At 10:30 am +0000 24/2/99, Luigi Borzacchini wrote:
> ----------
> Avinoam Mann wrote few days ago:
>
>> 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.
I've argued before that *all* of our early (say pre 2nd century BC)
evidence written in Greek is that surveyors, taxation officials,
mathematicians, astronomers, etc used the 'Egyptian' system, what I call
expressing division as sums of parts (things that other people call 'unit
fractions'). Beyond that, it is interpretation, and there is a lot of that.
My own interpretation is that expressing division like this simply excludes
our idea of rational numbers from the story.
> 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.
Our current earliest evidence in Greek of Babylonian sexagesimal numbers,
only a very fragmentary inscription, dates from the 2nd century BC. The
abundant evidence is later, in astronomy, where it is mixed in with the
expressions of sums of parts; see Ptolemy. (Hipparchus very probably used
them, but I don't think we have any direct surviving texts.) We also find
them in the Heronian corpus (I think!).
> 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.
The problem with the Pythagorean story is that most of our relevant
evidence is very late and, there, only found in manifestly unreliable
sources. So I think that attractive and appealing descriptions like this
fall under the heading of interpretation. I keep banging on about this kind
of thing because I want to offer another kind of interpretation.
> 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)
A lot of problems (elsewhere, not here) arise from not specifying which of
the many different meanings of 'number' is under discussion. But I do not
know the evidence for this distinction between 10 and 12 - or perhaps I do
not understand it.
The passage on doubling a square in Plato's Meno does flirt with the idea
of measuring the diagonal in numbers: "If you can't count it up, ...". But
here the slaveboy is not a mathematician, one of the points of the episode,
so this really comes under the description above by Avinoam of "the fact
that surveyors ... the diagonal of a unit square, as being measured by
(rational) numbers".
>> 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.
I've also argued that we have no early evidence (I don't know when the
first such statemment might be) that Elements V is about handling ratios
of, or proportions involving, *incommensurable* magnitudes. This, again, is
an interpretation - perhaps now a universal interpretation, but an
interpretation nevertheless.
Most things said about Elements II (including my own, in spades) are also
interpretation.
> 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
Again, my theme: what is the 'positive evidence' under consideration here?
> 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
I'm uneasy about that 'rigid opposition', 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.
> 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").
and I think that much the same can be said abundantly about other parts of
the Elements. (Interpretation!)
>> 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,
What we do find in Arabic mathematics is abundant fractional numerical
calculations, even bordering on decimal fractional calculations. (My own
interpretation in this story is that a component of the explanation of the
explosive develepmonents in algebra in the early 17th century, and the
strong move towards our kind of real numbers, was the introduction of
decimal fractional arithmetic in the late 16th century, by the very people
who play a leading role in these developments - but that is another story.)
> and the difficulties they had to face in
> dealing with continuity and infinite are in my opinion the best evidence
This my real ignorance: What evidence aout continuity and the infinite?
> of the fact that there was no idea of "real number" before Descartes
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!)
> (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
David Fowler
At 10:30 am +0000 24/2/99, Luigi Borzacchini wrote:
>
> Avinoam Mann wrote few days ago:
>
>> 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