48 A History ofMathematics
how they can be so sure; for example, there is nothing to suggest this in theElements. (Fowler, contribution to Historia
Mathematica mailing list, 1999)Today’s history, while usually avoiding picturesque stories about drowning, still has to cope with
the problem of presentism (referred to in the introduction) and that of reconstruction (see above,
Section 4). Indeed, if we use the word ‘problem’, it is because both have their uses, and the
historian’s difficulty is to decide when they are the right tools to reach for in the box. The case
of Euclid II.1 was already cited in the introduction; those who consider it to be equivalent to
the distributive law for multiplication are, one might argue, guilty of presentism. If Euclid had
wanted to state the distributive law, he was intelligent enough to do so. In this case, the problem
is complicated by evidence that informal Greek mathematics, continuing in the Egyptian tradition,
diduse precisely such a translation—see, for example, Heron (Chapter 3). This means that we have
two dividing lines to respect: between Euclid and his informal contemporaries on the one hand, and
between Euclid and ourselves, on the other. All this is part of a proper respect for differing historical
traditions. It makes the historian’s work harder, but no one said it had to be easy.
A quite different example is provided by Euclid’s theory of ‘ratios’ in book V. This complex
treatment of a ratio can be successfully modernized so that Euclid’s ratio (e.g. of the circumference
of a circle to its diameter) translates as the modern concept of a ‘real number’ (e.g.π—we now think
of the ratio as a number, and we take it for granted that it can be written as a decimal to as many
places as we like). That this is possible has been taken to mean that the Greeks, specifically Eudoxus
of Cnidus ‘invented’ the real number system over 2000 years before its development by Dedekind in
the nineteenth century. I have given an example of confident statement (in a mainstream textbook)
and scholarly doubt (in a listserv contribution) in the quotes at the beginning of this section.
Apart from doubts about Eudoxus’s role, the idea that he was concerned with what we call real
numbers in any sense is unhistorical, and is now out of favour, although the reasons for doubting
it are complex. Attacks on such ideas appear in the works of Knorr, Fowler, etc.^9 However, it
is interesting that it is in the nature of mathematics that such translation can be done; it is not
possible to make a similar translation of Aristotelian physics.
Question
See if you agree with this statement, and if you do, try to explain what features of mathematics
favour the translation.
As for ‘reconstruction’, historians feel the need for it most particularly when a source refers to
some mathematician’s work without indicating how the work was proved. One then proceeds to
present a plausible version of how the proof must have gone, with the hope of throwing some light
on the state of mathematics at the time, or of supporting a thesis about it. Here are two examples:- In Archimedes’sThe Methodhe states that the [volume of a] cone is the third part of the
[volume of a] cylinder having the same base and height. (This again was known to the
Egyptians, incidentally.) He attributes the discovery to Democritus ‘though he did not prove it’,
and the proof to Eudoxus. Archimedes may conceivably have had access to the works of
both Democritus and Eudoxus, supposing these to have been written down; in any case, the - One could also contrast the extremely complicated definition of ‘equal ratios’ in Euclid V with Dedekind’s ‘disappointingly’
simple definition of a real number (Chapter 9).