The chapter ‘Concepts in Greek Mathematics’ by Reviel Netz problematises a set of assumptions commonly encountered in the literature on Greek mathematics, which typically derive from a supposedly objective, a-historical conception of mathematical theory and practice. In sharp opposition to that tradition, Netz raises the possibility that the purpose of engaging with mathematical concepts may have been different in antiquity than what it has been taken to be. He asks central questions afresh, for instance: why do mathematical texts begin with definitions, and what is the purpose of mathematical definitions and of axioms. In connection to these issues, he highlights new aspects of the relationship between Greek mathematics and Greek philosophy, between engaging with mathematical concepts and philosophical thinking. He also advances the thesis that the relations between mathematics and philosophy changed through the various eras of antiquity, as did mathematical concepts and the role of mathematical definitions. We should seriously entertain the idea that even mathematical concepts need to be viewed within a given historical and cultural context.

The concept of analogy was first analysed in classical Greek thought. By 'analogy' was meant a four-term relation: A is to B as C is to D. Initially, within Greek mathematics, analogy expressed the equality of the relative magnitudes of two line pairs, when the ratio of line A to line B is identical with the ratio of line C to line D. An analogy asserted a proportionality. And the theory of similar triangles exhibits the basic form of argument by analogy, with a set of valid proofs showing which additional properties, equiangularity say, the two triangles must share. In Euclid are all the features of the analogical relationship relevant to our enquiry. For analogy was soon taken beyond its mathematical confines, especially by Aristotle, in exploring how these geometrical concepts can be applied in empirical contexts. These explorations kept the commitment to proportionality, which persists in every modern analyst of analogy knowingly upholding the Aristotelian tradition.

This chapter uses Aristotle’s account in the Nicomachean Ethics to reconstruct Eudoxus’ argument for the thesis that pleasure is the good. He sets out and explains Eudoxus’s argument from universal pursuit: pleasure must be the good because all animals pursue pleasure in all natural and fitting choices. Eudoxus’ naturalism is an important background assumption here. He assumes that each animal, by nature, successfully chooses in all situations what is good for itself. This allows him to move from an observation about the universal pursuit of pleasure to the claim that pleasure is a feature of all natural and good choices. The pleasure that features in such choices is overall pleasure. Thus, Eudoxus can allow that animals sometimes naturally choose things that are painful, provided that what they choose contains more pleasure than pain overall. Aufderheide ends by suggesting how Eudoxus might defend the claim that pleasure is not merely a good, but moreover the good. This is not to claim that pleasure is the only thing that is good, but rather that pleasure plays a unique role in relation to choice: it is the only thing that features in all natural and good choices as a good.

This chapter discusses the central question why Aristotle, in spite of having everything required to conceptualize a complex measure of speed in terms of time and space, did in the end not explicitly develop such a measure. It is first investigated whether contemporaries of Aristotle may have worked with such a complex measure of speed, and concluded that it cannot be found in either of the two thinkers most likely to have done so, namely Eudoxus and Autolycus. The second part of the chapter investigates what made Aristotle cling to a simple measure and suggests that there are mathematical and metaphysical reasons: metaphysically, Aristotle cannot explicitly accommodate a relation as a measure of motion, since relations are derivative and problematic for him; mathematically, the principle of homogeneity which derives from the realm of Greek mathematics makes it impossible to combine of different dimensions in a single measure in the way needed for measuring speed in a mathematically informed physics such as Aristotle’s.