On 5/27/2012 5:02 AM, Bruno Marchal wrote:
As Bruno said, "Provable is always relative to some axioms and rules of inference. It
is quite independent of "true of reality". Which is why I'm highly suspicious of
ideas like deriving all of reality from arithmetic, which we know only from axioms and
inferences.
We don't give axioms and inference rule when teaching arithmetic in high school. We
start from simple examples, like fingers, days of the week, candies in a bag, etc.
Children understand "anniversary" before "successor", and the finite/infinite
distinction is as old as humanity.
In fact it can be shown that the intuition of numbers, addition and multiplication
included, is *needed* to even understand what axioms and inference can be, making
arithmetic necessarily known before any formal machinery is posited.
But only a small finite part of arithmetic.
Brent
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