> I'm going to be away from Jerusalem, and off list, till the end
> of March, and I did not have the time to look deeply into prof.
> Borzacchini's thesis about the origin of real numbers. But I would
> like to make a couple of points anyway, before going away. First,
> while I agree that the Greek mathematicians did not have the
> concept of a real number, they could not have been ignorant of the
> fact that surveyors, or engineers, unhesitatingly considered all
> magnitudes, say the diagonal of a unit square, as being measured
> by (rational) numbers. I don't think there is any written evidence,
> but the mathematicians must have been aware of the problematics
> caused by this approach.
Greek numbers were only non-zero integers: "A number is a multitude
composed of units" (Euclid, Elements, def. VII,2). What we call
"rational numbers" was, in my opinion, only a patchwork of different
things: sexagesimal numbers in astronomy, ancient Egyptian techniques
based on unitary fractions in practical computations, Pythagorean
(commensurable) logoi theory in theoretical mathematics.
The Pythagorean attempt to unify geometry and arithmetic failed, and
in the history of Greek mathematics, from Pythagoras to Proclus (one
thousand years), they were always sharply separated, in a rigid
opposition between discrete and continuous, between infinite
divisibility and indivisible units, between commensurability and
incommensurability, but for the 'commensurable' geometrical practical
problems (surveyors, etc.), which played indeed no role in theoretical
mathematics, and give us no evidences of some 'rational numbers' theory.
In other words sometimes geometrical magnitudes could be numbers, but
this was accidental, because their opposition was one of the most rigid
during the whole Greek history.
Greeks employed 'numbers' for geometrical measures, but even the
numerical system opposed "numbers" (usually on base 10) and "parts"
(usually on base 12)
> What I mean is, they must have been aware that "magnitudes" possess
> "number-like" properties.
Had geometrical magnitudes "number-like" properties? The unique
candidate I can think for this connection between geometry and numbers
is the Eudoxian theory of proportions (V book of the Elements). In fact
we read (in Aristotle Anal. Post. 74a, in Proclus Commentary. 7) that
Greek mathematicians were aware that such theory was relative to a
common "genus", unifying numbers, lengths, solids, times. Another
positive evidence is the II book of the Elements, a sort of 'algebraic'
treatment of the surfaces, which, as "geometric algebra", could be the
background of the ancient approaches to discover and remember equations
solution techniques.
Thus we could think that this "common genus", even if it did not have a
'numerical' explicit definition, could implicitly get it.
I incline toward the negative answer for many reasons. First the above
positive evidence is mitigated by the little range of theoretical
problems it covered: only equality and strict proportions manipulation.
In fact even Euclid does not employ this theory to deal with arithmetical
(books VII-VIII) and geometrical (book VI) proportions theory. In
addition, Eudoxan theory seems built just for continuous and non
necessarily numerical magnitudes: ratio is not a number, but "a sort of
relation in respect of size (peelikoteeta: term usually referred to
geometrical magnitudes) between two magnitudes of the same kind" (Euclid
def.V.3).
The negative evidence is however most of all based on the above rigid
opposition, which is one of the most constant in Greek culture, so that
in Euclid's Elements the geometrical magnitudes do not display either
numerical or metrical aspect (in my opinion the 'strange' proof of
proposition I,2 can be explained underlining the 'substantial' and not
'metrical' nature of the "interval").
> My second point is that Borzacchini jumps straight from Aristotelean
> science to Renaissance one, ignoring the Arab (or Muslim) science. It
> seems to me that the Arabs (of whatever nationality, recall previous
> discussions) are the real (no pun intended) inventors of real numbers.
> Possibly influenced by the Indians; for all that I'm going to India by
> the end of the day, I know very little about ancient Indian mathematics.
I think there is no trace of real numbers in Arabs, Italian algebraists,
earlier nature philosophers, and the difficulties they had to face in
dealing with continuity and infinite are in my opinion the best evidence
of the fact that there was no idea of "real number" before Descartes
(only a physical 'flavour' in Galilee!). In my opinion "real numbers"
are the core of a brand new "symbolic form" (in Cassirer's terminology)
whose ingredients went beyond the notation of real numbers, including
the zero and the infinite, the relationship between geometry and algebra,
the breakdown of an opposition-based metaphysics for a coninuity-based
physics and then the extensional metaphor of the physical magnitudes.
Anyway I reached this conclusion just recently, so that they are more
a 'line of thought' than a 'strong belief'. Even for this reason I do
appreciate any criticisms and any suggestions against my theses, most
of all of if based on historical evidences.
Yours sincerely
Luigi Borzacchini