Ancient Mathematics

This excellent little book is part of a series called Sciences in Antiquity, under the editorship of Roger French, the aim of which is “to cover the subject matter of what we call science” (my emphasis—the first installment in the series is a volume called Ancient Astrology). Cuomo’s contribution to the series covers the period from the 5th century BC through the 6th century AD—a rather prodigious chunk of the history of mathematics but one that has been in dire need of precisely this sort of introductory text ever since courses in the history and philosophy of ancient science began to grow in popularity in the 1990s. Cuomo’s stated aim is to make the history of ancient mathematics more accessible, not only by treating such mundane aspects of mathematics as counting and measuring, but by putting the history of ancient mathematics into its cultural context. Although a book of this nature cannot hope to compete with Heath’s definitive history when it comes to Greek mathematics,¹ Cuomo offers a much more wide-ranging treatment of mathematics from around the Mediterranean basin, and hers is a refreshingly interdisciplinary approach (among her stated interests are the historiography of mathematics, the relation between mathematics and politics, and the self-image of mathematicians). She omits topics in advanced mathematics and the whole of pre-Hellenistic Egyptian and Mesopotamian mathematics, but these omissions are understandable in a survey of this scope, and there are other texts one could use to complement hers.²

The book is informally divided into four groups of two chapters, one group for each of the following topics: early Greek mathematics, Hellenistic mathematics, Graeco-Roman mathematics, and late ancient mathematics. Within each group the first chapter treats the literary and material (archaeological, papyrological, epigraphical, legal) evidence while the second poses two questions arising from the evidence gathered in the first. This scheme is quite promising and works well for the most part because Cuomo not only has an impressive command of the material and literary evidence but also understands what issues are historically and philosophically of the most interest to teachers of the history of mathematics, and why. For example, the “problems” chapter in the Graeco-Roman section addresses, first, the knotty historical problem of disentangling Greek from Roman mathematical perspectives on the basis of the literary evidence of Cicero and Plutarch and, second, the extremely important philosophical question of how the ancients viewed the distinction between pure and applied mathematics. Cuomo discusses both of these problems with great care and admirable clarity. Of special interest to classicists will be the first problem in the Hellenistic chapter group, which has to do with the status of the text of Euclid. Cuomo has a particular interest in the historiography of mathematics, and her account of the ways in which mathematicians practiced textual criticism in antiquity as a means of cleaning up disparities in proofs offered in various texts is not only extremely interesting in its own right but also illuminates in a particularly effective way the difficulties of the task of the historical reconstruction of ancient mathematical theories.

One of the virtues of Cuomo’s book is her willingness to raise questions about mathematics that contextualize the subject in provocative ways. This can be an important pedagogical approach to take in an age when interdisciplinary studies are highly valued and philistine administrators are persistently questioning the relevance of many subjects in the humanities. How successful this method is, however, will depend upon how carefully one articulates the relevant arguments on each side of a given issue. Cuomo has a tendency to put forward her interpretation of the evidence without contrasting her views with possible alternatives, and it seems rather counterproductive (in a pedagogical text) simply to assert a particular stance in this way without giving the reader the machinery to sort things out for herself. For example, in her treatment of “political mathematics” Cuomo offers an overly quick assessment of the politicization of Aristotle’s conception of mathematics. She begins with the claim that “he established a correspondence between epistemic and social hierarchy” (p. 47) within which mathematical knowledge was for the socially powerful while other, lesser sciences were for skilled workers. She here refers to the distinction between ἐπιστήμη and τέχνη. Now it is true that Aristotle offers normative claims about the relative values of these types of knowledge, but the valuation is based not upon political considerations but rather upon the fact that one sort of knowledge is necessary while the other is contingent. I suppose one might mount an argument designed to show that Aristotle’s “preference”, if that is the right word, for necessary truth over contingent truth, is itself an artifact of political considerations, but Cuomo does nothing to defend such a view. Instead she suggests only that, because τέχνη is the domain of the “manual worker” while mathematics is practiced by the idle rich, the political tension must be obvious. It does not help her case much that she mistakenly takes sense perception (“like animals have”, p. 47) to be a part of Aristotle’s epistemic hierarchy. Cuomo rightly notes that “for Aristotle, the pursuit of knowledge presupposed leisure, freedom from daily cares and independence, all of them prerogatives of the privileged classes” (p. 47), but it is a mistake to infer from this that Aristotle “privileged” mathematical knowledge for political reasons. The rational faculty and its objects are constituted in a certain way by nature, and it is an artifact of that constitution that entails that only those with sufficient leisure will be able to pursue every sort of knowledge. If Cuomo wants to argue that Aristotle first privileges the idle rich and then assigns types of knowledge to the social classes, a much more sophisticated argument will be necessary, one that takes into account not only the dialectical discussion at the start of the Metaphysics, but also the relevant passages from the De anima and the Posterior Analytics dealing with the various rational faculties and their objects. I suspect that the sort of argument that would be required to make such a case would be beyond the scope of a work such as this—a point that ought to give an author pause.

To complete her account of Aristotle’s politicization of mathematics Cuomo offers an interpretation of his conception of justice that has him “blithely associat[ing] the right sort of mathematics with the right sort of politics” (p. 47). Criminal justice, she writes, “can afford to apply full equality, or arithmetical proportion” (p. 48), to the distribution of justice while in the case of distributive justice and the economics of “the big divide between rich and poor”, by contrast, “the form of geometrical proportion rears its head.” (One almost cannot help but read “its ugly head.) I can’t say as how the treatment of justice in the Nicomachean Ethics strikes me as in any way “blithe”, but sadly no argument is offered for this interpretation. Cuomo seems to think that the distinction, while “subtle”, is nevertheless easy to find in the lengthy passage she quotes from Nicomachean Ethics V.2-3. As in the case of the “politicization” of )επιστήμη and τέχνη, however, a proper account of what is going on in NE V would need to pay more attention to the background constraints governing what Aristotle has to say—in particular the Greek concept of proportion and Aristotle’s communitarianism—in order to do justice to the idea that those who contribute more to society are, in some sense, owed more. It simply will not do to assert, without argument, that his view is a blithe and subtle importation of the politics of “rich and poor” into mathematics in a way that is unappealing to 21st century Western democrats. In an undergraduate text this sort of thing borders on pandering to the sensibilities of the uneducated. It is almost as if the author is suggesting that there is no need to read someone like Aristotle with any more care than one would read an op-ed piece in a newspaper: if once we detect a sense in which a passage may be interpreted from a political point of view, there’s no need to do the actual work of assessing the philosophical system in the sort of detail that would be necessary to gain a genuine understanding of Aristotle’s claims.

Happily Cuomo’s sloppy interpretation of Aristotle does not represent a problem that is endemic to the book as a whole. By and large her illustrative “questions” chapters are both illuminating and well-argued. In addition to her excellent treatment of the text of Euclid to which I allude above, she also does a fine job explaining the various ways in which mathematics impinged on culture generally, for example, in the influence of mathematics on developments within Christianity; and she gives a fascinating account of some of the histories of mathematics preserved from antiquity. A careful teacher ought to be able to make very good use of a text such as this.

Notes

1. Sir Thomas Heath, A History of Greek Mathematics. Oxford: Clarendon Press, 1921.

2. See, for example, W. R. Knorr, The Ancient Tradition of Geometric Problems, Boston: Birkhäuser, 1986; O. A. W. Dilke, Mathematics and Measurement, British Musem Publications, 1987; D. H. Fowler, The Mathematics of Plato’s Academy: A New Reconstruction, Oxford: Clarendon Press, 1999; R. Netz, The Shaping of Deduction in Greek Mathematics: A Study in Cognitive History, Cambridge: Cambridge University Press, 1999.