Editing Philosophy of mathematics (section)

===Epistemic argument against realism===
The [[anti-realist]] "[[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 concrete, 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 concrete, 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" — Katz, J. ''Realistic Rationalism'', 2000, p. 15</ref><ref>{{Cite web|url=http://www.philosophynow.org/issue81/|archiveurl=https://web.archive.org/web/20110207095054/http://philosophynow.org/issue81|url-status=dead|title=''Philosophy Now'': "Mathematical Knowledge: A dilemma"|archivedate=February 7, 2011}}</ref><ref>{{cite book| chapter-url = http://plato.stanford.edu/entries/platonism-mathematics/#EpiAcc| title = Standard Encyclopaedia of Philosophy| chapter = Platonism in the Philosophy of Mathematics| year = 2018| publisher = Metaphysics Research Lab, Stanford University| access-date = 2011-02-13| archive-date = 2010-12-04| archive-url = https://web.archive.org/web/20101204143629/http://plato.stanford.edu/entries/platonism-mathematics/#EpiAcc| url-status = live}}</ref> Another way of making the point is that if the Platonic world were to disappear, it would make no difference to the ability of mathematicians to generate [[mathematical proof|proofs]], etc., which is already fully accountable in terms of physical processes in their brains.

Field developed his views into [[#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.

The argument hinges on the idea that a satisfactory [[naturalism (philosophy)|naturalistic]] account of thought processes in terms of brain processes can be given for mathematical reasoning along with everything else. 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. A modern form of this argument is given by [[Sir Roger Penrose]].<ref>[http://www.c2.com/cgi/wiki?TheEmperorsNewMind Review] {{Webarchive|url=https://web.archive.org/web/20110514092121/http://c2.com/cgi/wiki?TheEmperorsNewMind |date=2011-05-14 }} of [[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|Realistic Rationalism]]''.

A more radical defense is denial of physical reality, i.e. the [[mathematical universe hypothesis]]. In that case, a mathematician's knowledge of mathematics is one mathematical object making contact with another.