Editing Anti-realism (section)

===Mathematical anti-realism===
{{main|Mathematical anti-realism}}
{{also|Post rem structuralism}}
In the [[philosophy of mathematics]], realism is the claim that mathematical entities such as 'number' have an observer-independent existence. [[Mathematical empiricism|Empiricism]], which associates numbers with concrete physical objects, and [[Mathematical Platonism|Platonism]], in which numbers are abstract, non-physical entities, are the preeminent forms of mathematical realism.

The "[[epistemic]] argument" against Platonism has been made by [[Paul Benacerraf]] and [[Hartry Field]]. Platonism posits that mathematical objects are ''[[abstract object|abstract]]'' entities. By general agreement, abstract entities cannot interact [[causal]]ly with physical entities ("the truth-values of our mathematical assertions depend on facts involving platonic entities that reside in a realm outside of space-time"<ref>Field, Hartry, 1989, ''Realism, Mathematics, and Modality'', Oxford: Blackwell, p. 68</ref>). Whilst our knowledge of 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.<ref>"Since abstract objects are outside the nexus of causes and effects, and thus perceptually inaccessible, they cannot be known through their effects on us" — [[Jerrold Katz]], ''Realistic Rationalism'', 2000, p. 15</ref><ref>{{Cite web |url=http://www.philosophynow.org/issue81/ |title=Philosophy Now: "Mathematical Knowledge: A dilemma" |access-date=2011-02-14 |archive-url=https://web.archive.org/web/20110207095054/http://philosophynow.org/issue81 |archive-date=2011-02-07 |url-status=dead }}</ref><ref>{{cite book| chapter-url = http://plato.stanford.edu/entries/platonism-mathematics/#EpiAcc| title = Stanford Encyclopedia of Philosophy| chapter = Platonism in the Philosophy of Mathematics| year = 2018| publisher = Metaphysics Research Lab, Stanford University}}</ref>

Field developed his views into [[Mathematical fictionalism|fictionalism]]. 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.

====Counterarguments<!--'Realistic rationalism' redirects here-->====
Anti-realist arguments hinge on the idea that a satisfactory, [[Naturalism (philosophy)|naturalistic]] account of thought processes can be given for mathematical reasoning. 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]], as in the argument given by [[Sir Roger Penrose]].<ref>{{cite web |title=Review of |url=http://www.c2.com/cgi/wiki?TheEmperorsNewMind}} [[The Emperor's New Mind]].</ref>

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''. In this book, he put forward a position called '''realistic rationalism'''<!--boldface per WP:R#PLA-->, which combines metaphysical realism and [[rationalism]].

A more radical defense is to deny the separation of physical world and the platonic world, i.e. the [[mathematical universe hypothesis]] (a variety of [[mathematicism]]). In that case, a mathematician's knowledge of mathematics is one mathematical object making contact with another.