Δευτέρα, 26 Δεκεμβρίου 2011

Badiou : Για τον Ευκλείδη και τους "πρώτους" αριθμούς (via Q Meillassoux)

Πηγή:Quentin Meillassoux, Histoire et événement chez Alain Badiou

Let us take the mathematical example, a very seminal procedure of thought for Badiou. That is to say the arithmetic theorem which states, in contemporary terms, that there are an infinity of prime numbers. It is known that Euclid had already demonstrated this theorem in the Elements, and one could thus deduce from this the restlessness of an eternal truth, intangible and unchanged by history, as true for a Greek as for a contemporary, and concealing the same kernel of significance for one as for the other. But the partisan of historical relativism, as self-styled "anthropologist of cultures" will underline our naivety, making apparent that the equivalence of two statements, present in two different cultural worlds, do not have a common truth - which already revealed by a difference in their formulation. Euclid, indeed, could not demonstrate the infinity of the prime numbers, since infinite arithmetic did not have any meaning for a Greek. It simply demonstrated that prime numbers were always higher in quantity than a given (finite) quantity of prime numbers. Other differences in formulation will end up convincing our relativist that the two statements support an incommensurable truth.

Badiou retorts that this naive illusion is on the side of the anthropologist, and not of the mathematician. Because the Greeks had discovered, via this theorem, a truth essential for number. The demonstration of Euclide, in effect, proceeds as a demonstration that any whole number is decomposable into prime factors. But Badiou insists that this truth always governs contemporary mathematics, in particular modern abstract algebra. This covers, in a given operational domain, the definition of operations similar to those of addition or multiplication, but also proceeds to break up its "objects" into primitive objects, in the same way that a number is always decomposable into prime numbers. There is thus, across the centuries and cultural and anthropological worlds, these truths which, though eternal, are not fixed but produce the sole authentic history: that of fertile theoretical gestures, always recommencing in diverse contexts, with the same fidelity, and yet at the same time the results of innovators.

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