Michael Detlefsen (1987).
Abstract
This paper aims to give a clear exposition of two critiques of logicism. The intuitionist, Brouwer, is best known for rejecting the law of the excluded middle (his “special critique.”) The paper argues that Brouwer’s “general critique” is deeper and more extensive than this. That is, Brouwer is arguing that mathematical knowledge requires a kind of direct mathematical experience, analogous to sensory experience and this cannot be attained simply by linguistic operations independently of experience. It also discusses Poincare’s critiques – that there is a huge difference between genuine mathematical insight and an ability to logically manipulate mathematical truths and that different areas of mathematics have distinctive forms of reasoning.
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Proof and Intuition