Tom Caylor wrote:
> 1Z wrote:
> > 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.
> >
>
> We're getting into the typical bifurcation of interpretation of terms.
> When you used the term "eternal" to describe math truth, I assumed you
> were talking about something that is independent of time.
Yes. Which physical truth isn't.
> > > > 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.
> >
>
> This agrees with what I am saying.
And mathematics per se does not tell us which
subset is physical. Empirical investigation has
to be used.
> > > > 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,
>
> This is a subject of debate.
If *you* can do it, I'd like to discuss next
week's race results with you.
> > you have to
> > get
> > out your telescope and look.
> >
>
> A telescope could be a way of looking at the state of the computation
> of the universe. This doesn't preclude being able to in theory compute
> the universe (in 3rd pov).
Fundamental indeterminism does.
> > > > 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.
> >
>
> Euclid's fifth postulate is an axiom.
..of Euclidean geometry. Which has "truths" -- such as "the sum of the
internal angles of a triangle is 180 degrees -- which aren't true in
non-Euclidean geometry.
> > > 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.
> >
>
> Discovery is not simply a matter of seeing where a particular set of
> axioms and rules of inference leads.
It is is mathematics.
> It's only when you see the truth
> from different perspectives, i.e. different sets of axioms and rules of
> inference, that you can start putting together a picture that gets
> closer and closer to reality.
So what about the conflicts between Euclidean and non-Euclidean
geometry? Maths alone will not tell you which is true of our universe.
> Tom
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Received on Wed Oct 25 2006 - 07:56:24 PDT