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.

Chapter 1 provides an overview of a new interpretation of Kant’s philosophy of mathematics and its relation to Kant’s critical philosophy. It argues that Kant’s philosophy of mathematics impacted his critical philosophy in an unappreciated way: not just as evidence of synthetic a priori knowledge, not just as a model for the role of intuition in all human cognition, but also for an account of experience of the world as constituted by magnitudes. Appreciating Kant’s understanding of magnitudes and their measurement requires recovering a way of thinking about mathematics prior to the nineteenth-century “arithmetization“ of mathematics. A view of mathematics as a science of magnitudes and their measurement has roots in ancient Greek mathematics, above all in Euclid’s Elements. It is only against the backdrop of this mathematical tradition that Kant’s philosophy of mathematics, and hence his account of experience, can be properly understood. Uncovering the presuppositions of the Euclidean theory of magnitudes also reveals the presuppositions of Kant’s own understanding of mathematics and the mathematical character of experience. At the same time, Kant reworked the theory of magnitudes and reformed the metaphysics of quantity found in eighteenth-century rationalism in ways that are crucial to understanding his views.

Chapter 7 returns to Kant for a deeper analysis of his views and their relation to the Euclidean mathematical tradition. Chapter 6 revealed that Euclid defined neither magnitude nor homogeneity, so that these notions are at best implicitly defined by the Euclidean-Eudoxian theory of proportions. Kant reworks the Euclidean theory of magnitudes, defining magnitude in terms of his own understanding of homogeneity, which admits of no qualitative difference of the manifold and which I call strict homogeneity. Most importantly, he thinks that intuition is required to represent either a continuous or a discrete manifold without qualitative difference. This role for intuition in Kant’s philosophy of mathematics and experience has not been appreciated, but finds support in various texts, especially Kant’s lectures on metaphysics and his criticisms of Leibniz’s views in the Amphiboly. Given Kant’s understanding of qualities, differences in dimension correspond to qualitative differences, so that Kant’s account corresponds to Euclid’s understanding of homogeneous magnitudes. Understanding the role of intuition allows us to appreciate the role of the categories of quantity and intuition in part–whole relations and the composition of magnitudes. The chapter closes with clarifying the sense in which intuition is required for the representation of magnitudes.

Chapter 10 briefly comments the influence of Euclid’s theory of magnitude and the Eudoxian theory of proportions on Kant’s thought in light of advances in mathematics from the early seventeenth century to the late eighteenth century. Despite these advances, mathematics was still often viewed as a science of magnitudes and their measure. The chapter also indicates further work to be done to understand Kant’s philosophy of geometry, arithmetic, algebra, and analysis.

Kant’s reworking of the Euclidean theory of magnitudes and reformation of the Leibnizian-Wolffian metaphysics of quantity are in service of his project of explaining the foundations of mathematical cognition and the mathematical character of experience. The previous chapters revealed that Kant’s account is fundamentally mereological. The categories of quantity allow us to represent the part–whole relations among magnitudes, and Kant’s understanding of the role of the categories of quantity, the nature of composition, and the definitions of extensive and intensive magnitudes are all mereological. This introduces a gap between Kant’s mereological account of magnitudes and Euclid’s notion of magnitude for the latter is implicitly defined by its role in the theory of proportions – a richer, mathematical notion of magnitude. This prompts a closer look at what makes Euclid’s understanding of magnitudes mathematical. This chapter argues that Euclid’s geometry presupposes a tacit theory of measurement that is general, pure, and concrete, a theory that crucially depends on the relation of equality. It traces these presuppositions through the Euclidean tradition. It then argues that Kant also tacitly assumed the theory of measurement, but that he was aware of the crucial role that equality plays in bridging the gap between mereology and mathematics.

The previous chapters established the centrality and importance of Kant’s theory of magnitudes. Before investigating Kant’s views more deeply, Chapter 6 provides critical background concerning the theory of magnitude, which is rooted in Euclid’s Elements. It describes the organization and deductive structure of the Elements to explain tensions between Euclid’s account of number and continuous spatial magnitude that persisted into the eighteenth century. It then explains the theory of proportions, in particular, the crucial definition of sameness of ratio. It also describes the mathematical nature of the continuous spatial magnitudes implicitly defined by that theory, including their homogeneity. Euclid also covers number and arithmetic and he regards numbers as collections of units. Much of the theory of proportions covers numbers as well as continuous spatial magnitudes, although Euclid gives numbers a separate treatment. This commonality inspired the search for a universal mathematics that covered continuous magnitudes more generally as well as number. The chapter traces the history and development of these themes in the Euclidean mathematical tradition, and describes its influence on three of Kant’s immediate predecessors: Wolff, Kästner, and Euler. It explains why these philosophers and mathematicians regarded mathematics as a science of magnitudes and their measurement.

This chapter challenges the binary contrast between ’myth’ and rational account (logos), reviewing the negative impact of the application of that dichotomy when used to draw contrasts between properly scientific modes of discourse and those to be dismissed as irrational. Ethnographic reports show that there is often no equivalent to our term ’myth’ in indigenous vocabularies, at least not one that carries similar pejorative undertones. The arguments of Lévi-Strauss that systems of myth may convey ’concrete science’ have the merit of taking those systems seriously, but still imply a pejorative binary judgement.

This chapter examines the varying roles that definitions may play in scientific investigations. Obviously they may laudably aim at clarifying the problem to be explored, but the demand and search for univocal definitions can have a limiting effect on the inquiry subsequently pursued. When a definition is presented as the goal of an investigation, for example of the characteristics of an animal species, that may have the effect of obscuring some of the complexities that may be uncovered along the way. The problem of the role of definitions in an axiomatic system such as Euclid’s lies in their presumed self-evidence.

Roberto Bonola, in his celebrated history of non-Euclidean geometry, imported a distinction between elementary and advanced mathematics commonly drawn around 1900 back to the discoveries of the 1820s and 1830s. In so doing he fell into anachronism and misrepresented the subject, most obviously in his treatment of the work of Lobachevskii.

The third chapter introduces Ghazālī’s and Rāzī’s responses to Ibn Sīnā’s theological and cosmological challenges to the occasionalist worldview. Ghazālī’s response is heavily influenced by Ashʿarite theology’s emphasis on the divine will and freedom. In this discussion, Ghazālī harkens back to the earlier Ashʿarite tradition, offers novel applications of old arguments, and raises important challenges to Ibn Sīnā. Rāzī, on the other hand, formulates a list of arguments for the defense of Ashʿarite cosmology based on a discrete and atomistic model of the universe. Rāzī’s atomistic arguments can be seen as a novel development in the occasionalist tradition. Rāzī’s use of Euclidian geometry for and against atomism also led to emergence of an occasionalist philosophy of science marked by pragmatic and sceptic attitude towards dominant scientific models.

By the turn of the twentieth century, many philosophers claimed that Kant’s philosophy of mathematics (and, indeed, his entire philosophy) was undermined by the real possibility of geometries in which Euclid’s axiom of parallels is false. It is surprisingly not well known that Kant – though never discussing parallel lines in his published work – did write a series of unpublished notes on the philosophical problems posed by parallel lines. This paper presents the argument and chief issues of these notes. Kant’s main point in these notes is that neither Euclid’s definition of parallel lines, nor the alternative definition proposed by Leibniz and Christian Wolff, fulfills the requirements on mathematical definitions that Kant explicates in the Critique of Pure Reason. Thus, these notes show that Kant did reflect on the problems posed by the theory of parallel lines and recognized that it was uniquely problematic. Moreover, Kant’s reflections yield new insights about his philosophy of mathematics. In particular, Kant’s attitude toward the theory of parallels demonstrates that he had a novel theory of mathematical definitions, axioms, and postulates – a theory that Kant’s contemporary readers have largely misunderstood.