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The philosophy of mathematics is the
branch A branch ( or , ) or tree branch (sometimes referred to in botany Botany, also called , plant biology or phytology, is the science of plant life and a branch of biology. A botanist, plant scientist or phytologist is a scientist who spe ...
of
philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about Metaphysics, existence, reason, Epistemology, knowledge, Ethics, values, Philosophy of mind, mind, and Philosophy of language, language. Such questio ...

philosophy
that studies the assumptions, foundations, and implications of
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
. It aims to understand the nature and methods of mathematics, and find 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. The philosophy of mathematics has two major themes; Mathematical realism and Mathematic anti-realism.


History

The origin of mathematics is subject to arguments and disagreements. Whether the birth of mathematics was a random happening or induced by necessity during the development of other subjects, like 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 Western philosophy encompasses the philosophical Philosophy (from , ) is the study of general and fundamental questions, such as those about existence Existence is the ability of an entity to interact with physical or mental reality ...
and
Eastern philosophy Eastern philosophy or Asian philosophy includes the various that originated in and , including , , , and ; all of these are dominant in East Asia and Vietnam. and (including , , , and ), which are dominant in , , , and . Indian philosophy ...
. Western philosophies of mathematics go as far back as
Pythagoras Pythagoras of Samos, or simply ; in () was an ancient and the eponymous founder of . His political and religious teachings were well known in and influenced the philosophies of , , and, through them, . Knowledge of his life is clouded b ...

Pythagoras
, who described the theory "everything is mathematics" ( mathematicism),
Plato Plato ( ; grc-gre, wikt:Πλάτων, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was an Classical Athens, Athenian philosopher during the Classical Greece, Classical period in Ancient Greece, founder of the Platonist school of thoug ...

Plato
, who paraphrased Pythagoras, and studied the
ontological status
ontological status
of mathematical objects, and
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher A philosopher is someone who practices philosophy Philosophy (from , ) is the study of general and fundamental quest ...

Aristotle
, who studied
logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents statements and ar ...

logic
and issues related to
infinity Infinity is that which is boundless, endless, or larger than any number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything t ...

infinity
(actual versus potential).
Greek philosophy Ancient Greek philosophy arose in the 6th century BC, at a time when the inhabitants of ancient Greece were struggling to repel devastating invasions from the east. Greek philosophy continued throughout the Hellenistic period The Hellenistic pe ...
on mathematics was strongly influenced by their study of
geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

geometry
. For example, at one time, the Greeks held the opinion that 1 (one) was not a
number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deduct ...

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 Irrationality is cognition Cognition () refers to "the mental action or process of acquiring knowledge and understanding through thought, experience, and the senses". It encompasses many aspects of intellectual functions and processes such as: ...
of the square root of two.
Hippasus Hippasus of Metapontum Metapontum or Metapontium ( grc, Μεταπόντιον, Metapontion) was an important city of Magna Graecia, situated on the gulf of Taranto, Tarentum, between the river Bradanus and the Casuentus (modern Basento). It wa ...
, a disciple of
Pythagoras Pythagoras of Samos, or simply ; in () was an ancient and the eponymous founder of . His political and religious teachings were well known in and influenced the philosophies of , , and, through them, . Knowledge of his life is clouded b ...

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 Simon Stevin (; 1548–1620), sometimes called Stevinus, was a Flemish mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as ...
was one of the first in Europe to challenge Greek ideas in the 16th century. Beginning with
Leibniz Gottfried Wilhelm (von) Leibniz ; see inscription of the engraving depicted in the "#1666–1676, 1666–1676" section. ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist, and diplomat. He is a promin ...

Leibniz
, the focus shifted strongly to the relationship between mathematics and logic. This perspective dominated the philosophy of mathematics through the time of
Frege Frege is a surname. Notable people with the surname include: * Carola Frege (born 1965), German scholar *Élodie Frégé, French singer and actress *Gottlob Frege (1848 – 1925), German philosopher, logician, and mathematician. * Livia Fre ...
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 Logic is an interdisciplinary field which studies truth Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth 2005 In everyday language, truth is typically ascribed to things that aim to re ...
,
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, i ...
(both
naive set theory Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sens ...
and
axiomatic set theory illustrating the intersection (set theory), intersection of two set (mathematics), sets Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although ob ...
), 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 Foundations of mathematics is the study of the philosophical Philosophy (from , ) is the study of general and fundamental questions, such as those about existence Existence is the ability of an entity to interact with physical or mental r ...
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 Epistemology (; ) is the concerned with . Epistemologists study the nature, origin, and scope of knowledge, epistemic , the of , and various related issues. Epistemology is considered a major subfield of philosophy, along with other major ...

epistemology
and
ontology Ontology is the branch of philosophy that studies concepts such as existence, being, Becoming (philosophy), becoming, and reality. It includes the questions of how entities are grouped into Category of being, basic categories and which of these ...

ontology
. Three schools,
formalism Formalism may refer to: * Form (disambiguation) * Formal (disambiguation) * Legal formalism, legal positivist view that the substantive justice of a law is a question for the legislature rather than the judiciary * Formalism (linguistics) * Scienti ...
,
intuitionism In the philosophy of mathematics Philosophy (from , ) is the study of general and fundamental questions, such as those about reason Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient ...
, and
logicism In the philosophy of mathematics The philosophy of mathematics is the branch The branches and leaves of a tree. A branch ( or , ) or tree branch (sometimes referred to in botany Botany, also called , plant biology or phytology, is ...
, emerged at this time, partly in response to the increasingly widespread worry that mathematics as it stood, and
analysis Analysis is the process of breaking a complex topic or substance Substance may refer to: * Substance (Jainism), a term in Jain ontology to denote the base or owner of attributes * Chemical substance, a material with a definite chemical composit ...
in particular, did not live up to the standards of
certainty Certainty (also known as epistemic certainty or objective certainty) is the property of s which a person has no rational grounds for doubting. One standard way of defining epistemic certainty is that a belief is certain if and only if the per ...
and
rigor Rigour (British English British English (BrE) is the standard dialect A standard language (also standard variety, standard dialect, and standard) is a language variety that has undergone substantial codification of grammar and usage ...
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 Euclid (; grc-gre, Εὐκλείδης Euclid (; grc, Εὐκλείδης – ''Eukleídēs'', ; fl. 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referre ...

Euclid
around 300 BCE as the natural basis for mathematics. Notions of
axiom An axiom, postulate or assumption is a statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' o ...

axiom
,
proposition In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, lab ...
and
proof Proof may refer to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Formal sciences * Formal proof, a construct in proof theory * Mathematical proof, a co ...
, as well as the notion of a proposition being true of a mathematical object (see Assignment), 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 In mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical ...
, propositions could be interpreted as referring to themselves or other propositions, enabling inquiry into the
consistency In classical logic, classical deductive logic, a consistent theory (mathematical logic), theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic def ...
of mathematical theories. This reflective critique in which the theory under review "becomes itself the object of a mathematical study" led
Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in man ...
to call such study ''
metamathematics Metamathematics is the study of mathematics itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories. Emphasis on metamathematics (and perhaps the creation of the term it ...
'' or ''
proof theory Proof theory is a major branchAccording to Wang (1981), pp. 3–4, proof theory is one of four domains mathematical logic, together with model theory In mathematical logic Mathematical logic is the study of formal logic within mathematics. Ma ...
''. At the middle of the century, a new mathematical theory was created by
Samuel Eilenberg Samuel Eilenberg (September 30, 1913 – January 30, 1998) was a Polish-American mathematician who co-founded category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph ...
and
Saunders Mac Lane Saunders Mac Lane (4 August 1909 – 14 April 2005) was an American mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ...
, known as
category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
, 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 Hilary Whitehall Putnam (; July 31, 1926 – March 13, 2016) was an American philosopher A philosopher is someone who practices philosophy. The term ''philosopher'' comes from the grc, φιλόσοφος, , translit=philosophos, meaning 'love ...

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 In the foundations of mathematics Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theori ...

Russell's paradox
comes to mind, as does 's attack on the actual
infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not exist in the standard real number system, but do exist in many other number systems, such a ...
—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 Realism, Realistic, or Realists may refer to: In the arts *Realism (arts), the general attempt to depict subjects truthfully in different forms of the arts Arts movements related to realism include: *Classical Realism *Literary realism, a movem ...
in general, holds that mathematical entities exist independently of the human
mind The mind is the set of faculties responsible for mental phenomena A phenomenon (; plural phenomena) is an observable fact or event. The term came into its modern philosophical Philosophy (from , ) is the study of general and fun ...

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;
triangle A triangle is a polygon In geometry, a polygon () is a plane (mathematics), plane Shape, figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The ...

triangle
s, 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 Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a renowned Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. He was known both for his ...

Paul Erdős
and
Kurt Gödel Kurt Friedrich Gödel ( , ; April 28, 1906 – January 14, 1978) was a logician Logic is an interdisciplinary field which studies truth Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dict ...
. 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
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 Platonism is the philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about Metaphysics, existence, reason, Epistemology, knowledge, Ethics, values, Philosophy of mind, mind, and Philosophy of ...
and
Aristotelianism Aristotelianism ( ) is a philosophical tradition inspired by the work of Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher A philosopher is someone who practices philo ...
.


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 Formalism may refer to: * Form (disambiguation) * Formal (disambiguation) * Legal formalism, legal positivist view that the substantive justice of a law is a question for the legislature rather than the judiciary * Formalism (linguistics) * Scienti ...
and
fictionalism Fictionalism is the view in philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaphysics, existence, Epistemology, knowledge, Ethics, values, Philosophy of mind, mind, and Phil ...
.


Contemporary schools of thought


Artistic

The view that claims that
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
is the
aesthetic Aesthetics, or esthetics (), is a branch of philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about Metaphysics, existence, reason, Epistemology, knowledge, Ethics, values, Philosophy of m ...

aesthetic
combination of assumptions, and then also claims that mathematics is an
art Art is a diverse range of (products of) human activities Humans (''Homo sapiens'') are the most populous and widespread species of primates, characterized by bipedality, opposable thumbs, hairlessness, and intelligence allowing the use ...

art
. A famous
mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ...

mathematician
who claims that is the British
G. H. Hardy Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematics, mathematician, known for his achievements in number theory and mathematical analysis. In biology, he is known for the Hardy–Weinberg principle, a basic prin ...
. For Hardy, in his book, ''
A Mathematician's Apology ''A Mathematician's Apology'' is a 1940 essay by British mathematician G. H. Hardy, which offers a defence of the pursuit of mathematics. Central to Hardy's " apology" — in the sense of a formal justification or defence (as in Plato ...
'', the definition of mathematics was more like the aesthetic combination of concepts.


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 Plato ( ; grc-gre, wikt:Πλάτων, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was an Classical Athens, Athenian philosopher during the Classical Greece, Classical period in Ancient Greece, founder of the Platonist school of thoug ...

Plato
's
Theory of Forms#REDIRECT Theory of forms {{Redirect category shell, 1= {{R from move {{Redirect from other capitalisation {{Redirect unprintworthy ...
and a "World of Ideas" (Greek: ''eidos'' (εἶδος)) described in Plato's
allegory of the cave As a literary device, an allegory is a narrative in which a character, place, or event is used to deliver a broader message about real-world issues and occurrences. Authors have used allegory throughout history in all forms of art to illustrate ...
: 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 Pythagoreanism originated in the 6th century BC, based on the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans. Pythagoras established the first Pythagorean community in Crotone, Italy. Early Pythagorean communities spr ...
'' of ancient Greece, who believed that the world was, quite literally, generated by
number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deduct ...

number
s. 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 In physics and cosmology Cosmology (from Ancient Greek, Greek κόσμος, ''kosmos'' "world" and -λογία, ''-logia'' "study of") is a branch of astronomy concerned with the studies of the origin and evolution of the universe, from the Big ...
, a theory that postulates that all structures that exist mathematically also exist physically in their own universe.
Kurt Gödel Kurt Friedrich Gödel ( , ; April 28, 1906 – January 14, 1978) was a logician Logic is an interdisciplinary field which studies truth Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dict ...
's PlatonismPlatonism in Metaphysics (Stanford Encyclopedia of Philosophy)
/ref> postulates a special kind of mathematical intuition that lets us perceive mathematical objects directly. (This view bears resemblances to many things
Husserl , thesis1_title = Beiträge zur Variationsrechnung (Contributions to the Calculus of Variations) , thesis1_url = https://fedora.phaidra.univie.ac.at/fedora/get/o:58535/bdef:Book/view , thesis1_year = 1883 , thesis2_title ...
said about mathematics, and supports
Kant Immanuel Kant (, , ; 22 April 1724 – 12 February 1804) was a German philosopher A philosopher is someone who practices philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about r ...

Kant
's idea that mathematics is
syntheticA synthetic is an artificial material produced by organic chemistry, organic chemical synthesis. Synthetic may also refer to: In the sense of both "combination" and "artificial" * Synthetic chemical or synthetic compress, produced by the process ...
''
a priori ''A priori'' and ''a posteriori'' ('from the earlier' and 'from the later', respectively) are Latin phrases used in philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaph ...
''.) Davis and 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 Formalism may refer to: * Form (disambiguation) * Formal (disambiguation) * Legal formalism, legal positivist view that the substantive justice of a law is a question for the legislature rather than the judiciary * Formalism (linguistics) * Scienti ...
. The mathematician
Alexander Grothendieck Alexander Grothendieck (; ; ; 28 March 1928 – 13 November 2014) was a mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbe ...

Alexander Grothendieck
was also a Platonist. 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 In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

axiom of choice
). It holds that all mathematical entities exist. 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 Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, i ...
is about a single universe of sets. This position (which is also known as naturalized Platonism because it is a
naturalized Naturalization (or naturalisation) is the legal act or process by which a non-citizen of a country may acquire citizenship Citizenship is the status of a person recognized under the law of a country (and/or local jurisdiction) of belongi ...
version of mathematical Platonism) has been criticized by Mark Balaguer on the basis of
Paul Benacerraf Paul Joseph Salomon Benacerraf (; born 26 March 1931) is a French-born American philosopher working in the field of the philosophy of mathematics Philosophy (from , ) is the study of general and fundamental questions, such as those about ...
's epistemological problem. A similar view, termed Platonized naturalism, was later defended by the
Stanford–Edmonton School In the philosophy of mathematics, logicism is a programme comprising one or more of the theses that — for some coherent meaning of 'logic' — mathematics is an extension of logic, some or all of mathematics is reduction (philosophy), reducible to ...
: 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 object In metaphysics Metaphysics is the branch of philosophy that studies the first principles of being, identity and change, space and time, causality, necessity and possibility. It includes questions about the nature of consciousness and the rela ...
s.


Mathematicism

Max Tegmark Max Erik Tegmark (born 5 May 1967) is a Swedish-American physicist, cosmologist and machine learning researcher. He is a professor at the Massachusetts Institute of Technology and the scientific director of the Foundational Questions Institute. ...

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 orldscomplex enough to contain self-aware substructures
hey Hey or Hey! may refer to: Music * Hey (band), a Polish rock band Albums * Hey (Andreas Bourani album), ''Hey'' (Andreas Bourani album) or the title song (see below), 2014 * Hey! (Julio Iglesias album), ''Hey!'' (Julio Iglesias album) or the ti ...

hey
will subjectively perceive themselves as existing in a physically 'real' world".


Logicism

Logicism In the philosophy of mathematics The philosophy of mathematics is the branch The branches and leaves of a tree. A branch ( or , ) or tree branch (sometimes referred to in botany Botany, also called , plant biology or phytology, is ...
is the thesis that mathematics is reducible to logic, and hence nothing but a part of logic. Carnap, Rudolf (1931), "Die logizistische Grundlegung der Mathematik", ''Erkenntnis'' 2, 91-121. Republished, "The Logicist Foundations of Mathematics", E. Putnam and G.J. Massey (trans.), in Benacerraf and Putnam (1964). Reprinted, pp. 41–52 in Benacerraf and Putnam (1983). 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 Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents statements and ar ...

logic
is the proper foundation of mathematics, and all mathematical statements are necessary
logical truth Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents statements and arg ...
s.
Rudolf Carnap Rudolf Carnap (; ; 18 May 1891 – 14 September 1970) was a German-language philosopher who was active in Europe before 1935 and in the United States thereafter. He was a major member of the Vienna Circle The Vienna Circle (german: Wiener Krei ...
(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 Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He worked as a mathematics professor at the University of Jena, and is understood by many to be the father of analy ...
was the founder of logicism. In his seminal ''Die Grundgesetze der Arithmetik'' (''Basic Laws of Arithmetic'') he built up
arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' or 'cr ...
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.
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell (18 May 1872 – 2 February 1970) was a British polymath A polymath ( el, πολυμαθής, , "having learned much"; la, homo universalis, "universal human") is an individual whose know ...
discovered that Basic Law V is inconsistent (this is
Russell's paradox In the foundations of mathematics Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theori ...

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 The axiom of reducibility was introduced by Bertrand Russell in the early 20th century as part of his Type theory, ramified theory of types. Russell devised and introduced the axiom in an attempt to manage the contradictions he had discovered in h ...
". Even Russell said that this axiom did not really belong to logic. Modern logicists (like Bob Hale,
Crispin Wright Crispin James Garth Wright (; born 21 December 1942) is a British British may refer to: Peoples, culture, and language * British people The British people, or Britons, are the citizens of the United Kingdom of Great Britain and Norther ...
, 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 In mathematics, a bijection, bijective function, one-to-one correspondence, or invertible function, is a function (mathematics), function between the elements of two set (mathematics), sets, where each element of one set is paired with exactly on ...

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 Euclidean geometry is a mathematical system attributed to Alexandria Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic Coptic may refer to: Afro-Asia * Copts, an ethnoreligious group mainly in the area of modern ...
(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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

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 (philosophy of mathematics), 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 Hilbert's program, program was intended to be a Gödel's completeness theorem, complete and consistency proof, 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 Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' or 'cr ...
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 Rudolf Carnap (; ; 18 May 1891 – 14 September 1970) was a German-language philosopher who was active in Europe before 1935 and in the United States thereafter. He was a major member of the Vienna Circle The Vienna Circle (german: Wiener Krei ...
, Alfred Tarski, and Haskell Curry, considered mathematics to be the investigation of formal system, 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 machine-readable medium, computer-readable formats, so as to facilitate proof checking, automated proof checking of mathematical proofs and the use of proof assistant, 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 A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ...

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 Euclidean geometry is a mathematical system attributed to Alexandria Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic Coptic may refer to: Afro-Asia * Copts, an ethnoreligious group mainly in the area of modern ...
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" (Luitzen Egbertus Jan Brouwer, 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 Reductio ad absurdum, proofs by contradiction. The
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

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 mathematical constructivism, constructivism, according to which a mathematical object does not exist unless it can be constructed from natural numbers in a finite set, 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: 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

Mathematical 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 and extrinsic properties (philosophy), 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 line (geometry), lines and plane (geometry), planes in geometry, or elements and operations in abstract algebra. Structuralism is an epistemologically realism (philosophy), 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 Ontology is the branch of philosophy that studies concepts such as existence, being, Becoming (philosophy), becoming, and reality. It includes the questions of how entities are grouped into Category of being, basic categories and which of these ...

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 Mathematical Platonism, 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 Philosophy of mathematics#Aristotelian realism, 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-realism, 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 A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deduct ...

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.Franklin, James (2014),
An Aristotelian Realist Philosophy of Mathematics
, Palgrave Macmillan, Basingstoke; Franklin, James (2021),
Mathematics as a science of non-abstract reality: Aristotelian realist philosophies of mathematics
" ''Foundations of Science'' 25.
Aristotelian realism is defended by James Franklin (philosopher), James Franklin and th
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).Penelope Maddy, Maddy, Penelope (1990), ''Realism in Mathematics'', Oxford University Press, Oxford, UK. 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 Christoph von Sigwart, Sigwart and Johann Eduard Erdmann, Erdmann as well as a number of psychologists, past and present: for example, Gustave Le Bon. Psychologism was famously criticized by
Frege Frege is a surname. Notable people with the surname include: * Carola Frege (born 1965), German scholar *Élodie Frégé, French singer and actress *Gottlob Frege (1848 – 1925), German philosopher, logician, and mathematician. * Livia Fre ...
in his ''The Foundations of Arithmetic'', and many of his works and essays, including his review of
Husserl , thesis1_title = Beiträge zur Variationsrechnung (Contributions to the Calculus of Variations) , thesis1_url = https://fedora.phaidra.univie.ac.at/fedora/get/o:58535/bdef:Book/view , thesis1_year = 1883 , thesis2_title ...
's ''Philosophy of Arithmetic''. Edmund Husserl, in the first volume of his ''Logical Investigations (Husserl), 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 evidence, 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 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 Hilary Whitehall Putnam (; July 31, 1926 – March 13, 2016) was an American philosopher A philosopher is someone who practices philosophy. The term ''philosopher'' comes from the grc, φιλόσοφος, , translit=philosophos, meaning 'love ...

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 existence, 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 Ontology is the branch of philosophy that studies concepts such as existence, being, Becoming (philosophy), becoming, and reality. It includes the questions of how entities are grouped into Category of being, basic categories and which of these ...

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 evidence, 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 Philosophy of mathematics#Embodied mind theories, 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 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 (logic), quantification over abstract models or deductions.


Social constructivism

Social constructivism sees 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 mathematical folklore, 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 Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a renowned Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. He was known both for his ...

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.


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 Proof may refer to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Formal sciences * Formal proof, a construct in proof theory * Mathematical proof, a co ...
, 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 of number statements

Realist and constructivist theories are normally taken to be contraries. However, Karl Popper argued that a number statement such as 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 as a language, 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 (linguistics), 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 grammatical tense, 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 type (mathematics), 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 Hilary Whitehall Putnam (; July 31, 1926 – March 13, 2016) was an American philosopher A philosopher is someone who practices philosophy. The term ''philosopher'' comes from the grc, φιλόσοφος, , translit=philosophos, meaning 'love ...

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.Putnam, H. ''Mathematics, Matter and Method. Philosophical Papers, vol. 1''. Cambridge: Cambridge University Press, 1975. 2nd. ed., 1985. The justification for the first premise is the most controversial. Both Putnam and Quine invoke naturalism (philosophy), 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 nominalism, nominalist who wishes to exclude the existence of Set (mathematics), 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 Paul Joseph Salomon Benacerraf (; born 26 March 1931) is a French-born American philosopher working in the field of the philosophy of mathematics Philosophy (from , ) is the study of general and fundamental questions, such as those about ...
and Hartry Field. Platonism posits that mathematical objects are ''abstract object, 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 perception, 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 mathematical proof, proofs, etc., which is already fully accountable in terms of physical processes in their brains. Field developed his views into
fictionalism Fictionalism is the view in philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaphysics, existence, Epistemology, knowledge, Ethics, values, Philosophy of mind, mind, and Phil ...
. 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 naturalism (philosophy), 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 (knowledge), intuition that involves contact with the Platonic realm. A modern form of this argument is given by Sir Roger Penrose.Review
of The Emperor's New Mind
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, 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 mathematical beauty, 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 . The first is the traditional proof by contradiction, ascribed to
Euclid Euclid (; grc-gre, Εὐκλείδης Euclid (; grc, Εὐκλείδης – ''Eukleídēs'', ; fl. 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referre ...

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 Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a renowned Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. He was known both for his ...

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 Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematics, mathematician, known for his achievements in number theory and mathematical analysis. In biology, he is known for the Hardy–Weinberg principle, a basic prin ...
's book ''
A Mathematician's Apology ''A Mathematician's Apology'' is a 1940 essay by British mathematician G. H. Hardy, which offers a defence of the pursuit of mathematics. Central to Hardy's " apology" — in the sense of a formal justification or defence (as in Plato ...
'', 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

*Definitions of mathematics *Formal language *Foundations of mathematics *Golden ratio *Model theory *Non-standard analysis *Philosophy of language *Philosophy of logic *Philosophy of science *Philosophy of physics *Philosophy of probability *Rule of inference *Science studies *Scientific method


Related works

*''The Analyst'' *Euclid's Euclid's Elements, ''Elements'' *"On Formally Undecidable Propositions of Principia Mathematica and Related Systems" *"On Computable Numbers, with an Application to the Entscheidungsproblem" *''Introduction to Mathematical Philosophy'' *"New Foundations, New Foundations for Mathematical Logic" *''Principia Mathematica'' *''Charles Sanders Peirce bibliography#CP, The Simplest Mathematics''


Historical topics

*History and philosophy of science *History of mathematics *History of philosophy


Notes


Further reading

*
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher A philosopher is someone who practices philosophy Philosophy (from , ) is the study of general and fundamental quest ...

Aristotle
, "Prior Analytics", Hugh Tredennick (trans.), pp. 181–531 in ''Aristotle, Volume 1'', Loeb Classical Library, William Heinemann, London, UK, 1938. *Paul Benacerraf, Benacerraf, Paul, and Hilary Putnam, 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. *George Berkeley, 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
*Nicolas Bourbaki, Bourbaki, N. (1994), ''Elements of the History of Mathematics'', John Meldrum (trans.), Springer-Verlag, Berlin, Germany. *Subrahmanyan Chandrasekhar, 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
*Philip J. Davis, Davis, Philip J. and Reuben Hersh, Hersh, Reuben (1981), ''The Mathematical Experience'', Mariner Books, New York, NY. *Keith Devlin, 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. *Michael Dummett, 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. *Paul Ernest, Ernest, Paul (1998), ''Social Constructivism as a Philosophy of Mathematics'', State University of New York Press, Albany, NY. *Alexandre George, George, Alexandre (ed., 1994), ''Mathematics and Mind'', Oxford University Press, Oxford, UK. *Jacques Hadamard, 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. *G. H. Hardy, Hardy, G.H. (1940), ''A Mathematician's Apology'', 1st published, 1940. Reprinted, C.P. Snow (foreword), 1967. Reprinted, Cambridge University Press, Cambridge, UK, 1992. *Wilbur Dyre Hart, Hart, W.D. (ed., 1996), ''The Philosophy of Mathematics'', Oxford University Press, Oxford, UK. *Vincent F. Hendricks, Hendricks, Vincent F. and Hannes Leitgeb (eds.). ''Philosophy of Mathematics: 5 Questions'', New York: Automatic Press / VIP, 2006

*H.E. Huntley, 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. *Jacob Klein (philosopher), 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. *Morris Kline, 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. *Gyula König, 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). *Stephan Körner, Körner, Stephan, ''The Philosophy of Mathematics, An Introduction''. Harper Books, 1960. *George Lakoff, Lakoff, George, and Rafael E. Núñez, 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 *Gottfried Wilhelm Leibniz, 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. *Edward A. Maziarz, Maziarz, Edward A., and Thomas Greenwood (philosopher), Greenwood, Thomas (1995), ''Greek Mathematical Philosophy'', Barnes and Noble Books. *Matthew Mount, Mount, Matthew, ''Classical Greek Mathematical Philosophy'', . * *Benjamin Peirce, Peirce, Benjamin (1870), "Linear Associative Algebra", § 1. See ''American Journal of Mathematics'' 4 (1881). *Charles Sanders Peirce, Peirce, C.S., ''Collected Papers of Charles Sanders Peirce'', vols. 1-6, Charles Hartshorne and Paul Weiss (philosopher), 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 Charles Sanders Peirce bibliography#Books authored or edited by Peirce, published in his lifetime, 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. *Michael Resnik, Resnik, Michael (1997), ''Mathematics as a Science of Patterns'', Clarendon Press, Oxford, UK, *Gilbert de Beauregard Robinson, 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
*Raymond M. Smullyan, 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. *Stewart Shapiro, 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. *N.I. Styazhkin, Styazhkin, N.I. (1969), ''History of Mathematical Logic from Leibniz to Peano'', MIT Press, Cambridge, MA. *William W. Tait, 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. *Stanislaw Ulam, 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. *Jean van Heijenoort, van Heijenoort, Jean (ed. 1967), ''From Frege To Gödel: A Source Book in Mathematical Logic, 1879-1931'', Harvard University Press, Cambridge, MA. *Eugene Wigner, Wigner, Eugene (1960), "The Unreasonable Effectiveness of Mathematics in the Natural Sciences", ''Communications on Pure and Applied Mathematics'' 13(1): 1-14.
Eprint
*Raymond Louis Wilder, 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

* * * * *
''Mathematical Structuralism'', Internet Encyclopaedia of Philosophy
*
''Abstractionism'', Internet Encyclopaedia of Philosophy
** *Th
London Philosophy Study Guide
offers many suggestions on what to read, depending on the student's familiarity with the subject:





*
The Philosophy of Real Mathematics – Blog
by David Corfield
Kaina Stoicheia
by C. S. Peirce {{DEFAULTSORT:Philosophy Of Mathematics Philosophy of mathematics,