The Shaping of Deduction in Greek Mathematics: A Study in Cognitive History

Reviel Netz has written an stimulating book about diagrams and mathematics, telling us facts that we all know, but hardly ever thought of. Thus he sets himself in the best of company, for isn't that what Euclid did from the very first proposition in the Elements? "The diagram is the metonym of mathematics" is RN's main claim. To understand what he means by that, think of two typical situations in the circus of conferences: if a philosopher or historian gives a talk, he will read aloud for half an hour, facing his audience without moving from his chair. If a mathematician gives a talk, he will dance around the platform talking to the blackboard while writing figures and letters on it, most of the time ignoring his audience and concentrating on his written deductions as they emerge out of sheer necessity.

Years ago David Fowler (of Plato's Academy) coined a motto: "Greek mathematics is to draw a figure and tell a story about it." RN has widened and deepened this into "Deductive mathematics grew out of the Greeks drawing lettered diagrams and telling stories by means of them, not only about them." The diagram and the argument live in such a close symbiosis that one cannot be understood without the other. The diagram is the metonym of mathematics.

RN presents his analysis in in seven well-structured chapters:

1. The lettered diagram
2. The pragmatics of letters
3. The mathematical lexicon
4. Formulae
5. The shaping of necessity
6. The shaping of generality
7. The historical setting

Beware: these chapters are more about linguistics and cognition than about mathematics proper. RN (in this book) is less interested in the mathematical tale than in the way it is told. It is only fair to stress this fact from the beginning, lest frustrated readers look in vain for details of Greek mathematics which are not there. The book was written (albeit RN would protest) for those who know the greater part of Euclid's Elements and some of Apollonius' Conics, if not by heart, then thoroughly. (Even better if they know them in Greek!)

The first two chapters discuss the lettered diagram and the pragmatics of letters:

A Greek mathematical proposition is, at face value, a discussion of letters: alpha, beta, etc. ... I argue that in this process the diagram is indispensable. This has the surprising result that the diagram is not directly recoverable from the text. ... there are assertions which are directly deduced from the diagram. This is a strong claim, as it seems to threaten the logical validity of the mathematical work. As I shall try to show, the threat is illusory. Then there is a large and vague field of assertions which are, as it were, 'mediated' via the diagram. ... The diagram [is] a vehicle for logic. This might be considered a miracle. Are diagrams not essentially misleading aids, to be used with caution? (p. 33)

RN joins Ian Mueller's view on Greek implicit assumptions that "these did not invalidate Greek mathematics, for they were true".

Having described the first tool of Greek deduction, the lettered diagram and its use, RN moves on to the second, the mathematical language, or as he prefers, "the mathematical lexicon". He demonstrates how definitions play a much lesser role in shaping the vocabulary than most scholars used to think. The lexicon is found to be governed by forces other than the definitions, one such force being the one-concept-one-word principle, leading to a lexicon which is "dramatically small -- not only in specifically mathematical words, but in any words, including the most common Greek grammatical words. It is strongly repetitive within authors and between authors" (p. 120). The smallness of the lexicon is an important factor in the shaping of deduction, especially because of the absence of ambiguity -- which absence, according to RN, is not produced through explicit codification, but by "self-regulating conventions".

"By the time one has learned the first book of Euclid's Elements, say, a considerable subset of the Greek mathematical lexicon must have been interiorised" (p. 125). This truth (acknowledged by many historians of maths who would hardly understand a line of Thucydides, but easily master the theory of parallels in Greek) is a consequence not only of the smallness of the lexicon, but of that particular feature which RN rightly calls "formulae". Not in the normal mathematical sense, but in Parry's sense of Homeric repetitions. There are differences, though: Greek mathematicians were not illiterate oral performers. RN gives a competent analysis of the characteristic features of Greek mathematical formulae.

By the end of chapter 4 we have finally found the Greek mathematician: "thinking aloud in a few formulae made up of a small set of words, staring at a diagram, lettering it. ... We now move on to see how deduction is shaped out of such material" (p. 167). The investigation falls in two parts, the shaping of necessity and the shaping of generality. From starting points, the "atoms of necessity", are built arguments in various ways, essentially by means of (what Ken Saito called) "the tool-box." There was such a unique set of tools used throughout Antiquity, the bulk of them made in the Elements. Book VI, about similarity geometry, "is the mainstay of Greek mathematics, the place where the visuality of geometry and the diagram meet the verbality of proportion and the formula. The four Books Elements I, III, V and VI account for what the Greek mathematician simply had to know" (p. 220). "Without the automatic use of the tool-box, no complicated results could be possible. The tool-box complements the formulae as the principal sources of necessity in Greek mathematics (p.236)".

If Greek proofs are about specific objects in specific diagrams, why do these proofs prove general results? RN answers this difficult question in chapter 6, "The Shaping of Generality," concluding: "Generality is the repeatability of necessity. The awareness of repeatability rests upon the simplification of the mathematical universe, as explained in the first four chapters". That is to say, arguments are general because the diagram and the mathematical lexicon reduce an infinity of possibilities to a small, manageable number of cases. To my view this chapter is rather vague, but I would be hard put to find how to add more detail. I am surprised, though, that while RN (on p. 117) comments on the rareness of 'ei' ('if'), used only 16 times in Book I of the Elements, he never mentions 'ean', the normal word for 'if' in a generalizing sense, and therefore the unrivalled opener of Greek propositions.

The last forty pages are devoted to "The Historical Setting of Greek Matematics," and will probably appeal more to lay readers than most of the rather difficult technicalities and statistics of the former chapters. RN has listed 144 individuals about whom we can make a guess that they may have been mathematicians; these cannot be a tiny fraction of the total number of ancient mathematicians, but must be a sizeable portion. So he settles at 1000, a convenient number. "Very few bothered at all in antiquity with mathematics. The quadrivium is a myth. ...The main consideration concerning the relative unpopularity of mathematics is quite simple: Mathematics is difficult" (p.290).

RN may be right, it is difficult, but not to all of us as difficult as some Studies in Cognitive History. I bet (though my wife forbids me to bet on the Net) that most readers of this review would rather read Euclid, Archimedes and Apollonius than familiarize themselves with the concepts and methods so dexterously handled by Reviel Netz. One (myself) who was once forced to put his feet into computational linguistics and managed to extricate them again before disappearing in the quicksand loops of Gödel, Escher, Bach, read this book 'dakryon gelasas', smiling through tears (Iliad 6.484). It's a nice demonstration of Plato's theory of anamnesis: I knew most of this in advance but did not know that I knew it. After my first reading of his book, I feel that I enjoyed it. I will try to find out why. It is bound to be uphill biking.

Christian Marinus Taisbak is Reader in History of Ancient Mathematics at Copenhagen University, after retiring in 1994 from a Chair of Classical Philology at the same university, which he held since 1964. His two books are his 1971 dissertation, Division and Logos on the arithmetical books of Euclid's Elements, and Coloured Quadrangles. A Guide to the Tenth Book of Euclid's Elements. After his retirement he has been preparing a translation with commentary on Euclid's Data, meant to appear in the XX'th century. His lenient (home made) motto as a reviewer is "Peccavit saepius qui librum scripsit quam qui nullum". He can be reached at marinus@private.dk.