Re: Does the Calculus rest on Euclid?


Hatto von Aquitanien wrote:

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.

Pi is not a member of the set of rationals. I do not accept your definition
of the real numbers.

Certainly pi uniquely determines a set of rational numbers--all the
rational numbers greater than pi. So we have associated to any real
number like pi a set S_pi = {x | x > pi}. We can define addition and
multiplication on these sets simply by adding or multiplying each
element of one with each of another, and inversion for nonzero elements
also member by member. Hence, whatever your personal definition of real
numbers may be, any number r in it is represented by a unique set S_r.
Of course we have also identified 1/2 with the set of rational numbers
greater than 1/2, and you can object that these clearly are not the
same; yet S_1/2 represents 1/2 perfectly well, so that isn't a problem,
or if it is it is a philosophical problem and not a mathematical
problem.

No assumptions of continuity in sight.

What do you want "continuity" to entail, and why won't this allow you
to deduce whatever that is? And if you insist continuity must be
assumed, then you are again arguing philosophy and are in the wrong
newsgroup.

However, I'll ask yet again why my "Euclidean convexity" along with
Euclid's axioms does not satisfy. What is it you want?

To say that you can order things implies a concept of a continuum.

Wrong. I can order the alphabet ABC...Z; that does not mean the
alphabet implies a continuum.

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

Which can be stated in terms of boolean algebra.

No, they cannot be so stated. We have Boolean algebras of truth values
and of sets inside of a universal set, but that hardly means either set
theory or logic is simply Boolean algebra.


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.

Levi-Civita's development is interesting. He sure likes that term
'a-priori'.

And the relevance of this comment is? Levi-Civita died before the
embedding theorem was even provn, incidentally.

What is the geometry of a locally tangent space?

To what kind of manifold?

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

Every discussion I have seen of this topic requires a concept of completed
infinity, and a notion of infinite repeatability. I have no concept of
repeatability without a concept of time. My concept of time is a-priori
continuous. Likewise for my concept of space. Describe the fundamental
axioms upon which these developments are based without appeal to words
dealing with space or time.

This is philosophy, not mathematics. If you want to argue that the only
philosophically acceptable way to do mathematics is by basing it on
Kantian intuitions of space and time, you are in the wrong newsgroup.
Unfortunately the correct newsgroups, talk.philosophy and
sci.philosophy.tech, are overrun by spammers and idiots these days, so
I don't know where, if anywhere, real philosophy is now located on the
net.

.

Relevant Pages

  • 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: counter example in analysis
    ... Cantor's interpretation of reals like more than infinitely many numbers ... I asked myself whether or not the singular point can have any meaning if ... at least unless EB can provide some definition of continuity of his own ... are restricted to only the rationals. ...
    (sci.math)
  • Re: counter example in analysis
    ... Cantor's interpretation of reals like more than infinitely many numbers ... I asked myself whether or not the singular point can have any meaning if ... at least unless EB can provide some definition of continuity of his own ... are restricted to only the rationals. ...
    (sci.math)
  • Re: Dedekind Cuts, Fundamental Sequences: why?
    ... With only rationals, you can get below Pi or above Pi but not exact. ... If you want the area of a circle, you need to patch the hole. ... seems to entail a necessary concept of continuity. ... The reals were designed to model physical measurements such as ...
    (sci.math)
  • Re: Computable functions/reasls: followup.
    ... The computable-function definition above still applies, ... Russian-style constructivism is BISH + MP, ... which were reals that you couldn't tell whether or not were rational. ... special about the rationals; they could be replaced by the integers, ...
    (sci.logic)