At 10:08 am +0000 1/3/99, Luigi Borzacchini wrote:
> ----------
...
[The complete mail is reproduced at the end]
...
> I would like to underline an additional point which seems to me
> worth to be noticed: the 'double' Greek mathematical tradition.
> It is rather well known and accepted that in Greek mathematics we
> can roughly recognize two different traditions. It is not easy to
> describe them, their differences and the border between. By and
> large I could say that one, the most celebrated, had a strong
> Platonic 'flavour' and was centred on the new geometry we know by
> Euclid's Elements, starting from the Pythagoreans and arriving to
> Archimedes. The second was more connected to the ancient Babylonian
> (and Egyptian) tradition, more inclined toward computation, and
> less celebrated today. It can be found for example in Heron and
> other 'minor' authors, but some fragments can be found even in
> Euclid, and its problems are also recognizable in Archimedes.
> Credibly the distinction between arithmetic and logistic was part of
> this opposition.
I am also trying to sustain an argument that there are two traditions,
perhaps the two referred to here but I am not quite sure about that:
(1) the 'numbers as natural numbers', sometimes starting with 2, Greek
arithmetike, that we find in formal mathematics, and (2) the 'numbers
as fractional quantities', which are universally 'Egyptian fractions',
my '(sums of) parts'. Even high class calculations like Archimedes'
Measurement of a Circle use these parts. (Beware: One must know that
our text of Archimedes is late, manifestly corrupt, and difficult to
understand in its Greek form. Most translators do a certain amount of
emendation and interpretation to extract sense from it.) So the
similarities are between the Egyptian and the *Greek* tradition.
There is no surviving explicit evidence of Babylonian sexagesimal
numerical calculation in Greek mathematics and astronomy before the 2nd
century BC. Of course, there may have been significant contact, as many
people propose, but that is at the moment speculative reconstruction.
We find sexagesmials in later authors, especially in numerical astronomy.
The Babylonian tradition may not be celebrated today, but we do make
everyday use of it in measuring time and angles!
> The second tradition had probably even points in common with the
> practical arithmetic employed by merchants and surveyors, etc., which
> usually also employed a different numerical system, more similar to
> the roman. It is also likely that, in the last centuries of the Ancient
> World in Rome and Alexandria, the second tradition had an overwhelming
> diffusion whereas the first tradition was substantially absent from
> the roman schools and intellectuals. Even stronger this phenomenon in
> the Middle Ages.
I know very little about the Roman tradition - indeed, is there a unique
such tradition? - but would be surprised to find it similar to the Greek
commercial tradition, which is found in great quantity in the papyrus
from what we now call Egypt.
> Probably the distinction between the two traditions was much stronger
> than, for example, our distinction between pure and applied mathematics,
> and the whole history of the second mathematical tradition, Diophantus
> included, could seem a smooth evolution of the earlier Babylonian and
> Egyptian ones, with very few influences from the first. Then the true
> problem could be the short and exceptional flourishing of the first
> tradition.
>
> The treatment of the "rational numbers" seems a paradigmatic point for
> this distinction, because the distinction between "part" (we could call
> it a 'fraction with unity as numerator') and "parts" (the other
> 'fractions') was typical of the second tradition, and appears in the VII
> book of the Elements and in Diophantus. I think this approach strongly
> different from the classical geometrical idea of the 'logoi', crucial
> in the Eudoxian and Euclidean tradition.
In my way of speaking, in this context, a 'part' is something like 'the
fifth', 'parts' is a sum of different such things, eg 'the fifth, the
seventh, the twelfth'. I prefer not to have anything to do with the
'fraction with unity as numerator' kind of description, as there is no
numerator: we are dealing with everyday things like 'the half, the third,
the fourth, the fifth, ...'. I would also like to develop the idea that
this does indeed lie behind part of the descriptions/definitions of
Elements VII, but cannot see anything that would take it beyond this as
a mere proposal.
> Yours sincerely Luigi Borzacchini
and your sincerely, David Fowler
==============
From: Luigi Borzacchini <gibi@pascal.dm.uniba.it>
To: David Fowler <david.fowler@warwick.ac.uk>
Cc: Historia Matematica <historia-matematica@chasque.apc.org>
Subject: Re: [HM] Greek arithmetic and the rational numbers
Date: Mon, 1 Mar 1999 10:08:24 +-100
At 12:27 pm -0500 26/2/99, Fernando Gouvea wrote:
> I learn a lot from David Fowler's postings to this list, and I
> particularly enjoyed his recent one about the non-existence of
> "rational numbers" in Greek arithmetic. I think his conclusion
>
>> Corollary. Our idea of the rational numbers was not part of the
>> Greek way of thinking.
>
> Is largely correct. I am curious, however, about how Diophantus fits
> into all this. From what I can tell, he does work with something
> quite close to rational numbers, for example when the solution to the
> problem of expressing 16 as a sum of two squares turns out to be
> 256/25 + 144/25.
>
> Judging from the selection of the Greek text reprinted in the Loeb
> library edition, these are even written as a sort of fraction (but
> "upside down" with respect to the way we do it). I don't have access
> to Heath's Diophantus right now, but the general histories I checked
> spend lots of time on Diophantus syncopated algebraic notation but
> not much (more often no time at all) on the issue of his use of (and
> notation for) fractions.
>
> David, could you comment? Does one know where Diophantus got his
> fractions?
>
> Thanks,
>
> Fernando
and David Fowler wrote:
A very good point. I should have added, and usually do add, "Apart
from Diophantus"!
And here are a few further remarks, but I must confess at the outset
that I haven't looked at him for quite a time and have never studied
him properly, so what follows shouldn't be taken as being too
reliable. I would welcome corrections and supplements.
In the following David Fowler sets out a series of very good points.
I would like to underline an additional point which seems to me worth
to be noticed: the 'double' Greek mathematical tradition.
It is rather well known and accepted that in Greek mathematics we can
roughly recognize two different traditions. It is not easy to describe
them, their differences and the border between.
By and large I could say that one, the most celebrated, had a strong
Platonic 'flavour' and was centred on the new geometry we know by
Euclid's Elements, starting from the Pythagoreans and arriving to
Archimedes. The second was more connected to the ancient Babylonian
(and Egyptian) tradition, more inclined toward computation, and less
celebrated today. It can be found for example in Heron and other
'minor' authors, but some fragments can be found even in Euclid, and
its problems are also recognizable in Archimedes. Credibly the
distinction between arithmetic and logistic was part of this opposition.
The second tradition had probably even points in common with the
practical arithmetic employed by merchants and surveyors, etc., which
usually also employed a different numerical system, more similar to
the roman. It is also likely that, in the last centuries of the Ancient
World in Rome and Alexandria, the second tradition had an overwhelming
diffusion whereas the first tradition was substantially absent from the
roman schools and intellectuals. Even stronger this phenomenon in the
Middle Ages.
Probably the distinction between the two traditions was much stronger
than, for example, our distinction between pure and applied mathematics,
and the whole history of the second mathematical tradition, Diophantus
included, could seem a smooth evolution of the earlier Babylonian and
Egyptian ones, with very few influences from the first.
Then the true problem could be the short and exceptional flourishing
of the first tradition.
The treatment of the "rational numbers" seems a paradigmatic point for
this distinction, because the distinction between "part" (we could call
it a 'fraction with unity as numerator') and "parts" (the other
'fractions') was typical of the second tradition, and appears in the
VII book of the Elements and in Diophantus. I think this approach
strongly different from the classical geometrical idea of the 'logoi',
crucial in the Eudoxian and Euclidean tradition.
-From this point of view Diophantus could be considered not as an
exceptional case but as the highest point of the second tradition.
Yours sincerely
Luigi Borzacchini
==============