FENG YE
WHAT ANTI-REALISM IN PHILOSOPHY OF MATHEMATICS MUST OFFER
ABSTRACT. I propose some tasks and constraints that anti-realists must accomplish and respect
in order to offer an acceptable anti-realistic philosophy of mathematics. The tasks require antirealists to account for meanings of mathematical statements as working mathematicians actually
understand them, to account for objective content of our mathematical knowledge, to account for
objective relationships between the mathematical and the physical that scientists discover and
take as objective reasons for applicability of mathematics, to account for the apparent a priority,
necessity and universality of arithmetic and logic, and finally to account for applicability of
infinite mathematics to a finite world, all without assuming that mathematical entities really exist,
of course. And the most critical constraint says that a coherent nominalist should not commit to
objectivity of infinity, and therefore has to embrace strict finitism in some sense. Then, I will also
present my suggestions on how to accomplish these tasks and respect the constraints, which
might seem impossible to some people.
1. THE TASKS AND CONSTRAINTS
By “anti-realism in philosophy of mathematics”, I mean the view that denies both
objective existence of abstract objects external to our minds and objective truth of pure
mathematical statements (not including applied statements such as “5 fingers plus 7
fingers are 12 fingers”, which are about fingers in this real world, not about numbers out
of this world). There are attempts to separate these two, but later I will actually argue that
they are inseparable. In particular, accepting objectivity of truths involving infinity will
contradict nominalism.
2
The most forceful objection to anti-realism in philosophy of mathematics is perhaps
this (see, for instance, Burgess 2004): As long as scientists seriously refer to
mathematical entities in their best scientific theories and seriously assert mathematical
theorems, they already justify that mathematical entities exist and mathematical theorems
are true in some proper sense, because scientists’ understanding of their own theories and
their judgments regarding the truth of their own theories should be respected, and there
are no other stronger or more superior standards for justifying truth1.
Since it is practically impossible that philosophers will invent better scientific
theories not referring to mathematical entities, recent anti-realistic arguments all focus on
arguing that scientists do not really commit to mathematical entities “just by referring to
them”. Strategies employed include resorting to the concept of empirical adequacy to
argue that mathematical theorems need not be true (Hoffman 2004), rejecting
confirmation holism and arguing that sciences do not confirm mathematical entities as
they do for atoms (Maddy 2005), and claiming that scientists and mathematicians are
using a figurative manner of speech when making mathematical statements and they
never really mean to refer to abstract mathematical entities and instead they mean
something else (Yablo 2002).
However, scientists’ references to mathematical entities should not be treated too
lightly. Their references to mathematical entities have real meaning for them. For
example, scientists discover that Riemann spaces (and not flat Euclidean spaces) are
approximately but more accurately isomorphic to large-scale space-time structures. They
therefore study Riemann spaces carefully, gain their experiences, intuitions and
knowledge about Riemann spaces, and use Riemann spaces to model space-time.
3
Riemann spaces really mean something for them. The structural similarity between
Riemann spaces and real space-time appears to be an objective fact to them and it is their
reason for choosing Riemann spaces to model space-time. It seems that Riemann spaces
must exist at least in some sense, because it could not be only nothingness there.
Nothingness has no structure and cannot resemble any real things in any way, and cannot
even be approximately isomorphic to anything in any meaningful sense, but Riemann
spaces obviously do have structures and do approximately resemble real space-time in
some way as scientists discover it. They use Riemann spaces because of something
special about Riemann spaces. Otherwise, why wouldn’t they just choose Euclidean
spaces or even just choose the number 2 to model space-time, if they all do not exist
anyway? May be Riemann spaces do not exist in the same sense as atoms and electrons
do, but Riemann spaces must exist at least in some sense. They must at least be something,
and not nothingness. When scientists assert that there are Riemann spaces, their
assertions must have some proper meanings and cannot be gibberishes or simply “literally
false”. If we do respect scientists’ understanding of their own theories and do respect
scientists’ judgments, it is perhaps philosophers’ duty to explore what is that ‘exist in
some sense’, and it is not philosophers’ right to simply claim that Riemann spaces do not
exist and scientists are wrong.
I am not trying to defend realism here. This simple argument certainly contains many
problems if viewed as a defense for realism. My concern is what it means for anti-realism.
At least it means that anti-realists should not simply say, “No, I do not have to commit to
mathematical entities”. The mathematical side of mathematical applications in sciences
could not be completely void. There must be something there on the mathematical side
4
and anti-realists must explain what are really there if not abstract mathematical entities.
Anti-realists must offer a positive account for mathematics. This should be their real and
major work in proposing anti-realism, instead of arguing that we do not have to commit
to mathematical entities. Moreover, as an instance, their account must agree with
scientists’ judgment that Riemann spaces are approximately but genuinely isomorphic to
real space-time structures and that this is the true and objective reason why Riemann
spaces can be used to model real space-time. Otherwise, it is unavoidable that antirealists will appear to be “intellectually dishonest” regarding our mathematical and
scientific knowledge.
There are attempts for giving anti-realistic answers to the question “what is
mathematics about?” For example, fictionalism, figuralism, modal structuralism and
verificationism are some recent ones. But they haven’t answered this challenge
satisfactorily yet or haven’t even tried. Fictionalism basically claims that Riemann spaces
do not exist and statements about them are either false or vacuously true. Therefore it
rejects scientists’ assertion that Riemann spaces are genuinely (although approximately)
isomorphic to real space-time structures, or at least it does not try to account for such
approximate isomorphism. Hoffman’s (2004) recent exposition of fictionalism implies
that scientists pretend that Riemann spaces are isomorphic to space-time structures, but
that obviously could not be the case. The isomorphism is genuine (although approximate)
and not a fake for scientists. Otherwise, scientists won’t succeed in their work in using
Riemann spaces to model space-time. Scientists could not work like kids in playing
games, pretending that a sofa is a mountain. They could not indulge in wishfully
pretending things. Figuralism (e.g. Yablo 2002) essentially claims that statements about
5
Riemann spaces have some real content that is not about Riemann spaces and is actually
not about any particular things, because they are logical truths. If that were indeed the
case, it would be unimaginable why scientists would judge that Riemann spaces in
particular are (approximately) isomorphic to real space-time structures. If the real
content of a statement about Riemann spaces were extracted in a similar way suggested
by Yablo (2002), it would be a sentence with an uncountable number of components.
Scientists could not have understood that alleged real content. They honestly ‘see’ only
Riemann spaces. Structuralism therefore does capture something essential about
mathematics. That is, in some cases, applicability of mathematical theories comes from
the structures that the theories describe and the fact that there are genuine relationships
between mathematical and physical structures. However, claiming that there are pure
mathematical structures, structures that could not have instances in the physical reality
(because the physical reality may be finite), faces the same epistemological difficulties
that entity realism faces, and resorting to the concept of modality in modal structuralism
only brings in an even less clear concept, the concept of mathematical possibility.
Verificationists may be right in saying that some of our knowledge is knowledge about
uses of language, not knowledge about external entities. However, it is quite unclear if
this can account for scientists’ knowledge about the genuine structural similarity between
Riemann spaces and space-time. At least much more work needs to be done there. More
importantly and unfortunately, instead of honestly and laboriously exploring what
scientists’ actual knowledge consists in when they use the language of classical
mathematics, some verificationists declare scientists’ successful uses of the language of
6
classical mathematics as illegitimate uses. This seems to be the current situation of antirealistic philosophies in mathematics.
This is only a brief summary of the challenges for anti-realism and my brief
assessments on the status of current anti-realistic philosophies. I will give a few more
comments on current anti-realistic philosophies in this paper, but I will not try to give full
analyses or to argue for my assessments in detail. This paper will focus on the challenges
for anti-realism. I will try to elaborate the challenges that are implied in Burgess’
criticism and are only briefly summarized above, and I will also try to add other
challenges and constraints that I believe to be critical for a coherent anti-realistic
philosophy of mathematics. However, I will present the challenges as positive tasks that
anti-realists must accomplish in order to provide a positive account for mathematics,
since in the end I still believe that anti-realism is a more promising position and is able to
motivate more new researches that could be interesting and fruitful.
(1) Anti-realism must explain what content or meaning of mathematical statements
consists in and what mathematicians’ knowledge, intuitions and experiences about
particular mathematical entities and structures consist in, if mathematical entities and
structures do not really exist.
Even if abstract mathematical entities and abstract, physically un-instantiated
mathematical structures do not really exist, mathematical statements still have content
and meaning, in the sense that mathematicians are not pronouncing gibberishes. When
mathematicians study Riemann spaces, they apparently ‘see’ some geometrical structures
and they actually resort to their geometrical intuitions in studying Riemann spaces.
Statements mathematicians make are meaningful and do communicate something. They
7
also seem to be about something. Mathematicians do have genuine knowledge,
experiences and intuitions about something, even if they are not literally about abstract
mathematical entities and structures. In other words, there must be something there. It
could not just be nothingness.
The challenge for anti-realists is: what are the things that are real if not abstract
mathematical entities and structures, and how can these real things constitute the content
of mathematical statements or account for the meanings of mathematical statements, and
how mathematicians’ knowledge, experiences and intuitions can be explained by
referring to these real things instead of abstract mathematical entities and structures?
This is a challenge for anti-realists, because anti-realists could not simply take the
content of mathematical statements as states of affairs about mathematical entities and
structures, and take mathematicians’ knowledge and experiences as knowledge and
experiences about mathematical entities and structures. This realistic answer has its own
difficulties, of course, and I will not discuss them here. However, I want emphasize here
that I am not setting a higher standard for anti-realists. It is the first question about
mathematics that any philosophy of mathematics should answer: What is mathematics,
what is it about, what are mathematicians saying and doing, and what do mathematicians’
understanding, knowledge, experiences or intuitions consist in? Realism at least has an
answer.
Moreover, an anti-realist account for these should focus on the current practices of
classical mathematics and should respect mathematicians’ actual understanding of
classical mathematics. It should not invent new mathematics or paraphrase mathematical
statements into something unrecognizable to mathematicians. Because, the real issue at
8
stake is: Do our actual mathematical practices and applications and do working scientists’
understanding of them and judgments regarding them imply realism?
For example, figuralism (Yablo 2002) tries to extract the real content of some
mathematical statements. This appears to be convincing for simple arithmetic statements
such as 5+7=12, but if we follow a similar strategy to get the real content of statements
about a Riemann space, what resulted will not be recognizable to mathematicians. No
geometrical intuition conveyed by the original statements could be preserved. That could
not be what mathematicians really mean. What it shows is the opposite: Mathematicians
do mean Riemann spaces themselves, and not any hidden real content. Similarly,
mathematicians are obviously not talking about ideal agents, possible concrete
inscriptions on papers, and so on. They do not understand these. They only mean and
understand Riemann spaces and other mathematical entities and structures.
This also means that what really exist there could not just be the sentences produced
by mathematicians and nothing else, because if there is really and completely nothing
else besides the sentences, such sentences could not be meaningful. They would just be
gibberishes. Formalists seem to be claiming exactly that mathematical statements are
meaningless strings of symbols. However, for ordinary mathematicians and scientists, the
theory of Riemann spaces is certainly not merely a randomly chosen bunch of
meaningless symbols. Mathematicians do not work on randomly chosen meaningless
formal systems. They study meaningful mathematical structures. No matter how one tries
to explicate the meaning of ‘meaningless’ or ‘meaningful’ here, it is a matter of fact that
mathematicians and scientists ‘see’ Riemann spaces in the theory, and anti-realists must
explain what that means and how it is possible, if Riemann spaces do not really exist.
9
They must respect mathematicians’ and scientists’ understandings of mathematical
theories, must give an acceptable account for such understandings, while not assuming
that mathematical entities really exist.
On the other hand, anti-realists are certainly entitled to give philosophical (but not
mathematical) hermeneutics on what mathematicians and scientists say on apparent,
because when it comes to metaphysical implications of their assertions, mathematicians
themselves frequently shy away. We have no reason to preclude all philosophical
hermeneutics. Many mathematicians as a matter of fact do hold various philosophical
views regarding the nature of mathematics. Physicists sometimes like to call mathematics
a language or formalism and like to talk as if mathematics is just manipulating symbols.
This is actually the presumption in Eugene Wigner’s well-known paper “The
Unreasonable Effectiveness of Mathematics in Natural Sciences”. Therefore, physicists
appear to have a different view about mathematics. There are genuine puzzles there. This
is essentially different from the issue of existence of atoms, about which perhaps no
working scientists have any doubt today, no matter in their “scientific moments” or in
their “philosophical moments”. There is a fine distinction between respecting
mathematicians’ and scientists’ understanding, knowledge, experiences and judgments on
the one side, and assigning the realistic interpretation to their words on the other side,
especially, considering the fact that mathematicians and scientists themselves are puzzled
when they think about the issue. The real important point is perhaps that anti-realists
must provide an account that is understandable and acceptable by working
mathematicians and scientists. In other words, a philosophy of mathematics must be
criticized and evaluated by working mathematicians and scientists, based on their
10
understandings of mathematics. If it is a good philosophy, it ought to be understandable
and acceptable by working mathematicians and scientists, and it should also resolve the
puzzles that working mathematicians and scientists themselves have.
(2) Anti-realism must account for the genuine relationships between some apparent
mathematical entities and structures on the one side and the physical reality on the other
side, without assuming that abstract mathematical entities and structures really exist.
Scientists choose Riemann spaces to model space-time structures for some good
reasons. Even if Riemann spaces do not really exist, it is still a matter of fact that
scientists’ geometrical intuition ‘sees’ that Riemann spaces and the real physical spacetime structures are structurally similar. Structural similarities between some mathematical
entities and some aspects of nature seem to be genuine. There are also other types of
relationships between the mathematical and the physical. For instance, a function may
approximately represent the states of a physics system in some way, and a stochastic
process may simulate some real random events. Such relationships all seem to be genuine
and are the objective reasons why mathematical theories are applicable in sciences.
The challenge for anti-realists is: If Riemann spaces, functions, stochastic processes
do not exist, what such relationships consist in? What our knowledge of such
relationships consists in? It could not be true that there is nothing on the mathematical
side. Nothingness could not structurally resemble any real things. Nothingness could not
be related to any real thing in any meaningful way. Anti-realists must explain what are
real on the mathematical side, and must show that such genuine relationships between the
mathematical and the physical are accountable based on what are real on the
11
mathematical side, and that our knowledge of such relationships are explainable,
although mathematical entities themselves do not exist.
Moreover, anti-realists must show how the content of a specific mathematical theory
could be relevant to the existence of relationships between some specific mathematical
structures and the physical reality. For example, the definition of Riemann spaces and the
content of Riemann space theory are certainly relevant to the fact that Riemann spaces
resemble real space-time structures, and the content of the theory of finite groups is
relevant to the fact that a finite group does not in any way resemble real space-time
structures. In other words, it is not enough to say generally that pragmatic consequences
decide which mathematical theory is useful to model reality in a particular aspect.
Scientists certainly do not randomly pick some literally false statements about nothing
and try to apply them in an arbitrary area in sciences. They choose (or discover, or define,
or invent) Riemann spaces to model large-scale space-time structures, because they really
discern some genuine relationship between these two in particular, based on their
understandings of Riemann spaces. Anti-realists have to admit scientists’ actual intuitions
and judgments, and have to explain them, without assuming that Riemann spaces really
exist.
There are anti-realistic approaches that resort to some general concepts that apply to
mathematics as a whole, such as conservativeness or empirical adequacy, to account for
usefulness of mathematics as a whole. These concepts may be of some interests in
themselves, but alone they are not enough to account for mathematical applications. They
alone cannot explain, for instance, what is special about Riemann spaces which makes
12
Riemann spaces applicable in modeling space-time and why scientists did not use the
theory of finite groups, or anything else, to model space-time.
There is an unfortunate misconception in various types of philosophical
instrumentalism including mathematical instrumentalism. Only real things can be used as
tools, such as screwdrivers or computers. Non-existent things could not be used as tools.
If abstract mathematical entities just do not exist, then they could not be what are really
instrumental. Moreover, we believe that there are objective reasons why a tool works. A
screwdriver may not work for driving a particular screw because they don’t fit each other
and a computer may mal function. We believe that there are explanations for how a tool
works and for applicability of a tool in a specific task, based the structure or design of the
tool as a real thing, and based on known natural laws or regularities among real things
including the tool and the things that the tool is applied to. Mathematical instrumentalists
should not simply claim that mathematics is instrumental and therefore mathematical
entities need not really exist, for if what are instrumental are just mathematical entities,
then the conclusion one ought to draw is that mathematical entities really exist just as
other instruments such as Geiger counters or screwdrivers in our labs, and that there must
be objective truths about mathematical entities and about fitness between mathematical
entities and the physical reality, just as the fitness between a screw driver and a screw, so
that the applicability of such mathematical entities as instruments is accountable. On the
other side, if what are instrumental are not mathematical entities, then instrumentalists
must answer what real things on the mathematical side are truly instrumental, and they
must provide an explanation for how such real things work as instruments.
13
To add to some complexity to the issue, the account for relationships between the
mathematical and the physical must be aware that in most cases scientists never really
assume that mathematical structures are exactly isomorphic to related physical structures.
While physicists use Riemann spaces to model large-scale space-time structures, they are
fully aware that they are glossing over microscopic details. It means that mathematical
structures and physical structures are independent. Since they do not have causal
connections either, some philosophers argue that ordinary scientific methodologies that
scientists employ in their daily work, for instance, those for confirming the existence of
atoms, are not relevant for confirming the existence of mathematical entities and
structures (see e.g. Maddy 2005, Sober 1993). This may be correct, but it is not enough,
for the real issue at stake here is: It appears that a Riemann space has at least to be
something so that it can even remotely resemble real space-time structures. Nothingness
cannot resemble anything in any way. If one can meaningfully and honestly claim that
mathematical entities and structures are indeed independent of physical structures and are
indeed not causally connected with the physical reality, then one already admits that
mathematical entities and structures exist in some sense. One would not say that
nothingness is independent of the physical reality.
Again, this is not setting a high standard for anti-realists. It is simply to gain a real
understanding of what the relationships between the mathematical and the physical really
consist in and what really makes mathematics applicable in sciences.
(3) Anti-realism must identify and account for objective content of our mathematical
knowledge and various types of objective facts regarding mathematical practices and
14
mathematical applications in sciences, without assuming that mathematical or other
abstract entities exist objectively.
Even if mathematical entities do not exist, our mathematical knowledge should still
have objective content. We are not making assertions out of our wishes in doing
mathematics. One could wish that Goldbach’s conjecture is true or false, but we know
that there is something objective and independent of our wishes there. Objectivity of our
mathematical knowledge in some proper sense is a matter of fact. Anti-realists must
explain what that ‘proper sense’ is and explain the source of such objectivity if it is not
from objective abstract mathematical entities.
A natural attempt to identify objectivity in mathematics from the anti-realistic
perspective is to focus on proof instead of literal truth in mathematics. Then, one must
show that correctness in following logical rules is an objective matter. There are
independent reasons for believing such objective correctness transcending human
agreements. When mathematicians examine proofs submitted to mathematical journals,
they obviously believe that correctness of the proof is an objective matter (given
commonly accepted axioms). A very mechanical proof can be extremely long and
complex so that no one is absolutely sure if it is correct and referees may have to vote to
decide if to publish it, but they obviously do not believe that correctness actually means
voted to be correct. They obviously believe that correctness transcends our agreements.
Here, the challenge for anti-realists is: Admitting such objective correctness appears to
commit to rules as abstract entities and commit to objective truths about abstract entities.
Another aspect of objectivity in mathematics is about the relationships between the
mathematical and the physical. Given scientists’ understanding of Riemann spaces, the
15
structural similarity between Riemann spaces and real space-time structures also seems to
be objective. Scientists do not pretend that Riemann spaces resemble real space-time
structures, for otherwise, why don’t they just pretend that flat and simpler Euclidean
spaces or even natural numbers resemble real space-time structures. For scientists, such
objective similarity is the objective reason for our successes in modeling space-time
structures by Riemann spaces. Scientists will not be successful if they indulge in any
wishful thinking or groundless imagination. Anti-realists must also show that such
relationships between the mathematical and the physical reality are accountable as
objective facts, without referring to mathematical entities as objective existences. In other
words, anti-realists can deny that Riemann spaces themselves exist objectively as abstract
objects, but they have to identify and account for such objective facts in some way, if
they really respect scientists’ judgments and they really want to research into the
objective reasons for applicability of mathematics in sciences.
Anti-realists who completely deny any objective realistic truths or seek to account
for mathematics (and our scientific knowledge in general) only as social-cultural
constructions or conventions may not accept these requirements. These include those who
take correctness in rule following as a social norm and nothing beyond, for instance,
Kripke’s Wittgenstein. Criticizing them is usually realists’ task and it is out of the scope
of this paper. I propose this as a task for anti-realists in mathematics because I take antirealism in mathematics as a clarification of and a defense for common sense realism and
scientific realism. It is perfectly compatible with ordinary scientists’ naïve realistic view
on truth, objectivity and sciences, as long as they are about things in this universe. It only
tries to clarify puzzles about the alleged mathematical truths about things out of this
16
universe, namely, infinity and abstract objects. For this, anti-realism in mathematics must
distance itself from views that deny realism and objectivity altogether, or deny common
sense realism, scientific realism, or realistic interpretations of our statements about things
in this universe. (See discussions on task (6) below for more on the basic ontological and
epistemological assumptions of anti-realism in mathematics, as I understand it.)
(4) Anti-realism must explain the apparent obviousness, universality, a priority and
necessity of simple arithmetic and set theoretical theorems, must provide a consistent
account for logic, and must explain the relationships between the two.
We have a strong intuition that “5+7=12” expresses some obvious, universal,
necessary and a priori truth. It does not help to say that “5+7=12” is “literally false”, as
some anti-realist philosophers seem to be saying, which only adds more puzzles.
“5+7=12” is certainly meaningful to everyone. It has content. Kids do learn something
when they learn “5+7=12”. There must be some truth in it even if numbers do not
literally exist and even if “5+7=12” is “literally false” in whatever sense. So the real
question is: What is the truth expressed by this statement and what is the truth about, if
numbers do not exist? Moreover, the truth expressed by “5+7=12” also appears to be
universal, a priori and necessary.
Anti-realists must explain what the content of “5+7=12” is and why it is obviously
true in some proper sense, and they must also explain how it is related to other obvious
truths such as “5 fingers plus 7 fingers are 12 fingers”. They must also answer questions
regarding universality, a priority and necessity of “5+7=12”. Their answers should either
confirm these, or give reasonable explanations as to why we strongly believe so.
Moreover, anti-realists’ answers should respect ordinary people’s understanding of
17
“5+7=12”. It should not, for example, claim that “5+7=12” means that the symbolic
formula is derivable from Peano axioms.
Mathematics and logic are tightly entangled. Some simple theorems in arithmetic
and set theory, such as “5+7=12” or “AB=BA”, appear to be logical truths in disguise,
and some logical concepts, such as the concept of logical consequence, are defined by
referring to mathematical entities, the models. The common wisdom is that logical truths
are universal, a priori and necessary truths. The universality, a priority and necessity of
arithmetic are obviously closely related to the same characteristics for logic.
Anti-realists must provide an account for logic consistent with their general ontology
and epistemology. They must explain what the concept of logical consequence is if
models (as mathematical entities) do not exist. They must also answer questions
regarding universality, a priority and necessity of logic, and explain how they are related
to answers to the same questions about arithmetic.
One anti-realistic attempt to explicate the truth in “5+7=12” is figuralism (Yablo
2002), which claims that the real content of the statement is expressed by a sentence
about numerical properties of two arbitrary predicates and it is a logical truth in the first
order logic. However, under this interpretation, the real content of arithmetic statements
with quantifiers has to be expressed by infinitely long sentences. We do not speak such
sentences and we do not have a disquotational truth predicate for such sentences. Such
sentences are actually mathematical constructions and are abstract entities. If such
sentences do not really exist as abstract entities, it is unclear if we do get real content by
such translations. It seems that what we actually have there is another mathematical
theory about infinitely long sentences as mathematical entities, which could be defined
18
using set theory with the axiom of infinity, and we have a “true” predicate for those
infinitely long sentences, which must be defined using transfinite inductions. It actually
requires either a transfinite induction up to the ordinal number 0 or mathematical
inductions on arbitrary quantified statements in the language of Peano arithmetic (that is,
n–induction for any n), in order to show that such a “true” predicate is well defined. The
translations actually translate statements about numbers into statements in this
mathematical theory about infinitely long sentences with an inductively defined “true”
predicate. It hasn’t explained the ‘real’ content of arithmetic statements.
(5) Anti-realism must account for insights about mathematics discovered by other
schools of philosophy of mathematics.
If a philosophy of mathematics is to provide a foundation for a new type of
mathematics, it can ignore what other philosophers are saying about mathematics, but if it
is to give a faithful account for our actual mathematical practices and to understand the
nature of real mathematics, it must take into account others’ insights. Each school of
philosophy of mathematics appears to focus on one aspect of our mathematical practices
and hold onto to one related insight. For example, structuralism seems to hold on to the
point that what really matter in mathematics are not entities themselves but the structures.
Formalism (and the so-called if-thenism) appears to hold on to the observation that
mathematicians’ daily work is deriving theorems from axioms. If anti-realism is to be a
faithful account for the actual mathematical practices, it must provide a framework under
which each such insight about mathematics can find its own proper position.
For example, anti-realists must explain whether or not there are such things as
mathematical structures and what they consist in if they exist but abstract entities do not
19
exist. Here, the challenge for anti-realists is that most mathematical structures such as
Riemann spaces are not exactly isomorphic to any physical structures. Therefore, as
structures, they are literally different from any physical structures (if they do exist). They
are pure mathematical structures. Now, if one admits pure mathematical structures, then
the view appears to be some sort of Platonism or realism, not anti-realism or nominalism
any more, and the well-known epistemological difficulties with realism (Benacerraf 1973)
will threaten it as well. However, if structures are only structures of things, then since
there are no mathematical entities, there must be no pure mathematical structures either,
for there could not be structures of nothingness. Finally, if mathematical structures do not
exist, then it appears that Riemann spaces cannot even be approximately structurally
isomorphic to real space-time structures, because nothingness cannot be approximately
isomorphic to anything.
Similarly, for formalism and if-thenism, anti-realism must accommodate the insight
that what is really essential in mathematical practices is deriving consequences from
axioms, not literal truth of the axioms themselves. However, anti-realists must also
explain what is special about the axioms we actually adopt. Scientists obviously do not
randomly select a bunch of ‘literally false’ sentences as axioms and then start deriving
theorems and applying them in sciences. Anti-realists must also explain how, for instance,
the formal theory of Riemann spaces in particular makes Riemann spaces approximately
isomorphic to space-time structures and makes such approximate isomorphism objective
facts.
20
(6) Anti-realists must maintain a coherent fundamental view regarding existence and
objectivity, especially, regarding existence and objectivity of infinity, and must provide
an explicit explanation for their position and defend its coherence.
Philosophers with inclination toward anti-realism or anti-Platonism may still hold
different views regarding existence and objectivity. For example, Field (1998) and Yablo
(2002) both accept objectivity of arithmetic truths involving infinity (when properly
paraphrased). Many philosophical accounts for mathematics rely on a realistic reading of
consistency assertions about formal systems. For instance, structuralism claims that a
structure exists as long as the formal system describing it is consistent, and full-bloodedrealism claims that for every consistent system, all entities that constitute any of its
models literally exist. Consistency assertions refer to arbitrarily long sentences, which
can only be abstract entities if they do exist and are understood literally. Therefore, either
these philosophers commit to arbitrarily long syntactic entities as abstract entities, or they
must have implicitly assumed a way of translating consistency statements into some other
objective truths not literally about arbitrarily long syntactic entities, and it is unclear how.
Frequently, it is unclear what are the fundamental assumptions regarding existence and
objectivity underlying a philosophical position and if the assumptions are coherent.
Field (1998) does explicitly assume infinity of the universe in defending objectivity
of arithmetic statements involving infinity. However, according to our sciences today, the
universe could be finite and there could be only finitely many concrete objects in the
universe in total. Modern physics does not give a definite opinion regarding if the
universe is infinite. Macroscopically, we believe that the universe is likely to be finite
spatially and finite in the past and we are less sure about if it is finite in the future time
21
direction. Microscopically, anything below the Planck scale (about 10-35 meters and 10-45
seconds) is obscure to modern physics and some physicists are entertaining discrete
space-time for quantum gravity. And more importantly, it seems that applicability of
classical mathematics has nothing to do with whether or not space-time is finite and
discrete, or infinite and continuous. In all areas of sciences so far, applicability of
mathematics seems to be independent of the physics conjectures about continuity or
discreteness of space-time. We apply infinite mathematics also in economics, which is
certainly about finite and discrete things. An acceptable anti-realistic account for our
today’s mathematical knowledge and mathematical practices should not literally depend
on a specific scientific assertion (namely, infinity of the universe or continuity of
microscopic space-time) that some scientists seriously doubt today, and that all scientists
may never have consensus opinions forever, and that does not appear to be relevant to
applicability of mathematics in almost all current areas in sciences. If an account does
depend on such a scientific assumption, it must have missed something essential about
the nature of our mathematical knowledge and must have missed the true the reasons for
applicability of mathematics. An account for applicability of mathematics is mostly just
an account for how infinite mathematics is applicable to a finite world.
Now, if some anti-realist philosophers accept objectivity of infinity but claim that it
does not mean infinity of this physical universe, then they must explain their fundamental
principles regarding objectivity, existence and our knowledge of them, because this
appears to imply some sort of realism and Platonism. They must explain where that
infinity is from if this physical universe is indeed finite and discrete. They also face a
similar epistemological problem as realists face, namely, explaining how knowledge
22
about objective truths involving infinity is possible, given that we are finite beings with
finite experiences, and assuming that the universe is also finite. They might try to exploit
some concept of scientific confirmation. Then, it is likely to be again some sort of holistic
confirmation about things that have no causal or other directly naturalistically
accountable connections with us as concrete and finite beings, and it is likely that realists
can simply take over their explanations to explain our knowledge about just any abstract
objects, just as Quine does. After all, as long as the door is open for accepting one
objective truth that appears to be essentially beyond and above this concrete universe, it
should not be too difficult to go ahead and accept others for similar reasons. Moreover, if
one admits that the physical universe is possibly finite but also admits that some
statements literally implying infinitely many objects are objectively and necessarily true,
then it seems that one already commits to some abstract entities not in this finite and
concrete physical universe. Therefore, I see rejecting objectivity of infinity as the only
way to be a coherent nominalist.
This assertion was already made long ago by Goodman and Quine (1947) and it
appears to be ignored by some contemporary philosophers. (But I must mention here that
I do NOT imply that anti-realists should object to scientists’ use of infinite classical
mathematics or should try to nominalize sciences, as Goodman and Quine tried in that
paper. I will discuss this later in connection with the next task, explaining applicability of
classical mathematics.)
This means that for any assertion containing quantifiers intended to range over
infinite domains, either it should subject to a proper anti-realistic interpretation, or one
has to admit that it can be vacuously false or vacuously true if understood as an assertion
23
about concrete things, or understood as a schematic assertion about concrete things.
These include assertions about consistencies of formal systems, assertions about Turing
machines, and assertions expressing simple arithmetic laws and so on. Notice that the
critical thing here is again infinity, which could be essentially beyond this concrete world.
An assertion about abstract entities can sometimes be interpreted as a schematic assertion
about related concrete objects. For example, an assertion about abstract sentence types
can be understood as a schematic assertion about sentence tokens, and an assertion about
an abstract Turing machine can be understood as a schematic assertion about its concrete
realizations. Thus interpreted, some assertions apparently about abstract entities can still
have realistic meanings and objective truth values for anti-realists. But if there are only
finitely many concrete objects in total, some assertions involving infinity can only be
vacuously true or vacuously false under such interpretations.
We do seem to have a strong intuition that some assertions involving infinity have
objective truth values, for instance, the general commutative law of addition (or the
commutative law expressed as an assertion about a Turing machine), and assertions about
consistencies of some very simple formal systems and so on. Therefore, anti-realists must
show that literal truth of such assertions involving infinity is not presumed in sciences or
implied by sciences, and is not needed in the anti-realistic account for mathematical
practices and mathematical applications either. On the other side, anti-realists must also
identify what are truly objective in our mathematical knowledge involving infinity if it is
not from an objective infinity, and they must also provide a satisfactory explanation for
our strong intuition on objectivity of some truths involving infinity.
24
Here, I like to suggest that anti-realism can take a species of naturalism consisting of
the following theses as its basic assumptions: (1) there is this objective world, the
universe, and space-time is our basic framework for describing things in this world, and
only things in real space-time really exist; (2) this universe could be finite and discrete, or
infinite and continuous, which is an empirical and contingent matter and is not
determined by our minds and the way we think or speak; (3) instead, our minds are a part
of this natural world and our knowledge (including mathematical knowledge) comes
from our brains’ and bodies’ interactions with finite concrete things in this universe,
either individually or programmed into our genes as a result of evolution; and (4) there
are no objective existences or objective truths beyond and above this concrete universe.
This basic worldview seems to be clear and coherent, although such a brand of naturalism
may sound too simplistic to some philosophers. It is not Quinean naturalism. It is
consistent with empiricism, nominalism in Goodman and Quine (1947), as well as
physicalism in contemporary philosophy of mind (e.g. Papineau 1993).2 It is also
consistent with ordinary scientists’ naturalistic world view and it is especially
emphasized by those cognitive scientists who contend that human intelligence is
embodied and try to account for human knowledge, including mathematical knowledge,
starting from embodied cognition (e.g. Lakoff & Núñez 2000, Lakoff & Johnson 1999).
This may also be close to Maddy’s naturalism (2005), but put in this way, it is a basic
ontological attitude that scientists seem to presuppose before they even start their
research work, prior to any conscientious practices of scientific methodologies, and it is
not a consequence of rejecting methodological confirmation holism. So far our scientists
25
never seem to have consciously tried to scientifically confirm existence of anything out
of this physical universe. Only theology does that.
I see anti-realism in mathematics as an effort to show that this brand of naturalism is
coherent in so far as our mathematical knowledge is concerned. Our mathematical
knowledge does generate some puzzles for such a naturalistic worldview, because
mathematics appears to include knowledge about things out of space-time. Quinean
pragmatism actually implies that modern extension of our mathematical knowledge and
successful applications of abstract mathematics in sciences force us to reject this (perhaps
a little simplistic) brand of naturalism and force us to accept a more sophisticated view on
existence that puts numbers on a par with other concrete things in this universe. I will not
discuss problems with Quinean realism here since my focus here is on tasks for antirealists. Other philosophers who reject Quinean realism but still accept objectivity of
infinity seem to be driven by our intuition about objectivity of simple arithmetic theorems
involving infinity (such as the commutative law of addition), and they seem to suffer
from inconsistencies as I have discussed above. I see anti-realism in mathematics as an
attempt to defend this ‘simplistic’ naturalistic worldview.
(7) Anti-realists must explain applicability of classical mathematics.
Successful applications of classical mathematics are simply facts. Any philosophical
account for mathematics must provide an explanation for applicability of classical
mathematics. However, it is not very clear what exactly need to be explained and what
type of explanations is expected. There may be an implicit theory about applicability
according to which when mathematics is applied to economics, for instance, there is a
first order axiomatic system of mathematics-cum-economics consisting of literally true
26
axioms about infinite mathematical entities as well as finite concrete things that we deal
with in economics and relationships between them. Perhaps no one ever explicitly
proposed such a theory but it was implicitly assumed in many discussions related to the
idea that applicability justifies literal truths about mathematical entities, for instance, in
Quine’s web-of-beliefs picture. This is an illusion. The part of such an alleged first order
axiomatic system that relates infinite mathematical entities to finite things can never be
literally true, even if one believes in literal truths about infinite mathematical entities. We
use infinity to approximate finite things in modeling nature with mathematics, and the
logic of such approximations is never clear. Within pure mathematics, scientists may
follow the logic rigorously, but when it comes to applying a mathematical model to real
things, they decide by intuitions and experiences of their trial-and-errors. They twist
interpretations of their mathematical consequences drawn from the model and they
simply throw away consequences that they judge as meaningless for the domain of
application, without considering the fact that this already refutes the axioms in the alleged
system of mathematics-cum-real things. Therefore it is an illusion that realism in
mathematics already has an explanation of applicability of mathematics. Otherwise,
Eugene Wigner would not have expressed his puzzlement in the well-known paper,
because even if there were this  up in the sky, it is still puzzling how that irrational
number  up in the sky is related to finite populations on earth. Even for realists there is
no literally true picture about infinite mathematical entities up in the sky together with
finite concrete things on the earth and their relationships. There may be two separate
literally true pictures for realists, one for abstract mathematical entities up in the sky and
27
the other for finite concrete things on the earth, but there are no literally true
correspondences linking them together.
The gap between infinity in mathematics and finitude of the real world is a genuine
puzzle about applicability of mathematics for any philosophy. It is more logical rather
than philosophical in nature. An answer to the puzzle must involve clarifications on the
logic of approximations in using infinity to model finite things. What anti-realists can
hope is that a complete clarification may turn out to favor anti-realism, because since the
real things that we apply infinite mathematics to are finite, it seems likely that a complete
clarification of such approximations will eventually mean showing that infinity can in
principle be eliminated in applications, and then it will follow that apparent references to
abstract entities can all in principle be eliminated as well. In other words, it may
eventually show exactly how mathematics functions as tools, helping to derive
nominalistic truths about finite concrete things from nominalistic premises about finite
concrete things.
This is the ideal explanation of applicability for anti-realists. It is instrumentalism,
but it is not merely proposing instrumentalism as a philosophical thesis. If such an
explanation is available, it will show exactly how the instruments work. If mathematics
does not consist of literal truths about mathematical entities, mathematics has to be
instrumental, but as I have mentioned above in discussing task (2), anti-realists’ real job
should be identifying what are the real instruments as real things there in mathematical
applications, and showing how exactly such real instruments work in terms of some very
realistic theory consisting of literal truths about those instruments as real things and their
relationships with other real things that the instruments are applied to. I have argued that
28
anti-realists must account for the objective relationships between the mathematical and
the physical, which can be a part of the explanation for applicability of mathematics. A
full explanation of applicability should certainly be more than that. It must describe and
explain how exactly those concrete real things on the mathematical side in mathematical
applications function as instruments for deriving true statements about other concrete real
things from premises about the same concrete real things. It has to be a realistic scientific
theory about human mathematical activities as real phenomena in the real world.
I must emphasize again here that this does NOT mean that we should nominalize
sciences and replace classical mathematics by something else. Anti-realists’ task is to
explain why and how exactly classical mathematics is applicable and is so effective, not
to invent a different type of mathematics. Ideally, their explanations should even include
explanations for working scientists’ intuitive judgments regarding why and how a
particular mathematical theory is applicable in a particular area in sciences. It is to
explain how exactly the instrument works ingeniously and effectively, not to suggest
abandoning the superb instrument and using our bare hands for the work.
These are the tasks that I propose for anti-realists. I have tried to excavate the
strongest objections to anti-realism from Burgess’ criticism, and tried to push the
principle of nominalism to its extreme regarding infinity. In summary, anti-realists must
be coherent regarding nominalism and infinity and must also seriously pursue a positive
account for mathematics, respecting all that working scientists intuitively understand and
assert but taken as evidence for realism by realists. They should not simply label some
troubling things by a name, e.g. “empirical adequacy” or “conservativeness”, and should
not be negative only (i.e. arguing that they do not have to commit to something). They
29
must explicitly say then what are real on the mathematical side and dive into those real
things and explain what exactly they are and how exactly they work, in terms of very
realistic truths again.
2. SOME SUGGESTIONS
To some people, these tasks and constraints may appear to have cornered antirealism to absurdity. However, accomplishing these tasks and respecting the constraints
is not entirely implausible. There are discourses in everyday life that are obviously
meaningful, but are also obviously not about and not meant to be about anything real, at
least not directly. These include telling stories or talking about idealized physical objects
such as mass points, ideal gases and so on. Fictional characters in stories or mass points
mentioned in physics textbooks do not literally exist. We seem to be imagining them or
just pretending that they exist. Fictional discourses are the most obvious non-realistic
discourses in everyday life, but they are also meaningful for us. Moreover, some
imaginary things seem to resemble some real things in some aspects objectively, given
how we imagine the imaginary things. We may have made up Romeo and Juliet in a story,
but given what are told about Romeo and Juliet, whether or not a real couple’s love story
is similar to Romeo and Juliet’s seems to be something objective. We do not and could
not make that up. That is, there are objective similarities between real things and fictional
characters although fictional characters themselves do not literally exist.
These seem to be facts, mere facts. Therefore, if our common sense is right in telling
us that fictional things such as Romeo and Juliet really do not exist, then notwithstanding
any complexities and puzzling confusions, we ought to be able to explain what the
30
content of statements about fictional characters is, and what the objective relationships
such as objective similarities between fictional characters and real things consist in,
without assuming that fictional characters really exist.
Therefore, for anti-realists in mathematics, a very natural thought is that
mathematicians are also imagining mathematical entities, deriving conclusions (the
theorems) from their assumptions about their imaginations (the axioms), and using
imaginary things to model real things in mathematical applications in sciences. To give
an account for mathematics along this line, anti-realists must first explain what is to
imagine something or to pretend that something exists. One natural thought is to
characterize imagining something as having relevant mental representations with the
same or similar formats and structures as mental representations of real external things,
but without any corresponding external things to be directly represented, namely, having
a subject and the subjects’ internal representations but without external objects. It means
that while there are no mathematical entities, there are mathematicians’ mental and
linguistic activities and their mental representations that they create and manipulate in
understanding mathematical statements and doing mathematics deductions. In other
words, anti-realists’ task will be describing mathematicians’ and scientists’ mental and
linguistic activities in doing and applying mathematics, exploring the relationships
between mathematicians’ and scientists’ mental representations and other real things in
the real physical world to explain applicability of mathematics. What are truly
instrumental in mathematical applications are mathematicians’ and scientists’ brains and
their mental representations, which are of course the most complex instruments in the
world. This should not be surprising. After all, since there are no external abstract
31
mathematical objects, only things in mathematicians’ brains (and word tokens that they
produce) could be the real things on the mathematical side. Anti-realists have to look into
mathematicians’ minds to account for mathematics. That appears to be their only choice.
Moreover, talking about ‘imaginary thing’ is only a manner of speech. What are truly
instrumental are not ‘imaginary mathematical entities’. They are our brains and mental
representations residing in our brains when we are imagining.
This is consistent with the brand of naturalism explained in the last section. That is,
only things in real space-time can be real, and sciences look for truths about things in this
real universe, and as for others, such as mass points, ideal gases, as well as numbers and
so on, we just talk as if they exist. What really exist in such talks are our mental
representations in such mental activities. We should look into our minds, not elsewhere,
to give a truthful account for our mathematical practices. An account for mathematical
practices and applications will be like other scientific descriptions and explanations,
addressing the real phenomena of our mental and linguistic activities in doing and
applying mathematics, and referring to real things in this real world, mental
representations and other external physical objects, to explain such real phenomena of
human mathematical practices. An account for mathematics will be a continuation and
extension of cognitive sciences, dealing specifically with human mathematical cognitive
activities. On the other side, faith in realism comes from attempts to project our mental
representations in imagination activities onto external, because they have similar formats
and structures as our mental representations of real external things and they appear to be
‘about’ something to our consciousness.
32
This idea is not new. As far as I can trace it, the earliest explicit exposition of
something like this is Renyi (1967). Renyi explicitly suggested that mathematical entities
are our imaginations. However, Renyi did not explain what exactly is to imagine
something and did not answer many questions that contemporary philosophers would ask,
including doubts and puzzles that I will briefly discuss below.
Linguists and cognitive psychologists Lakoff and Núñez (2000) explore a
psychological account for our mathematical imagination as metaphorical thinking
grounded in embodied cognition. They explore various types of metaphorical structures
that might have shaped our abstract concepts in mathematics. The suggestion here agrees
with their fundamental philosophical position in rejecting both Platonism and various
types of relativism. In my presentation, I only try to avoid literally committing to a
particular psychological theory regarding the internal mechanism of our mathematical
imagination and focusing on philosophical and logical issues. On the other side, from the
philosophical and logical point of view, there is still a missing component in Lakoff and
Núñez’s (2000) account for mathematics if it is to be a complete picture, because there is
the issue of objective reasons for applicability of those metaphorical abstract
mathematical concepts.
As it is well-known from Frege’s criticism on psychologism and Quine’s argument
for naturalized realism, realists contend that objective truth of our thoughts should be
distinguished from the actual psychological processes by which we come to have or to
assert those thoughts. For instance, the actual psychological process and mechanism by
which physicists come to have thoughts about atoms, quarks, or super-strings is a
different issue from whether or not such concepts and thoughts do directly correspond to
33
external real things in the universe, and whether or not successes of physics justify
existences of atoms, quarks, and super-strings as real things, and if atoms, quarks, or
super-strings do not really exist, then how our thoughts are actually related to real things
so that our physics theories are successful. Similarly, we might really come to have
abstract concepts and thoughts involving infinity by some metaphorical mappings from
some source domain of concepts grounded in our sensory-motor systems, as Lakoff and
Núñez suggest. However, realists argue that since infinity is indispensable in
mathematics for scientific applications, successes of mathematical applications justify
that infinity is objective, just as successes of physics theories justify that atoms really
exist. As anti-realists we do not believe that our thoughts involving infinity correspond to
another type of external reality, the reality of abstract objects different from atoms or
quarks but equally objective. And we do not want to claim that applicability is just a
brutal fact, a coincidence hit upon by scientists blindly, for scientists surely have their
intuitive understandings of objective reasons why a particular mathematical theory is
applicable in a particular area, based on the content of that theory. Therefore, there needs
to be a different type of explanation as to how our thoughts in mathematics are
objectively related to real things in this physical universe so as to make our mathematical
applications successful. That is the task of accounting for objective relationships between
the mathematical and the physical and accounting for applicability of mathematics
mentioned above. In other words, besides the question “where mathematics comes from”,
we have the question “how exactly such mathematics applies back”. Lakoff and Johnson
(1999) did mention aptness of metaphorical concepts, which is just applicability in the
case of mathematics, but the point is that applicability needs some further explanation in
34
order to counter the claim that applicability leads to naturalized realism about abstract
objects on a par with concrete things such as atoms. Moreover, if we really want to reject
verificationism, relativism, social-cultural constructivism, or a complete denial of
objective truth, as Lakoff and Johnson (1999) seem to want to do as well, the explanation
must identify objective grounds for applicability, since scientists intuitively believe that
there are such objective grounds. If one metaphor is apt and another is not, and both are
our inventions, there should be objective reasons behind it.
For simple abstract thoughts such as “5+7=12”, this may not be a big issue, because
such simple thoughts seem to directly correspond to real states of affairs about real things,
such as the fact that “5 fingers plus 7 fingers are 12 fingers”. However, when infinity is
involved, the logical connections between our thoughts in mathematics and states of
affairs about finite real things are not so straightforward. Therefore, we have that task of
explaining applicability of infinity in mathematics for describing this finite world. To be
more accurate, it is to explain how our metaphorical concepts of infinity as concrete
mental particulars are related to external real things in the concrete finite world, so as to
make our thoughts applicable. Such relationships will in the end be realized by our
sensory-motor systems, but for clarifying puzzles surrounding the gap between infinity
and finitude of the world and for gaining a clearer picture of how infinite mathematics
actually works, we can probably ignore and abstract away such psychological details and
focus on some structural relationships that can be analyzed logically and mathematically.
In other words, psychological descriptions will not have the required rigor and
transparency, because this is not about details of the cognitive processes within our brains.
It can and should be able to be explained at a more abstract level and therefore can be
35
simpler and more rigorous, closer to physics than to psychology. After all it is about
applicability of consciously recognized mathematical theories to the physical reality, not
about unconscious origins of our mathematical ideas, and scientists do have conscious
(though vague and intuitive) understandings of why infinite mathematical models can
correctly approximate some aspects of this finite reality. Such an explanation should be
able to take the form of logical analyses and will be itself logical and mathematical, not
psychological in nature. For instance, one might take an assumption about component
structures of our mathematical concepts and thoughts as mental particulars, and then
focus on logical and mathematical analyses on the relationships between such component
structures and external real things to which we apply mathematics.
However, I like to emphasize again that abstraction here only means ignoring noncritical (but very complex) details and simplifying the question about concrete things,
about our mental processes, internal representations and their relationships with external
real things. It does not mean committing to an external world of abstract entities or
committing to a disembodied reason. Moreover, a projection onto an external world of
abstract entities as realism does seems completely unhelpful and irrelevant for describing
and explaining such relationships between our concepts and thoughts as concrete mental
particulars and concrete external real things.
This is the same pattern of reasoning I have been repeating. That is, one can reject
mathematical realists’ arguments, but one still has to explain what needs to be explained.
One cannot just label what needs to be explained by a name different from “truth”, for
instance, by “conservativeness”, “empirical adequacy”, or “aptness”. We need to know
why it is so. Explanation does not stop there. And if we do not want to fall into various
36
types of relativism, we will have to identify what objective facts in nature and what
regularities in nature make something conservative, empirically adequate, or apt, and
make something else not so. Even if we put aside philosophical concerns, this should be a
valuable research topic in itself and should be able to shed some lights on our own
mathematical cognitive activities. In the end we may still stop at truth, but it will be truth
about real things in this universe accepted by common sense and sciences, not truth about
abstract objects and infinity out of this universe.
Missing such an explanation for applicability is perhaps the major reason why
researches by psychologists such as Lakoff and Núñez (2000) are mostly ignored by
philosophers of mathematics, which is very unfortunate, because it seems clear that, as
Lakoff and Johnson (1999) have argued, the new image of embodied minds suggested by
contemporary cognitive science does help to clear away many metaphysical illusions.
And I believe that it is almost indispensable for clearing up many puzzles and confusions
in philosophy of mathematics, especially, for clearing away the myth of the so-called
“semantic intuition about abstract entities”, which is the major psychological source of
our faith in realism about abstract objects.
Some contemporary philosophers also seem to have touched upon this idea for
accounting for mathematics. For instance, Maddy (2005) mentioned the possibility of a
complete scientific description of our mathematical practices. However, so far I haven’t
seen any explicit expositions along this line. Perhaps the major concern for philosophers
is still about the alleged indispensability of infinity and abstract objects in mathematical
applications, which deter naturalistically inclined philosophers from confidently and
plainly admitting that the only things that really exist are things in space-time, in this
37
universe, and that a truthful account for our mathematical practices will mean a
description of our mathematical mental representations in our brains and their
relationships with other real things in this universe.
Now, I will present some concrete suggestions regarding how to accomplish the
tasks proposed in the last section, following this idea of viewing our mathematical
practices as imagining things and drawing consequences from our own assumptions
about our imaginations, and I will also discuss the related conjecture that after all infinity
might not be strictly indispensable for mathematical applications, which suggests a way
to account for applicability. But before that, I want to clear up some possible confusions
and doubts about viewing our mathematical practices as imagining things and comparing
mathematics with fictions, which will also help to show that the idea is not intuitively
infeasible.
First, I must mention that our imagination activities do not literally create any
imaginary entities. Only our hands can create things out of preexisting materials. All our
minds do in our imagination activities is creating mental representations residing in our
brains. Saying that mathematical entities are imaginary entities can be misleading,
because we obviously cannot create uncountably many imaginary entities. If we look
around carefully, we see that what are real are things in this universe including things in
our brains3. We have a primitive and clear awareness about this universe where we live,
and if we are careful, we can always distinguish real things in this universe and things we
imagine4. Of course, “things we imagine” is only a manner of speech. There are no such
things.
38
Another obvious doubt is that mathematical theorems, at least some simple
arithmetic theorems, appear to be universal, objective, a priori, and necessary truths,
while fictions seem too arbitrary. For this, one can point to the fact that our mathematical
imaginations have very specific purposes for applications and are therefore highly
constrained. For instance, geometrical figures are imagined to simulate physical figures,
and numbers are imagined for counting, and sets are imagined to simulate collections of
stably individualized physical objects. Mathematical imaginations are not exactly like the
real things they mean to simulate, because we consciously exercise our mental powers of
abstraction and generalization (or metaphorical mapping). However, sometimes we also
say that a fiction can reveal human nature better and deeper than any meticulous
recordings of real human behaviors and mental activities, and we are also highly
constrained in telling a story with such a specific purpose. A story’s embodying truths
about real things does not make fictional characters we refer to in the story literally exist.
The critical point is that mathematics and fictions are similar on ontological and
epistemological aspects, while their differences are only in what they look to us and are
only a matter of degree. In other words, the difference between real things in this
concrete universe and imaginary things is much more critical ontologically and
epistemologically than the difference between the simple, abstract, rigorous, and crispy
entities that we imagine in mathematics and the complex, fuzzy fictional characters that
we imagine in ordinary fictions. Imagining a perfect circle and imagining Romeo and
Juliet as perfect lovers are indeed different, but the difference is rather a matter of degree
and both are our imaginations, not things in this real world. Moreover, the constraints in
39
imagining mathematical entities actually do not make mathematical imaginations unique,
especially when infinity is involved, as our set theory shows.
As for necessity and a priority of logic and arithmetic, one can point to the fact that
our minds and our imagination capabilities are constrained by the fact that we (and our
neurons) are stable macroscopic objects and we can directly observe stable macroscopic
objects only, for which classical logic and arithmetic are objectively valid (to some
limited finite extend). In other words, classical logic may not be directly applicable to the
quantum world as quantum logic suggests, but on the other hand when we imagine things
in doing mathematics, we can only follow classical logic and arithmetic, because our
minds and imagination capabilities are so structured by the macroscopic world. The
critical thing is still to distinguish the real from the imaginary carefully. While our
imaginations may necessarily follow classical logic and arithmetic because of our innate
mental capabilities, real things such as microscopic particles and the whole universe can
actually be beyond our imagination capability in some essential ways, as modern physics
tells us.
Another doubt may be that we do not add qualifications like “in Shakespeare’s
stories” when making assertions in mathematics. This can be explained by referring to the
fact that mathematics is a commonly accepted story for some common purposes, for the
purposes of counting, measuring, and so on. As a common story understood by everyone
in the language community, we do not have to add the qualification “in the story about
mathematical entities”. On the other hand, such qualifications are actually needed in
some special cases in order to clear up some puzzles. For instance, when we define
numbers as sets, we are embedding the story about numbers into the story about sets and
40
there are many reasonable ways to do this. This generates the well-known puzzle
(Benacerraf 1965). If we recognize that apparent references to imaginary entities in a
story are meaningful only relative that story and there are two stories that we want to
merge here, the puzzle will disappear. It is well-known that meanings of stories consist in
the structures that the stories describe. Therefore, this explanation is consistent with the
idea of structuralism. However, I must emphasize again that since this physical reality
could be finite, we only imagine that there are those infinite mathematical structures, and
what really exist are our words and our mental representations in imagining infinite
mathematical structures, not the infinite mathematical structures themselves.
Now, to explain the content of mathematical statements or statements in fictional
contexts in general, and to explain how we actually understand sentences in fictional
contexts and in mathematics, one can refer to cognitive studies of semantics and language
understanding. See, for instance, Croft and Cruse (2004), researches by the NTL group
http://www.icsi.berkeley.edu./NTL/index.php, Lakoff & Johnson (1999). What they
provide is a psychological description of our concepts as mental particulars and as
internal components of meaning. They show that lacking direct external references does
not make our concepts or linguistic terms meaningless. This helps to clear away the myth
of “semantic intuition about abstract entities”, which is based on the assumption that
external abstract entities must exist in order for our relevant terms to be meaningful.
Their detailed studies also help to clear away the illusion that abstract entities or concepts
external to our minds must exist in order for meanings to be objective and for
communications to be possible, because as all psychologists will insist, the true objective
basis for objective meaning and communication is similarity of our minds, grounded in
41
similarity of genes, and similarity of our environments. That is, the true objective basis
for objective meaning consists in regularities in nature, regularities among real things in
this universe, including our brains and our environments, not in existence of things out of
this universe.
From the philosophical point of view, these researches still need to be supplemented
by some linguistic and conceptual analyses. Psychological descriptions of our internal
processes in representing and processing meanings constitute only one side of the story
about meaning. The other side should be the semantic relationships between what are in
our brains and what are external. Here, one critical thing to notice is that although our
abstract concepts as mental particulars in our brains do not directly refer to particular real
things in this world as their references, they do relate to real things in some other ways.
For instance, “2” does not refer to any particular object in the real world, but it is
obviously related to physical properties such as “2 inches”, “2 pounds” and events such
as counting twice and so on. Such relationships are also semantic relationships although
they are not referential relationships in the ordinary sense. They are actually richer and
more flexible and are very useful for us. They are just the merits of abstract concepts (as
mental particulars in our brains) and are the basis for applicability of abstract concepts. A
competent understanding of “2” will include mental capacities in recognizing such
relationships. Characterizing such non-referential semantic relationships should also be
the central task of a semantic theory of mathematical language.
Moreover, to identify true objectivity and to avoid falling into relativism, we still
have to identify and characterize true objective content of our mathematical knowledge,
or what objective truth is really asserted and accepted when mathematicians prove a
42
theorem (if it is not objective truth about abstract mathematical entities). If we follow the
suggestion that to be true in mathematics is to be derivable from some assumptions about
our imaginations, then we must still show that there is objective correctness for such
derivations.
As for accounting for the structural similarities between the real and the imaginary, it
seems that one can refer to syntactic sentences describing the imaginary things or our
mental representations of the imaginary things, instead of imaginary things themselves.
Since it is objective as to what sentences are indeed included in a story or what mental
representations we actually have, characterizing structural similarities this way will refer
only to objective facts about real things, namely, stories or mental representations, and
other real objects to be compared with ‘imaginary entities’. This may account for the
relationships between the imaginary and the real as objective facts.
Applicability of mathematics in general is a bigger topic. First, in some simple cases,
especially in the cases where an exact structural isomorphism between the real and the
imaginary is available, applications of mathematics will be accountable by using the
ideas from if-thenism. That is, valid logical deductions in mathematics on sentences
about imaginary mathematical entities can be directly translated into valid logical
deductions on sentences about real things when mathematics is applied. Mathematical
proofs would actually function as schematic valid logical deductions, expressed as proofs
about imaginary things, to be instantiated as deductions on sentences about real things
when applied.
The situation is much more complex when advanced mathematics is applied. Ifthenism does not work because the “if” component may not come out true in applications.
43
For instance, consider the assumption that a function of time represents the physical
states of a system. This can be translated into a series of assumptions about physical
quantities of the system at various time moments, by evaluating values of the function.
These are the nominalistic content of the original assumption about the state function (as
a mathematical entity). Now, if we draw a conclusion about the mathematical state
function by a mathematical proof and we extract the conclusion’s nominalistic content in
a similar way, we would want that the conclusion’s nominalistic content is a logical
consequence of the assumption’s nominalistic content. If “if-thenism” were indeed
applicable, this would be the case. However, since the state function involves infinity and
is only an approximation, not an exact representation of all physical states of the system
(which must be finite if the universe is finite), and since we employ mathematical axioms
and logical rules involving infinity in the mathematical proof, there is no obvious
guarantee of this logical consequence relationship between nominalistic content of the
assumptions and that of the conclusion. It is not obvious that logical deductions in the
mathematical proof can also be translated into valid logical deductions on sentences
about physical quantities of the system (namely, about sentences with nominalistic
content only), unlike the simple cases where “if-thenism” is applicable. In other words,
there is no obvious explanation as to if and why using mathematical proofs will preserve
nominalistic truths.
Now, the idea I want to propose is that applications of advanced mathematics can
also be explained by reducing proofs in advanced mathematics to some logically simpler
(but could be much lengthier) proofs to which “if-thenism” does apply. First, there are
already some technical researches attempting to develop advanced applied mathematics
44
within finitism (Ye 2000). Finitism here means quantifier-free primitive recursive
arithmetic. Logical deductions in finitism are only applications of primitive recursive
definition schema, together with some propositional logical inferences and identity
equation substitutions. The idea is that because of such simplicity and transparency of
finitism, logical deductions within finitism can be directly translated into logical
deductions on sentences about finite real things when finitism is applied. Then, if all
practical applications of advanced mathematics can in principle be translated into
applications of finitism, we will have a plain explanation of applicability of advanced
mathematics, showing how exactly proofs in advanced mathematics in those practical
applications preserve nominalistic truths. Moreover, reducing applications of advanced
mathematics to applications of finitism will also show that references to abstract
mathematical entities can in principle be eliminated in mathematical applications.
On the technical side, so far some advanced analysis has been developed within
finitism, including basics of the theory of unbounded linear operators on Hilbert spaces,
the mathematical foundation for quantum mechanics (Ye 2000). This suggests the
conjecture that all applications of mathematics to this finite world can in principle be
translated into applications of finitism. More researches are still needed in order the get a
clearer view on this conjecture. On the other side, there are also intuitive reasons
supporting such a conjecture. For example, we use Riemann surfaces to model large-scale
space-time structures while we know that the microscopic structure of space-time is quite
radically different from the microscopic structure of a Riemann surface, which is
absolutely smooth (namely, infinitely differentiable). Now, if we draw a consequence
from our assumptions about space-time expressed as conditions on a Riemann surface,
45
and if some conditions such as continuity, differentiability, or compactness expressed in a
non-constructive format are strictly logically indispensable in our proof, we have good
reason to suspect that the consequence we draw is physically meaningless, because
conditions such as continuity, differentiability should not be taken too literally. They are
conditions that we assume exactly to gloss over microscopic details. Physicists will
quickly abandon mathematical consequences that appear to depend too literally on the
idealized conditions in their mathematical models. Now, if conditions like continuity,
differentiability or other conditions that appear to commit to infinity should not be taken
too literally and therefore should not be strictly logically indispensable in mathematical
proofs drawing physically meaningful consequences, we have intuitive reason to believe
that such proofs should be essentially constructive and even finitistic. That is, we have
intuitive reasons to believe that they can in principle be translated into proofs in finitism.
This provides some intuitive support for the conjecture of finitism.
If an application of advanced mathematics, for instance, some proof regarding a
differentiable function modeling the mass distribution of some fluids, can in principle be
reduced to an application of a proof in finitism, then an explanation of applicability may
go like this. (This is only a sketch of the basic ideas. Details will be given in a
forthcoming paper.) Scientists manipulate abstract mathematical concepts such as REAL
NUMBERS, FUNCTIONS,
and DIFFERENTIABILITY and so on as mental particulars in their
minds. Such concepts are in the end realized as neural circuitries, just like any other
concepts as mental particulars. Such concepts may origin from metaphorical mappings
from bodily-grounded sensory-motor concepts, as Lakoff and Núñez (2000) suggest, and
they do not directly represent anything in the external reality. Objective meaning and
46
successful communication is guaranteed by the fact that human beings share similar
intelligent capabilities and environments, including similar mental capability in forming
(metaphorical) abstract concepts, not by existence of some abstract concepts external to
our brains.
On the other side, such concepts are indeed related to the external physical reality in
some other ways. First, phenomenally they ‘look’ similar with our direct representations
of some real physical things to our consciousness, although real physical things are
actually finite and discrete. For instance, the mass distribution of real fluids ‘looks’
continuous. Such phenomenal resemblance is certainly not enough as scientifically solid
reason for successes in modeling the mechanical motions of fluids by continuous mass
distribution models, although it may indeed be the major psychological reason for
considering or discovering such a successful way of modeling. Realists actually argue
that since such a continuous model leads to true physics descriptions of motions of fluids,
our pure mathematical statements about the model must also be literally true
(notwithstanding the gap between the continuous models and discrete and finite physical
reality). However, if we research into the logic of such applications more carefully, we
will see that what is really critical for applicability is not the alleged literal truth about the
infinite models in pure mathematics. It is that when physicists draw mathematical
consequences from the assumptions about the models, they carefully not to stretch the
idealized assumptions too much, or not to take conditions such as continuity,
differentiability too literally, and not to draw physically meaningless consequences,
guided by their physics intuitions. For instance, they won’t draw the consequence that
fluids are therefore not composed of discrete atoms. In other words, scientists follow
47
some intuitive restrictions in their mathematical proofs so that the consequences they
draw are physically meaningful. This is another way how physicists’ manipulations of
mathematical concepts in their minds are truly related to the external physical reality.
From the logical point of view, we contend that at least one necessary restriction is
that the proof is in principle reducible into a finitistic proof. With the characterization of
finitism as quantifier-free primitive recursive arithmetic, the way to go beyond finitism is
induction on quantified statements, which commits to infinity if interpreted literally. On
the other side, if a proof of a quantified statement of the form x(x) x(x) (with, 
quantifier-free) can be reduced to a proof in finitism, it will become a proof of a
quantifier-free statement of the form y< f(x)(y)(x), where f is a concretely
constructed primitive recursive function (Ye 2000). This means that in a finitistic proof
the universal quantification in the antecedent x(x) is not taken literally, or that the
antecedent needs not be literally true for all numbers in order that the consequent is true
for some given number x. That seems to be how we really use conditions such as
continuity in applications. That is, intuitively we only need the fluids to be ‘sufficiently
continuous’, not literally continuous. (A continuity assumption can be expressed in the
format x(x) if the “rate of continuity” is known, and then y< f(x)(y) will mean “it is
‘sufficiently continuous’ to some degree depending on x.”) Therefore, there seems to be a
close connection between not committing to idealized assumptions (with infinity) too
literally and being finitistic.
Now, if a proof regarding the continuous mass distribution function is reduced to a
proof within finitism, the actual assumptions used in the finitistic proof will not be literal
continuity and could be literally true when translated into statements about discrete
48
particles that constitute the fluids. Similarly, the finitistic proof could itself be translated
into valid logical deductions with true premises on statements about concrete things, the
discrete particles that constitute the fluids. This way, it shows how exactly a
mathematical proof in advanced infinite mathematics helps to derive true statements
about finite real things from true premises about finite real things. It also shows how our
abstract mathematical concepts as mental particulars and our manipulations of them
following axioms and rules in classical mathematics are related to external finite real
things so as to make them applicable. This will be an explanation of how the real
instruments in mathematical applications, namely, scientists’ minds and their
manipulations of their mathematical concepts as mental particulars, work to retrieve truth
statements about finite real things from true premises about finite real things.
Here I must emphasize again that this is not to suggest replacing classical
mathematics by finitism in applied mathematics. The translation from applications of
classical mathematics to applications of finitism will certainly make mathematical proofs
and calculations extremely lengthy and tedious, but the idea is exactly that scientists
ingeniously imagine infinite mathematical entities and use them to construct greatly
simplified models to approximately simulate finite and discrete (but very complex) real
things, and that logicians can perhaps go there and figure out how exactly scientists’
ingenious inventions work. It is to explain how our proofs in infinite mathematics are
relevant to truths about finite things in the real world, not to suggest abandoning our
superior inventions.
This perhaps also refutes realists’ attempt to derive realism from applicability, for
what really matters in applications is not simply the literal truth about the alleged infinite
49
mathematical entities. It is much more complex than that. It is scientists’ ingenuous
inventions of relevant concepts and their ingenuous proofs and constructions that, on the
one side, greatly simplifies presentations of their theories and their calculations, but on
the other side, also assures that they do not take idealizations too literally. It is the
cleverly designed concepts (such as continuity for glossing over microscopic details) and
the cleverly restrained proofs (in order not to draw physically meaningless consequences)
that really matter. It is scientists’ ingenious manipulations of stuff in their minds that
really matter.
Moreover, such a research should have its own value for understanding the nature of
mathematics, independent of any philosophical purposes. While scientists courageously
invent and manipulate concepts, it is logicians’ and philosophers’ job to do some more
careful logical analyses. It is perhaps unfortunate that logicians and philosophers in the
past either attempt to invent a different type of mathematics based on some
‘philosophical prejudice’, or simply claim that mathematics is literally true since it is
applicable, while what could be more fruitful is to look into the logic of applications of
the actual mathematics more carefully.
The idea that mathematics is conservative over nominalistic content of scientific
assertions in sciences has been proposed and entertained by many people before, for
instance, by Hilbert long ago and by Hartry Field and some other philosophers more
recently. Compared with Hartry Field’s well-known program to nominalize physics
(Field 1980), and compared with other programs to nominalize mathematics, what is
special about the suggestion here is that it is based on strict finitism. As I have argued in
the last section, a consistent nominalist can assign realistic readings (namely, as
50
schematic assertions about real things) only to strict finitism, because the things we know
are finite. Moreover, from the technical point of view, only a reduction to finitism can
give us a really logically plain picture of how exactly mathematics is conservative over
nominalistic content about finite real things, that is, how mathematics helps to draw
nominalistic logical consequences about finite concrete things from nominalistic
premises about finite concrete things. Compared with Hilbert’s attempt to prove
conservativeness by meta-mathematical methods, here we only try a piece by piece
approach, focusing on applied mathematics and showing that it can actually be reduced to
finitism in itself. We know that Hilbert’s wholesale solution does not work, because
consistency cannot be proved. As a matter of fact, a wholesale approach is not likely to
be able to reveal how infinite mathematics really works in individual cases for deriving
truths about finite real things, because that is likely to depend on different types of
mathematical modeling on different types of subjects.
Finally, there are certainly beliefs about concrete things coming from mathematics
but cannot be derived from finitism. For example, designing a computer simulating
logical deductions in the ZFC axiomatic system, we may believe that the computer will
not output a contradiction, for instance, “0=1”. From a naturalist’s perspective, such
beliefs are actually inductive in nature. That is, after a long period of practices in
imagining sets and following more and more rigorous language rules in reasoning and
communicating about our imaginations, starting from Cantor, we become more and more
familiar with our concept of sets as mental particulars in our brains, and we come to
believe that we will not get paradoxes anymore. This is an inductive belief just like any
other inductive beliefs. It is a belief that we reach by observing (or reflecting upon) our
51
own mental activities in imagining things and reasoning about them in our imaginations.
It is an illusion that realism has a proof of consistency, for either that has to resort to
stronger axioms or it has to assume that our minds have some occult faculty for directly
perceiving infinitely many sets to constitute a model. Both certainly beg the question. For
a pragmatist realist like Quine, consistency can only be an inductive belief. However, it
seems that this particular belief comes more from our practices in imagining things and
playing with them in our heads, rather than from applications of mathematics, which so
far, logically speaking, has used only an extremely limited part of our imaginations in set
theory, if the conjecture of finitism mentioned above is correct. It seems that this point is
ignored by such pragmatist realists. As a matter fact, if one seriously recognizes our
superior imagination capabilities and our ability to associate our imaginations with
external real things, one will see that applications of mathematics do not necessarily lead
to realism about the alleged abstract entities.
This is only a brief summary of some ideas on how to accomplish those tasks for
anti-realism in philosophy of mathematics. Many ideas are sketchy and details are still
missing. But I hope this is already enough to attract some people, to convince some
people that this is perhaps a feasible program and could be more fruitful as a research
direction for philosophy of mathematics. In particular, for those naturalistically inclined
philosophers and for those who are averse to pure metaphysical speculations, this is
perhaps the natural way to do philosophy of mathematics.
And my last comment (which is perhaps nothing original) is that the Quinean
naturalism actually starts with a Mind wrapped by Language, alienated to nature, and
trying to catch nature through Language. It is quite different from our scientific image of
52
minds, according to which language is a natural phenomenon and it evolved among some
primates only recently. Therefore, Quinean naturalism is still a Cartesian solipsist First
Philosophy, trying to construct the world starting from a Mind out of the natural world. If
we truly take our scientific knowledge as our starting point, we will not talk about (our
Minds’) “posits” or “ontological commitments”, which is certainly a type of metaphysical
locution sounding very odd to working scientists. We will just talk about how our brains
ingeniously form concepts of real things that we see and touch in the world, and how our
brains even more ingeniously form abstract concepts and associate them with various
types of real things in very flexible ways extremely cleverly. We will look at our
mathematical cognitive activities as true phenomena in the natural world. We will still be
able to pretend taking a solipsist position and to conduct introspections, but it will only be
a supplementary method for understanding ourselves.
ACKNOWLEDGEMENTS
The research for this paper is financed by Peking University and by Chinese
National Social Science Foundation (grant number 05BZX049). My serious thinking
about philosophy of mathematics started from my graduate study at Princeton many years
ago. I am deeply indebted to Princeton University and my advisors John Burgess and
Paul Benacerraf for the scholarship assistance and for all the helps and encouragements
they offered, without which this paper would not have existed. An earlier version of this
paper was presented at Shanghai Conference on Philosophy of Mathematics and at
Beijing Conference on Analytic Philosophy, Philosophy of Science and Logic, both in
53
May 2005. I would like to thank participants for the comments that help to clarify a few
points.
NOTES
1
Burgess uses the name “anti-anti-realism” for his position and he favors Carnap more than
Quine’s more strongly realistic position. So, if I understand correctly, he is more concerned with
the apparent problems in current anti-realistic approaches and more concerned with respecting
mathematicians’ and scientists’ understandings and judgments, than with defending realism per
se. It seems to me that unfortunately the power of Burgess’ criticism on anti-realism was not well
recognized among philosophers. This paper is partially an attempt to spell out the details and
implications of that objection and to explore what it really means for anti-realism. It shows that
defending anti-realism is far from trivial, although I still believe that anti-realism is coherent and
is a more fruitful philosophical direction to work on.
2
Papineau (1993) does take Hartry Field’s fictionalism (Field 1980) as the basis in his physicalist
account for mathematics, and thus it assumes an objective infinity. This will be an inconsonant
chord in physicalism, since current physics theories do not commit to infinity of our universe.
The kind of naturalism exposed here is perhaps more consistent with the fundamental spirit of
physicalism.
3
I leave it open if there are the so-called qualia, or subjective feelings and so on. Anyhow they
are not abstract objects.
4
Well, philosophers who indulge in metaphysical imaginations too much can easily forget such
distinctions and may either put imaginary things on a par with real things in this universe or even
contend that imaginary things are realer than real things in this physical universe.
54
REFERENCES
Benacerraf, P.: 1965, ‘What numbers could not be’, Philosophical Review 74, 47-73.
Benacerraf, P.: 1973, ‘Mathematical truth’, Journal of Philosophy 70, 661-679.
Burgess, J. P.: 2004, ‘Mathematics and Bleak House’, Philosophia Mathematica (3) no.
12, 18-36.
Croft, W. and D. A. Cruse: 2004, Cognitive Linguistics, Cambridge University Press.
Field, H.: 1980, Science without Numbers, Basil Blackwell, Oxford.
Field, H.: 1998, ‘Which Undecidable Mathematical Sentences Have Determinate Truth
Values?’ in H. G. Dales and G. Oliveri (ed.), Truth in Mathematics, Oxford University
Press.
Goodman, N. and W. V. Quine: 1947, “Steps Toward a Constructive Nominalism”,
Journal of Symbolic Logic 12, 105-122.
Hoffman, S.: 2004, ‘Kitcher, Ideal Agents, and Fictionalism’, Philosophia Mathematica
(3) 12, 3-17.
Lakoff, G. and M. Johnson: 1999, Philosophy in the Flesh, Basic Books, New York.
Lakoff, G. and R. Núñez: 2000, Where Mathematics Comes From: How the Embodied
Mind Brings Mathematics into Being, Basic Books, New York.
Maddy, P.: 2005, ‘Mathematical Existence’, Bulletin of Symbolic Logic 11, 351-376.
Papineau, D.: 1993, Philosophical Naturalism, Basil Blackwell.
Renyi, A.: 1967, Dialogues on Mathematics, Holden-Day, San Francisco.
Sober, E.: 1993, ‘Mathematics and indispensability’, Philosophical Review 102, 35-57.
Yablo, S.: 2002, ‘Abstract Objects: A Case Study’, Noûs 36, 220-240.
55
Ye, F.: 2000, Strict Constructivism and the Philosophy of Mathematics, Ph.D.
dissertation, Princeton University, Princeton, NJ.
Department of Philosophy,
Peking University,
Beijing 100871, P. R. China
E-mail: yefeng@phil.pku.edu.cn