Dear Bob et al.
The question you raise goes right to the heart of my failure to understand
Klein. Of course it is possible that (a) he is wrong or (b) confused (or
both), but I don't think we should jump to either conclusion without
careful consideration.

Your questions

Just what is "general" number? In what sense are you using the word
"number"? Is a quaternion a number? An oction? A length, area? Is
there A modern concept of number? What do you mean by "intend"? For
that matter, what do you mean by "symbolic"?

go to the heart of the matter. Here are some possible answers:

1.) A number, as Newton says, is an abstracted ratio of (concrete)
quantities of the same kind. Consider two squares, A and B, and their ratio
A:B, and two lines, a and b, and their ratio a:b. We obtain the same ratio
on abstracting from both ratios if, and only if, the proportion
A:B::a:b
Holds according to the definition of "same ratio" in Euclid V, Definition 5.

2.) Contemporary mathematicians speak of natural numbers, integral numbers,
rational numbers, real numbers, and complex numbers. Quaternions and
octonions are not called numbers, I suppose because they don't satisfy
certain algebraic "laws" (e.g., quaternion multiplication doesn't satisfy
the commutative law: ab=ba)

3) Lines and triangles are not numbers, but their lengths and areas (with
respect to unit lines and unit areas) are. Lengths and areas are not
numbers if a line IS a length and a triangle IS an area, otherwise they are
positive real numbers, though in calculus it is convenient to allow
negative lengths and areas (in the theory of integration, for example).

4.) As to what Klein means by "intend" or "symbolic". I can only speculate.
By the mid eighteenth century the notion of real number was somehow
disconnected from Newton' geometrical account, and calculations using
"variables" and "constants" became the standard procedure in the Calculus.
It became more "algebraic" and symbol oriented. There's a detailed and
interesting story to be told here, but I am not sufficiently learned to
tell it.

5.) Newton's definition of number is not sufficiently rigorous for modern
mathematical taste because of its logic dependence on geometrical notions.
Now we are inclined to let the explanation run in the other direction, so
that we identify the line with the ordered system of real numbers, the
plane with the set of ordered pairs of real numbers, etc., and the real
numbers themselves are defined axiomatically a la Dedekind.

6.) I don't see how Euclid's numbers could be accounted "symbolic objects".
I think they are just what we would call finite sets composed of two or
more elements.

How much of this Klein would have accepted I don't know.

John

P.S. I think we have to take this discussion at a leisurely pace. Maybe
this is implied in the concept of a "slow read". Perhaps we should start by
reading sections A and B of Chapter 11.
Apologies to anyone who tried to access my website. Something has gone
wrong and I'll have to sort it out in the next few days.




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JP Mayberry, Department of Philosophy
University of Bristol
J.P.Mayberry@...
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