[HM] Marshack & When begins mathematics?
Robert Tragesser (RTragesser@compuserve.com)
Tue, 24 Aug 1999 14:35:25 -0400
HM list members,
I'd deeply like to see a discussion of how we should
characterize the threshold of mathematics, and the first mathematicals.
I am generally tremendously excited to learn on HM of the profound depth
and sensitivity and scholarly rigour with which ancient mathematics is
being investigated/elaborated. But at the same time I sense a lack of
what might be, albeit misleadingly, called philosophical depth, I mean
of the sort one finds in Marshack, which however he doesn't push as far
as he might.
I have a sort of meta-historical question that I suggest ought
to be something obsessively considered by historians of mathematics.
When does _mathematical thinking_ (and so mathematics in that
deepest-lying sense) begin? Like all things I don't expect an exact
answer (as when Kant could speak of the first proof"), but some sort
of rule of thumb.
What is the current rule of thumb by which the historian
rummaging around far down in the roots of the phylogenetic tree
supporting what we now call mathematics use to feel out early
mathematics.
I suspect that the inclination is to say: when there is evidence
of deploying or utilizing what we recognize now as mathematicals, or
akin to mathematicals. I was profoundly moved by Marshacks'
investigation of ubiquitous _Paleolithic_ curiously ordered stoned
scratchings and overscratchings, especially those prompting him to
speculate the development of a time-factoring culture and something
analogous to a lunar calendar speculatively decoded from some varieties
of those scratchings. I see the scratchings as something like an
ancestor of ordinal numerals.
THESIS: The proto-mathematicals were probably more complex and
elusive in their logic than geometric figures, numbers/numerals.
But are they mathematicals?
THESIS: Mathematics begins with problems carrying a logic that
enables them to be solved once and for all and principally "in the
head".
This suggests to me that we don't have mathematics until we
have the possibility of logical riddles, on the one hand, or something
like a proto-)algebra, as for example a proto-geometric algebra, on
the other.
QUESTION: Reading Marshack, the thought suggests itself that
simple counting, and in particular the cultural interests that makes
simple counting significant, came late; that it was preceded by
activities that were rather more complex and less ruleful and more vague
than counting. One thinks for example, of the tremendously complex
relations and interests that most have been in place before it made
sense to have "common coins" and standard, graded units, through which
transactions were formed. I mean for example, anyone intimately aware
of the tremendously complex differences between the members of a flock
of sheep that would be known to anyone who depended on sheep and sheep
by-products for their lives would have found some record of "how many
sheep" nigh on meaningless, and so cardinal numbers as measures
meaningless. Before cardinal numbers became meaningful, there would
have to be a rather complex commercial society and some needs for
measuring and accounting that so desperately needed measures that the
near vacuous measure provided by cardinal numbers were nevertheless in
high demand. Likewise for counting days or contriving units of time;
calendar time a result of the extremist abstraction; we are all surely
at war with numbers in the sense that we are at war with "time" measured
by clocks, periods measured out by calendars, etc., and even account
balances as reported by banks. I suspect we all live by other far more
complex and far more fragile but far more apt senses of time, date,
our current financial worth, etc. But what is more or less private to
us, was maybe what the "first" who tried to code and factor time,
wealth, etc. tried to manage, something that perhaps only a Borges or
a Marshack have the imagination, and the rigor of imagination, to
reflect upon.
Robert Tragesser
West(running)brook, Conn 06498
P.S. It is very heartening to have discovered in Conkey, Soffer,
Stratman, Jablonski, eds. BEYOND ART: PLEISTOCENE IMAGE AND SYMBOL,
San Francisco 1997 [recommended on HM by Marshack], the seeds of the
complete erasure of one of the most insidious and morally, spiritually
evil (because seriously misleading, misguiding) ideas entrenched in our
modern culture, that of "art". Calling something "art" is to display
it from the realm of serious thought, that is, my own "art", poetry,
I mean the real stuff, not the doggerel that lives in "poetry slams"
or the over-done stuff, precious, that tends to litter academic
literary mags, is wholly marginalized by being called "art"; such
would be quite otherwise if it were still, as it was so often in the
late Renaissance regarded as thought with a rigour and logic about it
barely equaled by mathematics (where "beautiful" meant "insightful;"
and not that rather vapid hormopnal beauty attached to landscapes,
say).