Plato’s involvement with mathematics has long attracted attention from scholars. His treatment of arithmetic, geometry, and astronomy acquired a considerable number of commentaries even among ancient students of philosophy (one may easily think of Theon of Alexandria, Iamblichus, Proclus, Simplicius, and even of the anonymous Theologoumena) and modern (think of the late John Cleary). Since the question of Plato’s understanding of mathematics (and its many branches) is profoundly linked to his understanding of true being as well as to his epistemology, and also has far-reaching consequences for political philosophy, this interest is hardly surprising.
As one can also observe in his earlier work, Kouremenos’s critical approach relies on focused, close reading of Plato’s text (or, to be precise, of selected passages of Plato’s work), with relatively few forays into the existing scholarship. The strict nature of this focus is reflected in how concise and selective the bibliography is (pp. 139-43). Still, the list appears representative where studies of Plato’s mathematics are concerned – although what it does not account for is the considerable literature on the cave and line similes; one also notices the absence of any references to Cleary’s magisterial Aristotle and Mathematics.
Kouremenos’s latest work is not easy. Its difficulty, however, lies not so much in the discussion it contains as in its reliance on previous publications by the author. First and foremost among these is Kouremenos’s study of mathematical sciences in the Republic: it is there that Kouremenos pays particular attention to the Platonic notion of astronomy as well as providing the necessary explanations for the identification of mathematical norms with forms (eidē). It is there, in fact, that Kouremenos asserts that “Plato seems to view what is studied in mathematics as forms approached in a particular way” (Unity, p. 12), and where he seeks to explore the critique aimed by Aristotle at the theory of forms. It is here that one finds the discussion of eidos as associated with the objects of mathematical knowledge—a discussion which appears to form a point of references in many places in Forms, Mathematics and Astronomy. In a manner of speaking, the 2015 monograph (and its 2010 predecessor, the Heavenly Stuff) is a logical antecedent of the present work, its prior reading in many ways presupposed by Kouremenos. Understandably, some knowledge of the preceding volumes facilitates the reading of his most recent publication.
The present volume falls into two parts of almost equal length. The first (pp. 8-76) focuses on mathematics (or on forms as forms of mathematical objects). Meanwhile, the second part (pp. 77-134) centers on complex issues concerning the relationship between philosophy and astronomy. Seen as propaedeutic to the cognition of the Good, astronomy emerges as an important component of Socrates’ own intellectual biography in the Phaedo, while playing a prominent part in other works by Plato (Kouremenos discusses Republic 7, Timaeus, Epinomis, and Laws). Impressively brief conclusions (pp. 135-7) reemphasize the importance of mathematics and astronomy as propaedeutics to philosophy, while simultaneously highlighting the contemporary relevance of Plato’s discussion of astronomy. Seen against the background of Eudoxus’ achievements, Plato’s discussion is seen as a voice in an ongoing debate over the observed phenomena, and an expression of a firm belief in the philosophical potential of mathematical sciences as source of true cosmological knowledge (p. 137).
As far as the contents of the monograph are concerned, it is certainly not an easy read: focusing on issues at the crossroads of the history of philosophy and the history of science, it is demanding, and requires prior knowledge of ancient astronomical theory. Furthermore, as in his previous volumes, Kouremenos challenges a number of generally accepted notions, while not always engaging with those inclined to defend them (as rightly pointed out by Zhmud, this includes M. Burnyeat and his ‘Plato on Why Mathematics is Good for the Soul’). Kouremenos’s lack of engagement (one could almost say: a policy of non-engagement) makes it especially hard going for an unprepared (or ‘uninitiated’) reader, who may easily take Kouremenos at his word, without realizing that a given issue can be (or, in fact, has been) interpreted in an alternative manner. This is not to say that Kouremenos’s position is untenable—it is only to note that the work cannot be read as an introduction to the study of Plato’s philosophy of mathematical sciences. But for someone well versed in both the matter and the relevant literature, Kouremenos is a valuable dissenting voice, the unorthodox opinion that makes his reader question the widespread opinion and re-appreciate the challenges of his study matter. And this is not a small achievement.
 J. J. Cleary Aristotle and Mathematics. Aporetic Method in Cosmology and Metaphysics, Philosophia Antiqua vol. 67, Leiden 1995. It is important to remember that considerable part of Cleary’s discussion in the work (indeed, Chapters 1-3) centers on Aristotle’s polemic with Plato and his views on mathematics, mathematical objects, etc.
 T. Kouremenos, The Unity of Mathematics in Plato’s Republic, Palingenesia vol. 102, Franz Steiner Verlag Stuttgart 2015.
 T. Kouremenos Heavenly Stuff: The Constitution of the Heavenly Objects and the Theory of Homocentric Spheres in Aristotle’s Cosmology, F. Steiner Verlag, Stuttgart 2010.
 Zhmud, as cited in n. 4 above; M. Burnyeat, ‘Plato on Why Mathematics is Good for the Soul’ in T. Smiley (ed.) Mathematics and Necessity. Essays in History of Philosophy (Oxford University Press 2000), pp. 1-81.