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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Κυριακή, 4 Δεκεμβρίου 2011

Constructible sets

Πηγή : Scribd Badiou-Alain-Politics-A-nonexpressive-dialectics

There is a very clear mathematical example of this relation between desire and law, between different forms of existence, which is really interesting. Don’t be afraid, it’s very simple. I think mathematics is very often something which is linked to ter- ror. And I am always speaking from a non-terrorist conception of mathematics…

Suppose that we are in the theory of sets – we have a theory of the pure multiplicity – and suppose we consider one set, no matter what set; a multiplicity absolutely ordinary. The interesting thing is that with some technical means we can for- malise the idea of a subset of this set which has a clear name. So the question of the relation between existence and clear name has a possible formalisation in the field of the mathematical theory of sets. Precisely, to have a clear name is to be defined by one clear formula. It was an invention of the greatest logician of the last century, Kurt Gödel. He named that sort of subset a con- structible subset; a constructible subset of a set is a set which has a clear description. And generally speaking we name con- structible set a set which is a constructible subset of another set.

So, we have here the possibility of what I name a great law. What is a great law? A great law is a aw of laws, if you want: the law of what is really the possibility of a law. And we have a sort of math- ematical example of what is that sort of law, which is not only a law of things or subjects, but a law for laws. The great law takes the form of an axiom, the name of which is the axiom of con- structibility and which is very simple. This axiom is: all sets are constructible. You know that is a decision about existence: you decide that exist only sets that are constructible, and you have as a simple formula a simple decision about existence. All sets are constructible, that is the law of laws. And this is really a possibil- ity. You can decide that all sets are constructible. Why

Because all mathematical theorems which can be demonstrated in the general theory of sets can also be demonstrated in the field of con- structible sets. So all that is true of sets in general is in fact true for only constructible sets. So – and it’s very interesting about the question, the general question of the law – we can decide that all sets are constructible, or if you like that all multiplicity is under the law, and we lose nothing: all that is true in general is true with the restriction to constructible sets. If you lose nothing, if the field of truth is the same under the axiom of constructibility, we can say something like: the law is not a restriction of life and thinking; under the law, the liberty of living and thinking is the same. And the mathematical model of that is that we don’t lose anything when we have the affirmation that all sets are constructible, that is to say all parts of sets are constructible, that is to say finally all parts have a clear definition. And as we have a general classifica- tion of parts, a rational classification of parts; classification of society if you want – without any loss of truth. At this point there is a very interesting fact, a pure fact. Practically no mathematician admits the axiom of constructibili- ty. It’s a beautiful order, in fact, it’s a beautiful world: all is con- structible. But this beautiful order does not stimulate the desire of a mathematician, as conservative as he might be. And why? Because the desire of the mathematician is to go beyond the clear order of nomination and constructibility. The desire of the mathematician is also the desire for a mathematical monster. They want a law, certainly – difficult to do mathematics without laws – they want a law but the desire to find some new mathematical monster is beyond this law.

The mathematical example is very striking. After Gödel, the def- inition of constructible sets, and the refusal of the axiom of con- structibility by the majority of mathematicians, the question of the mathematician’s desire becomes: how can I find a non-con- structible set? And you see the difficulty, which is of great politi- cal consequence. The difficulty is, how can we find some mathe- matical object without clear description of it, without name, with- out place in the classification: how to find an object the character- istic of which is to have no name, to not be constructible, and so on. And the very complex and elegant solution was found in the sixties by Paul Cohen. He found an elegant solution to name, to identify, a set which is not constructible, which has no name, which has no place in the great classification of predicates, a set which is without specific predicate. It was a great victory of desire against law, in the field of law itself, the mathematical field. And like many things, many victories of this type, it was in the sixties. And Paul Cohen gives the nonconstructible sets a very beautiful name: generic sets. And the invention of generic sets is something in the revolutionary actions of the sixties.