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.
"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
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Received on Tue Oct 24 2006 - 15:15:28 PDT