Re: Why real numbers and points can't model continuum

On Sat, 24 Nov 2007 21:46:27 -0800 (PST), mike3
<mike4ty4@xxxxxxxxx> wrote:

On Nov 24, 6:38 pm, Venkat Reddy <vred...@xxxxxxxxx> wrote:

The question is about the
attempt to model the continuum using points and real numbers, which
have zero measure and trying to fill up space using them.

Attempt to "model" the continuum using real numbers? The
real numbers *ARE* the continuum, not just a "model" of it.
You need to remember that: The Real Numbers ARE the
Continuum, or are "a" continuum. They are not merely a
"model" of it. "The Continuum", in the singular, refers
to the real numbers (R), not a model of it but the reals
themselves.

It's a pity those ancient Greeks were too dumb to realise this.

Clearly they should have recognised their limitations, given up
trying to do any mathematics at all, and just shut up shop and
waited humbly a couple of millennia for Weierstrass, Dedekind,
Cantor et al. to come along and explain it to them.

Of course Venkat is mistaken if he imagines that he has proved
that classical analysis is useless or self-contradictory!

But it doesn't follow that his own independent speculation is
itself useless or self-contradictory. To find a flaw in it,
one must accept his intuitions, in a sense "work within his
system", and see if he is thinking clearly about it, or has
missed some implication. Of course the same goes for him, and
to find a flaw in classical mathematics he must first learn it!
But the situation is not symmetrical. His speculations will
not deprive anyone of their access to established mathematics,
whereas, on the other hand, too premature an attempt to make
him recapitulate nearly three millennia of cultural development
might well deprive him (and indeed the rest of us) of access
to his own intuitions.

I actually can't see any need for this debate to be at all
heated. (A slightly disingenuous statement, as I confess at
once!) As you say yourself:

You have to quit thinking of
points as little BBs or something like that. Infinity
is _not_ an intuitively obvious concept that common sense
works well with. Quite probably because common sense is
just that -- COMMON sense, and infinity is far from our
common experience.

Aren't we all on the same footing, in that, however much or
however little we know about established mathematics, it is
hard for any of us to be intuitively clear about what it
means, and how it relates to our own experience? And isn't
there plenty of room for any of us to speculate, if we want
to? There is little point in having a cultural history at
all if we allow its massive weight to crush our curiosity!
And indeed, if it hadn't been for people refusing to do so,
we really wouldn't have a cultural history at all!

That said, mathematics has an underlying scientific impulse,
which requires us to test our intuitions against reality,
and the sense of the crushing weight of tradition may serve as
a reminder of the many lessons hard-learned during those three
millennia. To some extent, mathematics may reflect a world
which we absolutely cannot understand intuitively at all, but
we can only (was it von Neumann who said something like this
about mathematics?) "get used to it".

/That/ said, our intuitions still have to go somewhere, and
it can hardly be a good idea just try to crush them guiltily
out of existence, so why shouldn't we at least continue to
speculate playfully? Especially as some real advances in
mathematics are likely to come from what starts as playful
speculation, and so it shouldn't even be regarded as a mere
childish indulgence, but should be respected (even when it
itself fails to respect a tradition deserving of respect).
--
Angus Rodgers
(twirlip@ eats spam; reply to angusrod@)
Contains mild peril
.

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    ... Continuum, or are "a" continuum. ... not a model of it but the reals ... one must accept his intuitions, in a sense "work within his ... to find a flaw in classical mathematics he must first learn it! ...
    (sci.math)