Having developed the necessary mathematics in chapters 4 to 6, chapter 7 returns to physics Evidence for homogeneity and isotropy of the Universe at the largest cosmological scales is presented and Robertson-Walker metrics are introduced. Einstein’s equations are then used to derive the Friedmann equations, relating the cosmic scale factor to the pressure and density of matter in the Universe. The Hubble constant is discussed and an analytic form of the red-shift distance relation is derived, in terms of the matter density, the cosmological constant and the spatial curvature, and observational values of these three parameters are given. Some analytic solutions of the Friedmann equation are presented. The cosmic microwave background dominates the energy density in the early Universe and this leads to a description of the thermal history of the early Universe: the transition from matter dominated to radiation dominated dynamics and nucleosynthesis in the first 3 minutes. Finally the horizon problem and the inflationary Universe are described and the limits of applicability of Einstein's equations, when they might be expected to break down due to quantum effects, are discussed.

We give a broad-brush overview of cosmology, including a timeline of events starting from the Big Bang until the present day. We introduce the three pillars of the Big Bang cosmological model, the concepts of homogeneity and isotropy, as well as parsec as a unit of distance. We also introduce natural units, and develop intuition on how to adopt and use them.

Monolingualism, bilingualism, and multilingualism represent concepts of individual upbringing and social organization of extreme impact and scope. All in all, the book attempted to guide the reader from a multilingualism-as-problem to a multilingualism-as-resource perspective. However, it also argued that multilingualism cannot work wonders and should not be considered a goal in itself. Running a multilingual society can produce many beneficial effects, but maintaining several languages at the same time also incurs costs that a society must be prepared to burden and share. It is crucial to know which boundary conditions tip the balance from burden to benefit, or vice versa. The book further argues for a continuum from monolingualism to multilingualism based on the dimensions of homogeneity and heterogeneity. It further introduces a novel typology of English in multilingual contexts, distinguishing between English in heritage contexts, English in bilingual heritage contexts, English in contexts of balanced bilingualism, English in indigenous multilingual contexts, English in postcolonial multilingual contexts, and English as a lingua franca in modern multilingual immigrant contexts.

Critics of commercial country music say that the music is homogenous, cliché, and that the so-called bro-country subgenre has taken over. This chapter uses interviews with hit songwriters in Nashville to examine the social and structural factors that influence the way songwriters practice their craft. One such factor, the “360 deal,” is a type of recording contract introduced as a way for record labels to recoup some of the revenue lost with the decline of recorded music sales. Though these contracts are legal agreements between artists and their labels, they have entirely restructured the careers of professional songwriters and the music that they create. This analysis of country music in the twenty-first century is based on a deep understanding of the occupational arrangements that underlie the creation of songs to argue for understanding the structures that shape the songwriting community as critical to the formation of country songs.

GC II 7 supports Aristotle’s elemental theory, according to which the four elements possess a common matter that enables their inter-transformation, over the superficially similar Empedoclean one, by arguing that the former theory, and it alone, can accommodate the formation of homogenous stuffs like flesh and bone from the four elements. According to the interpretation offered here, these stuffs are mixtures in the sense spelled out in GC II 10, and appear to be exhibit the kind of strong uniformity that some interpreters have denied to Aristotelian mixtures. Special attention is devoted to bringing out the significance of elemental mixture for Aristotle’s twin projects in GC: understanding the causes of generation and destruction and establishing a theory of the elements. Explaining the formation of elemental mixtures is a crucial step in showing how the generation of more complex substances is possible and how the four elements, as he conceives them, function as elements of more complex substances.

During the seventeenth century, the advent of what were known as the “common” and “new” analyses fundamentally changed the landscape of European mathematics. The widely accepted narrative is that these analyses, analytic geometry and calculus (mostly due to Descartes and Leibniz, respectively), occasioned a transition from geometrical to symbolic methods. In dealing with the science of motion, mathematicians abandoned the language of proportion theory, as found in the works of Galileo, Huygens, and Newton, and began employing the Newtonian and Leibnizian calculi when differential and fluxional equations first appeared in the 1690s. This was the advent of a more abstract way of practicing mathematics, which culminated with the algebraic approach to calculus and mechanics promoted by Euler and Lagrange in the eighteenth century. In this chapter, it is shown that geometrical interpretations and mechanical constructions still played a crucial role in the methods of Descartes, Leibniz, and their immediate followers. This is revealed by the manner in which they handled equations and how they sought their solutions. The passage from proportions to equations did not occur in a single step; it was a process that took a century to reach completion.

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.

The plebiscitarian leader’s legitimacy is based on the tenets of populism: the people versus an elite opposition, and the promise that this time the government will follow the will of the authentic people. The primacy of popular will turns the regime against the self-constraining institutions of democracy. Populism as a movement and ideology is based on exclusion. Its anti-institutionalism and rejection of political mediation is contrary to democracy as rational decision-making. This rough democracy is deprived of its constitutional protection against the arbitrariness of the genuine will of genuine people. With this point of departure, the natural choice of the leader is to turn plebiscitarian. The leader claims that there is a direct relation between government and people. “The people,” as a concept used by populists is highly problematic for constitutional culture, even if the collective action of citizens (in specific circumstances) forms a bulwark of the constitutional system and the people’s properly institutionalized action plays an indispensable role in the system of modern checks and balances.

Chapter 4 collects and surveys arguments denying that will has any quantificational force.

Cosmology is the part of science concerned with the structure and evolution of the universe on the largest scales of space and time. Gravity governs the structure of the universe on these scales and determines its evolution. General relativity is thus central to cosmology, and cosmology is one of the most important applications of general relativity. Our understanding of the universe on the largest scales of space and time has increased dramatically in recent years – both observationally and theoretically. This book concentrates on the role of relativistic gravity in cosmology, introducing only the most basic observational facts and working out the simplest theoretical models. This chapter sketches the three basic observational facts about our universe on the largest distance scales: the universe consists of stars and gas in gravitationally bound collections of matter called galaxies, diffuse radiation, dark matter of unknown character, and vacuum energy; the universe is expanding; averaged over large distance scales, the universe is isotropic and homogeneous.

Chapter 4 scrutinises the notion of unity as an essential characteristic of the EU legal order and the internal market, and the aims of the multilateral agreements to achieve ‘homogeneity’ in the expanded internal market. The Chapter provides an examination of the notions of ‘unity’ and ‘homogeneity’ and the nature of the homogeneity provision in the acquis-exporting agreements. The Chapter seeks to establish the level of legislative commonality – as opposed to flexibility and differentiation – necessary in the extended internal market in order to be able to consider the third country market participants equal to their EU counterparts.