Re: Does the Calculus rest on Euclid?

On Thu, 08 Jun 2006 09:26:29 -0400, Hatto von Aquitanien
<abbot@xxxxxxxxxxxxxx> wrote:

David C. Ullrich wrote:

On Wed, 07 Jun 2006 13:44:23 -0400, Hatto von Aquitanien
<abbot@xxxxxxxxxxxxxx> wrote:

I suspect the answer to this may be 'Yes.', 'No.', and 'It depends.'. I
have never felt satisfactorily convinced that the transition from the
Riemann sum approximation to a smooth curve is logically founded upon
axioms I have assumed at the outset. These axioms are those of formal
logic and those of Euclid.

Well, those are not the axioms that are used in the
standard approach to a rigorous treatment of calculus.

I've checked several so-called advanced treatments of analysis, and to be
quite honest, they don't seem to state their assumptions very clearly. I
will concede that there is no need to appeal to Euclid in the calculus of
single variable functions. Nonetheless, I do believe there is an
assumption of continuity which is geometric in nature, and it is at the
foundation of all our mathematical reasoning.

Believe what you want. It's not so. You start with the integers,
construct the rationals from them. Now you define the reals as
follows:

A real number is a set S of rationals with the following properties:

(i) S is non-empty
(ii) S does not contain every rational
(iii) If s, t are rational, s is in S and t > s, then t is in S
(iv) S has no smallest element.

No assumptions of continuity in sight. But with that _definition_
of "real number" you can go on and give more definitions and
then prove that they have all the standard properties that the
real numbers are supposed to have.

Again, for the details look up "Dedekind cut" somewhere. (Note
that the definition you find may appear somewhat different from
what I wrote above; there are various "equivalent" ways of
saying what a Dedekind cut is.)

As regards the axioms typically assumed for analysis, they are, by
implication, those of axiomatic set theory which is isomorphic to symbolic
logic.

Uh no, set theory is symbolic logic _plus_ some axioms.

In differential geometry it is not infrequently stated that every
n-dimensional non-Euclidian space is described in terms of an n+m Euclidian
embedding space.

Uh, no. It's stated that every compact n-dimensional manifold can be
embedded in R^(n+m) for some m. And then it's _proved_ that this is
the case.

Where m is a positive integer such that m >= 1. Now,
that usually appears in the context of discussing physics, so there may be
a certain amount of poetic license taking place.

Um. Possibly you've _seen_ this in the context of physics.
If your knowledge of mathematics comes from expositions
in the context of physics it's not surprising that you
think that things are not justified as well as they should
be - physicists tend to just take mathematicians' word
for a lot of the finicky details. Find a _math_ book
on differential geometry and you'll find a _proof_ of
the theorem above (or not, in which case you need to
find a different book.)

For example, I do not
know if it would be more correct to say the embedding space is Lorentzian
in the case of general relativity.

Nonetheless, I believe when all is said and done, we cannot reason about any
of these ideas without appeal to our innate "Euclidian" space. The
developments I have seen for the real numbers beginning with Peano's axioms
seem to take a step of faith either explicitly through the continuum
hypothesis, or implicitly when they jump off the firm ground of the
rational numbers using the demonstrably irrational numbers as a
justification for the existence of a continuum of real numbers.

Have you ever seen a careful exposition of the basic properties
of reals, defined in terms of Dedekind cuts?

If yes, you should explain what the problem was with that
exposition - it didn't involve any of the leaps of faith
that you mention.

If no, then you're assuming that something doesn't exist just
because you've never seen it. And assuming that even though
you admit that you've never really studied the field.

I guess that's the whole point to the internet.

************************

David C. Ullrich
.

Relevant Pages

  • Re: Does the Calculus rest on Euclid?
    ... axioms I have assumed at the outset. ... Now you define the reals as ... A real number is a set S of rationals with the following properties: ... To say that you can order things implies a concept of a continuum. ...
    (sci.math)
  • Re: Dedekind Cuts, Fundamental Sequences: why?
    ... With only rationals, you can get below Pi or above Pi but not exact. ... seems to entail a necessary concept of continuity. ... The reals were designed to model physical measurements such as ... The numbers and their axioms came _after_ the underlying physical ...
    (sci.math)
  • Re: Does the Calculus rest on Euclid?
    ... axioms I have assumed at the outset. ... Now you define the reals as ... A real number is a set S of rationals with the following properties: ... To say that you can order things implies a concept of a continuum. ...
    (sci.math)
  • Re: Skolems Paradox and why is math the way it is?
    ... > Do you think constructable reals are a better class? ... through recursive calls to finite strings and the ZF axioms), ... > inside there are countably many members: ... seems large enough to do all of physics. ...
    (sci.math)
  • Re: Does the Calculus rest on Euclid?
    ... axioms I have assumed at the outset. ... Now you define the reals as ... A real number is a set S of rationals with the following properties: ... To say that you can order things implies a concept of a continuum. ...
    (sci.math)
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