"... intuitionistic mathematics is an essentially languageless activity of the mind having its origin in the perception of a move of time. This perception of a move of time may be described as the falling apart of a life moment into two distinct things, one of which gives way to the other, but is retained by memory."

Luitzen Brouwer

I can imagine somebody counting, beginning with 1 and then creating (more and more of) the natural numbers. But I cannot imagine the reverse process, somebody beginning with all of the natural numbers and then taking away one after the other.
This is the main motivation for intuitionism in mathematics.

The platonistic view assumes that the natural numbers are "just there" as described by Peano's axioms. I guess platonism is the majority view among mathematicians.

Perhaps this division mirrors the two different approaches I mentioned in my previous blog post and the Grundlagenstreit was/is another instance of the debate about it?

## 3 comments:

Wolfgang, as far as you know, is there any part of Cantor's work that an intuitionist could accept?

As far as L.B. himself is concerned I would guess no, but I am not an expert in constructive set theory and if/what parts of standard set theory it can recover...

On 2nd thought, both Cantor and Brouwer agreed that there is a fundamental difference between natural and real numbers - but of course L.B. would not have accepted Cantor's quantification of that difference.

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