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 only fromaxioms and inferences.We don't give axioms and inference rule when teaching arithmeticin high school. We start from simple examples, like fingers, daysof the week, candies in a bag, etc. Children understand"anniversary" before "successor", and the finite/infinitedistinction is as old as humanity.In fact it can be shown that the intuition of numbers, additionand multiplication included, is *needed* to even understand whataxioms and inference can be, making arithmetic necessarily knownbefore 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 a finitenumber of principle, but then we would be persuade that such finitetheory is consistent, and that intuition is not in the theory.I suspect that our intuition is full second order arithmetic, whichis not axiomatizable. In fact it is the very distinction betweenfinite and infinite that we cannot formalize. Like consciousness,we know very well what finite/infinite means, but we cannot definedit, without using implicitly that distinction. The natural numbersare *the* mystery, and it has to be like that: no machine will everbeen able to define what they are. Assuming comp, neither will we.Arithmetical truth per se, as no corresponding complete TOE. It isinexhaustible.BrunoOur intuition is that space is euclidean, the earth is stationaryand flat, and that there is only one world. It seems to me that theinfinity of arithmetic is just the intuition we should always beable 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 makes that`

`intuition inexhaustible.`

`Intuition is not entirely a given, it is something which develop with`

`the familiarity and life working. It is different for all of us, so it`

`nice that we can share some big initial segment of the arithmetical`

`truth. Comp does not need more than the sigma_1 intuition, at the`

`ontic level.`

But intuition fails us in precisely in questions like Hilbert's hotel.

`Why? Not sure, but it does not concern us, as comp builds on the`

`intuition of the finite things.`

Why should you be so trusting of your intuition is just thisparticular instance.

Do you doubt elementary arithmetic? Bruno http://iridia.ulb.ac.be/~marchal/ -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.