Mathematics is sometimes said to be the only discipline which “you could continue to do if you woke up tomorrow and discovered the universe was gone.”1 Would Claudius Ptolemy, one of the most celebrated scientific writers of antiquity, have agreed with that view? Yes and no. As Jacqueline Feke shows in her excellent monograph, Ptolemy situates mathematics in a privileged intermediate space between the physical sciences and metaphysical speculation.
On the one hand, for Ptolemy, mathematics considers its objects as separated from matter and, therefore, conceivable without sense perception; on the other hand, mathematical objects are also perceptible and, as such, they form part and parcel of the natural universe in which we live—in fact, their study is even capable of changing the way we live towards the better. By exploring Ptolemy’s delicate equilibrium between these two competing conceptions of mathematical learning and its philosophical implications, Feke’s book offers a welcome new addition to a growing succession of monographic studies devoted to the theoretical and intellectual-historical aspects of ancient scientific literature.2
The main body of Feke’s monograph is composed of seven chapters plus an Introduction (numbered as Chapter 1) and a Conclusion. In the first part of the book, up to and including Chapter 4, Feke lays out the foundations of Ptolemy’s philosophy of science, showing that Ptolemy’s classification of the sciences buttresses his ethical view of the value of mathematics. In the following four chapters, Feke takes a closer look at Ptolemy’s scientific practice in major mathematical disciplines such as harmonics, astronomy, and cosmology, elucidating their mutual interactions and the relationship of each with mathematics itself. Correspondingly, the primary sources Feke uses most frequently are Ptolemy’s treatise On the Kritērion and Hēgemonikon (which Feke considers authentic), his work on Harmonics, and the preface to the Almagest. Besides recapitulating the book’s main arguments, the Conclusion includes an extensive section on the reception and influence of Ptolemy’s thought after antiquity. A comprehensive Bibliography and an Index complete the volume.
The overarching thesis of Feke’s book is that Ptolemy’s philosophical approach to mathematics is a form of ‘art of living’ (in the Hellenistic sense): hence, mathematics is not just a discipline to be studied and pursued for its own sake, but it also fosters a “mathematical way of life”. In fact, through the virtuous habits promoted by the study of numbers and forms, mathematicians “transform their souls into a fine and well-ordered state” (p. 78). Thus, the epistemological preeminence of mathematics over other disciplines is also, importantly, the bearer of ethical value. By assimilating the human soul to the well-attuned celestial beings studied by astronomy and to the musical consonances studied by harmonics, mathematics provides an invaluable path towards ὁμοίωσις θεῷ, the ultimate goal of human thought and action in Ptolemy’s philosophical framework.
One of the main merits of Feke’s book is that it situates Ptolemy’s thought within the Hellenistic and Roman intellectual scene, with particular attention to Aristotelianism and (Middle) Platonism. In general, Feke’s presentation and discussion of the sources is commendable. The book includes numerous, illuminating analyses of Ptolemaic texts and parallel passages, which enrich the author’s argumentation while never diverting attention from it. In particular, as Feke shows, Ptolemy justifies his hierarchical ordering of disciplines in Almagest 1.1 through epistemological criteria that are fundamentally Aristotelian, yet the conclusions he reaches are completely different from Aristotle’s. Scientific disciplines, for Ptolemy, ought to be distinguished based on their different objects, methods, and claims to truth. As in Aristotle’s Metaphysics, theoretical knowledge is divided into three branches: mathematics, physics, and theology. For Ptolemy, however, mathematics is characterized by objects perceptible through multiple senses and thinkable without sense perception (such as shapes and numbers), whereas physics deals with objects perceived by only one sense (such as the celestial bodies, which sight alone can reach) and theology is solely concerned with imperceptible objects (such as the Prime Mover). With an anti-Aristotelian twist, Ptolemy goes on to argue that theology and physics are conjectural disciplines, since the objects of the former are imperceptible and those of the latter do not possess certainty or stability; mathematics, by contrast, attains clear, incontrovertible truths.
In Feke’s view, not only does Ptolemy refrain from espousing any single philosophical school in asserting the unique status of mathematical ἐπιστήμη, but this claim is quite original and even subversive. According to Feke, Ptolemy’s insistence on the scientific applicability of mathematics as a superior epistemic tool is helpful from a philological—as well as a philosophical—point of view, since it can aid the scholar in determining the relative chronology of Ptolemy’s works (thus, as Feke persuasively argues, On the Kritērion most probably predates the rest of Ptolemy’s œuvre). Throughout the volume, Feke persuasively shows that Ptolemy’s appeal to mathematization pervades his epistemic ordering of the sciences. In particular, astronomy and harmonics—both eminently mathematical—are united by the common attunement caused by ἁρμονία, which also helps them to mould the human souls that practise them.
Arithmetic and geometry, by contrast, are not sciences in and of themselves, but rather logical paths or tools that enable the mathematical disciplines to progress. It would be misguided, however, to conclude therefrom that Ptolemy regards mathematics merely as a supplier of hypothetical models consistent with the phenomena observed by the scientist. Thus, Feke rightly takes issues with the instrumentalist interpretation of Ptolemy’s astronomy, widespread since Pierre Duhem’s seminal work,3 and shows that for Ptolemy—in keeping with his Platonico-Aristotelian metaphysics—mathematical objects are real independently of the physical objects they are used to describe. At the same time, Ptolemy’s mathematical realism does not prevent him from acknowledging the limits of human observation and, ultimately, of astronomy as a science: astronomers, for him, inevitably have to allow for quantitative approximation and imperfect calculations based on limited observational records. The possible (and frequent) accumulation of observational error yields increasing uncertainty in the scientific predictions of planetary movements, which often forces astronomers to adopt hypothetical, counterintuitive models to makes sense of the phenomena (a case in point is the equant, a historically controversial issue to which Feke regrettably does not devote an extensive discussion).4
The notion of scientific conjecture, which Ptolemy uses to contrast physics and other applied sciences with mathematics, is certainly not exclusive to his thought, but the idea of conjectural discipline more typically applies to rhetoric and medicine, rather than physics or cosmology (τεχνικὸς στοχασμός plays a major role in Ptolemy’s contemporary Galen, while Philodemus had described παρρησία as a stochastic method). Employing the language of conjecture in relation to the physical and cosmological sciences (e.g. at Tetrabiblos 3.2.5-6) seems to be a fascinating innovation on Ptolemy’s part; it is also significant that, for Ptolemy, scientific conjecture appears to be closely tied to prognostication, as is especially apparent in the case of astronomy, whose task is to predict the motions and configurations of stars (p. 169). The relationship between conjecture and prediction in Ptolemy, and in ancient science at large, appears worthy of further inquiry.
Greek medicine, for instance, has a long history of defending its epistemological status from critics who deem its prognoses and conjectures too aleatory: this is especially evident in the Hippocratic treatise On Regimen in Acute Diseases (8.1-9.1), whose author argues against detractors that liken physicians to practitioners of divination, unable to agree with one another on their predictions. Feke refers to this Hippocratic work via Galen’s commentary on it (p. 43), which shows Galen’s awareness of the mathematicians’ claim to absolute certainty (à la Ptolemy). Yet the Hippocratic doctors’ approaches to predictions, conjectures, and their scientific value—in Acute Diseases, Prognostic, and elsewhere—would perhaps have been worthy of further comparative examination, particularly given Ptolemy’s tendency to emphasize astronomy’s predictive nature in connection with its epistemic weakness: is the very notion of prognostication an ‘Achilles heel’ of ancient Greek science across ages and disciplines?5
Not constrained by inherently imperfect observations and predictions, harmonics is for Ptolemy a more ‘exact’ science than astronomy. As astronomy shapes the mind through the study of the celestial bodies’ orderly motions, harmonics offers insights into the workings of the human soul insofar as they mirror those of musical pitches. Feke’s detailed examination of Ptolemy’s psychology is very useful to locate Ptolemy’s thought within the Greek philosophical tradition. The soul itself, according to Ptolemy, is an object of study pertaining to the physical sciences, as it is a material body and, as such, part of the natural world. Specifically, the mathematician’s approach to the soul is centered around the notion of harmonic attunement.
It is surprising, in this connection, that Feke does not discuss the ways in which Ptolemy’s account of the human soul may be compared to Simmias’ image of the ‘harmonic’ soul at Phaedo 85e. According to Simmias, Socrates’ argument that the soul survives the death of the body can be challenged by claiming that the mind-body relation is an epiphenomenal one, analogous to the relation between the strings of a lyre and the musical harmony generated by them. Simmias’ soul-lyre analogy has a rich reception history in the imperial age,6 which testifies to its influence on the Academy and beyond: this makes the absence of the Phaedo’s section on the harmonic soul from Feke’s treatment of Ptolemy’s psychology all the more conspicuous. This, however, not detract from an overall outstanding philosophical portrait of a towering figure in ancient science.
In this important and innovative work, Feke offers the first comprehensive treatment in English of the philosophical thought of Claudius Ptolemy, and particularly of his ethical and psychological approach to mathematical study. Feke’s book is primarily written for a specialized audience of intellectual historians and scholars of ancient philosophy and science. At the same time, the author’s decision to quote the source passages only in translation—with the original Greek usually provided in the footnotes—makes the volume very helpful for pedagogical purposes as well. The volume is impressively well edited and produced, free from major typographical infelicities, and affordably priced. In conclusion, Feke’s book deserves a place on the shelves of historians of science, philosophers, and classicists alike. It is to be hoped that future research on Ptolemy and his philosophical conception of mathematics will continue to bridge the gap between scientific and humanistic approaches to the life of the mind.
1. A statement of uncertain origin, though often attributed to David Rusin: see e.g. John D. Barrow, New Theories of Everything. The Quest for Ultimate Explanation (Oxford University Press, 2014), p. 236.
2. Note especially L. Taub, Ptolemy's Universe: The Natural Philosophical and Ethical Foundations of Ptolemy's Astronomy (Chicago, IL: Open Court, 1993); A. Barker, Scientific Method in Ptolemy’s Harmonics (Cambridge University Press, 2000); S. Berryman, The Mechanical Hypothesis in Ancient Greek Natural Philosophy (Cambridge University Press, 2009); C. A. Roby, Technical ekphrasis in Greek and Roman Science and Literature: the Written Machine between Alexandria and Rome (Cambridge University Press, 2016).
3. P. Duhem, To Save the Phenomena: An Essay on the Idea of Physical Theory from Plato to Galileo. Translated by E. Doland and C. Maschler (University of Chicago Press, 1969). (Originally: ΣΩΖΕΙΝ ΤΑ ΦΑΙΝΟΜΕΝΑ: Essai sur la notion de théorie physique de Platon à Galilée, Paris: A. Hermann, 1908.)
4. On Ptolemy’s equant see now J. L. Zainaldin, “The Philosophical Justification for the Equant in Ptolemy’s Almagest,” Phronesis 62 (2017): 417-42.
5. See further P. Struck, Divination and Human Nature. A Cognitive History of Intuition in Antiquity (Princeton University Press, 2016).
6. K. Corrigan, “Simmias’ Objection to Socrates in the Phaedo,” The International Journal of the Platonic Tradition 4 (2010): 147-62.