Intellectica 2009/1, n° 51

The Mathematical Continuum. New Conceptions, New Challenges

Michel De Glas

Getting away from the Cantorian Hell

 Abstract: Having briefly recalled the paradoxes conveyed by the mathematical continuum and the infinite, inherited from the founding works of Cantor and Dedekind, we show that the various mathematical constructivisms, despite their neo-aristotelian inspiration, prove to be unable to escape mathematical platonism. Thsi problem actually is, not only, one the key problems in the philosophy of mathematics, but also a problem to the solution of which cognitive science may greatly contribute. The role played by topology in the field of mathematics as well as in the elaboration of mathematical models in cognitive science is typical of this submission to this set-theoretical orthodoxy (punctiform continuum, actual infinite). Locology may be seen as an alternative to topology, a new Analysis situs. Localistic logic, which is based upon the locological substratum, aims at redefining the outline of logical and mathematical constructivism.

 Key words: Punctiform continuum, actual infinite, actualism/constructivism, topology, mereology, non standard analysis, locology, localistic logic, loci theory.