable to us.”23 The fact that planets do
not twinkle is more knowable to us; the fact that they are near is more
knowable in itself because it serves as a causal principle. But how do we get
from the former to the latter?
S1 makes use of that which we know (that the planets do not twinkle)
to arrive at a new truth: that the planets are near. The order of exposition
followed by Aristotle suggests that the “demonstration of the fact” provides
the needed minor premiss (S2B) for the demonstration proper displayed
in S2. Does this work? Everything depends on the major premiss S1A:
That which does not twinkle is near. This is where the choice of the
nontwinkling planets turns out to be a brilliant one. For one can plausibly
Ernan McMullin 151
point the causal arrow in either direction. Distance causes twinkling (which
yields S1A, if one allows the negation of the rather vague term “distant” to
be equivalent to “near”). Perhaps this is why Aristotle says so confidently
of S1A: “We must take this truth as having been reached by epagoge or by
perception.”24 (As we have already seen, however, something more than
generalization is required here, something /18/ like the attenuation theory
of On the Heavens.) If one grants that (great) distance “causes” twinkling,
then the (relative) nearness of the planets is established. But Aristotle clearly
takes the causal arrow to operate in the other direction also: nearness causes
(explains) nontwinkling (S2A). This is much more problematic. Distance
is, so far as one can see, not the only possible cause of the twinkle in starlight.
So nearness alone does not demonstratively explain why the planets do not
twinkle. This latter is not a deductive causal relationship, though Aristotle’s
choice of negative characteristics (nondistant, nontwinkling) that can be
thought of in a positive way (nearness, emitting steady light) conceals this.
The causal arrow does not really point in both directions here, though it
would be easy to miss this.
Aristotle was too good a logician not to realize what was going on here.
He recognizes that a condition is necessary in order for his analysis of proper
and improper demonstration to hold. The two attributes (nontwinkling
and nearness in the case of the planets) have to be reciprocally predicable,
or to put this in another way, the major premiss has to be convertible.
Whatever does not twinkle is near, and whatever is near does not twinkle.
This is equivalent to requiring that the cause postulated be the only possible
cause, so that one can infer in either direction. /19/ Now, of course, if this
can be taken for granted, the problem of constructing a demonstration is
greatly eased. There is still the question of how epagoge is to lead us to
the grasp of causal relationships. But at least we do not have the further
problem of dealing with alternative possible causes, each of them sufficient
to explain the effect. The normally hypothetical character of inference from
effect back to efficient cause (what we shall later call retroductive inference)
has been tacitly suppressed.
A glance at Aristotle’s other illustration of the two kinds of demonstration
bears this out:
S3 A Whatever waxes thus is spherical
B The moon waxes thus
Therefore the moon is spherical
This is demonstration of the fact that the moon is spherical, relying on
the knowledge that the moon goes through certain phases in relation to
the sun’s position, and, second, that waxing in this way implies a spherical
shape.
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S4 A Whatever is spherical waxes thus
B The moon is spherical
Therefore the moon waxes thus
This is demonstration proper because the spherical shape is the cause of
the waxing, and the truth of the minor premiss (S4B) is certified in advance
/20/ by S3. Aristotle remarks that a “quick wit” (agkinoia) is needed in order
to grasp that the lunar phases are due to a reflection of sunlight.25 But this
is a case where the perceived phenomenal correlation leads particularly
easily to a postulation of the causal connection. The wit does not have to
be especially quick!
More important, the major premiss once again looks more or less
convertible: spherical shape causes waxing thus, and waxing thus (more
or less) implies a spherical shape. It is true that other possible causes
of the waxing could not have been entirely ruled out in Aristotle’s day,
but it would have seemed overwhelmingly likely that sunlight falling on
a spherical moon was the cause. Geometrical optics lends itself nicely to
necessary-seeming claims and to simple inductive evidence. Thus, not only
do we have a cause but it can plausibly be represented as the only possible
cause. And this is needed for Aristotle’s account of demonstration to work.
Such cases are obviously not typical of the broad range of contexts in natural
science where demonstration/explanation is sought. And even these special
cases do not really enable necessity to be claimed for the premisses. The
transition from “better known to us” to “more knowable in itself ” must
remain tentative except in the most favorable of cases. Talk about the
“pores” that allow light to pass /21/ through glass, of “visual rays” and the
way they are attenuated over distance, of the manner in which nutrients
that would otherwise have been channeled to the incisors of the upper jaw
are diverted to the production of horns in horned animals, cannot be fitted
into the straitjacket of demonstration. Strict demonstration in cases such
as the astronomical ones above will work only when a causal relationship
between A and B can be “seen” to hold with necessity, and when B also
requires A, i.e., when A can be “seen” to be the only possible cause of B.
The Living World. This latter condition would be particularly
difficult to satisfy in the domain of living things, the domain to which
Aristotle devoted such an extraordinary effort. It has often been noted that
in his voluminous writings on animals there are few, if any, instances of the
demonstrative form laid down in the Posterior Analytics. This has, indeed,
furnished a major topic of scholarly research in recent years.26 In his review
of more than five hundred animal species, he lists for each species numerous
properties that could serve as differentiae. Scholars have tried hard to extract
a natural classification from this profusion, but it is clear that none is there:
Further, it seems doubtful that one was intended, since the divisions given
Ernan McMullin 153
are frequently criss-cross, as Aristotle himself notes.27 He does suggest pos-
/22/ sible causal connections between properties: the lungs temper the heat
of the body in warm-blooded animals;28 the kidneys carry more fat than
do other internal organs because the kidneys require a greater supply of
heat, being closer to the surface and having much “concoction” to perform,
and fat is a cause of heat.29
From our point of view, these would seem to be no more than causal
speculations, prompted by some fairly casual correlations and some very
general theories about the role of such causal agencies as heat and fat
in the animal body. How could they ever be made demonstrative? What
would the “primary premisses” in the study of living things look like?30
Functional explanations of the sort Aristotle relied on offer a particular
challenge in that regard: “The function of the lungs is to cool the blood
and hence the body.” Even if one could show that respiration is needed
to cool the body and that it does cool the body, how would one show
that it does not also have another function? This is the equivalent of the
problem about the convertibility of the major premiss in the case of the
astronomical examples earlier. What he needs, according to his theory of
demonstration, is definitions in which the essences of the various natural
kinds are expressed. But if these definitions are complex, containing a
list of differentiae (as he appears to envisage), and if /23/ many of the
terms used are equivocal, “said in many ways,” as he admits they are, how
could one ever trace a unique causal line with necessity from the genus or
from one or a cluster of the differentiae to the property to be explained?
The case to which later Aristotelians would always return was that of
man, with a conveniently simple single differentia. But this, as Aristotle
himself recognized, was altogether untypical of the inquiry into animal
natures generally. “If demonstration still remained an ideal in zoology, as
in mathematics, it was an ideal that had to recede the more Aristotle’s
zoological researches progressed.”31 To what extent was Aristotle himself
aware of the limited applicability of his teaching on demonstration?
Introducing an ingenious but highly speculative account of comets,
shooting stars, and the “fuel” that sustains them, he says: “We consider
a satisfactory explanation of phenomena inaccessible to observation to
have been given when our account of them is free of impossibilities.”32
A weak criterion, indeed, in comparison with the exacting demands of
demonstration! In On the Heavens, he frequently laments how far short of
demonstration his account falls: “When anyone shall succeed in finding
proofs of greater precision, gratitude will be due to him for the discovery,
but at present we must be content with what /24/ seems to be the case.”33
He obviously thought that proofs “of greater precision” were, in principle,
constructible; the significance to his doctrine of epagoge of “phenomena
inaccessible to observation” had not, to all appearances, sunk in. Our
later story will be devoted to the reweaving of these threads that he
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for the first time separated off. The patterns of inference/explanation
in contemporary science are strikingly different from those proposed in
the Posterior Analytics. But the quest for a causal explanation of natural
phenomena that should be as epistemically secure as possible still remains.
PART TWO: ENLARGING DEMONSTRATION
It would take us too far afield to comment on the medieval discussions of
the nature of demonstration in the detail they deserve. Our main goal in
this essay is to analyze the three primary patterns that have been proposed
over the years for explanation/ inference in natural science. We have already
seen one of these, and must rather summarily move on to the other two.
But it is worth asking first about the extent to which the transition to
other inference-patterns was already under way among medieval writers
on the nature of science. To what extent did medieval commentators on
the Posterior Analytics show an awareness of the difficulties involved in
finding “commensu- /25/ rately universal” relationships, the only kind
yielding demonstration proper? Did they, for example, relax the demand
for premisses that would be seen to be not only true but necessarily true,
once properly understood? Did they take steps to systematize the process
involved in epagoge in order to ensure that the regularities discovered
in nature would be genuinely universal ones? We shall see that on the
whole the requirements of strict demonstration were not relaxed, but that
significant efforts were made to grapple with the problems that these
requirements imposed. We shall also see that some contemporary attempts
to construe medieval enlargements of the doctrine of demonstration as
actively preparing the way for modern accounts of scientific method
are, though laudable in the generosity of their intention, considerably
overstated.
Looking at the Middle Ages as a whole, one would obviously have to
separate two quite diverse methodological traditions, the Aristotelian and
the nominalist. And one would have to bear in mind the great diversity
within both of those traditions themselves. The Aristotelians remained
faithful on the whole, to the ideal of demonstration se down in the
Posterior Analytics, while developing some aspects of that doctrine, the
distinction between demonstrations propter quid and quia, for example,
much more fully than Aristotle had /26/ done. The nominalists began to
shape the notion of inductive generalization, entirely rejecting the notion
of necessary connection between essence and property on which the older
notion of demonstration had been based. The idea of causal explanation
as tentative and “consequential,” resting in large part, that is, on the
verified observational consequences drawn from it, only sporadically made
its appearance and, among Aristotelians, never as any other than a stage on
the way to demonstration proper. In lieu of a detailed historical treatment,
Ernan McMullin 155
we shall pick out for brief comment three of the leading representatives of
the tradition of the Posterior Analytics, Grosseteste, Aquinas, and Zabarella,
each of whom has a special interest for our story.
Grosseteste. Forty years ago, Alistair Crombie in a widely discussed
book declared that:
As a result of their attempts to answer the Greek question: How is it possible
to reach with the greatest possible certainty true premisses for demonstrated
knowledge of the world of experience? the (sic) natural philosophers of Latin
Christendom in the thirteenth and fourteenth centuries created the experimental
science characteristic of modern times.34 /27/
This was the “continuity thesis,” announced earlier by the histo-
rian/philosopher of science. Pierre Duhem, now proposed in its strongest
possible form. More specifically, Crombie claimed that the appearance of
Robert Grosseteste’s commentary on the Posterior Analytics (c. 1225), which
made that difficult work accessible for the first time to Latin readers,35 in
conjunction with Grosseteste’s own works on optics and other scientific
topics, initiated a new approach to natural science. Grosseteste’s distinctive
contribution was “to emphasize the importance of falsification in the search
for true causes and to develop the method of verification and falsification
into a systematic method of experimental procedure.”36
Later historians of the period have, on the whole, been unsympathetic
to this reading of Grosseteste. In a recent study of that philosopher’s work,
James McEvoy writes:
If a broadly adequate methodology be regarded as a necessary and quasi-sufficient
basis for scientific advance, then it becomes at once essential and embarrassingly
difficult to account for the relative scientific sterility of the late medieval and early
Renaissance period, if one grants – and this is the force of Crombie’s main thesis –
that the requisite methodology was already available from around 1240 onwards.37
/28/
To assess the force of this objection to Crombie’s thesis, one would
have to distinguish between methodology, understood in terms of practical
prescriptions regarding working procedure, and conception of science, taken
to specify the sort of knowledge-claim that counts as science proper.
Crombie was, of course, claiming originality for Grosseteste and his
successors on both scores. It is only the second, his conception of science
that concerns us here. Being right about one’ conception of science does not
necessarily translate into practical success in one’s scientific work. Likewise,
failure in the latter regard does not necessarily connote failure in the
former, so we have to look at the matter more carefully. Did Grosseteste
transform the Aristotelian notion of demonstration into something akin
to the modern idea of experiment-based theoretical inference?
156 Zygon
To begin with, he certainly did stress the importance of experience as
the basis for scientific knowledge, even more perhaps than Aristotle had
done.38 His motive for this, it should be noted, was in large part theological.
Because of sin, man’s higher powers were corrupted but the humbler power
of perception, by standing firm in the rout, enabled the higher powers to
recover their proper function of apprehending the essences of things.39 In
practice, the features of the Posterior Analytics that he chooses to dwell on are
often the empir- /29/ ical examples that Aristotle uses to such good effect.
For example, when discussing the nontwinkling of the planets, he produces
his own explanation for the twinkling of starlight: the greater the distance
of any object from us, he suggests, the smaller the angle it subtends at the
eye and thus the greater the strain on vision. This strain and the tremor it
induces in the virtus visiva that goes out from the eye to the object causes the
appearance of twinkling.40 When he discusses, at considerable length, the
connections between lack of incisors and possession of multiple stomachs
in horned animals, he simply brings together what Aristotle and some of
his later commentators, like Themistius, have to say in different contexts
about this topic.41 There is no new observation involved, and Grosseteste
almost certainly had not himself seen all of the animals mentioned. He
does, however, draw attention to a consequence that Aristotle had left
unstated: though all horned animals lack upper incisors, not all those that
lack upper incisors possess horns (camels and hinds, for example). Thus
the properties are not mutually implicative, or to put this in the traditional
way, the major premiss in any attempted demonstration of a causal relation
between -the properties would not be convertible. In cases like that of the
camel, Grosseteste adds (again following Aristotle), the animals have other
means of defense so that horns /30/ are not needed. (This leaves the camel’s
lack of upper incisors unexplained; Aristotle and Grosseteste try to make
do with a teleological explanation for this by noting that the camel’s hard
palate substitutes for the lack of incisors.)
Had Grosseteste been of a mind to challenge or revise Aristotle’s account
of demonstration, this would have been an obvious context in which to do
so. Another context would have been where he comments on a particularly
difficult passage (II, 16 and 17) in the Posterior Analytics, and asks whether
the same effect might not be known from experience to be explainable in
terms of several different possible causes.42 Again, he forbears. The most
illuminating contexts are those in which he discusses different “opinions”
regarding the explanation of specific phenomena. Here one does find him
suggesting that the “opinions” may be tested by seeing whether inferences
drawn from them conflict with observation or general principle. Crombie
draws attention to two such examples, in particular.
In one, Grosseteste reviews four “opinions” regarding the nature of
comets and rejects all four of them on the grounds that their consequences
are at odds with observation, as well as with “the principles of the special
Ernan McMullin 157
sciences.”43 He advances a fifth view on his own account: a comet is a fire
of a transformed nonterrestrial sort linked to its /31/ present star or planet
by an attraction akin to that between magnet and iron. Though the theory
seems fanciful from our perspective, it is governed by the insight that
comets share in the revolutions of the heavenly bodies and hence cannot
be of a simply terrestrial nature. In another passage, he gives an example to
illustrate how the universal is reached by the mind through “the ministry of
the senses.” The taking of scammony (a purgative widely used in the ancient
Greek world) is commonly accompanied by the discharge of red bile.44 Is
the former the cause of the latter? We cannot see such a causal relation
directly. But “frequent observation” of the two events as co-occurring leads
us to suspect (estimare) a third factor which is itself not observable, that
is, a causal relation between the two. The power of reason (ratio) then
comes into play and suggests, by way of test, that scammony should be
administered when all other (known) causes of red bile have been excluded
from operating. If there is still a discharge of red bile in such cases, “a
universal is formed in the reason” affirming the causal connection between
the two sorts of occurrence, thus giving rise to a “universal experiential
principle.”
Though Crombie may have been overenthusiastic in his original
formulation of the continuity thesis, he was surely right to take passages
such as these to presage a manner of reasoning in /32/ science that differs
fundamentally from demonstration. But the extent of the change is by no
means clear, as the last example shows. Scientific inquiry still terminates
in the reason’s directly grasping a universal causal connection between
the administration of scammony and the discharge of red bile. Does this
claim to understanding continue to rest on the adequacy of the inductive
procedures followed, the exclusion of other relevant causal factors and the
testing of the invariability of the alleged correlation? In this case, how are
we to know that the connection between the two kinds of occurrence is
of an essential nature? Or, on the other hand, are we to suppose that the
grasp of the causal connection becomes at this point a direct intuitive one,
given that one has now an adequate familiarity with the concepts involved?
These are the alternatives that faced Grosseteste and his successors in one
guise or another. He did not himself, so far as one can discover from the
texts, resolve in favor of what we may call modernity. But the fact that the
issue itself can be so clearly documented in his work and in that of other
later natural philosophers, is already a major contribution in its own right.
Aquinas. Thomas Aquinas was assuredly one of the most perceptive
contributors to discussions of the nature of science in this period.
His interests in the issue were, of course, primarily theo- /33/ logical
and not physical. Unlike his mentor, Albertus Magnus, be devoted
little of his abundant energies to empirical inquiries into the natural
158 Zygon
world. Nonetheless, he was well-versed in the natural science of his day,
particularly in astronomy.45 His own most considerable “natural” work
was his commentary on the Physics of Aristotle. The Physics, it should be
remembered, is for the most part devoted to careful conceptual analysis.
It begins from the ways in which certain very general terms are used in
common speech and it systematically analyzes the consequences of these
legomena.46 It ranks itself, therefore, G.E.L. Owen remarks, “not with
physics in our sense of the word, but with philosophy. Its data are for
the most part the materials not of natural history but of dialectic, and its
problems are accordingly not questions of empirical fact but conceptual
puzzles.”47 In terms of the kind of evidence it relies on, it belongs rather
more to the tradition of Parmenides than to that of the “physicists,” as
Aristotle calls them, despite its frequent references to the latter. Physics I ,
for example, seeks the necessary and sufficient conditions for the correct
application of the term, “change,” and rests, not on a series of observations
of different sorts of natural change, but upon the simple ability to use
the term, “change,” correctly, an ability dependent /34/ on experience, of
course, but only in the most general way.
Analysis of this kind falls naturally into a deductive pattern once the
dialectical preparations are made. Aristotle sets out to prove his claims: that
an actual infinite cannot exist, that motion cannot have had a beginning,
and so on. But there are no demonstrations in the strict sense, since
he is not dealing here with essences or natural kinds. The problems we
have seen regarding twinkling stars and horned animals do not arise in
“philosophy of nature,” as this genre later came to be called.” (An analogue
of these problems may appear when causes are postulated, as for example
in the case of continuing “violent” motion.) The most general terms in
which we articulate our descriptions of the world around us (“motion,”
“place,” “time”) are assumed to refer in a straightforward way, and are given
precising definitions, if necessary. No need, then, to test hypothesis against
specific observations, or the like. As long as one stays with “philosophy of
nature” in this sense, the problems we have been discussing in regard to
demonstration in natural science do not appear. But they are apt to emerge,
in different guises elsewhere, as indeed they do for Aquinas.
There is a tension in his thinking, it has often been suggested, in regard to
our knowledge of sensible things.48 On the one hand, he is emphatic /35/
in claiming that knowledge of the quiddities of natural things constitutes
the appropriate object of the intellect: “The proper object of the human
intellect is the quiddity of a material thing, [when that thing has been]
apprehended by the senses and the imagination.”49 On the other hand,
he also maintains that the essences of physical things are hidden from us:
“Our power of knowing (cognitio) is so weak that no philosopher can ever
fully discover the nature of a single fly.”50 “The essences of things are not
known to us . . . .”51 Commenting on the text of Aquinas, Jacques Maritain
Ernan McMullin 159
expresses the contrast in dramatic fashion: Is it not scandalous that though
our intelligence has as its connatural object
the essences of corporeal things, in face of them it meets such serious impediments
that it has to be content, in a vast sector of our knowledge of nature, with the
imperfect intellection we call “perinoetic?”52
The notion of an imperfect (“perinoetic”) understanding is prompted by
texts like these: “In sensible things, the essential differences are unknown to
us, so they are signified by accidental differences which originate from the
essential ones, as a cause is signified by its effect.”53 “Substantial forms are
of themselves unknown to us; we learn about them from their proper acci-
/36/ dents.”54 “Since we do not know essential differences, sometimes . . .
we use accidents or effects in their place, and name a thing accordingly.”55
“Since substantial differences are unknown to us, or at least unnamed by
us, it is sometimes necessary to use accidental differences in their place . . .
for proper accidents are the effects of substantial forms and make them
known to us.”56
Aquinas was obviously far less optimistic than was Aristotle about the
ability of the human intellect to extract a knowledge of essence from the
regularities noted in perception. He recognizes that the features of sensible
bodies that are accessible to human powers of perception are not, in general,
part of essence; the forms abstracted by the mind in consequence of such
perception are not substantial forms. We may say that fire is a simple,
hot and dry body, but this is not to specify essence directly; rather it
is to designate in terms of phenomenal qualities which are the effect of
essence.57 Can one infer from these effects back to their cause, essence?
How do the proper accidents make the substantial forms known? Which of
the two strands in his thinking are we to emphasize here, the pessimistic one
(the essences of natural things are hidden from us) or the more optimistic
strand (we can discover essences through the clues afforded by the proper
accidents)? Or ought we combine these two: /37/ essences of physical
things are indeed initially hidden (i.e., we do not have immediate access to
them in the intellectual processes involved in perception) but they can be
progressively discovered by means of indirect modes of inference?
In support of this last suggestion:
The human intellect must of necessity understand by composition and division.
For since the intellect passes from potentiality to act, it has a likeness to generable
things, which do not attain to perfection all at once but acquire it by degrees.
In the same way, the human intellect does not acquire perfect knowledge of a
thing by the first apprehension; but it first apprehends something of the thing,
such as its quiddity, which is the first and proper object of the intellect; and
then it understands the properties, accidents, and various dispositions affecting
the essence. Thus it necessarily relates one thing with another by composition
160 Zygon
and division; and from one composition and division it necessarily proceeds to
another, and this is reasoning .58
But how exactly does this reasoning work? “Composition” is a linking
of attributes, “division” a separating. How is one to discover the /38/
necessary linkage he mentions? Are we back to postulating an ability of
mind simply to see the necessity of a causal linkage once the attributes
themselves are fully understood, the kind of ability that Aristotle had
ultimately to invoke in epagoge? Aquinas gives us very little to go on here,
and one reason is not far to seek. There were few, if any, plausible candidates
for inference from accidents to essence in the natural science of his day.
(Discussions of the nature of the rainbow might have afforded the best
clue.) In one text, he suggests that the senses must be the arbiter in such a
process:
Sometimes the properties and accidents of a thing revealed by the sense adequately
manifest its nature, and then the intellect’s judgement of the thing’s nature must
conform to what the sense reveals about it. All natural things, limited to sensible
matter, are of this sort. So the terminus of knowledge in natural science must be
in the sense, so that we judge of natural things as the sense reveals them, as is clear
in the De Caelo.59
What are the consequences of this initiative for the doctrine of
demonstration? In his Commentary on the Posterior Analytics, Aquinas
makes much of the distinction between demonstration /39/ propter quid
and demonstration quia, to which only a single passage in the Posterior
Analytics itself was devoted. A major reason for this shift in emphasis was
undoubtedly the importance of the distinction for Aquinas’ theology: we
can arrive at demonstrative knowledge of God’s existence, he argues, but the
demonstration can never be better than quia since we lack the knowledge
of essence, the appropriate definition, that a demonstration propter quid
would require as middle term:
From every effect the existence of its proper cause can be demonstrated [by a
demonstration quia] so long as its effects are better known to us . . . . Hence the
existence of God, since it is not self-evident to us, can be demonstrated from
those of His effects which are known to us . . . . When the existence of a cause is
demonstrated from an effect, this effect takes the place of the definition of the
cause in proving the cause’s existence. This is especially the case in regard to God,
because in order to prove the existence of anything, it is necessary to accept as a
middle term the meaning of the name, and not its essence, for the question of its
essence follows on the question of its existence.60 /40/
The problem here, of course, as with any demonstration quia, would
be (as we have seen) to prove convertibility, that is, to show that God
is the only possible explanation of the effects in question, and to show
this assertion itself to be necessary, to be something more than plausible
hypothesis.
Ernan McMullin 161
Aside from the elaboration of the quia-propter quid distinction, however,
the discussion of demonstration in Aquinas’ Commentary stays remarkably
close to the original. The examples of the nontwinkling planets and the
phases of the moon are presented without comment. First principles are
said to be arrived at by means of induction, understood as an abstraction
of universals from sensible particulars; many times experienced.61 There is
no hint as to how exactly the assertion of causal relationship is to be arrived
at. If we have to infer from perceived phenomenal qualities back to the
unperceived essence that is their cause, how is the universal corresponding
to this essence to be formed? Epagoge may work to relate perceivable
accidents, but how does it link those in turn with the substantial forms
Aquinas holds to be causally responsible for them?62 If the essences of
sensible things are hidden, as Aquinas says they are, is not the role of
strict demonstration in natural science so circumscribed as to be almost
nonexistent? /41/
It is puzzling, on the face of it, that someone who had so clearly realized
the seriousness of the barriers facing inquiry into natural essence would
not have allowed some hint of this challenge to appear in his exposition
of the canonical doctrine of demonstration in the Posterior Analytics. In
a recent Aquinas Lecture, Alasdair MacIntyre argued that demonstration
for Aristotle and Aquinas, as an “achieved and perfected knowledge,” is
an ideal constituting the goal of inquiry, one rarely perhaps reached.63
Rational justification can thus take two quite different forms:
Within the demonstrations of a perfected science, afforded by finally adequate
formulations of first principles, justification proceeds by way of showing of any
judgement either that it itself states such a first principle or that it is deducible
from [one] . . . . But when we are engaged in an inquiry which has not yet achieved
this perfected end state, that is, in the activities of almost every, perhaps every,
science with which we are in fact acquainted, rational justification is of another
kind.64
This second kind belongs first and foremost to dialectics. Principles
will be formulated provisionally; “apodictic theses” will be tested against
“empirical phenomena,” and reformu- /42/ lated, if necessary, in the light
of such tests.65 Even the very telos of the inquiry itself, the conception
of the sort of science that ought be aimed for, is open to modification in
the light of results achieved along the way. Progress will thus “often be
tortuous, uneven, move inquiry in more than one direction, and result
in periods of regress and frustration. The outcome may even be large-
scale defeat . . . .”66 “What had been taken to be a set of necessary apodictic
judgements, functioning as first principles, may always turn out to be false.”
Hence:
No one could ever finally know whether the telos/finis of some particular natural
science had been achieved or not. For it might well appear that all the conditions for
162 Zygon
the achievement of a finally-perfected science concerning some particular subject-
matter had indeed been satisfied, and yet the fact that further investigation may
always lead to the revision or rejection of what had previously been taken to be
adequate formulations of first principles suggests that we could never be fully
entitled to make this assertion.67
Such principles are “necessary,” then, only in the weak sense that
they are stages on the way to a true, but quite possibly unreachable,
judgment /43/ that “presents to us actually how things are and cannot
but be.”68
What are we to make of this account? It affords a perceptive description
of what might fairly be said to be the “received view” in contemporary
philosophy of science of the status of theory in natural science. But
can it claim a warrant in the text of the Posterior Analytics or of
Aquinas’ commentary on that work? MacIntyre allows that the Aristotelian-
Thomistic tradition has to be supplemented here by the insights of such
contemporaries as C. S. Peirce and Karl Popper in order to arrive at
this highly fallibilist conception of science, one which construes “first
principles” as tentative hypotheses open to continuing modification in
the light of new observational evidence. But surely this is rather radical
“supplementation?” What entitles us to call the resultant view “Aristotelian-
Thomistic?” The transformation that has come about in the conception
of natural science in modern times has been largely due to developments
in empirical inquiry itself, to an internal dynamic working within the
history of science; it is to be judged in the first place, then, by reference
to the history of science. MacIntyre remarks that it is in the spirit of the
Aristotelian-Thomist tradition to test a conception of inquiry against the
actual history of that form of inquiry. This may be so, but the conception
of inquiry that /44/ emerges from this testing may well diverge from the
Aristotelian tradition sufficiently to make the claim that it is a natural
extension of the doctrine of the Posterior Analytics a rather forced one.
There are two sticking points in the way of such a continuity thesis. The
warrant for an Aristotelian demonstration lies ultimately in the recognition
on the part of the intellect that the premisses, properly understood, are “self-
evident,” i.e., carry their own internal warrant. MacIntyre himself seems
to say as much. Argument to first principles, he notes:
cannot be a matter of dialectic and nothing more, since the strongest conclusions of
dialectic remain a matter only of belief, not of knowledge. What more is involved?
The answer is an act of the understanding which begins from but goes beyond
what dialectic and induction provide, in formulating a judgement as to what is
necessarily the case in respect of whatever is informed by some essence . . . . Insight,
not inference, is involved here . . . .69
This catches the rationalist aspect of epagoge, to be sure. But then
he imposes a significant qualification: the judgment of the intellect in
regard to essence still has to contend with “constraints” /45/ imposed
Ernan McMullin 163
by dialectical and inductive considerations, and the insight it affords
requires “further indication,” namely, a check as to whether the proposed
premisses/principles do, in fact, provide a “causal explanation of the known
empirical facts.”
Insight into “what is necessarily the case” is therefore, apparently
not of itself sufficient to warrant the first principle, the premiss of the
demonstration. Quite complex-sounding forms of inference, continuing
tests of the proposed principle against the empirical facts, are also needed.
It is difficult, however, to find a justification for this restriction in the text
of the Posterior Analytics itself. The claims made there for the nous that is
consequent upon epagoge show no such hesitation, as we have seen. But
perhaps it is to the text of Aquinas (though not, it would seem, to his
commentary on the Posterior Analytics) that we ought to be looking. One
relevant passage:
The ultimate end which the investigation of reason ought to reach is the
understanding of principles, in which we resolve our judgements. And when
this takes place, it is not called a rational procedure or proof but a demonstration.
Sometimes, however, the investigation of reason cannot arrive at the ultimate end,
but stops in the investigation itself, that is when two possible solutions /46/ still
remain open to the investigator. And this happens when we proceed by means
of probable arguments, which are suited to produce opinion or belief, but not
science. In this sense, rational method is contradistinguished to demonstrative
method, and we can proceed rationally in all the sciences in this way, preparing
the way for necessary proofs by probable arguments.70
This is much more promising. But it raises new questions. These
“probable arguments” can produce only “opinion.” How does the
transition to a first principle actually come about, then? How do the
probable arguments prepare the way for demonstration proper? Does a
demonstration arrived at in this way rest in any respect on the probable
arguments? It would seem not, for if it did, it would remain provisional.
And its warrant would no longer be self-evidence. But if it does not rest
on them, why were they needed? There is an unresolved difficulty here,
one that will surface again much later in Descartes’ Discourse on Method .
The divide between science and opinion, legacy of Greek ways of thinking,
represents a sharp dichotomy, not a continuum. MacIntyre suggests, as we
have seen, that demonstrative science is to be regarded as an ideal that lies
at the horizon of inquiry. This has the merit of deflecting the troublesome
problem of how the transition can /47/ ever be made from opinion to
science: in practice, it is never made or at least we can never know that it
has been. What the scientists of today would call “science” would, therefore,
have to be labeled “opinion,” or at least something other than science. And
the traditional Aristotelian claim that the human intellect can, on the basis
of sense-experience, directly grasp the relation between a particular kind of
164 Zygon
effect and its proper cause in a definitive way, would have to be either set
aside or at the very least forcefully reinterpreted.
There is a further sticking-point in the way of those who seek to establish
a strong continuity between the Aristotelian and the contemporary
conceptions of science. The phenomenalist cast of Aristotle’s account of
epagoge has already been noted. It imposes a severe restriction on the
concepts available for deployment in a demonstration. To demonstrate
in natural science is to discover a causal relationship between perceptible
features of sensible bodies, features that are regularly found together.
Aquinas comments:
If universals, from which demonstration proceeds, could be grasped apart from
induction, it would follow that someone could acquire scientia of things which
he could not sense. But it is impossible for universals to be grasped apart from
induction.71 /48/
Universals can come to be known only through induction (epagoge).
Or in the idiom of abstraction, the concepts in terms of which a science
of sensible bodies is to be constructed can only originate in perception:
the form can come to exist in the mind only if it be abstracted from
sensible instances of that form. An oft-quoted scholastic maxim made this
restriction quite explicit: nihil est in intellectu quod non prius fuerit in sensu.
But now let us return one further time to the nontwinkling planets.
As we have seen, Aristotle himself attempted a theoretical explanation of
the proposed causal link between distance and twinkling in terms of the
attenuation of visual rays emitted from the eye of the observer. These visual
rays are clearly not themselves observable, nor is their attenuation. And their
attenuation may itself require the introduction of further theoretical entities
to explain it. Explanations of this sort are dotted throughout Aristotle’s
work, particularly the Meteorology. The all-important spheres on which the
circular motions of the celestial bodies are said to depend would furnish
the most striking example.
When Aristotle is faced with the need to show that the link between
two kinds of feature is a genuinely causal one, he quite often in practice
postulates an underlying structure or process not itself observable, instead
of just relying on the /49/ assertion that the link can be intuitively seen to
be a necessary one. The necessity is thus mediated by the postulated entity;
the causal link is held to be necessary in virtue of this entity. But where
is the requisite universal to come from? There are universals for A and B
(twinkling and distance) but not, it would seem, for C (the visual ray). C has
to be somehow constructed in imagination, relying on elements drawn from
perception in other contexts, no doubt. But the nature of C is constructed
mentally in response to the request for explanation; it is not the result of
abstraction. The test of this constructed concept lies in its ability to explain,
not in its being properly abstracted from sensible particulars. The shift from
Ernan McMullin 165
abstraction to construction means that the resultant form of inference
cannot be demonstration, unless the requirements for demonstration be
greatly weakened, not to say transformed.
Neither Aristotle nor Aquinas addresses this problem. Aquinas’ detailed
account of the abstraction characteristic of natural science (the first degree
of abstraction from matter) gives no hint that something other than
straightforward abstraction may be needed at the crucial moment in
constructing causal explanations.72 Nevertheless, if one looks at what he
has to say, not about natural science, but about “divine science,” a possible
response suggests itself. We can come to /50/ know things that transcend
sense and abstraction-based imagination, he says, by beginning from the
level of sense and imagination and then arguing from this level as from an
effect to a cause which surpasses it.73 We cannot be said to know the nature
of such a cause (answering to quid est?), only that it exists (answering to an
est?). Still, in order to know that a thing is, something must be known of
what it is. Thus, in order to know that God and other immaterial beings
exist, we have to be able to postulate something, at least, of what they
are or, at least, of what they are not.74 When the cause so transcends the
effect as God does the physical universe, “we take the effect only as the
starting point to prove the existence of the cause and some of its conditions
[e.g., the power to create], although the quiddity of the cause is always
unknown.”75
The barrier here to knowability is difference of nature, and ultimately
transcendence of nature. And the response is to construct an incomplete
and imperfect definition of a cause that would be sufficient and (more
problematically) necessary to account for significant general features of the
world of sense. This is, in striking ways, similar to the sort of construction
that a retroduction from effect to cause in natural science might also
require. The visual ray does not transcend the sensible order as God does,
but it does differ enough /51/ from it in regard to its accessibility to human
modes of perception that a not-entirely-dissimilar constructive form of
inference has to be employed to reach it. One could say, then, that the
sort of retroduction that Aquinas employs to enable him to affirm the
existence of God and the angels might have suggested how to proceed
in natural science when causes inaccessible to sense appear to be required.
Admittedly, Aquinas specifically excludes this parallel, insisting that natural
science and divine science differ precisely on this point.76 Nevertheless, a
way out has been, if not opened, then at least indicated, and the tight
requirements of demonstration propter quid have been found incapable of
satisfaction in least one domain of science, and a weaker alternative has
been allowed.
It would be wrong, of course, to suggest that this relaxation is what
actually led to the later acceptance of theoretical and nondemonstrative
forms of inference in natural science. The change took a very long time,
166 Zygon
and the main inspiration for the change came from progress in the natural
sciences themselves. It gradually came to be realized that the causal agencies
underlying explanation in the natural sciences, if not as remote from
abstractive terms anchored in perception as Aquinas had declared God to
be, still required a /52/ new and nonabstractive approach to the concepts
required to define them.
Galileo and the Paduan Tradition. One other philosopher should
be mentioned in that context, Jacopo Zabarella, with whom yet another
continuity thesis has been linked. There is space here only to summarize
the argument; doing it justice would require a full-length study. J. H.
Randall suggested in a wide-ranging work, The School of Padua and the
Emergence of Modem Science (1961), that Galileo’s notions of scientific
method were heavily dependent on the traditions of the school where he
spent much of his teaching career, Padua, and particularly on the logical
works of Zabarella:
The logic and methodology taken over and expressed by Galileo and destined to
become the scientific method of the seventeenth century physicists . . . was even
more clearly the result of a fruitful critical reconstruction of the Aristotelian
theory of science, undertaken at Padua in particular . . . . [In] its completed
statement in the logical controversies of Zabarella . . . it reaches the form familiar
in Galileo . . . .77 /53/
The claim gave rise to a lively controversy. Two objections, in particular,
were raised. One was that the connections between Galileo and Zabarella
had not been clearly enough established, that although there were some
resonances between the logical terms used by Galileo here and there in his
scientific works, there was no real evidence of influence and no obvious
medium for it. The second objection was that Randall had conflated logic
and methodology. It was one thing to say that Galileo’s logic (more exactly,
his conception of science, his view of what kind of knowledge-claim scienza
makes) had some affinities with that of the Paduan tradition. But it was
quite another to claim that his methodology, the methodology that laid
such a distinctive stamp on the natural science of those who followed him,
also derived from Padua. This seemed far less plausible; indeed, it found
(and would still find) few defenders. Galileo gradually evolved a complex
methodology involving controlled experiment, repeated measurement,
mathematical idealization, and much more, which was strongly opposed
by his Paduan Aristotelian colleagues and certainly finds few resonances
in their tradition. Since it was his methodology and, of course, his actual
discoveries in mechanics and elsewhere, and not merely his concept of
science, that shaped what /54/ came after, the Randall thesis was generally
thought to fail, or at least to be greatly overstated.
Ernan McMullin 167
In the last ten years, it has been restated and strengthened by William
Wallace. Zabarella still retains his role, but the thesis has been broadened.
The “canons” of Galileo’s new science, and hence of science in the Galilean
tradition, Wallace suggests, “were those of Aristotle’s Posterior Analytics
read with the eyes of Aquinas, and appropriated by him from the Jesuits of
the Collegia Romano.”78 Wallace has shown, to most people’s satisfaction,
that two short commentaries on logical topics, dismissed by the editor
of the National Edition of Galileo’s works as juvenile school-pieces, were
written by Galileo in his mid-twenties when he was just beginning his
career as a teacher of mathematics and natural philosophy.79 Further, by
a remarkably painstaking piece of detective work, Wallace has also shown
that these two pieces are almost entirely derivative from lecture-notes
composed by some contemporary Jesuit teachers of natural philosophy
at the Collegia Romano, notably the notes of a certain Paolo Valla. And
Valla drew heavily on Zabarella as well as on the Thomist tradition. What
this establishes is that Galileo was cognizant of the Jesuit, and indirectly of
the Paduan, tradition of commentary on the Posterior Analytics, notably on
the topic of demonstration, to which one of the /55/ two sets of his notes
is devoted. Whether (as Wallace supposes) the notes represented Galileo’s
own views, then or later, is a quite different matter. That Galileo should
have carefully summarized and paraphrased portions of the lecture notes
of senior colleagues, who were teaching the courses he himself might be
called on to teach, does not give strong reason to describe these notes as
conveying his own “logical doctrine.”
Several features of this doctrine are said to mark the scientific work of
Galileo’s maturity. On this the continuity claim rests. First and foremost is
the alleged appearance of the regressus form of argument characteristic of
the Paduan Aristotelian tradition, and especially of Zabarella’s logic. The
regressus was the combination of the quia and propter quid inferences we
have already seen, the first establishing the existence and something of the
nature of the purported cause and the second demonstrating the effect
from this cause. Zabarella and his colleagues had developed an elaborate
analysis of this back-and-forward mode of explanation. Jardine describes
Zabarella’s theory as one of “vast and obsessive complexity”80 Crucial to
this analysis was the occurrence of a period of consideration or reflection
(negotiatio or investigatio) between the back-to-cause and forward-to-effect
phases. The first of these phases yields an indistinct (confusa) notion of
the cause. /56/ The intellect then wrestles with this notion, somehow
clarifies it, and finally comprehends the cause sufficiently to allow the
propter quid demonstration to be completed with assurance. In particular,
the intellect is said to have the ability to “see” that the crucial convertibility
condition holds (equivalently ruling out the possibility of alternative
causes).
168 Zygon
Though this intermediate phase cannot simply reduce to the epagoge
of the Posterior Analytics (which is required before the first phase can
even get under way), the attribution to the intellect of the ability to
discern the nature of the required cause is reminiscent of the traditional
doctrine of nous. There can be no doubt that the intention of the
Paduan Aristotelians was to show how the regressus could provide strictly
demonstrative explanations. It would be tempting to take the intermediate
phase as a forerunner of later hypothetical modes of explanation, but
Wallace and Jardine are surely correct in excluding this reading.81 Hence,
if Galileo is to be seen as a proponent of the regressus notion of proof, he
has also to be construed as a defender of Aristotelian demonstration as the
appropriate mode of proof in natural science.
Jardine, to the contrary, argues that far from promoting regressus, Galileo
was actively critical of its use as a model of proof: /57/
Galileo was well aware of the contemporary Aristotelian theory of scientific
demonstration, had a sure insight into its weaknesses, rejected it outright, and
set up in its place as a crucial part of his propaganda for the union of mathematics
and natural philosophy a method of inquiry modelled on a classical account of
the quest for proofs in geometry.82
The first thing to say here is that in the works of his scientific maturity
Galileo never alludes to the method of regressus one way or the other, either
to affirm it or reject it. True, he frequently uses the term “demonstration,”
but it carries with it the connotation of “convincing proof, no more.”
And he links it quite often with mathematics phrases like “the rigor of
geometrical demonstration,” “the purest mathematical demonstration”
support Jardine’s argument that Galileo’s notion of demonstration is
associated by him much more explicitly with geometry than with
the syllogism.83 To the extent that the properties of necessity and of
convertibility appear, it is because of the mathematical form in which his
arguments in mechanics are conveyed. Since he sets aside causal explanation
in terms of gravity, and confines himself to kinematical measures of space
and time only, the issue central to regressus (arguing from effect to efficient
cause) simply does not arise in /58/ his mechanics. Furthermore, he returns
again and again to the Platonic-Archimedean theme of “impediments,”
the various obstacles that arise when one tries to apply an idealized
mathematical system to the complexity of the material world.84 Such a
system applies only approximately, and approximation is something that
has no place in the classic conception of demonstration propter quid . A
necessary truth about nature cannot be just approximately true.
Galileo’s law of falling bodies affords a clear illustration. It is the
Aristotelian in the dialogues, Simplicio, who keeps objecting that the
necessity one finds in purely mathematical inference cannot readily be
transferred to claims about the material world. As long as Galileo’s account
Ernan McMullin 169
of uniformly accelerated motion be taken simply as a mathematical
definition, the issue of demonstration does not arise. It is the claim that
this is, in fact, the sort of motion that occurs were a body to fall in vacuo
at the earth’s surface that raises the problem. Galileo gives two sorts of
arguments in support of his claim. One is that uniform acceleration is
the simplest mathematical form that this motion could take and hence is
the one that Nature would employ. The other is that the assumption that
in vacuo fall is, in fact, uniformly accelerated is supported by the “very
powerful reason” that it “corresponds to that which physical experiments
show /59/ to the senses.”85 The first is reminiscent of strict demonstration:
Galileo is asking us to see that motion must take place in this way. (But, of
course, we know from Newton’s vantage-point that the law is not exact. The
acceleration of in vacuo fall gradually increases. Nature does not always act
in the simplest way.) Galileo relies also on a second line of argument, which
is that consequences drawn from the assumption of uniformly accelerated
motion can be experimentally verified. But, of course, this is no longer
a demonstrative form of argument: it rests on the extent to which, and
the precision with which, the consequences of the supposition have been
observationally verified. There is no suggestion that falling motion must
necessarily follow this law.
Wallace responds to this objection:
Galileo’s way of presenting and justifying this definition [of the motion of fall in
vacuo] has elicited criticism from some, who see him as employing a hypothetico-
deductive method such as characterizes modern scientific investigations, and
thus as falling into the fallacy of affirmatio consequentis when using the implied
consequences of his definition to support it as the antecedent. It is true that the
definition can be regarded as a suppositio, and therefore that the demonstrations to
follow are made /60/ ex suppositione . . . . From a formal point of view, moreover,
a suppositio has the character of an ipotesi [hypothesis], and thus its value might
be judged by its ability to save the appearances, regardless of whether or not it
describes a situation that is actually verified in the order of nature. It is for this
reason that Galileo repeatedly makes the distinction between supposizioni that are
true and absolute in nature, and those that are false and made purely for the sake
of computation . . . . Galileo’s principles, the definition of naturally accelerated
motion included, must stand or fall on their own merits, and not merely on the
basis of one or two consequences drawn from them.86
This long quotation is instructive. It helps us understand why Wallace is
so averse to allowing Galileo’s science to be called H-D and why he labors
so hard to drape it in the mantle of the Aristotelian-Thomist tradition.
Unless Galilean mechanics can be construed as demonstrative, it is reduced
to simply “saving the appearances,” thus separating it from what “is actually
verified in the order of nature.” There are echoes here of the distrust among
Aristotelian natural philosophers of the late Middle Ages for the epicycles
of the Ptolemaic astronomers, and the philosophers’ way of dismissing
as “fictive” /61/ (the alternative to “demonstrative” that Wallace often
170 Zygon
employs) anything which simply rests on “saving the appearances.” But
Galileo’s constant appeal to “the very powerful reason” that one of his
suppositions is verified by the consequences drawn from it is appealing to
the “appearances,” i.e., the experimental results. Those who see Galileo
as employing hypothetical forms of inference on occasion do not (as
Wallace suggests) take this to imply that this makes his reasoning fallacious.
They would reject the over-simple dichotomy between demonstrative and
fictive, and maintain that a hypothetical argument, one sup- ported by
the consequences drawn from it, can have any degree of likelihood up to
practical (not, however, absolute) certainty. Insofar as Galileo’s argument
for his law of fall did carry force, it was because of the fact that it did so
successfully “save the appearances,” i.e., fit the phenomena of experiment.
If there was fallacy, it might be said that it was in Galileo’s appeal to the
simplicity of Nature, in his vain attempt to present the appearance of strict
demonstration.
Galileo’s telescopic discoveries opened up for discussion a host of
questions about the natures of the beings now coming into view: sunspots,
comets, and the like. There were surprises too in regard to the lunar surface,
the variable illumination of Venus, and four small points of light that
/62/ seemed to accompany Jupiter. Galileo’s method in dealing with these
phenomena was to postulate a cause which might explain the observed
phenomena, and try to find as much evidence as possible in support of his
hypothesis. There was nothing particularly mysterious about this method;
it is, when all is said and done, little more than common sense. Would
it have been inhibited or favored by an appeal to the regressus tradition?
One could, perhaps, argue either way. But the negative side is surely the
stronger. The regressus tradition did insist on convertibility, on finding a
cause to which one could infer with necessity from the effect to be explained,
and on an interval of intellectual reflection on the nature of the proposed
cause. The Paduans had assuredly never taken negotiatio to designate a
period when alternatives are systematically explored, anomalies dissolved,
and further positive evidence accumulated.
In some favorable cases, like the inference that mountains are the cause
of certain variable shadows on the lunar surface, Galileo could infer
almost directly from effect to cause, that is, exclude alternative possible
explanations with a high degree of assurance. This does not, however,
make his proof demonstrative in the Aristotelian sense. What made his
supposition amount to certainty in his own mind was that it accounted so
neatly for the appearances. But it could not be /63/ deduced from them;
several (highly plausible) assumptions had to be made which in turn would
require support from the consequences drawn from them. Commenting,
Wallace allows that such assumptions are in fact present and remarks that
until Galileo could be assured of the truth of such auxiliary assumptions,
all he had was “opinion” and not “science.” (Once again, this is too sharp
Ernan McMullin 171
a dichotomy; instead, there is a spectrum of likelihood ascending as far as
practical certainty.87) Wallace goes on to note that Galileo was aware of an
objection to his claim about mountains on the moon. If there are lunar
mountains, how can the edge of the lunar disk be seen as quite smooth in
the telescope? He attempted to dissolve the objection, but it was not until
1664 that telescopes were sufficiently powerful to show that the moon’s
outline is in fact slightly irregular. Does this mean that until 1664, all
that astronomers had was opinion in regard to the lunar mountains? And
that it became science with the 1664 observation? And, in any event, does
not the dependence of the case for lunar mountains on this observational
consequence show how artificial it is to force Galileo’s argument into
the mold of Aristotelian demonstration, one purporting to yield “certain
knowledge based on true causes?”88 /64/
Nothing has been said about Galileo’s use of the phrase “ex suppositione,”
another supposed link to earlier logical doctrines and specifically to the
Thomist tradition of commentary on the Posterior Analytics.89 Nor has an
adequate distinction been drawn between what Galileo believed himself
to be doing and what, from our perspective, he was actually doing. When
tracing his links with earlier logical traditions, it is the former that is the
more important; in assessing his influence on his successors in regard to
this issue, it might be the latter that one would stress. Galileo was quick,
often too quick in our estimate, to claim certainty for his conclusions.
And in mechanics, at least, he sought principles which would, as far as
possible, carry conviction in their own right. On the issue that meant most
to him, that of the double motion of the earth, he sought for proof in such
supposed consequences as the ebb and flow of tides, and the curved paths
of the sunspots. The more acute kinematical arguments for the earth’s
motions, showing how much more “natural” it is to attribute motion to
the earth than to attach various sorts of motions to the heavenly bodies,
he called “plausible reasons,” and remarked: “I do not pretend to draw a
necessary proof from them, merely a greater probability.”90 /65/
In short, Galileo aimed when he could at demonstration, in the sense
of conclusive proof. But when this was not available, he would settle for
as high a degree of probability as the evidence would warrant, showing
no inclination to regard the resultant merely as “opinion.” He used
consequential modes of argument all the time, but never formulated a
“method of hypothesis” and would probably have been reluctant to regard
such a method as “the” method of science. All in all, then, even though
Galileo was fond of using the term “demonstration,” there is little to warrant
the claim that he was influenced in a significant way by the elaborations
of the notion of demonstration in the older tradition of the Posterior
Analytics.
As one looks at later seventeenth-century natural science, one imme-
diately notices a significant difference between mechanics and the other
172 Zygon
sciences. In the mechanics of Descartes as of Newton, there was an emphasis
on demonstration, on certainty, a suspicion of hypothesis. Perhaps this
might be seen as an echo of the Aristotelian requirements for episteme,
though in a mathematicized context remote from that of the Posterior
Analytics. On the other hand, in other parts of natural philosophy, in
optics, in chemistry, there was a growing realization that hypothesis is not
only unavoidable, but even respectable, and efforts were made to formulate
criteria in terms of /66/ which it should be judged. Both of these strands
are already found in different parts of Galileo’s work; there is thus no single
“Galilean” heritage in that regard.91
PART THREE: INDUCTIVISM
In deference to the occasion [the Aquinas Lecture], we have spent so long
on assaying the stability of some of the bridges that recent scholars have
thrown across the gap between the tradition of the Posterior Analytics and
modern views on what constitutes the basic form of scientific inference
that we are going to have to telescope these later views in a rather summary
way. The earlier emphasis on demonstration was not entirely lost; indeed,
it took daring form in Kant’s Metaphysical Foundations of Natural Science.
But two other kinds of inference more or less supplanted demonstration as
the paradigm of the scientist. (Demonstration remained a will-o’-the-wisp
for those who took mechanics as the paradigm of science; it is all too easy
to see its conceptual structure, whether in its Newtonian or more recent
relativistic forms, as so luminous as also to be necessary.) One of these types
of inference has a familiar label: “induction.” The other (which we shall
call “retroduction”) even still has not, which is rather extraordinary. /67/
Nominalism. “Modernity” began, as everyone knows, with the via
moderna of the fourteenth century, which was “modern” in part because of
its rejection of the notion of demonstration central to the Aristotelian
tradition it opposed. The notion of necessity that the possibility of
demonstration in natural science conveyed seemed to a great many
theologians, from the first introduction of the Aristotelian “natural”
works in the early thirteenth century onward, to require an unacceptable
restriction on God’s freedom in creating, and an equally unacceptable
determinism of causal action on the part of creatures. The fourth of the
219 propositions condemned by Bishop Tempier in 1277 rejects the claim
“that one should not hold anything unless it is self-evident or can be
manifested from self-evident principles.”92 The idea that the human mind
can, on the grounds of reflection, on what is perceived, affirm a necessary
relation between cause and effect was regarded as a challenge to the notion
of miracle so central to the Christian economy.
Ernan McMullin 173
The nominalism of Ockham, and especially the more extreme versions of
that nominalism in the works of Nicholas d’Autrecourt, Jean de Mirecourt
and Robert Holkot, presented an alternative to the Aristotelian scheme in
which the emphasis has shifted from the universal to the particular, from
demonstration to induction. /68/ Induction itself in one sense resembles the
epagoge of the earlier tradition because it begins from observed regularities
of co-occurrence in the sensible world. But it differs in a crucial way: instead
of the intellect’s going on to grasp the nature of the cause sufficiently clearly
to allow an unqualified affirmation of necessary connection between cause
and particular effect to be made, induction according to Ockham rests
simply on the evidence of the co-occurrences—and has the degree (and
only the degree) of logical force that this conveys. It is further dependent
on a principle of uniformity of nature which itself has to be regarded as
hypothetical, since it rests on the ordination of God’s will to a common
course of nature, which is not absolute but, in principle at least, open to
exception. One thing is said to be the efficient cause of another if in the
presence of the first the second follows, nothing more. A causal relation
between A and B cannot, therefore, be known a priori; it can be learnt
only from the repeated experience of their conjunction. Induction is thus
a matter of generalization from a limited set of instances of a regularity. It
is, if you will, a kind of sampling.
Nicholas goes on to draw a more skeptical conclusion than Ockham had
done. Since the existence of effects cannot strictly entail the existence of
corresponding causes, the best that one can /69/ aspire to in natural science
is a degree of probability. The attempt to infer to essence or substance from
perceived particulars must necessarily beg the question. In turning away
so decisively from the ideal of demonstration, Nicholas is not especially
advocating the importance in human terms of empirical investigation:
what little can be learned (he says) can be learned in a short while, but
it must be learned from things, not from the works of Aristotle.93 There
has been a great deal of discussion among historians of later medieval
thought in late years about the influence of nominalist ideas on the origins
of modern science. Our concern here is with the notion of induction only.
The nominalists advocated the substitution of induction (in the sense of
generalization) for demonstration as the paradigm mode of inference in
natural science, challenging the fundamental notion of nature on which
the earlier account had rested. But they did not work this up into a formal
account of method in natural science.
Bacon. That was left to Francis Bacon, and this brings us to the
second significant moment in the long story of inductivism. In the New
Organon (1620), Bacon proposed a new method which was to replace
174 Zygon
that of the Posterior Analytics. He saw the two methods as diametrically
opposed: /70/
There can be only two ways of searching into and discovering truth. The one
flies from the senses and particulars to the most general axioms, and from these
principles, the truth of which it takes for settled and immovable, proceeds to
judgement and to the discovery of middle axioms. And this way is now in fashion.
The other derives axioms from the senses and particulars, rising by a gradual and
unbroken ascent, so that it arrives at the most general axioms last of all. This is
the true way, but as yet untried.94
Bacon tries to find a middle position between the essentialism of Aristotle
and the more extreme forms of nominalism:
Though in nature nothing really exists besides individual bodies, performing pure
individual acts according to a fixed law, yet in philosophy this very law, and
the investigation, discovery, and explanation of it, is the foundation as well of
knowledge as of operation. And it is this law with its clauses that I mean when I
speak of forms, a name which I the rather adopt because it has grown into use and
become familiar.95 /71/
There is a shift here from forms, understood as intrinsic to natural things,
to forms understood as laws, as modes of action extrinsically imposed by
a Lawgiver. These latter can still be called “eternal and immutable,” and
hence the natural philosopher can still aim at certainty. But the mode of
attaining it is quite different.
Bacon sets out to construct natural histories organized by tables of
presence, absence and degree (from which J. S. Mill much later got his
methods of Sameness, Difference, and Concomitant Variation). These
tables link regularly co-occurring factors; this is what for Bacon defines
induction. The evidence for causal relationship comes from finding factors
either invariably linked in observation or co-varying in a significant way.
Evidence against is provided by absence, when presence might have been
expected. The method is thus one of generalization, with an element of
testing provided by the tables of absence.96 Such a method “leaves but little
to acuteness and strength of wits, but places all wits and under- standings
nearly on a level.”97 (There is some disagreement as to how seriously
he meant this.) And it provides “not pretty and probable conjectures,
but certain and demonstrable knowledge.”98 Again this would need to
be qualified. For one thing, he stresses the “dullness, incompetency and
deceptions of the senses.”99 For /72/ another, he treats his “axioms” as
plausible conjectures meant to be tested by the “trial by fire” that crucial
experiment can afford. Still, he does assume that at the end of that
sometimes laborious process, a causal link that is “sure and indissoluble”
can be found.100
Though Bacon is trying very hard to separate himself from the
Aristotelian tradition, one can still catch echoes of epagoge here. In
Ernan McMullin 175
particular, he is seeking to discover regular and reliable correlations; his
“laws,” though their ontological basis is quite different, loosely correspond
to the “natures” of the older tradition. The process of discovery, an “act
of the intellect” left unexplained by Aristotle, is not spelled out in terms
of the logical procedures to be followed. Bacon provides, then, not just a
conception of what counts as science, but a general methodology to enable
that goal to be attained.
If this were all, then Bacon could be presented as the inductivist par
excellence, his aphorisms constituting the “Ur” text for those concerned to
lay out the method of induction. But as we shall see in more detail later, a
second quite different sort of inference is also hinted at in the pages of the
New Organon. It is doubtful that Bacon was aware of the tension between
the two methods or of the incompleteness of induction without the other
mode of inference to back it up. /73/ Induction is a matter of noting
correlations between observables; unless both elements related by the “law”
are observable, a correlation between them obviously cannot be discovered,
on the basis of sense-evidence alone. Even if one were to extend the notion
of observation (and Bacon was surprisingly wary of such an extension,
recommending against a dependence on the new-fangled instruments just
then coming into use), it would still be true that inductive method is strictly
limited to factors that are observable in some sense. How, then, is the story
to be extended to unobservables? Bacon in his famous discussion of the
nature of heat in Book II of the New Organon showed himself perfectly
willing to assert that the “heat” of a body is to be understood in terms of the
motions of imperceptibly small parts of the body. We shall, however, leave
this question aside for the moment in order to complete this quick survey
of significant moments in the development of ideas about induction.
Hume. Hume’s contribution was of a different sort, closer in spirit
to that of Nicholas d’Autrecourt (whose name he had almost certainly
never heard of ) than of Bacon. Though in the introduction to the youthful
Treatise of Human Nature (1739), he had promised “a complete system of
the sciences, built on a foundation almost entirely new,” in practice, he
left the natural sci- /74/ ences to Newton and his heirs, content to take
Newton at his word that the method of these sciences is inductive:
And although the arguments from experiments and observations by induction be
no demonstration of general conclusions, yet it is the best way of arguing which
the nature of things admits of, and may be looked on as so much the Stronger, by
how much the induction is more general.101
Hume’s interest was not specifically in induction as it occurs in natural
science. His challenge was to inductive procedure generally, whether in
the routines of daily life or in the more technical pursuits of the natural
philosopher. “Reasoning concerning matters of fact” (he did not use the
176 Zygon
term, “induction”) is, according to him, founded on the relation of cause
and effect, which in turn reduces to a combination of constant conjunction,
contiguity and temporal succession. The idea of necessary connection,
which we also associate with the causal relationship, can be explained
as an expectation brought about by association or habit. But such an
expectation in no way justifies the prediction that C, which has been
constantly conjoined with E in the past, will once again be followed by
E on the next occasion. This skeptical undermining of the rationality of
belief /75/ in the logical force of inductive inference offers no threat to daily
living or to the natural sciences, according to Hume, but only because it is
powerless to overcome the natural sentiments and convictions that govern
daily life.
This is the famous “problem of induction” which went almost unnoticed
among Hume’s first readers but which has so intrigued twentieth-century
analytic philosophers.102 If one accepts Hume’s starting points, that all
our ideas are derived from sense-impressions or inner feelings, and that
causal relationship reduces to nothing more than the fact of constant
conjunction in the past of certain classes of sense-impressions lacking any
intrinsic connection, then indeed it would be difficult to warrant belief
in induction. (One can never, of course, conclusively prove that on a
given occasion E must follow C; Hume’s use of the phrase “demonstrative
reasoning” is part of the problem here.) But then, of course, these same
starting points would not enable us to distinguish between genuine laws
and accidental correlations. And causal inference to underlying structure
is excluded. Hume’s radical empiricism could take account neither of law
nor of theory, as these had come to be understood in the natural science of
the previous century. /76/
If the only form of nondeductive inference were to be the inductive
one, as empiricists have always tended to believe, then Hume’s problem
would still pose a troublesome challenge. It might perhaps be too easy
to say that the famous “problem” is an artifact, that it vanishes if this
faulty assumption be rejected. But, as we shall see, if the right form
of nondeductive inference be recognized, genuine laws and accidental
generalizations can be readily distinguished and the connection between
cause and effect becomes something more than constant conjunction.
J. S. Mill. The fourth moment in the tale of induction can be kept
a brief one. Mill’s System of Logic (1843) may be called the “Inductivist
Manifesto.” There is only one type of nondeductive inference, and that is
induction, understood as a straightforward procedure of generalization or
sampling: “It consists in inferring from some individual instances in which
a phenomenon is observed to occur, that it occurs in all instances of a certain
class.”103 This method, properly used, enables the “ultimate laws of nature”
to be discovered. Its validity rests upon a general principle of uniformity
Ernan McMullin 177
of nature. (His attempt to rest this principle in turn upon a broader
induction is clearly fallacious.) Since causal relations hold only between
observables, there can be no inference to unobservables. It need hardly be
said that at the /77/ very time Mill was writing, the growing reliance of
natural scientists on noninductive inference to and from unobservables was
leading to dramatic advances in fields like optics, chemistry, and theory of
gases.
Logical Positivism. The best-remembered moment in the story
occurred in our own century. A brief reminder of the salient points must
suffice. The logical positivists used the term, “induction,” to cover all forms
of nondeductive inference. Carnap attempted, unsuccessfully, to construct
a theory of logical probability, what he called an “inductive logic,” that
would link hypothesis and evidence by means of some sort of “credence
function.” Such a logic, he believed, would provide the rules for inductive
thinking, thus enabling rational choices to be made between hypotheses.
He rejected the traditional view of inductive reasoning that would make
it an inference from premisses (evidence) to conclusion (usually a law),
proposing instead that it is an assessment of the credibility of a particular
hypothesis in the light of specific evidence, however the hypothesis be
arrived at.104 The hypothesis itself might be a law, a singular prediction,
or a theory. And the logic, though a logic of induction (in Carnap’s sense
of the term), was deductive in form, enabling him (he hoped) to evade
Hume’s challenge. /78/
Much more commonly, however, the focus of positivist concern was on
how to get from the singular observation statements from which science
begins to the laws of which they believed “finished” science to consist.
Induction in this case would involve something like the traditional methods
of Sameness, Difference, and Concomitant Variation, that Mill had taken
over from the works of Bacon and John Herschel.105 It would essentially
be a special kind of sampling. The “laws” arrived at in this way would,
thus, be empirical generalizations. But not all laws are of this sort. Besides
empirical laws, there are also “theoretical” laws, those that make use of
“theoretical” terms, i.e., terms that refer to hypothetical (unobservable)
entities.106 But how are these “laws” to be arrived at? Not by means of
inductive generalization clearly. Carnap saw the difficulty:
How can theoretical laws be discovered? We cannot say: “Let’s just collect more
and more data, then generalize beyond the empirical laws until we reach theoretical
ones.” No theoretical law was ever found that way . . . . We never reach a point
at which we observe a molecule . . . . For this reason, no amount of generalization
from observations will ever produce a theory of molecular processes. Such a theory
must /79/ arise in another way. It is stated not as a generalization of facts but as a
hypothesis. The hypothesis is then tested in a manner analogous in certain ways
to the testing of an empirical law.107
178 Zygon
“Analogous,” perhaps, but exhibiting important differences. Carnap
was struggling toward a sharper distinction between theory and law, and
between the processes of inference involved in each of these. But it was
difficult to admit another mode of inference. An empiricist could never
feel entirely easy with theoretical terms. And the “logical empiricists,” as
the group preferred to be called in its later years, went to great lengths to
contrive devices like “correspondence rules” to get around the hard fact
that theoretical laws could simply not be derived from empirical laws,
indeed that the term “law” here is close to equivocal.108 Their ambivalence
toward a distinctively theoretical mode of inference led, in turn, to a
famous ambivalence in regard to the existence of theoretical entities. Their
instinct as positivists and empiricists was to regard theoretical terms simply
as heuristic devices. But a growing appreciation of the difficulties to which
this led would eventually encourage some of them, at least, to embrace a
somewhat hesitant realism.109 /80/
So far, explanation has not been mentioned in this discussion of
logical positivism. The well-known deductive-nomological (D-N) model
proposed by Hempel and Paul Oppenheim took laws to be the primary
explainers, and individual events to be the normal explananda. The
apparent symmetry between explanation and prediction to which this
led gave rise to problems severe enough to force the abandonment of the
model. Explanation could simply not be reduced to subsumption under
laws. Indeed, it appeared, laws are what have to be explained, rather than
the primary explainers. The gas laws do not explain the behavior of gases.
A theory of gases is needed, one that postulates an underlying structure of
entities, relations, processes.110
So here, once again, something other than the product of induction,
empirical laws, is needed if an adequate account is to be given of how
explanation functions in science. And there is one further context where
overreliance on induction also led to insuperable problems, as already
noted, and that was in finding an effective way to distinguish between
genuine laws and accidental generalizations. If all one has is empirical
generalization, à la Hume, this crucial distinction hovers out of reach.
/81/
PART FOUR: AND FINALLY TO RETRODUCTION
And so, at last, we arrive, by circuitous ways, at the account of inference
toward which all of this has been tending. By now, its outlines should be
moderately clear.
Beginnings. Let us return for a moment (only a moment!) to the
seventeenth century. We left Bacon an inductivist but with a hint that
this would not quite do. In Book II of the New Organon, he lays out a
Ernan McMullin 179
case-study in order to illustrate his new method. What is the nature of
heat? He eventually concludes that it reduces to motion. But in the case
of ordinary bodies, the motion of what? Bacon postulates minute particles
whose constrained motion is responsible for the impression of heat when
we feel a hot iron. But surely no inductive process of generalization could
arrive at such a conclusion? Bacon’s alchemical background leads him to
emphasize the importance of “latent process” that “escapes the sense”;
it is on this that the observed properties of things ultimately depend.
An understanding of the “latent configurations” of “things infinitely
small” is needed.111 He never explicitly recognizes that the induction-
by-generalization he has proposed in the opening aphorisms of the New
Organon will not suffice to reach such configurations. He does, however,
sketch a method of testing hypotheses, /82/ laying the groundwork for a
very different conception of science, one where hypothesis takes an honored
place, and the old ideal of demonstration is finally laid aside. Bacon himself
did not, however envision this denouement. For him, science still connoted
certainty, though he must have suspected that the configurations of “things
infinitesimally small” would not readily yield such a result.
The story of how this suspicion grew as the century progressed is
a fascinating one, too complex to follow here.112 Descartes saw that
hypothesis was the only way in which the motions and sizes of the
imperceptibly small corpuscles on which the observable properties of things
depend could be reached, but hoped that certainty might still be attained
either by eliminating all alternative explanations save one, or by finding
one that fits the phenomena of nature so well “that it would be an injustice
to God” to believe that it could be false.113 Boyle more realistically set
about formulating the criteria to be used in evaluating or comparing causal
hypotheses, where the causes are postulated, not directly observed. After
an acute analysis of the difficulties facing explanations that call upon
imperceptible corpuscles, Locke concluded that physics of the traditional
demonstrative sort is “forever out of reach,” but that the skillful use of
analogy may still allow the /83/ natural philosopher to attain the “twilight
of probability.”
Newton was misled by the quasi-demonstrative form he had been able
to impose upon his mechanics into supposing that hypothesis could be
dispensed with in science proper. Though he himself made extensive
and ingenious use of unobservable entities of all sorts in the Queries
appended to the Optics, he believed he could construct a mechanics and a
geometrical optics without their aid. (He was relying here on the plasticity
of his key concept, force: are forces unobservable causal agencies, or are
they merely dispositions to move in a certain way?114) Because of his
enormous influence on those who came after him, his restriction of science
proper to two modes of inference only, deduction and induction, was
to have negative repercussions for decades to come, until the atoms and
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ether-vibrations of the early nineteenth century once and for all showed
causal inference to underlying structure to be indispensable to the work of
the physical scientist.
This is much too rapid an excursion, but it may give some hint, at
least, of how long it took the practitioners of the new natural science to
realize how powerful a tool their causal hypotheses could become, how
far beyond the small world of the human senses they could reach, and
how secure a knowledge they could ultimately yield, /84/ despite their
apparent logical fragility. This was just as surely a discovery as was that of
the planet, Neptune, and like the latter, it was made in the first instance
by the scientists themselves, not by philosophers reflecting on the quality
of knowledge that scientists ought to aim at.
Whewell. The most perceptive nineteenth-century commentator on
these issues was probably William Whewell, whose Philosophy of the
Inductive Science, Founded upon their History (1840) made plain that only a
detailed study of the actual history and practice of science could allow one to
say how science is made.115 Though he recognized, and indeed emphasized
(against Mill) a third mode of inference in science besides deduction and
empirical generalization, he applied the old label, “induction,” to it which
may have obscured the importance of the point he was making. Induction
for him is, first and foremost, an untidy inventing of hypotheses meant
to “colligate,” or bind together, the known facts and to reveal new ones.
The first step is the crucial one of finding the appropriate “conceptions”
that will enable the facts to fall together in an intelligible order. This is the
distinctive contribution of mind , he notes, a contribution overlooked by
the empiricists (a touch of epagoge still?). /85/
But “induction” is not only invention, it is also verification. (In
deduction and even to a large extent in induction, to discover is to
verify; the fateful distinction between invention (discovery) and verification
comes into view only when a third mode of inference is recognized.)
A good hypothesis should explain the phenomena already observed, as
well as predicting new kinds. Successful prediction is already a measure
of truth. But when a “consilience of inductions” occurs, when hitherto
unrelated areas of inquiry fall together under a single hypothesis, this can
(he suggested) convince us of the truth of the hypothesis. Consilience
involves, then, both enlargement of scope and simplification of structure.
And it requires the scientist to follow the progress of a theory over time to
assess whether its growth has been coherent or ad hoc.
Peirce. Peirce was the first to say straightforwardly that to deduction
and induction, we must add a third (which he variously named abduction,
hypothesis, retroduction) if we are to categorize properly what it is that
makes science. Abduction is the move from evidence to hypothesis; it is
Ernan McMullin 181
“the provisional adoption of a [testable] hypothesis.”116 Unlike deduction
and induction, it may involve new ideas and thus may require new language
as one moves from known effect to unknown (and possibly unobservable)
cause. /86/ Unlike many of his contemporaries (Ernst Mach and William
James, for example), he had no hesitation about inferring to unobservable
entities. Criticizing James, for example, he notes that the sort of positivism
which would question the propriety, in general, of such inference is clearly
out of touch with the actual practice of physics. “Attempts to explain
phenomenally given elements as products of deep-lying entities” (using
molecules to explain heat is his example) are entirely legitimate; in fact,
this phrase may be said to describe “as well as loose language can, the
general character of scientific hypothesis.”117
A number of questions immediately suggest themselves. Is abduction the
inventing of the causal hypothesis, the hitting upon a plausible explanatory
account? Or does it in some way involve the evaluation of the proposed
explanation? In terms of a distinction later made famous in the philosophy
of science, ought it be regarded as discovery or as verification? One does
not need to ask this in the case of deduction and induction. But there seem
to be at least two (or perhaps even a spectrum of ) varieties of abduction,
depending on how much stress one puts on the term, “plausible,” when
defining it as a move from effect to “plausible” cause. Peirce gave more
stress to the inventive side, raising the further question as to why this
should be regarded as inference. “I /87/ reckon it as a form of inference,
however problematical the hypothesis may be held.”118
There has been a good deal of controversy among Peirce scholars as
to how, exactly, in the end he intended abduction to be understood.119
Some of this was prompted by the appearance of N. R. Hanson’s book,
Patterns of Discovery in 1958,120 since Hanson made use of Peirce’s
term, “retroduction,” in order to make his own point about the manner
in which discovery is “patterned” in science. To “discover” a causal
hypothesis is already to see certain phenomena as intelligible. He quotes
Peirce with approval: “Abduction, although it is very little hampered
by logical rules, nevertheless is logical inference, asserting its conclusion
only problematically or conjecturally, it is true, but nevertheless having
a perfectly definite logical form.”121 Hanson rejects the standard H-D
account of causal inference, claiming that (unlike the retroductive one) it
leaves the genesis of the hypothesis itself unaccounted for, focusing only
on the subsequent testing.
Peirce’s views on the triad, deduction, induction, retroduction, shifted in
marked fashion across the course of his long writing career. Early on, he saw
them as three more or less independent types of inference. Later, he presents
them as three linked phases of the same inquiry, part of a single complex
method. Thus, induction in his ear- /88/ lier account is more or less the
sort of generalization across particulars that we have already encountered in
182 Zygon
so many guises. It is basically a sampling technique, yielding an empirical
law. Whereas in his later writings, induction becomes the means by which
abductive hypotheses are tested, i.e., the final phase of inquiry. He is
critical of those who confuse abduction and induction, regarding them as
a single argument: “nothing has so much contributed to present chaotic
ideas of the logic of science.”122 One (abduction) is preparatory, the other
(induction) is the concluding step. They have in common that both lead
to “the acceptance of a hypothesis because observed facts are such as would
necessarily or probably result as consequences of that hypothesis. But for
all that they are the opposite poles of reason.” The method of one is, in fact,
the reverse of that of the other. Abduction begins from the facts without
having any particular theory in view, motivated only “by the feeling that a
theory is needed to explain the surprising facts.” Induction, on the other
hand, begins from a hypothesis already arrived at and seeks for facts to
support that hypothesis.
Whether, when induction is formulated in this way, it can so easily
be separated from the prior stage of abduction is a matter of debate. In
what way, and to what extent, is background /89/ knowledge to be taken
into account in the original abduction?123 Does abduction refer to the
manner in which a hypothesis is constructed, or the manner in which
a plausible hypothesis is selected from among the available alternatives?
Are we asking: what hypothesis is more likely to be true, or: which
one is more worth considering? Peirce himself had distinctive views on
what he called the “economy of research” which led him to hold that
one should ordinarily prefer the hypothesis that is more easily tested.124
It is not easy to disentangle the theme of abduction/retroduction from
the enormously complex and sometimes idiosyncratic metaphysical and
psychological system Peirce labored to build and rebuild. In the closing
paragraphs of this essay, we will leave this task aside to focus finally on a
relatively simple statement of “the inference that makes science.”
Proposal. The ambiguity we have noted above between the two
“sides” of abduction can be dealt with, in part at least, by allowing that there
are two quite different modalities to take into account. Let us restrict the
term, “abduction,” to the process whereby initially plausible and testable
causal hypotheses are formulated. This is inference only in the loosest
sense, but the extensive discussions of the logic of discovery in the 1970s
showed how far, indeed, it differs from mere guessing. The testing of
such hypotheses is /90/ of the most varied sort. It does, of course, involve
deduction in a central way, as consequences are drawn and tried out.
Some of these may be singular, others may be lawlike and hence involve
induction. But we shall not restrict induction to the testing of causal
hypotheses, as Peirce came to do. Much of experimental science is inductive,
in the sense of seeking interesting correlations between variables; all the
Ernan McMullin 183
factors are observable, in the extended sense of “observable” appropriate to
sophisticated instrumentation. (Aristotle’s restriction of the starting point
of natural science to features that are observable by the human senses has
long since been set aside. Virtually none of the properties on which such
sciences as physics and chemistry are built today can be perceived by ns, one
reason why the patterning involved in perception is such a poor analogue
for scientific explanation.) Induction is strictly limited to the observable
domain. And it is only in a very weak sense explanatory. Laws may explain
singular occurrences, of the sort that the D-N model was devised to handle.
But these are the material of history or of engineering, not of such natural
sciences as chemistry or physics. Laws are the explananda; they are the
questions, not the answers. /91/
To explain a law, one does not simply have recourse to a higher law
from which the original law can be deduced. One calls instead upon
a theory, using this term in a specific and restricted sense. Taking the
observed regularity as effect, one seeks by abduction a causal hypothesis
which will explain the regularity. To explain why a particular sort of thing
acts in a particular way, one postulates an underlying structure of entities,
processes, relationships, which would account for such a regularity. What
is ampliative about this, what enables one to speak of this as a strong
form of understanding, is that if successful, it opens up a domain that was
previously unknown, or less known.
The causal inference here is therefore not the abduction alone, which
is still a conjecture, even if a plausible conjecture. It is the entire process
of abduction, deduction, observational testing, and whatever else goes
into the complex procedure of theory appraisal. Recent philosophers of
science have stressed that the virtues to be sought in a good theory do
not reduce simply to getting the predictions right. One looks for “natural”
explanations, for example, avoiding ad hoc moves even if these latter lead
to correct predictions. One looks for “consilience,” in Whewell’s sense,
involving both unification and simplification. This issue of the criteria of
theory-appraisal is one of /92/ the most actively discussed topics in current
philosophy of science.125
Our concern here is not with the detail of this discussion. It is with the
process of theoretical explanation generally, the process by means of which
our world has been so vastly expanded. This is the kind of inference that
makes science into the powerful instrument of discovery it has become. It
allows us reach to the very small, the very distant in space, the very distant
in past time, and above all to the very different. As a process of inference,
it is not rule-governed as deduction is, nor regulated by technique as
induction is. Its criteria, like coherence, empirical adequacy, fertility, are of
a more oblique sort. They leave room for disagreement, sometimes long-
lasting disagreement. Yet they also allow controversies to be adjudicated
and eventually resolved.126
184 Zygon
It is a complex, continuing, sort of inference, involving deduction,
induction, and abduction. Abduction is generally prompted by an earlier
induction (here we disagree with Peirce). The regularity revealed by the
induction may or may not be surprising. Deductions are made in order that
consequences may be tested, novel results obtained, consistency affirmed.
The process as a whole is the inference by means of which we transcend
the limits of the observed, even the instrumentally observed. /93/
Let us agree to call the entire process retroduction. We are “led
backwards” from effect to cause, and arrive at an affirmation, not simply
a conjecture. Retroduction in this sense is more than abduction. It is not
simply the initial plausible guess. It is a continuing process that begins with
the first regularity to be explained or anomaly to be explained away. It
includes the initial abduction and the implicit estimate of plausibility this
requires. It includes the drawing of consequences, and the evaluation of the
match between those and the observed data, old or acquired in the light
of the hypothesis. Tentative in the first abduction, gradually strengthening
if consequences are verified, if anomalies are successfully overcome, if
hitherto disparate domains are unified, retroduction is the inference that
in the strongest sense “makes science.”
The product of retroduction is theory or causal explanation. It is distinct
from empirical law, the product of the simpler procedure of induction.
(This distinction is not entirely sharp, since the language in which laws
are expressed and the procedures by which observations are obtained are
likely to be to some degree theory-dependent.) The criticisms leveled
against the original Peircean account can be met, since the claims of
both “discovery” and “justification” are recognized, and an implausible
dichotomy /94/ between them avoided. Even the original abduction with
the modicum of assessment it requires (does the hypothesis, in fact, entail
the data prompting its creation? is it testable? is it coherent with background
knowledge?) can be called retroduction of a preliminary and tentative sort
since it already gives some hint of what the cause may be. But, let it be
emphasized again, retroduction is not an atemporal application of rule as
deduction is. It is extended in time and logically very complex. It is properly
inference, since it enables one to move in thought from the observation of
an effect to the affirmation with greater or lesser degree of confidence, of
the action of a (partially) expressed sort.
The language here is, of course, that of scientific realism. It is because
the cause is, in some sense however qualified, affirmed as real cause,
that retroduction functions as a distinct form of inference. Anti-realism
reduces effectively to instrumentalism; whether the anti-realist believes
that theories may in principle make truth-claims or not, if his or her view
does not permit one, in practice, to make an existence claim of any kind for
theoretical entities, the distinction between this view and instrumentalism
is a distinction with- out a real difference. (Many of those who are currently
Ernan McMullin 185
called anti-realists are given that label only because they reject the standard
arguments /95/ advanced in favor of realism. But most of these critics in
practice affirm the existence of the same entities as realists do, with very
similar qualifications; the difference is that they do not present arguments,
even bad arguments, in support of their realism.)
The instrumentalist account of explanation reduces retroduction to a
complicated form of induction, and theory to law. That is why in the
positivist tradition the distinctions between retroduction and induction
and between theory and law have been glossed over. If theoretical terms
are, in effect, no more than devices used to improve the scope and accuracy
of prediction, if “acceptance of a theory involves as belief only that it is
empirically adequate” (i.e., predicts observable results correctly), in the
influential form of anti-realism propounded by Bas van Fraassen,127 then
there really are only two sorts of inference after all. So that on the distinction
we have been laboring to draw in these pages, much depends. It is not just
a matter of logical convention. It is a question of the amplitude of our
world.
Conclusion. It is a far cry from the demonstrations of Aristotle to the
retroductions of modern theoretical science. Where they differ is, first, that
retroduction makes no claim of necessity, and it settles for less, much less,
than definitive truth. It can, under favorable conditions, when theories
/96/ are well-established, yield practical certainty. Recent discussions of
scientific realism show, however, how hedged this assertion must be, since
the truth of a theory requires the existence of the postulated cause under the
description given in the theory. Second, the inductions that retroduction
relies on are systematic and protracted, not simply a noticing of regularity.
Third, the observations from which retroduction begins are, for the most
part, performed by sophisticated instruments; the limited scope and lack of
precision of the human senses would never permit the range of retroduction
that is necessary if the “invisible realm,” as Newton called it, is to be opened
up. Fourth, abduction often requires the introduction of new concepts and
the testing of new language. The necessity for this was not appreciated
as late as Newton’s day; his Third Rule of Reasoning postulated that the
properties of all bodies would be the same as those of bodies accessible to
the human senses, i.e., would be the so-called primary properties, extension,
mobility, hardness, impenetrability, inertia. (He needed this restriction of
course, in order that induction might be, as he claimed, the all-sufficient
method of natural science.)128 Central to retroduction, as we know it, is the
imaginative modification of existing concepts or the creation of new ones
quite remote from the universals or forms that might be /97/ abstracted
from perceptual experience. Finally, though retroduction is, indeed, an
act of the intellect, as the epagoge underlying demonstration was asserted
to be, it is exceedingly complex, involving a whole series of discrete and
186 Zygon
well-defined operations, like the drawing and testing of consequences, the
assessing of anomalies, and so forth. And it is open-ended; it continues for
as long as the possibility of new and relevant evidence remains open. It
does not terminate in an act of intuitive insight wherein one sees that the
nature must be so.
Abduction would roughly correspond to the first part of the medieval
regressus and deduction to the second. But abduction differs fundamentally
from demonstration quia because it does not yield the certain starting
point that demonstration propter quid requires. Induction, in turn, would
roughly correspond to the intermediate phase of investigatio, but it is
methodic and involves experiment and the measurement of properties
inaccessible to the senses. Still, one can see something taking shape in the
sixteenth-century attempts to forge not just a conception of science, but
a method whereby it might be attained. The tripartite division gave an
inkling, at least, of things to come. /98/
What retroduction and demonstration have in common is the goal of
causal explanation, the discovery of natures. Both are realist, one direct,
the other much more guarded. Aristotle and Aquinas would marvel at
the complexity of retroduction and the degree of invention it requires;
they would be taken aback by the intricacies of structure that underlie
the “simple” bodies of our experience. They would, doubtless, regret the
inability of natural science to lay claim on “eternal and necessary truth,”
though Aquinas might not be altogether surprised at the news. But what
they would see as their own is the goal of a progressive and well-founded
understanding of the “natures” whose interlocking actions make our world
what it is.
NOTES
1. This familiar doctrine is found in the early chapters of the Posterior Analytics; see I, 2,
71b 19–22. I shall rely mainly on the standard Oxford translations (here that of G. R. G. Mure),
with occasional modifications. G. E. R. Lloyd remarks that Aristotle “was the first not just in
Greece, but so far as we know anywhere, explicitly to define demonstration in that way.” See
Demystifying Mentalities, Cambridge: Cambridge University Press, 1990, p. 74.
2. Three chapters in the Posterior Analytics (I, 18 and 31; II, 19) and two in the Nicomachean
Ethics (VI, 3 and 6) are the main sources.
3. Prior Analytics, I, 30, 46a 17–21.
4. Post. An., I, 18.
5. See, for example, Prior An., II, 23, 68b 29; Post. An., II, 7, 92a 37–92b 1, and especially
Post. An., I, 31, 88a 2–5, where he says that the “commensurate” or genuine universal requisite
for demonstration can be elicited on the basis of frequent recurrence.
6. See Deborah K. W. Modrak, Aristotle: The Power of Perception, Chicago: University of
Chicago Press, 1987, p. 175, and the references given there. See also D. W. Hamlyn, “Aristotelian
epagoge,” Phronesis, 21, 1976, 167–184.
7. Post. An., II, 19, 100a 4–5.
8. Some have argued that nous is a separate capacity for intuition subsequent to the process
of epagoge (construed as empirical generalization). We shall take it to refer to the state of
knowledge brought about by epagoge itself. See J. Lester, “The meaning of nous in the Posterior
Analytics,” Phronesis, 18, 1973, 44–68, and Modrak, 171–76.
Ernan McMullin 187
9. Post. An., II, 19; 100b 14–17. G. E. L. Owen reminds us that there is a parallel account
of epagoge in Aristotle’s work also, where the starting point is not perceptions but endoxa,
the common opinions on a particular subject. “Phainomena” in this account are not sense-
appearances, but what appear to people generally to be the case. This sense of epagoge is linked
with dialectic rather than with causal demonstration. See “Tithenai ta phainomena,” in Suzanne
Mansion, ed., Aristote et les problèmes de méthode, Louvain: Nauwelaerts, 1961, 83–103.
10. De Anima, III, 4, 429a 14–17.
11. De Anima, II, 5, 430a 15.
12. Metaphysics, IV, 5, 1010b 1–3. The passage continues with examples of this sort of
subjectivity: heaviness, for instance, will not be estimated in the same way by the weak and the
strong.
13. De Caelo, III, 7, 306a 16–17. See Owen, “Tithenai ta phainomena,” p. 90, and also
Terry Irwin, Aristotle’s First Principles, Oxford: Clarendon Press, 1988, pp. 33–34.
14. Post. An., I, 31, 88a 11–17.
15. Post. An., II, 11, 94b 27–30.
16. Indeed, he sometimes overlooks the (to us, obvious) possibility of bodies too small to be
seen by us, as in his lengthy discussion of the nature and origin of semen in the different kinds
of animals, and its role in generation (Generation of Animals, I, 17–23). His claim that in many
species the male emits no semen would be an instance of this kind of oversight.
17. Post. An., II, 14, 98a 16–19.
18. On the Parts of Animals, I, 5, 644b 25–645a 4.
19. Part. Anim., I, 5, 644b 35.
20. Post. An., I, 13, 78a 33–35.
21. Post. An., I, 4, 73b 25–28.
22. On the Heavens, II, 8, 290a 16–24.
23. “Knowable” is better than “known” here. “Better known in itself’ makes little sense:
known to whom? Jonathan Barnes in the new Oxford translation (Oxford: Clarendon Press,
1975) renders the distinction by “more familiar in itself” and “more familiar to.” “Familiar in
itself” is puzzling, since “familiar” make essential reference to individual experience. Another way
of expressing the distinction is found in Post. An. I, 3, 72b 28–29: things “prior from our point
of view” and things “simply prior.”
24. Post. An., I, 13, 78a 34–35.
25. Post. An., I, 34, 89b 10.
26. See, for instance, Jonathan Barnes, “Aristotle’s theory of demonstration,” Phronesis, 14,
1969, 123–152 and William Wians, Aristotle’s Method in Biology, PhD dissertation, University
of Notre Dame, Ann Arbor Microfilms, 1983.
27. See Hist. Anim., I, 6, 490b 19–27, and Gen. Anim. II, 1, 732b 15. D. M. Balme gives an
excellent review in “Aristotle’s use of differentiae in biology,” in Mansion, ed., Aristote, 195–212.
28. Part. Anim., III, 6.
29. “The formation of fat in the kidneys is the result of necessity, being, as explained, a
consequence of the necessary conditions accompanying the possession of such organs.” Part.
Anim., III, 9, 672a 13–15.
30. Allan Gotthelf attempts to answer this question positively in “First principles in Aristotle’s
Parts of Animals,” in Philosophical Issues in Aristotle’s Biology, ed. Allan Gotthelf and James G.
Lennox, Cambridge: Cambridge University Press, 1987, 167–98. For a more skeptical response,
see Lloyd, op. cit., p. 88, and “Aristotle’s zoology and his metaphysics: The status quaestionis,”
in Biologie logique et métaphysique chez Aristote, ed. Daniel Devereux and Pierre Pellegrin (Paris,
1990).
31. Lloyd, Demystifying Mentalities, p. 89.
32. Meteorology, I, 7, 344a 5–8.
33. On the Heavens, II, 5, 287b 34–288a 2.
34. A.C. Crombie, Robert Grosseteste and the Origins of Experimental Science, 1100–1700,
Oxford: Clarendon Press, 1953, p. 290.
35. The Posterior Analytics first became available in the Latin West in a translation by James
of Venice around 1140. But the obscurity of the text and the unattractiveness of its doctrine from
the perspective of the prevailing Augustinian theology delayed its impact. See R. W. Southern,
Robert Grosseteste, Oxford: Clarendon Press, 1986, 150–55.
36. Crombie, p. 84.
188 Zygon
37. James McEvoy, The Philosophy of Robert Grosseteste, Oxford: Clarendon Press, 1982,
p. 207. See his chapter 2: “Grosseteste’s place in the history of science.”
38. However, the “experimenta” or experiences he reports are for the most part culled from
Greek or Arabic sources, and are rarely firsthand. See Bruce Eastwood, “Medieval empiricism:
The case of Robert Grosseteste’s optics,” Speculum, 43, 1968, 306–321; “Robert Grosseteste’s
theory of the rainbow,” Arch. Intern. Hist. Sciences, 19, 1966, 313–322.
39. This particular way of describing the effects of sin on the human cognitive powers
was doubtless suggested by the famous description of the battle-rout in Aristotle’s discussion of
epagoge in Post. An. II, 19. Grosseteste disagrees with Aristotle, however, in allowing that God
can illuminate our minds directly, so that in principle not-all our knowledge need be based on
perception. See Southern, pp. 164–69.
40. Commentarius in Posteriorum Analyticorum Libros, ed. Pietro Rossi, Firenze: Olschki,
1981, pp. 190–91. Grosseteste also remarks in this passage that the planets are intrinsically
brighter than the farther-off stars, a claim that could be fatal to Aristotle’s attempt to make of
this example a propter quid demonstration.
41. Commentarius, pp. 381–83; see Post. An. II, 16, 98a 16–19; Part. Anim., III, 2, 662b
23–663a 17; III, 14, 674a 27–674b 14.
42. Commentarius, pp. 390–98. Crombie seems to be claiming in this passage that alternative
causal theories have to be tested “by further observation or experiment” but it is hard to find this
in the text.
43. S. H. Thomson, “The text of Grosseteste’s De cometis,” Isis, 19, 1933, 19–25, see p. 23;
Crombie, pp. 88–90.
44. Commentarius, pp. 214–15; Crombie, pp. 74, 83.
45. Thomas Litt lists fifteen different places in his works where Aquinas refers, in one
way or another, to the precession of the equinoxes, and no less than eleven passages where the
system of Ptolemy is mentioned (Les Corps Célestes dans l’Univers de S. Thomas d’Aquin, Louvain:
Publications Universitaires, 1963, chaps. 16 and 18). Aquinas was keenly aware of the differences
between the Ptolemaic and the Aristotelian astronomical systems, and shows sympathy for both
(p. 362).
46. For an extended discussion, see E. McMullin, “Matter as a principle,” in The Concept
of Matter in Greek and Medieval Philosophy, ed. E. McMullin, Notre Dame: University of Notre
Dame Press, 1963, 173–217, especially section 2, “‘Empirical’ versus ‘conceptual’ analysis.”
47. G. E. L. Owen, “Tithenai ta phainomena,” p. 88. Owen is critical of attempts to read
the Physics as empirical science, which to his mind necessarily makes it come out as “confused
and cross-bred,” p. 92.
48. For a recent comment, see John Jenkins, “Aquinas on the veracity of the intellect,”
Journal of Philosophy, 88, 1991, 623–32. I am indebted to my colleagues, John Jenkins, Eleanore
Stump, Alasdair MacIntyre, and Fred Freddoso, for discussions on this and related issues in
Aquinas scholarship.
49. Summa Theologica, I, q. 85, a. 5 ad 3 (Pegis translation, slightly modified); sec also, for
example, q. 85, a. 6 c and a. 8 c.; De Veritate, q. 1, a. 12 c.
50. In Symbolorum Apostolorum Expositio, in Opuscula Theologica, ed. R. M. Spiazzi, Rome:
Marietti, 1954, par. 864.
51. De Veritate, q. 10, a. 1 c.; see also ad 6.
52. Jacques Maritain, The Degrees of Knowledge, trans. Gerald B. Phelan, New York:
Scribners, 1959, p. 208.
53. De Ente et Essentia, c. 6.
54. De Spiritualibus Creaturis, a. 11 ad 3.
55. De Veritate, q. 4, a. 1 ad 8.
56. Summa Theologica, I, q. 29, a. 1 ad 3.
57. Summa Theologica, I, q. 29, a. 1 ad 3.
58. Summa Theologica, I, q. 85, a. 5, c. The distinction implied here between the knowledge
of quiddity, as the initial rough apprehension or the “whatness,” or distinctiveness, of a thing,
and the knowledge of essence, as the full understanding of the nature of the thing, is not always
maintained. More often Aquinas appears to use the terms equivalently.
59. In Boethium De Trinitate, q. 6, a. 2 c. (Maurer translation). The reference is to De Caelo,
III, 7, where Aristotle does indeed affirm that the test of general principles in natural science
(specifically in this case, regarding the manner in which mutual transformations of the elements
Ernan McMullin 189
take place is consistent with sense-observation. He criticizes his opponents’ views on the grounds
that the principles they rely on lead to consequences that are at odds with simple: observation, and
blames bias and carelessness for their error (306 a 7–18). This passage is compatible with either
of two very different readings of Aristotle’s point. One (the less likely) would be consequentialist:
the warrant for a principle in natural science lies in part at least in the verification of the empirical
consequences deductively derived from it; the other would still be “internalist”: epagoge has to
be properly performed (not disturbed e.g., by bias); if it is, the warrant or the principle is to lie
in its self-evidence. Here, of course, lies the Great Divide!
60. Summa Theologica, I, q. 2, a. 3 c and ad 2.
61. Commentarium, II, lect. 20, 14.
62. Summa Theologica, I, q. 29, a. 1 ad3.
63. Alasdair MacIntyre, First Principles, Final Ends and Contemporary Philosophical Issues,
Aquinas Lecture 1990, Milwaukee: Marquette University Press, pp. 24–27, 34–51. For an
opposed view, see Melvin A. Glutz, The Manner of Demonstrating in Natural Philosophy, River
Forest, IL: Pontifical Faculty of Philosophy, 1956. Glutz not only maintains that demonstration
can be relatively easily arrived at, but that the Aristotelian-Thomistic notion of demonstration
affords the proper model of proof for contemporary natural science.
64. MacIntyre, pp. 43–44.
65. MacIntyre, p. 38.
66. MacIntyre, p. 39.
67. MacIntyre, pp. 45–46.
68. MacIntyre, p. 44.
69. MacIntyre, pp. 35–36.
70. In Boethium De Trinitate, q. 6, a. 1 c.
71. Commentary, I, lect. 30; see also II, lect. 20.
72. “We judge of natural things as the sense reveals them”; in natural science, the sensible
properties of a thing “adequately manifest its nature.” In Boethium De Trinitate, q. 6, a. 2 c.
73. In Boethium, ibid. We can also proceed, he says, by simply affirming either transcendence
or negation of sensible properties. He is following the lead here of Psuedo-Dionysius, De Divinis
Nominibus.
74. In Boethium, q. 6, a. 3 c.
75. In Boethium, q. 6, a. 4 ad 2.
76. In Boethium, q. 6, a. 2 c.
77. J. H. Randall, “The development of scientific method in the School of Padua,” Journal
of the History of Ideas, 1, 1940, 177–206; partially reprinted in The Roots of Scientific Thought,
eds. Philip P. Wiener and Aaron Noland, New York: Basic Books, 1957; see p. 146.
78. “Aquinas, Galileo, and Aristotle,” Aquinas Medalist’s Address, Proceedings American
Catholic Philosophical Association, 57, 1983, 17–24; p. 22.
79. See W. A. Wallace, Galileo and his Sources: The Heritage of the Collegio Romano in Galileo’s
Science, Princeton: Princeton University Press, 1984; Tractatus de Praecognitionibus et Praecognitis
and Tractatio de Demonstratione, transcribed by William F. Edwards, and with Introduction,
Notes, and Commentary by W. A. Wallace, Padova: Antenore, 1988.
80. Nicholas Jardine, “Galileo’s road to truth and the demonstrative regress,” Studies in the
History and Philosophy of Science, 7, 1976, 277–318; see p. 296.
81. Randall yields to the temptation, and presents the Paduan regressus as the immediate
inspiration for seventeenth-century accounts of the “method of hypothesis.” Thus, despite their
both defending a strong continuity thesis, Randall and Wallace are entirely at odds as to the
content of what was supposedly transmitted to Galileo from Padua.
82. Jardine, p. 310.
83. Some of his uses of the term “demonstration” in the context of mechanics are discussed
in McMullin, “The conception of science in Galileo’s work,” in New Perspectives on Galileo, ed.
Robert Butts and Joseph Pitt, Dordrecht: Reidel, 1978, 209–57; see 229–40.
84. These obstacles are of quite different sorts and the texts are not always easy to harmonize.
See McMullin, “Galilean idealization,” Studies in the History and Philosophy of Science, 16, 1985,
247–273, and “Conception of science,” pp. 230–35.
85. Two New Sciences, transl. Stillman Drake, Madison: University of Wisconsin Press, 1974,
p. 153; Opere, 8, 197; see McMullin, “Conception of Science,” p. 229.
86. Wallace, Galileo and his Sources, p. 324.
190 Zygon
87. Wallace, “Galileo’s use of the Paduan regressus in his astronomical discoveries,” to appear.
88. In a footnote, Wallace asks his critic whether he is certain (on the basis of pre-spacecraft
evidence that the moon has mountains (“Galileo’s use . . . ,” note 7). If not, “he must hold that
planetary astronomy is not an apodictic science but only opinion, highly probable opinion,
but opinion nonetheless.” If he is certain, then he is committed to “a demonstrative regressus,
whether he recognizes it under that name or not.” This particular critic would respond that
planetary astronomy is not an apodictic science, indeed that no natural science is apodictic, that
the cut between the apodictic and the non-apodictic is not where science begins or ends. There
are theoretical claims in every science which might loosely be called “apodictic” because of the
degree of assurance with which we hold them. But the term is misleading because it tends to
conceal the degree to which there may be assumptions hidden in such claims. Much better to
acknowledge the presence of (highly supported) assumptions in quasi-apodictic claims on the one
hand and the value commonly attached to well-supported (though far from apodictic) theories
on the other. Further, that someone should attach practical certainty to the claim that there are
lunar mountains in no way implies that he or she has reached this by a regressus argument. A
standard consequential argument can come to have such overwhelming force as to yield a (more
or less) certain conclusion.
89. See W. Wallace, “Aristotle and Galileo: The uses of hypothesis (suppositio) in scientific
reasoning,” in Studies in Aristotle, ed. D. O’Meara, Washington: Catholic University of America
Press, 1981, 47–77. For an opposing view, see Winifred Wisan, “On argument ex suppositione
falsa,” Studies in the History and Philosophy of Science, 15, 1984, 227–36, and McMullin
“Conception of science,” pp. 234–37.
90. Dialogue on Two Chief World Systems, trans. Stillman Drake, Berkeley: University of
California Press, 1953, p. 118; Opere, 7, 144.
91. These last claims are developed in some detail in McMullin, “Conceptions of Science in
the Scientific Revolution,” in Reappraisals of the Scientific Revolution, eds. David Lindberg and
Robert Westman, Cambridge: Cambridge University Press, 1990, 27–92.
92. See E. L. Fortin and P. O’Neill, “The Condemnation of 1277,” in Philosophy in the
Middle Ages, ed. A. Hyman and J. J. Walsh, Indianapolis: Hackett, 1973, 540–9; Edward Grant,
“The Condemnation of 1277, God’s absolute power, and physical though in the late Middle
Ages,” Viator, 10, 1979, 211–44.
93. See J. Reginald O’Donnell, “The philosophy of Nicholas of Autrecourt and his appraisal
of Aristotle,” Medieval Studies, 4, 1942, 97–125; Julius Weinberg, Nicholaus of Autrecourt,
Princeton: Princeton University Press, 1948.
94. Francis Bacon, New Organon, transl. James Spedding, ed. Fulton Anderson, New York:
Bobbs-Merrill, 1960, Bk. I, aphorism 19.
95. New Organon, II, aph. 2.
96. For a fuller discussion, see McMullin, “Francis Bacon: Exemplar of inductivism?” pp.
45–54 in “Conceptions of science in the Scientific Revolution”; Peter Urbach, Francis Bacon’s
Philosophy of Science, LaSalle, IL: Open Court, 1987.
97. Novum Organon, I, aph. 61.
98. Preface to the New Organon, Anderson, p. 36.
99. New Organon, I, aph. 50.
100. New Organon, II, aph. 36; see also II, aph. 7.
101. Opticks, Query 31, New York: Dover, 1952, p. 404. In his earlier work, Newton was
much less guarded, and tended to insist, rather, on “deducing from the phenomena” as the
mode of inference proper to “experimental philosophy.” For a fuller discussion, see McMullin,
“Newton: Deducing from the phenomena,” in “Conceptions of science,” pp. 67–74.
102. For a useful survey of the various formulations of Hume’s problem, and a taxonomy
of the many different solutions that have been proposed, see Max Black, “Induction,” The
Encyclopedia of Philosophy, ed. Paul Edwards, New York: Macmillan, 1967.
103. A System of Logic, 8th ed., New York, 1874, p. 223.
104. Rudolf Carnap, “The aim of inductive logic,” in Logic, Methodology and Philosophy of
Science, eds. Ernest Nage, Patrick Suppe, and Alfred Tarski, Stanford, CA: Stanford University
Press, 1962, 303–18; see pp. 303, 316.
105. A clear presentation of this approach can be found in the most popular textbook of the
positivist era, Morris Cohen and Ernest Nagel, An Introduction to Logic and Scientific Method ,
New York: Harcourt, Brace 1934.
Ernan McMullin 191
106. C. G. Hempel, “The theoretician’s dilemma,” in Aspects of Scientific Explanation, New
York: Free Press, 1965, 173–226; see pp. 184–85.
107. R. Carnap, Philosophical Foundations of Physics, New York: Basic Books, 1966, p. 230.
108. Carnap, Philosophical Foundations, chap. 24: “Correspondence Rules.”
109. See for example, Hempel, “The theoretician’s dilemma.”
110. See McMullin, “Structural explanation,” American Philosophical Quarterly, 15, 1978,
139–47; Dudley Shapere, “Scientific theories and their domains,” in The Structure of Scientific
Theories, ed. Frederick Suppe, Urbana: University of Illinois Press, 1977, 518–65.
111. New Organon, II, aphs. 5–8. See McMullin, “Conceptions of science,” pp. 51–54.
112. See McMullin, “Conceptions of science,” for a detailed account.
113. Descartes, Principles of Philosophy, part III, sect. 43.
114. McMullin, “Realism in the history of mechanics,” to appear.
115. See Menachem Fisch, William Whewell: Philosopher of Science, Oxford: Clarendon,
1991; McMullin, “Philosophy of science, 1600–1900,” in Companion to the History of Modern
Science, ed. R. C. Olby et al., London: Routledge, 1990, 816–37.
116. C. S. Peirce, Collected Papers, vols. 1–6, eds. Charles Hartshorne and Paul Weiss; vols.
7–8 ed. By Arthur Burks, Cambridge, MA, 1931–35; 1958, vol. 1, par. 65.
117. Collected Works, 8, par. 60.
118. “Abduction, induction and deduction,” Collected Works, 7, pars. 202–07.
119. See, for example, K. T. Fann, Peirce’s Theory of Abduction, The Hague: Nijhoff, 1970.
120. N. R. Hanson, Patterns of Discovery, Cambridge: Cambridge University Press, 1958, see
pp. 85–92.
121. Collected Papers, 5, par. 188.
122. “Abduction,” Collected Papers, 7, par. 218–22; see par. 218 for this and the following
quotations.
123. Peter Achinstein argues that the “retroductivist” account given by Peirce and Hanson
does not square with examples drawn from the history of physics. See Particles and Waves:
Historical Essays in the Philosophy of Science, New York: Oxford University Press, 1991, pp.
168–69; 235–39; 247–48.
124. For a detailed treatment of the manner in which Peirce’s “economic” ideas influenced
his theory of research, see W. Christopher Steward, Social and Economic Aspects of Charles
Sanders Peirce’s Conception of Science, PhD dissertation, University of Notre Dame, Ann Arbor
Microfilms, 1992.
125. Thomas Kuhn’s “Objectivity, value judgement, and theory choice” (collected in The
Essential Tension, Chicago: University of Chicago Press, 1977, pp. 320–39) was in this respect a
seminal essay. See, for example, McMullin, “Values in science,” PSA 1982, eds. P. Asquith and T.
Nickles, E. Lansing, MI 3–25; McMullin, “Rhetoric and theory-choice in science,” in Persuading
Science: The Art of Scientific Rhetoric, ed. Marcello Pera and William Shea, Canton, MA: Science
History, 1991, 55–76.
126. McMullin, “Scientific controversy and its termination,” in Scientific Controversies: Case
Studies in the Resolution and Closure of Disputes, ed. H. T. Engelhardt and A. Capian, Cambridge:
Cambridge University Press, 1987, 49–91.
127. Bas van Fraassen, The Scientific Image, Oxford: Clarendon Press, 1980, p. 12.
128. McMullin, Newton on Matter and Activity, Notre Dame, IN: University of Notre Dame
Press, 1978, chap. 1.
Journal of Philosophy, Inc.
Indispensability and Practice
Author(s): Penelope Maddy
Source: The Journal of Philosophy, Vol. 89, No. 6 (Jun., 1992), pp. 275-289
Published by: Journal of Philosophy, Inc.
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THE JOURNAL OF PHILOSOPHY
VOLUME LXXXIX, NO. 6,JUNE 1992
+ ~ -I- 0 -4-
INDISPENSABILITY AND PRACTICE*
F or some time now, philosophical thinking about mathematics
has been profoundly influenced by arguments based on its
applications in natural science, the so-called “indispensability
arguments.” The general idea traces back at least to Gottlob Frege,’
but contemporary versions stem from the writings of W. V. Quine,2
and later, Hilary Putnam.3 Much contemporary philosophy of math-
ematics (including my own) operates within the parameters of the
indispensability arguments; they are called upon to motivate various
versions of nominalism, as well as to support various versions of
realism. Still, attention to practice, both scientific and mathematical,
has recently led me to doubt their efficacy. I shall try to explain
these doubts in what follows. If they are legitimate, we will be forced
to rethink much of current orthodoxy in the philosophy of mathe-
matics.
I. NATURALISTIC BACKGROUND
An argument based on scientific and mathematical practice can
only succeed from a sufficiently naturalistic perspective. Familiar
Quinean naturalism counsels us to reject prescientific first philo-
sophy in favor of an approach that begins within our current best
scientific theory:
* My thanks go to the NSF (DIR-9004168) and to the UC/Irvine Academic
Senate Committee on Research for their support.
. . . it is applicability alone which elevates arithmetic from a game to the rank
of a science,” P. Geach and M. Black, eds., Translations from the Philosophical
Writings of Gottlob Frege (Cambridge: Blackwell, 1970), p. 187. Although Frege
speaks of arithmetic, my focus here will be on analysis, broadly construed.
2 See, e.g., “On What There Is,” repr. in From a Logical Point of View, 2nd
ed. (Cambridge: Harvard, 1980), pp. 1-19; and “Carnap and Logical Truth,”
repr. in The Ways of Paradox, rev. ed. (Cambridge: Harvard, 1976), pp. 107-32.
3 See, e.g., “What Is Mathematical Truth?” and “Philosophy of Logic,” repr. in
Mathematics, Matter and Method, 2nd ed. (New York: Cambridge, 1979), pp.
60-78, 323-57.
0022-362X/92/8906/275-89 (?D 1992 The Journal of Philosophy, Inc.
275
276 THE JOURNAL OF PHILOSOPHY
… naturalism: abandonment of the goal of a first philosophy. It sees
natural science as an inquiry into reality, fallible and corrigible but not
answerable to any supra-scientific tribunal, and not in need of any
justification beyond observation and the hypothetico-deductive
method … The naturalistic philosopher begins his reasoning within
the inherited world theory as a going concern. He tentatively believes
all of it, but believes also that some unidentified portions are wrong.
He tries to improve, clarify, and understand the system from within.
He is the busy sailor adrift on Neurath’s boat.4
Epistemological studies, in particular, are to be carried out within
science, with the help of relevant psychological theories.
From the perspective of this scientific naturalism, a philosopher
can criticize scientific practice, but only on scientific grounds, as a
scientist might do, for good scientific reasons. This is enough to
ratify an appeal to scientific practice in philosophical contexts: be-
cause scientific practice can only be questioned on scientific
grounds, a conflict between scientific practice and philosophy must
be resolved by revising the philosophy. So, for example, if scientific
practice holds that p does or does not count as evidence for q, to
disagree on philosophical grounds is an offense against naturalism.
As we shall see, however, it is not clear that Quine intends to
extend this naturalistic faith in practice to the practice of mathemat-
ics. Leaving Quine aside for the moment, we must ask ourselves
what the role of the philosophy of mathematics should be. Mathe-
matics, after all, is an immensely successful enterprise in its own
right, older, in fact, than experimental natural science. As such, it
surely deserves a philosophical effort to understand it as practiced,
as a going concern. Indeed, as in any discipline, there remain con-
ceptual confusions in mathematics that might be clarified by philo-
sophical analysis, providing that analysis is sensitive to the realities of
actual mathematics. If it is to serve these purposes, a philosophical
account of mathematics must not disregard the evidential relations
of practice or recommend reforms on nonmathematical grounds.
These are, in my view, proper goals for the philosophy of mathe-
matics. We, as philosophers of mathematics, should provide an ac-
count of mathematics as practiced, and we should make a contribu-
tion to unraveling the conceptual confusions of contemporary math-
ematics. So it is against this backdrop that I shall assess the
indispensability arguments, or rather, the view of mathematics the
indispensability arguments generate. They will be judged by their
ability to account for actual mathematics as practiced.
4 Quine, “Five Milestones of Empiricism,” in Theories and Things (Cam-
bridge: Harvard, 1981), pp. 67-72; here p. 72.
INDISPENSABILITY AND PRACTICE 277
II. INDISPENSABILITY
The original indispensability arguments were aimed at those who
would draw a weighty ontological or epistemological distinction be-
tween natural science and mathematics. To those tempted to admit
the existence of electrons while denying the existence of numbers,
Quine5 points out that
Ordinary interpreted scientific discourse is as irredeemably committed
to abstract objects-to nations, species, numbers, functions, sets-as it
is to apples and other bodies. All these things figure as values of the
variables in our overall system of the world. The numbers and func-
tions contribute just as genuinely to physical theory as do hypothetical
particles (ibid., pp. 149-50).
But, Quine’s opponent insists, the scientific hypotheses in our
theory are tested by experiment, and the mathematical ones are not;
surely the two can be distinguished on these grounds. To which
Quine replies,
The situation may seem to be saved, for ordinary hypotheses in natural
science, by there being some indirect but eventual confrontation with
empirical data. However, this confrontation can be remote; and, con-
versely, some such remote confrontation with experience may be
claimed even for pure mathematics and elementary logic. The sem-
blance of a difference in this respect is largely due to overemphasis of
departmental boundaries. For a self-contained theory which we can
check with experience includes, in point of fact, not only its various
theoretical hypotheses of so-called natural science but also such por-
tions of logic and mathematics as it makes use of.6
Thus, the applied mathematics is confirmed along with the physical
theory in which it figures.
Of course, it is not enough for a piece of mathematics simply to
appear in a confirmed scientific theory. For any theory T, there is
another theory T’ just like T except that T’ posits a bunch of new
particles designed to have no affect on the phenomena T predicts.
Any experiment confirming T under these circumstances would also
(in some sense) confirm T’, but we do not take this as evidence for
the existence of the new particles because they are “dispensable,”
i.e., there is an equally good, indeed better theory of the same phe-
nomena, namely, T, that does not postulate them. The mathematical
apparatus of modern physics does not seem to be dispensable in this
way; indeed, Putnam has emphasized that many physical hypotheses
5″Success and Limits of Mathematization,” in Theories and Things, pp.
148-55.
6 “Carnap and Logical Truth,” p. 121.
278 THE JOURNAL OF PHILOSOPHY
cannot even been stated without reference to numbers, func-
tions, etc.7
So a simple indispensability argument for the existence of mathe-
matical entities goes like this: we have good reason to believe our
best scientific theories, and mathematical entities are indispensable
to those theories, so we have good reason to believe in mathematical
entities. Mathematics is thus on an ontological par with natural
science. Furthermore, the evidence that confirms scientific theories
also confirms the required mathematics, so mathematics and natural
science are on an epistemological par as well.
Unfortunately, there is a prima facie difficulty reconciling this
view of mathematics with mathematical practice.8 We are told we
have good reason to believe in mathematical entities because they
play an indispensable role in physical science, but what about mathe-
matical entities that do not, at least to date, figure in applications?
Some of these are admissible, Quine9 tells us,
… insofar as they come of a simplificatory rounding out, but anything
further is on a par rather with uninterpreted systems (ibid., p. 788).
So in.particular,
I recognize indenumerable infinites only because they are forced on
me by the simplest known systematizations of more welcome matters.
Magnitudes in excess of such demands, e.g. M,, or inaccessible numbers,
I look upon only as mathematical recreation and without ontological
rights. 10
The support of the simple indispensability argument extends to
mathematical entities actually employed in science, and only a bit
beyond.
The trouble is that this does not square with the actual mathemati-
cal attitude toward unapplied mathematics. Set theorists appeal to
various sorts of nondemonstrative arguments in support of their
customary axioms, and these logically imply the existence of :,,. Inac-
7 Hartry Field disputes this in his Science without Numbers (Princeton: Univer-
sity Press, 1980), and Realism, Mathematics and Modality (Cambridge: Black-
well, 1989), but the copious secondary literature remains unconvinced.
8 Versions of this concern appear in C. Chihara, Constructibility and Mathe-
matical Existence (New York: Oxford, 1990), p. 15; and in my Realism in Mathe-
matics (New York: Oxford, 1990), pp. 30-1.
9″Review of Charles Parsons’s Mathematics in Philosophy,” thisJOURNAL,
LXXXI, 12 (December 1984): 783-94.
10 Quine, “Reply to Charles Parsons,” in The Philosophy of W. V. Quine, L.
Hahn and P. Schilpp, eds. (La Salle, IL: Open Court, 1986), pp. 396-403; here p.
400.
INDISPENSABILITY AND PRACTICE 279
cessibles are not guaranteed by the axioms, but evidence is cited on
their behalf nevertheless. If mathematics is understood purely on
the basis of the simple indispensability argument, these mathemati-
cal evidential methods no longer count as legitimate supports; what
matters is applicability alone. Here simple indispensability theory
rejects accepted mathematical practices on nonmathematical
grounds, thus ruling itself out as the desired philosophical account
of mathematics as practiced.
So simple indispensability will not do, if we are to remain faithful
to mathematical practice. We insist on remaining faithful to mathe-
matical practice because we earlier endorsed a brand of naturalism
that includes mathematics. But it is worth noting that even a retreat
to purely nonmathematical naturalism (forgetting our commitment
to actual mathematical practice) will not entirely solve this problem.
From the point of view of science-only naturalism, the applied part
of mathematics is admitted as a part of science, as a legitimate plank
in Neurath’s boat; unapplied mathematics is ignored as unscientific.
But even for applied mathematics there is a clash with practice.
Mathematicians believe the theorems of number theory and analysis
not to the extent that they are useful in applications but insofar as
they are provable from the appropriate axioms. To support the
adoption of these axioms, number theorists and analysts may appeal
to mathematical intuition, or the elegant systematization of mathe-
matical practice, or other intramathematical considerations, but
they are unlikely to cite successful applications. So the trouble is not
just that the simple indispensability argument shortchanges unap-
plied mathematics; it also misrepresents the methodological realities
of the mathematics that is applied.
There is, however, a modified approach to indispensability consid-
erations which gets around this difficulty. So far, on the simple
approach, we have been assuming that the indispensability of (some)
mathematical entities in well-confirmed natural science provides
both the justification for admitting those mathematical things into
our ontology and the proper methodology for their investigation.
But perhaps these two-ontological justification and proper
method-can be separated. We could argue, first, on the purely
ontological front, that the successful application of mathematics
gives us good reason to believe that there are mathematical things.
Then, given that mathematical things exist, we ask: By what methods
can we best determine precisely what mathematical things there are
and what properties these things enjoy? To this, our experience to
date resoundingly answers: by mathematical methods, the very
methods mathematicians use; these methods have effectively pro-
280 THE JOURNAL OF PHILOSOPHY
duced all of mathematics, including the part so far applied in physi-
cal science.
From this point of view, a modified indispensability argument first
guarantees that mathematics has a proper ontology, then endorses
(in a tentative, naturalistic spirit) its actual methods for investigating
that ontology. For example, the calculus is indispensable in physics;
the set-theoretic continuum provides our best account of the cal-
culus; indispensability thus justifies our belief in the set-theoretic
continuum, and so, in the set-theoretic methods that generate it;
examined and extended in mathematically justifiable ways, this
yields Zermelo-Fraenkel set theory. Given its power, this modified
indispensability theory of mathematics stands a good chance of
squaring with practice, so it will be preferred in what follows.”
III. THE SCIENTIFIC PRACTICE OBJECTION
My first reservation about indispensability theory stems from some
fairly commonplace observations about the practice of natural
science, especially physics. The indispensability argument speaks of
a scientific theory T, well-confirmed by appropriate means and
seamless, all parts on an ontological and epistemic par. This seam-
lessness is essential to guaranteeing that empirical confirmation ap-
plies to the mathematics as well as the physics, or better, to the
mathematized physics as well as the unmathematized physics.
Quine’s’2 vivid phrases are well-known: “our statements about the
external world face the tribunal of sense experience not individually
but only as a corporate body” (ibid., p. 41).
Logically speaking, this holistic doctrine is unassailable, but the
actual practice of science presents a very different picture. Histori-
cally, we find a wide range of attitudes toward the components of
well-confirmed theories, from belief to grudging tolerance to out-
right rejection. For example, though atomic theory was well-con-
firmed by almost any philosopher’s standard as early as 1860, some
scientists remained skeptical until the turn of the century-when
certain ingenious experiments provided so-called “direct verifica-
tion” -and even the supporters of atoms felt this early skepticism to
be scientifically justified.’3 This is not to say that the skeptics neces-
This is more or less the position of my Realism in Mathematics.
12 “Two Dogmas of Empiricism,” repr. in From a Logical Point of View, pp.
20-46.
13 I trace the history in some detail in “Taking Naturalism Seriously,” in Pro-
ceedings of the 9th International Congress of Logic, Methodology and Philo-
sophy of Science, D. Prawitz, B. Skyrms, and D. Westerstahl, eds. (Amsterdam:
North Holland, forthcoming).
INDISPENSABILITY AND PRACTICE 281
sarily recommended the removal of atoms from, say, chemical
theory; they did, however, hold that only the directly verifiable con-
sequences of atomic theory should be believed, whatever the explan-
atory power or the fruitfulness or the systematic advantages of think-
ing in terms of atoms. In other words, the confirmation provided by
experimental success extended only so far into the atomic-based
chemical theory T, not to the point of confirming its statements
about the existence of atoms. This episode provides no comfort to
the van Fraassenite, because the existence of atoms was eventually
established, but it does show scientists requiring more of a theory
than the sort of theoretical virtues typically discussed by philo-
sophers.
Some philosophers might be tempted to discount this behavior of
actual scientists on the grounds that experimental confirmation is
enough, but such a move is not open to the naturalist. If we remain
true to our naturalistic principles, we must allow a distinction to be
drawn between parts of a theory that are true and parts that are
merely useful. We must even allow that the merely useful parts
might in fact be indispensable, in the sense that no equally good
theory of the same phenomena does without them. Granting all this,
the indispensability of mathematics in well-confirmed scientific the-
ories no longer serves to establish its truth.
But perhaps a closer look at particular theories will reveal that the
actual role of the mathematics we care about always falls within the
true elements rather than the merely useful elements; perhaps the
indispensability arguments can be revived in this way. Alas, a glance
at any freshman physics text will disappoint this notion. Its pages are
littered with applications of mathematics that are expressly under-
stood not to be literally true: e.g., the analysis of water waves by
assuming the water to be infinitely deep or the treatment of matter
as continuous in fluid dynamics or the representation of energy as a
continuously varying quantity. Notice that this merely useful mathe-
matics is still indispensable; without these (false) assumptions, the
theory becomes unworkable.
It might be objected that these applications are peripheral, that
they are understood against the background of more fundamental
theories, and that it is in contrast with these that the applications
mentioned above are “idealizations,” “models,” “approximations,”
or useful falsehoods. For example, general relativity is a fundamen-
tal theory, and when space-time is described as continuous therein,
this is not explicitly regarded as less than literally true. So the argu-
ment goes.
mbarrant
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mbarrant
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282 THE JOURNAL OF PHILOSOPHY
But notice, when those pre-Einsteinians were skeptical of atomic
theory, it was a fundamental theory in this sense; it was not pro-
posed against another background as a convenient idealization or
mere approximation. The skeptics were bothered, not by such pe-
ripheral simplifications, but by what they saw as the impossibility of
directly testing the core hypotheses of atomic theory. But consider
now the hypothesis that space-time is continuous. Has this been
directly tested? As Quine himself points out, “no measurement
could be too accurate to be accommodated by a rational number,
but we admit the [irrationals] to simplify our computations and gen-
eralizations.”‘4 Similarly, space-time must be regarded as continu-
ous so that the highly efficacious continuum mathematics can be
applied to it. But the key question is this: Is that continuous charac-
ter “experimentally verified” or merely useful? If it is merely useful,
then the indispensability argument sketched earlier, the one relying
on the role of continuum mathematics in science to support the
Zermelo-Fraenkel axioms, or ZFC, cannot be considered conclusive.
I shall not try to answer this question here; to do so would require
a more thorough study of the physics literature than I am capable of
launching just now. But until such a study is undertaken, until the
evidence for the literal continuity of space-time is critically exam-
ined,’5 I think the simple observations collected here are enough to
raise a serious question about the efficacy of this particular indis-
pensability argument.
IV. FOUNDATIONS OF SET THEORY
My second reservation about the indispensability arguments rests on
somewhat less familiar grounds; to reach it, I must review a bit of set
theory and (in the next section) a bit of physics.
It is well-known that the standard axioms of contemporary set
theory, ZFC, are not enough to decide every naturally-arising set
theoretical question.’6 The most famous independent statement is
the Cantor’s continuum hypothesis (CH), but there are others, some
of them more down-to-earth than CH. For example, between the
mid-seventeenth and the late nineteenth century, under pressure
from both physical and mathematical problems, the notion of func-
1 “Reply to Charles Parsons,” p. 400.
15 This issue of the continuity of space-time will take an unexpected turn in
sect. V.
16 Kurt Godel’s incompleteness theorem is enough to establish that there are
set-theoretic statements that can neither be proved nor disproved from ZFC, but
Godel’s later work on the inner model of constructible sets and Paul Cohen’s
forcing methods yield more: there are statements which mathematicians have
found it natural to ask which are likewise independent.
INDISPENSABILITY AND PRACTICE 283
tion became more and more general. Around the beginning of our
century, various mathematicians undertook to bring order to the
wild domain of discontinuous functions. It soon became clear that
the complexity of functions could be understood in terms of the
complexity of sets of real numbers-e.g., a function is continuous
if the inverse image of every open set is open-and this naturally
led to a serious study of the properties of definable sets of reals.
Among these, the Borel sets could easily be shown, for example, to
be Lebesgue measurable. This result generalized to projections’7 of
Borel sets (the analytic sets) and to the complements of these (the
coanalytic sets). One more application of projection produces the
sets we now call :2, but the question of their Lebesgue measurabi-
lity remained stubbornly unsolved. Sometime later, this question
was shown to be independent of ZFC.’8
In contrast with CH, this question concerns only a limited class of
definable sets of reals, sets whose definitions have concrete geomet-
ric interpretations, and it involves the intuitive analytic notion of
Lebesgue measurability rather than Cantor’s bold new invention,
the comparison of infinite cardinalities. In other words, it might be
said that this independent question, unlike CH, arose in the
straightforward pursuit of analysis-as-usual. And there are others of
this type.
There is a serious foundational debate about the status of these
statements. Despite their independence of ZFC, one might hold that
there is nevertheless a fact of the matter, that the statements are
nevertheless either true or false, and that it is the burden of further
theorizing to determine which.’9 At the other extreme, another
might insist that ZFC is all there is to set theory, that a statement
independent of these axioms has no inherent truth value, that the
study of extensions of ZFC that settle these questions one way or the
other are all equally legitimate. For future reference, let me attach
labels to crude versions of these positions: letfact be the bare as-
sumption that there is a determinate answer to our question, and let
the opposing view be no-fact.
Now let us pose this foundational question to the indispensability
theorist, taking the simple version of indispensability first, for pur-
17 The projection of a subset of the plane is its shadow on one of the coordinate
axes.
1 I discuss this history in more detail in ch. 4 of Realism in Mathematics and
in “Taking Naturalism Seriously.”
19 This was Godel’s view, and the one defended in Realism in Mathematics.
284 THE JOURNAL OF PHILOSOPHY
poses of comparison. Continuum mathematics, including every-
thing from real-valued measurement to the higher calculus, is
among the most widely applied branches of mathematics, and at
least some of the many physical theories in which it is applied are
extremely well-confirmed. Therefore, so the argument goes, we
have good reason to believe in the entities of continuum mathemat-
ics, for example, the real numbers. In these applications-for exam-
ple, in the theory of space (or space-time)-we also find quantifica-
tion over sets of reals, (or equivalently, over regions of space (or
space-time)), though particular instances are rarely as complex as
1 .21 If we believe in the reals and in those sets of reals definable in
our theory, then it seems we should accept the legitimacy of the
question: Are ‘ sets Lebesgue measurable?22 Thus the simple indis-
pensability theorist should endorse fact.
What follows from this for the practice of set theory? Should the
set theorist, as Godel suggests, seek an answer to this legitimate
question? Given that our independent question seems, for now, to
have no bearing on physical theory, and that it is not settled by the
most generous of “simplificatory rounding outs” (i.e., ZFC), the
simple indispensability theorist, who uses only the justificatory
methods of physical science, has no means for answering it. Further-
more, the exclusive focus on the needs and methods of physical
science hints at a lack of interest in any question without physical
ramifications. If so, the simple theorist may disagree with no-fact,
classifying our independent question as one with an unambiguous
truth value, without going so far as charging future theorists with
the task of answering it. Call this weak fact.
Fortunately, these subtle matters of interpretation are beside the
point here, because we have already identified the modified indis-
pensability argument as more promising than the simple. Like the
simple theorist, the modified indispensability theorist embraces the
ontology of continuum mathematics, on the basis of its successful
applicability, and thus the legitimacy of our independent question,
but she goes beyond the simple theorist by ratifying the set theorist’s
20 For now, I shall ignore the scientific practice objection and take fundamental
scientific theory at face value.
21 One theoretical proposal that involves nonmeasurable sets of reals is I. Pi-
towsky, “Deterministic Model of Spin and Statistics,” Physical Review D, xxvii,
10 (15 May 1983): 2316-26. (M. van Lambalgen called this paper to my atten-
tion.)
22 This might be avoided if we took mathematical entities to be somehow “in-
complete,” an idea of Parsons’s which Quine considers in passing. See Quine’s
“Reply to Charles Parsons,” p. 401, and references cited there.
INDISPENSABILITY AND PRACTICE 285
search for new axioms to answer the question. Call this strong fact.
Although the evidence for or against these axiom candidates will
derive, not from physical applications, but from considerations in-
ternal to mathematics, the modified theorist sees the past success of
such mathematical methods as justifying their contin-
ued use.
Either way, then, the indispensability theorist should adopt some
version of fact. Notice, however, that this acceptance of the legiti-
macy of our independent question and (for the modified theorist)
the legitimacy of its pursuit is not unconditional; it depends on the
empirical facts of current science. The resulting mathematical be-
liefs are likewise a posteriori and fallible.
V. FOUNDATIONS OF PHYSICS
Here the fundamental theories are general relativity and quantum
mechanics, and the central problem is to reconcile the two. Three of
the fundamental forces-electromagnetic, weak, and strong-have
yielded more or less workable quantum-field descriptions, but grav-
ity remains intractable. The mathematics of quantum-field theories
has an annoying habit of generating impossible (infinite) values for
some physical magnitudes, but the problem has been overcome by a
technical trick called “renormalization” in all cases but that of grav-
ity. So the problem remains: How to characterize the gravitational
force at quantum distances?
Physicists engage in fascinating speculations on the source of the
difficulty: for example, that it arises from the attempt to combine
the essentially passive space-time of quantum theory with the dy-
namic space-time of general relativity.23 But most important for our
purposes is the idea that the fault might lie in our conception of
space-time as a mathematical continuum. For example, Richard
Feynman24 writes:
I believe that the theory that space is continuous is wrong, because we
get these infinities and other difficulties . . . (ibid., p. 166).
And Chris Isham:
… it is clear that quantum gravity, with its natural Planck length, raises
the possibility that the continuum nature of spacetime may not hold
below this length, and that a quite different model is needed (op. cit.,
p. 72).
23 See Chris Isham, “Quantum Gravity,” in The New Physics, P. Davies, ed.
(New York: Cambridge, 1989), pp. 70-93, esp. p. 70. I review some of the popu-
lar literature on this problem in “Taking Naturalism Seriously.”
24 The Character of Physical Law (Cambridge: MIT, 1967).
286 THE JOURNAL OF PHILOSOPHY
Here the suggestion is not (as in the previous section) that the conti-
nuity of space-time is a “mere idealization,” but that it does not
belong in our best theory at all!
All this, as I have indicated, is quite speculative; no one yet knows
what a reasonable theory of quantum gravity might be like. But the
very suggestion that space-time may not be continuous is enough to
add previously unimagined poignancy to our earlier conclusion of a
posteriority and fallibility. What seemed a rather small concession at
the end of the last section-that the grounds for the indispensability
theorist’s adherence to fact could be overthrown by progress in
physics-now looms as a real possibility.
Of course, the (potential) falsity of continuum mathematics in its
application to space-time would be only part of the story; there are
other successful scientific uses for the calculus and higher analysis.
But, if science were to change so that all fundamental theories were
thoroughly quantized, so that no continuum mathematics appeared
there, if all the remaining applications of continuum mathematics
were explicitly understood as “approximations” or “idealizations”
or “models,” then even the modified indispensability theorist would
retreat to some version of no-fact. The case of quantum gravity
should keep us from dismissing this possibility out of hand.
VI. THE MATHEMATICAL PRACTICE OBJECTION
For the modified indispensability theorist, the choice between
strong fact and no-fact hinges on developments in physics (and per-
haps the rest of science). We should now ask the impact of this
choice: How would the pursuit of our independent question be af-
fected by it? In other words, we want to know if the metaphysical
distinction between strong fact and no-fact has methodological con-
sequences.
If no-fact is correct, if there is no pre-existing fact to discover
about the Lebesgue measurability of I’ sets of reals, then what
approach should the set theorist take? Many observers would hold
that no-fact is the end of the story, that mathematicians are in the
business of discovering truths about mathematical reality, and that,
if there is no truth to be found, the mathematician should reject the
question. From this point of view, all (relatively consistent) set the-
ories extending ZFC are equally legitimate, there is no call to chose
between them, and indeed, no grounds on which to do so apart
from subjective aesthetic preferences. Once our question is shown
to be independent, and developments in science undercut the claim
to inherent truth value, there is nothing more of serious import to
be said about the Lebesgue measurability of I’ sets. Call this end of
the story no-fact.
INDISPENSABILITY AND PRACTICE 287
This position makes nonsense of the contemporary search for new
set-theoretic axioms to settle independent questions like ours. In-
deed, in our case, there are two competing candidates: V = L
(G6del’s “axiom of constructibility”) and MC (the existence of a
measurable cardinal). If V = L, then there is a non-Lebesgue mea-
surable I’ set; if MC, then all I’ sets are Lebesgue measurable. Set
theorists offer arguments for and against these axiom candidates,25
and in this debate, MC is strongly favored over V = L.26 If we are
not to reject this activity as inconsequential mutterings-an espe-
cially unappealing move, given that the original axioms of ZFC are
supported by arguments of a similar flavor-we must instead reject
end of the story no-fact.
But there is another version of no-fact. Even if there is no pre-ex-
isting fact of the matter to be discovered, the process of extending
the axioms of set theory might well be governed by nonarbitrary
principles. This idea turns up, not only in the study of set theory,
but when ontological decisions are made in other branches of mathe-
matics as well. For example, Kenneth Manders27 describes the theo-
retical norms at work in the expansion of the domain of numbers to
include the imaginary or complex numbers, and Mark Wilson28 un-
covers the rationale behind the move from affine to projective geom-
etry. In such cases, despite lip service to the notion that any consis-
tent system is as good as any other, mathematicians actually insist
that a given mathematical phenomenon is correctly viewed in a cer-
tain (ontological) setting, that another setting is incorrect.
One need not assume fact to endorse these practices. Even if
there is no fact of the matter, no pre-existing truth about the exis-
tence or nonexistence of complex numbers or geometric points at
infinity or nonconstructible sets, the pursuit of these mathematical
topics might be constrained by mathematical canons of “correct-
ness.” For our case, one might hold that there is no fact of the
matter about the Lebesgue measurability of I’ sets, but that there
are still good mathematical reasons to prefer extending ZFC in one
way rather than another, and perhaps, good mathematical reasons
to adopt a theory that decides our question one way rather than
another. From this point of view, no-fact is just the beginning of the
25 See my “Believing the Axioms. I-II,” The Journal of Symbolic Logic, LIII, 2
(June 1988): 481-511, and 3 (September 1988): 736-64.
26 I discuss part of the case against V = L in “Does V equal L?” in The Journal
of Symbolic Logic (forthcoming).
27 “Domain Extension and the Philosophy of Mathematics,” this JOURNAL,
LXXXVI, 10 (October 1989): 553-62.
28 “Frege: The Royal Road from Geometry,” in Nouis (forthcoming).
288 THE JOURNAL OF PHILOSOPHY
story; it opens the door on the fascinating study of purely mathemat-
ical canons of correctness. Call this beginning of the story no-fact.
The question before us is this: What are the methodological con-
sequences of the choice between strong fact and no-fact? If strong
fact is correct, the set theorist in search of a complete theory of her
subject matter should seek out additional true axioms to settle the
Lebesgue measurability of I’ sets (and so on). If end of the story
no-fact is correct, the set theorist left with any interest in the matter
should feel free to adopt any (relatively consistent) extension of ZFC
she chooses, or even to move back and forth between several mutu-
ally contradictory such extensions at will. And finally, if beginning
of the story no-fact is correct, the set theorist should use appropriate
canons of mathematical correctness to extend ZFC and to decide
the question.29
Obviously, the method prescribed by strong fact differs from that
prescribed by end of the story no-fact; that much is easy. But what
about strong fact and beginning of the story no-fact? Does the pur-
suit of truth differ from the pursuit of mathematical correctness? In
fact, I think it does. Consider, for example, a simple argument that
V = L should be rejected because it is restrictive. A supporter of this
argument owes us an explanation of why restrictive theories are bad.
A beginning of the story no-fact-er might say, “because the point of
set theory is to realize as many isomorphism types as possible, and
set theory with MC is richer in this way.”30 A strong fact-er might
agree that the world of MC has desirable properties, while insisting
that desirability (notoriously!) is no guarantee of truth.3′ Faced with
the beginning of the story no-fact-er’s argument, a strong fact-er
would reply, “Yes, MC is nice in the way you indicate, but if V does
29 The mathematical canons invoked in beginning of the story no-fact could
ultimately recommend that several different set theories be accorded equal status.
The methodology at work would still be different from that of end of the story
no-tact, and the range of theories endorsed would almost certainly be narrower.
3 Spelling out this line of thought precisely is no simple exercise, but I shall
leave that problem aside here. The point is just that the beginning of the story
no-fact-er appeals to some attractive feature of set theory with MC.
Philip Kitcher touches on this point in his reply to Manders, “Innovation and
Understanding in Mathematics,” thisJoURNAL, LXXXVI, 10 (October 1989): 563-
4, when he writes, p. 564: “Suppose this is a way in which mathematical knowl-
edge can grow. What kinds of views of mathematical reality and mathematical
progress are open to us? Can we assume that invoking entities that satisfy con-
straints we favor is a legitimate strategy of recognizing hitherto neglected objects
that exist independently of us? From a realist perspective, the method of postulat-
ing what we want has (in Bertrand Russell’s famous phrase) ‘the advantages of
theft over honest toil.’ If that method is, as Richard Dedekind supposed, part of
the honest trade of mathematics, is something wrong with the realist perspec-
tive?”
INDISPENSABILITY AND PRACTICE 289
equal L, L does contain all the isomorphism types possible. What’s
needed is an argument that V = L is false.” So strong fact will differ
methodologically from beginning of the story no-fact as well as end
of the story no-fact.
We have reached this point: a methodological decision in set
theory-namely, that between the methodologies proper to strong
fact and to beginning of the story no-fact-hinges on developments
in physics. If this is correct, set theorists should be eagerly awaiting
the outcome of debate over quantum gravity, preparing to tailor the
practice of set theory to the nature of the resulting applications of
continuum mathematics. But this is not the case; set theorists do not
regularly keep an eye on developments in fundamental physics. Fur-
thermore, I doubt that the set-theoretic investigation of indepen-
dent questions would be much affected even if quantum gravity did
end up requiring a new and different account of space-time; set
theorists would still want to settle open questions about the mathe-
matical continuum. Finally, despite the current assumed indispensa-
bility of continuum mathematics, I suspect that the actual approach
to the Lebesgue measurability of V sets, to V = L versus MC, is
more like that prescribed by beginning of the story no-fact than that
prescribed by strong fact,32 and I see no mathematical reason to
criticize this practice. In short, legitimate choice of method in the
foundations of set theory does not seem to depend on physical facts
in the way indispensability theory requires.
VII. CONCLUSION
I have raised two doubts about indispensability theory, even modi-
fied indispensability theory, as an account of mathematics as prac-
ticed. The first, the scientific practice objection, notes that indispen-
sability for scientific theorizing does not always imply truth and calls
for a careful assessment of the extent to which even fundamental
mathematized science is “idealized” (i.e., literally false). The second,
the mathematical practice objection, suggests that indispensability
theory cannot account for mathematics as it is actually done. If these
objections can be sustained, we must conclude that the indispensa-
bility arguments do not provide a satisfactory approach to the ontol-
ogy or the epistemology of mathematics. Given the prominence of
indispensability considerations in current discussions, this would
amount to a significant reorientation in contemporary philosophy of
mathematics.
PENELOPE MADDY
University of California/Irvine
32 I hope to argue this in some detail elsewhere.
- Article Contents
- Issue Table of Contents
p. 275
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The Journal of Philosophy, Vol. 89, No. 6 (Jun., 1992), pp. 275-330
Front Matter
Indispensability and Practice [pp. 275 – 289]
The Skeleton in Frege’s Cupboard: The Standard Versus Nonstandard Distinction [pp. 290 – 315]
Book Reviews
untitled [pp. 316 – 320]
untitled [pp. 321 – 326]
untitled [pp. 326 – 329]
Notes and News [p. 330]
Back Matter
McMullin’s Inference: A Case for Realism?
with Bas C. van Fraassen, “Scientific Realism and the Empiricist Challenge: An
Introduction to Ernan McMullin’s Aquinas Lecture”; and Ernan McMullin, “The
Inference that Makes Science”
SCIENTIFIC REALISM AND THE EMPIRICIST
CHALLENGE: AN INTRODUCTION TO ERNAN
MCMULLIN’S AQUINAS LECTURE
by Bas C. van Fraassen
Abstract. In The Inference That Makes Science, Ernan McMullin
recounts the clear historical progress he saw toward a vision of the
sciences as conclusions reached rationally on the basis of empirical
evidence. Distinctive of this vision was his view of science as driven by
a specific form of inference, retroduction. To understand this properly,
we need to disentangle the description of retroductive inference from
the claims made on its behalf. To end I will suggest that the real rival to
McMullin’s vision of science is not the methodologies he criticizes so
successfully but a more radical empiricist alternative in epistemology.
Keywords: abduction; empiricism; induction; Ernan McMullin;
retroduction; scientific realism
In The Inference That Makes Science, Ernan McMullin takes us on a fabulous
journey through the history of philosophy of science, displaying clear
progress toward a vision of the sciences as conclusions reached rationally
on the basis of empirical evidence.1 This is McMullin’s vision, distinctively
his, though in large outlines shared by the twentieth-century philosophers
to whom he refers, in the last few pages, as scientific realists. And surely, in
large outlines, though with characteristic qualifications, it is also shared by
those to whose contrasting points of view he refers there as instrumentalist.
For on all hands, the empirical sciences are accepted as a paradigm of
rational inquiry into what our world is like.
But the title itself announces what is distinctive of McMullin’s view:
the sciences are driven by a specific form of inference that accounts for
Bas C. van Fraassen is a professor of philosophy at San Francisco State University and
may be contacted at the Department of Philosophy, San Francisco State University, 1600
Holloway Avenue, San Francisco, CA 94132, USA; e-mail: fraassen@sfsu.edu.
Unless otherwise noted, page references will be to the text of The Inference That Makes
Science (originally McMullin 1992) which is reprinted in this issue of Zygon: Journal of
Religion and Science.
[Zygon, vol. 48, no. 1 (March 2013)]
C© 2013 by the Joint Publication Board of Zygon ISSN 0591-2385 www.zygonjournal.org
131
132 Zygon
their success, and is indeed the hallmark of the scientific approach to
any subject. As the history unfolds we see the attempts, one after the other
found wanting, to identify that form of inference, until its final articulation
as a process of (as McMullin decides to call it) retroductive inference.
Aristotle, Grosseteste, Aquinas, Galileo, Zabarella, the late me-
dieval nominalists and Francis Bacon, Isaac Newton’s methodological
vacillations, . . . the story reads as well and as fluently as a mystery novel,
and is as engaging. What I shall comment on here is not McMullin’s
excursions into history, however, though they were certainly for me the
most fascinating part. My concern will instead be with McMullin’s project,
the project to characterize the sciences as, in essence, a practice identified
by a form of inference.
CONTRASTING MCMULLIN’S VISION WITH RIVAL
EPISTEMOLOGIES
McMullin does enough to discredit some alternative projects with a similar
aim, such as attempts to define induction as a method for science. Today
other projects of that sort exist as well, drawing in one way or another on
the concept and theories of probability, notably varieties of Bayesianism
or a more liberal probabilism. It would be of interest to ask how, or
to what extent, such alternatives could do justice to the insights that
support McMullin’s concept of retroduction as the crucial or central form
of scientific inference. I will leave that aside as well. The more interesting
question, for me, is rather whether scientific practice, the enterprise of
science, is best characterized in that sort of form at all.
McMullin does not have an overriding ambition in this project. He
emphasizes that it was “not intended to furnish a criterion of demarcation
between science and non science [ . . . .] retroductive inference makes use
of ingredients that are commonplace in human reason generally” (144).
In good human reasoning to be sure; McMullin mentions approvingly the
detective and the journalist. But retroduction is easily discerned in not so
good human reasoning as well, when conspiracy theorists are retroductively
inferring from the facts in evidence to their weird or wonderful causal
explanation. So it seems at least at first blush as if the hallmark of scientific
inquiry will not be that the form of inference is different, but rather how
well it is employed:
What is distinctive about the way in which explanatory theories are constructed
and tested in natural science is the precision, as well as the explicitness, with which
retroductive inference is deployed. (146)
But that is too modest. It is not just a matter of doing it better, not just a
matter of greater precision and explicitness, because McMullin emphasized
features of the practice that are not captured by such earlier accounts as
Bas C. van Fraassen 133
were focused on deduction, induction, or even Peircean abduction. The
details emerge for McMullin after a long scrutiny of errors and insights
accumulating through some twenty centuries of reflection on the matter,
and they are not simple or neat, let alone algorithmic.
As a process of inference, retroduction “is not rule-governed as deduction
is, nor regulated by technique as induction is” (183). McMullin elaborates
on this elsewhere, indicating a strong difference from another rival that
was much in the limelight in the closing decades of the twentieth century:
retroduction [ . . . ] is not a strict form of rule-governed reasoning, or at least,
it is not as long as it isn’t equated with the easily-criticized “inference to best
explanation.” [ . . . .] The vulnerability of such an inference need hardly be
emphasized. (McMullin 2007, 175)
These are important differences, and it is a characteristically twentieth-
century insight that rational change in view is not a matter of rule following,
that rules of right reason cannot be dictates, only guidelines. But something
is needed beyond this negative point.
MCMULLIN’S ACCOUNT OF RETRODUCTIVE INFERENCE
It is in fact not easy to disentangle the points that allow us to recognize
a process of retroductive inference from the claims McMullin makes
concerning this sort of inference. We must concentrate on the definitive
account that McMullin provides in the last 5 pages (in the reprint that
follows) of The Inference That Makes Science, but it may help to look
first at a formulation McMullin provided in a later publication, as a short
summary:
Retroduction, argument from observed data to an explanatory causal structure
which may itself be unobserved though not necessarily unobservable is of its
essence tentative. It terminates in likelihood (in the everyday sense of that term,
not the sense given it in probability theory). It allows for the gradual mounting of
evidence of all sorts: increasingly troublesome anomalies eliminated, ambiguities
resolved, new evidence successfully incorporated, and the rest. Above all, under
certain circumstances it encourages more and more persistent questioning of the
assumption that the paradigm in possession is beyond challenge or that a potential
rival is, on the face of it, absurd. There is a lot of room here between strict reason
and credo quia absurdum, the room afforded by an ever-increasing likelihood that
may begin from a very low level indeed. (McMullin 2007, 176)
To what extent is this a description, such as a neutral observer of scientific
practices might give, and to what extent does it involve claims about the
adequacy or rationality or truth-conduciveness of this form of inference?
First: that in such an inference we are “led backwards” from effect
to cause, for example, we can read as merely describing the form (from
premises about what happens to conclusions about what causes them).
But we can also read it as a claim that what happens is always in fact an
134 Zygon
effect, that is, an event that has causes, and in addition that these causes
are discovered by retroductive inference. That this composite claim is in
fact part of what McMullin maintains becomes quickly evident toward the
end of his Aquinas lecture.
That McMullin is making a strong claim on behalf of this form appears
also earlier in his critique of Newton, whom he describes as having been
misled by the “quasi-demonstrative” form of his own writings, and as
having had a distorting influence on eighteenth century methodological
reflection, which
was to have negative repercussions for decades to come, until the atoms and ether-
vibrations of the early nineteenth century once and for all showed causal inference
to underlying structure to be indispensable to the work of the physical scientist.
(180)
Second: one feature McMullin lists, which clearly distinguishes this
retroductive inference, is the creation of new concepts. We can imagine a
situation in which all attempts at explanation fail, within the conceptual
framework that has been actualized so far. In that case—and surely there
are famous historical cases of this sort—a smaller or larger conceptual
revolution is the only way forward. As a distinguishing mark of retroductive
inference, though, it has its limits; for this feature is one that may be present,
and certainly is not always involved.
Once again, Newton furnishes the bad example of a misdirected
empiricism. The need for new concepts and new language appears to
be ruled out by Newton’s Third Rule of Reasoning which postulated that
the relevant properties of all bodies would be those accessible to the human
senses. And this was not incidental, Newton “needed this restriction . . . in
order that induction might be, as he claimed, the all-sufficient method of
natural science” (185).
Third: that the product of retroduction is a theory which presents a
causal explanation, distinct from the sort of empirical law that registers a
regularity, is crucial. We can perhaps typically see the feature of causal
“explanatoriness” at a glance, and if so it can serve as a hallmark to
recognize retroduction. But even here a claim of adequacy or efficacy, not
just something offered as description, is entangled with the description:
The language here is, of course, that of scientific realism. It is because the cause is,
in some sense however qualified, affirmed as real cause, that retroduction functions
as a distinct form of inference. (184)
Here, after all, Newton appears as on the side of the angels. For this phrasing
echoes Newton’s First Rule of Reasoning, the “vera causa” principle. What
I will suggest though is that inductivism in the naı̈ve form that Newton
may have preached, if not practiced, is in any case not the most important
rival to McMullin’s view of science.
Bas C. van Fraassen 135
AN ANALOGY, TO ARRIVE AT WHAT MAY BE DISTINCTIVELY
DIFFERENT
As an analogy, suppose that someone wanted to construct an account not of
what science is but of business, commerce. If someone starts a business, he
will begin by amassing some capital, acquire a place of business, equipment,
inventory, employees, and begin to advertise. As the business gets going he
has to look ahead, plan replenishing his stock, have reserve funds for repair
and for salary, including his own, when receipts are lagging. What is the
inference that makes business?
Certainly inference is involved. Evidence of demand for his goods or
services needs to be available before he can set out at all. A record of the
expenses and receipts, and the timing of each, forms a growing base of
evidence that he needs to consult continually, not simply to assess how
well he is doing but to assess what is needed to go on. This assessment is a
process of arriving at some conclusion that, though perhaps not logically
derivable from that evidence, is at least sufficiently likely to him in the light
of that evidence. That process is a process of inference. So yes, inference is
involved.
But this we could say of almost any form of intentional activity or
practice. In order to characterize business in a way that distinguishes it
from other human practices, is looking for a distinctive form of the sort
of inference involved the right thing to do? Is business distinguished by
a special form of inference? Is engaging in that sort of inference precisely
what it is to do business?
McMullin’s concentration on inference in developing his view of science,
in continuation with the tradition he explores, suggests that we should
assume science to be distinguished from such other practices as business
and commerce in these terms. Science, not business or commerce or the
like, is distinguished by a special form of inference. But it takes patience
and willingness to look for differences, partly differences of degree and
partly of kind, to elucidate what is special about that special form.
We can go back at this point to the early pages of McMullin’s Aquinas
lecture and remember that the ingredients of retroductive inference, as
present in science, are commonplace in human reason generally. That all
sorts of rational ways to reach conclusions are involved in business, and that
this should be a common feature of business practice and scientific practice,
should come as no surprise. It may well be in addition that in business
sometimes the way to victory over rivals, to commercial success, can only
come through the creation of new, novel concepts. A new invention,
conceptually novel, may open an opportunity for a business to take on
a whole new form. The concept of the computer was with us since Pascal,
but not the concepts involved in electronic information storage; that may
be a case in point.
136 Zygon
But then, if the differences are to be discerned about much that is in
common, all the weight of McMullin’s project comes to rest on some
specific claims he makes, even if implicitly in the main, that we need to
disentangle from whatever counts as a neutral description of the form.
MCMULLIN’S CLAIMS ON BEHALF OF RETRODUCTION
Those claims I introduced briefly above; let me quote the entire passage
now:
retroduction is not an atemporal application of rule as deduction is. It is extended
in time, and logically very complex. It is properly inference, since it enables one to
move in thought from the observation of an effect to the affirmation, with greater
or lesser degree of confidence, of the action of a cause of a (partially) expressed
sort.
The language here is, of course, that of scientific realism. It is because the cause is,
in some sense however qualified, affirmed as real cause, that retroduction functions
as a distinct form of inference. (184)
McMullin goes on to indicate clearly the position in epistemology that
provides the epistemic side of this scientific realism:
It is a far cry from the demonstrations of Aristotle to the retroductions of modern
theoretical science. Where they differ is, first, that retroduction makes no claim
of necessity, and it settles for less, much less, than definitive truth. It can, under
favorable conditions, when theories are well-established, yield practical certainty.
(185)
What we see here, I think, is that by “inference” McMullin means
something more than something that “enables one to move in thought
from the observation of an effect to the affirmation, with greater or lesser
degree of confidence, of the action of a cause of a (partially) expressed sort”
(184). Even adding, for example, “rationally” after the word “move” would
not be enough to complete what he means. Logically speaking, there is
still a gap between the statement that this is a rational sort of move from
evidence to an affirmed conclusion and the claim that this sort of move
is “truth-conducive,” that it is likely to lead to true conclusions. When
McMullin speaks here of a degree of confidence, and of practical certainty,
he is surely not just describing a subjective state of mind of the person
engaged in retroduction. He is claiming that retroductive inference, to the
unobserved causes of observed events classified as effects, leads to truth.
MCMULLIN’S SOPHISTICATED SCIENTIFIC REALISM
McMullin did not much like the use of “true” and “truth” in this context.
In his well-known “A Case for Scientific Realism,” he ends by apparently
eschewing claims to truth altogether:
Bas C. van Fraassen 137
I do not think that acceptance of a scientific theory involves the belief that it is
true. Science aims at fruitful metaphor and at ever more detailed structure. To
suppose that a theory is literally true would imply, among other things, that no
further anomaly could, in principle, arise from any quarter in regard to it. [ . . . ]
Scientists are very uncomfortable at this use of the word “true,” because it suggests
that the theory is definitive in its formulation . . . .
The realist would not use the term “true” to describe a good theory. He would
suppose that the structures of the theory give some insight into the structures of
the world. But he could not, in general, say how good the insight is. He has no
independent access to the world, as the antirealist constantly reminds him. His
assurance that there is a fit, however rough, between the structures of the theory
and the structures of the world comes not from a comparison between them but
from the sort of argument I sketched above, which concludes that only this sort
of reasoning would explain certain contingent features of the history of recent
science. (McMullin 1984, 35)
These passages make it clear, however, that the discomfort with “true”
signals only an implicature: use of “true,” in ordinary contexts, will convey
a possibly quite unwarranted certainty. That contextual import is not
typical in philosophical debate. Indeed, it was the scientific realists of the
1950s and 1960s who convinced the philosophical community to sever
all logical connection between certainty or verification (on the side of the
subject or speaker) and truth or reference. Nor does McMullin shrink back
from language that we are entirely used to connecting with knowledge,
and thereby to truth and reference. To have some insight into X is to know
something true about X, and implies the reality of X; to have assurance of
a fit between model and world is to have assurance that there is in fact a
fit between the two. These points are not affected by the degree of caution
with which any of this should be asserted.
But McMullin’s caution here is also alerting us to significant differences
between his scientific realism and that of other realists he mentions, such as
Hilary Putnam (circa. 1980) or Richard Boyd. For McMullin’s historically
and philosophically informed view of science is nuanced and sophisticated
in ways that much writing of the time was not. A remarkable feature
of his view is its emphasis on the metaphorical power of theory, with
its implications for intellectual fertility and the adventure of scientific
exploration, both experimental and theoretical. Discussion of this aspect of
his scientific realism—grounded in a conception of science quite different
from that found in naı̈ve realism—would warrant a separate article, and
I will leave it aside here. Instead we can look closely at how McMullin
characterizes our current topic, the character of retroductive inference in
science, in the above-mentioned article that makes his distinctive case for
scientific realism. For there too we see a remarkable twist away from the
more traditional epistemology’s conception of rules of right reason that
one might have suspected of McMullin’s allegiance.
138 Zygon
Here McMullin introduces the claim that retroduction is the inference
that makes science as, first of all, a historical conclusion about the
sciences, and does immediately enmesh that conclusion in claims about its
epistemic efficacy, but only to expose the radical contingency of any such
claim.
First, the historical conclusion:
A third consequence one might draw from the history of the structural sciences
is that there is a single form of retroductive inference involved throughout.
(McMullin 1984, 29)
Then the claim of a warrant for truth:
As C. S. Peirce stressed in his discussion of retroduction, it is the degree of success
of the retroductive hypothesis that warrants the degree of its acceptance as truth.
(1984, 29)
which is classified as a logical point:
What the history of recent science has taught us is not that retroductive inference
yields a plausible knowledge of causes. We already knew this on logical grounds.
(1984, 29)
But then comes the twist: if that is not what we learned from the history,
what precisely was learned is an actual, contingent, empirical fact about
retroductive inference:
What we have learned is that retroductive inference works in the world we have
and with the senses we have for investigating that world. This is a contingent
fact, as far as I can see. [ . . . ] There could well be a universe in which observable
regularities would not be explainable in terms of hidden structures, that is, a world
in which retroduction would not work. Indeed, until the eighteenth century, there
was no strong empirical case to be made against that being our universe. (1984,
29–30)
The sentence I omitted in this last quote is: “This is why realism as I have
defined it is in part an empirical thesis.” And McMullin emphasizes this in
several ways: “The realist seeks an explanation for the regularities he finds
in science, just as the scientist seeks an explanation for regularities he finds
in the world” (1984, 34), the realist’s “assurance that there is a fit, however
rough, between the structures of the theory and the structures of the world
comes not from a comparison between them but from the sort of argument
I sketched above, which concludes that only this sort of reasoning would
explain certain contingent features of the history of recent science” (1984,
35).
But at this point, it seems to me, McMullin has severely undermined
his own argument. For this means that the argument for the efficacy of
retroductive inference, and indeed for the claim that it is the inference
that makes science, is an argument that is itself an instance of retroductive
inference.
Bas C. van Fraassen 139
McMullin was quite right to dismiss naı̈ve forms of inference to the
best explanation in the current literature as easily criticized (McMullin
2007, 175). But at this point his form of defense of retroduction certainly
recalls the more familiar “that the practice of science as inference to the
best explanation accounts for the success of science is true, for it is the best
explanation of that success.”
Not every circle is vicious. To show that one may refer to the soundness
and completeness proofs for classical deductive logic, which are indeed
rigorous proofs in the sense that they themselves follow the rules of
classical logic. We gain real understanding of logic by going through those
arguments, and this might be offered as a parallel for the use of a form of
inference in the study of that form. But there is a great disanalogy. In the
case of logic we do not add, as McMullin did for retroduction, that there
could well be a universe in which logical inference would fail to preserve
truth.
If the warrant for the claim that retroductive inference accounts for
scientific success is itself the conclusion of a retroductive inference, what
is the warrant for that? For its warrant, it would have to return to the
purported logical connection between such inference and the warrant for
belief, indicated by McMullin’s rather offhand claim that “retroductive
inference yields a plausible knowledge of causes” is something we “already
knew . . . on logical grounds.” And whatever those logical grounds are
meant to be, they would have to be something that is very reassuring
after all, offsetting the admission of pure contingency, about how assured
we can be of having true insight into the causes behind the phenomena.
But that is where realist and empiricist part ways.
EMPIRICIST DEMURRAL
In McMullin’s description of retroductive inference as the process by which
science arrives at theories, we encounter a great liberalization of traditional
epistemology’s view of the sciences. As McMullin moves to his conclusion,
he shows us how much had to be progressively discarded. First, the
Aristotelian tradition’s pretension that we can arrive at contingent empirical
truths with certainty. Then on the other side the classical empiricist’s claim
that the general truths about nature can be deduced from the phenomena,
or arrived at by a straight induction from the evidence. Going even further,
scrutinizing his own heroes of epistemology, McMullin takes his distance
from William Whewell and Charles Sanders Peirce. The kind of confidence
in this sort of inference, expressed by Whewell (1847, vol. 2, 67, 284, 286)
with his proud claim that consilience is the mark of truth, and that no
truly consilient theory has ever been found false, McMullin eschews.
But McMullin is equally adamant that the kind of rule-following
paradigm of such lively contemporaneous movements as formal
140 Zygon
epistemology is wide off the mark. Retroductive inference is a creative,
innovative, often conceptually revolutionary, risk-taking, at the same time
severely self-policing, epistemic enterprise.
All of this must be music to any would-be empiricist’s ears today.
The emphasis on choice and practical decision in scientific progress that
entered early on in Hans Reichenbach’s, Rudolf Carnap’s, and other logical
empiricists’ writings, are here just as much evident in McMullin’s scientific
realist epistemology. So what sets McMullin still clearly opposite to that
tradition?
I think we can see the clues first of all in McMullin’s brief reference to
the first great schism he notes in the history he is retelling:
Looking at the Middle Ages as a whole, one would obviously have to separate two
quite diverse methodological traditions, the Aristotelion and the nominalist [ . . . .]
The Aristotelians remained faithful on the whole, to the ideal of demonstration
set down in the Posterior Analytics, while developing some aspects of that doctrine,
the distinction between demonstrations propter quid and quia, for example, much
more fully than Aristotle had done. The nominalists began to shape the notion
of inductive generalization, entirely rejecting the notion of necessary connection
between essence and property on which the older notion of demonstration had
been based. (Inference, 154)
That nominalist turn was also the first step on a course leading to Hume’s
critique of a concept of causality involving any sort of necessary connections
in nature.
There is no countering McMullin’s critique in The Inference That Makes
Science of Bacon, Hume, Mill and other such figures in the history of
the empiricist tradition who tried to raise naı̈ve inductive methods to the
status of scientific methodology. But that critique, however well it does
in demolishing those attempts, does not end the story for a more radical
empiricist view of our epistemic situation. McMullin sets those mistakes
aside only to return to the tradition that assures us of epistemic safety, of a
proper handling of evidence that we can be assured will be likely to lead to
truth about nature’s deepest structure.
We cannot be entirely sure, from McMullin’s actual text, what all is
involved in this. I think we can be sure that the “likely” does not signify
mere subjective probability on the side of the scientist, or for that matter, the
scientific realist. In addition, we have McMullin’s own qualifying comment
quoted above: “likelihood (in the everyday sense of that term, not the sense
given it in probability theory)” (McMullin 2007, 176). Whatever that sense
may be, it is to be understood as involving sufficient objectivity to give
bite to the professed realism. The entire tradition recounted by McMullin,
including the classical empiricism and the inductivism he associates with
it, consists in the unfolding of an ever more desperate search for epistemic
safety. If not demonstration then induction, if not evidence from recondite
experimental or observational procedures then the sense data of immediate
Bas C. van Fraassen 141
perception, if not inductive generalization from such data then retroductive
inference to the vera causa . . . safety for our beliefs about the natural world
can be gained.
So in this respect McMullin and the sorts of epistemology that he
submits to his severe critique are the same. The real opposition emerges in
a rival strand never given such ample critical attention: the vision of our
epistemic situation perhaps most clearly found in Blaise Pascal, but always
initially evoked in the recurring empiricist reactions to realism, before the
temptation to seek safety defeats it again.
That other reaction, the one that I would think proper to what
empiricism can be today, is to set aside any such illusory safety (see also
van Fraassen 2002). Recall that McMullin characterized retroduction as
“properly inference” because it “enables one to move in thought from the
observation of an effect to the affirmation, with greater or lesser degree of
confidence, of the action of a (partially) expressed sort.” From this, we can
glean a concept of inference in general, as a practice enabling one to move
in thought from given or assumed information to a conclusion. There are
applicable criteria of rationality, to ensure that at least consistency, perhaps
some stronger standard of coherence, even plausibility, are preserved. But
what they can guarantee only is to avoid inevitable or necessary failure
to reach truth. There is in the satisfaction of such criteria no guarantee,
however much we would like to have one, of reaching truth, with anything
more than subjective certainty or probability.
The real rival to McMullin’s vision is not the classical empiricist program
of induction as objective road from certainty in the deliverances of sense
to certainty going beyond the concrete individual fact. That was certainly
a philosophical illusion; McMullin is right about that. When McMullin
states, in “A Case for Scientific Realism,” that the success of retroductive
inference is a contingent matter, he realizes a crucial point, but does not go
far enough. That the contingent conditions for the success of retroduction
actually obtain could only be inferred by retroductive inference: an insight
that sweeps the rug from under any assurance of epistemic safety to be
found there.
Rationality and criteria of rationality for our epistemic life will rule out
self-sabotage, reject any procedures with built-in failure. Apart from that,
there remains just the admission that our procedures will work for us, and
give us the wherewithal to live and act in this world, only if the world
continues to be hospitable to them—and the hope, or faith, that it shall
be so.
NOTE
1. I am personally as well as academically deeply indebted to Ernan McMullin from whom
I learned much over the years; and I treasure especially the copy of The Inference That Makes
Science that Ernan gave me.
142 Zygon
REFERENCES
McMullin, Ernan. 1984. “A Case for Scientific Realism.” In Scientific Realism, ed. Jarrett Leplin,
8–40. Los Angeles: University of California Press.
———. 1992. The Inference That Makes Science. Milwaukee, WI: Marquette University Press.
Reprinted in Zygon: Journal of Religion and Science 48:143–191.
———. 2007. “Taking an Empirical Stance.” In Images of Empiricism, ed. Bradley Monton,
167–82. Oxford: Oxford University Press.
van Fraassen, Bas C. 2002. The Empirical Stance. New Haven, CT: Yale University Press.
Whewell, William. 1847. The Philosophy of the Inductive Sciences, 2nd ed. London. Reprinted
New York: Johnson Reprint, 1967.
80 Scientific American, August 2011
Fractals, such as this stack
of spheres created using
3-D modeling software, are
one of the mathematical
structures that were invent-
ed for abstract reasons yet
manage to capture reality.
© 2011 Scientific American
August 2011, ScientificAmerican.com 81Illustration by Tom Beddard
Mario Livio is a theoretical astrophysicist at the Space Telescope
Science Institute in Baltimore. He has studied a wide range of
cosmic phenomena, ranging from dark energy and super nova
explosions to extrasolar planets and accretion onto white dwarfs,
neutron stars and black holes.
Is math invented or discovered?
A leading astrophysicist suggests that the answer
to the millennia-old question is both
By Mario Livio
M
ost of us take it for granted
that math works—that sci
entists can devise formulas
to describe subatomic events
or that engineers can calcu
late paths for space craft. We
accept the view, initially es
poused by Galileo, that mathematics is the language of
science and expect that its grammar explains experi
mental results and even predicts novel phenomena.
The power of mathematics, though, is nothing short of
astonishing. Consider, for example, Scottish physicist
James Clerk Maxwell’s famed equations: not only do
these four expressions summarize all that was known
of electromagnetism in the 1860s, they also anticipat
ed the existence of radio waves two decades before
German physicist Heinrich Hertz detected them. Very
few languages are as effective, able to articulate vol
umes’ worth of material so succinctly and with such
precision. Albert Einstein pondered, “How is it possi
ble that mathematics, a product of human thought
that is independent of experience, fits so excellently
the objects of physical reality?”
As a working theoretical astrophysicist, I encoun
ter the seemingly “unreasonable effectiveness of math
ematics,” as Nobel laureate physicist Eugene Wigner
called it in 1960, in every step of my job. Whether I am
struggling to understand which progenitor systems
produce the stellar explosions known as type Ia super
novae or calculating the fate of Earth when our sun ul
timately becomes a red giant, the tools I use and the
models I develop are mathematical. The uncanny way
I N B R I E F
The deepest mysteries are often the things
we take for granted. Most people never
think twice about the fact that scientists
use mathematics to describe and explain
the world. But why should that be the case?
Math concepts developed for purely ab-
stract reasons turn out to explain real phe-
nomena. Their utility, as physicist Eugene
Wigner once wrote, “is a wonderful gift
which we neither understand nor deserve.”
Part of the puzzle is the question of wheth-
er mathematics is an invention (a creation
of the human mind) or a discovery (some-
thing that exists independently of us). The
author suggests it is both.
Math
P H I L O S O P H Y O F S C I E N C E
Works
Why
© 2011 Scientific American
82 Scientific American, August 2011
ED
W
A
RD
C
H
A
RL
ES
L
E
G
RI
CE
G
et
ty
Im
ag
es
that math captures the natural world has fascinated me through
out my career, and about 10 years ago I resolved to look into the
issue more deeply.
At the core of this mystery lies an argument that mathemati
cians, physicists, philosophers and cognitive scientists have had
for centuries: Is math an invented set of tools, as Einstein be
lieved? Or does it actually exist in some abstract realm, with hu
mans merely discovering its truths? Many great mathemati
cians—including David Hilbert, Georg Cantor and the group
known as Nicolas Bourbaki—have shared Einstein’s view, associ
ated with a school of thought called Formalism. But other illustri
ous thinkers—among them Godfrey Harold Hardy, Roger Pen
rose and Kurt Gödel—have held the opposite view, Platonism.
This debate about the nature of mathematics rages on today
and seems to elude an answer. I believe that by asking simply
whether mathematics is invented or discovered, we ignore the
possibility of a more intricate answer: both invention and dis
covery play a crucial role. I posit that together they account for
why math works so well. Although eliminating the dichotomy
between invention and discovery does not fully explain the un
reasonable effectiveness of mathematics, the problem is so pro
found that even a partial step toward solving it is progress.
INVENTION AND DISCOVERY
mathematics is unreasonably effective in two distinct ways, one I
think of as active and the other as passive. Sometimes scientists
create methods specifically for quantifying realworld phenome
na. For example, Isaac Newton formulated calculus for the pur
pose of capturing motion and change, breaking them up into in
finitesimally small framebyframe sequences. Of course, such ac
tive inventions are effective; the tools are, after all, made to order.
What is surprising, however, is their stupendous accuracy in some
cases. Take, for instance, quantum electrodynamics, the mathe
matical theory developed to describe how light and matter inter
act. When scientists use it to calculate the magnetic moment of
the electron, the theoretical value agrees with the most recent
experimental value—measured at 1.00115965218073 in the ap
propriate units in 2008—to within a few parts per trillion!
Even more astonishing, perhaps, mathematicians sometimes
develop entire fields of study with no application in mind, and yet
decades, even centuries, later physicists discover that these very
branches make sense of their observations. Examples of this kind
of passive effectiveness abound. French mathematician Évariste
Galois, for example, developed group theory in the early 1800s for
the sole purpose of determining the solvability of polynomial
equations. Very broadly, groups are algebraic structures made up
of sets of objects (say, the integers) united under some operation
(for instance, addition) that obey specific rules (among them the
existence of an identity element such as 0, which, when added to
any integer, gives back that same integer). In 20thcentury phys
ics, this rather abstract field turned out to be the most fruitful
way of categorizing elementary particles—the building blocks of
matter. In the 1960s physicists Murray GellMann and Yuval
Ne’eman independently showed that a specific group, referred to
as SU(3), mirrored a behavior of subatomic particles called had
rons—a connection that ultimately laid the foundations for the
modern theory of how atomic nuclei are held together.
The study of knots offers another beautiful example of passive
effectiveness. Mathematical knots are similar to everyday knots,
except that they have no loose
ends. In the 1860s Lord Kelvin
hoped to describe atoms as knot
ted tubes of ether. That misguid
ed model failed to connect with
reality, but mathematicians con
tinued to analyze knots for many
decades merely as an esoteric
arm of pure mathematics. Amaz
ingly, knot theory now pro vides
important insights into string
theory and loop quantum gravi
ty—our current best attempts at
articulating a theory of space
time that reconciles quantum
mechanics with general relativi
ty. Similarly, English mathemati Similarly, English mathematiSimilarly, English mathemati
cian Hardy’s discoveries in num
ber theory advanced the field of
cryptography, despite Hardy’s
earlier proclamation that “no one
has yet discovered any warlike purpose to be served by the theo
ry of numbers.” And in 1854 Bernhard Riemann described non
Euclidean geo met ries— curious spaces in which parallel lines
converge or diverge. More than half a century later Einstein in
voked those geometries to build his general theory of relativity.
A pattern emerges: humans invent mathematical concepts
by way of abstracting elements from the world around them—
shapes, lines, sets, groups, and so forth—either for some specific
purpose or simply for fun. They then go on to discover the con
nections among those concepts. Because this process of inventing
and discovering is manmade—unlike the kind of discovery to
which the Platonists subscribe—our mathematics is ultimately
based on our perceptions and the mental pictures we can conjure.
For instance, we possess an innate talent, called subitizing, for in
stantly recognizing quantity, which undoubtedly led to the con
cept of number. We are very good at perceiving the edges of indi
vidual objects and at distinguishing between straight and curved
lines and between different shapes, such as circles and ellipses—
abilities that probably led to the development of arithmetic and
geometry. So, too, the repeated human experience of cause and ef
fect at least partially contributed to the creation of logic and, with
it, the notion that certain statements imply the validity of others.
SELECTION AND EVOLUTION
michael atiyah, one of the greatest mathematicians of the 20th
century, has presented an elegant thought experiment that re
veals just how perception colors which mathematical concepts we
embrace—even ones as seemingly fundamental as numbers. Ger
man mathematician Leopold Kronecker famously declared, “God
created the natural numbers, all else is the work of man.” But
imagine if the intelligence in our world resided not with human
kind but rather with a singular, isolated jellyfish, floating deep in
the Pacific Ocean. Everything in its experience would be continu
ous, from the flow of the surrounding water to its fluctuating tem
perature and pressure. In such an environment, lacking individu
al objects or indeed anything discrete, would the concept of num
ber arise? If there were nothing to count, would numbers exist?
Like the jellyfish, we adopt mathematical tools that apply to
The universe
has regularities,
known as
symmetries, that
let physicists
describe it
mathematically.
And no one
knows why.
© 2011 Scientific American
August 2011, ScientificAmerican.com 83
our world—a fact that has undoubtedly contributed to the per
ceived effectiveness of mathematics. Scientists do not choose an
alytical methods arbitrarily but rather on the basis of how well
they predict the results of their experiments. When a tennis ball
machine shoots out balls, you can use the natural numbers 1, 2, 3,
and so on, to describe the flux of balls. When firefighters use a
hose, however, they must invoke other concepts, such as volume
or weight, to render a meaningful description of the stream. So,
too, when distinct subatomic particles collide in a particle accel
erator, physicists turn to measures such as energy and momen
tum and not to the end number of particles, which would reveal
only partial information about how the original particles collid
ed because additional particles can be created in the process.
Over time only the best models survive. Failed models—such
as French philosopher René Descartes’s attempt to describe the
motion of the planets by vortices of cosmic matter—die in their
infancy. In contrast, successful models evolve as new information
becomes available. For instance, very accurate measurements of
the precession of the planet Mercury necessitated an overhaul of
Newton’s theory of gravity in the form of Einstein’s general rela
tivity. All successful mathematical concepts have a long shelf life:
the formula for the surface area of a sphere remains as correct to
day as it was when Archimedes proved it around 250 b.c. As a re
sult, scientists of any era can search through a vast arsenal of for
malisms to find the most appropriate methods.
Not only do scientists cherrypick solutions, they also tend to
select problems that are amenable to mathematical treatment.
There exists, however, a whole host of phenomena for which no
accurate mathematical predictions are possible, sometimes not
even in principle. In economics, for example, many variables—the
detailed psychology of the masses, to name one—do not easily
lend themselves to quantitative analysis. The predictive value of
any theory relies on the constancy of the underlying relations
among variables. Our analyses also fail to fully capture systems
that develop chaos, in which the tiniest change in the initial condi
tions may produce entirely different end results, prohibiting any
longterm predictions. Mathematicians have developed statistics
and probability to deal with such shortcomings, but mathematics
itself is limited, as Austrian logician Gödel famously proved.
SYMMETRY OF NATURE
this careful selection of problems and solutions only partially
accounts for mathematics’s success in describing the laws of na
ture. Such laws must exist in the first place! Luckily for mathema
ticians and physicists alike, universal laws appear to govern our
cosmos: an atom 12 billion lightyears away behaves just like an
atom on Earth; light in the distant past and light today share the
same traits; and the same gravitational forces that shaped the
universe’s initial structures hold sway over presentday galaxies.
Mathematicians and physicists have invented the concept of sym
metry to describe this kind of immunity to change.
The laws of physics seem to display symmetry with respect to
space and time: They do not depend on where, from which an
gle, or when we examine them. They are also identical to all ob
servers, irrespective of whether these observers are at rest, mov
ing at constant speeds or accelerating. Consequently, the same
laws explain our results, whether the experiments occur in Chi
na, Alabama or the Andromeda galaxy—and whether we con
duct our experiment today or someone else does a billion years
from now. If the universe did not possess these symmetries, any
attempt to decipher nature’s grand design—any mathematical
model built on our observations—would be doomed because we
would have to continuously repeat experiments at every point in
space and time.
Even more subtle symmetries, called gauge symmetries,
prevail within the laws that describe the subatomic world. For
instance, because of the fuzziness of the quantum realm, a giv
en particle can be a negatively charged electron or an electri
cally neutral neutrino, or a mixture of both—until we measure
the electric charge that distinguishes between the two. As it
turns out, the laws of nature take the same form when we inter
change electrons for neutrinos or any mix of the two. The same
holds true for interchanges of other fundamental particles.
Without such gauge symmetries, it would have been very diffi
cult to provide a theory of the fundamental workings of the
cosmos. We would be similarly stuck without locality—the fact
that objects in our universe are influenced directly only by their
immediate surroundings rather than by distant phenomena.
Thanks to locality, we can attempt to assemble a mathematical
model of the universe much as we might put together a jigsaw
puzzle, starting with a description of the most basic forces
among elementary particles and then building on additional
pieces of knowledge.
Our current best mathematical attempt at unifying all inter
actions calls for yet another symmetry, known as supersymme
try. In a universe based on supersymmetry, every known parti
cle must have an as yet undiscovered partner. If such partners
are discovered (for instance, once the Large Hadron Collider at
CERN near Geneva reaches its full energy), it will be yet another
triumph for the effectiveness of mathematics.
I started with two basic, interrelated questions: Is mathemat
ics invented or discovered? And what gives mathematics its ex
planatory and predictive powers? I believe that we know the an
swer to the first question. Mathematics is an intricate fusion of
inventions and discoveries. Concepts are generally invented, and
even though all the correct relations among them existed before
their discovery, humans still chose which ones to study. The sec
ond question turns out to be even more complex. There is no
doubt that the selection of topics we address mathematically has
played an important role in math’s perceived effectiveness. But
mathematics would not work at all were there no universal fea
tures to be discovered. You may now ask: Why are there univer
sal laws of nature at all? Or equivalently: Why is our universe
governed by certain symmetries and by locality? I truly do not
know the answers, except to note that perhaps in a universe
without these properties, complexity and life would have never
emerged, and we would not be here to ask the question.
M O R E T O E X P L O R E
The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Eugene Wigner
in Communications in Pure and Applied Mathematics, Vol. 13, No. 1, pages 1–14; February 1960.
Pi in the Sky: Counting, Thinking, and Being. John D. Barrow. Back Bay Books, 1992.
Creation v. Discovery. Michael Atiyah in Times Higher Education Supplement; Septem-
ber 29, 1995.
Is God a Mathematician? Mario Livio. Simon & Schuster, 2010.
SCIENTIFIC AMERICAN ONLINE
Is mathematics invented, discovered, both or neither? See examples of remarkable math-
ematical structures that invite this question at ScientificAmerican.com/aug11/livio
© 2011 Scientific American
—–GROVER MAXWELL——
The Ontological Status of Theoretical Entities
That anyone today should seriously contend that the entities referred
to by scientific theories are only convenient fictions, or that talk about
such entities is translatable without remainder into talk about sense con-
tents or everyday physical objects, or that such talk should be regarded
as belonging to a mere calculating device and, thus, without cognitive
con tent-such contentions strike me as so incongruous with the scientific
and rational attitude and practice that I feel this paper should turn out
to be a demolition of straw men. But the instrumentalist views of out-
standing physicists such as Bohr and Heisenberg are too well known to
be cited, and in a recent book of great competence, Professor Ernest
Nagel concludes that “the opposition between [the realist and the in-
slrumentalist] views [of theories] is a conflict over preferred modes of
sp cch” and “the question as to which of them is the ‘correct position’
ha s only terminological interest.” 1 The phoenix, it seems, will not be
laid to rest.
The literature on the subject is, of course, voluminous, and a compre-
lt nsive treatment of the problem is far beyond the scope of one essay.
I sl1all limit myself to a small number of constructive arguments (for a
r lically realistic interpretation of theories) and to a critical examination
of s me of the more crucial assumptions (sometimes tacit, sometimes
· pli it) that seem to have generated most of the problems in this area.
2
‘ fo: . Nngcl, TJ1c Structure of Science (New York: Harcourt, Brace, and World,
l ‘) il), h . 6.
1 l•’or th e ge nes is and part of the content of some of the ideas expressed herein,
I 11n ind ·bled to a number of sources; some of the more influential are H. Feig!,
” 11: IN! ·11t inl llypotheses,” PI1ilosophy of Science, 1
7
: 35-62 ( 1950); P . K. Feyerabend ,
”
11
Alt · 111pt nt n Rcnlistic Interpretation of Experience,” Proceedings of the Aristo-
1 / 1111 Soi ty, 58 :144- 170 (1958); N . R . Hanson, Patterns of Discovery (Cam-
111 ii 1: ,n111hridgc University Press, 1958); E. Nagel, Joe . cit.; Karl Popper, The
I 11 c• of S ·i 11tilic Dis ovcry (London : Hutchinson, 1
9
59); M. Scriven, “Definitions,
f1,ph11111 t i11 11 s 1 :ind Th ·ori ·s,” in Miuneso ta Studies in tlie Philosophy of Science,
3
Grover MaxweII
The Problem
Although this essay is not comprehensive, it aspires to be fairly self-
contained. Let me, therefore, give a pseudohistorical introduction to the
problem with a piece of science fiction (or fictional science).
In the days before the advent of microscopes, there lived a Pas teur-
like scien tist whom, following the usual custom, I shall call Jon es. Re-
fl ecting on the fact that certain diseases seemed to be transmitted from
one person to another by means of bodily contact or b y contact with
articles handled previously by an afHicted person, Jones began to specu-
late about the mechanism of the transmission. As a “heuristic crutch,”
he recalled that there is an obvious observable mechanism for transmis-
sion of certain afHictions (such as body lice), and he postulated that all,
or most, infectious diseases were spread in a similar manner but that in
most cases the corresponding “bugs” were too small to be seen and, pos-
sibly, that some of them lived inside the bodies of their hosts . Jones pro-
ceeded to develop his theory and to examine its testable consequences .
Some of these seemed to be of great importance for preventing the
spread of disease.
After years of struggle with incredulous recalcitrance, Jones managed
to get some of his preventative measures adopted. Contact with or prox-
imity to diseased persons was avoided when possible, and articles which
they handled were “disinfected” (a word coined by Jones) either by
means of high temperatures or by treating them with certain toxic prepa-
rations which Jones termed “disinfectants.” The results were spectacular:
within ten years the death rate had declined 40 per cent. Jones and his
theory received their well-deserved recognition.
However, the “crobes” (the theoretical term coined by Jones to refer
to the disease-producing organisms) aroused considerable anxiety among
many of the philosophers and philosophically inclined scientists of the
day. The expression of this anxiety usually began something like this:
“In order to account for the facts, Jones must assume that his crobes
are too small to be seen. Thus the very postulates of his theory preclude
V~I. IT , TT . Feig!, M . Scri~en ~ and G . ~axw~l.1 ‘. eds . (Minneapolis : University of
!”111111. s tn Pre~s. I ?58); W1lfn~ Sellars, Empmc1sm and the Philosophy of Mind,”
111 M11111 so tn t11d1 cs m the Philosophy of Science, Vol. I, H. Feig! and M . Scriven
, ~ I s. ( ,M i::n. npo lis: niversity of M in.nesota Press, .
19
56), and ” The Language of
111 ·0 11 ~. 111 urr ·11t I ssues m the P/11losophy of Science, H . Feig! and G . Maxwell,
Is . ( N ·w 0 1 : ll olt, Rin hart, and Win ton, 1961) .
4
THE ONTOLOGICAL STATUS OF THEORETICAL ENTITIES
their being observed; they are unobservable in principle.” (Recall that
no one had envisaged such a thing as a microscope.) This common prefa-
tory remark was then followed by a number of different “analyses” and
“interpretations” of Jones’ theory. According to one of these, the tiny
organisms were merely convenient fictions-fa~ons de parler-extremely
useful as heuristic devices for facilitating (in the “context of discovery”)
the thinking of scientists but not to be taken seriously in the sphere of
cognitive knowledge (in the “context of justification”). A closely related
view was that Jones’ theory was merely an instrument, useful for organ-
izing observation statements and (thus) for producing desired results,
and that, therefore, it made no more sense to ask what was the nature
of the entities to which it referred than it did to ask what was the nature
of the entities to which a hammer or any other tool referred .3 “Yes,” a
philosopher might have said, “Jones’ theoretical expressions are just
meaningless sounds or marks on paper which, when correlated with ob-
servation sentences by appropriate syntactical rules, enable us to predict
successfully and otherwise organize data in a convenient fashion .” These
philosophers called themselves “instrumentalists.”
According to another view (which, however, soon became unfashion-
able), although expressions containing Jones ‘. theoretical terms were
g nuine sentences, they were translatable without remainder into a set
(perhaps infinite) of observation sentences. For example, ‘There are
robes of disease X on this article’ was said to translate into something
like this: ‘If a person handles this article without taking certain pre-
autions, he will (probably) contract disease X; and if this article is
fir t raised to a high temperature, then if a person handles it at any time
afterward, before it comes into contact with another person with disease
, he will (probably) not contract disease X; and . . .’
Now virtually all who held any of the views so far noted granted, even
in istecl, that theories played a useful and legitimate role in the scientific
·ntcrprise. Their concern was the elimination of “pseudo problems”
whi ch might arise, say, when one began wondering about the “reality
f upraempirical entities,” etc. However, there was also a school of
th llght, founded by a psychologist named Pelter, which differed in an
• 1 hove borrowed the h ammer analogy from E. Nagel, “Science and [Feigl’s]
.’n111nnti Rc:ilism,” Philosophy of Science,
17
:174- 181 (1950), but it should be
p11l111 cd 11 t thnt Professor Nagel makes it clear that he does not necessarily subscribe
lo th vi w whi h he is explaining.
5
Grover Maxwell
interesting manner from such positions as these. Its members held that
while Jones’ crobes might very well exist and enjoy “full-blown reality,”
they should not be the concern of medical research at all. ‘They insisted
that if Jones had employed the correct methodology, he would have dis-
covered, even sooner and with much less effort, all of the observation
laws relating to disease contraction, transmission, etc. without introduc-
ing superfluous links (the crobes) into the causal chain.
Now, lest any reader find himself waxing impatient, let me hasten to
emphasize that this crude parody is not intended to convince anyone,
or even to cast serious doubt upon sophisticated varieties of any of the
reductionistic positions caricatured (some of them not too severely, I
would contend) above. I am well aware that there are theoretical en-
tities and theoretical entities, some of whose conceptual and theoretical
statuses differ in important respects from Jones’ crobes. (I shall discuss
some of these later.) Allow me, then, to bring the Jonesean prelude to
our examination of observability to a hasty conclusion .
Now Jones had the good fortune to live to see the invention of the
compound microscope. His crobes were “observed” in great detail, and
it became possible to identify the specific kind of microbe (for so they
began to be called) which was responsible for each different disease.
Some philosophers freely admitted error and were converted to realist
positions concerning theories . Others resorted to subjective idealism or
to a thoroughgoing phenomenalism, of which there were two principal
varieties. According to one, the one “legitimate” observation language
\
had for its descriptive terms only those which referred to sense data. 1 he
other maintained the stronger thesis that all “factual” statements were
translatable without remainder into the sense-datum language. In either
case, any two non-sense data (e.g., a theoretical entity and what would
ordinarily be called an “observable physical object”) had virtually the
same status. Others contrived means of modifying their views much less
drastically. One group maintained that Jones’ crobes actually never had
been unobservable in principle, for, they said, the theory did n t imply
the impossibility of finding a means (e.g., the mi r op ) f b erving
them . A more radical contention was that th r b s w r not b erved
at all ; it wa s argued that what was seen by rn an s of th 111i ros pe was
just a sk1dow or an image rather than a rp r ·n l or •t111i m .
6
THE ONTOLOGICAL STATUS OF THEORETICAL ENTITIES
The Observational-Theoretical Dichotomy
Let us turn from these fictional philosophical positions and consider
some of the actual ones to which they roughly correspond. Taking the
last one first, it is interesting to note the following passage from Berg-
mann: “But it is only fair to point out that if this … methodological
and terminological analysis [for the thesis that there are no atoms] …
is strictly adhered to, even stars and microscopic objects are not physical
things in a literal sense, but merely by courtesy of language and pictorial
imagination. This might seem awkward. But when I look through a
microscope, all I see is a patch of color which creeps through the field
like a . hadow over a wall. And a shadow, though real, is certainly not
a physical thing.” 4
I should like to point out that it is also the case that if this analysis
is strictly adhered to, we cannot observe physical things through opera
glasses, or even through ordinary spectacles, and one begins to wonder
about the status of what we see through an ordinary windowpane. And
what about distortions due to temperature gradients-however small
and, thus, always present-in the ambient air? It really does “seem awk-
ward” to say that when people who wear glasses describe what they see
they are talking about shadows, while those who employ unaided vision
talk about physical things-or that when we look through a window-
pane, we can oply infer that it is raining, while if we raise the window,
we may “observe directly” that it is. The point I am making is that there
is, in principle, a continuous series beginning with looking through a
vacuum and containing these as members: looking through a window-
pane, looking through glasses, looking through binoculars, looking
through a low-power microscope, looking through a high-power micro-
ope, etc., in the order given. The important consequence is that, so
far, we are left without criteria which would enable us to draw a non-
nrbitrary line between “observation” and “theory.” Certainly, we will
ften find it convenient to draw such a to-some-extent-arbitrary line; but
il position will vary widely from context to context. (For example, if
w are determining the resolving characteristics of a certain microscope,
w would certainly draw the line beyond ordinary spectacles, probably
‘ . Bergmann , ” Outline of an Empiricist Philosophy of Physics,” American Jour-
111 1 of Pliysics, 11 : 248- 258; 335-342 (1943), reprinted in Readings in the Philoso-
/lli y f Science, JI. Feig! and M. Brodbeck, eds. (New York : Appleton-Century-
:1orl , I 953 ) , pp . 262-287.
7
Grover Maxwell
beyond simple magnifying glasses, and possibly beyond another micro-
scope with a lower power of resolution.) But what ontological ice does
a mere methodologically convenient observational-theoretical dichotomy
cut? Does an entity attain physical thinghood and/or “real existence” in
one context only to lose it in another? Or, we may ask, recalling the con-
tinuity from observable to unobservable, is what is seen through pecta-
cles a “little bit less real” or does it “exist to a slightl y less extent” than
what is observed by unaided vision? 5
However, it might be argued that things seen through sp tacles and
binoculars look like ordinary physical objects, while those seen through
microscopes and telescopes look like shadows and patches of li ght. I can
only reply that this does not seem to me to be the case, p::irticularly
when looking at the moon, or even Saturn, through a telescope or when
looking at a small, though “directly observable,” physical objc t thro ugh
a low-power microscope. Thus, again, a continuity appears.
“But,” it might be objected, “theory tells us that wh::it we ce by
means of a microscope is a real image, which is certainly clistin t from
the object on the stage.” Now first of all, it should be remark d that it
seems odd that one who is espousing an austere empiri ism which re-
quires a sharp observational-language/theoretical-langua g di tinction
(and one in which the former language has a privileged ta u ) hould
need a theory in order to tell him what is observable . But, l lling this
pass, what is to prevent us from saying that we still ob erve th object
on the stage, even though a “real image” may be involved? therwise,
we shall be strongly tempted by phenomenalistic demons, and at this
point we are considering a physical-object observation language rather
than a sense-datum one. (Compare the traditional puzzles: o I see one
physical object or two when I punch my eyeball? Does one obj ec t split
into two? Or do I see one object and one image? Etc.)
Another argument for the continuous transition from the observable
to the unobservable (theoretical) may be adduced from theoretical con-
• r. am not attributing to Professor Bergmann the absurd views sugges ted by th ese
q11 cs l1ons. Ile seems to take a sense-datum language as his observation language (the
!ins’ or wl ~nt he called ” the empirical hi~rarchy”) , and, in some ways, such a position
is mor’~ difTi ult to refu te than one which purports to take an “observable-physical-
ohjt•< I" vi ·w. I low •vcr, I beli eve that demolishing the straw men with which I am
11 ow
cli 111H I 1111 I 111 11 p 11 11 d ont ologi ally crucial seem to me to entail positions which
< 111 1 !'~ po11tl lo 11 < It f1 uw 111 · 11 rn th r losely.
THE ONTOLOGICAL STATUS OF THEORETICAL ENTITIES
siderations themselves. For example, contemporary valency theory tells
us that there is a virtually continuous transition from very small mole-
cules (such as those of hydrogen) through “medium-sized” ones (such
as those of the fatty acids, polypeptides, proteins, and viruses) to ex-
tremely large ones (such as crystals of the salts, diamonds, and lumps
of polymeric plastic). The molecules in the last-mentioned group are
macro, “directly observable” physical objects but are, nevertheless, genu-
ine, single molecules; on the other hand, those in the first mentioned
group have the same perplexing properties as subatomic particles (de
Broglie waves, Heisenberg indeterminacy, etc.). Are we to say that a
large protein molecule (e.g., a virus) which can be “seen” only with an
electron microscope is a little less real or exists to somewhat less an ex-
tent than does a molecule of a polymer which can be seen with an
optical microscope? And does a hydrogen molecule partake of only an
infinitesimal portion of existence or reality? Although there certainly is
a continuous transition from observability to unobservability, any talk
of such a continuity from full-blown existence to nonexistence is, clearly,
nonsense.
Let us now consider the next to last modified position which was
adopted by our fictional philosophers. According to them, it is only
those entities which are in principle impossible to observe that present
special problems . What kind of impossibility is meant here? Without
going into a detailed discussion of the various types of impossibility,
about which there is abundant literature with which the reader is no
doubt familiar, I shall assume what usually seems to be granted by most
philosophers who talk of entities which are unobservable in principle-
i.e., that the theory ( s) itself (coupled with a physiological theory of
perception, I would add) entails that such entities are unobservable.
We should immediately note that if this analysis of the notion of un-
observability (and, hence, of observability) is accepted, then its use as
n means of delimiting the observation language seems to be precluded
for those philosophers who regard theoretical expressions as elements of
calculating device-as meaningless strings of symbols. For suppose they
wi bed to determine whether or not ‘electron’ was a theoretical term .
Fir t, they must see whether the theory entails the sentence ‘Electrons
r unobservable.’ So far, so good, for their calculating devices are said
to be able to select genuine sentences, provided they contain no theo-
ti l terms. But what about the selected “sentence” itself? Suppose
9
Grover Maxwell
that ‘electron’ is an observation term. It follows that the expression is a
genuine sentence and asserts that electrons are unobservable. But this
entails that ‘electron’ is not an observation term. Thus if ‘electron’ is
an observation term, then it is not an observation term. Therefore it is
not an observation term. But then it follows that ‘Electrons are un-
observable’ is not a genuine sentence and does not assert that electrons
are unobservable, since it is a meaningless string of marks and does not
assert anything whatever. Of course, it could be stipulated that when a
theory “selects” a meaningless expression of the form ‘Xs are unobserv-
able,’ then ‘X’ is to be taken as a theoretical term. But this seems rather
arbitrary.
But, assuming that well-formed theoretical expressions are genuine
sentences, what shall we say about unobservability in principle? I shall
begin by putting my head on the block and argue that the present-day
status of, say, electrons is in many ways similar to that of Jones’ crobes
before microscopes were invented. I am well aware of the numerous
theoretical arguments for the impossibility of observing electrons. But
suppose new entities are discovered which interact with electrons in
such a mild manner that if an electron is, say, in an eigenstate of posi-
tion, then, in certain circumstances, the interaction does not disturb it.
Suppose also that a drug is discovered which vastly alters the human
perceptual apparatus-perhaps even activates latent capacities so that
a new sense modality emerges. Finally, suppose that in our altered state
we are able to perceive (not necessarily visually) by means of these new
entities in a manner roughly analogous to that by which we now see by
means of photons. To make this a little more plausible, suppose that
the energy eigenstates of the electrons in some of the compounds pres-
ent in the relevant perceptual organ are such that even the weak inter-
action with the new entities alters them and also that the cross sections,
relative to the new entities, of the electrons and other particles of the
gases of the air are so small that the chance of any interaction here is
negligible. Then we might be able to “observe directly” the position and
possibly the approximate diameter and other properties of some elec-
tron s. It would follow, of course, that quantum theory would have to
b alt r d in some respects, since the new entities do not conform to
rill ils prin ipl s. But however improbable this may be, it does not, I
111ni11Lni11 , involv any logical or conceptual absurdity. Furthermore, the
10
THE ONTOLOGICAL STATUS OF THEORETICAL ENTITIES
modification necessary for the inclusion of the new entities would not
necessarily change the meaning of the term ‘electron.’ 6
Consider a somewhat less fantastic example, and one which does not
involve any change in physical theory. Suppose a human mutant is
born who is able to “observe” ultraviolet radiation, or even X rays, in
the same way we “observe” visible light.
Now I think that it is extremely improbable that we will ever observe
electrons directly (i.e., that it will ever be reasonable to assert that we
have so observed them). But this is neither here nor there; it is not the
purpose of this essay to predict the future development of scientific
theories, and, hence, it is not its business to decide what actually is ob-
servable or what will become observable (in the more or less intuitive
sense of ‘observable’ with which we are now working). After all, we are
operating, here, under the assumption that it is theory, and thus science
itself, which tells us what is or is not, in this sense, observable (the ‘in
principle’ seems to have become superfluous). And this is the heart of
the matter; for it follows that, at least for this sense of ‘observable,’ there
are no a priori or philosophical criteria for separating the observable from
the unobservable. By trying to show that we can talk about the possi-
bility of observing electrons without committing logical or conceptual
blunders, I have been trying to support the thesis that any ( nonlogical)
term is a possible candidate for an observation term.
There is another line which may be taken in regard to delimitation
f the observation language. According to it, the proper term with which
l work is not ‘observable’ but, rather ‘observed.’ There immediately
mes to mind the tradition beginning with Locke and Hume (No idea
without a preceding impression!), running through Logical Atomism
nd the Principle of Acquaintance, and ending (perhaps) in contempo-
ry positivism. Since the numerous facets of this tradition have been
tensively examined and criticized in the literature, I shall limit myself
re to a few summary remarks.
Again, let us consider at this point only observation languages which
ntnin ordinary physical-object terms (along with observation predi-
s, etc., of course). Now, according to this view, all descriptive terms
tu observation language must refer to that which has been observed.
• F r nrgumcnts that it is possible to alter a theory without altering the meanings
t t m1 , see my “Meaning Postulates in Scientific Theories,” in Current Issues in
1 I lJ1il p l1 y of Science, Feig! and Maxwell, eds .
11
Grover Maxwell
How is this to be interpreted? Not too narrowly, presumably, otherwise
each language user would have a different observation language. The
name of my Aunt Mamie, of California, whom I have never seen, would
not be in my observation language, nor would ‘snow’ be an observation
term for many Floridians. One could, of course, set off the observation
language by means of this awkward restriction, but then, obviously, not
being the referent of an observation term would have no bearing on the
ontological status of Aunt Mamie or that of snow.
Perhaps it is intended that the referents of observation terms must be
members of a kind some of whose members have been observed or in-
stances of a property some of whose instances have been observed. But
there are familiar difficulties here. For example, given any entity, we can
always find a kind whose only member is the entity in question; and
surely expressions such as ‘men over 14 feet tall’ should be counted as
observational even though no instances of the “property” of being a man
over 14 feet tall have been observed. It would seem that this approach
must soon fall back upon some notion of simples or determinables vs.
determinates. But is it thereby saved? If it is held that only those terms
which refer to observed simples or observed determinates are observation
terms, we need only remind ourselves of such instances as Hume’s no-
torious missing shade of blue. And if it is contended that in order to be
an observation term an expression must at least refer to an observed de-
terminable, then we can always find such a determinable which is broad
enough in scope to embrace any entity whatever. But even if these diffi-
culties can be circumvented, we see (as we knew all along) that this
approach leads inevitably into phenomenalism, which is a view with
which we have not been concerning ourselves .
Now it is not the purpose of this essay to give a detailed critique of
phenomenalism. For the most part, I simply assume that it is untenable,
at least in any of its translatability varieties.7 However, if there are any
unreconstructed phenomenalists among the readers, my purpose, insofar
as they are concerned, will have been largely achieved if they will grant
what I suppose most of them would stoutly maintain anyway, i.e., that
theoretical entities are no worse off than so-called observable physical
object.
‘ Th r ad r is no doubt familiar with the abundant literature concerned with this
i ~ 11 • S ·,, f r xnmple, Sellars’ “Empiricism and the Philosophy of Mind,” which
nl , o 011 l n 111 ~ r f r ·nee to other pertinent works.
12
THE ONTOLOGICAL STATUS OF THEORETICAL ENTITIES
Nevertheless, a few considerations concerning phenomenalism and re-
lated matters may cast some light upon the observational-theoretical
dichotomy and, perhaps, upon the nature of the “observation language.”
As a preface, allow me some overdue remarks on the latter. Although I
have contended that the line between the observable and the unobserv-
able is diffuse, that it shifts from one scientific problem to another, and
that it is constantly being pushed toward the “unobservable” end of the
spectrum as we develop better means of observation-better instruments
-it would, nevertheless, be fatuous to minimize the importance of the
observation base, for it is absolutely necessary as a confirmation base for
statements which do refer to entities which are unobservable at a given
time. But we should take as its basis and its unit not the “observational
term” but, rather, the quickly decidable sentence. (I am indebted to
Feyerabend, Joe. cit., for this terminology.) A quickly decidable sentence
(in the technical sense employed here) may be defined as a singular,
nonanalytic sentence such that a reliable, reasonably sophisticated lan-
guage user can very quickly decide 8 whether to assert it or deny it when
he is reporting on an occurrent situation. ‘Observation term’ may now .
be defined as a ‘descriptive (nonlogical) term which may occur in a
quickly decidable sentence,’ and ‘observation sentence’ as a ‘sentence
whose only descriptive terms are observation terms.’
Returning to phenomenalism, let me emphasize that I am not among
those philosophers who hold that there are no such things as sense con-
tents (even sense data), nor do I believe that they play no important
role in our perception of “reality.” But the fact remains that the refer-
nts of most (not all) of the statements of the linguistic framework
u ed in everyday life and in science are not sense contents but rather
l hysical objects and other publicly observable entities. Except f~r pains:
dors, “inner states,” etc., we do not usually observe sense contents; and
lthough there is good reason to believe that they play an indispensable
l in observation, we are usually not aware of them when we visually
tactilely) observe physical objects. For example, when I observe a
‘storted, obliquely reflected image in a mirror, I may seem to be seeing
l nby elephant standing on its head; later I discover it is an image of
11 le harles taking a nap with his mouth open and his hand in a
· uliar position. Or, passing my neighbor’s home at a high rate of
‘ W may . say “~oninferent~lly” decide, provided this is interpreted liberally
n h to avoid startmg the entire controversy about observability all over again.
13
Grover Maxwell
speed, I observe that he is washing a car. If asked to report th ·s · oh · r
vations I could quickly and easily report a baby elephant and n wn sliin
of a car; I probably would not, without subsequent obsc rvaliou s, h • uhl
to report what colors, shapes, etc. (i.e., what sense data) w •r · involv •<1.
Two questions naturally arise at this point. How is ii tl111l w · ran
(sometimes) quickly decide the truth or falsity of a pert i 11 •111 oh1.t•1 vn
tion sentence? and, What role do sense contents play in Iii · app1opdnl ·
tokening of such sentences? The heart of the matter is tlrnl 111 . ., · 111
primarily scientific-theoretical questions rather than “p 111 ·ly 1011it1 1′ ,”
“purely conceptual,” or “purely epistemological.” If the r I i · ii I l1 y~i<'s,
psychology, neurophysiology, etc., were sufficiently advan •cl , W(' ('()11 ld
give satisfactory answers to these questions, using, in all Ii k ·lil1ood , 111
physical-thing language as our observation language and tr "" i 111: ,\(" 11 ~ :1 ·
tions, sense contents, sense data, and "inner states" as t/1 ·c11 ·t i(':i/ ( l'S,
theoretical!) entities.9
It is interesting and important to note that, even bcf or · W(‘ i:iw <'<1 111
pletely satisfactory answers to the two questions consid ·1 ·cl 1hoV<', w ·
can, with due effort and reflection, train ourselves lo "obs ·1 v<• di11 ·c t ly"
what were once theoretical entities-the sense conlcnls (C'o lo1 M' 11 \1 d 1011 s,
etc.)-involved in our perception of physical things. As !i ns 1>1· ·11 po11il ·d
out before, we can also come to observe other kind s of 1·111ii1<' wl11d1
were once theoretical. Those which most readil y c 111 • lo 111111d 111volv ·
the use of instruments as aids to observation . Ind · d, """I: 011 1 pain·
fully acquired theoretical knowledge of the world, w co 1111· lo I'<' lliul
we "directly observe" many kinds of so-called thcor ·li ·1 1' 111111 1: . All ·r
listening to a dull speech while sitting on a hard h ncli , W<' liq:111 Io h •
come poignantly aware of the presence of a consid •rahly . 11011111:111viln·
tional field, and as Professor Feyerabend is fond of poi111i111: rnll , ii w
were carrying a heavy suitcase in a changing gravi tali o11 11 I flC'lcl , we• c 011ld
observe the changes of the Gµ.v of the metric tensor.
I conclude that our drawing of the obscrvatio1111l I li c011·1111 1′ li111• at
any given point is an accident and a function of 0111 pl1 wloi: r 11 11111k .
•C f. Sellars, “Empiricism and the Philosophy of Mi11 I ” A 1 1 111f1,~111 S •llnrs
points out, this is the crux of the “other-minds” p1ohlc111 , Sr n 111 1111 1111tl 111111 tut e
(r~lativc to nn i’.~tcrsubjc~ti~~ observation l:mgu:iric, I w1111ld 11 dd) 1111 11111111 t 111 1 t•n.
~111.cs (n 11d 1·h.’Y. really exist ) and not mer ly o ·111 11 l 1111d/111 \ “‘~~l tit l11l111v111 S111 ly
1t is 11 ~ · 1111w1ll111gncss to countena nce th or lira I ·111 it 1·~ I 11 1111111 t 11111 1·vny ·n·
t ·1.1c· 1 ~ I 11111slatnhl ‘. 11 01’. on ly int? som ohsc1~11 t i ~> 11 li111111111H• li11t 11111 t 111 phyNi1. l·
I h111 g !:11.11:11111: wh1 ·h 1s r ·spons1blc for th 10111< 111 lir l111 v 111 - 111 " 111 1111 11ro Witt·
fll' n ~ I t•11 11 1 111 ~ .
THE ONTOLOGICAL STATUS OF THEORETICAL ENTITIES
up, our current state of knowledge, and the instruments we happen to
have available and, therefore, that it has no ontological significance what-
ever.
What If We COULD Eliminate Theoretical Terms?
Among the candidates for methods of eliminating theoretical terms,
three have received the lion’s share of current attention: explicit defin-
ability, the Ramsey sentence,10 and implications of Craig’s theorem.11
Today there is almost (not quite) universal agreement that not all theo-
retical terms can be eliminated by explicitly defining them in terms of
observation terms . It seems to have been overlooked that even if this
could be accomplished it would not necessarily avoid reference to un-
observable (theoretical) entities. One example should make this evident.
Within the elementary kinetic theory of gases we could define ‘mole-
cules’ as ‘particles of matter (or stuff), not large enough to be seen even
with a microscope, which are in rapid motion, frequently colliding with
each other, and are the constituents of all gases.’ All the ( nonlogical)
terms in the definiens are observation terms, and still the definition it-
self, as well as kinetic theory (and other theoretical considerations), im-
plies that molecules of gases are unobservable (at least for the present).
It seems to me that a large number-certainly not all, however; for
example, ‘photon,’ ‘electromagnetic field,’ ‘if-function’-of theoretical
terms could be explicitly defined wholly in terms of observation terms,
but this would in no way avoid a reference to unobservable entities. This
important fact seems to have been quite generally overlooked. It is an
important oversight because philosophers today are devoting so much
nltention to the meaning of theoretical terms (a crucially important
pr blem, to be sure), while the ontological stomach-aches (ultimately
unjustifiable, of course) concerning theories seem to have arisen from
I h • fact that the entities rather than the terms were nonobservational.
Implicit, of course, is the mistaken assumption that terms referring to
1111ob crvable entities cannot be among those which occur in the ob-
c’IVnlion language (and also, perhaps, the assumption that the referent
of 11 defined term always consists of a mere “bundle” of the entities
~ Iii ·h nrc referents of the terms of the definiens).
‘” l•’ 111nk P. Ramsey, The Foundations of Mathematics (New York: Humanities
11111 ) . ,
” Wlllinrn rnig, “Replacement of Auxiliary Expressions,” Philosophical Review
r1 Ill ~~ ( 1956). ‘
15
Grover Maxwell
Surprisingly nough, both the Ramsey sentence and Craig’s theorem
provid e us with gen uine (in principle) methods for eliminating theo-
reti al term s provided we are interested only in the deductive “observa-
ti nal” onscq uences of an axiomatized theory. That neither can provide
a viable method for avoiding reference to theoretical entities has been
pointed out clearly by both Hempel and Nagel.1 2 I shall discuss these
two devices only briefly.13
The first step in forming the Ramsey sentence of a theory is to take
the conjunction of the axioms of the theory and conjoin it with the
so-called correspondence rules (sentences containing both theoretical
and observational terms-the “links” between the “purely theoretical”
and the observational). This conjunction can be represented as follows:
—P—Q— . . .
where the dashes represent the sentential matrixes (the axioms and C-
rules) containing the theoretical terms (which are, of course, almost
always predicates or class terms) ‘P,’ ‘Q,’ ‘ .. .’;the theoretical terms are
then “eliminated” by replacing them with existentially quantified vari-
ables . The resulting “Ramsey sen tence” is represented, then, by
(3f)(3g) … (—f—g— . .. ) .
Or, consider an informal illustration. Let us represent schematically an
oversimplified axiomatization of kinetic theory by
All gases are composed entirely of molecules. The molecules are
in rapid motion and are in frequent collision, etc., etc.
And for simplicity’s sake, suppose that ‘molecules’ is the only theoretical
term. The Ramsey sentence would be something like the following:
There is a kind of entity such that all gases are composed entire-
ly of these entities . They are in rapid motion and are in frequent
collision, etc., etc.
Now it is a simple matter to demonstrate that any sentence containing
only observation (and logical) terms which is a deductive consequence
of the original theory is also a deductive consequence of its Ramsey
sentence (see, for example, Rozeboom’s article in this volume); thus,
:is for as any deductive systemization is concerned, any theory may be
” nrl ,. IT mp I, “The Theoretician’s Dilemma,” in Minnesota Studies in Phi-
losoi11ty I. i nee, Vol. II, Feig!, Scriven, and Maxwell, eds. Nagel, Joe. cit.
‘ For nn · l nd d nsiderntion of the Ramsey sentence see Professor William
Hoi ·hoo111’ s ny 111 thi volume.
16
THE ONTOLOGICAL STATUS OF THEORETICAL ENTITIES
eliminated and its Ramsey sentence used instead. However, it is also
easy to prove (if indeed it is not obvious) that if a given theory (or a
theory together with other considerations, theoretical or observational)
entails that there exist certain kinds of unobservable entities, then the
appropriate Ramsey sentence will also entail that there exist the same
number of kinds of unobservable entities.14 Although, insofar as deduc-
tive systemization is concerned, the Ramsey sentence can avoid the use
of theoretical terms; it cannot, even in letter, much less in spirit (Hem-
pel, Ioc. cit., was too charitable), eliminate reference to unobservable
(theoretical) entities.
The Craig result, like the Ramsey sentence, provides a ” method” of
reaxiomatizing a postulate set so that any arbitrarily selected class of
terms may be eliminated, provided one is interested only in those theo-
rems which contain none of these terms . Its “advantages” over the
Ramsey sentence are that it does not quantify over predicates and class
terms and that its final reaxiomatization eliminates reference both in
spirit and in letter to unobservable entities. However, its shortcomings
(for the purposes at hand) render it useless as an instrument of actual
scientific practice and also preclude its having, even in principle, any
implications for ontology. The resulting number of axioms will, in gen-
eral, and particularly in the case of the empirical sciences, be infinite in
number and practicably unmanageable.
But if the practical objections to the use of Craig’s method as a means
for elimination of theoretical terms are all but insurmountable, there are
objections of principle which are even more formidable. Both Craig’s
method and the Ramsey device must operate upon theories (containing,
of course, theoretical terms) which are “already there.” They eliminate
theoretical terms only after these terms have already been used in inter-
” The proof may be sketched as follo ws: Let ‘T’ designate the theory (conjoined,
i( nc essary, with other statements in the accepted body of knowledge ) which entails
Iii 1t the kinds of entities C, D, … are not observable, i.e ., T entails that
(3 x)( 3 y) … (Cx·Dy .. . xis not observable•y is not observable
. . . ) which in turn entails
(3f)(3 g) . .. (3x)(3y) … (fx•gy … x is notobservable•yisnot
observable . . . ) .
Ramsey result holds for any arbitrary division of nonlogical terms into two
1 l.1, r , 6 we may put ‘observable’ into the class with the observation terms, so that
lltr h1 tt <•r formalized statement may be treated as an "observational" consequence
111 ' I' ( I 111 itivity of entailment). But then it is also a consequence of the Ramsey
111 l 1 1H' f T . Q. • .D.
17
Grover Maxwell
mediary steps. Neither provides a method for axio1n ut izu ti n ab initio
or a recipe or guide for invention of new theo ri es. 0 11 . qn •11tly neither
provides a method for the elimination of th co rcli ul I •rn1 s in the all-
important “context of discovery.” 1 5 It might be argu cl th ut t-hi s objec-
tion is not so telling, after all, for we also lack any re ·i I f r lh c inven-
tion of theories themselves, and it is logically poss ibl lh:it w should
discover, without the use of theories as intermediari es, Rams ·y se n tences
or Craig end products which are just as useful for xpl ai nin a nd pre-
dicting observations as the theories which we happ 11 to ha ve (acci-
dently) adduced. It might be added that it is al so logica ll y poss ible that
we should discover just those observation statement ( in cl ncl iu g pre-
dictions, etc.) which happen to be true without th e use of a 11 y i nstru-
mental intermediaries.
We must reply that the accomplished fact that it is th eo ri es, referrin g to
unobservables, which have been invented for thi s pmposc and tl1 at many
of them serve it so admirabl y-this fa ct, itself, cri es out for expl anation.
To say that theories are designed to accompli sh thi task is no r pl y un-
less at least a schema of an instrum entalist recipe for su h cl signin g is
provided. As far as I know thi s has not been don . Th e th sis t·hat theo-
retical entities are “really” just “bundles” of obscrva hl bj ts or of
sense data would, if tru e, p~ovicle an explanati on; but it is not taken very
seriously by most philosophers today- for th e very good r ason tha t it
seems to be fal se. The only reasonable explanati on for th e success of
theories of which I am aware is that well-confirmed th eo ri es arc con-
junctions of well-confirmed, genuine statements and that th e cn ti ti c to
which they refer, in all probability, exist. That it is psychologica ll y pos-
sible for us to invent such theories is explained by the fa ct that many
of th e entities to which they refer resemble in many respects (although
. ‘,: T he Ramsey sentence is intuitively tractable enough so that ve ry simple ” theo-
ri es . might be mvented as full-blown Ramsey sentences without the use of inter-
mec!1ary ~crms. However’. Craig’s theorem provides no means of operating ab initio.
ra 1g points out ( Joe. cit. ) that once the original theory is ” there,” reference in 18
THE ONTOLOGICAL STATUS OF THEORETICAL ENTITIES they may differ radically from them in others) the entities which we It should also be remembered, at this point, that theories, even as We have seen that the elimination of theoretical terms, even by ex- -‘ me up with a gimmick-a prediction machine or “black box” -into 1y the likelihood-of the existence of unobserved causes for the ob- lL li ction machine or ” calculating device” in terms of the established … k. f r much mischief in considerations concerning the cognitive status ” tutful” orga ni za tion of observational data or, more specifically, with 19 Grover Maxwell physicist involve such things as the actual properties and vari ties of “Criteria” of Reality and i nstrum en talism Ernest Nagel considers the dispute between realists and instrumentalists 0 1111 1·i n between criteria and th e “sen es [ i I] f ‘ r nl’ r ‘exist.’ ” 18
•• p. it., pp . 141- 152 . 2 n fore proceeding to a criticism of these arguments, let me point out that terns,” although he does say that he “could instead have spoken of There are two main points that I wish to make regarding this kind of Id mistake of confusing meaning with evidence. To be sure, the fact The second point is even more serious . One would hope that (Pro- predicate (and existence a property) . ‘ an be made into a merely verbal one only by twisting the meanings of 1hnt in “ordinary language,” the most usual uses of these terms are such . are real =dr . exist
md that
. exist = dr there are 8
ll l l that the meanings of these definiens are clear enough so that no ”l’h are iJl.’ would, of course, be expressed by ‘ ( 3x) ( x) .’) Thus, if 21 Grover Maxwell we have a well-confirmed set of statements (laws or theories plus initial In summary, let us recall three points concerning instrumentalism. The Ontological Status of Entities-Theoretical and Otherwise the key to the solution of all significant problems in ontology can be Although wide latitude in choosing and constructing frameworks is • 111 this volume, the reader 1s referred to it for an interesting critique of instrumen- ” R. a map, Meaning and Necessity, 2nd ed. (Chicago: Univers ity of Chicago ” fo’or n mor cl ctailccl discussion of linguistic frameworks as well as th eir releva nce 22
THE ONTOLOGICAL STATUS OF THEORETICAL ENTITIES I h ‘ f II owing features: ( 1) the usual L ( ogical )-formation and L-trans- 23 Even a term such as ‘red’ has ptual or epistemological and not ontological. Just what relations are this point, two views may be mentioned . I will omit consideration · · rcling to one view, it is always a proper subset of the lawlike sen- 11 •• Maxwell and H. F eig!, “Why Ordinary Language Needs Reforming,” Journal r. nlso the writi ngs of Wilfrid Sellars, for example in “Some Reflections on 1 S • · ll. amnp, ” Bcobachtun gsprache und theoretisch Sprache,” Dialectica, 23 Grover Maxwell a sentence which is A-true in one context may be contingent in another The complication just mentioned, however, has led many philoso- Now when we engage in any considerations about any kinds of en- • S h is essay in this volume.
24
THE ONTOLOGICAL STATUS OF THEORETICAL ENTITIES I he other properties of ordinary physical objects. Molecules, for exam- ‘ some but not all of the properties of everyday physical things.
A : Do molecules exist? which when conjoined with other true sentences such as A: But are they real? mond in your ring. As for those which are submicroscopic A: How about electrons? sary to try several reformations, taking into account many A: Perh aps you are right. However, I am genuinely puzzled 25 Grover Maxwell B: Take the last first. We would probably never call them “°Cf. B. Mayo, “The Existence of Theoretical Entities,” Science News, 32: 7-18 In connection with convenient fictions, we might consider such entities as ideal 26
THE ONTOLOGICAL STATUS OF THEORETICAL ENTITIES well-confirmed sentences) entails that there are lines of force, ne of the exciting aspects of the development of science has been 111any philosophers and philosophically inclined scientists to despair of r •nl entities are the good old familiar ones which we sense directly every- j ·t ion that there are very few ontologically legitimate kinds of entities, 27
I ·11 ·r, 1·0 theoret ical e.ntities in the application of his method may be avoided by u; ing
l!ll’ ,11 11 111 ·s of t~1 coret1 ca l terms rather t~an using the terms themselves (i.e . by men-
l1011111 g, lh ·or l1 cnl term s ra ther than usmg them) . But surely only a diehard instru-
11 1 c· 11 l 1 il 1 ~ I • 111 l:ik : more than very scant comfort from this. The qu es tion would still
11′ 11 1111 11 : h ·r did th theory come fr om in th e first place, and why are the names
ol th C’ ~ ci p11il i(o 11l 11 I ·n 11s arra ng cl in this particu lar ma nn er such admirable “instru-
11 11•11I H” 101 <• p l 11 11 1 ~ 1 ion . nn I pr ·di tio1.1 of ob crvat ions? Whatever ontological im-
pl 111 11011 I I ii ~ 111 odif1 c11 11 011 of t he ra1g m ·thod may have, they seem to be exactly
t li 1 11 11 1 11 lh
have already observed.
instruments, are important not only for deductive systemization but also
for inductive systemization (see Hempel, Joe. cit.) . We often reason
theoretically using induction, and the conclusions may be either obser-
vational or theoretical. Thus we might infer from the facts that a certain
substance was paramagnetic, that it catalyzed the recombination of free
radicals, and that it probably contained a ” one-electron” bond; and we
might go on to infer, again inductively, that it would probably catalyze
the conversion of orthohydrogen to parahydrogen. The Craig result ap-
plies only to deductive systemization and, thus, cannot, even in its Pick-
wickian fashion, eliminate theoretical terms where inductive theoretical
reasoning is involved . Although Craig’s theorem is of great interest for
formal logic, we must conclude, to use Craig’s (Joe. cit.) own words,
“[as far as] the meaning [and, I would add, the referents] of such ex-
pressions [auxiliary expressions (theoretical terms)] . . . [is concerned]
the method . . . fails to provide any . . . clarification.”
plicit definition, would not necessarily eliminate reference to theoretical
(unobservable) entities. We have also seen that, even if reference to
theoretical entities could be eliminated after the theories themselves
have been used in such an elimination (for example, by a device such
n Craig’s), the reality (existence) of the theoretical entities is not there-
by militated against. But the most crucial point follows. Even if we do
whi ch we can feed data and grind out all the completely veridical ob-
N rvational predictions which we may desire, the possibility-I should
•rv cl events would still remain . For unless an explanation of why any
ul · of explanation, confirmation, etc., were forthcoming, the task of
·i •n c would still be incomplete.
‘fltis brings us to another mistaken assumption that has been responsi-
tl1 · ries-th e assumption that science is concerned solely with the
· ·ss fnl predi ction. Surel y the main concerns of, say, a theoretical
subatomic particles rather than the mere predictions abo ut where and
how intense a certain spectral line will be. The instrum ntalist has the
picture entirely reversed; as far as pure science is con crned, most ob-
servational data-most predictions-are mere instrum ents and are of
value only for their roles in confirming theoretical principles. Even if
we obtain the prediction machine, many of the theoric ex tant today
are well confirmed enough to argue strongly for the reality of th eoretical
entities. And they are much more intellectually satisfactory, for they pro-
vide an explanation of the occurrence of the observational events which
they predict. And-equally important-an explanation for th e fact that
theories “work” as well as they do is, as already noted, also forth coming;
it is simply that the entities to which they refer exist.
It was pointed out in the beginning of this article that Professor
to be merely a verbal one.16 There follows here a brief and what I hope
is a not too inaccurate summary of his argument. Various criteria of
‘real’ or ‘exist’ (runs the argument) are employed by scientists, philoso-
phers, etc., in their considerations of the “reality problem.” (Among
these. criteria-some of them competing, some compatible with each
other-are public perceivability, being mentioned in a generally accepted
law, being mentioned in more than one law, being mentioned in a
“causal” law, and being invariant “under some stipulated set of trans-
formation, projections, or perspectives.” 17 ) Since, then (it continues)
any two disputants will, in all probability, be using ‘real’ or ‘exist’ in two
different senses, such disputes are merely verbal. Now someone might
anticipate the forthcoming objections to this argument by pointing out
that the word ‘criteria’ is a troublesome one and that perhaps, for Nagel,
the connection between criteria and reality or existence is a con tingent
one rather than one based on meaning. But a mom ent’s reA tion makes
it obvious that for Nagel’s argument to have for e, ‘ rit ri,’ must be
tak n in the latter sense; and, indeed, Nagel xpli ·itl y 1 ak for the
“Nn11 ·I, op. it., pp . 145- 150.
,. p. it., p. 151.
THE ONTOLOGICAL STATUS OF THEORETICAL ENTITIES
Professor Gustav Bergmann, completely independently, treats ontologi-
al questions in a similar manner. Rather than criteria, he speaks of “pat-
riteria,” and he makes explicit reference to various “uses” of ‘exist.’ 19
approach to ontological issues. First, it seems to me that it commits the
that a kind of entity is mentioned in well-confirmed laws or that such
entities are publicly perceptible, etc.-such facts are evidence (very good
evidence!) for the existence or “reality” of the entities in question. But
I cannot see how a prima-facie-or any other kind of-case can be made
for taking such conditions as defining characteristics of existence.
fess or Norman Malcolm notwithstanding) over nine hundred years of
debate and analysis have made it clear that existence is not a property.
Now surely the characteristics of being mentioned in well-confirmed
laws, being publicly perceptible, etc ., are properties of sorts; and if these
omprised part of the meaning of ‘exists,’ then ‘existence’ would be a
Thus it is seen that the issue between instrumentalism and realism
‘existence’ and ‘reality,’ not only beyond their “ordinary” meaning but,
al o, far beyond any reasonable meanings which these terms might be
•iven. In fact, it seems not too much to say that such an interpretation
f the “reality problem” commits a fallacy closely akin to that of the
ntological Argument.
What can be said about the meanings of ‘real’ and ‘exists’? I submit
Iha
111 lh r xplication is seriously needed . (In most “constructed languages,”
••” Phy i ond Ontology,” Pl1ilosopl1y of Science, 28 : 1-14 ( 196 1).
conditions) which entail the statement ‘There are cf!.’ (or ‘ ( 3x) ( cf!x )’),
then it is well confirmed that cf!. are real-full stop!
First, as is shown above, it cannot be excused on the grounds that it
differs from realism only in terminology. Second, it cannot provide an
explanation as to why its “calculating devices” (theories) are so success-
ful. Realism provides the very simple and cogent explanation that the
entities referred to by well-confirmed theories exist. Third, it must be
acutely embarrassing to instrumentalists when what was once a ” purely”
theoretical entity becomes, due to better instruments, etc., an observ-
able one.20
As I have stated elsewhere (see the second reference in footnote 22),
found in Carnap’s classic article, “Empiricism, Semantics, and Ontol-
ogy.”21 Taking this essay as our point of departure, we may say that in
order to speak at all about any kind of entities whatever and thus, a for-
tiori, to consider their existence or nonexistence, one must first accept
the “linguistic framework” which … troduces the entities.” 22 This sim-
ply means that in order to understand considerations concerning the
existence of any kind of entities one must understand the meanings of
the linguistic expressions (sentences and terms) referring to them-and
that such expressions have no meaning unless they are given a place in
a linguistic framework which “talks about the world” and which has at
least a minimum of comprehensiveness . (Since I am interested, here,
primarily in empirical science, I neglect universes of discourse containing
only “purely mathematical” or “purely logical” entities.)
permissible, any satisfactory framework will embody, at the very least,
20 “.’-!though I cannot agree with all the conclusions of Professor Feycrabend ‘s essay
talism .
Pr ss, 19 59 ) .
for nn tn lop i(•nl probl ms, sec Carnap, ibid .; and G . Maxw II , “Theories, Frameworks,
11 11 <1 )11tolor,y,' Pl1ilosop hy of Science, vol. 28 (1961 ). For an elaboration of the
li11 11 11i NI i · I Ii ·scs pr •s 11ppos d by th e fatter article and, to some extent, by this essay,
fo1 n1ation rules and the corresponding set of L-true sentences which
I Ii y generate; ( 2) a set of confirmation rules, whose nature I shall not
di · n s here but which I shall assume are quite similar to those actually
11 , d in the sciences; ( 3) a set of sentences whose truth value is quickly
d idable on other than purely linguistic grounds-these correspond to
“s i11 gular observation statements,” but, of course, as we have seen, it is
11 ithcr necessary nor desirable that such statements be incorrigible or
i11dubitable or that a sharp distinction between observation and theory
h ‘ drawn; and ( 4) a set of law like sentences, which, among other things,
provide that component of meaning which is nonostensive for every
d s riptive ( nonlogical) term of the framework. (I have argued in the
1· •f rcnces given in footnote 22 that every descriptive term has a mean-
i11 omponent which is nonostensive.
I art of its meaning provided by, for example, the lawlike sentence ‘No
snrfoce can be both red and green all over at the same time.’ Such a
vi •w is sometimes stigmatized by the epithet ‘holism.’ But if there is
111 y holi sm involved in the view I am advocating, it is completely con-
pr ‘ nt, or absent, between the actual entities of the “real world” is an
·111pirical question and must be decided by considerations within a de-
s ·ri ptivc linguistic framework rather than by consideration about such
f1t11n works.)
o xpli citly defined terms, since they are, in principle, always eliminable.
t · 11 ontaining a given term which contributes to the term’s meaning.
‘ I ‘Ii · s n tcnces in this subset are A-true 24 (analytic in a broad sense)
111d :ir · totally devoid of any factual content-their only function is to
I 1 ovid part of the meaning of the term in question. The situation is
111111 ·n cly com plicated by the fact that when actual usage is considered,
11/ l ‘/1ilosop hy, 58: 488-498 (1961); G. Maxwell, “Meaning Postulates in Scientific
‘ l’ll(‘oii ·s,” in Current Issues in the Philosophy of Science, Feigl and Maxwell, eds.; and
111 hd •f nrti le, “Th e N ecessary and the Contingent,” in this volume.
I .111111111H a mes,” Pl1iloso pl1y of Science, 21 : 204-228 ( 1954 ).
I ~ 2<18 ( 1957); as we ll as the referen ces in fn . 22 .
and that even in a given context it is, more often than not, not clear,
unless the context is a rational reformation, whether a given sentence
is being used as A-true or as contingent. This confusion can be avoided
by engaging in rational reformation, i.e., by stipulating (subject to cer-
tain broad and very liberal limitations) which sentences are to be taken
as A-true and which as contingent. Needless to say, this is the viewpoint
which I prefer.
phers, including Professor Putnam 25-to say nothing of W. V. Quine-
to the other viewpoint. According to it, no segregation of the relevant
lawlike sentences into A-true and contingent should be attempted; each
law like sentence plays a dual role : ( 1) it contributes to the meanings
of its descriptive terms and (2) it provides empirical information. For-
tunately, we do not have to choose between these two viewpoints here,
for the thesis of realism which I am advocating is (almost) equally well
accommodated by either one.
tities and, a fortiori, considerations about the existence of theoretical
entities, it is to the lawlike sentences mentioning the entities-for theo-
retical entities, the theoretical postulates and the so-called correspond-
ence rules-to which we turn. These sentences tell us, for example, how
theoretical entities of a given kind resemble, on the one hand, and differ
from, on the other, the entities with which we happen to be more fa-
miliar. And the fact that many theoretical entities, for example those of
quantum theory, differ a great deal from our ordinary everyday physical
objects is no reason whatever to ascribe a questionable ontological status
to them or to contend that they are merely “calculating devices.” After
all, the very air we breathe as well as such things as shadows and mir-
ror images are entities of quite different kinds from chairs and tables,
but this provides no grounds for impugning their ontological status. The
fa ct that molecules, atoms, etc., cannot be said in any non-Pickwickian
sense to have a color has given some philosophers ontological qualms.
But, of course, the air has no color (unless we invoke the color of the
sky); and a transparent object whose refractive index was the same as
t li a t of air would be completely invisible, although it would have all
pl e, are in about the same category; they are physical things which pos-
B: Certainly. We have an extremely well-confirmed theory,
‘There are gases’ entails that there are molecules.
B: What do you mean?
A: Well, I’m not sure. As a starter : Are they physical objects?
B: Certainly the large ones are. Take, for example, that dia-
but still large enough to have large quantum numbers, it
seems that in almost any reasonable reformation they would
be classified as physical objects. It would seem unjustifiable
to withhold from them this status simply because they can-
not be said to have a color in any straightforward fa shion .
In fact, I would even be inclined to call the smallest, the
molecule of hydrogen, a physical object. It has mass, a
reasonably determinate diameter, and, usually, something
which approximates simple location, etc.
B : The decision here is more difficult. We might find it neces-
facets of contemporary physical theory, before we arrived
at the most satisfactory one. It would also be helpful to
have a more specific problem in view than the one which
we are now considering. At any rate, we might begin by
pointing out that electrons do have mass, even rest mass.
111ey can be simply located at the expense of refraining
from ascribing to them a determinate momentum . They
ca~ be said to causally interact with “bona fide” physical
ob1 ec ts, even by those who have a billiard-ball notion of
ca usality. The important point is that the question ‘Are elec-
tron s physical objects?’ is a request for a rational reformation
of a very thoroughgoing variety. For most purposes, a ra-
tion al reformation would not need to answer it. For your
purposes, wh y not be content to learn in what ways elec-
trons are similar to, and in what ways they differ from , what
yo u would call “ordinary physical objects”? This will enable
you to avoid conceptual blunders.
nhoul fi eld s, and even photons.
physical objects. For example, they have no rest mass and
it would be a conceptual mistake to ask, except in a Pick-
wickian sense, What is their color? However, it would be
reasonable to say that they are a sort of physical continuant;
and they can even interact with electrons in a billiard-ball
manner. At any rate, we must agree, speaking loosely, that
they are “every bit as real” as electrons. The concepts of field
theories are so open textured that it is difficult to decide
what kinds of reformations one should adopt here. And it
is virtually impossible to find similar kinds of entities with
which one is prescientifically familiar. Perhaps these theories
will someday be enriched until decisions concerning the
most appropriate rational reformations are easier to make-
perhaps not. But even here, the meanings of the terms in-
volved are usually sufficiently clear to avoid conceptual
blunders and ontological anxieties. You might like to con-
sider the “lines of force,” which are often spoken of in con-
nection with fields. These are often used as a paradigm of
the “convenient fiction” by those who hold such a view of
theories. 26 But though convenient, lines of force are not
fictions. They “really exist.” Let me try to make this a little
more plausible. Consider the isobars of meteorology, or the
isograms which connect points of equal elevation above sea
level. Now at this very moment, the 1017 millibar isobar,
i.e., the line along which the barometric pressure is 1017
millibars, exists right here in the United States. Its location
can even be determined “operationally.” And all of this is
true whether anyone ever draws, or ever has drawn, a weather
map. Since a well-confirmed theory (plus, perhaps, other
(1954), and “More about Theoretical Entities,” ibid., 39:42-55 (1956) . For a
critique of these articles and for excellent constructive remarks concerning theoretical
entities, see J. J. C . Smart, “The Reality of Theoretical Entities,” Austrafasian Journal
of Philosophy, 34 :1-12 (1956).
gases and bodies uninfluenced by external forces. These actually are fictions. But no
theory (or theory plus true sentences) entails that there are such things . To under-
sta nd their function, we need only recourse to the notion of a limit, often used in
mathematics. Roughly speaking, what we actually do when we use theories involving
s11 ch “fictions” is to assume, for example, that the influence of external forces on the
body in question is very, very small, or that the behavior of the gas with which we
:11 011 ·crnccl is approximately given by ‘PV = nRT,’ or, in early kinetic theory, that
t Ii ~· di:1111 ·1 ·r of ~ molecule is very, very small compared to the distance between mole-
c·11l<·s. Nol· th nt lind van clcr Waals taken the calculating-device or convenient-fiction
vi •w, Ii · p1ohahly wou ld not ha ve developed his equation which embodies a correction
for th · d i n ·! d11l0 lo Ili c finite (greater than zero) diameter of molecules.
lines of force exist. To be sure, they are very different from
everyday physical objects. But as long as we are clear about
this, what metaphysical-what ontological-problems re-
main?
I It emergence of reference to strikingly new kinds of entities. This is
p:irticularly true in field theories and quantum theory. The great differ-
•ncc between these and the old, familiar categories seems to have caused
·ff cting a satisfactory conceptual analysis of these powerful new con-
. ptual tools. The attitude too often has been, “Let us proceed to use
I ho e new devices and, if necessary for heuristic reasons, even to behave
is it they consisted of genuine sta~ements about real entities. But let us
1 ·member that, in the last analysis, they are only meaningless calculating
I ‘vices, or, at best, they talk only of convenient fictions, etc. The only
da y.” To turn the purpose of a saying of Bertrand Russell’s almost com-
pl Lely about-face: such a view has advantages-they are the same as
I hose of theft over honest toil. The compulsion toward metaphysical
1s ·p is which appears to have been the motivation for the espousal of
1rn1n y of these reductionistic philosophies seems, itself, to have arisen
I 10111 a preoccupation with metaphysical pseudo problems, e.g., the con-
l •rhaps only one.
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