Overview
At minimum, mathematical realism is the view that the truth value of mathematical statements is independent of human minds.
Related notes:
- Mathematical anti-realism and non-Platonism
- Mathematical pluralism
- Realism is the conjunction of aptness, belief, truth, independence, and face-value, after Clarke-Doane
Variants of mathematical realism
| Name of theory or theorist | Description | Problems | Further reading |
|---|---|---|---|
| Gödel’s “straightforward” account | ”According to Gödel there really are mathematical objects, and the human mind has a faculty different from but not totally disanalogous to perception with the aid of which it acquires better and better intuitions concerning the behavior of mathematical objects” (@1994putnamWhy). | ”Flatly incompatible with the simple fact that we think with our brains, and not immaterial souls” (@1994putnamWhy); that is, not sufficient justification for realism of math? | - K. Gödel (1964), “What is Cantor’s Continuum Problem?” |
| Holism (after Quine) | Mathematics should be viewed as a necessary part of science: “Sets and electrons are alike for Quine, in being objects we need to postulate if we are to do science as we presently do it” (@1994putnamWhy). | If scientific theory is justified as a whole by its efficacy for explaining sensations, then “what the mathematician is doing is contributing to a scheme for explaining sensations” (@1994putnamWhy); doesn’t seem to reflect mathematical practice. | |
| Quasi-empirical realism (after Putnam) | Extends Quine’s holism to include combinatorial facts among things that mathematics is about: “A sophisticated quasi-empirical realist can grant that mathematical truths attain the status of being ‘a priori relative to our body of knowledge’, as some physical laws do” (@1994putnamWhy). Further, mathematical facts are constrained by agreement with mathematical intuitions. | Unclear what satisfying the non-experimental constraint of “agreeing with intuitions, whatever their source” has to do with truth: “Having accepted the stance of realism—which means that we do regard mathematical statements as true or false—and having given a description, however vague, of how mathematical statements come to be accepted, we cannot duck the question: what is the link between acceptability and truth?” (@1994putnamWhy). | - @1994putnamWhy, “Philosophy of Mathematics: Why Nothing Works.” |
Quotations
[D]espite their remoteness from sense experience, we do have a perception also of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true.