Editing Philosophy of mathematics (section)

===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]].}}