Editions of the single treatises composing Sextus Empiricus’ Against the Mathematicians (M) are not many[1] and before Lorenzo Corti’s work, none had been devoted to Against the Arithmeticians, namely M IV. As the acknowledgements make clear, the author’s preparation for this book lasted more than fifteen years, through post-doctoral researches carried out at some of the world’s most prestigious Philosophy and Classics departments and under the advice of the most renowned specialists in ancient scepticism. The book comprises the following four sections, which will be singularly discussed below: (a) an ample introduction; (b) an English translation based on the Greek text printed in Mau 1961 but revised in some passages where Corti adopts readings appearing in other editions or critical-text studies; (c) an extensive commentary; (d) a detailed summative conclusion. If we add that two commentary chapters include the results of two previous articles that Corti wrote on the notion of number in Sextus,[2] all ingredients seem to be in place for a momentous contribution in Sextus scholarship. And yet, that turns out not to be the case due to the structural and interpretive flaws with which the work is littered and which will be examined here. But even so, this book remains unmissable both for Sextan scholars, given its uniqueness on the shelves, and for those in ancient mathematics, given the vast and meticulous comparison that Corti performs between Sextus and the several preceding philosophers who had already written on number and arithmetic before him.
The introduction is divided into four sections. In the first (‘Sextus’ Life and Works’), Corti addresses the difficulty of determining Sextus’ time and place and reviews the vexed issue of the relative chronology of Sextus’ three extant philosophical works, namely Outlines of Pyrrhonism (PH), M VII-XI, and M I-VI, the last of which being also known as Against the Professors and consisting of six essays directed against the six arts of grammar, rhetoric, geometry, arithmetic, astrology, and music. What strikes here is that soon after opening the book with the correct observation of the non-coextensiveness of ‘Pyrrhonism’ and ‘ancient scepticism’—the former being a variety of the latter—Corti refers to PH as Outlines of Scepticism instead of Outlines of Pyrrhonism (see pp. 1 and 3), as one would have also expected from its original Greek title Πυρρώνειοι Ὑποτυπώσεις. As Corti himself explains, Outlines of Scepticism is what PH was retitled in Annas/Barnes 1994.[3] Influential as that edition might have been, however, that particular translation was based on admittedly weak reasons of audience’s understanding; hence[4] the Sextan scholar Richard Bett has always opted for Outlines of Pyrrhonism.[5]
The second paragraph of the introduction covers Sextus’ targets, method, and sources in the whole Against the Professors. Here, Corti shares Jonathan Barnes’ views that in this work Sextus is an excerptor and a compiler drawing from Pyrrhonian, Epicurean, Peripatetic, Academic, and Stoic sources.[6] Then, narrowing his focus down to Against the Arithmeticians, the third paragraph starts by arguing for a tripartite argumentative structure of the text, consisting of M IV.1 (where Sextus announces his aim of destroying arithmetic by destroying its subject, number), M IV.2-10 (where Sextus introduces what he calls a Pythagorean philosophy of number), and M IV.11-34 (where Sextus objects to the principles of that system). The third paragraph continues with Corti’s agreement with Walter Burkert’s thesis that despite Sextus calling his adversaries ‘Pythagoreans’, his real target are philosophers making use of ideas originating from Plato and the Old Academy.[7] Finally, the reader is briefed on the distinction between two major ancient Greek theoretical approaches to number, namely the mathematical one, represented by the so-called ‘arithmetical books’ of Euclid’s Elements, and the philosophical one, belonging to neo-Pythagorean and neo-Platonic authors, such as Nichomachus of Gerasa, Iamblichus of Chalcis, and Theo of Smyrna, and followed by Sextus in M IV.
The introduction ends with a fourth paragraph entitled ‘Originality and Interest of the Present Work’, where Corti observes, among other things, that M IV is a ‘dense and rather cryptic’ text, and so the primary goal of his commentary is ‘to provide a philosophical elucidation and understanding of Sextus’ arguments’ (p. 21). Alas, such expectations end up not being met, for two reasons. First, Corti’s Commentary is not as successful in explaining specific elements of Sextus’ exposition as it is in reconstructing and assessing his arguments as wholes. The following are two examples: (a) at M IV.5, Sextus says that out of the number four ‘the pyramid comes to be, a solid body and shape’ (emphasis mine). Here, it would have been the task of the commentator to explain why the pyramid is taken to be the best example of a solid built upon four units, maybe by means of the illuminating loci similes of S.E. M X.278-280 and Xenocrates Fragm. 178 Isnardi Parente2, of which Corti makes also use at p. 65, n. 29; (b) how is it that at M IV.21 Sextus even takes into account the possibility that once a unit is juxtaposed to another unit something might be subtracted? The verb ‘juxtapose’ (παρατίθεσθαι) makes one think only of an addition of two units. But if Sextus uses it in connection with subtraction, then there must be some (figurative?) sense in which it is taken and which the commentator should have clarified. Second, the very reconstruction and assessment of Sextus’ argumentation is often submerged and overshadowed by pages and pages of Quellenforschung, which it would have been better to conclude in the third section of the introduction, where generous space was already given to it. This second drawback is particularly noticeable in the first two chapters of the commentary.
In the first chapter, commenting on M IV.1, Corti spends the eleven-page section ‘Arithmetic, Philosophy of Number, and Sextus’ Strategy in M IV’ (pp. 42-52) answering why Sextus, who wants to show the inexistence of number, focuses on treatises of philosophy of arithmetic rather than arithmetic. Besides having already appeared in introduction, this question is an obvious one, since one just needs to know what a mathematician and a mathematics philosopher do to understand that only the latter is interested in the problems of the existence and nature of numbers.
Likewise, throughout the second chapter (on M IV 2-10) Corti resumes the issue of the identity of the Pythagoreans who are attacked by Sextus, and he confirms his support for Burkert’s views that the doctrines of the Two Principles and of the Derivation System here described actually have a Platonic origin. This takes up a whopping twenty-seven pages (pp. 57-83) where several loci similes from Sextus himself and other, mostly Platonic, authors are dissected, while the reader is never given an answer to the natural question why if by ‘Pythagorean mathematicians’ (οἱ ἀπὸ τῶν μαθημάτων Πυθαγορικοί) at M IV.2 Sextus is actually referring to ‘scholars who come after and are different from the Pythagoreans’ (p. 57), he nonetheless calls them like that.
As for the rest of the commentary, chapter 3 is devoted to M IV 11-20, where Sextus introduces and counters two definitions of the one, i.e. ‘that without which nothing is called one’ and ‘that by sharing in which each thing is called both one and many’, which he attributes to Plato. By contrast, Corti believes that neither of them can be traced back in Plato’s extant works, but wrongly. For both appear, although differently phrased and in reverse order, at Prm. 144a-c, where it is argued: first, that the existence of the one entails that of number, which, in turn, entails that of a plurality of beings (144a); then, that each singular being must possess the one in order to exist (144c). The henological Parmenides, of which Corti himself makes use only in connection with M IV.18-20, is therefore the natural dialogue to look at also for the rest of this passage, and that would have saved him a few pages (pp. 86-91) of great interpretive feats excavating the Philebus and Phaedo in search of something helfpful.[8]
Chapter 4 comments on M IV 21-2, where Sextus opposes the idea that the number two is originated from the one through juxtaposition, a scenario causing a puzzle that Plato had already discussed in the Phaedo. This is the first of a series of sophisms that will occupy the rest of the work and which Corti judges ‘instructive’, insofar as concerning ‘serious and deep metaphysical issues’ (p. 20), which he accurately analyses in this and the last two chapters of the commentary. More precisely, the fifth chapter is devoted to M IV.23-30, where Sextus refutes the idea that a number can result from the subtraction of a unit from another number, while chapter 6 to M IV 30-34, where it is the addition of a unit to another number to be shown impossible. Corti argues that Sextus borrows the arguments of the whole M IV.23-34 from Pyrrhonian predecessors, who had drawn from second-century BCE Academics, who, in turn, had used Epicharmus’ so-called Growing Argument to prove that the Stoic notions of increase and decrease are not intelligible (pp. 188-189). However, in the search for ancestors of this kind of puzzles, Corti forgets at least three other candidates which are arguably closer in thought and time to Sextus’ one than that by Epicharmus,[9] who lived in the early fifth century BCE. These are Pl. Cra. 432, Arist. Soph. el. 178a, and, especially, Dissoi Logoi 5.14, which I date between 355 and 338 BCE[10] and which reads ‘if someone should take one from ten, there would not be either ten or one anymore’, in complete assonance with M IV.25.
Finally, all the points established by Corti in the commentary are summed up in the conclusion, where he also outlines the kind of arithmetic Sextus attacks in M IV. It appears to be a monistic system (i.e. based on the tenet that the one is the principle of all the other numbers) to be ascribed to Academics either of the first century BCE or the first century CE, who were somehow indebted to the Old Academy. Attestations of this system are in Mathematics Useful for Understanding Plato, a work by Theo of Smyrna (second century CE), who seems to have inherited such notions through Moderatus of Gades (first century CE). Furthermore, according to Corti, Sextus’ arguments against the existence of number are likely to come from his Pyrrhonian predecessor Aenesidemus (first century BCE). This hypothesis perfectly squares with what I have been arguing about the relationship between Dissoi Logoi and Sextus Empiricus for a while—but which Corti seems to have overlooked, also considering his silence about of the locus similis from this sophistic text that I discussed above. For Dissoi Logoi has been handed down at the end of Sextus’ manuscripts because it worked as a source for him, who presumably found it among now-lost materials relating to Aenesidemus, who too had already known and studied it.[11]
Coming to the translation, it stands out as the best part of the book and is based on a Greek text that has improved from Mau’s one thanks to Corti’s editorial choices, which are always justified in specific paragraphs of textual remarks within the commentary. However, one wonders why the reader is provided with the original text of almost all loci similes,[12] even when it seems to be of no use (see e.g. pp. 136-137, nn. 9-11), but they cannot have Corti’s Greek of the work in parallel to the translation. It would have taken five more pages, but it would have allowed the reader an easier check on the translation and spared them the effort of keeping an eye on Mau 1961 to understand philological notes starting like ‘At M IV 14, line 18-19…’ (p. 84, emphasis mine).
A commented translation like this would have also deserved a bibliography distinguishing primary and secondary sources, as well as a general index where the appearances of figures such as Theo of Smyrna and Walter Burkert, so relevant for Corti’s discussion, are fully recorded. However responsible the author might be for these editorial inaccuracies, they too contribute to making of this book an overall good but not excellent work.
References
Annas, J., and Barnes, J. (1994), Sextus Empiricus. Outlines of Scepticism, Cambridge, Cambridge University Press.
Annas, J., and Barnes, J. (2000), Sextus Empiricus. Outlines of Scepticism, Cambridge, Cambridge University Press (2nd edn).
Barnes, J. (2014), ‘Scepticism and the Arts’, in J. Barnes, Proof, Knowledge, and Scepticism. Essays in Ancient Philosophy III, Oxford, Clarendon Press: 512-535.
Bett, R. (1997), Sextus Empiricus. Against the Ethicists, Oxford, Clarendon Press.
Bett, R. (2005), Sextus Empiricus. Against the Logicians, Cambridge, Cambridge University Press.
Bett, R. (2012), Sextus Empiricus. Against the Physicists, Cambridge, Cambridge University Press.
Bett, R. (2018), Sextus Empiricus. Against Those in the Disciplines, Oxford, Oxford University Press.
Blank, D. (1998), Sextus Empiricus. Against the Grammarians, Oxford, Clarendon Press.
Burkert, W. (1972), Lore and Science in Ancient Pythagoreanism, Cambridge (Mass.), Harvard University Press.
Corti, L. (2015a), ‘Sextus, the Number Two and the Phaedo’, in S. Delcomminette, P. d’Hoine, M.-A. Gavray (eds.), Ancient Readings of Plato’s Phaedo, Leiden, Brill: 90-106.
Corti, L. (2015b), ‘Scepticism, Number and Appearances: the ἀριθμητικὴ τέχνη and Sextus’ Targets in M I-VI’, Philosophie Antique 15: 123-147.
Davidson Greaves, D. (1986), Sextus Empiricus. Against the Musicians, Lincoln and London, University of Nebraska Press.
Mau, J. (1961), Sexti Empirici Opera, vol. 3: Adversus Mathematicos I-VI, Leipzig, Teubner (2nd edn).
Molinelli, S. (2018), Dissoi Logoi: A New Commented Edition, Doctoral dissertation at the University of Durham. http://etheses.dur.ac.uk/12451/
Molinelli, S. (2024), Dissoi Logoi: Introduction, Critical Text, Translation, and Commentary, Cham, Springer.
Spinelli, E. (1995), Sesto Empirico. Contro gli Etici, Napoli, Bibliopolis.
Spinelli, E. (2000), Sesto Empirico. Contro gli Astrologi, Napoli, Bibliopolis.
Notes
[1] Blank 1998 on M I; Spinelli 2000 on M V; Davidson Greaves 1986 on M VI; Bett 2005 on M VII-VIII; Bett 2012 on M IX-X; Bett 1997 and Spinelli 1995 on M XI.
[2] Corti 2015a, Corti 2015b.
[3] Annas/Barnes 1994.
[4] ‘We should add a note on the title: a decent translation of Sextus’ Greek title would be Pyrrhonian Outlines or Outlines of Pyrrhonism. We decided to substitute ‘Scepticism’ for ‘Pyrrhonism’, fearing (perhaps wrongly) that the latter word might be misunderstood or not understood at all. To critics, we offer the sophistical defence that Outlines of Scepticism is the title of our translation, not the translation of Sextus’ title’ (Annas/Barnes 2000, xxxiv, n. 1).
[5] Cf. e.g. Bett 2005, Bett 2012, Bett 2018 passim.
[6] Barnes 2014, 515-516.
[7] Burkert 1972, 53-83.
[8] The very Phlb. 14b-17a, which Corti resorts to on pp. 86-87, is actually a lower-level version of Prm. 144b-e, as is clear just from comparing the two theses of both passages, namely ‘one is many and many is one’ (Phlb.14c) and ‘not only what is one is many, but also the one itself, which is divided by being, must be many’ (Prm.144e).
[9] Inexplicably and unusually of him, Corti never literally quotes Epicharmus’ argument, but he just references (fr. 276 Kassel-Austin) and paraphrases it (cf. p. 159 n. 43, p. 187).
[10] Molinelli 2024, 45.
[11] See Molinelli 2018, 286-295, which has now evolved into Molinelli 2024, 57-62.
[12] But see n. 9 above for an (unfortunate) exception.