Download Confirming Mathematical Theories: an

ANTHONY PERESSINI
CONFIRMING MATHEMATICAL THEORIES:
AN ONTOLOGICALLY AGNOSTIC STANCE
ABSTRACT. The Quine/Putnam indispensability approach to the confirmation of mathematical theories in recent times has been the subject of significant criticism. In this paper
I explore an alternative to the Quine/Putnam indispensability approach. I begin with a van
Fraassen-like distinction between accepting the adequacy of a mathematical theory and
believing in the truth of a mathematical theory. Finally, I consider the problem of moving
from the adequacy of a mathematical theory to its truth. I argue that the prospects for
justifying this move are qualitatively worse in mathematics than they are in science.
1. INTRODUCTION
Of late it has become customary for philosophers to speak of the “confirmation” of pure mathematical theories. This notion derives from the
Quine/Putnam indispensability account which maintains that the confirmation of mathematical theories involves nothing beyond ordinary scientific
confirmation: pure mathematical theories, in virtue of being an indispensable part of our accepted scientific theories, enjoy the same confirmation.
Quine rarely applies the concept of “confirmation” to scientific theories,
and even less often to mathematical theories. My discussion here, consequently, does not target strict Quineans (if any such exist); rather, my
remarks are directed toward the contemporary discussion of mathematical
Platonism that has grown out of the Quine/Putnam approach. In this arena,
the contemporary import of the Quinean move to accept the truth of mathematics because mathematics plays an indispensable role in our scientific
theories can be fairly characterized as a move to understand mathematical
theories as being confirmed because they participate (indispensably) in
confirmed scientific theories. Finally, my project here is not to argue directly against contemporary indispensability approaches – that can be found
elsewhere.1 In this paper, I seek to offer an alternative to indispensability
considerations as way of thinking about the reasons we have to believe
that mathematical theories are true in a sense that entails the existence of
mathematical objects.2
In spite of the problems with the indispensability account, it seems
promising to approach the confirmation of mathematical theories in a way
Synthese 118: 257–277, 1999.
© 1999 Kluwer Academic Publishers. Printed in the Netherlands.
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parallel to the confirmation of scientific theories. Care must be taken, however, not to force mathematics into the mold of science. Serious attention
must be paid to the particular mathematical methodologies involved.
A fundamental problem with certain versions of the indispensability
approach is that they depict mathematical theory as being confirmationally
dependent on science in an implausible way. Unlike discarded scientific
theories, pure mathematical theories do not “dry up and blow away” if
their usefulness in science ceases; rather, they retain whatever status they
had qua pure mathematical theory. On the other hand, when a scientific
theory is no longer useful in science, it does in an important sense, “dry
up and blow away”. This is because its role in empirical science was its
(sole) source of confirmation. Pure mathematical theories, on the other
hand, appear to have their own internal sources of confirmation – and this
confirmation is independent of how/whether the theory is used in science. I
propose to pay particular attention to this confirmational independence and
not merely “chalk it up” to appearance. In what follows, respecting this
confirmational independence of pure mathematical theories will be taken
as an adequacy condition for an account of mathematical confirmation.
In developing my account, I begin with a van Fraassen-like distinction
between accepting the adequacy of a theory and believing the truth of a
theory; I then work out an appropriate sense of “adequacy” for the mathematical setting. This mathematical adequacy will be determined by the
confirmational methodology of pure mathematics. In particular, the mathematical adequacy of a theory is what is being directly confirmed by the
methodology of pure mathematics – just as empirical adequacy is what is
being directly confirmed by the methodology in science. I argue that the
confirmational independence of pure mathematics is due to the fact that
the criterion for accepting a theory of pure mathematics, its mathematical
adequacy, is independent of the evidential support at work in science. I
next consider the problem of moving from the mathematical adequacy of
a theory to its truth. The question becomes: at what point, if ever, does
a theory’s mathematical adequacy commit us to its truth? I argue that the
prospects for justifying this move are qualitatively worse in mathematics
than they are in science.
2. ACCEPTING AND BELIEVING
The debate between scientific realists, and empiricists (led by van
Fraassen) has suggested a subtle shift in how we think about confirmation.
It is no longer to be taken for granted that in confirming a scientific theory,
we simply confirm the truth of the theory’s statements. The idea that we
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may accept a scientific theory as empirically adequate without believing
all of it to be true is a possibility that has enough merit to have sustained
considerable discussion. At the very least, this possibility has forced philosophers to recognize that the empirical adequacy of a scientific theory is
what is most directly confirmed by the activity of scientists, and that to pass
from empirical adequacy to the truth of the theory requires an additional
step.3
2.1. Some Terminology
The propositional attitudes picked out by the terms “believe” and “accept”
are really not as different as might be thought. In fact, the latter may be
analyzed in terms of the former. To believe a theory is to believe that the
statements that make it up are true.4 To accept a theory is to believe that
the theory is “adequate” in some relevant sense, but not necessarily that all
of the statements that make it up are true. We may also speak of accepting
individual statements: to accept a statement p (in the context of an accepted theory of which it is a part) is to believe it to be “adequate” in some
sense, but not necessarily to believe that p is true. Thus the statements of a
theory T have a natural partition into the statements that are believed (Tb )
and the statements that are merely accepted (Ta ). In describing a statement
as merely accepted the implication is, of course, that it is not believed.
Similarly, when a theory is described as merely accepted, the implication
will be that it is not believed and thus that Ta is non-empty.
Analyzing the concept of acceptance in terms of belief avoids an
important criticism that has been successfully leveled against certain
presentations of it. It has been argued that dichotomous (either/or) accounts
of propositional attitudes like acceptance are too coarsely grained. We do
not accept or reject hypotheses rather, we assign degrees of belief to them.5
Notice, however, that since acceptance of T is analyzed as believing T
empirically adequate, if belief comes in degree, then so does acceptance.
The essential feature of the idea of acceptance is not that it be an either/or
affair, but rather that it be an attitude toward an hypothesis distinct from
believing it true. In this discussion, belief, and consequently acceptance,
will be understood as being a “matter of degree”.
2.2. Adequacy in Science
To accept an hypothesis is to believe it “adequate” in some relevant sense.
In the confirmational setting of science, “adequate” gets fleshed out in
terms of empirical adequacy. What it means for a scientific theory to be
empirically adequate is that it “gets things right” with respect to what we
can (even potentially) observe. The idea is that an empirically adequate
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theory will correctly describe and predict any empirical result we may
ever obtain. A rough but helpful way of thinking about what it means
to be an empirically adequate theory is that such theories say true things
and only true things about observables.6 The important feature of this idea
for the purposes of this discussion is that, as long as a theory gets things
right about everything observable, then it will be empirically adequate –
regardless of how it happens to relate the observables to unobservables
and whatever it says about unobservables.7 Thus an empirically adequate
theory need not get anything right other than what it says about what we
can observe; in particular, what it says about unobservables need not be
true.
In what sense, then, is the empirical adequacy of a theory most directly
confirmed? Consider what scientists actually do in the process of confirming a theory – they work to establish, by way of ingenious experiments and
subtle observations, that the theory gets things right about the available empirical data. This empirical data is, of course, data regarding observables.
In this sense, the immediate object of confirmation is the theory’s empirical
adequacy. It may well be that scientists are, with the same work, also confirming the truth of statements about unobservables. This is done, however,
by way of observables: whatever further confirmation a scientific theory
accrues grows out of the confirmation of its empirical adequacy. Science is
such that whatever it confirms, be it about observables or unobservables, it
does so by way of observables, i.e., by way of establishing empirical adequacy. It requires an additional step to move from the empirical adequacy
of a theory to the truth of its statements regarding unobservables, (i.e.,
the truth of the theory).8 Empiricists such as van Fraassen insist that this
additional step is never epistemologically justified. According to this view,
there is an unbridgeable epistemological chasm separating claims about
observables from claims about unobservables: the line separating claims
we are justified in accepting from claims we are justified in believing falls
precisely along the line separating claims about observables from claims
about unobservables. Our empirical methods justify belief only in claims
about observables; claims about unobservables are categorically relegated
to the class of the accepted. The standard realist criticism of this line has
been to deny the epistemological significance of the distinction between
observable and unobservable.
As the debate has developed, it seems the realist is right about the
epistemological significance of the distinction between observable and unobservable. We clearly can be justified in believing statements about things
we have not observed based on things we have observed. For example,
when I see tiny footprints and gnawed baseboards, and when I hear tiny
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footsteps and little squeaking sounds, I am justified in believing in the
existence of the mouse that I have not observed. In just the same way, I am
justified in believing in the electron I have not directly observed, but whose
tracks in cloud chambers I have seen, and whose impact on Geiger sensors
I have heard. It is true that I cannot, even in principle, observe the electron,
whereas I could observe the mouse if I were in the right place at the right
time. It seems mistaken, however, to maintain that the unobservability of
the electron makes a qualitative difference in whether we have reason to
believe it exists, especially when the structure of the reasoning is otherwise so similar. Furthermore, we lack even the beginning of an effective
argument that observability should make an epistemic difference in such
cases of “abductive” reasoning. As Clark Glymour (1984, 187) put it, “the
argument for atoms is as good as the argument for orbits”.
On the other hand, it is not an easy task to justify a realist position; in
fact, it is not clear that it has been done successfully. I am not going to try
to sort out the realism debate in science. Instead I am going to describe, in
rough terms, two features of scientific confirmation, at least one of which
is employed in the standard arguments for scientific realism. In the next
section I will consider the extent to which these two features carry over to
the setting of pure mathematics.
2.3. Explanatory Value and Evidential Relevance
The two features of confirmation that are typically used in justifying
the move from empirical adequacy to truth are explanatory value and
evidential relevance. One general line argues that the truth of a theory’s
claims regarding unobservables is well supported by the explanatory value
such assumptions have. The other approach attempts to make out some
kind of evidential relationship which shows how observations can be
confirmationally relevant to the truth of claims regarding unobservables.
Philosophers such as Clark Glymour (1980a, 1984), Ernan McMullin
(1984), and Richard Boyd (1984, 1985) have argued that the explanatory
value of a theory provides reason to believe it, independent of the reason
provided by its passing tests of empirical adequacy. The rough idea behind
these approaches is that, while it may be true that scientific tests of a theory
only give us reason to accept a theory, the explanatory considerations give
us further, independent reason to believe that the theory is true.
Glymour’s approach follows that of Michael Friedman’s (1974). The
basic idea is that good theories reduce the number of independently accepted hypotheses.9 Specifically, we should believe the theory that makes the
most extensive use of established regularities in explaining the phenomena
(Glymour 1984, 184). Consider for example, the atomic theory, that is, the
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theory that asserts the existence of atoms. An early form of this theory
was developed by Dalton (1808) to help explain the established regularities embodied in the law of definite proportions. A contending theory,
the theory of equivalences, maintained that the regularity of the law of
definite proportions is a brute tendency in no need of explanation. The
atomic theory is better on explanatory grounds because it uses the further
regularity of the additivity of masses. This explanatory value is supposed
to warrant belief in the theory.
McMullin’s and Boyd’s approach differs from Glymour’s in that the
explanatory value is not internal to the particular theories involved, but has
to do with explaining the success of science in general. The idea is that
we should accept realism because realism provides the best explanation
of the instrumental reliability of scientific methodology (Boyd 1984; 66).
Science has come up with, and continues to come up with, amazingly
effective theories as far as empirical adequacy goes. Quite often it does
so by positing unobserved or unobservable entities like planets, genes, dinosaurs, and electrons. What explains this remarkable success? According
to the argument, the best explanation of this success is that the posited
entities actually exist and behave (approximately) as the theory describes.
Scientific realism is the best explanation of scientific success.
The second strategy for moving from the empirical adequacy of a theory to its truth involves establishing the evidential relevance of claims
about observables for claims about unobservables. This general move can
be found in most accounts that work with formal scientific epistemologies
(e.g., Bayesian, boot-strapping, likelihood). The strategy is to establish
first the legitimacy of the account of confirmation, and then show that the
account allows for evidence regarding observables to confirm claims about
unobservables.
Consider the likelihood framework for characterizing how evidence
bears on hypotheses.10 The central idea is the likelihood principle:
(LP)
observation O favors H1 over H2 if and only if p(O|H1 ) >
p(O|H2 ).
The underlying intuition is that if H1 says that the observation is to be
expected, while H2 says that it is miraculous, then the observation strongly
favors H1 over H2 . This account allows for observations to be evidentially
relevant for hypotheses about unobservables, since to deny this would
mean that for some reason we cannot make sense of the conditional probability p(O|H ), where H is an hypothesis about unobservables. This,
however, is clearly not the case: our ability to make sense of this probability does not depend on whether H is about an observable like a continent
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or an unobservable like an electron. If we can make sense of one of these,
and we can, then we can make sense of the other. At bottom, what gets
an account like this off the ground is the clear intuition that we can have
observable evidence for things we do not (or cannot) observe; the empirical
adequacy of a theory is evidentially relevant to the truth of the theory.
This allows us to move rationally from belief in statements about observables (empirical adequacy of the theory) to belief in statements about
unobservables (truth of the theory).
3. MATHEMATICAL ADEQUACY
In the setting of science, the adequacy involved in the distinction between
accepting and believing is empirical adequacy. This same distinction
between acceptance and belief is appropriate in the setting of pure mathematics as well. In this setting, we exchange empirical adequacy for mathematical adequacy. In considering what mathematical adequacy amounts to,
I will naturally look to the methodology of mathematics itself. I will here
provide only a sketch of what a fully developed account of mathematical
adequacy would consist of, since this is a project (or more) in itself.11
What do mathematicians do, confirmationally speaking? As an oversimplified sketch of what mathematicians do, consider the following.
Given an area of interest – generally specified by some intuitive, pretheoretic principles and definitions – mathematicians seek to characterize
formally and explore the logical relationships that obtain among the constituents of the formal characterization. Just as in science, the factors that
determine which areas of pure mathematics are of interest seem to be of
an essentially pragmatic nature. In fact, the pragmatic factors often come
indirectly to mathematics through science. For example, the interest in being able to navigate effectively fueled the interest in celestial mechanics,
which in turn led to the discovery and development of the calculus. But
again, as in science, once pragmatics fixes an area, it makes sense to speak
of non-pragmatic epistemological principles. More specifically, once an
area is fixed, mathematicians seek to develop, in the loosely specified area,
formal theories that have consistent and manageable foundations, sound
proof schemes, unambiguous definitions, and internal consistency. I will
take this as a rough specification of what it means to have a mathematically adequate theory, namely, that such theories will have consistent and
manageable foundations, a sound proof scheme, well-defined definitions,
and that they will be internally consistent. I will say a bit about each of
these below.
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In the case of empirical adequacy, as van Fraassen (1980, 12, 69) points
out, the phenomena that must be “saved” is vast.
I must emphasize that this refers to all the phenomena; these are not exhausted by those
actually observed, nor even by those observed at some time, whether past, present or future.
. . . empirical adequacy goes far beyond what we can know at any given time. (All the results
of measurement are not in; they will never all be in; and in any case, we won’t measure
everything that can be measured.)
Similarly, establishing that a mathematical theory has consistent foundations, that all the statements of the theory are internally consistent, that the
proof scheme is sound, etc., is not usually something that it is humanly
possible to do. We develop relative consistency proofs. We continue to
prove theorems and explore the logical relations as a way of helping to
establish internal consistency. We analyze and explore our proof schemes.
By these means, we obtain good evidence that a theory is adequate, but
as in science, it is not something that we can know with certainty. The adequacy of a theory, mathematical or empirical, is what is directly confirmed
by the activity of mathematicians or scientists, but it is neither easily nor
absolutely confirmed.
I note also that these conditions characterize a minimally acceptable
pure mathematical theory.12 A theory that satisfies these conditions will be
an adequate pure mathematical theory. This does not mean, however, that
the theory will have any further virtues such as applicability, usefulness,
richness, depth, fecundity, simplicity, elegance, connections with other
mathematical theories, or explanatory power. Pure theories lacking some
or all of these “further virtues” may not be deemed worthy of investigation, but this does not mean that they are not legitimate pure mathematical
theories. Only inconsistency, ambiguity, or unsoundness seem sufficient to
condemn a pure mathematical theory.
3.1. Consistent Foundations
An essential feature of a pure mathematical theory is that it have a manageable set of fundamental definitions and rules; it must be axiomatizable in
some acceptable way, the axiomatization must be “manageable” in some
way. An infinite (or large finite) number of axioms with no clear recursive
pattern or scheme would not be manageable, for example, whereas a small
finite number or recursively enumerable set of axioms would be. Of course,
in the early stages of a theory’s development, it is often not well enough
understood to be axiomatized. A classic example is the calculus. It took
over a hundred years for the calculus, present in the late 17th century work
of Newton and Leibniz, to be put on secure foundations. It was not until
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the latter half of the 19th century that Weierstrass, Cauchy, and others were
able to work out consistent foundations for analysis.
Even in later stages, certain features of the mathematical theories
may not make mathematical sense. It may take refinements in the theory, or even an entirely new theory, to make mathematical sense of
things. The Dirac delta function was used in Fourier analysis for decades before a mathematically consistent description was worked out. To
this day the “renormalization” procedure in quantum electrodynamics is
what its inventor, Richard Feynman, called a “dippy process! . . . I suspect
that renormalization is not mathematically legitimate” (Feynman 1985,
128). Working out these foundations is part of the ongoing process of
establishing the mathematical adequacy of the mathematical theory.
3.2. Sound Proof Scheme
The proof scheme that a pure mathematical theory employs is an essential
feature of it. By “proof scheme” nothing more is meant than the rules for
inferring the truth of certain statements from the truth of others. It must be
sound in that the method of inference must not give rise to inconsistency.
Mathematical proof is generally taken to be deductive. While deductive
proof may not be the only means of justification in pure mathematics, it is
the predominant type. A theory that did not incorporate (at least) a sound
deductive proof scheme could hardly be an adequate pure mathematical
theory. As we have come to see, proof schemes are closely linked to the
choice of logic used to formalize the mathematical theory. The particular types of proof utilized by a mathematical theory, (e.g., direct proofs,
proofs by contradiction, proofs by mathematical induction, etc.), are determined by how it is formalized. There are pure theories that allow only
certain types of proof, e.g., constructivist theories. Constructivist theories
are distinct from their classical counterparts, which allow a wider range of
proof. At bottom this difference stems from the different logics employed
by each: constructivist theories employ an intuitionist logic and classical
theories employ classical logic.
A branch of mathematics itself, mathematical logic, provides a formal
setting for investigating the various properties (e.g., soundness, completeness, etc.) of formal systems in general. In the last one hundred years
or so, mathematical logic has provided us with a clearer understanding
of the virtues and limitations of particular formalizations of mathematics.
While the informal logic actually used in mathematics has become rather
universal and uncontroversial, its formal rendering is still a topic of debate.
The debate over whether the formal logic behind mathematics should be
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first-order or higher is still being waged, for example, and there continue
to be responsible calls for alternative logics.
3.3. Well-Defined Definitions
Another necessary condition for a theory’s mathematical adequacy is that
the concepts it employs must be well-defined. Definitions can fail to be
“good” definitions (well-defined) by being ambiguous or by “failing to
refer”. Often it is not obvious whether a definition is unambiguous, and establishing it requires non-trivial proof. As a simple example of the “failure
to refer” problem, consider the following definition:
Let M be the maximum value of the continuous function f (x)
on the open interval (a, b).
M is not well-defined because continuous functions on finite open intervals
need not have maximum values. (It would be well-defined if the functions
were continuous on the closed interval.) The definition of M “fails to refer”
in the sense that the procedure for assigning a value to M fails to pick out
a value for some functions over which the definition implicitly quantifies.
As an example of an ambiguous definition, consider the following. Let
G be a group and N a subgroup of G. The left coset of N in G containing
a ∈ G is defined to be Na := {xa|x ∈ N}. The factor group G/N
is defined to be {Na|a ∈ G}, with the factor group operation defined
as (Na)(Nb) := N(ab). The questionable definition here is the factor
group operation. The problem arises because there are different ways of
representing a given coset Na; in particular, if a " = b and b ∈ Na, then
Nb = Na. So care must be taken in defining the factor group operation as
(Na)(Nb) := N(ab) since, if it depends on the particular representative (a
or b), the definition will be ambiguous. It needs to be established that if a
and a # are in Na, and b and b# are in Nb (i.e., Na = Na # and Nb = Nb# ),
then it must be the case that (Na)(Nb) = (Na # )(Nb# ). This is not easy to
prove, and it turns out that the factor group operation is well-defined if and
only if the subgroup N is normal. (See Herstein 1986, 90–92 for details.)
3.4. Internal Consistency
Naturally, the statements of a pure mathematical theory must be mutually
consistent. One might think (mistakenly) that mathematical theories are
obviously internally consistent, since mathematicians start with consistent
axioms and from these deduce the remaining part of the theory. This, however, is not how mathematics usually gets done. Theories typically develop
in isolated bits and pieces, often initially with no explicit foundations; thus,
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267
internal consistency is a genuine concern. Also, pure mathematical theories
are often taken to include statements that have not yet been proven, but for
which a proof is believed to be forthcoming (e.g., the Riemann Hypothesis,
and prior to a few years ago, Fermat’s Last Theorem). The implications
of such conjectures are often explored well before they themselves are
proven, if in fact they ever are. In the end, the conjecture may turn out
to be independent of the main body of the theory, inconsistent with it, or
provable from it. The size, complexity, and vitality of pure mathematical
theories make the issue of internal consistency one that requires ongoing
attention. The continuing process of expanding the theory by proving new
results can be seen as helping to confirm a theory’s internal consistency. If
internal inconsistencies are present, then the process of proving new results
might eventually expose a contradiction. The more a mathematical theory
is explored and expanded without contradiction, the more reason we have
to believe it is internally consistent, and hence mathematically adequate.
3.5. Summary
As developed above, the work of mathematicians directly confirms a
pure theory’s mathematical adequacy (consistent foundations, sound proof
scheme, unambiguous definitions, and internal consistency). As with scientific theories, however, the truth of the theory is not directly confirmed.
Thus on this conception, whether a pure theory is mathematically adequate
is a question internal to mathematics; in particular, it is independent of
whatever application the pure theory might have in scientific theories.13 In
the next section, I examine the prospects of moving from the adequacy of
pure mathematical theories to their truth.
4. FROM ADEQUACY TO BELIEF
Given that mathematicians are directly confirming the mathematical adequacy of pure mathematical theories, we may ask whether we are (at least
sometimes) justified in believing the truth of the pure theory. Following
van Fraassen and in accordance with what I have said above, what I have
in mind with respect to “the truth of the pure theory” is the literal truth
of the theory taken at face-value. This is a full-blown sense of “true” –
with ontological commitments and all. At this point, I am not concerned
with the various interpretations of mathematics (e.g., if-then, modal, opensentence, Field-style nominalistic), which (rightly) are said to “muddy the
water” as far as ontology is concerned.
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In pure mathematics, the question becomes whether we ever have
reason to believe in the existence of the theoretical ontology based on
the fact that the theory is mathematically adequate. For example, does the
fact that we have good reason to believe that the theory of real analysis
is internally consistent, consistently founded, well-defined,
√ etc., give us
further reason to believe that there exist numbers (0, 1, 2, e, π , . . . ),
functions (x n , |x|, sin(x), ex , . . . ), operators (d/dx, dx), and sets (of numbers, of functions, of sets, of physical objects, . . . )? I begin by examining
the prospects for justifying the move from acceptance to belief in a way
analogous to how it is (putatively) justified in science. Finally, I briefly
discuss a few other possibilities.
4.1. Explanation and Evidence
Empirical adequacy has a natural link to the truth of scientific theories – a
causal link. We are able to make sense of evidential relationships between
statements about observables and statements about unobservables. This is
because the truth of both kinds of statements is grounded in the physical
world, and we have clear intuitions (at least some of the time) about such
evidential relationships. In general, these intuitions are grounded in causal
relationships between observable and unobservable.
The observable tracks in a cloud chamber, for example, or audible
clicks of a Geiger counter, have a chance of constituting evidence for
the existence of electrons because the tracks and clicks are causally related to the existence and behavior of electrons. This causal relationship
is generally what allows us to make sense of conditional probabilities
like P r(H |e) and P r(e|H ) (Bayesian, likelihood, etc.) that quantify the
evidential relationship. Consider for example, the likelihood, P r(vestigial
organs | natural selection took place). We understand intuitively that this
probability is rather high; this is because of causal background assumptions relating the process of natural selection and its effects (vestigial
organs, etc.).
Even Bayesian accounts seem to make use of causal ideas. While they
do not depend on particular causal relationships between hypothesis and
evidence, causal ideas seem inevitably to be present in the basic intuitions
on which Bayesianism is grounded. For example, Bayesians reject the idea
that “positive instances” in general confirm hypotheses. A positive instance
of a generalization may or may not be confirming – depending on background assumptions having to do with the sampling process.14 It is in the
assumptions about the “sampling process” that causal concepts enter, since
actual sampling processes will be causal processes. In mathematics, there
appears to be no such link between the mathematical adequacy of a theory
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and its truth – surely not a causal link. What is the evidential relationship
between the internal consistency (etc.) of number theory and the existence
of numbers? How can the consistency (etc.) of the theory be evidence that
the theory is true, i.e., that numbers exist?
That a theory is mathematically adequate is grounded in the nature of
logic. Whatever grounds the truth of mathematical existence claims (if
they are true) – presumably the nature of the mathematical world – does
not have any obvious connection with the laws of logic. We have learned
from Frege, Russell and other logicists that mathematics bears an intimate
relation to logic. The second part of their lesson, however is that mathematics is not merely logic. Unless one subscribes to some over-zealous
Meinongian principle like
(MP) any mathematically adequate theory describes an actually existing mathematical ontology,
the move from believing the mathematical adequacy of a theory to believing the truth of the theory (in an ontologically significant way) remains
philosophically unjustified. What is more, unlike the situation in science,
we seem to lack an intuitive sense that it should be justified.
Perhaps, then, the inference can be justified by the explanatory value
of posited entities. In science there is a clear sense in which the existence
of electrons, atoms, and genes help explain certain physical, chemical, and
hereditary phenomena. According to some philosophers, this explanatory
value is itself confirming; the posits’ explanatory value justifies belief in
their existence. But the same move cannot be made for mathematical theories. In fact, even granting that explanatory value is confirming, there is a
problem with this move for pure mathematical theories.
First of all we need to understand how to think about explanatory value
– whether in the internal sense of Glymour, or the reliability sense of Boyd
and McMullin. In other words, what exactly is the explanandum? Is it the
fact that theorem T follows from assumptions A? Is it the fact that theory
R is mathematically adequate? Or is it rather the remarkable success of
mathematical methodology in coming up with mathematically adequate
theories that is to be explained by the existence of mathematical objects?
Unfortunately, the existence of mathematical entities has no explanatory
value for these explanandums. The first is explained by the logical relationships that obtain among the mathematical concepts and definitions
involved. As a concrete example, it is a theorem of real analysis due to
Lindelöf that
(LT)
any collection of open sets of real numbers has a countable
subcollection that has the same union.
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The two key ideas in the proof of (LT) are that between any two real numbers exists a rational number, and that the rational numbers are countable.
The first idea follows from the completeness axiom for real numbers and
the second idea is a consequence of the fact that the rationals can be put
in a 1-1 correspondence with the natural numbers. The logical connections
between the steps of the proof of (LT) explain why it is that (LT) follows
from the axioms for real numbers. Nothing is gained, explanatorily, by the
assumption that real numbers exist.
In a similar way, the existence of real numbers does not help explain
the mathematical adequacy of the theory of real analysis. This adequacy is
explained by general facts about what is required to be a mathematically
adequate theory, along with the specifics of how the standard theory of
real analysis satisfies these requirements. Even if further explanation were
required, it is not at all clear how the fact (if it is a fact) that the theory is
about (ontologically) real mathematical entities helps explain anything. In
general, the fact that a theory is about real things does not explain why it
is consistent.
What then explains the success of mathematical methodology in coming up with mathematically adequate theories? Could not the existence of
the posited mathematical entities (i.e., the truth of the theory) be the best
explanation of the ability of the methods of pure mathematics to come up
with mathematically adequate theories?15 No. There is a crucial disanalogy
lurking here. The prima facie plausibility of the scientific version of this
argument stems from the intuitive idea that, if I continue to deduce many
and varied statements that I directly verify as true from a provisionally
assumed statement whose truth I do not (or cannot) directly verify, then
this gives me reason to believe that my provisional assumption is true.
Science does seem to have a knack of (somehow and eventually) coming
up with “provisional assumptions” that entail (in the context of the theory)
many and varied verifiably true statements. It does seem initially plausible
that what explains this is that the provisional assumptions that science is
coming up with are true.
For pure mathematics, however, this picture does not work. Mathematicians do not come up with “provisional assumptions” (e.g., that numbers
exist and obey Peano’s axioms) whose consequences can be verified to
be true. The proper analogy here is at the level of adequacy. There is
a way of looking at things in which mathematicians and scientists both
have a knack for (somehow and eventually) coming up with “provisional
assumptions” that ground a theory whose adequacy (empirical or mathematical) can be verified. Due to the different senses of adequacy, the analogy
breaks down. In science, since empirical adequacy amounts to the truth
CONFIRMING MATHEMATICAL THEORIES
271
of the theory’s statements about observables, verifying adequacy amounts
to establishing the truth of a certain class of statements. In mathematics,
however, verifying adequacy does not amount to verifying the truth of the
theory’s non-axiomatic statements (theorems). In the mathematical setting,
verifying adequacy amounts to verifying that the theorems do in fact follow from the provisional assumptions (axioms), not that the theorems are
themselves true. Thus, in science we have provisional assumptions that
lead to statements that are verified to be true, whereas in mathematics
we have provisional assumptions that lead to statements that are verified
to be consistent with or consequences of each other and the provisional
assumptions. It is prima facie plausible that the truth of the provisional
assumptions is required to explain why the resulting statements are true;
it is not plausible, however, that the truth of the provisional assumptions
is required to explain why the resulting statements are consistent with or
consequences of the provisional assumptions.
In science, the connection between what is directly confirmed (empirical adequacy) and the truth of the existence claims are causal and
constitutive.16 This sort of connection does support explanation. For
example, the empirical adequacy of the atomic theory has causal and
constitutive connections with the existence claims regarding atoms and
molecules. The observable macroscopic properties of chemical reactions
are explained (causally) by the various atomic-level transactions (borrowing/lending/sharing of electrons, etc.). The fact that this liquid is not water
is explained (constitutively) by the fact that water is H2 O and this liquid
is H2 O2 . Without an analogous explanatory connection between what it
is that is directly confirmed by mathematicians (mathematical adequacy)
and the truth of the existence claims, the existence claims will have no
explanatory relevance.17
4.2. Other Possibilities
The standard considerations which in the scientific setting, help justify
the inference from the acceptance of a theory to its belief, are unavailable
in the mathematical setting. The evidential and explanatory relationships
that intuitively ground such arguments in the setting of science do not
carry over to mathematics. It seems, then, that two possibilities for such
justification remain:
1. distinct internal (to mathematics) means, or
2. external means, (e.g., Quine’s indispensability approach, metaphysical
or conceptual considerations).
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As mentioned above, the standard argument for external justification from
science, the Quine/Putnam indispensability argument, is beginning to
appear to be problematic. Furthermore, the desiderata of capturing the
confirmational independence of pure mathematics, while perhaps not inconsistent with an external justification, renders the prospects dim. Even
so, it is possible that there is an external metaphysical justification, one
that does not suffer the problems of the indispensability approach. More
effort should be expended investigating this possibility.
As for other internal means of justifying the move, again the prospects
are not good. The obvious strategy would be to justify belief in the axioms,
which would in turn justify belief in the theory – assuming the soundness
of the system. How does one justify belief in an axiom? If one makes use
of criteria like explanatory power, fecundity, ability to unify, etc., then one
will again be faced with showing that satisfying these criteria counts as
evidence for the truth of an axiom. This is why we are trying to justify
belief in the axioms in the first place, so no obvious gain will have been
made. Another possibility is that an axiom is somehow self-evident, that
“once you understand it, you see that it is true”.18 But not even “universally” accepted axioms like the axiom of choice enjoy this status, much
less the more controversial ones.19
Fueled in part by these poor prospects for justifying the move from
acceptance to belief, I next explore an ontological “agnosticism” for mathematical theories. Before doing that, however, I explain briefly how this
project differs from Hartry Field’s (1989) fictionalist account, which makes
use of the thesis that “mathematics need not be true to be good”.
Field is wedded to a particular notion of “good” for a mathematical
theory, conservativeness, and as I discussed briefly above, his notion could
certainly be subsumed under my general notion of mathematical adequacy.
This affinity between Field’s notion of “good” and my notion of “adequacy” might suggest that our conclusions are more similar than they
in fact are. Field’s position seems to be that if mathematical entities play
an indispensable role in science (i.e., if no substitute for the literal truth
of the theory could be had), then mathematical entities would have to be
taken as confirmed (Field 1989, 20, esp. note 15). His notion of conservativeness is thus an effort to show how the truth of the mathematical theory
is not indispensable for science, but rather only its conservativeness. As I
have argued elsewhere, it is not at all clear that such an indispensability
argument is valid, and thus I do not accept the above if/then as the main
staging ground for issues of mathematical ontology. I attempt here to begin
to develop an alternative way of thinking about mathematical ontology.
In particular, I make use of a general notion of mathematical adequacy
CONFIRMING MATHEMATICAL THEORIES
273
in order to capture what mathematicians are doing (confirming) in doing
their work; I then argue that the move from mathematical adequacy to the
truth of the theory (in an ontologically significant sense) is unwarranted. In
doing this, I consider whether the truth of the mathematical theory is (indispensably) required to explain its mathematical adequacy. This is quite
distinct from Field’s question of whether the truth of the mathematical
theory is (indispensably) required for the scientific theory (in which it is
applied) to do its explanatory work.
5. ONTOLOGICAL AGNOSTICISM
As we have seen, van Fraassen’s categorical agnosticism with respect
to the truth of scientific theories is extravagant; however, an analogous agnosticism with respect to mathematical theories is more tenable.
Van Fraassen’s agnosticism requires that the line of demarcation between
acceptance and belief coincide with the line between observable and
unobservable; but as has been (I believe) convincingly argued, the observable/unobservable distinction is not up to this epistemological task.20 The
general consensus seems to be that in certain situations, with respect to
certain scientific theories, we may have reason to believe the truth of statements about unobservables; in other situations and with respect to other
theories, we may not. The acceptance/belief line is unclassifiably situationdependent; how it is drawn seems to depend in an ungeneralizable way on
the specific confirmation situation.
In a similar way, in mathematics the line demarcating acceptance and
belief is situation-dependent. Precisely what statements we should believe
will depend on the epistemic situation and particular features of the theory.
For example, whether we should believe the statement,
(S)
Fermat’s Last Theorem follows from the basic principles of
number theory,
depends on what we take the principles of number theory to be and how
much we know about the theory. Five or six years ago we would have been
much less justified in believing (S) than we are now!21 But in mathematics,
this situation-dependence is not quite as inscrutable. There is at least one
important generalization that may be made regarding acceptance and belief: categorical existence claims are always on the acceptance side of the
line. Statements like “numbers exist” or “sets exist” are not to be believed
on the basis of the mathematical adequacy of mathematical theories.
The statements that the mathematical adequacy of a theory licenses one
to believe are statements like: p follows from q, p is inconsistent with
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q, p is independent of q, x is well-defined, if A then T , etc. These are
statements that we are justified in believing based on a theory’s mathematical adequacy – categorical existence claims, however, are not similarly
licensed.22 It seems, then, that there is a qualitative difference between the
scientific and mathematical setting with respect to moving from adequacy
to truth. Consequently, an agnostic stance with respect to the existence of
mathematical entities is defensible.
ACKNOWLEDGEMENTS
I am grateful to Mike Byrd, Malcolm Forster, Penelope Maddy, Michael
Resnik, Elliott Sober, Mark Steiner, and anonymous referees for this
journal for various comments and discussion.
NOTES
1 See Maddy (1992), Sober (1993b), Vineberg (1996), and Peressini (1997) for some of
the problems being raised; see Resnik (1995) and Colyvan (1998) for defenses.
2 Throughout this paper, when I write of a mathematical theory being true, I intend this to
be understood in the natural way that implies the existence of mathematical objects. As is
well-known, there have been numerous attempts to reinterpret mathematical theory or the
notion of (mathematical) truth in such a way as to avoid such ontological implications, my
remarks here are obviously not directed at such accounts.
3 Volumes have been devoted to justifying this step; see, for example, Leplin (1984),
Churchland and Hooker (1985).
4 I will not enter here into the debate about how precisely to explicate the notion of
“believing a statement to be true”. One may safely understand this notion in terms of any
of the standard accounts of belief and truth since our concern here, the difference between
believing and accepting, does not depend on any particular account of belief or of truth.
5 This has been thought to be one lesson of Kyburg’s (1970) lottery paradox; see Sober
(1993a, 44–49) for discussion.
6 In general, this characterization will not be strictly correct, since most scientific theories will contain statements relating observables to unobservable posits. These statements,
while about observables, need not be true for the theory to be empirically adequate.
7 As van Fraassen (1980, 12) puts it,“. . . a theory is empirically adequate exactly if what
it says about the observable things and events in this world is true – exactly if it ‘saves the
phenomena’. A little more precisely: such a theory has at least one model that all the actual
phenomena fit inside. I must emphasize that this refers to all the phenomena: these are not
exhausted by those actually observed, nor even by those observed at some time, whether
past, present, or future”.
8 See Sober (1993a, 52–56) for a likelihood analysis of the idea that the knowledge
science provides us of observables is more direct than the knowledge it provides us of
unobservables.
CONFIRMING MATHEMATICAL THEORIES
275
9 Friedman (1974 and esp. 1983) associates “reducing the number of independently ac-
cepted hypotheses” with theory unification. While I will not get further into this here,
recent work has suggested that unification tends to have the opposite effect on the number
of independent hypotheses; see Forester and Sober (1994) for discussion.
10 The basic idea behind this analysis is Edwards’ Likelihood Principle (1972). The current
applications of the likelihood idea to the scientific realism debate come from Sober (1990,
1993).
11 See Maddy (1998) for extended attempt to understand mathematics on its own methodological terms. Also Hartry Field (1989, 57 ff), in developing his thesis that “mathematics
need not be true to be ‘good’, argues that what makes a mathematical theory good” is it
being conservative (i.e., consistent with any internally consistent theory about the physical
world). These insights (and many others no doubt) can be used to supplement, refine, and
replace parts of my sketch of mathematical adequacy; this will only serve to strengthen
what I say about the general notion of mathematical adequacy in subsequent sections.
12 See Peressini (1999) for a discussion of the pure/applied distinction and its relevance to
the debate.
13 Strange enough results in science might lead scientists (and mathematicians) to question,
say, the consistency of a pure theory employed in science. But ultimately the fate of the
pure theory qua pure theory would turn on whether there was enough “pure mathematical”
evidence to show the theory mathematically inadequate. I note also that while one might be
tempted to think of this scenario as “empirical” evidence for the inconsistency of the pure
theory, there is nothing essential about the empirical observations here. That the predictions
of an applied mathematical theory fail to match observations is no grounds for believing
the pure theory to be mathematically inadequate; rather, the predictions would have to be
such that they could not match observations (i.e., inconsistent, incoherent, fail to behave as
required by measurement theory, etc.).
14 See Horwich (1982, 54–63) for the details of a Bayesian rejection of the “positive
instance” account of confirmation.
15 In the next few paragraphs I will lapse into an untenable “positivistic dialect” regarding
science and an equally untenable “deductivist dialect” regarding mathematics. This is only
to simplify the discussion; it should be clear that the points I make survive translation to a
more tenable “dialect”.
16 For a discussion of the different roles played by causal and constitutive relations in
scientific explanation, see Enç (1986, esp. 404–408).
17 There is a distinct but important sense of explanatory value at work in mathematics:
namely, the idea that some proofs are more explanatory than others. It is not at all clear
that this sort of explanatory value makes the proven statement more secure. At any rate, this
explanatory value will favor one proof over another, rather than one belief over another.
18 See Chisholm (1977, 40) as quoted in Maddy (1990, 144).
19 See Maddy (1990, 143 ff.) for discussion.
20 See, for example, Sober (1993a).
21 At the time this was initially written, Wile’s proof was still incomplete.
22 Relative existence claims may be believed on the basis of a theory’s adequacy. In analysis it is proven that, given a function f (x), there exists a function f + (x) that is zero
where f (x) < 0 and is f (x) where f (x) ≥ 0. We are indeed justified in believing this theorem. But this is not a categorical existence claim in that its truth does not entail that there
are functions in the first place. In a similar fashion, model theoretic results establishing the
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existence of a model meeting certain conditions, do so only relative to the assumption that
certain other mathematical entities (numbers or sets) exist.
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Department of Philosophy
Marquette University
Milwaukee, WI 53201
E-mail: [email protected]
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