naturalism in mathematics and the authority of philosophy

Brit. J. Phil. Sci. 56 (2005), 377–396

Naturalism in Mathematics and

the Authority of PhilosophyAlexander Paseau

ABSTRACT

Naturalism in the philosophy of mathematics is the view that philosophy cannot legiti-

mately gainsay mathematics. I distinguish between reinterpretation and reconstruction

naturalism: the former states that philosophy cannot legitimately sanction a reinter-

pretation of mathematics (i.e. an interpretation different from the standard one); the

latter that philosophy cannot legitimately change standard mathematics (as opposed

to its interpretation). I begin by showing that neither form of naturalism is self-refuting.

I then focus on reinterpretation naturalism, which comes in two forms, and

examine the only available argument for it. I argue that this argument, the so-called

Failure Argument, itself fails. My overall conclusion is that although there is no self-

refutation argument against reinterpretation naturalism, there are as yet no good

reasons to accept it.

1 Naturalism in mathematics

2 The consistency of mathematical naturalism

3 The failure argument

4 Objections to the failure argument

5 Philosophy as the default

1 Naturalism in mathematics

Naturalism in the philosophy of mathematics is the view that philosophy

cannot legitimately gainsay mathematics. Perhaps ‘quietism’ or ‘conservatism’

would be more suggestive labels, but ‘naturalism’ is the entrenched term of

art.1 Whatever we call it, the view rests on the distinction between norms

governing mathematics and norms governing philosophy, a distinction

which is nicely illustrated by different justifications for the Axiom of Choice.

1 Many other recent ‘naturalisms’ are discussed in Kitcher ([1992]), in Wagner and Warner (eds.)

([1993]), and elsewhere. In general, the term ‘naturalism’ is overworked in contemporary

philosophy. Vague orientation aside, most philosophical naturalisms have little in common

with one another. Doctrines going by the name of ‘mathematical naturalism’ prior to the

establishment of current usage in the 1990s also tend to be unconnected to our present topic.

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A mathematical reason for accepting the axiom might be that it leads to a more

unified and tractable arithmetic of infinitary cardinals (e.g. the well-ordering

of cardinals follows from it).2 A philosophical reason for accepting it might be

that sets are abstract entities structured in such a way that they obey the

axiom.3 As a second example, consider the norm of ontological economy,

which gives preference to theories with smaller ontologies. Ontological eco-

nomy certainly isn’t a mathematical norm, and perhaps isn’t even a scientific

one.4 Yet it is accepted as a philosophical norm by many philosophers.5 On

the basis of these and similar examples, naturalism more generally assumes a

distinction between philosophical and mathematical norms.6 Both philosoph-

ical and mathematical norms may of course be thought of as grounded in

overarching norms common to all forms of inquiry, such as the norm of the

pursuit of truth. Norms in the various spheres of inquiry, however, constitute

different proposals as to precisely how to pursue truth, and this leaves open

the possibility of a clash of norms from the different spheres.

The rough statement of the naturalist argument to be examined below is

that reinterpretations of mathematics should not be carried out on the basis of

philosophical norms because these norms are less credible than mathematical

ones. A reinterpretation of mathematics is here defined as an interpretation

different from the standard interpretation, which is the interpretation of

mathematics accepted by mathematicians when doing mathematics. Reinter-

pretations are to be contrasted with reconstructions, which change the actual

mathematics as opposed to its interpretation.7 An exemplary case of recon-

struction is Brouwer’s proposal to rebuild mathematics along intuitionistic

lines. When Brouwer proved that any function on a bounded interval

is (uniformly) continuous, he was writing new mathematics, not just

2 I will speak of norms, reasons, principles, etc., interchangeably. The norm of ontological

economy, for example, is equivalent to a principle recommending ontological economy; and

all the norms or principles discussed function as reasons for or against some position.3 See Maddy ([1997], p. 191; [1998], p. 164) for a similar way of distinguishing between

mathematical and philosophical reasons. Maddy also gives the example of the acceptability

of impredicative definitions. A philosophical reason against such definitions would be the

constructivist one that mathematical objects are mental constructs, and hence that such

definitions are constructively circular. A mathematical reason for accepting such definitions

might be that they are needed for a classical theory of the real numbers.4 See Burgess ([1998]) for why ontological economy may not be a scientific norm.5 I shall take no stand on where philosophical norms come from, to what extent they are handed

down by tradition, or grow out of common sense, or are continuous with science, or should

be thought of as communal rather than individual, etc. The delineation and genealogy of

philosophical norms are fascinating subjects, but tangential here.6 Naturalists need not be committed to the existence of a principle that precisely demarcates this

boundary. I return to the distinction between philosophical and mathematical norms in Section 4

below.7 One may propose to reinterpret or reconstruct some individual branches of mathematics rather

than all of it, but I omit qualifications of this kind since they don’t affect the substance of the

discussion.

378 Alexander Paseau

reinterpreting old (i.e. classical) mathematics.8 Most forms of mathematical

constructivism more generally involve reconstruction of one sort or another,9

and the past century has seen many a mathematical Bolshevik urging the

reconstruction of mathematics on philosophical grounds.10 In contrast, a

reinterpretation keeps the mathematical content of the mathematics intact

and changes only the interpretation, replacing the standard interpretation

with some preferred alternative.

?Here are a few examples of possible interpretations of mathematics (the list is

not exhaustive). Minimal realism has it that mathematical statements are

to be taken at logico-grammatical face value, but that they say nothing about

the nature of mathematical objects, properties and relations. Singular terms

denote objects of some domain over which the quantifiers range, and predicates

denote properties or relations on this domain. In particular, according to min-

imal realism, mathematical statements say nothing about whether mathemat-

ical entities, properties and relations are abstract or concrete. Platonism adds to

minimal realism the claim that these entities, properties and relations are

abstract. Eliminative structuralism, on the other hand, has it that mathematical

statements are general claims about any structure that satisfies the (particular

branch in question’s) axioms or principles. Finally, according to fictionalism,

mathematical statements are to be understood as prefaced by (something like) a

fictional operator ‘According to the story of mathematics, . . .’.11

In arguing that philosophy does not have the authority to replace the

standard interpretation of mathematics, the naturalist assumes that the latter

exists. Yet mathematics is a sprawling subject, and it may be that some of

its branches are understood one way, and some of them in quite another.

Furthermore, it may be that mathematical practice within an individual

branch is indeterminate between two (or more) interpretations. There may

also be live disagreement within mathematics about how best to understand

many statements. For these and other reasons, mathematics or any of its

branches may not have a single standard interpretation. The naturalist

8 Brouwer’s so-called continuity theorem does not hold for classical mathematics. A nice example

of a classically discontinuous function on an interval [a, b] (where a< b) is the function that has

value 1 for rational numbers and value 0 for irrational numbers.9 Assuming that they are understood as advocating the one true mathematics rather than as

offering another kind of mathematics that can peacefully co-exist alongside classical

mathematics (of interest, for example, to a mathematician who wishes to see what can be

done with fewer than the usual available tools).10 See Shapiro ([1997], pp. 21–5) or ([2000], pp. 7–10) for useful potted summaries of various

reconstructive movements in the history of philosophy of mathematics. Frank Ramsey called

both Brouwer and Hermann Weyl mathematical Bolsheviks; in response, Wittgenstein intoned

that Ramsey was a ‘bourgeois philosopher’ (see Mancosu [1998], p. 66).11 These definitions are not intended to capture all the different positions ever denoted by these

labels. In particular, fictionalism as defined here is different from Hartry Field’s well-known

position of the same name.

Naturalism in Mathematics and the Authority of Philosophy 379

argument to be examined below should therefore be understood in a condi-

tional sense: if there is a standard interpretation of mathematics (or any of its

branches), then philosophical norms cannot overturn it. Having said that, the

success of the naturalist argument concerning philosophy’s authority either to

ordain or to overturn the standard interpretation of mathematics does not

depend on the fact that some one or other of these positions happens to be the

correct standard interpretation. Whether it is minimal realism, platonism,

eliminative structuralism, fictionalism, or some other doctrine that fills that

bill is irrelevant.12

I have so far distinguished reconstructions of mathematics from reinter-

pretations. A second distinction is needed in order to present the naturalist’s

argument. Let us say that scientific philosophy is philosophy that consists only

of scientific norms, i.e. norms enshrined in science (excluding mathematics).

And let us call general philosophy any philosophy that consists more broadly

of norms that may be drawn either from within science or from outside it

(again, excluding mathematics). As an illustration, let us suppose that onto-

logical economy is not a norm of scientific theory choice, whereas empirical

adequacy is. Then any philosophy that accepts empirical adequacy as a rel-

evant norm in cases of philosophical theory choice is both scientific and

12 A specific recent example of reinterpretation without reconstruction, which further serves to

illustrate the distinction between the two, is Geoffrey Hellman’s ‘modal-structuralism’ (see

esp. Hellman [1989]). Hellman offers a ‘translation scheme’ (whose details don’t matter here)

between his modal-structural interpretation of arithmetic and what he takes to be the standard

interpretation of arithmetic, stressing that ‘the modal-structuralist aims at much more [than

the recovery of proofs]: in some suitable sense, the translates must be mathematically

equivalent to their originals’ ([1989], p. 26). Most of the first chapter of Hellman ([1989]) is

taken up with explaining the sense in which Hellman’s proffered translation scheme provides a

‘mathematical equivalence’—in my terminology, a reinterpretation rather than a reconstruction.

In general, many philosophies of mathematics are intended as either reinterpretations or

reconstructions. For instance, Hilbert’s philosophy of mathematics aims to reinterpret

finitary mathematics as stroke arithmetic and ideal mathematics formalistically. Had it been

successful, this philosophy would have been reinterpretative (Hilbert’s interpretations of finitary

and ideal mathematics do not appear to be the standard ones), but not reconstructive (although it

adds consistency proofs to the body of mathematics, no mathematics is supplanted). A few

philosophies of mathematics, however, appear to fit neither category, for example: (1) Any

philosophy of mathematics that sees mathematics as positing a network of entities of

unspecified nature that play such-and-such a conceptual role and which advocates some

particular entities as the role-fillers. (Compare physicalist functionalism in the philosophy of

mind.) Such a philosophy of mathematics is clearly not reconstructive. And if, say, minimal

realism is the standard interpretation, such a philosophy is reinterpretative only in the weak sense

that it fills out, rather than opposes, the standard interpretation. (2) A philosophy that abides by

the Wittgensteinian precept that philosophy should leave everything as it is. One way of

understanding this is as the conservative attitude that crowns the correct standard

interpretation as ipso facto the correct philosophical one. (This attitude is what I define as

strong reinterpretation naturalism in the next paragraph.) Alternatively, and more faithfully

to Wittgenstein, his precept can be taken as saying that there should be no prescriptive

philosophies. This second, subtly different, form of naturalism—a subtly different ‘anti-

philosophy’ philosophy—would claim that philosophy cannot even sanction any

interpretation of mathematics, be it standard or not.


general (in this respect). On the other hand, any philosophy that accepts

ontological economy as a relevant norm in cases of philosophical theory

choice (e.g. to argue for nominalism over platonism) is general but not sci-

entific, or, in other words, is non-scientific philosophy. More generally, the

inclusion of scientific philosophy within general philosophy entails that the

naturalist doctrine according to which general philosophy cannot overturn

the standard interpretation of mathematics is stronger than the doctrine

that non-scientific philosophy cannot overturn the standard interpretation.

I accordingly distinguish between two types of reinterpretation naturalism:

Analogously, one can distinguish between strong and weak versions of

reconstruction naturalism. Since my focus is on the interpretation of math-

ematics, however, I shall speak of reconstruction naturalism tout court when

discussing it. Likewise, I will speak of reinterpretation naturalism where I do

not need to distinguish between its strong and weak variants.

I am now able to state my article’s twofold aim more precisely. My first aim is

to examine an argument by Hilary Putnam that claims to undermine any view

that privileges some set of norms as the correct ones for inquiry. In Section 2, I

show that this argument, which threatens to cut the ground from under the

naturalist’s feet, undermines neither reconstruction nor reinterpretation nat-

uralism. My article’s second and main aim is to examine the Failure Argument

(my label) on behalf of reinterpretation naturalism. The Failure Argument

infers from philosophy’s poor historical track record that philosophical rein-

terpretations of mathematics are highly dubious. I examine the argument in

Sections 3 and 4, where I show that it supports neither version of reinterpreta-

tion naturalism. I conclude in Section 5 that we have been given no reason to

accept either strong or weak reinterpretation naturalism. My overall conclu-

sion, accordingly, is that although there is no self-refutation argument against

reinterpretation naturalism, there are as yet no good reasons to accept it.

2 The consistency of mathematical naturalism

Putnam introduces his self-refutation argument with a reminder that the

logical positivists’ equation of rational believability with scientific verification

was self-refuting, the reason being that this very equation is itself not scien-

tifically prescribed, hence not rationally believable ([1981], pp. 105–6).

(Strong Reinterpretation

Naturalism)

General philosophy cannot legitimately sanction

a reinterpretation of mathematics.

(Weak Reinterpretation

Naturalism)

Non-scientific philosophy cannot legitimately

sanction a reinterpretation of mathematics.


He urges that the positivists’ mistake contains a ‘deep’ or general lesson

([1981], p. 106; see also p. 111). As he sees it, the self-refutation objection

applies to any ‘criterial’ conception of rationality, where the latter is defined as

‘any conception according to which there are institutionalised norms which

define what is and what is not rationally acceptable’ ([1981], p. 110). He

defends this claim with the following argument:

[I]f it is true that only statements that can be criterially verified can be

rationally acceptable, that statement itself cannot be criterially verified,

and hence cannot be rationally acceptable. If there is such a thing as

rationality at all [. . .] then it is self-refuting to argue for the position

that it is identical with or properly contained in what the institutionalized

norms of the culture determine to be instances of it. For no such argument

can be certified to be correct, or even probably correct, by those norms

alone. ([1981], p. 111)

Putnam’s argument thus rests on two premises. The first premise is that a

statement of the form ‘A statement is rightly assertible if and only if it is

assertible according to X-norms’ has to be assertible according to X-norms to

avoid self-refutation. The second premise is that no such X-norms exist.

The second premise is very speculative, even if restricted to substantive

X-norms. I know of no argument to support it in Putnam’s writings (including

Putnam [1983], where the argument is reprised), or indeed elsewhere. The

inductive evidence Putnam himself adduces for it in this chapter is not com-

pelling. It emerges there by a bold extrapolation from the failure of some sorry

proposals for a universal norm of rationality that have cropped up in the

history of philosophy (e.g. the failed recommendation of empirical verification

as the universal norm). The general evaluation of Putnam’s second premise,

however, is of no concern to me here. What is of interest is the particular value

of X relevant to my discussion: mathematics.13 This instance does indeed fall

to Putnam’s self-refutation objection. The self-refutation objection clearly

applies to an extreme mathematical naturalism that states that only mathem-

atical norms can be appealed to in epistemic inquiry. The reason is that it is no

part of the claims of mathematics that it aims to settle every possible question.

Mathematics doesn’t condone the instance of the general premise ‘A statement

is rightly assertible if and only if it is assertible according to mathematical

norms’. So the self-refutation objection knocks down the extravagantly global

claim that mathematics provides the only norms or criteria of inquiry. This

global naturalism is untenable.

The question is whether there is room for a more local form of naturalism.

Can one consistently advocate a naturalism that allows only mathematical

13 Everything I say below also applies to some other values of X, e.g. X¼ science or

X¼mathematics-cum-science (i.e. mathematics and science considered as a whole).


norms to be appealed to in the evaluation of a statement about the

interpretation of mathematics, such as, say, ‘numbers exist and are abstract’?

And can one consistently advocate a naturalism that allows only mathemat-

ical norms to be appealed to in the evaluation of any mathematical statement,

such as, say, ‘2þ 3¼ 5’? By looking at the two respective modifications of

Putnam’s schematic first premise, I show that these more restricted natural-

isms are not subject to self-refutation.

Take first the claim ‘A statement about the interpretation of mathematics is

rightly assertible if and only if it is assertible according to mathematical

norms’. This biconditional is an epistemological thesis, a thesis about which

norms decide the correct interpretation of mathematics. It is a higher-order

claim about the norms that should be used to settle claims about the inter-

pretation of mathematics rather than a first-order claim about what the inter-

pretation of mathematics is. It therefore does not fall within its own ambit,

which includes only first-order statements about the interpretation of math-

ematics. Hence it is not vulnerable to the self-refutation charge. Another

way of putting the same point is that the stated biconditional avoids incon-

sistency precisely by being understood as applying to first-order, but not

higher-order, statements about the interpretation of mathematics. In particu-

lar, it follows that both strong and weak reinterpretation naturalism survive

the self-refutation objection, since the biconditional (so understood) is logic-

ally consistent.

Now consider the claim ‘A mathematical statement is rightly assertible if

and only if it is assertible according to mathematical norms’. Is this claim itself

a mathematical or a philosophical statement? Reasoning similar to that in the

previous paragraph suggests the latter; but in fact it does not matter either

way. Suppose first that it is a mathematical statement. In that case, it is

vindicated by mathematics and not undermined by it. For it is part of math-

ematics’ self-image that only its norms can be appealed to in the investigation

of a strictly mathematical question (as opposed to, say, a question about how

to settle the correct interpretation of mathematics). Mathematics condones

the equation of mathematical correctness with assertibility according to math-

ematical norms. This, after all, is no more than a methodological tautology:

any discipline, not just mathematics, will claim jurisdiction for the questions

definitive of its domain. On this alternative, then, the biconditional at the start

of this paragraph is self-vindicating rather than self-undermining. Suppose, on

the other hand, that the biconditional in question is a philosophical statement.

In this case, the statement is shielded from self-application. It is no longer a

mathematical statement, hence mathematical norms don’t apply to it. Either

way, the self-refutation charge fails. Our reasoning shows in particular that

reconstruction naturalism survives the self-refutation objection, since the

biconditional at the start of this paragraph is logically consistent.


The self-refutation argument is thus incapable of undermining mathe-

matical naturalism. Neither reinterpretation nor reconstruction naturalism

has the sweeping—megalomaniac—ambitions that casualties of Putnam’s

argument have.

3 The failure argument

Reinterpretation naturalism survives the self-refutation charge unscathed,

with the result that we have to examine whether it is rationally supported.

Now our current practice as an epistemic community creates a presupposition

in favour of reconstruction naturalism and against reinterpretation natural-

ism. Mathematicians use mathematical norms to decide what the correct

mathematics should be, and philosophers are by and large happy to let them

do so.14 But it is typically up to philosophers (or mathematicians when they

wear a philosophical hat) to trouble themselves with the question of its correct

interpretation. Or at least so philosophers think, which is what matters given

that the naturalist attack on the authority of philosophy is an internal one.

Thus an argument is called for if philosophers are to be persuaded to reject

reconstruction naturalism, and likewise an argument is called for if they are to

be persuaded to accept reinterpretation naturalism. In the absence of convin-

cing arguments, philosophers have no good reason to change their ways and

may continue to enjoy the current allocation of intellectual rights.

In light of this, I shall present the only argument for reinterpretation nat-

uralism to be found in the literature.15 I call it the Failure Argument. The

argument is contained in the following well-known passage by David Lewis:16

Mathematics is an established, going concern. Philosophy is as shaky as

can be. To reject mathematics for philosophical reasons would be absurd.

14 Strictly speaking, my discussion applies to the mathematics justified by today’s mathematical

norms rather than to today’s accepted mathematics. I equate the two merely to simplify

exposition. These norms, it should be noted, are justificatory, that is, they are the standards

by which to judge whether a piece of reasoning is mathematically justified or not. Thus they do

not necessarily capture the psychology of mathematical evaluation, nor do they necessarily add

up to an algorithm to effectively decide between the claims of different mathematical theories

(compare Lycan [1988], pp. 130–3, on scientific norms).15 I henceforth set aside the doctrine of reconstruction naturalism. Although much of what I say

below is relevant to it, its assessment is a large task for another occasion.16 Mark Colyvan endorses this argument verbatim ([2001], p. 33); Stewart Shapiro ([1997], p. 30)

also backs it but wonders whether this commits him to the principle of philosophy-last-if-at-all

(see next note); and John Burgess ([1998], p. 197) also offers a version of the argument. Many

others have commented on the apparent differential success of philosophy and mathematics,

including the mathematician G. H. Hardy in this colourful passage: ‘A metaphysician, says

Bradley, will be told that ‘‘metaphysical knowledge is wholly impossible’’, or that ‘‘even if

possible to a certain degree, it is practically no knowledge worth the name’’. ‘‘The same

problems’’, he will hear, ‘‘the same disputes, the same sheer failure. Why not abandon it and

come out? Is there nothing else more worth your labour?’’ There is no one so stupid as to use this

sort of language about mathematics. The mass of mathematical truth is obvious and imposing;


If we philosophers are sorely puzzled by the classes that constitute

mathematical reality, that’s our problem. We shouldn’t expect mathem-

atics to go away to make our life easier. Even if we reject mathematics

gently—explaining how it can be a most useful fiction, ‘good without being

true’—we still reject it, and that’s still absurd. Even if we hold onto some

mutilated fragments of mathematics that can be reconstructed without

classes, if we reject the bulk of mathematics, that’s still absurd.

That’s not an argument, I know. Rather, I’m moved to laughter at the

thought of how presumptuous it would be to reject mathematics for philo-

sophical reasons. How would you like the job of telling the mathematicians

that they must change their ways, and abjure countless errors, now that

philosophy has discovered that there are no classes? Can you tell them, with

a straight face, to follow philosophical argument wherever it may lead? If

they challenge your credentials, will you boast of philosophy’s other great

discoveries: that motion is impossible, that a Being than which no greater

can be conceived cannot be conceived not to exist, that it is unthinkable

that anything exists outside the mind, that time is unreal, that no theory

has ever been made at all probable by evidence (but on the other hand that

an empirically ideal theory cannot possibly be false), that it is a wide-open

scientific question whether anyone has ever believed anything, and so on,

and on, ad nauseam?

Not me! ([1991], pp. 58–9)

Lewis’ echo of the Ciceronian plaint against philosophy (‘there is nothing so

absurd but some philosopher has said it’) reads as straightforward advocacy

of reconstruction naturalism; and it can be argued that this passage together

with others in Parts of Classes provides evidence that Lewis endorses strong

reinterpretation naturalism as well (see note 17). Since my focus here is on the

Failure Argument rather than Lewisian exegesis, however, I leave the inter-

pretation of Lewis to one side and concentrate on the Failure Argument itself,

understood here as an argument on behalf of reinterpretation naturalism. The

argument can be regimented thus:

The Failure Examples premise abbreviates the comic battery of absurdities

and errors to which the history of philosophy is supposed to bear witness in

the quotation from Lewis. Reinterpretation naturalism follows directly from

its practical applications, the bridges and steam-engines and dynamos, obtrude themselves on

the dullest imagination. The public does not need to be convinced that there is something in

mathematics’ ([1992], pp. 64–5).

(Failure Examples) Philosophical norms have a very poor historical

track record.

; (Failure Thesis) Philosophical norms are highly dubious.

; (Conclusion) A reinterpretation of mathematics based on

philosophical norms is highly dubious.


the argument’s conclusion by comparing philosophical with mathematical

norms. Strong reinterpretation naturalism in particular follows from the claim

that general philosophical norms are more dubious than mathematical ones.

Weak reinterpretation naturalism follows from the claim that non-scientific

philosophical norms are more dubious than mathematical ones. In short, the

Failure Argument advises us to trust mathematics about the nature of its

subject matter—since its norms have considerable epistemic authority—rather

than to seek a reinterpretation on the basis of the epistemically less secure

norms of philosophy.17

17 Weak reinterpretation naturalism has its roots in Quine’s general advocacy of the ‘abandonment

of the goal of a first philosophy’ and his recommendation that science should not be ‘answerable

to any supra-scientific tribunal, and not in need of any justification beyond observation and the

hypothetico-deductive method’ (Quine [1981], p. 72). More recently, Burgess and Rosen adopt

weak reinterpretation (as well as reconstruction) naturalism in their book on nominalism: ‘The

naturalists’ commitment is [. . .] to the comparatively modest proposition that when science

[understood to include mathematics] speaks with a firm and unified voice, the philosopher is

either obliged to accept its conclusions or to offer what are recognizably scientific reasons for

resisting them’ ([1997], p. 65); see also ([1997], p. 205); for earlier statements of these views by

John Burgess, see Burgess ([1983]) and ([1990]). Burgess and Rosen argue against nominalistic

‘reconstruction’ or ‘reformulation’ of mathematics, umbrella terms for what I distinguish as

reconstruction or reinterpretation. That reinterpretations as well as reconstructions are within

the scope of their critique is apparent in several passages: see e.g. ([1997], p. 210) where they reject

‘reconstructions’ that are ‘notational variants’, that is to say, reinterpretations. And they intend

their point to generalise beyond nominalistic reinterpretations. Another philosopher in this

mould is Alan Baker, who in a recent article presupposes the ‘insight’ of naturalism right from

the start: ‘There is one theme that [. . .] I want to stress at the outset. It derives from the insight

that—given the naturalistic basis of the Indispensability Argument, which rejects the idea of

philosophy as a higher court of appeal for scientific judgments—the only sensible way of judging

alternatives to current science [including mathematics] is on scientific grounds’ (Baker [2001],

p. 87). Since Baker also considers reinterpretations of mathematics as scientific alternatives,

he too subscribes to the ‘insight’ of weak reinterpretation naturalism. Maddy ([1997]) is a

naturalist manifesto on behalf of strong reconstruction naturalism: ‘Mathematical

naturalism, as I understand it, is just a generalisation of this conclusion, namely, that

mathematical methodology is properly assessed and evaluated, defended or criticised, on

mathematical, not philosophical (or any other extra-mathematical) grounds’ ([1998], p. 164);

see also ([1997], p. 184); ([1996], p. 502), and ([1998], p. 165) for other crisp statements. (In a more

recent article, Maddy reformulates her naturalism as an ‘approach’ rather than a doctrine

([2001], p. 37); see also ([2000], p. 108). However, this relabelling, even if accurate, does not

exempt her naturalism from criticism, since a stance or approach can perfectly well be criti-

cised.) Throughout her book, Maddy also assumes weak reinterpretation naturalism, enjoining

that ‘naturalistic philosophy of mathematics takes place within natural science’ ([1997], p. 202);

see also ([1997], p. 183, and pp. 200–3 for more citations; note that qualifications of the kind

made in Maddy ([1998], p. 178) and discussed in Colyvan ([2001], pp. 96–8) show that Maddy’s

weak reinterpretation naturalism is sensitive to the point made in note 14 of my article). As for

Lewis, the footnote to his quoted credo ([1991], p. 58 note 16) demonstrates that his adoption of

naturalism is preceded by some hesitation. The tempting structuralist interpretation of set

theory (a reinterpretative philosophy, as Lewis sees it) is explored in depth in the book’s

appendix and the succour it offers in the face of the mysteries of the singleton function is

only reluctantly renounced. Yet Lewis does eventually disown his flirtation with structura-

lism in Parts of Classes (even if he ends up endorsing it in Lewis [1993], which represents a

significant shift in this respect). Indeed, one can consistently read strong reinterpretation

naturalism as his official view. Witness several of the book’s summarising position

statements: ‘I mock those philosophers who refuse to take mathematics as they find it [. . .]. I

would much prefer a good answer to my [epistemological] worries about primitive singleton, so


I shall present four objections to the Failure Argument. As we will see, the

first two are (at best) inconclusive; the last two, however, sink the argument.

Before that, a preliminary point is in order. An observation that must precede

any critique of the argument is that philosophical standards often—indeed,

usually—get absorbed in successful science or give rise to a successful new

science. A stock example is psychology, which grew out of philosophy.

Psychology as a modern autonomous discipline may be said to have begun

in the second half of the nineteenth century following the establishment of

experimental laboratories. Many of the doctrines and programmes which the

modern discipline fruitfully explores today had their roots in its prehistory,

when psychology was still a part of speculative philosophy. And of course

empirical psychology has not been wholly independent of philosophy ever

since the late nineteenth century, and still today is much influenced by philo-

sophy (and reciprocally influences it). Another recent example is that of

Chomskyan linguistics, which emerged from the confluence of logic, philo-

sophy of mind, and philosophy of language in the mid-twentieth century

(witness its reliance on twentieth-century philosophical doctrines), and whose

pioneer was himself personally influenced by philosophy and philosophers

(e.g. Nelson Goodman). More specific examples abound: Einstein’s use of a

verificationist heuristic in developing his special theory of relativity, or the

influence of Popper’s falsificationism on Hermann Bondi and many other

scientists; and so on. And of course there are many older examples, going

back to the Greeks: (parts of ) natural science, logic . . .. In fact, most of the

central theoretical disciplines have historically been intermingled with philo-

sophy; arguably, several of them emerged directly out of philosophy itself.

The preliminary point is thus that the historical track record of past prin-

ciples we now consider philosophical is an unreliable guide to whether present

philosophical principles will prove theoretically successful.18 Their very des-

ignation as distinctly philosophical prejudices the issue of these principles’

that I could in good conscience take mathematics as I find it’ ([1991], pp. viii–ix); ‘Despite all my

misgivings over the notion of singleton, I am not fully convinced that structuralist revolution is

the right response. I want to carry on examining set theory as we find it. Therefore I leave

structuralism as unfinished business’ ([1991], p. 54). But I do not want to push this interpretation

of Lewis ([1991]) too far. The very least we can say is that the tenor of Lewis ([1991]) is friendly

towards strong reinterpretation naturalism, urging that any reinterpretation of mathematics

deserves considerable suspicion. Many other philosophers are in the position of Stewart

Shapiro, who is sympathetic without being fully committed to what he calls the philosophy-

last-if-at-all principle, viz. ‘philosophers must wait on the mathematician (perhaps in two senses)

and be prepared to reject their own work out of hand if developments in mathematics come into

conflict with it’ ([1997], p. 28). In general, Shapiro flirts with (this undifferentiated form of )

naturalism without actually espousing it ([1997], pp. 30–2; [2000], pp. 10–15).18 Part of what the theoretical success of a past principle amounts to is that what are in some sense

the principle’s successor principles (or theories, etc.) are highly credible today. (If a historical

theory’s success amounted to its being strictly acceptable today in its original form, all previous

scientific theories would be classified as unsuccessful; thus, according to this style of argument,

they would be disqualified as unworthy of belief at the time.)


eventual historical success. To correct for this inherent bias in the selection

process, one has to look at all principles or norms that were in the past

considered philosophical. So the Failure Thesis must be reformulated accord-

ingly so as to assert that philosophical norms that in the past were considered

philosophical are highly dubious. Otherwise the historical induction at best

tells us that those of today’s norms that will in the future be deemed philo-

sophical are dubious, a claim which goes no way towards establishing

reinterpretation naturalism. With this preliminary point out of the way, I

examine the four objections to the Failure Argument.

4 Objections to the failure argument

The first objection is that subject demarcations are a relatively recent phe-

nomenon. We cannot so easily project back to, say, even as relatively recent a

time as the eighteenth century (recent relative to the history of philosophy) a

distinction between philosophy and many disciplines we now consider distinct

from it. Yet the Failure Argument crucially depends on the ability to read

these disciplinary distinctions into the past. Take one of Lewis’ comic exam-

ples: Zeno’s argument for the impossibility of motion. In Zeno’s time (c. 495–

c. 430 BCE), would it have been considered a philosophical, mathematical, or

physical argument? Or some combination of all three? The desired conclusion

follows only if the argument was considered exclusively philosophical in

Zeno’s own era. Yet that isn’t clearly so,19 and this suggests that the only

reason that Lewis was inclined to classify Zeno’s reasoning as paradigmati-

cally philosophical in the first place is precisely its failure. But if that were true

more generally, the Failure Argument would be viciously circular. If the only

measure by which previous principles are classified as philosophical is just

their presently perceived success or failure, then the Failure Argument tells us

nothing about the credibility of what we currently call philosophical norms.

Notice in passing that the example of Zeno’s paradox is actually poorly

chosen, for the very reason that Zeno’s little piece of reasoning was under-

stood by most philosophers as issuing in a paradox, not a true conclusion.

That in itself undermines any argument for the incredibility of philosophy

based on this particular example. Whatever the founder of dialectic may have

19 Michael White ([1992], p. 168) points out that Zeno’s paradoxes have often been thought to

constitute an important early influence on Greek mathematics, the implication being that they

were seen as part of the mathematics of the time. Although both White and the main scholar he

relies on, Wilbur Knorr, themselves subscribe to the opposing view, they clearly see the issue as a

subtle matter of scholarly debate—Knorr for instance remarks that his own arguments ‘cannot

presume to be confirmed explicitly by our documentary evidence’ ([1982], p. 113). To the best of

my knowledge, Lewis nowhere defended the claim that Zeno’s paradox was considered

exclusively philosophical in its day.


thought, philosophy itself never included the impossibility of motion as one of

its official theorems.

The first objection attempts to defuse the Failure Argument by questioning

the disciplinary demarcation between mathematics and philosophy. As it

stands, however, the objection is inconclusive. After all, the defender of

the Failure Argument can reply that whatever the past may have been like,

at this point in history much clearer demarcations between philosophy and

other disciplines—especially mathematics—have emerged. The norm of onto-

logical economy and the other examples from Section 1 may be thought to

clearly illustrate the difference. The reinterpretation naturalist might even go

on the offensive at this point and remark that the demarcation issue is a

problem for everyone. If the existence of a dividing-line between mathematics

and philosophy cannot be taken for granted, every philosopher of mathem-

atics owes us a precise account of what mathematical activity consists in. Yet

surely, the reinterpretation naturalist will add, this is supererogatory: we can

all distinguish the theorem-proving from, say, the throw-away asides or the

coffee-sipping that usually accompany it.

This claim about the historically recent emergence of demarcations is apt to

get rather murky. Philosophical norms have indeed been intertwined with

mathematical ones in the past, and it is a moot question just how clear the

separation between the two contemporary disciplines is. The philosophical

principles mentioned in Section 1 may be relatively clear-cut examples; but not

all cases are as clear. The boundary between philosophy and mathematics is

undoubtedly firmer than that between philosophy and other disciplines, such

as, say, cognitive science or physics—after all, the philosophy of physics looks

a lot more like physics than the philosophy of mathematics looks like math-

ematics; but again, that is not the same as the boundary’s actually being firm.

Without further development, therefore, the first objection looks indecisive,

and achieves no more than to shift the burden of proof onto the naturalist.

Recognising the complexity of the issues, let us set aside this first objection and

look for a more decisive way to undermine the argument.

The second objection claims that mathematics itself could hardly withstand

an analogous failure argument. After all, mathematics has had its fair share of

circle-squarers, of erroneous ‘proofs’ of renowned theorems not just by crack-

pots but by eminent mathematicians, of two-volume-long erroneous ‘proofs’

of the fifth postulate from the rest of the Euclidean axioms, and so on.20

Berkeley perhaps made too much of the muddle in the mathematics of his

time, but there was after all some good reason, in examining the infinitesimal

20 See e.g. Davis ([1972]), or Hersh ([1979], p. 41; [1997], pp. 43–5) for some specific examples of

mathematical errors. Crowe ([1988], sec. 4, pp. 263–5) also contains further examples and

relevant references.


calculus (real analysis) of his day, to answer in the negative, as he did, the title

question of his tract The Analyst: ‘whether the object, principles, and infer-

ences of the modern Analysis are more distinctly conceived, or more evidently

deduced, than religious Mysteries and points of Faith’ ([1734], p. 60).

This second objection, however, strikes me as rather weak. At the end of the

day, we should admit that mathematics is clearly different from philosophy

(or science) in this respect. The most important difference is that almost all

problems, confusions, mistakes—in a word, failures—eventually get cleared

up in mathematics. I cannot establish this assertion in its whole generality

here, to be sure; but I can offer two representative instances. The mathem-

atician Philip Davis has remarked that: ‘A mathematical error of international

significance may occur every twenty years or so. By this I mean the conjunc-

tion of a mathematician of great reputation and a problem of great notoriety’

([1972], pp. 174–5). A famous example occurred in 1945 when the distin-

guished mathematician Hans Rademacher thought that he had solved the

Riemann Hypothesis. It turned out, however, that Rademacher was wrong:

his proof was erroneous. A more recent example is that of Fermat’s Last

Theorem, whose initial proof by Andrew Wiles also contained a flaw. Yet

surely the moral of these examples isn’t that mathematics is just as replete with

failure as philosophy. After all, both mistakes were quickly detected and made

public (in the case of Fermat’s Last Theorem, the flaw was soon fixed). The

moral, rather, is that mistakes in mathematics are usually brought to light.

Following the nineteenth-century revolution in rigour in mathematics—

following the historical ‘rigorisation’ of the subject we owe to Bolzano,

Cauchy, Abel, Weierstrass and many others—mathematics has more generally

become a more secure discipline. As attested, indeed emphasised, in histories

of mathematics, today’s mathematics is significantly more rigorous than the

mathematics prior to the mid-nineteenth century.21 What is sanctioned by

today’s mathematical norms can therefore hardly be said to be of dubious

epistemic standing. Now that’s not to say, first, that a result certified by

today’s mathematical community is guaranteed to be true. We can all agree

that mathematicians, even in their collective wisdom, are fallible. Nor is it to

say, secondly, that mathematics has reached the acme of human rigour. More

than a century later, no one today can seriously endorse Poincare’s opinion

expressed to the 1900 International Congress of Mathematicians, ‘One may

say today that absolute rigour [in mathematics] has been attained’ (quoted in

Kline [1970], p. 265, and [1980], pp. 172 and 195). Nor is it to say, thirdly, that

21 Kline remarks, ‘One can safely say that no proof given up to at least 1850 in any area of

mathematics, except in the theory of numbers, and even there the logical foundation was

missing, would be regarded as satisfactory by the standards of 1900, to say nothing about

today’s standards’ ([1970], p. 270). See also Stein ([1988]) for a good discussion of the

‘second birth’ of mathematics.


every miscalculation, mistake, misapprehension, or misconception is

eventually weeded out. We need not go as far as Rene Thom, who claimed

that ‘never has a significant error slipped into a conclusion without almost

immediately being discovered’ ([1971], p. 697). Nor is it to say, finally, that

mathematics is cumulative, in the strict sense that each generation’s work is

eventually built upon by subsequent ones. We may acknowledge that math-

ematical branches, methods, concepts, etc., are forever abandoned by the

mathematical community, and for good reason.22 What it is to say, rather,

and more simply, is that mathematics’ epistemic credit rating is currently

impressively high. Hence the second objection that mathematical norms are

just as dubious as philosophical ones seems wrong-headed, or at best unproven.

A third objection to the Failure Argument begins by pointing out that not

all philosophical principles can be absurd in the way that the Failure Argu-

ment suggests: after all, many of them contradict each other. Take a couple of

characteristically philosophical principles (particularly relevant to the philo-

sophy of mathematics). One principle states that we can know about some

domain only if we’re somehow causally connected to it, where ‘causal con-

nection’ is construed very inclusively, to include for instance an indirect causal

connection mediated through other knowers. A second principle says that, on

the contrary, we can know about some subject area without having had any

causal contact with it. The second principle is the negation of the first. Yet

philosophers have defended both these principles, but—and here’s the rub—

one of them has got to be true.23

One reply to this is to say: ‘Well, alright, one of the principles has to be true,

but the point is that knowing which one is true is so beyond our powers that

we cannot rely on either of them.’ But this reply does no more than beg the

question. The epistemic unreliability of philosophical principles is precisely

what the Failure Argument sets out to show: it cannot assume it.

Another reply is: ‘Well, alright, philosophy cannot be all wrong, since it

contains contradictories, but what this shows is precisely that it can at best

only be half-right, and hence unreliable.’ Now this second reply is very con-

cessive, since it can be defeated simply by arguing that there are often better

reasons to prefer one rather than the other from a pair of philosophical

contradictories. In other words, the reply can be defeated by straightforward

philosophical argumentation.24 What such argumentation would achieve is to

fill the supposedly half-empty glass of philosophical success to the brim.

22 Rejecting the rosy cumulative conception does not entail buying into the tendentious

doctrines that subsequent mathematics is incommensurable with previous mathematics or

that it falsifies it.23 I ignore desperate replies such as the claim that they are meaningless, or that they are both false

despite being contradictories.24 Starting from some shared set of assumptions, which needn’t all be philosophical.


Furthermore, the Failure Argument as it stands leaves it open that some—

however minimal—part of philosophy is untainted by historical controversy

and is thus epistemically secure, indeed secure enough to mandate some

interpretation or other of mathematics. And finally, at this point, we are

entitled to ask a question which has been waiting in the wings for some time

now. Which norms, after all, have been successful when it comes to questions

of interpretation? Patently, mathematical norms have been effective in the

generation of successful mathematics; but the question is whether they have

been effective in the generation of successful interpretations of mathematics.

That they have a better track record here than philosophical norms remains to

be shown. This observation leads to my fourth objection against the Failure

Argument.25

Whatever plausibility the examples in Lewis’s credo have stems from their

being cases of reconstruction as opposed to reinterpretation. Indeed therein

lies their comic effect. Of course any philosophy that claims that motion is

impossible thereby receives a very low credit rating. At best, however, that

only shows that philosophy cannot credibly challenge entrenched common

sense. Yet the standard interpretation of mathematics does not command the

same allegiance as these cherished tenets of our worldview. Only the most

deluded philosopher will deny the Moorean observation that we are more

certain that 2þ 3¼ 5 (and other mathematical truths) than any philosophy

that tells us otherwise. But certainty about the mathematical content of the

statement that 2þ 3¼ 5 does not extend to certainty about any metaphysical

claim the statement might make. It would not extend, for instance, to the claim

that there really exist non-spatiotemporal, acausal objects standing in the

addition relation.

Each of Lewis’ examples can more generally be seen to fall into one of two

categories. Either it comically exposes the dubiety of a certain line of philo-

sophical reasoning but is reconstructive (about the area in question). Or it

must be taken seriously and consequently does not discredit philosophy.

Either way, examined more carefully, the examples turn out not to support

25 The third objection suggests a related objection against the Failure Argument: the fact that

philosophers have successfully refuted their predecessors shows, it might be said, that

philosophy cannot be all wrong. The proponent of the Failure Argument may attempt to

parry this further objection by distinguishing between positive and negative philosophy in

some way and then insist that, say, Aristotle’s valid criticisms of Plato, or Hegel’s of Kant,

etc., constitute negative philosophy, so that the existence of successful positive philosophy

remains to be shown. I suspect, however, that a suitable distinction between positive and

negative philosophy will be very hard to make out. So it looks like the burden of proof here

rests with the proponent of the Failure Argument. Notice, by the way, that if the distinction

between positive and negative philosophy can be satisfactorily made out, that would not imply

that contradictory statements—such as the causal theory of knowledge and its negation—must

be of different polarity. The causal theory of knowledge is presumably a positive philosophical

claim; and it is presumably also a positive philosophical claim to say that one can know about

some domain without being causally connected to it.


reinterpretation naturalism. Which of the two categories a particular example

falls under depends, of course, on how exactly it is understood. For example,

philosophers who maintain that no evidence has strictly speaking probabil-

ified a theory shouldn’t deny, if they are sensible, that in a lax, everyday sense,

theories have been made probable by evidence. Denying this commonsensical

datum commits one to an absurd reconstructive theory. Accounting for it,

however, grants the theory a hearing. Closely parallel considerations apply to,

say, eliminativism about belief. In general, which of the two categories Lewis’

examples fall under depends on how they are spelled out (the exception may be

Zeno’s argument that motion is impossible, which seems to fall squarely in the

first category). Bald statements of these claims are absurd precisely because

they are reconstructive; more sophisticated accounts of the philosophical

positions the slogans crudely summarise command at least initial respect.

This general strategy, then, offers another line of defence against the Failure

Argument. It deals with Lewis’ examples and—since the Failure Argument is

liable to grow more heads as soon as one is severed—it also promises to deal

with other putative examples of philosophical failure.

5 Philosophy as the default

Given the paucity of arguments for reinterpretation naturalism, and the

failure of the only one to be found in the literature, it begins to look as if

reinterpretation naturalism is a bad bet. To conclude that, of course, would be

too swift. But unless they have been hiding all their best arguments up their

sleeves, philosophers who embrace either form of reinterpretation naturalism

have succumbed to a besetting temptation, namely, plumping for a position on

the basis not of argument but of temperament. Suppose, then, that there are

no reasons to embrace reinterpretation naturalism. Where does that leave us?

Does the decision of how much authority we invest in philosophy and its

norms on the question of the interpretation of mathematics become a personal

one, an acte gratuit unconstrained by argument? Liberalism of this form may

be tempting. There may be no arguments for reinterpretation naturalism,

we are assuming; but, it will be said, where are the arguments on behalf of

philosophy? If there are no good reasons one way or the other to abide by

philosophical norms, perhaps rationality recommends suspension of belief, or

even arbitrary choice, when it comes to a philosophical question such as the

correct interpretation of mathematics.

Intellectual liberalism of this form, however, is misguided, as is shown by

keeping firmly in mind the nature of our own current practice. In a world

without philosophy, what we call philosophical norms would of course have

no say in the correct interpretation of mathematics. In such a world, anyone

who introduces specifically philosophical considerations and proposes to


overturn the entrenched naturalism owes her peers an argument. But in our

world, the situation is reversed. Philosophy as we philosophers understand it is

a highly fallible discipline; but it is nevertheless an established truth-seeking

practice, internally bearing the approved hallmarks of intellectual authority.

In the absence of an argument to overturn it, we are thus entitled to acquiesce

in our practice in epistemic good conscience. So the burden of proof lies with

anyone who would dislodge us from our accustomed ways, who would foist

reinterpretation naturalism upon us, and, in doing so, entreat us that when it

comes to answering the characteristically philosophical question of mathem-

atics’ interpretation, we cannot appeal to philosophical norms.

Acknowledgements

Thanks to audiences at Cambridge and UC Berkeley for discussion. I am also

grateful to Alex Oliver, Dominic Gregory, Dorothy Edgington, Gideon Rosen,

Michael Potter, Paul Benacerraf, Peter Smith, Simon Blackburn, Tim Gowers,

Timothy Williamson and three anonymous BJPS referees for comments on

earlier drafts.

Jesus College

Cambridge CB5 8BL

United Kingdom

[email protected]

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