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/ --~--~---------~--~----~------------~-------~--~----~ 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 [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~----------~----~----~----~------~----~------~--~---
