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

Description

MR1683176

This book is a highly welcome and original addition to the study of Greek mathematics. It is a study in cognitive history, that is, about the means and methods of Greek mathematics, and not so much about its mathematical content. In many respects it is what one might call an eye-opener, pointing to features in the history of mathematics that are immediately clear once they are stated, but which had never before received their due attention.
One of the author's theses is that "perhaps the most enduring of them all [i.e. intellectual practices from the Greeks until today] has been the Greek mathematical practice'' (p. 4). He develops his arguments in seven chapters.
1. The lettered diagram. For Greek geometry—and to a certain extent, arithmetic—the lettered diagram supplies the universe of discourse. By using lettering—one of the central achievements of the early Greek mathematicians—the diagram is turned into a finite, manageable system about which it is possible to speak in a meaningful way. We do not know when and by whom the lettered diagram was introduced into mathematics. It is present in the oldest mathematical texts. Netz thinks (p. 60) that the mathematical diagram did not evolve as a reflection on architectural or other practical diagrams. One may, however, suggest that the practice of drawing as well as the mathematical vocabulary owes a lot to architecture. For instance, Plato [The Republic, 501 a-c] mentions "pinaka'', wooden tablets, as a means of representation of ground plans of cities, which are partly erased and redrawn. (For a survey of architectural drawings see [J. P. Heisel, Antike Bauzeichnungen, Wissensch. Buchgesellschaft, Darmstadt, 1993].)
2. The pragmatics of letters. This chapter is complementary to the first one. Greek mathematical practice does not use specific letters to represent stereotypical objects (like $r$ for radius). Rather, the principle of baptism is alphabetical. Letters are introduced in the diagram according to the sequence in which they appear in the proof.
3. The mathematical lexicon. The Greek mathematical lexicon is much more comprehensive than containing just definitions. Definitions are there, to be sure, but to a large extent they play roles different from modern definitions. Partly, they indicate what one is talking about. (A point is that which has no part.) On the other hand, they may serve as starting points of deductions, like the implicitly defined parallelograms of Euclid. More important than definitions is the mathematical vocabulary as a whole. In that sense, the Greek mathematical lexicon is dramatically small, not only in specifically mathematical words, but in any words. It follows, on the whole, a principle of one-concept–one-word and is thus able to avoid any ambiguities. That mathematics has also changed very little in this respect may be confirmed by any mathematician who is able to read a mathematical text, but not an article in a newspaper, in a certain foreign language.
One big difference between the modern and the ancient mathematician is, however, that the modern mathematician will recognize international mathematical termini like "orthogonal'' at first glance, but the Greeks had to invent the geometric meaning of "right knees'' in the first place.
4. Formulae. What is meant here are not mathematical formulae in the modern sense, which the Greeks did not have, but words or groups of words which are used in a rigid and repetitive way in mathematical texts. Especially the theory of proportions abounds with stereotyped expressions (like enallax = alternately) for manipulations of proportions (pp. 139–140). Mathematical proofs find their expression in a web of formulae. At the end of this chapter, Netz draws a short portrait of the Greek mathematician: He is "thinking aloud, in a few formulae, made up of a small set of words, starting at a diagram, lettering it. This is the material reality of Greek mathematics'' (p. 167).
5. The shaping of necessity. This is the central and most extensive chapter of the book. It has the subsections (1) starting points, the atomic necessary producing elements, which are combined in necessity-preserving ways by (2) the arguments in (3) the structure of proofs where (4) the toolbox plays an essential role. The diagram and the technical language are the elementary tools, yielding the atoms of necessity. They are combined in the proofs in a necessity-preserving way. The arguments in the proofs are studied carefully by the author, most of them being in the simple form $P\to Q$. The structure of derivation is fully explicit. The logical architecture of the proofs is much simplified by the use of what has been called by Ken Saito the toolbox of the Greek mathematician. Let us have a closer look at this. There was such a unique set of tools used throughout antiquity (p. 217). Most of its contents came from Euclid's Elements. Pythagoras' theorem is a typical example. (Euclid himself constructs a specific toolbox for the investigation of incommensurable segments in his Book X, propositions 10–20, and on other occasions. Invariants like the sum of the interior angles of a triangle or the size of an angle in a segment of a circle are other typical examples.) More generally, Book VI of the Elements, 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. Books I, III, V, and VI of the Elements account for what the Greek mathematician simply had to know. Without the automatic use of the toolbox, no complicated results could be possible. The toolbox complements the formulae as the principal sources of necessity in Greek mathematics (p. 236). The author concludes this chapter with a short remark about the stagnation and decline of Greek mathematics, essentially after Archimedes and Apollonius. He says more about that in his last chapter, but here he points to one essential reason: Greek mathematics was not backed by any institution. Mathematicians were few and lived far apart. As far as we know, no schools were established. Only when printed books became available and the construction of the toolbox was done much more methodically in the Renaissance did mathematics explode exponentially (p. 238).
6. The shaping of generality. If Greek proofs are about specific objects in specific diagrams, why do these proofs prove general results? It is because the diagram (and the mathematical lexicon) reduces an infinity of possibilities to a small, manageable number of cases. If a diagram includes, say, a bisected line and another point somewhere on the extended line, there are only a few possibilities for the diagram, which can easily be mastered via the lettering of the diagram. Arithmetic is somewhat different, but even here Euclid's use of segments as a sort of variables for numbers leads us back to diagrams. "In short, then, the simplification of the universe, both in terms of the qualitative diagram and in terms of the small and well-regulated language, makes inspection of the entire universe possible. Hence generality is made possible'' (p. 268).
7. The historical setting. The main task of the book being completed, the author moves on to inspect the historical situation of Greek mathematics in a more general way. When did Greek mathematics begin? Who were the Greek mathematicians? How many were they and in what cultural context did they live? As for the chronological question, Netz suggests that the origin of Greek mathematics could have been a sudden explosion of knowledge which he dates around 440 BC (pp. 272–275). To the reviewer, this date seems to be a little too late. If Hippocrates of Chios wrote the first "Elements'' in about 430 and, moreover, tried to square his lunes, then it is implausible that Greek mathematics started in earnest only ten years earlier. A date of about 480–470 seems to be more convincing, even if there is no explicit reference to mathematics by the pre-Socratic philosophers of this time. At least Aeschylus, in his "Prometheus bound'' of about 465, lets Prometheus say: "And numbers, too, the chiefest of sciences, I invented for them'' (lines 459–460). Certainly Netz is right when he says, "whatever the first communication act containing Greek mathematical knowledge was, it included something worth communicating, something impressive and surprising'' (p. 273). The reviewer would add that probably the construction of the dodecahedron and, in this same context, the discovery of the first pair of incommensurable segments by the Pythagorean Hippasus were such impressive achievements. Moreover, at least in Pythagorean circles in the first half of the 5th century BC, there may have existed something like a coherent group of people interested in mathematics. Again the reviewer would agree with the author in his statement that the mathematical subject matter was recognized and organized as a unity essentially during the years 440 until 360.
As for the demographic aspect, Netz has counted all persons in antiquity who possibly qualify as mathematicians and are mentioned in one of the surviving texts (of any nature). He found 144, and certainly this can only be a tiny fraction of all people who did mathematics in any sensible way. A rough estimate, then, may be 1000 mathematicians in the 1000 years from 500 BC to 500 AD (p. 283). Almost all of them seem to have been members of the rich class and the literate elite. He sums up: Very few bothered at all in antiquity with mathematics. The quadrivium is a myth. On the whole, Greek culture, excluding the Platonic-Aristotelian tradition, knew no mathematics (p. 289). It is difficult to argue against such a strong statement. One argument against it may be the relatively large number of manuscripts that survive from Nicomachus' arithmetic, admittedly a book on the border of mathematics and numerology. Clearly the author is right when he says that the main consideration concerning the relative unpopularity of mathematics is quite simple: Mathematics is difficult (p. 290). (And, one may add, this has not changed over the millennia. Nicomachus is not difficult—this explains his success.)
The form of mathematics, a literate style originating from verbal arguments and combining the lettered diagram with stereotypic arguments, survived throughout antiquity. With considerable modification, this form became the distinctive feature of modern science as well. Netz seems to be right when he thinks that perhaps the introduction of letters to the diagram was the most important single step in the early development of mathematics.
The more linguistic parts of the book may be underrepresented in this review. The author has carefully studied selected parts of the mathematical literature for their linguistic characteristics, mostly pieces from Euclid, Archimedes and Apollonius. His book is very well written. The author uses a lively style. He speaks directly to his audience. Especially useful guides for the reader are the outlines and summaries of the various chapters.
On the whole, Reviel Netz has written a highly original book that opens up several new perspectives onto a subject that only three decades ago had been thought to be overworked. His work once more confirms the essential identity of mathematics from its birth in ancient Greece until today. All that is said about the style of Greek mathematics looks so modern that one might think it was projected backwards. But Netz's book is philologically soundly researched, and a short glance into Euclid's Elements will convince the sceptical mathematician that the author is right. One feature, however, of the book does not seem right to the reviewer. Its title should be "The shape of deduction$\ldots$'' rather than "The shaping of deduction$\ldots$''. The basis of Netz's investigations is authors such as Euclid and later ones, whose mathematical style is already fully developed. The shaping of deduction took place more than a hundred years before Euclid, and with the sole exception of a small (and philologically dubious) piece by Hippocrates we have no surviving document from this early time. Reviewed by B. Artmann

p. 269 on mathematical induction ("Fermatian inference")