Many thanks to Dick Tahta, Manoel Almeida, F Fasanelli, Cabillon,
Alison Roberts, ...
[1] But are any of the knobs regular (spheroid/convex) solids?
The informative report from Ms. Roberts on the Ashmolean
collection of "knobs" still leaves open whether there is among them
those having the symmetries of the five Platonic solids? And if not,
where those that do are to be found (if only in some irresponsible New
Ager's scam)?
[2] _What is the mathematical?_
I don't recall a discussion on [HM] about what ought to be
counted as mathematical. In any case, should there indeed be those
Neolithic artifacts exhibiting all the symmetries of the Platonic
solids, is there anything to suggest that their original fascination
and discovery should owe something essential to, say, Mesopotamia,
or, better put, to an already emerged and distributed mathematical
tradition?
I should think that the mathematical does not lie in the
character of the things, but in how the things are thought about, how
they are conceived. I don't think that practical commercial arithmetic
ought to be counted as mathematics save that it is connected with some
considerations of why this rather than that, and so on, involving
definitive (ineluctable) answers. Likewise in the case of the Neolithic
regular solids (should they exist), I would not regard them as
distinctively mathematical products unless there was behind them some
attempt to anticipate (and so clearly conceive so that one can
anticipate) something like logically (that is, with an air of
ineluctability) further regular solids than one already has
found/constructed. Marshack's studies of Paleolithic scratching suggest
an anticipation of mathematics there, but whether the scratching
emerging out of time factoring phenomena, say, have mathematical
thought behind them is a tough call (which I don't think Marshack says
enough to answer).
I don't imagine that one can specify the "beginning" of
mathematical thinking in an exact and wholly satisfactory way. But I
have found it fruitful and helpful to think of mathematics in this way:
mathematical thinking begins when problems are posed that are solved
definitively -- once and for all (the solutions are "necessary") -- by
thought, by using one's head. For example, a combinatorial problem
concerning physical arrangements might for a long time be
satisfactorily solved empirically or inductively (in the empirical
sense), but there is always the open question of whether one has really
considered all the possibilities. Then one day some bright person
might notice that inherent in as it were the logic of the problem is
inherent a necessary solution that can be reached, or in principle
reached, by thought alone, and it can be seen that the solution of
exact and exhaustive.
I am very much against calling any empirical or practical process
(counting, measuring, constructing) "mathematical" if it anticipates
mathematics but does not involve the posing and effort to decisively
solve ( in principle by thought alone) problems (such as why is such and
such a formula or formulaic procedure exactly and inevitably right? --
Formulaic procedures for which this question makes no sense are not
mathematical.).
With best regards,
Robert Tragesser
West(running)brook, Connecticut 06498/usa
(860)399-6305
RTragesser@compuserve.com