Greeks and‘Origins’ 47
should naturally also come across a theory of proportions, in which no arithmetical parts remain. (Hasse and Scholz
1928, p. 13)
It is not accidental that they were writing at the time of Russell’s paradox, when (modern)
pure mathematics was experiencing just such a crisis. Extrapolating backwards, they read the
anxiety of the ancient Greeks as a version of fin-de-siècle angst about mathematical certainty.
Although the Hasse-Scholz thesis was criticized in the years which followed its appearance (notably
by Freudenthal 1966) and never became the only accepted view, it has survived well, partly because
it does explain the problems referred to above, and because it is easy to adapt and revise. The most
recent sustained attack is by David Fowler (1999). Fowler rehearses many arguments which are
now standard: that Plato and Aristotle, who are the nearest to contemporary sources, refer to
the irrational without in any way suggesting that it was a problem; that the subject is not even
mentioned by Proclus, who is supposed to be summarizing an earlier history; that the time of the
supposed ‘crisis’ is also a time when many major mathematical discoveries were made, apparently
without trouble; and that Iamblichus is notoriously inconsistent and unreliable. (In passing, this
basic difference of opinion drives home the point about the difficulty of arriving at conclusions
about the period.) Fowler contrasts Aristotle’s hard-headed assessment (in one of many allusions to
the problem):
A geometer, for instance, would wonder at nothing so much as that the diagonal should prove to be commensurable.
(Metaphysics983a, in Fauvel and Gray 2.H6)
with Pappus’ (third centuryce) apparent confusion:
the soul...wanders hither and thither on the sea of non-identity...immersed in the storm of the coming-to-be and
the passing-away, where there is no standard of measurement. (Commentary on Book X of Euclid’s Elements, I.2)
and concludes that ‘the discovery [of the irrational] was no more than an incidental event in
the early development of mathematics’ (1999, p. 362.) While his thesis certainly advances some
useful arguments, it depends strongly on a particular reconstruction of how Greek mathematics
worked in the pre-Euclid period; while it fails to deal with a feeling that what is unusual in the
Euclidean approach (as detailed above) must have come fromsomewhere. And despite all arguments
against it, the discovery of incommensurability is still often thought to betherevolution in Greek
mathematics, as opposed to the simple introduction of deductive method; two of the contributors
to Gillies’ (1992) consider it as such.
Exercise 6.What is ‘regular’ about the five solids in Fig. 3?
Exercise 7.Why are there no others?
8. On modernization and reconstruction
Eudoxus’s general theory of proportions, which, from our vantage, amounts to a theory of real numbers, resolved the
anomaly that the discovery of several incommensurables had introduced into Greek mathematics. (Calinger 1999,
p. 110)
For example, the significant content of the proportion theory ofElementsV is almost universally acknowledged to be
due to [Eudoxus], in some form or another, though the explicit evidence for this is very tenuous indeed...[A]lmost
everybody says that Eudoxus’ aim and achievement in Book V was to handle incommensurable ratios. I do not know