Editing Analytic philosophy (section)

=== Philosophy of mathematics ===
[[File:Kurt gödel.jpg|thumb|130px|Kurt Gödel]]

Since the beginning, analytic philosophy has had an interest in the [[philosophy of mathematics]]. [[Kurt Gödel]], a student of Hans Hahn of the Vienna Circle, produced his [[Gödel's incompleteness theorems|incompleteness theorems]] showing that Russell and Whitehead's ''Principia Mathematica'' also failed to reduce arithmetic to logic. Gödel has been ranked as one  of the four greatest logicians of all time, along with [[Aristotle]], Frege, and Tarski.<ref name="Restall">{{cite web |last=Restall |first=Greg |date=2002–2006 |title=Great Moments in Logic |url=http://consequently.org/writing/logicians/ |url-status=live |archive-url=https://web.archive.org/web/20081206052240/http://consequently.org/writing/logicians/ |archive-date=6 December 2008 |access-date=2009-01-03}}</ref> [[Ernst Zermelo]] and [[Abraham Fraenkel]] established [[Zermelo–Fraenkel set theory|Zermelo Fraenkel Set Theory]]. Quine developed his own system, dubbed [[New Foundations]].

Physicist [[Eugene Wigner]]'s seminal paper "[[The Unreasonable Effectiveness of Mathematics in the Natural Sciences]]" poses the question of why a formal pursuit like mathematics can have real utility.<ref>{{Cite journal|last1=Wigner|first1=E. P.|author-link=Eugene Wigner|year=1960|title=The unreasonable effectiveness of mathematics in the natural sciences. Richard Courant lecture in mathematical sciences delivered at New York University, May 11, 1959|url=http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html|journal=Communications on Pure and Applied Mathematics|volume=13|issue=1|pages=1–14|bibcode=1960CPAM...13....1W|doi=10.1002/cpa.3160130102|s2cid=6112252|archive-url=https://web.archive.org/web/20210212111540/http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html|archive-date=2021-02-12}}</ref> [[José Benardete]] argued for the reality of [[infinity]].<ref>Infinity: An Essay In Metaphysics</ref>

Akin to the medieval debate on universals, between realists, idealists, and nominalists; the philosophy of mathematics has the debate between logicists or platonists, conceptualists or [[Intuitionism|intuitionists]], and [[Formalism (philosophy of mathematics)|formalists]].<ref>Quine, On What There Is</ref>

==== Platonism ====
Gödel was a platonist who postulated a special kind of mathematical intuition that lets us perceive mathematical objects directly. Quine and Putnam argued for platonism with the [[Quine–Putnam indispensability argument|indispensability argument]]. [[Crispin Wright]], along with [[Bob Hale (philosopher)|Bob Hale]], led a Neo-Fregean revival with his work ''Frege's Conception of Numbers as Objects''.<ref>The Reason's Proper Study: Essays Towards a Neo-Fregean Philosophy of Mathematics</ref>

===== Critics =====
[[Structuralism (philosophy of mathematics)|Structuralist]] [[Paul Benacerraf]] has an epistemological objection to mathematical platonism.

==== Intuitionism ====
The intuitionists, led by [[L. E. J. Brouwer]], are a [[Constructivism (philosophy of mathematics)|constructivist]] school of mathematics that argues that mathematics is a [[Cognition|cognitive]] [[Construct (philosophy)|construct]] rather than a type of [[objective truth]].

==== Formalism ====
The formalists, best exemplified by David Hilbert, considered mathematics to be merely the investigation of [[Formal system|formal axiom systems]]. [[Hartry Field]] defended [[mathematical fictionalism]].