Re: Does the Calculus rest on Euclid?


Hatto von Aquitanien 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. 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.

I think a Euclidean approach to calculus would be possible, and to some
extent you find Newton doing it. To do it right you need to nail down
the ordering properties which Euclid assumed, and you need to add an
axiom something like this:

(Euclidean convexity) Any convex collection of points on a line is
either the interval between two points, a ray extending from one point,
or the entire line.

By a convex collection is meant a set of points such that any point
between two points in the set is in the set. From this starting point,
without ever defining the field of real numbers and sticking to the
language of Euclid, you could do calculus--albeit from a peculiar point
of view.

By the way, it would be interesting to hear an argument that Euclidean
convexity could have been proven by Euclid from his assumptions; one
hears that Eudoxus basically constructed the positive reals, but it
seems to be there are problem with that view. If it's actually all in
Euclid, Euclidean convexity (is there an actual name for this axiom?)
should be provable.

.

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