liza_kozner at yahoo.co.uk
Thu Dec 8 16:37:27 PST 2005
"it gradually became clear that the proper business of the pure mathematician is to derive theorems from postulated assumptions, and that it is not his concern as a mathematician to decide whther the axioms he assumes are actually true. An finally , these successful modifications of orthodox geometry stimulated the revision and completion of the axiomatic bases for many other mathematical systems " etc
Moreover, intuition is not a safe guide: it cannot properly be used as a criterion of either truth or fruitfulness in scientific explorations.
However, the increased abstractness of mathematics raised a more serious problem. It turned on the question whether a given set of postulates serving as foundation of a system is internally consistent, so that no mutually contradictory theorems can be deduced from the postulates. The problem does not seem to be pressing when a set of axioms is taken to be about a definite and familiar domain of objects; for then it is not only significant to ask, but it may be possible to ascertain, whether the axioms are indeed true of these objects. Since the Euclidean axioms were generally supposed to be true statements about space (or objects in space), no mathematician prior to the 19th century ever considered the question whether a pair of contradictory theorems might some day be deduced from the axioms. The basis for this confidence in the consistency of Euclidean geometry is the sound principle that logically incompatible statements cannot be simultaneously true; accordingly , if a set of
statements is true (and this was assumed of the Euclidean axioms), these statements are mutually consistent.
The non-Euclidean geometries were clearely in a different category. Their axioms were initially regarded as being plainly false of space, and, for that matter, doubtfully true of anything; thus the problem of establishing the internal consistency of non-Euclidean systems was recognized to be both formidable and critical. In Riemannian geometry, for example, Euclid's parallel postulate is replaced by the assumption that through a given point outside a line _no_ parallel to it can be drawn. Now suppose the question: Is the Riemannian set of postulates consistent? The postulates are apparently not true of the space of ordinary experience. How then, is their consistency to be shown? How can one prove they will not lead to contradictory theorems? Obviously the question is not settled by the fact that the theorems already deduced do not contradict each other-- for the possibility remains that the very next theorem to be deduced may upset the apple cart. But until the question is settled,
one cannot be certain that Riemannian geometry is a true alternative to the Euclidian system, ie., equally valid mathematically. The very possibility of non-Euclidean geometries was thus contingent on the resolution of this problem. A general method for solving it was devised. [etc]
Harald, as you see, the start of the Nagel and Newmann book on Gödel's Theory, deals with the discovery of assumption since the start of Mathematics. ( previously to these statements they dealt with how assumption discovery led to a kind of panic in the 19th century, where it was feared: mutually contradictory theorems, they have taken the parallel definition of Euclid and demonstrated this fear from the discovery that it was all based on assumption and could be contradictory and thus not self consistent) To me assumption is then a sign of a system with a blocked line of deterritorialisation towards the plane of consistency. And what I am asking myself, is: indeed the question lay interesting question: how to make up a space of consistency from all this. How to use the ressource , the power of this system with relative line of deterritorialisation (blocked positivity that could lead to consistency) in non Deleuzian language this is: how to deterritorialize mathematics, and give it
absolute freedom , no need for extra referent, like ordinary space. something valid for everybody who would not assume, but create? thus we would escape the Paul Klee's malediction, that the people lacks? It's very very difficult, but it would lead us away from the black holes, and make a new Earth. Be capable to affect with intensities, not to look for intensities, is the artist mission. The consumers of Art want intensities, the masses want intensities. It's all reversed!
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