## 1. Methodological Naturalism




Methodological naturalism has three principal and related senses in
the philosophy of mathematics. The first is that the only
authoritative standards in the philosophy of mathematics are those of
natural science (physics, biology, etc.). The second is that the only
authoritative standards in the philosophy of mathematics are those of
mathematics itself. The third, an amalgam of the first two, is that
the authoritative standards are those of natural science and
mathematics. We refer to these three naturalisms
as *scientific*,
*mathematical*, and *mathematical-cum-scientific*. Note
that throughout this entry 'science' and cognate terms encompass only the
natural sciences.


### 1.1 Mathematical Anti-Revisionism




Naturalism--'methodological' and 'in the
philosophy of mathematics' hereafter understood--seems to
have anti-revisionary consequences for mathematics. The
mathematician-philosopher L.E.J. Brouwer developed intuitionistic
mathematics, which sought to overthrow and replace standard
('classical') mathematics. Brouwer attempted to motivate
intuitionistic mathematics philosophically with an intuition-based
account of mathematical truth. A more recent exponent of
intuitionistic mathematics is Michael Dummett, who has developed
arguments from the philosophy of language, in particular the theory of
meaning, on its behalf. Yet scientific standards arguably condone
classical over intuitionistic mathematics: even if present-day science
could be entirely recast intuitionistically--a
big if--it would be less simple and more cumbersome than
its current classically-based version. Mathematicians also tend to
view intuitionistic mathematics as inferior to classical mathematics
if construed as a rival to it. Hence neither Brouwer nor Dummett's
intuitionism is apparently sanctioned by scientific or mathematical
standards, so naturalism rules them out of court. Indeed, for
many of its adherents, that is precisely naturalism's point. Its
point, it is often thought, is to block fanciful attacks on
established disciplines such as mathematics by disciplines with less secure
methodologies.


## 2. Recent History of Methodological Naturalism


### 2.1 Recent Background




Contemporary interest in naturalism stems from Quine, whose naturalism
is prominent in his later works. A representative quotation is that
naturalism is 'the recognition that it is within science itself,
and not in some prior philosophy, that reality is to be identified and
described' (Quine 1981, 21). Another major influence is Hilary Putnam, who articulates his scientific naturalism as follows:



> 
> 
> ...it is silly to agree that a reason for believing that
> *p* warrants accepting *p* in all scientific
> circumstances, and then to add 'but even so it is *not good
> enough*.' Such a judgement could only be made if one
> accepted a trans-scientific method as superior to the scientific
> method; but this philosopher, at least, has no interest in
> doing *that*. (Putnam 1971, 356)




Thus from this perspective mathematics is judged by scientific standards because
everything is. Moreover, Quine and Putnam maintain that these
standards sanction platonist mathematics because mathematics and its
platonist construal are an indispensable part of our best scientific
theories.




Although naturalism as a self-conscious position in the philosophy of
mathematics emerges most fully with Quine there are, as always,
precursors. The empiricist tradition in its various guises (logical
positivism, Mill, Hume, etc.) is the most obvious precursor, though
there are significant differences between pre-Quinean empiricists and
contemporary naturalists. The rise of scientific naturalism in the
philosophy of mathematics also coincides with a rise in a broader
scientific naturalism, also in part attributable to Quine, which sees
all philosophy--not just philosophy of mathematics--as
taking place within natural science. Naturalism also goes hand in hand
with a now generally prevalent pessimism about traditional
philosophical modes of argumentation.




Some version of naturalism is attractive to virtually all
philosophers today. That the methodologies of mathematics and science are the
best we have seems a platitude, which philosophy should try to
acknowledge and build on rather than ignore. The question is how
exactly to do so.


### 2.2 Present context




The past couple of decades have seen a great surge of interest in
naturalism. 1997 was an important year in recent philosophy of
mathematics, as it saw the publication of four books articulating the
positions of five leading philosophers of mathematics: John Burgess
and Gideon Rosen's *A Subject with No Object,* Penelope
Maddy's *Naturalism in Mathematics*, Michael
Resnik's *Mathematics as the Science of Patterns*, and Stewart
Shapiro's *Philosophy of Mathematics*. All four books are to
various degrees and amongst other things naturalistic: the first two
are naturalist manifestos, the third advocates a Quinean scientific
naturalism, and the last, though mainly concerned with other matters,
is sympathetic to naturalism. John Burgess's naturalism, first set out
in his (1983) and in the past few years elaborated with his colleague
Gideon Rosen (1997, 2005), is perhaps most naturally construed as a
version of mathematical-cum-scientific naturalism (1997,
211). Penelope Maddy's naturalism is a heterogeneous form of
naturalism which distinguishes between mathematics proper and the
philosophy of mathematics, embracing mathematical naturalism about the
former and scientific naturalism about the latter (section 5). Another
position suggested by Maddy (1997) is a thoroughgoing mathematical
naturalism that takes mathematical standards as authoritative within
both mathematics and its philosophy.




Ignoring qualifications, the main contemporary versions of
naturalism, with their representative advocates, may be tabulated as
follows:



> 
> 
> 
> |  |  |  |
> | --- | --- | --- |
> |  | **Mathematics Proper** | **Philosophy ofMathematics** |
> | Scientific(Quine) | Scientific | Scientific |
> | Mathematical-Cum-Scientific(Burgess) | Mathematical-cum-Scientific | Mathematical-cum-Scientific |
> | Mathematical(?) | Mathematical | Mathematical |
> | Heterogeneous(Maddy 1997) | Mathematical | Scientific |
> 
> 
> 




To illustrate the difference between statements in mathematics proper
and the philosophy of mathematics, consider as an example of the
former the Green-Tao theorem, proved in 2004, which states that the
sequence of prime numbers contains arbitrarily long arithmetic
progressions (but of course no infinitely long one); as examples of
the latter take the platonist claim that the number two exists and is
abstract or the claim that because mathematical entities are created
rather than discovered, impredicative definitions are not permissible.
For any given philosopher, the Green-Tao theorem and its proof are to
be assessed by the standards of the first column. For example, Quine
accepts the theorem if and only if it is part of the best
systematisation of science (which it is assuming that the principles
from which it is deduced and the logic by which it is deduced from those principles are). Similarly, for any given
philosopher, platonism and the stricture regarding impredicativity are
to be assessed by the standards of the second column. For example,
Quine accepts platonism because he takes it to be the scientifically
sanctioned interpretation of mathematics: in his view, the mathematics
contained in the best systematisation of science is
platonic.




There are no clear exemplars of the view that mathematical standards
should be authoritative in philosophy of mathematics, hence the
question mark. David Lewis flirted with this position in his monograph
on set theory (1991, viii-ix, 54), but by the time of his (1993)
had already repudiated it. The position is suggested by remarks in
Maddy (1997), though as we will see in section 5, Maddy (1997) is more
naturally interpreted as advancing a heterogeneous form of
naturalism. Several other philosophers of mathematics are also
professed naturalists, notably Alan Baker (2001) and Mark Colyvan
(2001).


## 3. Elucidating Methodological Naturalism




Like many -isms, naturalism is perhaps better thought of as an
orientation with doctrinal implications than as a doctrine per se.
Nevertheless, we may attempt to elucidate it along several
dimensions.


### 3.1 Authoritative




How should we understand the claim that some set of standards ought
to be authoritative in the philosophy of mathematics? We may preserve
generality by referring to *X*-standards, the instances of interest to us
being 'scientific', 'mathematical' or
'scientific-cum-mathematical'. Here are some
(non-exhaustive) readings of the authority claim:


1. Biconditional Naturalism: accept *p* iff *p* is
sanctioned by *X*-standards
2. Trumping Naturalism: accept *p* if *p* is sanctioned by
*X*-standards.
3. Emphasis
Naturalism: In assessing whether *p*, give more emphasis to
*X*-standards.
4. Compatibility Naturalism: do not accept *p* if *p* is
incompatible with *X*-standards.




The biconditional reading is the strongest of the four. It expresses
the thought that valid standards just are *X*-standards. In contrast with
biconditional naturalism, the trumping version apparently allows that a
statement in the philosophy of mathematics may be acceptable even if
*X*-standards do not sanction it. For instance, Burgess and Rosen express
their naturalism as follows:



> 
> 
> 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, 65).




This is naturally glossed as a trumping version of
scientific-cum-mathematical naturalism. It seems to allow that if
mathematics and science do not speak with a firm voice on a question,
we may accept the verdict of some other discipline.




Emphasis Naturalism is a vaguer doctrine. Various versions arise depending on
how much emphasis is placed on *X*-standards. A modest version captures the position of naturalistically-inclined but not outright naturalist philosophers. We recover biconditional naturalism when *X*-standards are emphasised to the point where no others matter.




Compatibility naturalism is not as strong as trumping
naturalism. If *p* is sanctioned by *X*-standards then
compatibility naturalism enjoins the non-acceptance of
!*p* but it does not necessarily enjoin the acceptance
of *p*. Non-acceptance of !*p* falls short of
acceptance of *p* since there is always the option to suspend judgment on
*p*. The compatibility view is generally accepted by most contemporary
philosophers. To take an example from outside the philosophy of
mathematics, most philosophers would reject a philosophy of time that
clashes with the theory of relativity (in this case *X* = science).
Within the philosophy of mathematics, most philosophers would reject a
philosophy of mathematics that implies, for instance, that complex
analysis should be jettisoned (in this case *X* could be science,
mathematics, or mathematics-cum-science).




Various weaker methodological theses are sometimes labelled
'naturalist', for instance the rejection of Cartesian
foundationalism, but here we understand the term more robustly. Which
of the above variants is the right way to develop naturalism depends
of course on how naturalism is motivated (section 4).


### 3.2 The boundary problem and the scientific method




Naturalism sets up an opposition between *X*-standards (scientific,
mathematical or scientific-cum-mathematical) and other types of
standards, e.g., astrological, theological, or the standards of common
sense. Another example of standards that the naturalist sees as
on the wrong side of the tracks are 'fundamental'
philosophical ones. Goodman and Quine (in his pre-naturalist phase)
once began an article by declaring that the basis for their nominalism
was a fundamental philosophical intuition irreducible to scientific
grounds (1947). Naturalists reject appeal to such standards.




An apparent problem with naturalism is that there do not seem to be
sharp boundaries between science and non-science and between
mathematics and non-mathematics. For example, the transition from
physics to philosophy of physics to physics-heavy metaphysics to
common-or-garden metaphysics seems to be gradual. When a mathematician
writes a research article, an undergraduate textbook, a popular book on
mathematics, a book expounding his personal philosophy of mathematics
and his psychological associations with various mathematical theories,
at what point exactly has he stopped doing mathematics? When research
mathematicians get together after the seminar and agree over coffee
that the Riemann Hypothesis is the most important outstanding problem
in mathematics, is this a strictly mathematical claim or a personal
judgement recognised by mathematicians as outside the province of
mathematics proper?




Many philosophers follow Quine by citing a
standard cluster of principles putatively constitutive of scientific
standards: empirical adequacy, ontological economy, simplicity,
fertility, and so on (Quine 1955, 247; Quine and Ullian 1970, chapter
5). However, lists of this kind are unsatisfactory for several
reasons. For one thing, the principles come in different
versions. Yet it is doubtful whether the general versions are the
scientifically sanctioned ones. Some writers claim, for example, that
scientific appeal to ontological parsimony does not extend to the
postulation of abstract objects (Burgess 1998). Others maintain that
scientific appeals to simplicity are not best captured by the utterly
general slogan 'Prefer *any* theory *T*1 to
any less simple theory *T*2 (in this respect)'
(Paseau 2007). Moreover, lists of this kind do not tell us how to
balance desiderata against one another.




Since the explosion of science studies in the 1960s, more attention
is now paid to the nuances of scientific practice. Yet a more
precise articulation of scientific standards and their weights remains
elusive. The existence of an algorithm encapsulating the scientific
method is generally questioned (though many human beings apparently
manage to implement the scientific method, so if the method is not
algorithmic then neither are our minds). What is not questionable,
however, is that its exhibition currently escapes us.




Having said that, it is not clear how serious the boundary problem
is for naturalists. Perhaps they can argue that there is a fairly
sharp boundary, though one that is hard to define. Perhaps
mathematicians implicitly know when something counts as a piece of
mathematics and when it is a non-mathematical commentary on
mathematics. In any case, naturalism seems to survive the absence of a
sharp boundary. Naturalism can apparently rest its claims on a set of
standards with fuzzy boundaries.


### 3.3 Methodological unity?




What if there are no global scientific standards, but just the
standards of this or that part of science (e.g., physics or biology, or
particle physics or fluid mechanics), or even just the standards of
this or that group of scientists? In that case, scientific naturalism
would fragment into several versions (e.g., physics-naturalism or
biology-naturalism). If the motivation for scientific naturalism points
to one of these finer-grained naturalisms over the others (e.g.,
physics-naturalism), or if they all return the same verdicts in the
philosophy of mathematics, the potential fragmentation is not worrying.
But if not, the scientific naturalist is in trouble, as all these
competing naturalisms are by assumption equally valid. So far
scientific naturalists have tended to assume that science operates with
a single set of standards. Plausible though that assumption is, a rigorous naturalist will want to shore it up with detailed
case studies. Similarly for mathematical naturalism and
mathematical-cum-scientific naturalism.


### 3.4 Degree of sanction




Scientific standards sanction propositions to a certain degree
rather than outright. The latest hypothesis at the research coalface
adopted tentatively by a few experts does not have the same standing as
a long-entrenched theory. Thus the black-or-white notion of scientific
standards sanctioning or not sanctioning *p* will not do. (This is an idea which Bayesian confirmation theory takes seriously.) It might also be argued that
this is also true for mathematical standards, for example by considering non-deductive grounds for belief in mathematical
propositions. It seems reasonable to see Goldbach's
Conjecture, the claim that every even number greater than 4 is the sum
of two primes, as supported to a high degree by the non-deductive
evidence currently available for it. In the absence of a
proof of Goldbach's Conjecture, however, this degree falls short
of 1. A more precise statement of the naturalist's creed must
therefore issue guidelines for apportioning
credence in *p* in line with the degree and type of commitment
to *p* scientific or mathematical standards recommend.


### 3.5 Standards and practitioners




It would be wrong to equate what *X*-standards sanction with what
*X*-practitioners believe (for *X* = science or
mathematics or more generally). For one
thing, *X*-practitioners may not have given a particular
question any thought. For another, *X*-practitioners might all
be mistaken. Moreover, *X*-practitioners might self-consciously
maintain the opposite of or at any rate something distinct from
what *X*-standards sanction. For instance, a scientist may
believe *p*, perhaps on the basis of a 'gut
feeling' or on account of some overriding religious convictions
or for whatever non-scientific reason, whilst still recognising that
scientific standards support not-*p*. Or a mathematician may
believe that the number 7 has mystical properties but not believe
it *qua* mathematician. A tighter connection between practitioners and standards, then, might
be this: what *X*-standards sanction is
what *X*-practitioners are disposed to correctly
believe *qua* *X*-practitioners. (This is not intended
as a reductive analysis.)




Having said that, it takes special pleading to attribute widespread
error to the community of *X*-practitioners regarding
what *X*-standards sanction. Thus what *X*-practitioners
in fact believe usually serves as good, though defeasible, evidence
for what *X*-standards sanction.


### 3.6 Broader naturalisms




Scientific naturalism as here defined encompasses only
*natural* science (and likewise for scientific-cum-mathematical
naturalism). Broader naturalisms take in not just traditional natural
science but also some of the other sciences: perhaps all of social
science, perhaps only some of it, perhaps linguistics, perhaps
cognitive science. Note that in later writings Quine himself adopts a
broad version of naturalism (1995, 49). Penelope Maddy has more
recently made it clear that the form of scientific naturalism she
espouses--'Second Philosophy' as she now prefers to
call it--is very broad indeed, taking in not only the natural
sciences but also psychology, linguistics, sociology, etc. (2007,
2).




Broadening scientific or scientific-cum-mathematical naturalism to
include these disciplines has potentially significant consequences for
the philosophy of mathematics. For instance, if semantics falls under
the naturalist umbrella it may provide a naturalistic sanction for
mathematical realism or platonism, as the face value semantics for
mathematics seems to assimilate it to the uncontroversially literal
parts of language--a point famously made in Benacerraf (1973).




Whether to construe naturalism broadly or narrowly depends on its
motivation. The attractions of scientific or mathematical or
scientific-cum-mathematical naturalism lie in the disciplines'
incomparable success (on some understanding of what this success comes
to--see section 4). At this point in time, however, the
non-natural sciences are less successful than the natural
sciences. The less successful one deems the non-natural sciences
compared to the natural ones, the less attractive a broader scientific
or scientific-cum-mathematical naturalism becomes in comparison with a
strictly natural-scientific one.




All naturalists, especially those of the broader variety, have to
balance potentially competing standards. A broad naturalist might
decide, say, to give the natural sciences 2/3-weighting and semantics
1/3-weighting. Or she might hold that a proposition in the philosophy
of mathematics is acceptable if (or even: if and only if) all the
sciences, natural or non-natural, sanction it--that is to say,
when all the sciences speak with one voice. These balancing questions
have unfortunately not been much addressed by naturalists. Perhaps
that is because questions of this kind arise for everyone: whichever
justificatory standards one accepts, the problem of how to adjudicate
between them arises. But to the extent that naturalism is prescriptive
and cannot rely on an implicit procedure that is already in place, it
owes us an articulation of how to balance different sets of
standards.




The balancing issue is particularly pressing for
mathematical-cum-scientific naturalists. A scientific naturalist is in
principle happy to say that if a mathematical theory *M* is
scientifically superior but mathematically inferior to another
mathematical theory *M*\* then *M* should be
preferred to *M*\* (the next paragraph contains an
example). Mathematical-cum-scientific naturalists, however, may come
down on either side, depending on the details of the particular
theories: it all depends on whether *M*'s scientific advantages
compared to *M*\* are outweighed by its mathematical
disadvantages. Now the philosophy of mathematics does not have an
established tradition of weighing mathematical theories' scientific
and mathematical pros and cons against one another; nor does any other
discipline. So the problem of how to balance mathematical and
scientific standards is particularly pressing for the
mathematical-cum-scientific naturalist.




To illustrate this problem, suppose, as many philosophers maintain,
that the general principle of ontological economy--posit as few
entities as possible--is a scientific standard. Suppose also, as
Penelope Maddy maintains, that a set-theoretic version of ontological
profligacy--posit as many sets as possible--is a
mathematical standard (this is one way of cashing out the
set-theoretic maxim Maddy calls *MAXIMIZE*). These two
standards clash, as Maddy recognises (1997, 131). Thus given these
assumptions a predicativist set theory which posits a relatively small
number of sets, say of the kind developed in Hermann Weyl's *Das
Kontinuum*, might be scientifically superior to ZFC, which posits
more sets. Yet ZFC is generally thought to be mathematically superior
to a predicativist set theory. Perhaps the right diagnosis is
that the clash is only superficial, since the correct scientific
version of ontological economy is 'posit as
few *concrete* entities as possible' and the correct
mathematical version of ontological profligacy is 'posit as
many *abstract* entities as possible'. However that may
be in this instance, the mathematical-cum-scientific naturalist has to come up with a
general policy for dealing with potential clashes, or argue that no
such clashes are possible.


## 4. Motivating Naturalism




One may simply adopt naturalism without providing an argument or
motivation for it. But motivations for naturalism buttress it, making
it internally stronger, and give it dialectical strength, increasing
its appeal to non-naturalists. They answer the fundamental question:
why exactly *those* standards?




(The same goes for naturalisms primarily conceived as approaches or
stances rather than doctrines. Since the publication of her 1997 book,
Penelope Maddy has clarified that her version of naturalism is a
stance (Maddy 2001) or a piecemeal approach/method of inquiry (2007,
see e.g., p. 19 fn. 15, p. 403). However, stances and approaches to
inquiry are attractive to the extent that they are well motivated.)


### 4.1 Naturalism is Revolutionary in some Respects




Naturalism is often thought to be a conservative philosophy of
mathematics, as we suggested in section 1. But actually things are more
complicated. Each of the three naturalisms of interest is revolutionary in some
respect.




Our default view is that mathematical standards decide questions in
mathematics proper, for example questions such as whether
Fermat's Last Theorem or the Axiom of Choice is true. Scientific
standards are thought not to affect this: when Andrew Wiles proved
Fermat's Last Theorem in the mid-1990s, he was not particularly
concerned with how his proof would go down in physics departments, or
more generally its impact on empirical science. Similarly, the claim
that if some large cardinal axiom is not scientifically sanctioned
(perhaps because it leads to no new empirical consequences) then there
is no good reason to accept it is, as Maddy points out, 'out of
step with the actual practice of set theory' (1997, 132), and
indeed out of step with the actual practice of much philosophy. We
usually do not judge large cardinal axioms by scientific standards; we
judge them by mathematical standards. Quine was self-consciously going
against the grain when he rejected the higher flights of set theory on
scientific grounds (1986, 400).




Scientific naturalism about mathematics proper is thus a
philosophically revolutionary view, since it advocates a different set
of standards with which to judge mathematics (scientific ones) from
the traditional ones (mathematical). It is also *potentially*
revolutionary about mathematics itself, as it might lead to a revision
of mathematics. (Notice that even if scientific naturalism does not
entail a revision of mathematics, it still counts as philosophically
revolutionary: advocating the replacement of *Y*-standards with
*X*-standards as the proper arbiters in some domain is
philosophically revolutionary even if the *Y*-standards
and *X*-standards happen to endorse the same claims in that
domain.) Having said that, recent scientific naturalists have tended
to be mathematically conservative in temperament and have advocated
little or no revision of mathematics.




These morals also apply to mathematical-cum-scientific naturalism,
but to a lesser extent, as the latter gives some weight to mathematical
standards in mathematical justification.




The third of the three naturalisms of interest, mathematical naturalism, is philosophically
but not mathematically revolutionary. Mathematically, it is as
conservative as can be: no standards other than mathematical ones are
relevant to the assessment of mathematical claims. Thus no accepted
mathematics is overturned from without. Yet mathematical naturalism is
a revolutionary stance in the philosophy of mathematics. To see this,
suppose that platonism is part of standard, accepted mathematical
practice. In that case, mathematical naturalism entails that there is
no further question of its truth. Loosely put, simply because
mathematicians (*qua* mathematicians) are platonists, platonism
is the correct philosophy of mathematics. This is clearly out of step
with philosophical practice: philosophers look to mathematicians'
views (*qua* mathematicians) as defeasible data for the
philosophy of mathematics, not its conclusion. Thus we see that the
simple characterisation of mathematical naturalism as conservative is
not quite right: though mathematically conservative, it is philosophically
revolutionary.




In sum, scientific, mathematical, and mathematical-cum-scientific
naturalism are each revolutionary in some respect and face a
corresponding burden of proof. We may now appreciate the sense in which
the general claim, expressed in section 1, that naturalism is
anti-revisionary is true and the sense in which it is not.





This also shows that contemporary naturalism is different from the
latter Wittgenstein's superficially similar metaphilosophy.
Wittgenstein's anti-philosophy, like naturalism, bars philosophy
from changing mathematics: "Philosophy may in no way interfere
with the actual use of language...It leaves everything as it is.
It also leaves mathematics as it is" (1953, SS163).
Naturalism's revolutionary tenor, however, means that it does not
leave everything as it is.


### 4.2 Dearth of Arguments in the Literature




Arguments for naturalism are lacking in the literature. Most
naturalists simply posit their naturalism and work downstream from it,
hoping that its consequences will prove attractive to the susceptible
(Maddy 2007, 3). Naturalism thus effectively becomes a personal
credo with little direct attempt to bring anyone else on board:
*I* accept only *X*-standards in some domain because *I*
find them more credible than others. Now perhaps at the end of the day one cannot do better. But we shouldn't assume that at the outset. That is
all the more important given naturalism's revolutionary features,
as explained earlier. A conservative theory of justification might
sanction naturalism given a broadly naturalist starting point; but our
starting point, as we have seen, is not that of trumping
naturalism: it is at most compatibility naturalism. So an argument for
anything more than the mildest version of naturalism would be
welcome.


### 4.3 Current Success




Naturalists are motivated by the thought that scientific or
mathematical standards are the most successful standards we possess.
But what does success come to? Very roughly, the following could be
taken to be a mark of a discipline's success: (i) among its
practitioners, there is a widely shared conception of the
discipline's guiding questions and permissible methodology; and
(ii) there is progress within the discipline in addressing its guiding
questions. One may then try to argue along these lines that physics is
more successful than metaphysics, that psychology is more successful
than parapsychology, and that astronomy is more successful than
astrology.




An approach of this kind faces a twofold problem. If anything goes
when it comes to methodology, all sorts of disciplines can achieve
success without thereby achieving credibility. Consider Guru-ology, the
discipline that takes its guiding questions to be those enunciated by
some guru, and lays down as its methodology that acceptable answers are
all and only the guru's pronouncements (perhaps assuming
consistency--so let us assume for good measure that the guru is
consistent and more generally probabilistically coherent). These
answers could be as fanciful as you like: we leave it to the
reader's imagination to devise examples of outlandish claims made
by the guru. If we suppose that the guru answers each of the questions
that he raises, Guru-ology is therefore progressive--it answers
all the questions it raises--and it is therefore successful. But
its success speaks nothing of its credibility.




In general, if the success of any given set of standards *S*
is gauged on its own terms, i.e., by using *S*-standards,
several self-supporting but intuitively unacceptable sets of standards
count as successful. This relativism is clearly not what the naturalist
wants. Likewise for the thought that success is to be determined by how
well the standards help us to 'cope with reality'; several
non-scientific and non-mathematical naturalisms likewise self-vindicate
under this criterion. Perhaps success should instead be measured by how
well the standards explain and predict natural phenomena, i.e., with how
they cope with the subject matter of natural science. But adopting our
usual view of how to judge success in this respect would be
question-begging in favour of scientific naturalism, since scientific
standards are precisely the standards we have developed to cope with
this portion of reality. Compare an astrological naturalism motivated
by the idea that success is to be measured by how well the standards
explain and predict 'astrological phenomena', understood as
the astrologers do. So if a naturalist argument based on success is to
succeed, some other naturalistically acceptable but non-question
begging understanding of 'success' must be found.




The second problem with a success argument is that success in one
sphere is no indication of success in another. Biology is rather
successful at explaining and predicting biological phenomena. But why
should that give it authority over questions in mathematics or the
philosophy of mathematics? Likewise for other natural sciences. As we
shall see, this point generalises.


### 4.4 Agreement




Traditional philosophy, a naturalist might say, leads to endless
disagreement. Science and mathematics, on the other hand, usually reach
broad agreement--often consensus--on questions within
their domain. Hence scientific or mathematical standards are to be
preferred to others. (Such arguments from disagreement and lack of
convergence in opinion have featured prominently in other areas of
philosophy, notably meta-ethics.)




However, the moral that the naturalist wishes to draw from patterns of
agreement and disagreement seems unwarranted. Agreement or disagreement
in a community is a contingent matter. A totalitarian state could
achieve community-wide agreement with chilling efficacy by imposing
some preferred set of standards on its subjects. In general, there are
countless non-epistemic reasons for agreement or disagreement. It is
therefore not agreement that matters, but rather the explanation of why
agreement obtains.




A more sophisticated version of this argument might therefore be
based on the tractability of disagreement rather than on its mere
presence. Disagreement is rife in both philosophy and science, but only
in the latter, it might be said, is disagreement tractable. At the very
least, progress can be made and perhaps agreement can in principle
always be reached. Actual patterns of agreement and disagreement might
then be cited as evidence for the respective tractability or
intractability of debates umpired by, say, scientific and
non-scientific philosophical standards.




It is beyond the scope of this entry to assess this more
sophisticated version of the argument, which in one form or
another has recently garnered considerable attention outside the
philosophy of mathematics. However, note a couple of prima facie difficulties.




In order to abstract from the contingencies of our epistemic
situation, arguments about tractability usually proceed by considering
highly idealised subjects, in particular subjects whose factual, logical, etc.
knowledge far exceeds ours. But the problem with such idealisations is
that they seem question-begging. For example, a theological
anti-naturalist would maintain that factually highly-informed subjects will
be apprised of facts about supernatural reality. Our grip on
idealised subjects and how disagreement between them is likely to
resolve itself may therefore be too loose to draw any substantive morals from such
thought experiments. Either that or such arguments are likely to be question-begging. 




Second, we may grant that considerations of tractability reveal that
scientific and mathematical standards are more truth-conducive in their
respective spheres. Yet that does not seem to provide a warrant for
thinking that they will be successful in other spheres. (This is the
same point we made in connection with the argument from success.)


### 4.5 Historical Success




Perhaps the most promising argument for naturalism is based on
historical success. Scientific and mathematical standards have a better
track record than others; therefore scientific and mathematical
standards should be taken as authoritative, in the philosophy of
mathematics and elsewhere. Observe that like the previous two arguments, this
argument for naturalism in the philosophy of mathematics is an argument
for global naturalism.




Some naturalists have explicitly relied on this motivation. For
example, Lewis uses it to reject structuralism as the correct
philosophy of set theory in *Parts of Classes* (1991,
58-9) even if he disclaims it as an argument; see also Colyvan
(2001, 33), Shapiro (1997, 30) and Burgess (1998, 197). There is much
to say about the argument, explored in Paseau (2005). Here we
content ourselves with two critical observations.




Since philosophers deploy apparently different standards, it is not
clear what it means to say that philosophy as a whole has a poor
historical track record. Consider the early Popper (1935), who held
that no amount of evidence can make an unfalsified theory probable (or at least no
more probable than any other unfalsified theory). This is the kind of
example David Lewis wields to poke fun at philosophy:
surely--*surely*--the standards that led to this
conclusion are not trustworthy. The theory of relativity is undoubtedly
more probable than the as yet unfalsified hypothesis that the world
will end in the year 2525. Yet if I do not share Popper's 1935
standards, the fact that his then philosophy of science is from my
point of view manifestly wrong does nothing to shake my faith in my own
philosophical standards. Similarly, take Thomas Aquinas, whose
philosophical standards included consonance with the Bible and more
generally with the tenets of the Christian faith. If I am not a
Christian, the fact that Aquinas' philosophical theology is from
my point of view wrong does nothing to shake my faith in my
philosophical standards. Thus, that Sir Karl or St Thomas' standards
were, as I see it, no good does not tend to undermine my faith in my
own.




I may therefore agree with the naturalist that philosophy has a
worse track record than science and mathematics. But it does not follow
that the (non-scientific or non-mathematical) standards that *I*
use have a poor track record. *If* philosophers have
consistently assumed a more or less uniform set of standards over the
course of history, and *if* I too follow in that tradition by
accepting them as mine, and furthermore *if* these standards
demonstrably have a poorer track record than scientific or mathematical
standards, that would be a reason for me to turn naturalist. But
the first assumption is at least questionable, and it is not clear what
remains of the argument from history if *my* standards'
antecedents do not have a poor track record.




A second problem with the argument has to do with its application to
philosophical questions. Let us agree that scientific standards have a
good track record when it comes to answering scientific questions, that
mathematical standards have a good track record when it comes to
answering mathematical questions, and moreover that these track records
are better than philosophical standards' track record in
answering philosophical questions. These facts, however, do not seem to
be relevant to the issue of which standards should be treated as
authoritative when it comes to philosophical questions. A good track
record in one sphere is not in itself evidence for truth-conduciveness
in another.




This by now familiar objection may be illustrated by
considering the debate between platonists and structuralists in the
philosophy of mathematics. Platonists interpret '1 + 2 =
3' as a claim about abstract objects. Structuralists on the
other hand interpret '1 + 2 = 3' as a claim about what is
the case in any structure that satisfies the axioms of
arithmetic. (Here we are thinking of structuralism as the type of view
often labelled 'eliminative structuralism' following
Charles Parsons, and whose most sophisticated book-length--and
modal--version may be found in Hellman (1989).) Naturalists claim
that mathematical standards have been more successful in the past and
hence should be trusted on this issue. But it is by no means clear
that this is the
*type* of question concerning which mathematical standards have
a better track record than philosophical ones. Mathematical standards
have a good track record when it comes to questions such as whether 1
+ 2 is equal to 3, or what the series 1 + 1/4 + 1/9 + 1/16 + ...
converges to, or whether the Poincare Conjecture (concerning
the classification of 3-manifolds) is true; but they do not have a
proven track record when it comes to questions such as whether these
truths are platonist or structuralist. (A related objection to the
argument is that scientific standards do not speak to questions of
interpretation, such as the question of which of platonism or
structuralism to prefer. Cf. Paseau (2007).)




In sum, the need for naturalists to develop these arguments or to
produce better ones remains pressing.


## 5. Heterogeneous Naturalism




So far we have considered three uniform types of methodological naturalism relating
to mathematics: scientific, mathematical, and
mathematical-cum-scientific. Consider now a heterogeneous methodological naturalism
that accepts mathematical standards when it comes to mathematics proper
but adopts scientific standards for the philosophy of mathematics and
for philosophy more generally. Heterogeneous naturalism has been
advanced by Penelope Maddy (1997), whose rich contributions have
animated and incomparably influenced the debate on naturalism in the
philosophy of mathematics over the past twenty years. (Maddy now
prefers to call her naturalism 'Second Philosophy', as in
the title of her 2007 book, but here we maintain the label
'naturalism' in keeping with the rest of the entry.)




To begin with, a representative quotation from Maddy:



> 
> 
> Where Quine holds that science is 'not answerable to any
> supra-scientific tribunal, and not in need of any justification beyond
> observation and the hypothetico-deductive method' ..., the
> mathematical naturalist adds that mathematics is not answerable to any
> extra-mathematical tribunal and not in need of any justification
> beyond proof and the axiomatic method (1997, 184).




In our terminology, Maddy's heterogeneous naturalism is a
trumping thesis. As she puts it, 'if our philosophical account of
mathematics comes into conflict with successful mathematical practice,
it is the philosophy that must give' (1997, 161). Neither philosophy nor science can overturn mathematics' 'methodological
verdicts' (2007, 361), since they are both extra-mathematical tribunals. However, Maddy takes the philosophy of mathematics as opposed to mathematics
proper to be a branch of natural science, as she explains in the following passage:



> 
> 
> So naturalistic philosophy of mathematics takes place within natural
> science, like naturalistic philosophy of science, but unlike
> naturalistic philosophy of science, it takes a hands-off attitude
> toward the naturalized model of mathematical practice (1997,
> 202).




These and similar passages (especially 1997, 200-203) indicate that Maddy takes the philosophy of mathematics to be answerable
to (natural-) scientific standards.




The distinctiveness of heterogeneous naturalism is thus that it
recommends one set of standards (mathematical) for settling some
questions about mathematics--mathematical ones, such as which
axioms to choose--and another set of standards (scientific) for
settling other questions about mathematics left open by the practice
itself--philosophical ones, such as how to interpret mathematics.
This is in contrast to uniform naturalisms in the philosophy of
mathematics, for example Quinean scientific naturalism, or Burgessian
mathematical-cum-scientific naturalism, or a uniformly mathematical
naturalism (also suggested by Maddy (1997) but in our view ultimately not advocated there).




To explain how this bivalent attitude might work in practice, take
Maddy's favourite example: set theory. Suppose that ZFC + LCA is our
current accepted set theory, where LCA is some collection of large
cardinal axioms. Given mathematical naturalism about
mathematics proper, there is no question of rejecting ZFC + LCA in
favour of, say, ZFC + the axiom that there are no inaccessibles, or
some other set theory, say Quine's New Foundations. Mathematical
standards sanction ZFC + LCA, so that is the set theory we must
accept. But how should we interpret ZFC + LCA? Platonistically, structurally or in some other way? That is a philosophical question, so given scientific naturalism about the philosophy of mathematics (and philosophy more generally) its correct
answer is that sanctioned by scientific standards. For example,
if an overriding scientific criterion of simplicity favours platonism
over all other interpretations, that must be the correct
interpretation of ZFC+LCA.




Maddy motivates her naturalism by appeal to the "fundamental
spirit that underlies all naturalism: the conviction that a successful
enterprise, be it science or mathematics, should be understood or
evaluated on its own terms, that such an enterprise should not be
subject to criticism from, and does not stand in need of support from,
some external, supposedly higher point of view" (1997, 184).
One strained reading of this sentence is that naturalism's fundamental
conviction by definition applies only to natural science and
mathematics. A more natural reading is that it applies to any
successful science whatsoever. Maddy then claims that as a matter of
fact non-mathematical grounds have not impinged on mathematics. We can
express this as the thesis that mathematics
is *autonomous*.




The assessment of Maddy's heterogeneous naturalism consists
principally in the assessment of the autonomy thesis regarding
mathematics and its implications. One question is whether the thesis is
true. Another is whether, if true, the thesis supports heterogeneous
naturalism.


### 5.1 Is Mathematics Methodologically Autonomous?




Maddy claims that the correct model of mathematical justification
will vindicate the hypothesis that traditionally scientific or
philosophical arguments do not enter into the justification of
mathematical statements. For example, the French analysts Baire, Borel
and Lebesgue criticised the Axiom of Choice on the basis of a
definabilist methodology, according to which the existence of an object
depends on its definability: functions should be definable rules,
membership of a set should given in a definable way, etc. But
definabilism ended up having no influence on the practice of
mathematics, and its sympathisers were silenced or gave up on it when
its mathematically undesirable features became clear. For example,
the Axiom of Choice allows the taking of maximal ideals in rings and
other structures; it entails the kinds of maximality principles that even the analysts were
already using; it simplifies transfinite arithmetic; and despite its
suspicious abstractness it turns out to be equivalent to
'concrete' and 'mathematical' statements such
as the claim that every vector space has a basis. The reasons for
accepting the Axiom of Choice were in the end purely mathematical.




Maddy also offers an admirably detailed account of the kinds of
mathematical reasons that militate against the 'Axiom' of
Constructibility, V = L, in particular the fact that its adoption goes
against the set-theoretical maxim *MAXIMIZE,* which enjoins that
the universe of sets should be as expansive as possible (by containing
as many isomorphism types as possible).




Most commentators have let the descriptive component of
Maddy's naturalism, the autonomy thesis, pass, focusing criticism
instead on its normative implications. Yet the autonomy thesis is very
radical. The usual picture of the interaction between mathematics and
philosophy is of a two-way street. In particular, it is usually thought
that philosophy impinges to some degree on mathematical practice.
Intuitionists for example take the dependence to be profound: they think that all of mathematical practice rests on a false philosophical
foundation and that once that foundation is removed, mathematics
totters.




Maddy willingly concedes that philosophical doctrines such as
definabilism or realism are important inspirations for mathematical
development even if they fall short of justifications (1997, 192).
And she accepts that "theories of mathematical truth or
existence or knowledge do in fact appear in most mathematical debates
over proper methods, alongside more typically mathematical
considerations" (2007, 348). But she maintains that such
theories have not in the end played an 'instrumental role'
(2007, 359), and that they have racked up a 'track record of
irrelevance' (2007, 366) in the development of mathematics.




So do philosophical (or more generally
non-mathematical) considerations always fall on the side of inspiration
rather than justification? Well, philosophical considerations are
hardly ever advanced in mathematical journals or books. And whenever
they are Maddy sees them as amounting, when one scratches the surface,
to 'intra-mathematical' arguments (1997, 193), of which a
naturalized model of the practice would be 'purified',
because of their methodological irrelevance (1997, 197). If she is right, this reply generalises from this particular, rather restricted, type of context to all mathematical contexts.




One of the implications of this view is that the kinds of
maxims Maddy sees as purely internal to mathematics, such as
*MAXIMIZE* or *UNIFY*, are never themselves underlain by
philosophical beliefs. *UNIFY* is a methodological consequence
of set theory's foundational ambition of providing 'a
single system in which all objects and structures of mathematics can be
modelled or instantiated' (1997, 208-9). Yet perhaps
*UNIFY* and set theory's foundational ambition--the
fact that as Maddy correctly observes, "set theory is (at least
partly) designed to provide a foundation for classical
mathematics" (2007, 355)--are themselves in some small
way, perhaps to some slight degree, underpinned by set-theoretic
realism, i.e., the view that set theory is about a single universe of
sets. Likewise for the Axiom of Choice, which on the surface owes its
place in the set-theoretic canon partly--perhaps only to a small
degree--to an ingrained realism about set theory. Several set
theorists are on record as making claims to that effect, so the burden
of proof is on Maddy to explain away these remarks.




Maddy also employs the following problematic style of argument. She
thinks that the fact that philosophical debates (e.g., about realism)
are wide open but that mathematics has developed in a particular way
(e.g., to allow impredicative methods) shows that philosophical debates
have not affected the outcome of modern mathematics (e.g., 2007, 
348). But the fact that a debate about realism is wide open in
philosophy does not mean that it is wide open in mathematics. Perhaps
mathematicians have implicitly taken a realist stance, which in part
underlies their acceptance of impredicativity, even as philosophers
continue to debate the correctness of realism as a philosophy of
mathematics. Mathematics may be philosophically *parti pris*,
which is why it has developed the way it has.




These points against the autonomy thesis are hardly conclusive. To
evaluate it, greater clarity is required on where the boundary between
justification and inspiration lies and what exactly it means to say that something falls on one side or the other. And of course we then need to identify more precisely
which of the factors in the evolution of mathematics are justificationally operative and
which idle. However, even if the autonomy thesis is not true,
perhaps it is almost true. And perhaps nothing quite as strong as an
autonomy thesis is required to support trumping as opposed to
biconditional naturalism.


### 5.2 If Mathematics is Methodologically Autonomous, does that establish Maddy's Naturalism?




That a statement-issuing practice is as a matter of fact autonomous
does not entail that its statements are thereby acceptable. Practices
may be hermetically sealed from outside influence (e.g., astrology,
dogmatic theology), but that does not in itself make their claims
acceptable. What makes mathematics different?




Maddy recognises this problem for her position (1997, 203-5;
2005, 449; 2007, 346-7), which reviews and discussions of her
work have seized upon (e.g., Dieterle 1999, Rosen 1999, Roland 2007;
only Tappenden (2001) is more sympathetic). She addresses it by making
a distinction between pure and applied disciplines. Taking astrology as
a foil, she notes that applied astrology may be construed as making
claims about the empirical world. Using her ordinary scientific
standards, the scientific naturalist appreciates that applied astrology
is false (as it diverges from the accepted scientific story). Hence
applied astrology does not deserve the naturalist's respect. Pure
astrology in contrast is astrology interpreted as treating of
'certain supernatural vibrations that don't interact
causally with ordinary physical phenomena' (1997, 204).
However, there is no reason to believe in pure astrology, since it does
not feature in our best scientific theory of the world. On either
interpretation, then, astrology is disanalogous to mathematics.




Maddy seems to want to have her cake and eat it. The reason for
mathematics' credibility is supposed to be its application in
science. But why should the fact that mathematics features in our best
science be a reason for believing mathematicians'
utterances--that is, a reason for taking them to be true? The
suspicion is that if featuring in best science is the mark of
credibility, it should be scientific standards that ultimately
determine mathematical theories' acceptability. Maddy has indeed
cited a feature distinguishing mathematics from pure astrology, as she
points out (2007, 390); but what still remains unclear is why
*this* feature should make mathematics rather than science
authoritative with respect to questions in mathematics proper. Thus it
seems that she has not explained how one can consistently be a
mathematical naturalist about mathematical theories but a scientific
naturalist about everything else, including the truth and nature of
mathematics. More generally, given that a practice issues in
truth-evaluable statements, it seems that one cannot advocate
*X*-standards as the arbiters of these statements' acceptability
while simultaneously advocating a different set of standards, the
*Y*-standards, as arbiters of whether or not the statements should be
taken as true, how they should be interpreted, etc. Notice that this is a problem
faced exclusively by heterogeneous as opposed to uniform 
naturalism.




The only way to avoid this apparent inconsistency is to suppose that
to 'accept' a statement sanctioned by mathematical
practice--a statement to which the autonomous practice of
mathematics gives the thumbs-up--does not amount to taking it as
true. Although suggested by a couple of passages in Maddy (1997), this
interpretation is not one that can seriously be put on her
book. Besides, it is tantamount to scientific naturalism in all but
name. It amounts to the claim that whatever mathematicians spend their
time saying, doing, and publishing should not be interfered with, but
that we should be mindful only to believe those mathematical claims
sanctioned on scientific grounds, whether or not they are given the
thumbs-up in mathematicians' language-game.




This picture has recently been complicated by Maddy's
claims that what she calls Arealism--not taking set theory and
more generally mathematics to be true--may be just as
scientifically respectable as Thin Realism--roughly, the view that
sets have only the properties ascribed to them by set theory (2007,
IV.4). This entry does not have space to do justice to this twist in
Maddy's metaphilosophy. Suffice it to note the following. The issues
just discussed seem to arise for the Thin Realist just as much as any
other kind of realist. And the Arealist construal seems to transform
Maddy's position into something quite different from the
heterogeneous naturalism under consideration here.


## 6. Ontological Naturalism




Turning away from methodological naturalism, consider now ontological
naturalism, the view that all entities are natural. One way to read
this is that only the entities posited by the natural sciences
exist. A second and perhaps more natural reading is that only
spatiotemporal entities exist. We address both readings briefly in
this final section, and take in epistemological naturalism briefly in 6.3.


### 6.1 Natural Science as the Arbiter of Ontology




On its first reading, ontological naturalism in the philosophy of
mathematics is a straightforward consequence of methodological
scientific naturalism. It states that the ontology of mathematics is
the mathematical ontology of our best natural science. Scientific
platonists claim, following Quine and Putnam, that this ontology is
platonist, as do mathematical-cum-scientific platonists (e.g., Burgess
and Rosen (1997)). Resistance to scientific platonism and the
associated indispensability argument has been mounted on several fronts
(e.g., Field 1980, Sober 1993, Maddy 1997, ch. II.6, Paseau 2007).
Consult Colyvan (2011) for a detailed discussion.


### 6.2 All Entities are Spatiotemporal




The second reading of ontological naturalism, according to which all
entities are spatiotemporal, amounts to a version of anti-platonism in
the philosophy of mathematics.




The position subdivides. On a reductionist view, mathematics is
taken at logico-grammatical face value but its objects (numbers,
functions, sets, etc.) are taken to be spatiotemporal. This view is
advocated for sets in Armstrong (1991) and more generally in Bigelow
(1988). Non-reductionist views are manifold. They include taking
mathematics as meaningless symbol-manipulation (formalism), or as the
exploration of what is true in all structures obeying the axioms
(structuralism), or as the exploration of what is true in all possible
structures obeying the axioms (modal-structuralism). Bueno
(forthcoming) discusses various nominalisms, i.e., views which
countenance only spatiotemporal entities. Since many of these
nominalisms are compatible with non-naturalist as well as ontologically
naturalist motivations, we do not discuss them here. We concentrate on
a handful of issues relating mainly to reductionist versions of
ontological realism.




Reductionist ontological naturalism and non-modal structuralism about
set theory face an immediate problem: there are apparently far fewer
entities in spacetime than there are sets. Even on the most liberal
assumptions (spacetime points and arbitrary regions thereof exist,
some smallish infinity of entities may be collocated at any of these
points or regions), the size of spacetime and the objects in it is a
relatively low infinite cardinality (surely no more than
![Beth](beth.jpg)o--even that is generous). Thus there are
not enough spatiotemporal entities to interpret set theory literally
nor to make a structural interpretation of set theory non-vacuously
true, and hence to ensure that set-theoretic falsehoods come out false
rather than true. See Paseau (2008) for discussion of this and other
problems for set-theoretic reductionism.




Another problem is that even if spacetime were large enough to
provide either a model for a literal interpretation of set theory or an
instance for its structural interpretation, this would be a contingent
fact about our universe. Set theory would be true but contingent. Since
we typically think of mathematics as necessary, however, this is an
untoward consequence for a philosophy of mathematics. Some might even
call it a *reductio*.




Versions of these problems also affect Mill's empiricism
(1843). For Mill, mathematics and logic are natural sciences, and their
principles are laws of nature. Arithmetic, for instance, is the theory
of aggregates, i.e., the theory of collections of concrete entities.
Geometry is the theory of idealised limits of concrete
entities--lines, points, planes and so on--whose principles
are "real facts with some of their circumstances exaggerated or
omitted" (Mill 1843, vol. 1, bk. II, ch.v). Millian philosophy of
mathematics is susceptible to the cardinality problem just given. (Of
course this is an anachronistic criticism, since infinitary set theory
had yet to be developed in Mill's time.) As for the
contingency of mathematics, Mill bit the bullet and accepted it.




A related problem for a Millian view, which arises even for the
mathematics of Mill's day, is a dilemma concerning the existence
of aggregates of aggregates, aggregates of aggregates of
aggregates.... Higher-order aggregates, if they exist, can only be
abstract--what else? But if they do not exist--if there are
only first-order aggregates--then in particular there are no
numbers of numbers, for instance it is meaningless or false to say that
there are two primes between 20 and 30.




Kitcher (1983) is an attempt to resuscitate Mill's philosophy
of mathematics by modalising it. It accounts for mathematical truth in
terms of the operations of a possible but non-actual ideal agent, and
thus falls under the heading of modalist philosophies of mathematics.
(Though Kitcher himself is not fond of the label (1983, 
121-2).)




Other apparent problems for reductionist ontological naturalism
include the problem of arbitrariness and the fact that it goes
profoundly against mathematical method. Suppose arithmetic is the study
of some particular spatiotemporal entities. Very well; but which ones?
Surely it is arbitrary which spatiotemporal entity is chosen as the
number zero. This criticism is a version of a general anti-reductionist
argument presented in Benacerraf (1965). The response to it is usually
that reductionism does not seek to uncover the meaning of number terms
but instead proposes a theoretical identification (Paseau 2009). The charge of contradicting mathematical method is a serious one too. If mathematical objects are
spatiotemporal, why do mathematicians not perform experiments to
discover their properties? If mathematics were genuinely concerned with
the spatiotemporal, surely its methodology would be more
empirical.




Ontological naturalist views of the type discussed are for these and
related reasons seen as problematic, and are consequently
unpopular.


### 6.3 Naturalist Anti-Platonism and Epistemological Naturalism




Whatever their motivation, ontological naturalists are by definition
(on this second reading of the doctrine) agreed that platonism is
false. Sometimes ontological naturalism is motivated by metaphysical
doctrines, for instance by the principle that everything that exists
has causal powers. Subscribers to this principle include Armstrong
(1997) who calls it the Eleatic Principle; for criticism, see Colyvan
(2001 ch. 3) and Papineau (2009).




The most popular argument for ontological naturalism is
epistemological, and consequently ontological naturalism is often
allied with epistemological naturalism. If there are abstract entities
then it seems we cannot know nor form reliable beliefs about them
(Benacerraf 1973, Field 1989), because of their causal isolation from
us. Most philosophers take this to be the main problem afflicting
platonism. Note that the argument typically leads to agnosticism
rather than denial of abstract mathematical objects' existence. This
is not the place to engage with the argument--for more details,
see Balaguer (2009)--save to sketch how a platonist who is also a
scientific or mathematical-cum-scientific naturalist--e.g.,
Quine--responds to it.




The naturalist-platonist response is two-pronged (Burgess and Rosen
1997, 2005; for criticism, see Linnebo 2006, Chihara 2006). The first
prong is that no simple criterion for knowledge (or reliable belief or
justified belief) has ever been devised that succeeds in ruling out
knowledge of the abstract without thereby ruling out types of knowledge
most naturalists would accept (Steiner 1975). A couple of examples: (i)
the condition that *p* is a cause of the belief that *p*
is too strong, as it rules out knowledge of the future; (ii) as the
naturalist-platonist sees it, that abstract mathematical reality is
thus-and-so is in fact part of the best explanation for the belief that
*p*; hence an explanatory condition of this kind turns out to be
compatible with platonism. Moreover, naturalist-platonists complain
that even if a criterion were found that draws the line where the
nominalist wishes it to be drawn, it would beg the question against
platonism.




Secondly, naturalist-platonists run a standard Quinean line by
construing any challenge to the reliability of our beliefs about
platonic mathematical objects as a general challenge to the scientific
method's reliability. (This is from the point of view of the
scientific naturalist; the scientific-cum-mathematical naturalist can
run the same line with corresponding adjustments.) However, by our best
lights--according to our best theory of the world, i.e., natural
science, which posits abstract mathematical objects--belief in
abstract mathematical entities is arrived at by a reliable method,
namely the scientific method. This is not simply self-vindication, since the
scientific method is here being used to explain the reliability of
mathematical beliefs, albeit holistically. But of course if the
reliability of the scientific method itself is put into question, the
naturalist has no choice but to use the scientific method itself to
explain its own reliability. The naturalist-platonist may add that we
can do no better, and that anyone who questions the scientific
method's reliability has thereby abandoned the naturalist camp.
From this perspective, then, there is no epistemological problem for
platonism once it is settled that platonist mathematics is part of best
science.