Undergraduate research in mathematics is growing and has become a standard practice in some countries. However, for a novice there is much to learn about mentoring students in mathematics research. In this chapter, we discuss the state of undergraduate research in mathematics and detail a set of best practices for successfully mentoring undergraduate students. Also, we explore some needs and future directions that would help improve undergraduate research in mathematics. Throughout the chapter, we include resources for more information on various topics.

Cavendish displayed a lifelong fascination with one of the hardest of the “hard” problems, the nature of infinity. In an age which saw the birth of calculus as well as revolutionary developments in cosmology, a consistent theory of infinity was generally regarded as an illusory goal. Cavendish tackled this vexing scientific problem, which represented a radical departure from the cosmological and theological consensus of the 1660s; it anticipates a new worldview which emerged toward the end of the century, in which biblical revelation was eventually subordinated to empirical science, the Copernican hypothesis triumphed over rival theories, and the notion of a plurality of worlds became commonplace rather than shocking. From the playful speculations of the 1650s, Cavendish’s confident analysis of the nature of infinity had evolved into an essential ingredient in her prescient “theory of everything.”

This chapter charts the way in which the study of nature was made increasingly less philosophical between 1500 and 1700. At the start of the period, natural philosophy was largely conducted as a form of ‘metaphysical physics’. The erosion of this approach was driven by three factors: 1) the impact of humanist critique; 2) The colonisation of natural philosophy by physicians; 3) The colonisation of natural philosophy by mixed mathematicians. Despite a spirited fightback from the metaphysicians, by the middle of the seventeenth century the anti-metaphysical physicians and mixed-mathematicians – often operating in tandem – had won. A major concomitant of this is that the idea that most of seventeenth-century natural philosophy was grounded in ontological mechanism is wrong. To the extent that natural philosophers were mechanists, they were operational mechanists, who modelled nature on machines but refused to commit to an ontological reductionism, and often directly opposed it. In this and other respect, Descartes and his followers, far from being representative of seventeenth-century natural philosophy, were outliers.

In the years after the publication of the second edition of the Principia, Newton further elaborated his vision of the genealogy of knowledge, and his subsequent conception of the limits of ‘legitimate’ knowledge. Metaphysics now emerged for him as the unifying force that explained all the evils of intellectual life, above all pagan idolatry; the hubristic rationalism in theology that gave birth to odium theologicum and persecution; and the unwarranted search for speculative, causal explanations in natural philosophy. In a set of elaborate writings, ranging from the ecclesiastical-historical ‘Of the Church’ (in which we see the influence of Bayle’s close friend Jacques Basnage) to further polemical writings against various followers of Leibniz and Malebranche, he developed his mature vision of a Kingdom of Darkness at the centre of which lay speculative, metaphysical philosophy. The manuscript ‘Tempus et Locus’ should be dated to this period, rather than to the 1690s. Finally, it is shown that Newton’s earliest followers understood perfectly the broad methodological message which he was trying to advance, and continued to disseminate it aggressively in their writings. The earliest decades of the eighteenth century were devastating for the practice of ‘philosophy’ as it had been conducted for much of Western history.

This paper is detective work. I aim to show that the brilliant Pythagorean mathematician Archytas of Tarentum was the founder of ancient Greek mathematical optics. The evidence is indirect. (1) A fragment of Aristotle preserved in Iamblichus is one of two doxographical notices to mention Pythagorean work in optics. (2) Apuleius credits Archytas with a theory of visual rays which saves the principle that the angle of reflection is equal to the angle of incidence. I argue that the source from which Apuleius got this information was the Catoptrics of Archimedes, the genuineness of which I defend against Knorr’s hypothesis that it is the Euclidean Catoptrics, which had been misattributed to Archimedes. (3) The omission of optics from the mathematical curriculum in Plato’s Republic, and the Timaeus’ wholly physical account of mirror images, can be explained as polemical, for it is well attested that optics was practised in the Academy. The reason Plato does not mention optics is that he objected to Archytas using mathematics to understand the physical world rather than to transcend it.

Why does Plato in the Republic attach such importance to mathematics? Not for its practical utility, nor for the transferable skills acquired by the mathematician, nor because of the rigour of the formal procedures of mathematical proof. It is rather that the five mathematical sciences described and explained in Book VII convert the soul from merely human perspective, and tell us how things are objectively speaking. Their content is what counts. They convey knowledge or understanding of the context-invariant truth of unqualified reality. In contrast with modern conceptions of mathematics and its relation to reality, these sciences are conceived as themselves sciences of value. Above all, they enlarge ethical understanding. Crucial here is harmonics, which incorporates principles first studied through the first four sciences that Plato specifies. Mathematical proportion is what underpins the musical structures – the concords – that form the subject matter of harmonics. Such mathematical structures, when internalised by the philosopher, function as abstract schemata for applying their knowledge of the Good in the social world. Plato values them so highly because they create and sustain unity: unity is for him the highest value.

This chapter introduces Newton’s intellectual biography before the publication of the Principia, and provides a new account of his methods as a natural philosopher. From the 1660s onwards, Newton – in line with his mentor Isaac Barrow and with other mixed mathematicians discussed in I.1 – sought a phenomenological science of properties, actively disdaining conjecture concerning the underlying causes of phenomena. The famous ‘De gravitatione’ manuscript is shown to stem from hydrostatical lectures delivered in 1671; contrary to most of the literature, it contains no elaborate metaphysics of divine omnipresence. Newton’s interest in revealed theology developed when he had to perform disputations in 1677; he did not become an antitrinitarian until the late-1680s, and there is no evidence that his theological views influenced the Principia. For all its mathematical sophistication, that work was very much the product of a methodology not much different from that which mixed mathematicians had been advocating for the previous century. In particular, Newton’s ideas at this time bear a strong conceptual resemblance to those developed by other English mixed mathematicians, such as John Wallis. The very first ‘Newtonians’ recognised the anti-metaphysical thrust of his ideas.

In 1713 Newton finally published the second edition of the Principia. Its changes included not only the new Rules of Philosophising discussed in III.1, but also the very famous General Scholium. This chapter provides the fullest ever contextualisation and interpretation of that text. It charts in detail how Newton’s dispute with Leibniz led him to double-down on his anti-metaphysical stance, and to declare many questions – not least concerning causation – to be beyond the boundaries of legitimate natural philosophy (now described as ‘experimental’). Second, it shows that Newton’s talk of the ‘God of Dominion’ was derived from Samuel Clarke’s recent writings – in line with Clarke’s position, Newton had now moved away from his earlier neo-Arianism into a position of trinitarian nescience in which all speculation on the subject, with consubstantialist or Arian, was dismissed as ‘metaphysical’. The role played by the ‘God of Dominion’ was simply the standard refrain of Newtonian natural theology: that the true conception of the divine that could be predicated from nature was not of an impersonal metaphysical first principle, but of a living God. Finally, Newton’s talk of God’s ‘substantial’ omnipresence was again not a concession to metaphysical thinking, but a residue of yet another polemic devised by his friend Samuel Clarke, this time against the freethinker Anthony Collins.

A brief recap of some mathematical topics.

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.

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 chapter looks at the ways in which the analytic method adopted in Parts I and II, where Kant addresses the possibility of mathematics and natural science, bears on the status of metaphysics. The essay canvasses two possible accounts of how mathematics and science relate to metaphysics as a priori cognition – the ‘Necessary Conditions’ view, and the ‘Examples First’ proposal – and rejects each. Rather, Kant denies that metaphysics can be a science not because it fails to achieve the necessity that we find in mathematics and natural science, but instead because metaphysics does not amount to cognition at all. The analytic method Kant adopts does not lead to a quick rejection of metaphysics as not being something we in fact possess, but requires a subtler and more complex case to show that metaphysics cannot have any cognition of an a priori object, though it still has some other methodological value to offer.

As a result of New Zealand’s colonial history, the indigenous Maori language was excluded from schooling (formally) and from a number of other language domains (informally) for over 100 years. By the late 1970s, Maori was considered an endangered language, heading towards extinction. In response, various grassroots Maori communities initiated Maori-medium education, which required, amongst other linguistic challenges, the rapid development of a corpus of terms to enable the teaching of all subjects in the Maori language. Eventually, Maori-medium schooling became state funded, which was accompanied by a requirement by the state agency which controlled education to standardize the corpus of terms for schooling. In this paper, we explore the challenges associated with the (re)development of te reo Maori in the 1980s and 1990s as an educationally standardized indigenous language in relation to one key curriculum area: pangarau (mathematics). This includes analysing the key role of top-down agents and agencies in the standardization of the pangarau lexicon and register. The chapter also examines the influence of the agents’ linguistic ideologies on subsequent corpus development that still determines the codification of terms thirty years later.

The main aim of Part 2 is to explain how the form of the good gives rise to knowledge of forms, the forms in question being of virtues and virtue-related things. This ramifies into discussions of dialectic and mathematics, the ambiguous property 'clearness' (saphēneia), hypotheses, and the non-hypothetical principle. It is proposed that the form of the good is interrogative. This position is defended against philosophical and textual objections, and argued to be preferable to alternatives. There is discussion of why Plato excludes the use of diagrams from dialectic and whether he can allow input from experience. The role of context in the rulers' dialectic is explained, and becomes the basis for explaining why Plato's treatment of dialectic in the Republic remains at the level of a sketch. There is an exploration of the difference between true philosophers and sight-lovers, and of the criteria and scope of 'good' in dialectic. This last discussion encounters the classic problem of the connection between Plato's 'justice in the soul' and just conduct as ordinarily recognized, and a solution to this problem is proposed.

Stoppard's theatre regularly engages with science, both in works that specifically deal with issues like chaos theory, quantum physics, and consciousness, and in those which more generally raise questions about the limits of human abilities to understand and control the natural world.

The role played by mathematics in Stoppard’s theatre is both formal and epistemological. It is a tool for visualisation, providing shape and structure for the quandaries faced by his characters; it also serves as a model for ways of thinking that can integrate contradiction, instability, and unpredictability into a frame of knowledge.