On 29 May 2012, at 08:46, meekerdb wrote:

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On 5/28/2012 1:13 PM, Bruno Marchal wrote:On 28 May 2012, at 18:02, meekerdb wrote:On 5/28/2012 12:36 AM, Bruno Marchal wrote:On 27 May 2012, at 20:59, meekerdb wrote:On 5/27/2012 5:02 AM, Bruno Marchal wrote:As Bruno said, "Provable is always relative to some axioms andrules of inference. It is quite independent of "true ofreality". Which is why I'm highly suspicious of ideas likederiving all of reality from arithmetic, which we know onlyfrom axioms and inferences.We don't give axioms and inference rule when teachingarithmetic in high school. We start from simple examples, likefingers, days of the week, candies in a bag, etc. Childrenunderstand "anniversary" before "successor", and the finite/infinite distinction is as old as humanity.In fact it can be shown that the intuition of numbers, additionand multiplication included, is *needed* to even understandwhat axioms and inference can be, making arithmetic necessarilyknown before any formal machinery is posited.But only a small finite part of arithmetic.I don't think so. Our arithmetical intuition is already notformalizable. If it was, we would be able to capture it by afinite number of principle, but then we would be persuade thatsuch finite theory is consistent, and that intuition is not inthe theory.I suspect that our intuition is full second order arithmetic,which is not axiomatizable. In fact it is the very distinctionbetween finite and infinite that we cannot formalize. Likeconsciousness, we know very well what finite/infinite means, butwe cannot defined it, without using implicitly that distinction.The natural numbers are *the* mystery, and it has to be likethat: no machine will ever been able to define what they are.Assuming comp, neither will we. Arithmetical truth per se, as nocorresponding complete TOE. It is inexhaustible.BrunoOur intuition is that space is euclidean, the earth is stationaryand flat, and that there is only one world. It seems to me thatthe infinity of arithmetic is just the intuition we should alwaysbe able to add one more.Not really. I think there are complete theories of (N, successor).But we have an intuition of adding and multiplying and this makesthat intuition inexhaustible.Intuition is not entirely a given, it is something which developwith the familiarity and life working. It is different for all ofus, so it nice that we can share some big initial segment of thearithmetical truth. Comp does not need more than the sigma_1intuition, at the ontic level.But intuition fails us in precisely in questions like Hilbert'shotel.Why? Not sure, but it does not concern us, as comp builds on theintuition of the finite things.Why should you be so trusting of your intuition is just thisparticular instance.Do you doubt elementary arithmetic?I doubt infinities.

`I can doubt actual infinities. Not potential infinities, which gives`

`sense to any non stooping program notion.`

`Comp is ontologically finitist. As long as you don't claim that there`

`is a biggest prime number, there should be no problem with the comp`

`hyp. Infinities can be put in the epistemology, or at the meta-level:`

`they are mind tool, souls attractor etc.`

Bruno

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