This chapter explores narratives that informed two influential attempts to automate and consolidate mathematics in large computing systems during the second half of the twentieth century – the QED system and the MACSYMA system. These narratives were both political (aligning the automation of mathematics with certain cultural values) and epistemic (each laid out a vision of what mathematics entailed such that it could and should be automated). These narratives united political and epistemic considerations especially with regards to representation: how will mathematical objects and procedures be translated into computer languages and operations and encoded in memory? How much freedom or conformity will be required of those who use and build these systems? MACSYMA and QED represented opposite approaches to these questions: preserving pluralism with a heterogeneous modular design vs requiring that all mathematics be translated into one shared root logic. The narratives explored here shaped, explained and justified the representational choices made in each system and aligned them with specific political and epistemic projects.

It is Zheng Xuan (127–200), more than any other commentator, to whom we owe our current understanding of the Confucian Classics – Zheng Xuan, his subcommentators, Kong Yingda (574–648), and Jia Gongyan (fl. 637), who canonised his readings in the early Tang (618–907). Zheng Xuan made mistakes, however, and this chapter offers a case in point: the calculation of physically impossible target geometries for the royal archery meet reputedly held in the Western Zhou (1045–771 BCE), the ‘Big Shoot’ (dashe). In transforming the ambiguous language of the Classics into an elaborate mathematics problem with definite quantities, Zheng Xuan neglects to account for the targets being in three dimensions, and his subcommentators supply mathematical proofs for numbers that are ‘flat wrong’. It is not as if they could not have known, I argue, as the Big Shoot was living practice in their respective days, and as Zheng and Kong were trained in mathematics. Faced with this curious dissonance between exegesis, experience, and mathematical training, I suggest that we might better understand the work of the exegete as reconstructing abstract, ideal forms which were necessarily divorced from their reality and for which mathematics suggested itself as the perfect tool.

In the early decades of the Tang dynasty (618–907), many scholars were involved in official projects whose aim was to produce authoritative editions of canonical texts. As a result, The Ten Canonical Texts of Mathematics and The Proper Meaning of the Five Canonical Texts were presented to the Tang throne, respectively, in 656 and 653, with, in both cases, selected ancient commentaries and subcommentaries composed at the time. In both contexts, there existed mathematical commentaries, and mathematical subcommentaries were composed on them. The aim of this chapter is to analyze and contrast different scholars’ mathematical commentaries and subcommentaries on both the mathematical and Confucian canons. The authors show that in the framework of both projects, exegetes aimed at highlighting the meaning/intention (yi) of the text commented upon. More importantly, mathematical commentaries and subcommentaries allow the authors to show that far from being a loose concept, yi referred to a quite precise notion of “meaning.” However, interestingly, this notion of “meaning,” and the part of the text commented upon in which exegetes located it, show striking differences, depending on whether exegetes operated in the context of the Confucian canon or in that of the mathematical canon. On the one hand, in the context of the mathematical canon, Liu Hui (fl. 263) and Li Chunfeng (602–670) both focused on looking for the meaning/intention yi of the procedures. In the commentary and subcommentary on The Nine Chapters on Mathematical Procedures, in order to fulfill this task, the exegetes relied on diagrams (tu), blocks (qi), and problems (wen), while, in his subcommentary on Zhen Luan’s (fl. sixth century) commentary on Mathematical Procedures of the Five Canonical Texts, Li Chunfeng added problems and reformulated procedures to reveal the yi of Zhen Luan’s procedures, where he believed Zhen Luan failed to do it. Conversely, in the context of the Confucian canon, although Kong Yingda (574–648) and Jia Gongyan (fl. 650–655) also looked for the yi, for them, the yi on which they focused was located in quantities stated by previous commentators (such as Zheng Xuan, 127–200). Additionally, the two sets of exegetes did not practice mathematics in the same way. For instance, by contrast with exegetes who composed commentaries for the mathematical canon, Jia and Kong did not rely on any computational tool. The authors conclude that, although scholars’ aims were partly the same in the two contexts, their understandings of, as well as their commentarial and mathematical practices in relation to, the yi differed. This chapter also explains why the differences established between the mathematical commentaries and subcommentaries in each of the two contexts could also provide promising tools to distinguish the two layers of texts where they were blurred.

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 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.

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.

Different approaches to past mathematical texts are reviewed. The question addressed is: should we stress the continuity of past mathematics with the mathematics practiced today, or should we emphasize its difference, namely what makes it a product of a distant mathematical culture?

The last half century has seen great advances in our understanding of Kant’s philosophy of mathematics, but we have not yet arrived at a satisfactory interpretation of Kant’s philosophy of arithmetic. In fact, it will be argued that on a key point – whether Kant’s understanding of number is fundamentally cardinal or ordinal – we face a standstill in which competing interpretations appear to have equal support. A new approach is needed in order to overcome this standstill and adjudicate between interpretations. This paper does not argue for a particular interpretation; instead, it outlines what that new approach requires, one that I believe will deepen our understanding of Kant’s philosophy of arithmetic and permit a more nuanced account.