May I paste in here a bit of some of my notes that bear on this kind of
thing. It relates mainly to the Greeks and Babylonians, and only to
mathematics and 'algorithmatics', but it may be relevant.
Knuth has argued that algorithmatics and mathematics [that word needs much
refining; for the moment I'll just put a small m], are different things,
and those who are good or able in one may not be good at the other; see
D.E. Knuth, Algorithmic thinking and mathematical thinking, American
Mathematical Monthly 9 (1985) 171-181.
Generalising this, I think of Mathematics in the wide sense, big M, as a
great multidimensional cloud, in which the cluster of
Euclidean-and-later-type deductive procedures are a long way from the
cluster of algorithmatics.
Examples of algorithmatics are:
* Chinese arithmetical manipulations(?)
* Babylonian arithmetical manipulations(?)
* Babylonian numerical astronomy, as opposed to Greek geometrical
astronomy
* Table computing throughout the ages. (E.g., see the shift in
Ptolemy's style between his geometrical proofs, his tabulations and
calculations, and his experimental procedures.)
* Computer programmes, and other things associated with them
Mathematics, big M, started with a bias towards algorithmatics, then
mathematics, little m, became dominant, though algorithmatics was always
there, along with a lot of other Mathematical kinds of reasoning, but now
the money and the balance are shifting away from mathematics and back
towards algorithmatics!
An important example: Contrast
* A Babylonian procedure (i.e. algorithm) for solving some
numerically posed problem,
with
* Some Greek proposition and its proof (i.e. a bit of mathematics,
species geometry), that can be adapted to solve this numerical problem, and
which can also be used in many other different contexts,
and then with
* Our formula expressing the solution (i.e. a bit of Mathematics,
species algebra), which we can also use to solve the problem numerically,
or interpret as a description of the geometrical configuration, or adapt to
yet other ends different from these.
Final note: Babylonian and Greek 'mathematics' are very different things.
As well as speculating about whether the Babylonians affected the early
Greeks, we might just as well investigate whether the very extensive
Babylonian accountancy procedures influenced the extensive Greek
accounting. (I personally doubt that they did, and think that this puts
the proposal about mathematics in a wider context.).
David Fowler