Dear John and all,

Sorry to be so slow in responding.

In deference to those who are here to study Klein I think we should
avoid far ranging discussions of other issues. I mentioned Newton only
because Klein ignores him, as also Dedekind. He says explicitly: "the
modern concept of "number", . . . is identical with Vieta's concept of
species". (If so isn't he through? Why then does he go on to Steven ,
Descartes and Wallis?) Since clearly moderns have more than one concept
of number I take Klein to mean there is only one _legitimate_ modern
concept of number and it is the same as Species. And what is that
"legitimate" concept? And what makes it legitimate and others illegitimate?

I mentioned quaternions and octions, (which form algebras and are
referred to as hypercomplex numbers), only to emphasize the
arbitrariness of the distinction. If Klein was even aware of the
"Newtonian" concept he may have absorbed the widespread disdain, which
you have displayed, for it, which eliminates it from consideration.
Which leaves us with what?

Klein places great emphasis on Steven, who seems to have conceived
number as a decimal expansion. (All Steven's chatter about zero being
the source of number seems to have to do with adding the digit zero to
the end of a number to create the next rank of decimals) But I don't
see much evidence of Steven in Klein's discussion of Vieta's species, do
you?

As I mentioned, Newton uses the term "species" to refer to the _letters_
which, the way he uses them, are place holders and not numbers at all.
One has numbers, which he has defined as ratios, i.e. relationships of
quantities, which have names, 2, 3/4, pi, 3.628 etc. and can be
manipulated in certain ways, added, subtracted, etc. and one has
"species", letters, which can be manipulated in similar ways. This
latter technique is what he calls Universal Arithmetic.

Vieta is not so clear, to me at least. He distinguishes magnitudes,
ratios and numbers. His ladder scheme allows that magnitudes of
different sorts, lengths, areas, etc., are related. Modern
mathematicians find his scheme tedious and pointless, but when they try
to translate into modern terms (which they invariable do) they run into
some difficulty because modern math. takes no account of magnitude
(megethos). (Modern mathematicians are indeed dissatisfied with
Newton's definition of number, because in fact he doesn't "define"
number, he says what he _means_, a no-no for mathematicians.)

More to the point, what is it that Klein understands?

Klein does not exactly ignore Descartes, just the good part. Descartes,
in La Geometrie, provides an ingenious solution to many of the problems
Vieta struggled with, but Klein ignores that. Perhaps again, for the
same reason, he doesn't recognize what Descartes does.

Your wrote:
<< I don't see how Euclid's numbers could be accounted "symbolic
objects". >>

I don't either because I don't know what is meant by "symbolic objects",
but the question is what is it Klein sees different about Vieta's
numbers which make them "symbolic objects"?

I am trying to find some meaning for the statement I quoted earlier:
The modern concept of "number", as it underlies symbolic calculi, is
itself, as is that which it intends, _symbolic in nature -- it is
identical with Vieta's concept of species._
It is of course not the concept which is symbolic, but the thing
conceived, i.e. number, which is symbolic. "Symbolic calculi" I take to
mean the algebraic symbols, a, b, c, +, = and rules, and any such
algebraic system will have objects, numbers, whatever, underlying it .
It is numbers which underlie the symbolic algebra, and he is saying
these numbers are themselves symbolic. What do I do with "it intends"?
What intends? Number according to modern concept, I guess. And what
is "that which it intends"? And what does intend meant? Signify. So
according to the modern concept, number is symbolic and signifies
something symbolic. Now I am completely lost in my own thoughts. Is
Klein really worth this, or would it be better to spend the time on Vieta?

<<Perhaps we should start by reading sections A and B of Chapter 11.>>

Section A is a biography of Vieta. He never says anything about any of
the other of the swarm of characters which populate his book, why only
Vieta?

Section B discusses analysis and synthesis. I don't have any thoughts
on that.

He makes several statements which indicate his awareness that Vieta
distinguishes magnitude and number, which adds to the mystery what Klein
means by number in the passage we discussed.

Klein seems slightly mystified by Vieta's emphasis on equations and
proportions, as indeed I was, but I think I have made some progress.
It allows Vieta to sneak around the fact that there are no
"square-squares". The ladder corresponds on the one hand to "powers"
of the length or width which forms its basis, and on the other to a
continued proportion in the base, i.e.

A is to A square as A square is to A cube as A cube is to A square-square

To us a "proportion" is nothing but an equation in which both sides are
"ratios" i.e. fractions or divisions. To V. an equation is a "making
level" of two expressions for magnitudes of the same kind, or of
numbers, whereas a proportion is a _likeness_ of ratios, i.e. a sameness
of relationships. Vieta is emphasizing that these two very different
sorts of statements correspond.

By the way, the entire 557 page collected works of Vieta are available
in a pdf from
http://visualiseur.bnf.fr/CadresFenetre?O=NUMM-107597&M=telecharger
It is a 39 meg download and the web site is entirely in French.
Evidently the French language police are on guard lest the librarians
answer a question in an inferior language. But Vieta wrote in medieval
Latin, so I guess if you can deal with the one, you can deal with the
other. I am working on English translations of interesting chunks of
this stuff, if anyone cares.

Speaking of translation, I sometimes wonder what the underlying German
looks like. Take the last sentence I quoted: "This appears most
clearly in the species of the first degree, where the designation of the
'rung' is not appended to the letter sign and may therefore be said to
coincide with it."

To what does the pronoun "it" refer? "Designation" I suppose. So
substitute:
"designation of the 'rung' coincides with the letter sign ". Of course
the rung doesn't have a designation, i.e. not square, or cube or
anything. So we have:
" coincides with the letter sign ".

Whatever.

Now you understand why no one invites me to parties.

Bob