Re: Does the Calculus rest on Euclid?

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.

When using a geometric argument to justify the
transition from the chord-length approximation to a continuous curve - for
example in the typical proof of the arclength theorem - there seems to be
an unacknowledged step of faith.

Is this a subject which has been discussed in mathematics?


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

David C. Ullrich
.

Relevant Pages

  • Re: Does the Calculus rest on Euclid?
    ... 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. ...
    (sci.math)
  • Re: Does the Calculus rest on Euclid?
    ... 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. ...
    (sci.math)
  • Does the Calculus rest on Euclid?
    ... 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. ...
    (sci.math)
  • Re: Does the Calculus rest on Euclid?
    ... Riemann sum approximation to a smooth curve is logically founded upon ... the transition from the chord-length approximation to a continuous curve ... but the concept of continuity cannot be ...
    (sci.math)
  • Re: Does the Calculus rest on Euclid?
    ... Riemann sum approximation to a smooth curve is logically founded upon ... the transition from the chord-length approximation to a continuous curve ... is meaningful to transcend the concept of discrete partitions as these ...
    (sci.math)
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