Besides his teachers and mentors, Pierre Boulez was surrounded by a circle of friends at the turn of the 1950s with whom he shared artistic and political interests and whom he often met in the more personal context of his social life. His interest in contemporary painting and interdisciplinary relations connected him with the painter Bernard Saby who, like Boulez, had pursued mathematical studies. Armand Gatti and Pierre Joffroy (pseudonym of Maurice Weil) were engaged journalists and writers, marked by the terror of the German occupation and the political turmoil of the post-war period. From this circle of friends emerged significant stimulations and influences in the transition from the composerʼs youthful works to the first phase of maturity

Despite the fact that Boulez was criticised by many of his contemporaries insofar as they perceived him as having an excessively mathematical bent, some recent scholars have tended to minimise the significance of mathematical thinking for his compositional approach. This chapter posits that Boulez’s engagement with mathematical thinking cannot be so quickly dismissed. It disentangles the history of ideas and brings a new perspective to Boulez’s relationship with mathematics. After summarising the references to mathematical thinking in the literature on Boulez, it discusses the transformation of the field of mathematics that provided the context for Boulez’s engagement with the discipline and teases out the significance of mathematical thinking in Boulez’s compositional approach. Ultimately, it argues that there is an intimate relationship between the technical and aesthetic basis of his compositional approach and contemporary developments in the field of mathematics.

Chapter 1 presents a brief overview of the book and the basics on inpainting, visual perception and Gestalt laws, together with a presentation of the Fitzwilliam Museum dataset of illuminated manuscripts, selected to represent different types of damage and consequent restoration challenges, which will be used throughout the book.

The emergence of a systematic literature around land-surveying in the late first century AD affords an ideal opportunity to study the development of an ars within the scientific culture of specialized knowledge in the early Roman Empire. The variegated methods that belonged to the historical inheritance of surveying practice challenged the construction of a discrete and coherent disciplinary identity. The surveying writings of Frontinus and Hyginus evince several strategies intended to produce a systematic and explanatory conception of the ars. These include rationalizing explanations of key surveying terminology and practice with a view to natural first principles and an accounting of surveying methods in interdisciplinary perspective with astronomy, natural philosophy, and mathematics. While these earliest surveying works pose several unique challenges, they ultimately provide a precious window onto the challenges and opportunities that greeted the emergence of an ars in the fervid scientific culture of the period.

Chapter 1 will examine the ontological and epistemological questions surrounding music in the knowledge system of the medieval Islamic world by exploring the philosophical system of Ibn Sina and his later followers, all of whose works laid the foundations for scholars of music in the centuries to come. In particular, I will address how mathematics was conceptualized vis-à-vis the cosmology of the falsafa tradition as the discipline that examined the existents whose existence was dependent on physical matter but could be conceptualized without the said matter. Through this conceptualization of music and mathematics, scholars of music were able to broaden their subject matter to cover topics from the melodic modes in vogue in their time to the poetics of music. At the same time, since everything in the universe was connected to one another, music was linked with many other scientific disciplines such as astronomy and medicine.

Chapter 4 considers another major actor in the learning of musical knowledge, besides the patrons: professional scholars. While it is true that musical treatises were for the most part commissioned for the elites, once a text was out in the market, anyone with an interest in the subject and a small amount of money in their pocket could acquire a copy. Professional scholars pursued music as a part of their training in mathematics. I center my discussion around the studies of one such scholar of music at the madrasa of Mustansiriyya, who was a student of al-Urmawi himself. I analyze a rare manuscript that contains marginal notes written by this scholar who studied the subject matter under the master. This rare manuscript grants us a unique perspective into how scholars actually went about learning their subject matter.

Focusing on Menippus’ description of his celestial journey and the great cosmic distances he has travelled, I argue that Icaromenippus is a playful point of reception for mathematical astronomy. Through his acerbic satire, Lucian intervenes in the traditions of cosmology and astronomy to expose how the authority of the most technical of scientific hypotheses can be every bit as precarious as the assertions of philosophy, historiography, or even fiction itself. Provocatively, he draws mathematical astronomy – the work of practitioners such as Archimedes and Aristarchus – into the realm of discourse analysis and pits the authority of science against myth. Icaromenippus therefore warrants a place alongside Plutarch’s On the Face of the Moon and the Aetna poem, other works of the imperial era that explore scientific and mythical explanations in differing ways, and Apuleius’ Apology, which examines the relationship between science and magic. More particularly, Icaromenippus reveals how astronomy could ignite the literary imagination, and how literary works can, in turn, enrich our understanding of scientific thought, inviting us to think about scientific method and communication, the scientific viewpoint, and the role of the body in the domain of perhaps the most incorporeal of the natural sciences, astronomy itself.

We briefly offer the reader a sense of what “logic” is supposed to be: its scope, its goals, and the kind of tools logicians use. We discuss the relationship between logic and the rest of mathematics, outline various conceptions of logic and ways it has been applied, and offer a concrete example of the kind of reasoning one might wish to “formalize” and how this might look.

The economics used by governments is based on ideas from the 1870s, when economists adopted the language of science, but not the method. To make the maths easy to solve, they assumed the economy was simple, predictable, and static. Nobody believes these assumptions are true, but they still shape analysis that informs policy. When the economy is complex, uncertain, and changing, this kind of analysis can lead us to bad decisions.

This chapter is divided into three main sections. The first proposes the Ten Teacher Questions framework. This set of questions is designed to provide you with a generic framework for critical enquiry into all your pedagogical choices, and to connect your pedagogical knowledge to what you have learned in previous chapters. The second section provides the curriculum context structures ‒ that is, the ACARA Cross-curriculum Priorities and General Capabilities, which inform our work. The third section presents Teaching Ideas in Mathematics, The Arts and English. Our key message is not that you must implement every Teaching Idea! Instead, we hope the examples will consolidate a practical approach to harnessing the linguistic diversity of your students. We hope that you will grasp the principles which you can see at work in the Teaching Ideas, and the way that they respond to one or more of the Ten Teacher Questions.

In ‘Early Learning in Plato’s Republic 7’, James Warren provides an analysis of Socrates’ account of the sort of early learning needed to produce philosopher-rulers in Republic 7 (521c–525a), namely a passage describing a very early encounter with questions that provoke thoughts about intelligible objects and stir up concepts in the soul. Warren explains how concepts of number, more specifically the concepts ‘one’, ‘two’, ‘a pair’, and so on, play an essential role in these very early stages of the ascent towards knowledge, and he stresses the continuity between the initial and very basic arithmetical concepts and the concepts involved in more demanding subjects taught in later stages of the educational curriculum. On this account, Socrates is prepared to ascribe to more or less everyone an acquaintance with some, albeit elementary, intelligible objects. This, in turn, can shed some light on broader debates in Platonic epistemology about the extent to which all people – not just those whom Socrates calls philosophers – have some conceptual grasp of intelligibles.

The Introduction provides an overview of the book’s argument about how novels in nineteenth-century Britain (by George Eliot, Wilkie Collins, William Thackeray, and Thomas Hardy) represented modes of thinking, judging, and acting in the face of uncertainty. It also offers a synopsis of key intellectual contexts: (1) the history of probability in logic and mathematics into the Victorian era, the parallel rise of statistics, and the novelistic importance of probability as a dual concept, geared to both the aleatory and the epistemic, to objective frequencies and subjective degrees of belief; (2) the school of thought known as associationism, which was related to mathematical probability and remained influential in the nineteenth century, underwriting the embodied account of mental function and volition in physiological psychology, and representations of deliberation and action in novels; (3) the place of uncertainty in treatises of rhetoric, law, and grammar, where considerations of evidence were inflected by probability’s epistemological transformation; and (4) the resultant shifts in literary probability (and related concepts like mimesis and verisimilitude) from Victorian novel theory to structuralist narratology, where understandings of probability as a dual concept were tacitly incorporated.

Chapter 11 focuses on EC STEM education. It describes what STEM looks like in EC settings and identifies ways in which STEM elements can be incorporated into children’s learning. The chapter describes how STEM-related play can enhance young children’s appreciation of the world and provides a range of examples that have potential for STEM learning. Digital tools and applications for STEM learning are featured in this chapter.

The Tokugawa period saw a transformation in the systematic inquiry into nature. In the seventeenth century scholars were engaging in discrete fields of study, such as astronomy or medicine. But over the course of the next two centuries the fields that initially seemed distant and unrelated gradually converged into one enterprise that we now call “science.” Although Japanese scholars were not isolated from European science, it was not the outside influence that caused this transformation. Rather, the new conceptualization of science came from within, as different scholars came to align themselves along different lines. What brought them together was no longer social status, practical goals, or even their respective disciplines, but the kind of questions they asked, the kind of evidence they considered acceptable, and the sources they deemed authoritative. Together, they now engaged in Science, with a capital S, that was greater than the sum of its parts.