BMCR 2000.02.17

The Shaping of Deduction in Greek Mathematics. “Ideas in Context”, 51

, The Shaping of Deduction in Greek Mathematics. "Ideas in Context", 51. Cambridge: Cambridge University Press, 1999. 327. $64.95.

The Shaping of Deduction in Greek Mathematics represents a very new approach to a set of texts which, at least as far as classical scholarship and pedagogy are concerned, have been regarded as marginal. N. attempts to redress this marginality in a number of ways. He approaches the mathematical corpus (primarily Euclid, the first four books of Apollonius, and Archimedes) in a deliberately naive way. That is, he casts off many of the preconceptions about how we should read these texts in favour of a broad set of rather unusual approaches. N. borrows from Homeric studies, for example, an insistence on the importance of ‘formulae,’ and applies it (somewhat modified, of course) to Euclid and company; he uses Peircean semiotic classification to understand the role of letters in the mathematical diagrams; and he takes a psychological approach to the idea of ‘necessity’ in Greek geometrical arguments. He ultimately concludes that the most important defining feature of Greek mathematics is its form rather than its content, a conclusion which is closely tied to his method of reading the texts.

N. is never thinking in small terms. In this book he is constantly working to keep his arguments applicable to more general and larger questions than just the immediate and particular ones he is raising. He wants his book to “be read on three levels: first as a description of the practices of Greek mathematics; second as a theory of the emergence of the deductive method; third, as a case-study for a general view on the history of science” (p. 1). N. admirably succeeds in juggling this multiplicity, although his book works best, I think, on the second and third of the three levels.

The work will be a necessary read for anyone interested in the history of Greek mathematics but will also be interesting to a wider audience, particularly philosophers of science and intellectual historians. Many non-specialists will want to skip over some of the long technical arguments, particularly in the middle chapters, and N. has accordingly provided clear summaries at the beginnings and ends of each chapter for those wishing to move on more quickly. The first and last chapters, however, should be read in their entirety as they do a great deal (indeed, most) of N.’s argumentative work.

One word of caution: N.’s book purports to be about Greek mathematics, but his arguments typically are applied and proved only with respect to Greek geometry, and in particular the ‘big three’ of Greek geometry: Euclid, Apollonius and Archimedes. In this summary I will, for the sake of consistency, follow N.’s terminology, referring to Greek ‘mathematics,’ but the reader should understand that this almost always means ‘third-century geometry’ — a nontrivial difference.

The book begins by focusing on the role of diagrams in Greek mathematics, specifically on how they are used in both the writing and the reading of the texts. He argues that in the presentation of a theorem, even orally, the diagram was assumed as complete and given from the outset, rather than constructed along the way. The text and the diagram are mutually interdependent in subtle and complex ways. One of the surprising things that emerges from N.’s study is his demonstration of how frequently Greek geometrical diagrams are ‘underspecified’ or ‘unspecified’: that is, points or lines (letters) turn up in the diagrams which have been nowhere defined (or at best sketchily defined) in the body of the text. This leads to the conclusion that, first, many theorems cannot stand on their textual presentation alone but require their diagrams to make sense, and, second, that the diagram is not directly constructable from the text alone.

The interdependence of text and figure goes beyond the specification of points and objects; it applies equally to mathematical argument and derivation. N. concedes that some assertions do not require a diagram, but he argues at some length that many do require both text and diagram. Moreover, diagrams tend overwhelmingly to be specific to particular propositions: few diagrams are used twice unchanged (although this does happen, a fact I consider to be more significant than N. does). One of N.’s legitimate criticisms of Heath’s “translation” of Apollonius and Archimedes is that Heath forces the diagrams to be as identical as possible across propositions. N. ties the (almost) one-to-one association of diagram and proposition to the claim that “diagrams are the metonyms of propositions,” a claim which seems to operate on three levels: first, that the Greek word ‘diagramma’ frequently seems to mean ‘proposition’ rather than ‘diagram’; second, that specific diagrams can individuate specific propositions (almost) as uniquely as, say, the phrase ‘Elements III.5’ identifies a particular proposition (e.g., see p. 40); and third, that the diagram serves as the working idea around which the proposition is finalized (see esp. p. 167). These last two claims use “metonym” in an unusual, extended sense. But of course, any claims N. makes about the diagrams are really only demonstrable for the diagrams as they are preserved in the MSS, and may tell us little about the relationship between text and diagram in the original texts.

N. then moves to a consideration of the semiotic practices of the lettered diagrams. In particular he argues, I think correctly, that the letters serve as Peircean indices. The move to Peirce here is effective, in that it quite sensibly vaults over much of modern semiotics, which has been mostly concerned, since Saussure, with linguistic signs as paradigmatic. The letter, N. argues, simply sits in the diagram on the object it represents, indicating it in exactly the same way a pointed index finger would. N. does not consider whether there is a semiotic function to the letters as they are used in the text apart from their function of diagrammatic specification or reference. This I think would make an interesting follow-up to his study. N. argues that the semiotic function of a Peircean index necessitates by definition that the letters in Greek mathematics refer always to determinate, rather than general, objects. And this is not a trivial claim. N. argues that it implies, for example, that Klein was right when he said that Greek mathematics does not employ variables.1 It also ties in with one of N.’s subthemes: that Greek mathematics is always about particular concrete diagrams rather than general or ideal representations, and this fact, N. thinks, effectively seals Greek mathematics off from any second-order ontological discourse.

He next moves on to a look at how letters function as names of mathematical objects, and how strings of those letters are used in the text. N. spends several pages looking at how a line called AB, for example, can be alternately called BA. He concludes that a convention of naming objects existed, though it was always unintentional. His examination of the (mostly) alphabetically-ordered process of labeling a diagram brings him back to the conclusion that, contrary to what we might naively assume on reading the proposition, the diagram is not drawn or labeled as the proposition unfolds, but that it was drawn by the author when there was only a “rough idea for the proposition” (p. 85) in his head. Moreover, terminological switches (e.g., referring to a triangle first as ABD then as DAB) can be attributed to the failure of the author’s short-term memory during the act of transcribing the text from what N. thinks was an essentially oral origin. Again, this all assumes that the sequence of letters in the text, the order in which objects are labeled, and the final appearance of the diagrams all appear in the MSS in the original form in which they were first put to papyrus by their authors, an assumption I am not so willing as N. to make. I am doubly wary because N. neither discusses the problems of MSS transmission nor cites textual apparatus at all.

But of course, one could subvert this criticism by simply substituting “Byzantine” or “Medieval” where N. has “Greek,” and “copyist” where N. has “author,” and N.’s conclusions would often remain curiously unchanged: after all it was the Byzantine and Medieval MSS which shaped our mathematics, not the invisible and idealized Ur-texts. This strange robustness of N.’s overall argument is what leads me to claim that his book works better as a general thesis about deductive practices rather than as a specific (historical) argument about Greek mathematics.

To return to N.’s book: his next move is to look at the mathematical lexicon, and he shows that the lexicon was not mainly structured through definitions. That is, although abstract objects such as “point,” or “line” are defined, the majority of objects discussed and manipulated in the texts, the real meat and potatoes of the propositions (e.g.: ” τὸ ΑΒ” as a manipulable object), are never defined. N. argues that this lexicon was self-regulating. The most important feature of the lexicon is perhaps its small size, allowing it a precision, a lack of ambiguity, which is critical to the deductive functioning of mathematical arguments and which plays a role in the development of the mathematical form as the very paradigm of rigor. However, his claim that the “science of mathematics [as opposed to, e.g., anatomy] seems practically to have been born armed with its terminology” (p. 121) is again symptomatic of N.’s problematic approach to the historical material, where he does not take a hard enough look at the uncertain state of our knowledge of early Greek mathematics. Indeed his comparison of the mathematical and anatomical lexica (p. 120-122) is problematic exactly insofar as we have a significantly better record of the early history of anatomical terminology than we do of mathematical terminology, a fact which makes a comparison of the two misleading. This being said, I do concede that the mathematical lexicon was, as N. argues, the first to attain such a high degree of precision and unambiguity. N. ascribes this precision partly to the nature of mathematical objects themselves and partly to the social and communicative function of the mathematical texts, where second-order discourse is always separate and less important that the first-order discourse of the propositions themselves.

N. complements this lexical analysis with an examination of the overall form of Greek mathematical propositions. N.’s analysis begins by contrasting mathematical formulae to Homeric formulae, where the latter are supposed to have arisen out of the needs of an oral performer and the former in the context of an undeniably literate tradition. N. defines ‘formula’ as a word or group of words that is either semantically marked, or is very markedly repeated. Thus rather than calling a point ” τὸ σημεῖον,” mathematical texts usually use the formula ” τὸ Α.” As throughout the book, N. here distinguishes first-order from second-order formulae, where first-order formulae are ways of describing mathematical objects and constructions, and second-order ones are such things as conjunctive phrases, introductions, and ‘Q.E.D.’ One of the important features he argues for in the formulae is the specificity of their signification: they designate particular rather than general objects, a point which ultimately derives from N.’s semiotic analysis in chapter one. One thing worth noting is that the formulae are not slavishly adhered to; they undergo what N. calls ‘transformations’ and ‘ellipses’. Like the lexicon, there is a tendency to a certain repetitiveness, but there is as much deviation as repetition, particularly across propositions. N. argues that this points to the conventions of formulae like those of the lexicon being self-regulating rather than rigidly enforced. They have what N. calls an internalized “generative grammar,” a set of implicit ‘ways of doing things’ learned and either consciously or unconsciously adopted by the mathematician. This conformity to a particular style is part of the acceptance or emulation of a set of professional norms commonly held among the community of mathematical practitioners.

N. next explores how deductive necessity functions in the mathematical texts. He shows that mathematical arguments take as their starting point a great many more ‘givens’ than are explicitly enunciated in the axioms and definitions of the works. From these starting points, arguments develop whose necessity is grounded in one of several ways: 1) reference to previous work (an explicit statement of some previous result); 2) ‘intuition’ (here meaning something like mathematical “obviousness”); 3) visual necessity (where the diagram plays a role); and 4) use of a “tool box” (implicit use of previous results).

The penultimate chapter solves a crucial problem for N.: if Greek mathematical arguments are always about specific objects and specific diagrams, how can they be generally applicable, as they were surely seen to be? His answer is straightforward: the invariance of formulae allowed for a repeatability (sometimes specifically stated: “and so for any X…” and sometimes implicit) which guaranteed generality. That is: since a proof about a particular object was structured according to the (internalized and habitual) formulae of Greek mathematical discourse, then it could be used as a general conclusion about a class of objects if one could obviously and easily repeat the proof, invariantly, for any similar object.

Finally, N. brings together a number of themes that he has been playing with throughout the book. He looks at the way that mathematical discourse developed in the context of the professionalization of mathematics, concluding that mathematicians wished to distance themselves as much as possible from practical applications. I would counter that this is only demonstrable if we take (as N. does) Apollonius, Archimedes, and the Euclid of the Elements as being our paradigmatic mathematicians. If we are willing to include as “mathematical” Euclid’s Optics, or Phaenomena, or authors such as Autolycus, Ptolemy, or Heron, then such an argument becomes impossible to sustain. I note also that N.’s big three were all working within (probably) a century of each other, an extremely narrow temporal window, and one which is not totally representative of much of Greek mathematical practice either before or after. It would be interesting to see how the inclusion of, say, specific propositions from Ptolemy or Nicomachus of Gerasa would play themselves out in N.’s close textual analysis. This being said, N.’s discussion of the community of mathematicians in chapter seven is one of the most interesting and novel parts of the book. Here he looks into the question of how many mathematicians would have been working at any one time, where they stood in their societies, how they communicated with each other, and how they published. This kind of work has been seldom seen in the history of ancient mathematics, and it is a most welcome addition to the modern literature.

A few minor points:

I remain unconvinced by N.’s argument that Aristotle took his use of letters in the Organon from mathematics. An interesting twist can be put on N.’s thesis about how mathematical letters function as indices by pointing out that the first use of epi + dative as a way of specifying the object to which a letter applies in an argument shows up in Aristotle in both logical and mathematical contexts. Of course Simplicius uses this construction in a fragment of Hippocrates of Chios (5th c. B.C.), but this passage is such a philological mess as to be at best highly problematic evidence for the fine points of fifth-century mathematical grammar. Now, Aristotle’s use of epi + dative in, e.g., the Analytics, is clearly not implying a spatial relationship, a point N. concedes, and, since Aristotle’s logical use cannot be shown to postdate a mathematical use, one could argue that mathematics in fact borrows this construction from logic rather than, as N. does, the other way around. This would complicate N.’s argument that the letters are seen by mathematicians to stand in a strictly spatial relation to the points and lines they indicate and possibly also compromise N.’s support for Klein’s thesis about variables (see above).

A minor mistake, but one which lets me hop up onto one of my favorite soapboxes, occurs in a passage where N. is speculating about why no one before the Greeks invented the lettered diagram. He says that the Phoenician alphabet filled a necessary precondition of the lettered diagram and that previous writing systems prevented the lettered diagram because “pictograms suggest their symbolic content,” (p. 58) here meaning that a “pictogram,” e.g., that for “bird,” could not have stood as a label in a mathematical text since it too immediately suggests “bird” rather than “point” or “line.” He is here making the same mistake as semioticians such as Bottero and Manetti.2 The real problem with this argument is that there has never been a pictographic script. Sumerian, Akkadian, Egyptian, Chinese and even Mayan all are complex scripts with plenty of room for polyvalence, abstraction, and innovation (a good explanation of this can be found in Michael Coe’s Breaking the Maya Code).3 The Sumerians had no problems naming their children, they should have had no problems naming triangles.

And a final small correction: N.’s claim that “none of the many Greek civic calendars of antiquity paid the least attention to astronomical science” (p. 301) rests on an unfortunate misreading of Bickerman and is untrue.4

To sum up: this is a novel work, and should not be dismissed by historians of mathematics simply because it treats of the form of geometry rather than its content. Although I do not concede to N. that the form is more important than the content to the shaping of deduction, I think his claim does have its merits and that to ignore the form is to miss out on an important part of the very structure of Greek mathematics. His attempt to situate mathematics and mathematicians in the more general framework of Greek intellectual culture will be of interest to many classicists. N. has made an important contribution to intellectual history and has asked a diverse set of questions whose answers, while difficult, will broaden our understanding of the development of deductive practices.


1. J. Klein, Greek Mathematical Thought and the Origins of Algebra, (Cambridge, MA, 1968). See Netz, p. 50; also, p. 56 n. 118.

2. J. Bottero, ‘Symptômes, signes, écritures en Mésopotamie ancienne’ in J.P. Vernant, ed., Divination et rationalité (Paris, 1974); G. Manetti, Le teorie del segno nell’antichità classica (Milan, 1987).

3. M. Coe, Breaking the Maya Code (London, 1992).

4. E. Bickerman, Chronology of the Ancient World (London, 1968), p. 29 f.