Editing Philosophy of mathematics (section)

==Contemporary schools of thought==
Contemporary schools of thought in the philosophy of mathematics include: artistic, Platonism, mathematicism, logicism, formalism, conventionalism, intuitionism, constructivism, finitism, structuralism, embodied mind theories (Aristotelian realism, psychologism, empiricism), fictionalism, social constructivism, and non-traditional schools. 

However, many of these schools of thought are mutually compatible. For example, most living mathematicians are together Platonists and  formalists, give a great importance to [[aesthetic]], and consider that axioms should be chosen for the results they produce, not for their coherence with human intuition of reality (conventionalism).<ref name="Borel-1983" />

===Artistic===
The view that claims that [[mathematics]] is the aesthetic combination of assumptions, and then also claims that mathematics is an [[art]]. A famous [[mathematician]] who claims that is the British [[G. H. Hardy]].<ref>{{Cite web|url=https://www.goodreads.com/work/quotes/1486751-a-mathematician-s-apology|title=A Mathematician's Apology Quotes by G.H. Hardy|access-date=2020-07-20|archive-date=2021-05-08|archive-url=https://web.archive.org/web/20210508052133/https://www.goodreads.com/work/quotes/1486751-a-mathematician-s-apology|url-status=live}}</ref> For Hardy, in his book, ''[[A Mathematician's Apology]]'', the definition of mathematics was more like the aesthetic combination of concepts.<ref>{{cite journal |last1=S |first1=F. |title=A Mathematician's Apology |journal=Nature |date=January 1941 |volume=147 |issue=3714 |pages=3–5 |doi=10.1038/147003a0 |bibcode=1941Natur.147....3S |s2cid=4212863 }}</ref>

===Platonism===
{{excerpt|Mathematical Platonism}}

===Mathematicism===
{{Main|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".<ref>{{cite journal |last=Tegmark |first=Max |date=February 2008 |title=The Mathematical Universe |journal=Foundations of Physics |volume=38 |issue=2 |pages=101–150 |doi=10.1007/s10701-007-9186-9 |arxiv=0704.0646 |bibcode=2008FoPh...38..101T|s2cid=9890455 }}</ref><ref>Tegmark (1998), p. 1.</ref>

===Logicism===
{{Main|Logicism}}
[[Logicism]] is the thesis that mathematics is reducible to logic, and hence nothing but a part of logic.<ref name=Carnap>[[Rudolf Carnap|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.&nbsp;41–52 in Benacerraf and Putnam (1983).</ref>{{rp|41}} 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 proposition|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 truth]]s.

[[Rudolf Carnap]] (1931) presents the logicist thesis in two parts:<ref name=Carnap/>
#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. [[Bertrand 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 [[Alfred North Whitehead|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 much of mathematics, such as the "[[axiom of reducibility]]". Even Russell said that this axiom did not really belong to logic.

Modern logicists (like [[Bob Hale (philosopher)|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 [[bijection|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===
{{Main|Formalism (philosophy of mathematics)}}
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 known as [[deductivism]].<ref>{{cite book |author1=Alexander Paseau |author2=Fabian Pregel |title=Deductivism in the Philosophy of Mathematics |date=2023 |publisher=Metaphysics Research Lab, Stanford University |url=https://plato.stanford.edu/entries/deductivism-mathematics/}}</ref> In deductivism, the Pythagorean theorem is not an absolute truth, but a relative one, if it follows deductively from the appropriate axioms. The same is held to be true for all other mathematical statements.

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.

[[File:Hilbert.jpg|thumb|[[David Hilbert]]]]
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.<ref>{{Citation|last=Zach|first=Richard|title=Hilbert's Program|date=2019|url=https://plato.stanford.edu/archives/sum2019/entries/hilbert-program/|encyclopedia=The Stanford Encyclopedia of Philosophy|editor-last=Zalta|editor-first=Edward N.|edition=Summer 2019|publisher=Metaphysics Research Lab, Stanford University|access-date=2019-05-25|archive-date=2022-02-08|archive-url=https://web.archive.org/web/20220208161851/https://plato.stanford.edu/archives/sum2019/entries/hilbert-program/|url-status=live}}</ref> 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 theorem]]s, 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 system|formal axiom systems]]. [[Mathematical logic]]ians 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{{Who|date=July 2012}} 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—{{Crossreference|see [[QED project]] for a general overview}}.

===Conventionalism===
{{Main|Conventionalism|Preintuitionism}}
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 equation]]s 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===
{{Main|Mathematical 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.<ref>[[Robert Audi|Audi, Robert]] (1999), ''The Cambridge Dictionary of Philosophy'', Cambridge University Press, Cambridge, UK, 1995. 2nd edition. Page 542.</ref>

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]] 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 [[algorithm]]s are meaningful and should be investigated in mathematics. This has led to the study of the [[computable number]]s, first introduced by [[Alan Turing]]. Not surprisingly, then, this approach to mathematics is sometimes associated with theoretical [[computer science]].

====Constructivism====
{{Main|Constructivism (philosophy of mathematics)}}
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 using proof by contradiction when showing the existence of an object or when trying to establish the truth of some proposition. 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.''<ref>{{Citation|last=Bishop|first=Errett|author-link=Errett Bishop|year=2012|orig-year=1967|title=Foundations of Constructive Analysis|publisher=Ishi Press|location=New York|edition=Paperback|isbn=978-4-87187-714-5}}</ref>

====Finitism====
{{Main|Finitism}}
[[File:Leopold Kronecker (ca. 1880).jpg|thumb|[[Leopold Kronecker]]]]
[[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 number]]s 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]],<ref>From an 1886 lecture at the 'Berliner Naturforscher-Versammlung', according to [[H. M. Weber]]'s memorial article, as quoted and translated in {{cite web
|url=http://www.cs.nyu.edu/pipermail/fom/2000-February/003820.html
|title=FOM: What were Kronecker's f.o.m.?
|access-date=2008-07-19
|author=Gonzalez Cabillon, Julio
|date=2000-02-03
|archive-date=2007-10-09
|archive-url=https://web.archive.org/web/20071009235907/http://cs.nyu.edu/pipermail/fom/2000-February/003820.html
|url-status=live
}}
Gonzalez gives as the sources for the memorial article, the following: Weber, H: "Leopold Kronecker", ''Jahresberichte der Deutschen Mathematiker Vereinigung'', vol ii (1893), pp. 5-31. Cf. page 19. See also ''Mathematische Annalen'' vol. xliii (1893), pp. 1-25.</ref> who said:
{{Blockquote|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''.<ref name="Mayberry-2001">{{cite book |first=J.P. |last=Mayberry |author-link=John Penn Mayberry |title=The Foundations of Mathematics in the Theory of Sets |year=2001 |publisher=[[Cambridge University Press]]}}</ref> 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===
{{Main|Mathematical 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]]). 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.<ref>{{cite book |last=Brown |first=James |title=Philosophy of Mathematics |publisher=Routledge |location=New York |year=2008 |isbn=978-0-415-96047-2}}</ref>

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 {{Crossreference|(see [[Benacerraf's identification problem]])}}.

The ''in re'' structuralism ("in the thing") is the equivalent of [[#Aristotelian realism|Aristotelian 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]] springs from the experience of counting discrete objects (requiring the human senses such as sight for detecting the objects, touch; and signalling from the brain). 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.

The cognitive processes of pattern-finding and distinguishing objects are also subject to [[neuroscience]]; if mathematics is considered to be relevant to a natural world (such as from [[Philosophical realism|realism]] or a degree of it, as opposed to pure [[solipsism]]).

Its actual relevance to reality, while accepted to be a trustworthy approximation (it is also suggested the [[evolution]] of perceptions, the body, and the senses may have been necessary for survival) is not necessarily accurate to a full realism (and is still subject to flaws such as [[illusion]], assumptions (consequently; the foundations and axioms in which mathematics have been formed by humans), generalisations, deception, and [[hallucination]]s). As such, this may also raise questions for the modern [[scientific method]] for its compatibility with general mathematics; as while relatively reliable, it is still limited by what can be measured by [[empiricism]] which may not be as reliable as previously assumed (see also: 'counterintuitive' concepts in such as [[quantum nonlocality]], and [[action at a distance]]).

Another issue is that one [[numeral system]] may not necessarily be applicable to problem solving. Subjects such as [[complex number]]s or [[imaginary number]]s require specific changes to more commonly used axioms of mathematics; otherwise they cannot be adequately understood.

Alternatively, computer programmers may use [[hexadecimal]] for its 'human-friendly' representation of [[binary code|binary-coded]] values, rather than [[decimal]] (convenient for counting because humans have ten fingers). The axioms or logical rules behind mathematics also vary through time (such as the adaption and invention of [[zero]]).

As [[perception]]s from the human brain are subject to [[illusion]]s, assumptions, deceptions, (induced) [[hallucination]]s, cognitive errors or assumptions in a general context, it can be questioned whether they are accurate or strictly indicative of truth (see also: [[Ontology|philosophy of being]]), and the nature of [[empiricism]] itself in relation to the universe and whether it is independent to the senses and the universe.

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''. {{Crossreference|For more on the philosophical ideas that inspired this perspective, see [[cognitive science of mathematics]].}}

====Aristotelian realism<!--linked from 'Structuralism (philosophy of mathematics)'-->====
{{Main|Aristotelian realist philosophy of mathematics}}
{{See also|In re structuralism|Immanent realism}}
[[Aristotelian realist philosophy of mathematics|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.<ref>{{cite book |last=Franklin |first=James |date=2014 |title=An Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure |url=https://web.maths.unsw.edu.au/~jim/franklinaristotelianrealistphilosophyofmathematics.pdf |publisher=Palgrave Macmillan |isbn=9781137400727}}</ref><ref>{{cite journal |last1=Franklin |first1=James |date=2022 |title=Mathematics as a science of non-abstract reality: Aristotelian realist philosophies of mathematics |url=https://rdcu.be/chatd |journal=Foundations of Science |volume=27 |issue=2 |pages=327–344|doi=10.1007/s10699-021-09786-1 |s2cid=233658181 |access-date=30 June 2021}}</ref> Aristotelian realism is defended by [[James Franklin (philosopher)|James Franklin]] and the [http://web.maths.unsw.edu.au/~jim/structmath.html 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).<ref>[[Penelope Maddy|Maddy, Penelope]] (1990), ''Realism in Mathematics'', Oxford University Press, Oxford, UK.</ref> 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''<ref name="Mayberry-2001"/> 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====
{{Main|Psychologism}}
{{See also|Anti-psychologism}}

[[Psychologism]] in the philosophy of mathematics is the position that [[mathematical]] [[concept]]s 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 [[Gottlob Frege|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 (Husserl)|Logical Investigations]]'', called "The Prolegomena of Pure Logic", criticized psychologism thoroughly and sought to distance himself from it. The "Prolegomena" is considered{{By whom|date=April 2025}} a more concise, fair, and thorough refutation of psychologism than the criticisms made by Frege, and also it is considered today by many{{By whom|date=April 2025}} as being a memorable refutation for its decisive blow to psychologism. Psychologism was also criticized by [[Charles Sanders Peirce]] and [[Maurice Merleau-Ponty]].

====Empiricism====
{{Main|Quasi-empiricism in mathematics|Postmodern mathematics}}
[[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&nbsp;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,<ref>{{cite book |last1=Ayer |first1=Alfred Jules |title=Language, Truth, & Logic |date=1952 |publisher=Dover Publications, Inc. |location=New York |isbn=978-0-486-20010-1 |page=[https://archive.org/details/languagetruthlog00alfr/page/74 74 ff] |url-access=registration |url=https://archive.org/details/languagetruthlog00alfr/page/74 }}</ref> it makes statements like {{nowrap|"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.

[[Karl Popper]] was another philosopher to point out empirical aspects of mathematics, observing that "most mathematical theories are, like those of physics and biology, hypothetico-deductive: pure mathematics therefore turns out to be much closer to the natural sciences whose hypotheses are conjectures, than it seemed even recently."<ref>{{cite book |first=Karl R. |last=Popper |author-link=Karl Popper |title=In Search of a Better World: Lectures and Essays from Thirty Years |location=New York |publisher=Routledge |chapter=On knowledge |year=1995 |page=56 |isbn=978-0-415-13548-1 |bibcode=1992sbwl.book.....P |url-access=registration |url=https://archive.org/details/insearchofbetter00karl}}</ref> Popper also noted he would "admit a system as empirical or scientific only if it is capable of being tested by experience."<ref>{{cite book |last=Popper |first=Karl |year=2002 |orig-year= 1959 |title=The Logic of Scientific Discovery |publisher=Routledge |location=Abingdon-on-Thames |isbn=978-0-415-27843-0 |page=18}}</ref>

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 [[electron]]s 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]] 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''.<ref>[[Thomas Tymoczko|Tymoczko, Thomas]] (1998), ''New Directions in the Philosophy of Mathematics''. {{isbn|978-0691034980}}.</ref> 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 [[#Embodied mind theories|embodied mind theory]].

{{Crossreference|For experimental evidence suggesting that human infants can do elementary arithmetic, see [[Brian Butterworth]].}}

===Fictionalism<!--'Mathematical fictionalism' redirects here-->===
{{See also|Fictionalism}}
'''Mathematical fictionalism'''<!--boldface per WP:R#PLA--> was brought to fame in 1980 when [[Hartry Field]] published ''Science Without Numbers'',<ref>[[Hartry Field|Field, Hartry]], ''Science Without Numbers'', Blackwell, 1980.</ref> 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 field]]s. 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 {{nowrap|"2 + 2 {{=}} 4"}} is just as fictitious as "[[Sherlock Holmes]] lived at 221B Baker Street"—but both are true according to the relevant fictions.

Another fictionalist, [[Mary Leng]], expresses the perspective succinctly by dismissing any seeming connection between mathematics and the physical world as "a happy coincidence". This rejection separates fictionalism from other forms of anti-realism, which see mathematics itself as artificial but still bounded or fitted to reality in some way.<ref>{{cite book |last=Leng |first=Mary |date=2010 |title=Mathematics and Reality |publisher=Oxford University Press |page=239 |isbn=978-0199280797}}</ref>

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.{{Citation needed|date=March 2023}}

===Social constructivism<!--'Postmodern mathematics' redirects here-->===
{{Main|Social constructivism}}
[[Social constructivism]] sees mathematics primarily as a [[Social constructionism|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 [[subculture]]s. 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 conjecture]]s 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.{{Clarify|date=August 2010}} More recently [[Paul Ernest]] has explicitly formulated a social constructivist philosophy of mathematics.<ref>{{cite web |first=Paul |last=Ernest |url=http://www.people.ex.ac.uk/PErnest/pome12/article2.htm |title=Is Mathematics Discovered or Invented? |publisher=University of Exeter |access-date=2008-12-26 |archive-date=2008-04-05 |archive-url=https://web.archive.org/web/20080405035604/http://www.people.ex.ac.uk/PErnest/pome12/article2.htm |url-status=live }}</ref> 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,<ref>{{cite interview |first=Reuben |last=Hersh |interviewer=John Brockman |url=http://edge.org/documents/archive/edge5.html |title=What Kind of a Thing is a Number? |publisher=Edge Foundation |date=February 10, 1997 |access-date=2008-12-26 |archive-url=https://web.archive.org/web/20080516103111/http://edge.org/documents/archive/edge5.html |archive-date=May 16, 2008 |url-status=dead }}</ref> similar to but not quite the same as that associated with Alvin White;<ref>{{cite web |title=Humanism and Mathematics Education |work=Math Forum |url=http://mathforum.org/mathed/humanistic.math.html |publisher=Humanistic Mathematics Network Journal |access-date=2008-12-26 |archive-date=2008-07-24 |archive-url=https://web.archive.org/web/20080724071728/http://mathforum.org/mathed/humanistic.math.html |url-status=live }}</ref> 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 [[mathematical proof|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 of number statements<!--Linked from 'Karl Popper'-->====
Realist and constructivist theories are normally taken to be contraries. However, [[Karl Popper]]<ref>Popper, Karl Raimund (1946) Aristotelian Society Supplementary Volume XX.</ref> argued that a number statement such as {{nowrap|"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.<ref>Gregory, Frank Hutson (1996) "[[s:Arithmetic and Reality: A Development of Popper's Ideas|Arithmetic and Reality: A Development of Popper's Ideas]]". City University of Hong Kong. Republished in Philosophy of Mathematics Education Journal No. 26 (December 2011)</ref>

====Philosophy of language====
{{original research|section|date=February 2023}}
Innovations in the philosophy of language during the 20th century renewed interest in whether mathematics is, as is often said,{{citation needed|date=February 2023}} the ''language'' of science. Although some{{who|date=February 2023}} mathematicians and philosophers would accept the statement "mathematics is a language" (most consider that the [[language of mathematics]] is a part of mathematics to which mathematics cannot be reduced),{{citation needed|date=February 2023}} linguists{{who|date=February 2023}} 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.<ref name="Ganesalingam-2013">{{cite book |last1=Ganesalingam |first1=Mohan |title=The Language of Mathematics: A Linguistic and Philosophical Investigation |volume=7805 |date=2013 |publisher=Springer |isbn=978-3-642-37011-3 |doi=10.1007/978-3-642-37012-0 |series=Lecture Notes in Computer Science |s2cid=14260721 }}</ref> 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 grammar]]s). 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."<ref name="Ganesalingam-2013" />{{rp|251}}