Re: Surviving Einstein.

From: Douglas A. Gwyn (DAGwyn_at_null.net)
Date: 07/18/03 

Date: Fri, 18 Jul 2003 19:18:52 GMT

Nicol So wrote:
> The questions Simon posed are all meaningful and have well-known
> answers. Natural numbers, unlike physical objects, are not "produced" by
> a "process". You seem to think that a natural number does not exist
> until it is "produced" by an execution of a "process". That's not true.

Actually this is tied up with what you mean when you say that a given
thing (number in this case) "exists". There are multiple forms of
existence, and one relevant for this discussion is tanatamount to
"(conceptually) producible". In fact a whole school of mathematics
was founded during the 20th century that took essentially that
approach. (Brouwer et al.)

Yet even so, logical meaning can be given to many uses of the term
"infinite". This is made precise in a (what used to be) standard
course in calculus.

E.g., the set of natural numbers is unbounded => there are an
infinite quantity of natural numbers.

One of the oldest mathematical proofs was of the infinitude of
primes, and proceeded by deriving a contradiction to the contrary
assumption. Proof by contradiction is an ancient, well established
technique.

> Not true. It is meaningful to talk about the cardinality of the set of
> natural numbers. Finite cardinals have names, quite obviously. Some
> infinite cardinals have names, too. The cardinality of N happens to be a
> named one, aleph_0, which is (and can be easily shown to be) the
> smallest infinite cardinal.

A logical alternative would be to maintain that cardinality is
not applicable to infinite sets. That is evidently a consistent
position, whereas there are troublesome issues associated with
the other (Cantorian) position. The development that the
continuum hypothesis is independent of ZF set theory could be
taken as showing a disconnect between the things we have real
use for and some other things that mathematicians can dream up.
That is not to say that mathematical inventions cannot be useful
in unexpected ways; history shows that often they can.

I would say that the important thing for this thread is having
a whole logical (consistent) system in agreement with
experience, not just carping at particular facets of others'
systems. So far, Stonelock's ideas haven't appeared to be
justified by the accumulated evidence of experience.

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