For intuitionists like L.E.J. Brouwer (1881-1966) the subject matter of mathematics is intuited non-perceptual objects and constructions, these being introspectively self-evident. Indeed, mathematics begins with a languageless activity of the mind which moves on from one thing to another but keeps a memory of the first as the empty form of a common substratum of all such moves. Subsequently, such constructions have to be communicated so that they can be repeated — i.e. clearly, succinctly and honestly, as there is always the danger of mathematical language outrunning its content.
How does this work in practice? Intuitionist mathematics employs a special notation, and makes more restricted use of the law of the excluded middle (that something cannot be p' and not-p' at the same time). A postulate, for example, that the irrational number pi has an infinite number of unbroken sequences of a hundred zeros in its full expression would be conjectured as undecidable rather than true or false. But the logic is very different, particularly with regard to negation, the logic being a formulation of the principles employed in the specific mathematical construction rather than applied generally. But what of the individual, self-evident experiences which raise Wittgenstein problems of private languages? Do, moreover, we have to construct and then derive a contradiction for a proposition like a square circle cannot exist rather than conceive the impossibility of one existing? And wouldn't consistency be more easily tested by developing constructions further rather than waiting for self-evidence to appear?
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