Re: Does the Calculus rest on Euclid?


Hatto von Aquitanien wrote:
Nathan wrote:

Hatto von Aquitanien wrote:

If a real number is a "set of rationals", and there is only one real
number designated by the concept indicated by the phrase "a real number",
then, in the case of pi either the set called pi is identical to the
object called pi, or we are avoiding (or more correctly, evading) the
original topic.

It doesn't make sense to talk about what the object called pi "is".
Numbers don't exist in the same sense as the pen sitting on my desk.
Certain properties characterize pi, and especially its properties as a
member of the set of all real numbers. In different models of the real
numbers, pi might be a set of rational numbers, an equivalence class of
sequences of rational numbers, or even a point on a line. Whether or
not these objects are "identical" is irrelevant. The point is that all
these models behave equivalently in the ways that are significant for
real numbers. What color is pi? It doesn't matter.

Well, at the risk of posting on low blood caffeine, I believe it does
matter. Again, this is a topic to which I simply cannot direct my full
attention at present. What appears to be happening is a crossing of levels
of abstraction. This is how Russell tried to climb out of his paradox. My
instincts tell me that there is a contradiction to be found by reviewing
carefully all the statements and assumptions. Or perhaps there is a
necessary step missing which, if completed, would result in a
contradiction.

Carefully reviewing to check for contradictions, eh? Why did nobody
ever think to try doing that before? I suppose they didn't have your
intuition to guide them.

--
mike.

.

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