The philosophy of mathematics is the branch
of philosophy that studies the assumptions,
foundations, and implications of mathematics.
It aims to understand the nature and methods
of mathematics, and finding out the place
of mathematics in people's lives. The logical
and structural nature of mathematics itself
makes this study both broad and unique among
its philosophical counterparts.
== Recurrent themes ==
Recurrent themes include:
What is the role of humankind in developing
mathematics?
What are the sources of mathematical subject
matter?
What is the ontological status of mathematical
entities?
What does it mean to refer to a mathematical
object?
What is the character of a mathematical proposition?
What is the relation between logic and mathematics?
What is the role of hermeneutics in mathematics?
What kinds of inquiry play a role in mathematics?
What are the objectives of mathematical inquiry?
What gives mathematics its hold on experience?
What are the human traits behind mathematics?
What is mathematical beauty?
What is the source and nature of mathematical
truth?
What is the relationship between the abstract
world of mathematics and the material universe?
== History ==
The origin of mathematics is subject to arguments
and disagreements. Whether the birth of mathematics
was a random happening or induced by necessity
duly contingent upon other subjects, say for
example physics, is still a matter of prolific
debates.Many thinkers have contributed their
ideas concerning the nature of mathematics.
Today, some philosophers of mathematics aim
to give accounts of this form of inquiry and
its products as they stand, while others emphasize
a role for themselves that goes beyond simple
interpretation to critical analysis. There
are traditions of mathematical philosophy
in both Western philosophy and Eastern philosophy.
Western philosophies of mathematics go as
far back as Pythagoras, who described the
theory "everything is mathematics" (mathematicism),
Plato, who paraphrased Pythagoras, and studied
the ontological status of mathematical objects,
and Aristotle, who studied logic and issues
related to infinity (actual versus potential).
Greek philosophy on mathematics was strongly
influenced by their study of geometry. For
example, at one time, the Greeks held the
opinion that 1 (one) was not a number, but
rather a unit of arbitrary length. A number
was defined as a multitude. Therefore, 3,
for example, represented a certain multitude
of units, and was thus not "truly" a number.
At another point, a similar argument was made
that 2 was not a number but a fundamental
notion of a pair. These views come from the
heavily geometric straight-edge-and-compass
viewpoint of the Greeks: just as lines drawn
in a geometric problem are measured in proportion
to the first arbitrarily drawn line, so too
are the numbers on a number line measured
in proportion to the arbitrary first "number"
or "one".These earlier Greek ideas of numbers
were later upended by the discovery of the
irrationality of the square root of two. Hippasus,
a disciple of Pythagoras, showed that the
diagonal of a unit square was incommensurable
with its (unit-length) edge: in other words
he proved there was no existing (rational)
number that accurately depicts the proportion
of the diagonal of the unit square to its
edge. This caused a significant re-evaluation
of Greek philosophy of mathematics. According
to legend, fellow Pythagoreans were so traumatized
by this discovery that they murdered Hippasus
to stop him from spreading his heretical idea.
Simon Stevin was one of the first in Europe
to challenge Greek ideas in the 16th century.
Beginning with Leibniz, the focus shifted
strongly to the relationship between mathematics
and logic. This perspective dominated the
philosophy of mathematics through the time
of Frege and of Russell, but was brought into
question by developments in the late 19th
and early 20th centuries.
=== Contemporary philosophy ===
A perennial issue in the philosophy of mathematics
concerns the relationship between logic and
mathematics at their joint foundations. While
20th-century philosophers continued to ask
the questions mentioned at the outset of this
article, the philosophy of mathematics in
the 20th century was characterized by a predominant
interest in formal logic, set theory, and
foundational issues.
It is a profound puzzle that on the one hand
mathematical truths seem to have a compelling
inevitability, but on the other hand the source
of their "truthfulness" remains elusive. Investigations
into this issue are known as the foundations
of mathematics program.
At the start of the 20th century, philosophers
of mathematics were already beginning to divide
into various schools of thought about all
these questions, broadly distinguished by
their pictures of mathematical epistemology
and ontology. Three schools, formalism, intuitionism,
and logicism, emerged at this time, partly
in response to the increasingly widespread
worry that mathematics as it stood, and analysis
in particular, did not live up to the standards
of certainty and rigor that had been taken
for granted. Each school addressed the issues
that came to the fore at that time, either
attempting to resolve them or claiming that
mathematics is not entitled to its status
as our most trusted knowledge.
Surprising and counter-intuitive developments
in formal logic and set theory early in the
20th century led to new questions concerning
what was traditionally called the foundations
of mathematics. As the century unfolded, the
initial focus of concern expanded to an open
exploration of the fundamental axioms of mathematics,
the axiomatic approach having been taken for
granted since the time of Euclid around 300
BCE as the natural basis for mathematics.
Notions of axiom, proposition and proof, as
well as the notion of a proposition being
true of a mathematical object (see Assignment
(mathematical logic)), were formalized, allowing
them to be treated mathematically. The Zermelo–Fraenkel
axioms for set theory were formulated which
provided a conceptual framework in which much
mathematical discourse would be interpreted.
In mathematics, as in physics, new and unexpected
ideas had arisen and significant changes were
coming. With Gödel numbering, propositions
could be interpreted as referring to themselves
or other propositions, enabling inquiry into
the consistency of mathematical theories.
This reflective critique in which the theory
under review "becomes itself the object of
a mathematical study" led Hilbert to call
such study metamathematics or proof theory.At
the middle of the century, a new mathematical
theory was created by Samuel Eilenberg and
Saunders Mac Lane, known as category theory,
and it became a new contender for the natural
language of mathematical thinking. As the
20th century progressed, however, philosophical
opinions diverged as to just how well-founded
were the questions about foundations that
were raised at the century's beginning. Hilary
Putnam summed up one common view of the situation
in the last third of the century by saying:
When philosophy discovers something wrong
with science, sometimes science has to be
changed—Russell's paradox comes to mind,
as does Berkeley's attack on the actual infinitesimal—but
more often it is philosophy that has to be
changed. I do not think that the difficulties
that philosophy finds with classical mathematics
today are genuine difficulties; and I think
that the philosophical interpretations of
mathematics that we are being offered on every
hand are wrong, and that "philosophical interpretation"
is just what mathematics doesn't need.
Philosophy of mathematics today proceeds along
several different lines of inquiry, by philosophers
of mathematics, logicians, and mathematicians,
and there are many schools of thought on the
subject. The schools are addressed separately
in the next section, and their assumptions
explained.
== Major themes ==
=== Mathematical realism ===
Mathematical realism, like realism in general,
holds that mathematical entities exist independently
of the human mind. Thus humans do not invent
mathematics, but rather discover it, and any
other intelligent beings in the universe would
presumably do the same. In this point of view,
there is really one sort of mathematics that
can be discovered; triangles, for example,
are real entities, not the creations of the
human mind.
Many working mathematicians have been mathematical
realists; they see themselves as discoverers
of naturally occurring objects. Examples include
Paul Erdős and Kurt Gödel. Gödel believed
in an objective mathematical reality that
could be perceived in a manner analogous to
sense perception. Certain principles (e.g.,
for any two objects, there is a collection
of objects consisting of precisely those two
objects) could be directly seen to be true,
but the continuum hypothesis conjecture might
prove undecidable just on the basis of such
principles. Gödel suggested that quasi-empirical
methodology could be used to provide sufficient
evidence to be able to reasonably assume such
a conjecture.
Within realism, there are distinctions depending
on what sort of existence one takes mathematical
entities to have, and how we know about them.
Major forms of mathematical realism include
Platonism.
=== Mathematical anti-realism ===
Mathematical anti-realism generally holds
that mathematical statements have truth-values,
but that they do not do so by corresponding
to a special realm of immaterial or non-empirical
entities. Major forms of mathematical anti-realism
include formalism and fictionalism.
== Contemporary schools of thought ==
=== Platonism ===
Mathematical Platonism is the form of realism
that suggests that mathematical entities are
abstract, have no spatiotemporal or causal
properties, and are eternal and unchanging.
This is often claimed to be the view most
people have of numbers. The term Platonism
is used because such a view is seen to parallel
Plato's Theory of Forms and a "World of Ideas"
(Greek: eidos (εἶδος)) described in
Plato's allegory of the cave: the everyday
world can only imperfectly approximate an
unchanging, ultimate reality. Both Plato's
cave and Platonism have meaningful, not just
superficial connections, because Plato's ideas
were preceded and probably influenced by the
hugely popular Pythagoreans of ancient Greece,
who believed that the world was, quite literally,
generated by numbers.
A major question considered in mathematical
Platonism is: Precisely where and how do the
mathematical entities exist, and how do we
know about them? Is there a world, completely
separate from our physical one, that is occupied
by the mathematical entities? How can we gain
access to this separate world and discover
truths about the entities? One proposed answer
is the Ultimate Ensemble, a theory that postulates
that all structures that exist mathematically
also exist physically in their own universe.
Kurt Gödel's Platonism postulates a special
kind of mathematical intuition that lets us
perceive mathematical objects directly. (This
view bears resemblances to many things Husserl
said about mathematics, and supports Kant's
idea that mathematics is synthetic a priori.)
Davis and Hersh have suggested in their 1999
book The Mathematical Experience that most
mathematicians act as though they are Platonists,
even though, if pressed to defend the position
carefully, they may retreat to formalism.
Full-blooded Platonism is a modern variation
of Platonism, which is in reaction to the
fact that different sets of mathematical entities
can be proven to exist depending on the axioms
and inference rules employed (for instance,
the law of the excluded middle, and the axiom
of choice). It holds that all mathematical
entities exist, however they may be provable,
even if they cannot all be derived from a
single consistent set of axioms.Set-theoretic
realism (also set-theoretic Platonism) a position
defended by Penelope Maddy, is the view that
set theory is about a single universe of sets.
This position (which is also known as naturalized
Platonism because it is a naturalized version
of mathematical Platonism) has been criticized
by Mark Balaguer on the basis of Paul Benacerraf's
epistemological problem. A similar view, termed
Platonized naturalism, was later defended
by the Stanford–Edmonton School: according
to this view, a more traditional kind of Platonism
is consistent with naturalism; the more traditional
kind of Platonism they defend is distinguished
by general principles that assert the existence
of abstract objects.
=== Mathematicism ===
Max Tegmark's mathematical universe hypothesis
(or mathematicism) goes further than Platonism
in asserting that not only do all mathematical
objects exist, but nothing else does. Tegmark's
sole postulate is: All structures that exist
mathematically also exist physically. That
is, in the sense that "in those [worlds] complex
enough to contain self-aware substructures
[they] will subjectively perceive themselves
as existing in a physically 'real' world".
=== Logicism ===
Logicism is the thesis that mathematics is
reducible to logic, and hence nothing but
a part of logic. Logicists hold that mathematics
can be known a priori, but suggest that our
knowledge of mathematics is just part of our
knowledge of logic in general, and is thus
analytic, not requiring any special faculty
of mathematical intuition. In this view, logic
is the proper foundation of mathematics, and
all mathematical statements are necessary
logical truths.
Rudolf Carnap (1931) presents the logicist
thesis in two parts:
The concepts of mathematics can be derived
from logical concepts through explicit definitions.
The theorems of mathematics can be derived
from logical axioms through purely logical
deduction.Gottlob Frege was the founder of
logicism. In his seminal Die Grundgesetze
der Arithmetik (Basic Laws of Arithmetic)
he built up arithmetic from a system of logic
with a general principle of comprehension,
which he called "Basic Law V" (for concepts
F and G, the extension of F equals the extension
of G if and only if for all objects a, Fa
equals Ga), a principle that he took to be
acceptable as part of logic.
Frege's construction was flawed. Russell discovered
that Basic Law V is inconsistent (this is
Russell's paradox). Frege abandoned his logicist
program soon after this, but it was continued
by Russell and Whitehead. They attributed
the paradox to "vicious circularity" and built
up what they called ramified type theory to
deal with it. In this system, they were eventually
able to build up much of modern mathematics
but in an altered, and excessively complex
form (for example, there were different natural
numbers in each type, and there were infinitely
many types). They also had to make several
compromises in order to develop so much of
mathematics, such as an "axiom of reducibility".
Even Russell said that this axiom did not
really belong to logic.
Modern logicists (like Bob Hale, Crispin Wright,
and perhaps others) have returned to a program
closer to Frege's. They have abandoned Basic
Law V in favor of abstraction principles such
as Hume's principle (the number of objects
falling under the concept F equals the number
of objects falling under the concept G if
and only if the extension of F and the extension
of G can be put into one-to-one correspondence).
Frege required Basic Law V to be able to give
an explicit definition of the numbers, but
all the properties of numbers can be derived
from Hume's principle. This would not have
been enough for Frege because (to paraphrase
him) it does not exclude the possibility that
the number 3 is in fact Julius Caesar. In
addition, many of the weakened principles
that they have had to adopt to replace Basic
Law V no longer seem so obviously analytic,
and thus purely logical.
=== Formalism ===
Formalism holds that mathematical statements
may be thought of as statements about the
consequences of certain string manipulation
rules. For example, in the "game" of Euclidean
geometry (which is seen as consisting of some
strings called "axioms", and some "rules of
inference" to generate new strings from given
ones), one can prove that the Pythagorean
theorem holds (that is, one can generate the
string corresponding to the Pythagorean theorem).
According to formalism, mathematical truths
are not about numbers and sets and triangles
and the like—in fact, they are not "about"
anything at all.
Another version of formalism is often known
as deductivism. In deductivism, the Pythagorean
theorem is not an absolute truth, but a relative
one: if one assigns meaning to the strings
in such a way that the rules of the game become
true (i.e., true statements are assigned to
the axioms and the rules of inference are
truth-preserving), then one must accept the
theorem, or, rather, the interpretation one
has given it must be a true statement. The
same is held to be true for all other mathematical
statements. Thus, formalism need not mean
that mathematics is nothing more than a meaningless
symbolic game. It is usually hoped that there
exists some interpretation in which the rules
of the game hold. (Compare this position to
structuralism.) But it does allow the working
mathematician to continue in his or her work
and leave such problems to the philosopher
or scientist. Many formalists would say that
in practice, the axiom systems to be studied
will be suggested by the demands of science
or other areas of mathematics.
A major early proponent of formalism was David
Hilbert, whose program was intended to be
a complete and consistent axiomatization of
all of mathematics. Hilbert aimed to show
the consistency of mathematical systems from
the assumption that the "finitary arithmetic"
(a subsystem of the usual arithmetic of the
positive integers, chosen to be philosophically
uncontroversial) was consistent. Hilbert's
goals of creating a system of mathematics
that is both complete and consistent were
seriously undermined by the second of Gödel's
incompleteness theorems, which states that
sufficiently expressive consistent axiom systems
can never prove their own consistency. Since
any such axiom system would contain the finitary
arithmetic as a subsystem, Gödel's theorem
implied that it would be impossible to prove
the system's consistency relative to that
(since it would then prove its own consistency,
which Gödel had shown was impossible). Thus,
in order to show that any axiomatic system
of mathematics is in fact consistent, one
needs to first assume the consistency of a
system of mathematics that is in a sense stronger
than the system to be proven consistent.
Hilbert was initially a deductivist, but,
as may be clear from above, he considered
certain metamathematical methods to yield
intrinsically meaningful results and was a
realist with respect to the finitary arithmetic.
Later, he held the opinion that there was
no other meaningful mathematics whatsoever,
regardless of interpretation.
Other formalists, such as Rudolf Carnap, Alfred
Tarski, and Haskell Curry, considered mathematics
to be the investigation of formal axiom systems.
Mathematical logicians study formal systems
but are just as often realists as they are
formalists.
Formalists are relatively tolerant and inviting
to new approaches to logic, non-standard number
systems, new set theories etc. The more games
we study, the better. However, in all three
of these examples, motivation is drawn from
existing mathematical or philosophical concerns.
The "games" are usually not arbitrary.
The main critique of formalism is that the
actual mathematical ideas that occupy mathematicians
are far removed from the string manipulation
games mentioned above. Formalism is thus silent
on the question of which axiom systems ought
to be studied, as none is more meaningful
than another from a formalistic point of view.
Recently, some formalist mathematicians have
proposed that all of our formal mathematical
knowledge should be systematically encoded
in computer-readable formats, so as to facilitate
automated proof checking of mathematical proofs
and the use of interactive theorem proving
in the development of mathematical theories
and computer software. Because of their close
connection with computer science, this idea
is also advocated by mathematical intuitionists
and constructivists in the "computability"
tradition—see QED project for a general
overview.
=== Conventionalism ===
The French mathematician Henri Poincaré was
among the first to articulate a conventionalist
view. Poincaré's use of non-Euclidean geometries
in his work on differential equations convinced
him that Euclidean geometry should not be
regarded as a priori truth. He held that axioms
in geometry should be chosen for the results
they produce, not for their apparent coherence
with human intuitions about the physical world.
=== Intuitionism ===
In mathematics, intuitionism is a program
of methodological reform whose motto is that
"there are no non-experienced mathematical
truths" (L. E. J. Brouwer). From this springboard,
intuitionists seek to reconstruct what they
consider to be the corrigible portion of mathematics
in accordance with Kantian concepts of being,
becoming, intuition, and knowledge. Brouwer,
the founder of the movement, held that mathematical
objects arise from the a priori forms of the
volitions that inform the perception of empirical
objects.A major force behind intuitionism
was L. E. J. Brouwer, who rejected the usefulness
of formalized logic of any sort for mathematics.
His student Arend Heyting postulated an intuitionistic
logic, different from the classical Aristotelian
logic; this logic does not contain the law
of the excluded middle and therefore frowns
upon proofs by contradiction. The axiom of
choice is also rejected in most intuitionistic
set theories, though in some versions it is
accepted.
In intuitionism, the term "explicit construction"
is not cleanly defined, and that has led to
criticisms. Attempts have been made to use
the concepts of Turing machine or computable
function to fill this gap, leading to the
claim that only questions regarding the behavior
of finite algorithms are meaningful and should
be investigated in mathematics. This has led
to the study of the computable numbers, first
introduced by Alan Turing. Not surprisingly,
then, this approach to mathematics is sometimes
associated with theoretical computer science.
==== Constructivism ====
Like intuitionism, constructivism involves
the regulative principle that only mathematical
entities which can be explicitly constructed
in a certain sense should be admitted to mathematical
discourse. In this view, mathematics is an
exercise of the human intuition, not a game
played with meaningless symbols. Instead,
it is about entities that we can create directly
through mental activity. In addition, some
adherents of these schools reject non-constructive
proofs, such as a proof by contradiction.
Important work was done by Errett Bishop,
who managed to prove versions of the most
important theorems in real analysis as constructive
analysis in his 1967 Foundations of Constructive
Analysis.
==== Finitism ====
Finitism is an extreme form of constructivism,
according to which a mathematical object does
not exist unless it can be constructed from
natural numbers in a finite number of steps.
In her book Philosophy of Set Theory, Mary
Tiles characterized those who allow countably
infinite objects as classical finitists, and
those who deny even countably infinite objects
as strict finitists.
The most famous proponent of finitism was
Leopold Kronecker, who said:
God created the natural numbers, all else
is the work of man.
Ultrafinitism is an even more extreme version
of finitism, which rejects not only infinities
but finite quantities that cannot feasibly
be constructed with available resources. Another
variant of finitism is Euclidean arithmetic,
a system developed by John Penn Mayberry in
his book The Foundations of Mathematics in
the Theory of Sets. Mayberry's system is Aristotelian
in general inspiration and, despite his strong
rejection of any role for operationalism or
feasibility in the foundations of mathematics,
comes to somewhat similar conclusions, such
as, for instance, that super-exponentiation
is not a legitimate finitary function.
=== Structuralism ===
Structuralism is a position holding that mathematical
theories describe structures, and that mathematical
objects are exhaustively defined by their
places in such structures, consequently having
no intrinsic properties. For instance, it
would maintain that all that needs to be known
about the number 1 is that it is the first
whole number after 0. Likewise all the other
whole numbers are defined by their places
in a structure, the number line. Other examples
of mathematical objects might include lines
and planes in geometry, or elements and operations
in abstract algebra.
Structuralism is an epistemologically realistic
view in that it holds that mathematical statements
have an objective truth value. However, its
central claim only relates to what kind of
entity a mathematical object is, not to what
kind of existence mathematical objects or
structures have (not, in other words, to their
ontology). The kind of existence mathematical
objects have would clearly be dependent on
that of the structures in which they are embedded;
different sub-varieties of structuralism make
different ontological claims in this regard.The
ante rem structuralism ("before the thing")
has a similar ontology to Platonism. Structures
are held to have a real but abstract and immaterial
existence. As such, it faces the standard
epistemological problem of explaining the
interaction between such abstract structures
and flesh-and-blood mathematicians (see Benacerraf's
identification problem).
The in re structuralism ("in the thing") is
the equivalent of Aristotelean realism. Structures
are held to exist inasmuch as some concrete
system exemplifies them. This incurs the usual
issues that some perfectly legitimate structures
might accidentally happen not to exist, and
that a finite physical world might not be
"big" enough to accommodate some otherwise
legitimate structures.
The post rem structuralism ("after the thing")
is anti-realist about structures in a way
that parallels nominalism. Like nominalism,
the post rem approach denies the existence
of abstract mathematical objects with properties
other than their place in a relational structure.
According to this view mathematical systems
exist, and have structural features in common.
If something is true of a structure, it will
be true of all systems exemplifying the structure.
However, it is merely instrumental to talk
of structures being "held in common" between
systems: they in fact have no independent
existence.
=== Embodied mind theories ===
Embodied mind theories hold that mathematical
thought is a natural outgrowth of the human
cognitive apparatus which finds itself in
our physical universe. For example, the abstract
concept of number springs from the experience
of counting discrete objects. It is held that
mathematics is not universal and does not
exist in any real sense, other than in human
brains. Humans construct, but do not discover,
mathematics.
With this view, the physical universe can
thus be seen as the ultimate foundation of
mathematics: it guided the evolution of the
brain and later determined which questions
this brain would find worthy of investigation.
However, the human mind has no special claim
on reality or approaches to it built out of
math. If such constructs as Euler's identity
are true then they are true as a map of the
human mind and cognition.
Embodied mind theorists thus explain the effectiveness
of mathematics—mathematics was constructed
by the brain in order to be effective in this
universe.
The most accessible, famous, and infamous
treatment of this perspective is Where Mathematics
Comes From, by George Lakoff and Rafael E.
Núñez. In addition, mathematician Keith
Devlin has investigated similar concepts with
his book The Math Instinct, as has neuroscientist
Stanislas Dehaene with his book The Number
Sense. For more on the philosophical ideas
that inspired this perspective, see cognitive
science of mathematics.
==== Aristotelian realism ====
Aristotelian realism holds that mathematics
studies properties such as symmetry, continuity
and order that can be literally realized in
the physical world (or in any other world
there might be). It contrasts with Platonism
in holding that the objects of mathematics,
such as numbers, do not exist in an "abstract"
world but can be physically realized. For
example, the number 4 is realized in the relation
between a heap of parrots and the universal
"being a parrot" that divides the heap into
so many parrots. Aristotelian realism is defended
by James Franklin and the Sydney School in
the philosophy of mathematics and is close
to the view of Penelope Maddy that when an
egg carton is opened, a set of three eggs
is perceived (that is, a mathematical entity
realized in the physical world). A problem
for Aristotelian realism is what account to
give of higher infinities, which may not be
realizable in the physical world.
The Euclidean arithmetic developed by John
Penn Mayberry in his book The Foundations
of Mathematics in the Theory of Sets. also
falls into the Aristotelian realist tradition.
Mayberry, following Euclid, considers numbers
to be simply "definite multitudes of units"
realized in nature—such as "the members
of the London Symphony Orchestra" or "the
trees in Birnam wood". Whether or not there
are definite multitudes of units for which
Euclid's Common Notion 5 (the Whole is greater
than the Part) fails and which would consequently
be reckoned as infinite is for Mayberry essentially
a question about Nature and does not entail
any transcendental suppositions.
==== Psychologism ====
Psychologism in the philosophy of mathematics
is the position that mathematical concepts
and/or truths are grounded in, derived from
or explained by psychological facts (or laws).
John Stuart Mill seems to have been an advocate
of a type of logical psychologism, as were
many 19th-century German logicians such as
Sigwart and Erdmann as well as a number of
psychologists, past and present: for example,
Gustave Le Bon. Psychologism was famously
criticized by Frege in his The Foundations
of Arithmetic, and many of his works and essays,
including his review of Husserl's Philosophy
of Arithmetic. Edmund Husserl, in the first
volume of his Logical Investigations, called
"The Prolegomena of Pure Logic", criticized
psychologism thoroughly and sought to distance
himself from it. The "Prolegomena" is considered
a more concise, fair, and thorough refutation
of psychologism than the criticisms made by
Frege, and also it is considered today by
many as being a memorable refutation for its
decisive blow to psychologism. Psychologism
was also criticized by Charles Sanders Peirce
and Maurice Merleau-Ponty.
==== Empiricism ====
Mathematical empiricism is a form of realism
that denies that mathematics can be known
a priori at all. It says that we discover
mathematical facts by empirical research,
just like facts in any of the other sciences.
It is not one of the classical three positions
advocated in the early 20th century, but primarily
arose in the middle of the century. However,
an important early proponent of a view like
this was John Stuart Mill. Mill's view was
widely criticized, because, according to critics,
such as A.J. Ayer, it makes statements like
"2 + 2 = 4" come out as uncertain, contingent
truths, which we can only learn by observing
instances of two pairs coming together and
forming a quartet.
Contemporary mathematical empiricism, formulated
by W. V. O. Quine and Hilary Putnam, is primarily
supported by the indispensability argument:
mathematics is indispensable to all empirical
sciences, and if we want to believe in the
reality of the phenomena described by the
sciences, we ought also believe in the reality
of those entities required for this description.
That is, since physics needs to talk about
electrons to say why light bulbs behave as
they do, then electrons must exist. Since
physics needs to talk about numbers in offering
any of its explanations, then numbers must
exist. In keeping with Quine and Putnam's
overall philosophies, this is a naturalistic
argument. It argues for the existence of mathematical
entities as the best explanation for experience,
thus stripping mathematics of being distinct
from the other sciences.
Putnam strongly rejected the term "Platonist"
as implying an over-specific ontology that
was not necessary to mathematical practice
in any real sense. He advocated a form of
"pure realism" that rejected mystical notions
of truth and accepted much quasi-empiricism
in mathematics. This grew from the increasingly
popular assertion in the late 20th century
that no one foundation of mathematics could
be ever proven to exist. It is also sometimes
called "postmodernism in mathematics" although
that term is considered overloaded by some
and insulting by others. Quasi-empiricism
argues that in doing their research, mathematicians
test hypotheses as well as prove theorems.
A mathematical argument can transmit falsity
from the conclusion to the premises just as
well as it can transmit truth from the premises
to the conclusion. Putnam has argued that
any theory of mathematical realism would include
quasi-empirical methods. He proposed that
an alien species doing mathematics might well
rely on quasi-empirical methods primarily,
being willing often to forgo rigorous and
axiomatic proofs, and still be doing mathematics—at
perhaps a somewhat greater risk of failure
of their calculations. He gave a detailed
argument for this in New Directions. Quasi-empiricism
was also developed by Imre Lakatos.
The most important criticism of empirical
views of mathematics is approximately the
same as that raised against Mill. If mathematics
is just as empirical as the other sciences,
then this suggests that its results are just
as fallible as theirs, and just as contingent.
In Mill's case the empirical justification
comes directly, while in Quine's case it comes
indirectly, through the coherence of our scientific
theory as a whole, i.e. consilience after
E.O. Wilson. Quine suggests that mathematics
seems completely certain because the role
it plays in our web of belief is extraordinarily
central, and that it would be extremely difficult
for us to revise it, though not impossible.
For a philosophy of mathematics that attempts
to overcome some of the shortcomings of Quine
and Gödel's approaches by taking aspects
of each see Penelope Maddy's Realism in Mathematics.
Another example of a realist theory is the
embodied mind theory.
For experimental evidence suggesting that
human infants can do elementary arithmetic,
see Brian Butterworth.
=== Fictionalism ===
Mathematical fictionalism was brought to fame
in 1980 when Hartry Field published Science
Without Numbers, which rejected and in fact
reversed Quine's indispensability argument.
Where Quine suggested that mathematics was
indispensable for our best scientific theories,
and therefore should be accepted as a body
of truths talking about independently existing
entities, Field suggested that mathematics
was dispensable, and therefore should be considered
as a body of falsehoods not talking about
anything real. He did this by giving a complete
axiomatization of Newtonian mechanics with
no reference to numbers or functions at all.
He started with the "betweenness" of Hilbert's
axioms to characterize space without coordinatizing
it, and then added extra relations between
points to do the work formerly done by vector
fields. Hilbert's geometry is mathematical,
because it talks about abstract points, but
in Field's theory, these points are the concrete
points of physical space, so no special mathematical
objects at all are needed.
Having shown how to do science without using
numbers, Field proceeded to rehabilitate mathematics
as a kind of useful fiction. He showed that
mathematical physics is a conservative extension
of his non-mathematical physics (that is,
every physical fact provable in mathematical
physics is already provable from Field's system),
so that mathematics is a reliable process
whose physical applications are all true,
even though its own statements are false.
Thus, when doing mathematics, we can see ourselves
as telling a sort of story, talking as if
numbers existed. For Field, a statement like
"2 + 2 = 4" is just as fictitious as "Sherlock
Holmes lived at 221B Baker Street"—but both
are true according to the relevant fictions.
By this account, there are no metaphysical
or epistemological problems special to mathematics.
The only worries left are the general worries
about non-mathematical physics, and about
fiction in general. Field's approach has been
very influential, but is widely rejected.
This is in part because of the requirement
of strong fragments of second-order logic
to carry out his reduction, and because the
statement of conservativity seems to require
quantification over abstract models or deductions.
=== Social constructivism ===
Social constructivism see mathematics primarily
as a social construct, as a product of culture,
subject to correction and change. Like the
other sciences, mathematics is viewed as an
empirical endeavor whose results are constantly
evaluated and may be discarded. However, while
on an empiricist view the evaluation is some
sort of comparison with "reality", social
constructivists emphasize that the direction
of mathematical research is dictated by the
fashions of the social group performing it
or by the needs of the society financing it.
However, although such external forces may
change the direction of some mathematical
research, there are strong internal constraints—the
mathematical traditions, methods, problems,
meanings and values into which mathematicians
are enculturated—that work to conserve the
historically-defined discipline.
This runs counter to the traditional beliefs
of working mathematicians, that mathematics
is somehow pure or objective. But social constructivists
argue that mathematics is in fact grounded
by much uncertainty: as mathematical practice
evolves, the status of previous mathematics
is cast into doubt, and is corrected to the
degree it is required or desired by the current
mathematical community. This can be seen in
the development of analysis from reexamination
of the calculus of Leibniz and Newton. They
argue further that finished mathematics is
often accorded too much status, and folk mathematics
not enough, due to an overemphasis on axiomatic
proof and peer review as practices.
The social nature of mathematics is highlighted
in its subcultures. Major discoveries can
be made in one branch of mathematics and be
relevant to another, yet the relationship
goes undiscovered for lack of social contact
between mathematicians. Social constructivists
argue each speciality forms its own epistemic
community and often has great difficulty communicating,
or motivating the investigation of unifying
conjectures that might relate different areas
of mathematics. Social constructivists see
the process of "doing mathematics" as actually
creating the meaning, while social realists
see a deficiency either of human capacity
to abstractify, or of human's cognitive bias,
or of mathematicians' collective intelligence
as preventing the comprehension of a real
universe of mathematical objects. Social constructivists
sometimes reject the search for foundations
of mathematics as bound to fail, as pointless
or even meaningless.
Contributions to this school have been made
by Imre Lakatos and Thomas Tymoczko, although
it is not clear that either would endorse
the title. More recently Paul Ernest has explicitly
formulated a social constructivist philosophy
of mathematics. Some consider the work of
Paul Erdős as a whole to have advanced this
view (although he personally rejected it)
because of his uniquely broad collaborations,
which prompted others to see and study "mathematics
as a social activity", e.g., via the Erdős
number. Reuben Hersh has also promoted the
social view of mathematics, calling it a "humanistic"
approach, similar to but not quite the same
as that associated with Alvin White; one of
Hersh's co-authors, Philip J. Davis, has expressed
sympathy for the social view as well.
A criticism of this approach is that it is
trivial, based on the trivial observation
that mathematics is a human activity. To observe
that rigorous proof comes only after unrigorous
conjecture, experimentation and speculation
is true, but it is trivial and no-one would
deny this. So it's a bit of a stretch to characterize
a philosophy of mathematics in this way, on
something trivially true. The calculus of
Leibniz and Newton was reexamined by mathematicians
such as Weierstrass in order to rigorously
prove the theorems thereof. There is nothing
special or interesting about this, as it fits
in with the more general trend of unrigorous
ideas which are later made rigorous. There
needs to be a clear distinction between the
objects of study of mathematics and the study
of the objects of study of mathematics. The
former doesn't seem to change a great deal;
the latter is forever in flux. The latter
is what the social theory is about, and the
former is what Platonism et al. are about.
However, this criticism is rejected by supporters
of the social constructivist perspective because
it misses the point that the very objects
of mathematics are social constructs. These
objects, it asserts, are primarily semiotic
objects existing in the sphere of human culture,
sustained by social practices (after Wittgenstein)
that utilize physically embodied signs and
give rise to intrapersonal (mental) constructs.
Social constructivists view the reification
of the sphere of human culture into a Platonic
realm, or some other heaven-like domain of
existence beyond the physical world, a long-standing
category error.
=== Beyond the traditional schools ===
==== Unreasonable effectiveness ====
Rather than focus on narrow debates about
the true nature of mathematical truth, or
even on practices unique to mathematicians
such as the proof, a growing movement from
the 1960s to the 1990s began to question the
idea of seeking foundations or finding any
one right answer to why mathematics works.
The starting point for this was Eugene Wigner's
famous 1960 paper The Unreasonable Effectiveness
of Mathematics in the Natural Sciences, in
which he argued that the happy coincidence
of mathematics and physics being so well matched
seemed to be unreasonable and hard to explain.
==== Popper's two senses ====
Realist and constructivist theories are normally
taken to be contraries. However, Karl Popper
argued that a number statement such as "2
apples + 2 apples = 4 apples" can be taken
in two senses. In one sense it is irrefutable
and logically true. In the second sense it
is factually true and falsifiable. Another
way of putting this is to say that a single
number statement can express two propositions:
one of which can be explained on constructivist
lines; the other on realist lines.
==== Philosophy of language ====
Innovations in the philosophy of language
during the 20th century renewed interest in
whether mathematics is, as is often said,
the language of science. Although some mathematicians
and philosophers would accept the statement
"mathematics is a language", linguists believe
that the implications of such a statement
must be considered. For example, the tools
of linguistics are not generally applied to
the symbol systems of mathematics, that is,
mathematics is studied in a markedly different
way from other languages. If mathematics is
a language, it is a different type of language
from natural languages. Indeed, because of
the need for clarity and specificity, the
language of mathematics is far more constrained
than natural languages studied by linguists.
However, the methods developed by Frege and
Tarski for the study of mathematical language
have been extended greatly by Tarski's student
Richard Montague and other linguists working
in formal semantics to show that the distinction
between mathematical language and natural
language may not be as great as it seems.
Mohan Ganesalingam has analysed mathematical
language using tools from formal linguistics.
Ganesalingam notes that some features of natural
language are not necessary when analysing
mathematical language (such as tense), but
many of the same analytical tools can be used
(such as context-free grammars). One important
difference is that mathematical objects have
clearly defined types, which can be explicitly
defined in a text: "Effectively, we are allowed
to introduce a word in one part of a sentence,
and declare its part of speech in another;
and this operation has no analogue in natural
language."
== Arguments ==
=== Indispensability argument for realism
===
This argument, associated with Willard Quine
and Hilary Putnam, is considered by Stephen
Yablo to be one of the most challenging arguments
in favor of the acceptance of the existence
of abstract mathematical entities, such as
numbers and sets. The form of the argument
is as follows.
One must have ontological commitments to all
entities that are indispensable to the best
scientific theories, and to those entities
only (commonly referred to as "all and only").
Mathematical entities are indispensable to
the best scientific theories. Therefore,
One must have ontological commitments to mathematical
entities.The justification for the first premise
is the most controversial. Both Putnam and
Quine invoke naturalism to justify the exclusion
of all non-scientific entities, and hence
to defend the "only" part of "all and only".
The assertion that "all" entities postulated
in scientific theories, including numbers,
should be accepted as real is justified by
confirmation holism. Since theories are not
confirmed in a piecemeal fashion, but as a
whole, there is no justification for excluding
any of the entities referred to in well-confirmed
theories. This puts the nominalist who wishes
to exclude the existence of sets and non-Euclidean
geometry, but to include the existence of
quarks and other undetectable entities of
physics, for example, in a difficult position.
=== Epistemic argument against realism ===
The anti-realist "epistemic argument" against
Platonism has been made by Paul Benacerraf
and Hartry Field. Platonism posits that mathematical
objects are abstract entities. By general
agreement, abstract entities cannot interact
causally with concrete, physical entities
("the truth-values of our mathematical assertions
depend on facts involving Platonic entities
that reside in a realm outside of space-time").
Whilst our knowledge of concrete, physical
objects is based on our ability to perceive
them, and therefore to causally interact with
them, there is no parallel account of how
mathematicians come to have knowledge of abstract
objects. Another way of making the point is
that if the Platonic world were to disappear,
it would make no difference to the ability
of mathematicians to generate proofs, etc.,
which is already fully accountable in terms
of physical processes in their brains.
Field developed his views into fictionalism.
Benacerraf also developed the philosophy of
mathematical structuralism, according to which
there are no mathematical objects. Nonetheless,
some versions of structuralism are compatible
with some versions of realism.
The argument hinges on the idea that a satisfactory
naturalistic account of thought processes
in terms of brain processes can be given for
mathematical reasoning along with everything
else. One line of defense is to maintain that
this is false, so that mathematical reasoning
uses some special intuition that involves
contact with the Platonic realm. A modern
form of this argument is given by Sir Roger
Penrose.Another line of defense is to maintain
that abstract objects are relevant to mathematical
reasoning in a way that is non-causal, and
not analogous to perception. This argument
is developed by Jerrold Katz in his 2000 book
Realistic Rationalism.
A more radical defense is denial of physical
reality, i.e. the mathematical universe hypothesis.
In that case, a mathematician's knowledge
of mathematics is one mathematical object
making contact with another.
== Aesthetics ==
Many practicing mathematicians have been drawn
to their subject because of a sense of beauty
they perceive in it. One sometimes hears the
sentiment that mathematicians would like to
leave philosophy to the philosophers and get
back to mathematics—where, presumably, the
beauty lies.
In his work on the divine proportion, H.E.
Huntley relates the feeling of reading and
understanding someone else's proof of a theorem
of mathematics to that of a viewer of a masterpiece
of art—the reader of a proof has a similar
sense of exhilaration at understanding as
the original author of the proof, much as,
he argues, the viewer of a masterpiece has
a sense of exhilaration similar to the original
painter or sculptor. Indeed, one can study
mathematical and scientific writings as literature.
Philip J. Davis and Reuben Hersh have commented
that the sense of mathematical beauty is universal
amongst practicing mathematicians. By way
of example, they provide two proofs of the
irrationality of √2. The first is the traditional
proof by contradiction, ascribed to Euclid;
the second is a more direct proof involving
the fundamental theorem of arithmetic that,
they argue, gets to the heart of the issue.
Davis and Hersh argue that mathematicians
find the second proof more aesthetically appealing
because it gets closer to the nature of the
problem.
Paul Erdős was well known for his notion
of a hypothetical "Book" containing the most
elegant or beautiful mathematical proofs.
There is not universal agreement that a result
has one "most elegant" proof; Gregory Chaitin
has argued against this idea.
Philosophers have sometimes criticized mathematicians'
sense of beauty or elegance as being, at best,
vaguely stated. By the same token, however,
philosophers of mathematics have sought to
characterize what makes one proof more desirable
than another when both are logically sound.
Another aspect of aesthetics concerning mathematics
is mathematicians' views towards the possible
uses of mathematics for purposes deemed unethical
or inappropriate. The best-known exposition
of this view occurs in G.H. Hardy's book A
Mathematician's Apology, in which Hardy argues
that pure mathematics is superior in beauty
to applied mathematics precisely because it
cannot be used for war and similar ends.
== Journals ==
Philosophia Mathematica journal
The Philosophy of Mathematics Education Journal
homepage
== See also ==
=== Related works ===
=== Historical topics ===
History and philosophy of science
History of mathematics
History of philosophy
== Notes ==
== Further reading ==
Aristotle, "Prior Analytics", Hugh Tredennick
(trans.), pp. 181–531 in Aristotle, Volume
1, Loeb Classical Library, William Heinemann,
London, UK, 1938.
Benacerraf, Paul, and Putnam, Hilary (eds.,
1983), Philosophy of Mathematics, Selected
Readings, 1st edition, Prentice-Hall, Englewood
Cliffs, NJ, 1964. 2nd edition, Cambridge University
Press, Cambridge, UK, 1983.
Berkeley, George (1734), The Analyst; or,
a Discourse Addressed to an Infidel Mathematician.
Wherein It is examined whether the Object,
Principles, and Inferences of the modern Analysis
are more distinctly conceived, or more evidently
deduced, than Religious Mysteries and Points
of Faith, London & Dublin. Online text, David
R. Wilkins (ed.), Eprint.
Bourbaki, N. (1994), Elements of the History
of Mathematics, John Meldrum (trans.), Springer-Verlag,
Berlin, Germany.
Chandrasekhar, Subrahmanyan (1987), Truth
and Beauty. Aesthetics and Motivations in
Science, University of Chicago Press, Chicago,
IL.
Colyvan, Mark (2004), "Indispensability Arguments
in the Philosophy of Mathematics", Stanford
Encyclopedia of Philosophy, Edward N. Zalta
(ed.), Eprint.
Davis, Philip J. and Hersh, Reuben (1981),
The Mathematical Experience, Mariner Books,
New York, NY.
Devlin, Keith (2005), The Math Instinct: Why
You're a Mathematical Genius (Along with Lobsters,
Birds, Cats, and Dogs), Thunder's Mouth Press,
New York, NY.
Dummett, Michael (1991 a), Frege, Philosophy
of Mathematics, Harvard University Press,
Cambridge, MA.
Dummett, Michael (1991 b), Frege and Other
Philosophers, Oxford University Press, Oxford,
UK.
Dummett, Michael (1993), Origins of Analytical
Philosophy, Harvard University Press, Cambridge,
MA.
Ernest, Paul (1998), Social Constructivism
as a Philosophy of Mathematics, State University
of New York Press, Albany, NY.
George, Alexandre (ed., 1994), Mathematics
and Mind, Oxford University Press, Oxford,
UK.
Hadamard, Jacques (1949), The Psychology of
Invention in the Mathematical Field, 1st edition,
Princeton University Press, Princeton, NJ.
2nd edition, 1949. Reprinted, Dover Publications,
New York, NY, 1954.
Hardy, G.H. (1940), A Mathematician's Apology,
1st published, 1940. Reprinted, C.P. Snow
(foreword), 1967. Reprinted, Cambridge University
Press, Cambridge, UK, 1992.
Hart, W.D. (ed., 1996), The Philosophy of
Mathematics, Oxford University Press, Oxford,
UK.
Hendricks, Vincent F. and Hannes Leitgeb (eds.).
Philosophy of Mathematics: 5 Questions, New
York: Automatic Press / VIP, 2006. [1]
Huntley, H.E. (1970), The Divine Proportion:
A Study in Mathematical Beauty, Dover Publications,
New York, NY.
Irvine, A., ed (2009), The Philosophy of Mathematics,
in Handbook of the Philosophy of Science series,
North-Holland Elsevier, Amsterdam.
Klein, Jacob (1968), Greek Mathematical Thought
and the Origin of Algebra, Eva Brann (trans.),
MIT Press, Cambridge, MA, 1968. Reprinted,
Dover Publications, Mineola, NY, 1992.
Kline, Morris (1959), Mathematics and the
Physical World, Thomas Y. Crowell Company,
New York, NY, 1959. Reprinted, Dover Publications,
Mineola, NY, 1981.
Kline, Morris (1972), Mathematical Thought
from Ancient to Modern Times, Oxford University
Press, New York, NY.
König, Julius (Gyula) (1905), "Über die
Grundlagen der Mengenlehre und das Kontinuumproblem",
Mathematische Annalen 61, 156-160. Reprinted,
"On the Foundations of Set Theory and the
Continuum Problem", Stefan Bauer-Mengelberg
(trans.), pp. 145–149 in Jean van Heijenoort
(ed., 1967).
Körner, Stephan, The Philosophy of Mathematics,
An Introduction. Harper Books, 1960.
Lakoff, George, and Núñez, Rafael E. (2000),
Where Mathematics Comes From: How the Embodied
Mind Brings Mathematics into Being, Basic
Books, New York, NY.
Lakatos, Imre 1976 Proofs and Refutations:The
Logic of Mathematical Discovery (Eds) J. Worrall
& E. Zahar Cambridge University Press
Lakatos, Imre 1978 Mathematics, Science and
Epistemology: Philosophical Papers Volume
2 (Eds) J.Worrall & G.Currie Cambridge University
Press
Lakatos, Imre 1968 Problems in the Philosophy
of Mathematics North Holland
Leibniz, G.W., Logical Papers (1666–1690),
G.H.R. Parkinson (ed., trans.), Oxford University
Press, London, UK, 1966.
Maddy, Penelope (1997), Naturalism in Mathematics,
Oxford University Press, Oxford, UK.
Maziarz, Edward A., and Greenwood, Thomas
(1995), Greek Mathematical Philosophy, Barnes
and Noble Books.
Mount, Matthew, Classical Greek Mathematical
Philosophy,.
Parsons, Charles (2014). Philosophy of Mathematics
in the Twentieth Century: Selected Essays.
Cambridge, MA: Harvard University Press. ISBN
978-0-674-72806-6.
Peirce, Benjamin (1870), "Linear Associative
Algebra", § 1. See American Journal of Mathematics
4 (1881).
Peirce, C.S., Collected Papers of Charles
Sanders Peirce, vols. 1-6, Charles Hartshorne
and Paul Weiss (eds.), vols. 7-8, Arthur W.
Burks (ed.), Harvard University Press, Cambridge,
MA, 1931 – 1935, 1958. Cited as CP (volume).(paragraph).
Peirce, C.S., various pieces on mathematics
and logic, many readable online through links
at the Charles Sanders Peirce bibliography,
especially under Books authored or edited
by Peirce, published in his lifetime and the
two sections following it.
Plato, "The Republic, Volume 1", Paul Shorey
(trans.), pp. 1–535 in Plato, Volume 5,
Loeb Classical Library, William Heinemann,
London, UK, 1930.
Plato, "The Republic, Volume 2", Paul Shorey
(trans.), pp. 1–521 in Plato, Volume 6,
Loeb Classical Library, William Heinemann,
London, UK, 1935.
Resnik, Michael D. Frege and the Philosophy
of Mathematics, Cornell University, 1980.
Resnik, Michael (1997), Mathematics as a Science
of Patterns, Clarendon Press, Oxford, UK,
ISBN 978-0-19-825014-2
Robinson, Gilbert de B. (1959), The Foundations
of Geometry, University of Toronto Press,
Toronto, Canada, 1940, 1946, 1952, 4th edition
1959.
Raymond, Eric S. (1993), "The Utility of Mathematics",
Eprint.
Smullyan, Raymond M. (1993), Recursion Theory
for Metamathematics, Oxford University Press,
Oxford, UK.
Russell, Bertrand (1919), Introduction to
Mathematical Philosophy, George Allen and
Unwin, London, UK. Reprinted, John G. Slater
(intro.), Routledge, London, UK, 1993.
Shapiro, Stewart (2000), Thinking About Mathematics:
The Philosophy of Mathematics, Oxford University
Press, Oxford, UK
Strohmeier, John, and Westbrook, Peter (1999),
Divine Harmony, The Life and Teachings of
Pythagoras, Berkeley Hills Books, Berkeley,
CA.
Styazhkin, N.I. (1969), History of Mathematical
Logic from Leibniz to Peano, MIT Press, Cambridge,
MA.
Tait, William W. (1986), "Truth and Proof:
The Platonism of Mathematics", Synthese 69
(1986), 341-370. Reprinted, pp. 142–167
in W.D. Hart (ed., 1996).
Tarski, A. (1983), Logic, Semantics, Metamathematics:
Papers from 1923 to 1938, J.H. Woodger (trans.),
Oxford University Press, Oxford, UK, 1956.
2nd edition, John Corcoran (ed.), Hackett
Publishing, Indianapolis, IN, 1983.
Ulam, S.M. (1990), Analogies Between Analogies:
The Mathematical Reports of S.M. Ulam and
His Los Alamos Collaborators, A.R. Bednarek
and Françoise Ulam (eds.), University of
California Press, Berkeley, CA.
van Heijenoort, Jean (ed. 1967), From Frege
To Gödel: A Source Book in Mathematical Logic,
1879-1931, Harvard University Press, Cambridge,
MA.
Wigner, Eugene (1960), "The Unreasonable Effectiveness
of Mathematics in the Natural Sciences", Communications
on Pure and Applied Mathematics 13(1): 1-14.
Eprint
Wilder, Raymond L. Mathematics as a Cultural
System, Pergamon, 1980.
Witzany, Guenther (2011), Can mathematics
explain the evolution of human language?,
Communicative and Integrative Biology, 4(5):
516-520.
== External links ==
Philosophy of mathematics at PhilPapers
Philosophy of mathematics at the Indiana Philosophy
Ontology Project
Horsten, Leon. "Philosophy of Mathematics".
In Zalta, Edward N. (ed.). Stanford Encyclopedia
of Philosophy.
"Philosophy of mathematics". Internet Encyclopedia
of Philosophy."Ludwig Wittgenstein: Later
Philosophy of Mathematics". Internet Encyclopedia
of Philosophy.
The London Philosophy Study Guide offers many
suggestions on what to read, depending on
the student's familiarity with the subject:
Philosophy of Mathematics
Mathematical Logic
Set Theory & Further Logic
R.B. Jones' philosophy of mathematics page
Philosophy of mathematics at Curlie
The Philosophy of Real Mathematics Blog
Kaina Stoicheia by C.S. Peirce.
