Re: Roger Ebert comes out of the closet!!

On Sun, 06 Sep 2009 01:31:34 +0000, Garamond Lethe wrote:

On 2009-09-06, Walter Bushell <proto@xxxxxxxxx> wrote:
In article <QNnom.284960$E61.140928@xxxxxxxxxxxx>,
Garamond Lethe <cartographical@xxxxxxxxxxxxxx> wrote:

<q>
I'm not qualified to say whether or not God exists. I kind of doubt He
does. Nevertheless, I'm always saying that the SF [Supreme Fascist,
Erdos's title for God] has this transfinite Book that contains the
best proofs of all mathematical theorems, proofs that are elegant and
perfect... You don't have to believe in God, but you should believe in
the Book.
</q>

Paul Erdos

But one of the things that support the real existence of mathematical
objects is that the axiomatic method has been shown to be a failure. No
matter how many axioms we have there remain things that cannot be
proven or disproved by the axiom for the natural numbers.

I'd also consider the underwhelming success of automated theorem proving
as an indictment of the axiomatic approach.

Some of my professors
were quite adamant[1] about the independent reality of some
mathematical objects.

Poking snex aside, I would really prefer to believe that the natural
numbers are an entirely human construct, that several other rich and
viable systems could exist and we just happened to luck into the one
we're familiar with.

But of the many people I know who do math for a living, not one of them
believes this. One in particular trained as a physicist and is now a
theoretical biologist, and she emphatically considers herself a
neoplatonist.

I do have to admit, though, that I don't know any coders or computer
scientists who this this way. Programs (and algorithms) are still
thought to be created, not discovered. That may be due to the field
still being in its infancy, or maybe it's because our thinking is still
largely guided by the limitations of the machines. Or maybe CS folks
are much more comfortable with arbitrary manipulation of meaningless
symbols....

Time to reread "The Unreasonable Effectiveness of Mathematics", I think.


[1] Not the first ant as the word seems to shape.


I posted this here a few years ago, but I like it enough to post it
again:

"Maxwell's equations and the principles of quantum mechanics led to
the idea of gauge invariance. Attempts to generalize this idea, motivated
by physical concepts of phases, symmetry, and conservation laws, led
to the theory of non-Abelian gauge fields. That non-Abelian gauge
fields are conceptually identical to ideas in the beautiful theory of
fiber bundles, developed by mathematicians *without reference to the
physical world*, was a great marvel to me. In 1975 I discussed my feelings
with Chern, and said 'this is both thrilling and puzzling, since you
mathematicians dreamed up these concepts out of nowhere'. He immediately
protested: 'No, no. These concepts were not dreamed up. They were natural
and real'." - C.N. Yang, quoted in John Baez and Javier Muniain, "Gauge
fields, Knots, and Gravity", World Scientific, 1994.

I work as a computer programmer, but my academic background is in math,
and I certainly consider myself a Platonist (not a Neoplatonist, that's
a different thing) about mathematical objects. Dieudonne has it
right, formalism is a defensive posture to make the philosophers
go away and leave us alone. I don't think of algorithms as mathematical
objects, though, so you can believe what you like about those.

I wish I could be around when we make contact with some alien form
of intelligent life, to find out about their mathematics. Will they
know about logarithms and conic sections and imaginary numbers?
Groups and fields and vector spaces? Differential forms? Projective
geometry? Is it possible to have radio and space flight if you
haven't discovered calculus?

John

.

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