[1] My question originated with the (supposed -- this is still not
confirmed) discovery of Neolithic sets of stone-carved knobs in Scotland
that have among them all the symmetries (!@) of the five Platonic
solids. Although these are "mathematical objects", the question is
did mathematical thinking play a role in their discovery and
construction and appreciation?
[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!
[3] As an analogous, historically problematic phenomenon (that I came
to in private correspondence on these matters), there is the issue in
the history of ancient Greek music whether or not there were so-to-speak
self-standing tunes or melodies, or whether the music had no
independent life or sense without reference to recitation of a poetical
text, say, or dance or ceremony, where the music helped to accentuate
or unify or intensify, but it would make no sense, really, to try to
"enjoy" it on its own, in contrast (supposedly!) to much of our music
which means often to self-standing. "Pure music" and "Pure mathematics"
[e.g., dimensionless numbers, as for example "the real numbers"] were
hard won. So there is the question of when -- and deeply if -- any
"self-standing" mathematics came into being. It is all too easy for us
to look at scraps of ancient mathematics and treat it as self-standing.
One has a recent case in the emergence of a mathematics of chance into a
mathematics of stochastic processes into Probability Theory which is
itself now a subject of pure mathematics in which it's connection with
the stochastic is increasingly a matter of history; for example Rota's
use on "geometric probability" or Paul Erdo"s' use in non-constructive
existence proofs (relative to combinatorial problems, say).
[4] Mathematical pedagogy. Here the issue arises, also. There is some
justice in the complaint that mathematics is rarely taught or learned.
The rote, formulaic, legalistic, cook-book problems, false divided
subjects that most student's encounters leaves most without an inkling
of what it is to think mathematically (that is, without an inkling of
mathematics) [a favorite conversational theme of Gian-Carlo Rota]. As
the educationist Howard Gardner has put it, however, there is no
reason but institutional stupidity [and perhaps an excess of
mathematicians and teachers of mathematics who have no sense of what
mathematical thinking/understanding in Rota's sense is] that keeps the
schools and students vaccinated against mathematics. [The move to
Calculus Lite is definitely in the wrong direction, even if it is not
as wrong as the New Math direction was].
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?
Greetings from
Babbling West(running)brook, Ct. u.s.a.
robert tragesser
RTragesser@compuserve.com