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.

> 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?

> 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.

> 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.  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.

Tom


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