This chapter examines a literary critical ‘methodological moment’ from the middle of the nineteenth century to modernism. It argues that the re-emergence of the scientific method in this period was key to the normal scientific study of poetry. By returning to a series of forgotten critical debates about the relevance of the scientific method to the study of poetry, the chapter demonstrates how the nineteenth-century revival of method introduced a technical vocabulary into twentieth-century poetics, an epistemologically and politically charged discourse that centred on concepts of method, hypothesis and scientific law. The second half of this chapter goes on to examine published and unpublished poetry by George Oppen to show how he offered a new way of conceptualising the relationship between poetry and the scientific method. It suggests that Oppen turned to mathematics and set theory to create a new nominalist method that could create rather than explain. However, it is also argued that Oppen’s employment of the mathematical method actually ends up illustrating the epistemological power of poetic artifice: its ability to create the sights and sounds of the invisible but not inexistent multitude that Oppen’s poetry sought to bring into being.

This chapter explores different strands of the theory of two-player zero-sum games and equilibrium concepts for general multiplayer games. The conventional viewpoint is that equilibrium is an extension of the concept of value (and its associated optimal strategies) to non-zero-sum games, and the value is just a special case of an equilibrium payoff. However, it is argued that a number of important concepts apply only to one of these concepts.

The equilibrium notion of Nash has been the primary tool for predicting strategies and outcomes of games with rational players. But the Nash equilibrium is a weak criterion for games with dynamic interactions and/or private information among the players. Stronger criteria called equilibrium refinements are intended to remedy deficiencies that stem from these features. This chapter summarizes motives for refinements, the main refinements themselves, and reports progress on characterizing the strongest refinement, called stability, via axioms that express basic properties of rational behavior.

We briefly survey the state of the art for mean field games without entering any technical/mathematical details. We review both the existing mathematical results and the modeling toolbox. We also mention a few applications. After describing a new numerical approach, we conclude with a few perspectives.

This chapter introduces the three contributions that constitute Part II, “Mathematics of Game Theory and Its Foundations.” Those concern (1) mean field games, (2) value and equilibrium in zero-sum games, and (3) refinements of Nash equilibrium.

A reading of the best-known experimental work of 1976, Einstein on the Beach, that traces the sources of its imagery in mass media, popular culture, and art history, and that studies how the kinetics and contingency of live performance complicate the classical decorum associated with Robert Wilson’s theater. The chapter also discusses the performance styles of Lucinda Childs and Sheryl Sutton, the relationship of the opera to mathematics, the value of error and the handmade, and the persistence of emotion despite the production’s apparent coolness.

This chapter lays out the ways in which Hans Christian Ørsted (1777–1851) influenced the development of the concept of thought experiment. Ernst Mach (1838–1916) is currently more often credited with laying the foundations of contemporary views, and he is sometimes thought to have been little (if at all) influenced by Ørsted. Against these standard accounts, I will show that Ørsted’s and Mach’s descriptions have key features in common. Both thinkers hold that thought experiments: (1) are a method of variation, (2) require the experimenter’s free activity, and (3) are useful in educational contexts for guiding students to arrive at certain conclusions on their own (i.e., to genuinely appropriate new concepts). The process of variation is guided by the search for invariants, some of which do not directly appear in experience. Since it is important that teachers and students be able to bring the same ideal objects to mind, thought experiments play a key role for both Ørsted and Mach in math education. While Ørsted’s emphasis on the role of thought experiments in math has been proposed as a reason why his descriptions are not relevant for contemporary use of thought experiments, I will show how their role in mathematical thinking – stemming from Kant’s descriptions of the method of construction in geometry – are part of a wider account of thought experiments that encompasses their role in the sciences and also philosophy.

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.