At 12:58 PM 12/21/98 -0700, David Fowler wrote:
[deletion]
>Gordon Fisher said:
>
>> I'm interested in why you think the discovery of incommensurables
>> was not a crucial event in the history of early Greek mathematics?
>> Do you hold, against some (many!) historians that it didn't lead to
>> a Grundlagenkrisis? Or are you saying that came in classical Greek
>> mathematical thought not "early" but sometime later?
For example, I find it hard to keep from thinking that what Euclid chose to
put in the *Elements* reflects some concerns among his predecessors over
the century or so preceding the time of Euclid about the nature of
incommensurables, and notions about mathematical proofs. Again, I find it
hard to motivate Aristotle's connection of notions of proof, foundations of
logic, etc., with the topic of incommensurables.
Maybe "foundations crisis" isn't a good term to use here, suggesting as it
does a certain panic, so I call attention to a substitute term "concerns
about foundations" in this connection. Surely Euclid's *Elements* provided
a kind of foundations for a kind of ancient Greek mathematics, not to
mention subsequent kinds of mathematics over a couple of thousand years
(right up to now). (Cf. complaints about calling what happened in the 16th
and 17th centuries the "scientific revolution", suggesting as it does
something along the order of political revolutions.)
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I believe there is a crucial aspect in the discovery of
incommensurability which it is necessary to underline: it is probably
the first proof "by absurd".
In fact I believe (I do not give here the supporting elements) that
pre-Euclidean geometry was essentially visual and constructive, and then
there was no room for a discovery whose proof can not be 'positive'.
The fact that there was no root of 2 was already known, as a
numerical fact about Pythagorean triples, to Babylonians, but it appears
as a numerical fact, analogous to the lack of the inverse of 7. The
difference between the two cases, from a numerical point of view, is in
that the first does not depend on the numerical basis while the second
does. But this was scarcely appreciable in the alphabetic Greek numerical
system. The other difference is in that the first can be 'proved', but
only 'by absurd', and then it was not recognizable in a visual and
constructive geometry.
If this kind of argument is right, it means that the crucial fact
was credibly the failure of some kind of "geometrical" construction,
which extended the problem beyond the numerical fact ("antifairesis"
on the square or on the golden ratio or on the pentagon).
Where did the proof 'by absurd' come from?
I did not ever find an employment of the proof 'by absurd' before
Archytas' proof of the infinity of the cosmos, and this supports the
idea of a basically Pythagorean innovation. On the other side this kind
of proof is rooted in the Parmenidean philosophy, and here I must only
remind Szabo's analysis about this Eleatic connection.
Among the Sophists and then in Plato and Aristotle, the proof by
absurd was well established and many ancient constructive proofs based
on visual evidence were transformed in Euclidean proofs by absurd by
employing the axiomatic structure of the idea of "equality".
Was this a "foundational crisis"?
I think it was something more: it was the beginning of our "formal
thinking" enterprise, which is the unique environment wherein it will
become possible to speak about "foundational crises".
Wonderful topic!
Luigi Borzacchini