[deletion]
> It is a tremendous embarrassment to be thought to be asking a What
> question.
[further deletion]
>
> Clearly I'm not asking one question. AND I DO NOT MEAN TO BE
> ASKING A PHILOSOPHY OF MATHS QUESTION, EITHER!
> When a historian of mathematics encounters a bit of
> mathematical text/signs/record, the question I don't find asked often
> enough for my satisfaction is: WHAT SORT OF THINKING DID OR MIGHT HAVE
> GIVEN RISE TO THIS? Finding a way of cogently placing/framing this
> question is what I am after; not, absolutely not, What is mathematics?
>
Looks to me like you are asking a "what" question after all. Furthermore
I don't see a lot of difference other than emphasis and perhaps
particularization between "what is mathematics" and "what sort of thinking
might have given rise to mathematical texts/signs/records and Marshack's
carved stones". In order to answer the last question, I propose that you
are inquiring, in part, about whether or not Marshack is justified in
applying the word "mathematics" to what he presumed to have been the
meaning of the scratched bones in connection with lunar cycles. This seems
to involve the question, what *is* mathematics (and what isn't). From what
you wrote earlier, I suspect you feel that maybe mere attempts at measuring
and assigning numerical values to lunar cycles, perhaps for calendrical
purposes, shouldn't be labeled "mathematics" because there may possibly have
been no notion of mathematical certainty of some sort involved. And you seem
in a rather haughty manner to accuse historians of mathematics of not being
aware of this aspect of mathematics. I think, on the contrary, that most
historians of mathematics are acutely aware of this aspect, and also that
most such historians have wider definitions of what constitutes mathematics.
You write:
> [2] I then connected this question with the matter of the Paleolithic
> scratched bones that Marshack interpreted as attempts at something like
> a lunar calendar. (As not only an astronomer, but a Muslim can, too,
> the cycles of the moon as a time structuring/keeping device is complex,
> because of the irregularity in, say, the visibility of the phases of
> the moon (independently of the condition of the atmosphere) because of
> the tilt of the orbit of the moon, etc etc.). The question is whether
> the symbolic systems of scratchings, which could be considered a kind
> of measure or mensuration ["men" derives from the word for moon, I
> think], and so a kind of number, but whether or not it reaches the
> status of a mathematical is another matter!
You seem to have accidentally left some words out here, e.g. in the
parenthetical remark concerning Muslims, whose import I hope I don't
understand, and in the last phrase: "of a mathematical ..." A mathematical
what? (Is that a what question which should be banned, as for example, a
question like "what is gravity?" and "what's your name?" and "what is the
chemical formula for water?" -- maybe you meant to ban "why" questions,
such as "why does gravity behave the way it does? and "why is the chemical
formula for water what it is?", and "why did you ask this question about
how historians judge something to be 'mathematical'?". I know Newton is
often quoted to the effect that he didn't propose to say, inside a certain
work, *what* gravity is, like Descartes did with his faulty hydrodynamical
theory, but only to give a (you should pardon the word) *mathematical*
formula for describing its effects numerically. However, I personally take
this to have been a kind of programmatic statement by Newton meant to
indicate he wasn't going to give a mechanical theory in the manner of
Newton, but rather to describe a way to deal with gravity numerically.
In any case, in your remarks about the difficulties of keeping track of
lunar cycles, I'm enlightened in a certain way by the fact that you make
no mention in this connection of the notorious 3-body problem of astronomy
or celestial mechanics or mathematics or whatever discipline you want to
assign it to -- not to mention perturbations due to other planets and maybe
asteroids or other matter in our solar system. Eventually, I suppose, we
could extend the problem to take in the whole universe (whatever that may
be), by way of Mach's principle. Or would that be carrying things too far
(pun intended)?
This will be my last reply to you, Robert. I detect in you a certain
disdain for my attempts at clarifying this issue, and I try not to stay
where I'm not wanted. However, I reserve the right to submit further
messages about "what is mathematics?" (which seems a sensible question to
me, as it evidently did to Courant and Robbins), based on other people's
messages.
Peace be with you.
Gordon Fisher gfisher@shentel.net