The implicit philosophical belief of the working mathematician is today the Hilbert-Bourbaki formalism. Ideally, one works within a closed system: the basic principles are clearly enunciated once for all, including (that is an addition of twentieth century science) the formal rules of logical reasoning clothed in mathematical form. The basic principles include precise definitions of all mathematical objects, and the coherence between the various branches of mathematical sciences is achieved through reduction to basic models in the universe of sets. A very important feature of the system is its non-contradiction ; after Gödel, we have lost the initial hopes to establish this non-contradiction by a formal reasoning, but one can live with a corresponding belief in non-contradiction. The whole structure is certainly very appealing, but the illusion is that it is eternal, that it will function for ever according to the same principles. What history of mathematics teaches us is that the principles of mathematical deduction, and not simply the mathematical theories, have evolved over the centuries. In modern times, theories like General Topology or Lebesgue’s Integration Theory represent an almost perfect model of precision, flexibility and harmony, and their applications, for instance to probability theory, have been very successful.
My thesis is: there is another way of doing mathematics, equally successful, and the two methods should supplement each other and not fight.
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