Tom Caylor wrote: > 1Z wrote: > > Tom Caylor wrote: > > > > > > David and 1Z: > > > > > > How is exploring the Mandelbrot set through computation any different > > > than exploring subatomic particles through computation (needed to > > > successively approach the accuracies needed for the collisions in the > > > linear accelerator)? Is not the only difference that in one case we > > > have a priori labeled the object of study 'matter' and in the other > > > case a 'set of numbers'? Granted, in the matter case we need more > > > energy to explore, but couldn't this be simply from the sheer quantity > > > of "number histories" we are dealing with compared to the Mandelbrot > > > set? > > > > > > Tom > > > > > > > > A number of recent developments in mathematics, such as the increased > > use of computers to assist proof, and doubts about the correct choice > > of basic axioms, have given rise to a view called quasi-empiricism. > > This challenges the traditional idea of mathematical truth as eternal > > and discoverable apriori. > > In either case, with math and matter, our belief is that there is an > eternal truth to be discovered, i.e. a truth that is independent of the > observer.

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"Eternal" doesn't mean "independent of the observer". Empirically-detectable facts are often fleeting. > > According to quasi-empiricists the use of a > > computer to perform a proof is a form of experiment. But it remains the > > case that any mathematical problem that can in principle be solved by > > shutting you eye and thinking. Computers are used because mathematians > > do not have infinite mental resources; they are an aid. > > In either case, an experiment is a procedure that is followed which > outputs information about the truth we are trying to discover. Math > problems that we can solve by shutting our eyes are solvable that way > because they are simple enough. As you point out, there are math > problems that are too complex to solve by shutting our eyes. In fact > there are math problems which are unsolvable. I think Bruno > hypothesizes that the frontier of solvability/unsolvability in > math/logic is complex enough to cover all there is to know about > physics. Therefore, what role is left for matter? Physical truth is a tiny subset of mathematical truth. > > Contrast this > > with traditonal sciences like chemistry or biology, where real-world > > objects have to be studied, and would still have to be studied by > > super-scientitists with an IQ of a million. In genuinely emprical > > sciences, experimentation and observation are used to gain information. > > In mathematics the information of the solution to a problem is always > > latent in the starting-point, the basic axioms and the formulation of > > the problem. The process of thinking through a problem simply makes > > this latent information explicit. (I say simply, but of ocurse it is > > often very non-trivial). > > The belief about matter is that there are basic properties of matter > which are the starting point for all of physics, and that all of the > outcomes of the sciences are latent in this starting point, just as in > mathematics. You can't deduce the state of the universe at time T in any detailed way from the properties of matter, you have to get out your telescope and look. > > The use of a computer externalises this > > process. The computer may be outside the mathematician's head but all > > the information that comes out of it is information that went into it. > > Mathematics is in that sense still apriori. > > Having said that, the quasi-empricist still has some points about the > > modern style of mathematics. Axioms look less like eternal truths and > > mroe like hypotheses which are used for a while but may eventualy be > > discarded if they prove problematical, like the role of scientific > > hypotheses in Popper's philosophy. > > > > Thus mathematics has some of the look and feel of empirical science > > without being empricial in the most essential sense -- that of needing > > an input of inormation from outside the head."Quasi" indeed! > > I'd say that the common belief of mathematicians is that axioms are > just a (temporary) framework with which to think about the invariant > truths. The "truths" are not invariant with regard to choice of axioms. Consider Euclid's fifth postulate. > And one of the most important (unspoken) axioms is the > convenient "myth" that I don't need any input from outside my head, so > that I can have "total" control of what's going on in my head, an > essential element for believing the outcome of my thinking. However, > the fact is that a mathematician indeed would not be able to discover > anything about math without external input at some point. This is the > process of learning to think. You need to learn axioms and rules of inference. Everything else is implicit in them. > Tom --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~----------~----~----~----~------~----~------~--~---