Editing Philosophy of mathematics (section)

====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]].}}