thanks for you comments and "firm" criticism. I very appreciate them
because I am not first and foremost an historian: I am more interested
in history-based foundations of science.
Then I incline to "divine" from history to its foundational
interpretation (the term "divine", in its italian and congruent form
"divinazioni", was employed by Gino Loria at the end of the last century
to critically describe the free attempts to conjecture about past, as
Maurolico's and Viviani's reconstruction of lost Apollonious' books).
Hence I am a "consumer" of history of mathematics, and I employ to a
large extent even themes, concepts and aims from philosophy and
foundational studies.
The risk of this approach is clear: to fancy something which does not
hold any supporting evidence. And it is also clear the remedy: to submit
the "divination" to public criticism, most of all of professional
historians. For the same reason I very often add to my interpretations
the words "in my opinion".
Then it is clear that my disposition to be "firmly" criticized is far
from rhetoric, but is an essential ingredient of my "research program".
About David's specific remarks I have just to carefully think about his
words, and I agree that I can not give often a strong support to my
assertions, even though it could be greater than the few words I could
and can write in an e-mail. I do not quote all David's remarks:
otherwise this mail would become a book, but substantially I completely
accept his critiques about the lack of explicit evidence for many
assertions of mine.
I can now only precise some minor points and a general question, and I
hope to get other firm critiques.
My general question about Greek mathematics is the "quadrivium"'s role:
it is something absent in other ancient mathematics environment, not
only in Babylonian or Egyptian, but also in Chinese one. Quadrivium is
in my opinion the real Greek mathematical Grundlagen, and it completely
fits with the discovery of incommensurability: for that reason I do not
believe in the "grundlagenkrisis" thesis.
Its best descriptions (as far as I know) are in Proclus (In Eucl. Comm.
35-36) and in a text ascribed to Architas reproduces in Jamblichus and
other authors (DK 47 b1). "Quadrivium" in this view becomes, through
Aristotle's theory of infinite and discrete/continuous, and through
Euclid's "Elements", nothing less than the first and basic foundation of
our mathematics. It is not casual that it will last completely untouched
till Renaissance.
The "quadrivium" is based on a double opposition: the first between
<multiplicity> ("poson", arithmetic+music) and <magnitude>
("peelikotes", geometry+astronomy), the second inside both pairs between
rest and motion, but even maybe between theory and model. This last
aspect can be another "divination", but in the above Archytas' fragment
(and in other his arithmetical-musical fragments) the approach to the
music/arithmetic connection is model-driven, whereas in Philolaus' ones
the approach is in 'intrinsic' and 'numerologic' (I can try, if needed,
to give some supports to these statements).
In addition this model-driven approach is completely out of earlier
mathematical-astronomical traditions (for example Babylonian astronomy),
and some names, as Eudoxus, are crucial both in the model-based Greek
astronomy and in the genesis of the Greek new mathematical tradition.
Where does the "quadrivium" come from? I do not know, but I am very
interested in 'divining' about Pythagoreanism. Then I recognize, with
David's remarks, that "The problem with the Pythagorean story is that
most of our relevant evidence is very late and, there, only found in
manifestly unreliable sources.", but I cannot avoid this step in order
to deal with my foundational aims.
Other minor points.
1. David remarks "I'm uneasy about that 'rigid opposition' [between
multitudes and magnitudes], and would prefer something more neutral like
'absence'. And is it completely absent? Do we not find moves in that
direction in Aristotle? Also I think that you cannot make sense of much
of Pappus' much later proportion theory without it having some
numerical-like meaning."
I incline toward the 'rigid opposition'. Probably David is right about
Pappus, but he is very late; I do not agree about Aristotle: beyond the
positive evidence of the 'common genus', he always sharply distinguishes
discrete and continuous, even in his theory of the infinite (Is there
opposite evidence?).
2. Another specific point I would like to face is David's remark "I do
not know the evidence for this distinction between 10 and 12 - or
perhaps I do not understand it.", to my statement "Greeks employed
'numbers' for geometrical measures, but even the numerical system
opposed "numbers" (usually on base 10) and "parts" (usually on base 12)"
There are a lot of examples of the base-12 employed by Babylonians,
Greeks and Romans for their pantheon or musical or measuring systems,
even sometimes with an autonomous system of numerical graphical
representation (as in Vitruvius), there is the Greek abacus quoted by
Heath in the first pages of his "Greek Mathematics" with its double
system of division (5 for integers, 6 for parts), but the best is, in my
opinion, the history of Athens' organization, as performed by Clistenes,
Pericles, etc. and 'sublimated' in Plato's "Laws", where base 12 appears
throughout when there is a whole to divide and base 10 when there is a
group to build. (The easiest and trivial rationale could be the base-12
in astronomy in the passage from the lunar to the solar calendar, and
the base-10 in hands-counting.)
3. Another interesting point:
I wrote:
>> and the difficulties they had to face in dealing with continuity and
>> infinite are in my opinion the best evidence
and David:
> This my real ignorance: What evidence about continuity and the
> infinite?
I continued:
>> of the fact that there was no idea of "real number" before Descartes
and David:
> I keep arguing that Descartes' underlying justification for his algebra
> is geometry, not real-numerical, but he encourages his readers, very
> successfully indeed, to think of it all as numerical. (Indeed, perhaps
> he himself later forgets that it really is geometrical!)
No disagreement (at first view) about Descartes. On the Greek troubles
about continuity and infinite I can give some evidence. The best is
Archimedes' lemma (for the area of the parabola's segment) where he
substantially discovers the sum of the geometrical series, but is not
able to assert it.
Anyway, there is no real disagreement on the historical points, because
I am not an historian whereas David is a very good one. The point is in
the mere possibility of a history-based foundational study of
mathematics. If this is possible, I must risk interpretations and
related errors. The alternative is to accept the idea of mathematics as
a long journey from darkness to light or the progressive conquer of a
continent. It is the prevalent view, it sounds platonic, but, in my
opinion, Plato first would have rejected it.
'firmly' and sincerely yours
Luigi Borzacchini