The reputation Theologoumena Arithmeticae has acquired is largely that of being an odd, and frequently opaque, compilation of arithmological lore. As a sourcebook for this aspect of the Pythagorean tradition it is, of course, invaluable. However, its poor reputation is increased, and its historical value lessened, by the depredations time has wrought on the text. ThA was never great prose: it is a compilation, largely from the lost Theologoumena Arithmeticae of Nicomachus of Gerasa and from Anatolius' Peri Dekados; and the (...) text which cannot be safely or conjecturally assigned to these two sources often reads like little more than the written-up notes of some student. However, the treatise contains passages of considerable lucidity, within the context of its peculiar framework. (shrink)

On Zeno Beach there are infinitely many grains of sand, each half the size of the last. Supposing Aristotle denied the possibility of Zeno Beach, did he have a good argument for the denial? Three arguments, each of ancient origin, are examined: the beach would be infinitely large; the beach would be impossible to walk across; the beach would contain a part equal to the whole, whereas parts must be lesser. It is attempted to show that none of these arguments (...) was Aristotle’s. Indeed, perhaps Aristotle’s finitism applied to magnitude only, not plurality. (shrink)

In Plato's Phaedrus, Socrates offers two speeches, the first portraying madness as mere disease, the second celebrating madness as divine inspiration. Each speech is correct, says Socrates, though neither is complete. The two kinds of madness are like the left and right sides of a living body: no account that focuses on just one half can be adequate. In a recent paper, Hugh Benson gives a left-handed speech about a psychic condition endemic among mathematicians: dianoia. Benson acknowledges that his account (...) is one-sided, but only hints at the virtues of right-handed dianoia. This note sketches a somewhat fuller picture. (shrink)

This paper examines a passage in the Theaetetus where Plato distinguishes knowledge from true belief by appealing to the example of a jury hearing a case. While the jurors may have true belief, Socrates puts forward two reasons why they cannot achieve knowledge. The reasons for this nescience have typically been taken to be in tension with each other . This paper proposes a solution to the putative difficulty by arguing that what links the two cases of nescience is that (...) in neither case do the jurors act from an epistemic virtue and that doing so is a necessary condition of knowledge. Appreciating that it is a necessary condition of knowledge that it be the result of an epistemic agent's agency in a distinctive way provides a satisfying solution to the difficulty Burnyeat detected and also does justice to an otherwise neglected aspect of Plato's epistemology: his talk of cognitive capacities and virtues and his focus on what it is that is active and passive in epistemic processes. (shrink)

The thirty year long friendship between Imre Lakatos and the classic scholar and historian of mathematics Árpád Szabó had a considerable influence on the ideas, scholarly career and personal life of both scholars. After recalling some relevant facts from their lives, this paper will investigate Szabó's works about the history of pre-Euclidean mathematics and its philosophy. We can find many similarities with Lakatos' philosophy of mathematics and science, both in the self-interpretation of early axiomatic Greek mathematics as Szabó reconstructs it, (...) and in the general overview Szabó provides us about the turn from the intuitive methods of Greek mathematicians to the strict axiomatic method of Euclid's Elements. As a conclusion, I will argue that the correct explanation of these similarities is that in their main works they developed ideas they had in common from the period of intimate intellectual contact in Hungarian academic life in the mid-twentieth century. In closing, I will recall some relevant features of this background that deserve further research. (shrink)

The purpose of this paper is to reassess some mathematical examples in Plato’s dialogues which at a first glance may appear to be nothing more than trivial puzzles. In order to provide the necessary background for this analysis, I shall begin by sketching a brief overview of Plato’s mathematical passages and discuss the criteria for aptly selecting them. Second, I shall explain what I mean by ‘mathematical examples,’ and reflect on their function in light of the discussion on παραδείγματα outlined (...) in theSophistand theStatesman. Against this framework, I shall move to a close examination of specific examples drawn from theTheaetetus(154c–55d), theRepublic(523c–24d), and thePhaedo(96d–97b, 100e–101d). By placing these examples in the broader context of pre-Euclidean mathematics, I shall show that their mathematical content is often less elementary than might appear at first sight. Moreover, by placing emphasis on the specific philosophical concerns that motivated their introduction, I shall argue that the examples are not merely nonsensical jokes. Even if their illustrative purpose is not always immediately clear, and even if they can sound playful and bizarre, they in fact fulfil an important function: by virtue of their power to generate wonder or confusion, they serve as exercises and act as a trigger with respect to the difficult philosophical issues they are intended to clarify. (shrink)

The discovery of incommensurability by the Pythagoreans is usually ascribed to geometric or arithmetic questions, but already Tannery stressed the hypothesis that it had a music-theoretical origin. In this paper, I try to show that such hypothesis is correct, and, in addition, I try to understand why it was almost completely ignored so far.

I argue against a formidable interpretation of Plato’s Divided Line image according to which dianoetic correctly applies the same method as dialectic. The difference between the dianoetic and dialectic sections of the Line is not methodological, but ontological. I maintain that while this interpretation correctly identifies the mathematical method with dialectic, ( i.e. , the method of philosophy), it incorrectly identifies the mathematical method with dianoetic. Rather, Plato takes dianoetic to be a misapplication of the mathematical method by a subset (...) of practicing mathematicians. Thus, Plato’s critique of dianoetic is a not a critique of mathematics, as such, but of mathematicians. (shrink)