On 25 Nov 2012, at 20:53, Stephen P. King wrote:

On 11/25/2012 9:22 AM, Bruno Marchal wrote:

On 24 Nov 2012, at 14:00, Roger Clough wrote:

Hi Bruno Marchal

Poincare had a lot to say on intuition vs logic in mathematics.
For one thing, the idea of continuity is an intuition.

http://www-history.mcs.st-and.ac.uk/Extras/Poincare_Intuition.html

Poincaré, like Kronecker were intuitionist before and after Brouwer (the founder of intuitionism).

It is the philosophy of the first person of the machine, but it is an error to extrapolate it to reality (at least in the comp theory). It is epistemologically correct, but ontologically disastrous, as it leads to different form of solipsism.

Bruno



Dear Bruno,

Please elaborate in these remarks as to how intuitionism is, among other things, 'ontologically disastrous'.

It is the logical-mathematical version of solipsism. What exist is what you can construct. It forbid the proof by absurdum of existence and negation.

But with comp, we need only arithmetic, and the intuitionist double negation ◊p = ~~p, (not not p), can be used to recover the notion of non constructive proof, and so are technically equivalent. It is in Analysis that intuitionism and classical logic differ substantially.




It seems to me that intuitionism is a more general logical framework that can, if constructed carefully, allow us to make correct predictions when we are considering finite approximations to the 'perfect' Platonic notions.

There is just no Platonia for an intuitionist. It makes no sense. In comp intuitionism is really the logic of the constructive self- extending self, and it plays the role of the first person, which is coherent with the fact that the 1p is solipsist de facto, even if he has been civilized and believe in the other people, but that belief is not intuitionnistic.




It is one thing to have ontological theories that are based on 'ideal' conditions and it is another thing to have the means to make local calculations and approximations.

I can be OK with this. Such differences arguably appears in the comp theories too.




Let us never make Perfection the enemy of the effective.

God and arithmetical truth can be said to be perfect. A lot of imperfections exists and that is part of what we have to explained. And comp leads to an explanation close to the neoplatonist "fall of the soul".

Bruno


http://iridia.ulb.ac.be/~marchal/



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