Dear John and All,

Let me say I am a strong believer in the principles of slow read laid
out by Lance Fletcher in his little essay. I think they go beyond the
study of certain philosophic texts and apply more generally. I am
reminded of this when someone replies to an e-mail having read only the
first sentence, or when observing a political discussion which consists
of more or less deliberate misinterpretation of what the other persons
mean. For the slow read, they require us to accept, at least for the
time, that whatever the author says is true and make an effort to find a
true understanding. So, at least in this discussion, if I seem to
disagree with Klein, it is only because I do not understand him.

Klein seems to be saying that there is a modern view of number upon
which everyone agrees. p. 175 he says:

<<As soon as "general number" is conceived and represented in the medium
of species as an "object" in itself, that is symbolically, the modern
concept of "number" is born. 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._
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.>>

Recalling Socrates plaint in the Phaedrus that writing is an inferior
way of communicating ideas because one cannot ask the author questions,
there are a swarm of questions I would like to ask Klein.

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"?

But my real problem here is the interpretation of Vieta. Klein seems to
think Vieta the culmination of the developing concept of number,
ignoring Newton, to say nothing of Dedekind, Cantor, etc. Perhaps this
is so. At least I suspect that Newton (in Universal Arithmetic) was
only codifying and clarifying developments that had taken place
previously. Newton used the term "species" to refer simply to the
letters used in algebra as place holders, which is not, I think, what
Vieta meant, nor what Klein takes it to mean. But the "rungs" of
Vieta's "ladder" are _non-homogeneous_ magnitudes, i.e. lengths, areas,
not _homogeneous_ numbers. (In modern terminology, they have dimensions
as contrasted with dimensionless numbers.) Vieta is very clear about
that. His system can handle both numbers and magnitudes but he makes a
clear distinction and says (in Isagoge) explicitly that numbers are
homogeneous. What Vieta does not make clear, but I tend to assume, is
that these numbers are still multitudes of unities.

Does then Klein's "general" number include magnitudes also?

And he says that modern numbers are conceived as symbolic objects, as
contrasted with the older conception, whatever that may have been. But
is Vieta thinking any differently, or that much differently, than
Euclid? Are Euclid's numbers and magnitudes not also "symbolic objects"?

Hidden in the bushes is the problem of precisely how to interpret e.g. a
square-square. Vieta ignores this problem, but puts great emphasis on
the correspondence of the ladder of magnitudes with a continued proportion.

Regards,
Bob
Robert Eldon Taylor
philologos at mindspring dot com