Mathematical Discourse vs. Mathematical Intuition

In Carlo Cellucci & Donald Gillies (eds.), Mathematical Reasoning and Heuristics. College Publications. pp. 137-165. (2005)

Carlo Cellucci
Università degli Studi di Roma La Sapienza (PhD)
The aim of this article is to show that intuition plays no role in mathematics. That intuition plays a role in mathematics is mainly associated to the view that the method of mathematics is the axiomatic method. It is assumed that axioms are directly (Gödel) or indirectly (Hilbert) justified by intuition. This article argues that all attempts to justify axioms in terms of intuition fail. As an alternative, the article supports the view that the method of mathematics is the analytic method, a method originally used by the mathematician Hippocrates of Chios and the physician Hippocrates of Cos, and first explicitly formulated by Plato. The article examines the main features of the analytic method, and argues that, while intuition plays an essential role in the axiomatic method, it plays no role in the analytic method, either in the discovery or in the justification of hypotheses. That the method of mathematics is the analytic method involves that mathematical knowledge is not absolutely certain but only plausible, but the article argues that this is the best we can achieve.
Keywords intuition  axiomatic method  analytic method  certainty
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How to Think About Informal Proofs.Brendan Larvor - 2012 - Synthese 187 (2):715-730.
Philosophy of Mathematics: Making a Fresh Start.Carlo Cellucci - 2013 - Studies in History and Philosophy of Science Part A 44 (1):32-42.
The Nature of Mathematical Explanation.Carlo Cellucci - 2008 - Studies in History and Philosophy of Science Part A 39 (2):202-210.
Theological Underpinnings of the Modern Philosophy of Mathematics.Vladislav Shaposhnikov - 2016 - Studies in Logic, Grammar and Rhetoric 44 (1):147-168.

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