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Topic: Mathematics as a language
Replies: 35   Last Post: Nov 8, 2010 1:53 AM

 Messages: [ Previous | Next ]
 herb z Posts: 1,187 Registered: 8/26/06
Re: Mathematics as a language
Posted: Nov 7, 2010 2:15 AM

Daryl McCullough wrote:

> Now, there might be a way to weasel out of it by talking
> about "possible universes": Hercules exists in one [logically] possible
> universe, and Heavystone exists in a different [logically] possible
> universe. But then you're waffling about the meaning of the word "exists".

The meaning of the term "mathematically exists" is indeed at issue here.

> Do you mean exists within *one* universe, or exists within *any* universe?

I really am having difficulty understand what is being asked, so I don't
know how to answer it.

Hercules exists in one logically possible universe, Heavystone exists in
a different logically possible universe. Both possible universes exist
within the universe of logically possible universes -- I don't see the
problem.

It's like this -- suppose we find in a text on number theory that
there "exists" a unique prime decomposition for every natural
number n > 1.

If we have some doubt about the nature of the existence of the
naturals themselves, how much more so about these more complicated
objects, prime decompositions of naturals? What sort of "existence"
do they have?

But really, in ordinary use, there's no mystery at all. For the
purpose of doing number theory, we accept the existence of naturals
with their (axiomatically?) given properties, whether as hypothetical
entities or as eternal Platonic objects, or whatever, and it then
follows that for these entities, whatever their nature, there will
be a unique prime decompostion for each one greater than one. That's
all that we mean or need to mean about "existence" to do number theory.

There is no question, unless we're pretty confused, that the existence
of these "prime number decompositions" is like the material existence
of the Empire State building, having a definite position in space and
time, etc. It's just that given that one thing "exists", it follows
that another thing "exists". This is an utterly ordinary usage:
nothing deep is going on here. When we study number theory, we
don't worry about the ontology of naturals -- their "existence"
is simply accepted as posited.

And that's really all there is to it. We can posit anything we want
(e.g. Conway's "surreal" numbers") and reason about what properties
these posited objects "have" -- that is, what properties of these
objects "exist".

It's just a way of talking -- we suspend disbelief, and take them
as having "real" existence of a sort. And until a contradiction
turns up, the game is a good game. You never know, it might usefully
model some phenomena in the physical world -- it's happened before.

That's what I mean when I say that mathematical existence is logically
possible existence. It's really very ordinary.

--
hz

Date Subject Author
11/2/10 Aatu Koskensilta
11/3/10 herb z
11/3/10 Herman Jurjus
11/3/10 Marshall
11/3/10 Herman Jurjus
11/4/10 herb z
11/4/10 Marshall
11/5/10 herb z
11/5/10 Herman Jurjus
11/6/10 herb z
11/6/10 James Dolan
11/6/10 Tim Little
11/6/10 Daryl McCullough
11/6/10 Marshall
11/6/10 Brian Chandler
11/6/10 Tim Little
11/7/10 lwalke3@lausd.net
11/8/10 Brian Chandler
11/7/10 herb z
11/7/10 Daryl McCullough
11/8/10 herb z
11/3/10 lwalke3@lausd.net
11/3/10 Marshall
11/4/10 herb z
11/4/10 herb z
11/4/10 herb z
11/3/10 Daryl McCullough
11/4/10 Bill Taylor
11/4/10 Daryl McCullough
11/5/10 herb z
11/4/10 herb z
11/4/10 Daryl McCullough
11/5/10 herb z
11/5/10 Daryl McCullough
11/4/10 VK