Re: Platonism
examachine_at_gmail.com
Date: 03/09/05
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Date: 9 Mar 2005 04:45:28 -0800
Gerry Myerson wrote:
> In article <422db205$0$556$b45e6eb0@senator-bedfellow.mit.edu>,
> tchow@lsa.umich.edu wrote:
>
> > In article <gerry-621401.11284208032005@sunb.ocs.mq.edu.au>,
> > Gerry Myerson <gerry@maths.mq.edi.ai.i2u4email> wrote:
> > >They teach you how to add fractions in high school. You don't
think
> > >adding fractions is mathematics? 99.99% of the people who have
been
> > >through high school probably do, and when it comes to what words
> > >mean, I think you have to go by what the overwhelming majority of
> > >speakers mean by those words.
> >
> > I don't think that the "majority rules" argument works here.
Different
> > people mean different things by the word "mathematics." When
someone is
> > trying to pin down one of those meanings, it's not a valid
objection to say
> > that most people use the same word with a different meaning.
>
> I take your point. I got into this discussion in response
> to Dave Rusin writing
>
> Math is sort of like the game of Nomic: there are just a few
initial
> hard-and-fast rules (like "you have to prove what you claim" and
> "you have to make clear what your axioms are") and everything else
> is just a consequence of that.
>
> Perhaps I should have contented myself by saying, that's what Math is
> to some of us, but to more people Math is adding fractions and
> solving quadratics.
I don't think Tim's explanation is satisfactory.
The only interesting thing about mathematics may be that the set of
concepts isn't fixed. There was no such thing as set theory just a
short time ago, but there was mathematics. Or maybe Tim wants to mean
that with axiomatization of set theory everything got fixed? I don't
think so.
In fact, it is our drive to find better concepts and methods that makes
mathematics interesting at all. A good example: category theory. How do
you get to a new theory like that?
The answer of course, if you're talking in the framework of axioms,
well, it's figuring out the axioms! A second answer: finding
intelligent ways to reach theorems. Tim's answer addresses this second
point, but I don't see how it addresses the first. Especially, figuring
out the axioms, it seems, is possible only in a larger context than a
few papers. So what makes an axiom of set theory "correct"? Depending
on what?
Also I suggest you to think why we have such a thing as graph theory.
Well, it seems graph theory simply follows from set theory, right?
Formally, a graph is just G=(V,E) a pair of sets, that's all? Then it
can't be too interesting or meaningful? Well, it's not that simple.
Graph theory has a lot of useful concepts of its own. And graph theory
is a theory of *something*, but that *something* is not some
meaningless toy like a board game or whatnot. It's part of our lives,
and something we cannot ignore. Graph theory is a theory of what? Why
did we construct something like graph theory? What is the cause of its
invention? These are the kinds of questions you have to answer, you
have to explain the evolution of mathematics if you are going to
explain it at all.
So, think about it, where can you "use" graph theory? Contrast this to
where you can use the cardinality of the continuum with. Which one is
more useful? In which context? Wittgenstein tells us that meaning is
use. For a moment, believe in him, and tell me how the meaning of
"graph" differs from "the number of entities in the continuum" (is it
not a "number", oh!). Contrast these now with the concept of "whole
number", and "fourier transform". Also put in "quadratics" if you want.
Now put in something like "proof theory". How do the uses change?
Also, we know that mathematics is basically open: there is space for
extension...
Otherwise, Tim's explanation fits to any modality where intelligence is
at work, not just mathematics. There is regularity *everywhere*, and
our common sense is responsible for figuring out a working description
of the world. Figuring out how to win the game. When you prove to
yourself that if you don't get out in five minutes, you are going to
miss the bus, this is not so much different than proving something in
mathematics. In fact, a well-known AI researcher, J. Schmidhuber makes
this case in his Godel machine:
http://www.idsia.ch/~juergen/goedelmachine.html
Godel machine proves itself such things to improve itself. Well, maybe
humans are not godel machines, but we still prove things to ourselves,
that's considered to be a hallmark of intelligence: finding out things
without having to do them.
So, regardless of what mathematicians dismiss as mere arithmetics or
simple numerical calculations, many logical thoughts fit into what Tim
describes as mathematical.
As a philosopher, I would say that if you are doing an action like
"counting", then I think you are investing in a mathematical activity.
It would be absurd to say that is not mathematical.
But there is a certain kind of *work* that is considered mathematics in
the mathematical community. That can be very well defined indeed, but
it has little to do with the general sense of intelligence or what
mathematical thought is (it seems). That is to say, most professional
mathematics does not carry this philosophical weather with it. It's
only a few people who're interested in foundations that ever cares
about these issues. (Precisely the point of this thread. Almost no
working mathematician cares whether the inherent Platonism in their
education is right or wrong :)
Regards,
-- Eray
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