Mathematical Intuition pp 66-91 | Cite as

# Mathematical Intuition

## Abstract

We are now in a position to launch more fully into a phenomenological account of mathematical intuition. We shall view mathematical intuition as a process which produces evidence for M’s mathematical beliefs, much as straightforward perception is a process which produces evidence for M’s beliefs about the physical world. We have already noted that from a phenomenological point of view mathematical objects are recognized to be of a different type from physical objects. Reductive enterprises that would seek to explain away mathematical objects are not viewed as true to mathematical experience. Acts of mathematical intuition and their objects are, however, said to be founded on acts of straightforward intuition and their objects. The intuition of mathematical objects will be founded on intuition of their parts, but mathematical intuition is also founded on straightforward intuition in the sense that the structure of perceptual acts provides the (representing) content for acts of mathematical intuition. Reflection and abstraction will be involved in acts of mathematical intuition.

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## Notes

- 4.See John Nolt, “Mathematical Intuition” [100].Google Scholar
- 10.See, e.g., Kitcher in
*The Nature of Mathematical Knowledge*[77], Chapter 3, and “Hilbert’s Epistemology” [75]. The objections that follow are discussed by Kitcher. Tait raises somewhat similar objections in his “Finitism” [133].Google Scholar - 11.Becker, “Mathematische Existenz” [4]. Heyting, “The Intuitionist Foundations of Mathematics” [49].Google Scholar
- 12.Kolmogorov, “Zur Deutung der Intuitionistischen Logik” [78]. See Sundholm’s paper “Constructions, Proofs, and the Meaning of the Logical Constants” [130].Google Scholar
- 13.See Martin-Löf [90].Google Scholar
- 15.Kreisel, “Foundations of Intuitionistic Logic” [80]; “Mathematical Logic” [81]. N. Goodman, [40], [41], [42], [43].Google Scholar
- 18.See Sundholm’s paper [130].Google Scholar