Re: Successor Axiom: on what grounds TF?

From: Jeffrey Ketland (ketland_at_ketland.fsnet.co.uk)
Date: 02/07/05 

Date: Mon, 7 Feb 2005 02:38:28 -0000

<examachine@gmail.com> wrote
> Jeffrey Ketland wrote:
>> We construct scientific theories as follows
>> (i) we assume that our mathematics (at least enough to do arithmetic,
>> analysis, diff. equations, linear algebra, group theory, differential
>> geometry, some topology, etc.) is true. This allows us to formulate
>> our theories, and express the concepts we need;
>> (ii) *Then* we go on to see if our theories can predict and explain.
>> And they do, quite successfully.
>
> First, this doesn't seem to be a methodology to construct scientific
> theories.

It is known as Hypothetico-Deductivism. That's only a part of the story, but
an important part.

> Computer Science, for instance, was not born because we assumed our
> mathematics is true and complete.

Why do you introduce the notion of "complete", an entirely different
concept? Turing certainly did assume that prior mathematics (e.g.,
arithmetic and analysis) was true. Assuming that our mathematics is true
means assuming it *sound*.
Actually, soundness *contradicts* completeness (for axiomatizable theories
of sufficient richness). This is Goedel's First Incompleteness Theorem.

>New mathematical concepts were
> needed. Same goes for I suppose, say, nanotechnology, molecular biology
> or complexity sciences etc.

Of course. But you are mixing up soundness and completeness, a common
confusion.

> Likewise, physics has required physicists and mathematicians to come up
> with innovations to adequately address physicists' conceptual
> requirements, that is in fact how calculus was born. It did not fall
> out of thin air AFAICT.

Again, this confuses soundness and completeness.

>> If the mathematics we routinely assume to be true isn't true, then
> this is
>> an astonishing miracle. Why would a grossly false fiction lead to
> correct
>> predictions?
>
> I have an objection to this point of view.
>
> You assume that there is such a thing as objective truth about
> mathematics.

You mean the notion of objective truth per se? Yes, despite the incoherence
of various relativists, Alfred Tarski defined it rather precisely (at least
for certain kinds of interpreted languages). The notion of objective truth
also played a central role in Goedel's discoveries of the incompleteness
theorems, etc. (See, e.g., Wang's books on Goedel and Feferman's article
"Kurt Goedel: Conviction and Caution" in Feferman 1998 _In the Light of
Logic_.)
So yes, I'm perfectly happy with the notion.

If you mean, "do I assume that mathematics is true"? Of course.

> How do you decide that, by assuming mathematical realism?
> At one level, mathematics is all about rigorous, meaningful
> definitions, but we can't assume some mathematical realism I suppose.

I assume mathematical realism every day. I assume that "7+5 = 12" is true,
and I assume that when I apply this in any given situation, the results will
be in accord with the associated fact that the union of 7 objects with 5
distinct objects gives a set with 12 objects.

> Furthermore, at least we can't say, "Ok we now have something called
> mathematical geometry which uses this set theoretical notion of R^n.
> Then it follows that this is the one and only true geometry also in the
> real world". I don't see how that follows. Not at all.

Thanks to Albert Einstein and others, we have a very good theory of
spacetime, a theory that works, that makes successful predictions and
explains a great deal. I say, "The theory's probably a good approximation to
the truth. Perhaps it will be revised in future. If it is revised, then the
future theory will have our present theory as a limiting case". You're
saying, "No, it's all false". I say, "If it's false, why does it make so
many correct predictions?".

> That is relying on incomplete information in my eyes.

All information is incomplete. This is the realist view.

> Also, not all mathematical conclusions are supposed to make physical
> sense (some mathematicians say!),

Of course, there's no obvious reason why, say, the axiom of choice should be
directly applicable in physics. There's no reason to suppose that the whole
of V is exemplified physically. Actually, AC is indeed used in some abstract
theorems in General Relativity. For example, to prove that certain kinds of
3-space have a unique maximal Cauchy development. So, the initial value
problem in GR uses the axiom of choice (actually Zorn's lemma).

But you are claiming that R^n doesn't make physical sense, and I am merely
pointing out that it's assumed in every physical theory we have.
This is like claiming that unobservable entities like genes don't make
sense, despite the fact that they are assumed in our highly successful
biological theories of heredity and the like.

>so you don't really think cardinal
> numbers, etc., reveal facts about our physical world, do you?

Yes, I do. Cardinal numbers are exemplars of applicability par excellence.
When I have N physical objects, and M distinct physical objects, I do indeed
expect that their union will have cardinality N+M. In fact, this is roughly
how a Turing machine program that computes N+M actually works (more exactly,
it concatenates two sequences, one of N strokes and one of M strokes).
Similarly, if I try to arrange N+1 objects into N slots in a one-to-one
manner, I do expect that I can't do it. Etc.

>If not,
> why the privilege for another arbitrary mathematical construct: R^n?
> Where are the points with no extension in physical space, how would you
> show somebody who asks you?

How do we "show" someone the Big Bang? Or how do we "show" them the fibre
bundle structure attached to space-time (assuming standard gauge theories).
How do we "show" them the Poincare group? Or the Riemann tensor? How do we
"show" them a Dirac spinor?
"Showing", it seems, is much over-rated.

In any case, we can easily point out that the points in physical spacetime
(understood as having a manifold structure) are in one-to-one correspondence
with quadruples (x_0, x_1, x_2, x_3) of reals. If you're worried about
points, then think about the properties of these quadruples, and transfer
what you know about them to how you expect points to behave.

>Would you say, well, read this set
> theory/topology textbook, it tells all about it?

Actually, yes. I'd say, "If you want to learn about the geometry of physics,
here's a nice textbook: Theodore Frankel 1997, _The Geometry of Physics_
(CUP)".
Just checking, I see no discussion of ultra-finitist or nominalist
philosophy in this textbook. Seems like the physicists are not much
interested in sceptical philosophical fairy tales. Rightly so. They're
trying to understand the world, and will happily use any mathematics that
happens to have been developed (even if it contradicts
finitist/nominalist/constructivist scruples).

--- Jeff
 

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