Realism and Naturalism, Mathematical
Many versions of realism in mathematics are intimately related to versions of naturalism. The purpose of this article is to explore relationships between the various views, and, briefly, the main opposition to them. The focus here is exclusively on mathematics. So, for example, "Platonism" is to be read as "Platonism about mathematics." This entry does not claim to do justice to the subtle and detailed works of everyone who works in the philosophy of mathematics, or even everyone who defends versions of realism and/or naturalism. Instead, this entry seeks to provide a useful road map of an important part of the territory.
In broad terms, realism is the view that mathematics is objective: independent of the lives, customs, language, and form of life of mathematicians. This statement is deliberately indeterminate. What aspects of mathematics are being discussed? What, exactly, is it independent of? And what is it to be independent? What is it to be objective? In philosophy there is little that one can take for granted.
Realism
There are at least two forms of realism: realism in ontology, which concerns mathematical objects, and realism in truth value, which concerns mathematical truth. Realism in ontology is the view that mathematical objects, such as numbers, sets, functions, and geometric points exist independently of the mathematician.
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