Le 20-oct.-06, à 17:04, 1Z a écrit :
> As usual, the truth of a mathematical existence-claim does not
> prove Platonism.
By Platonism, or better "arithmetical realism" I just mean the belief
by many mathematician in the non constructive proof of "OR" statements.
Do you recall the proof I have given that there exists a couple of
irrational numbers a and b such that a^b is rational? The proof was not
constructive and did show only that such a number was in a two element
set without saying which one. AR means we accept such form of
reasoning. Formally it means I accept that the principle of excluded
middle holds for the arithmetical propositions (that is those build in
first order predicate calculus + the symbols =, 0, s, +, *).
For example I believe that either every positive integer bigger than
four can be expressed as the sum of two primes or there is a positive
integer which is bigger than four and which cannot be written as the
sum of two primes. This is exactly what I mean by being "platonist" or
better "realist" about numbers and their relations. I put it explicitly
in the hypotheses for avoiding sterile debates with
ultra-constructivists or ultra-intuitionist. Note that I have no
problem with moderate constructivism/intuitionism: non classical
results can be recasted there through the use of the double negation.
Without AR I am not sure CT makes any sense.
Bruno
http://iridia.ulb.ac.be/~marchal/
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Received on Mon Oct 23 2006 - 07:40:55 PDT