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This chapter focuses on the ancient Greek tradition of geometrical proof in light of recent studies by Kenneth Manders and others. It advances the view that the borderline of elementary mathematics is strictly linked with the adoption of hypotheses. To this end, the chapter considers Euclidean geometry, which elaborates on both the problems and the proof methods based on diagrams. It argues that Euclidean geometry can be understood as a theoretical, idealized analysis (and further development) of practical geometry; that by way of the idealizations introduced, Euclid's Elements builds on hypotheses that turn them into advanced mathematics; and that the axioms or “postulates” of Book I of the Elements mainly regiment diagrammatic constructions, while the “common notions” are general principles of a theory of quantities. The chapter concludes by discussing how the proposed approach, based on joint consideration of agents and frameworks, can be applied to the case of Greek geometry.

This chapter proposes the idea that advanced mathematics is based on hypotheses—that far from being a priori, it is based on hypothetical assumptions. The concept of quasi-empiricism is often linked with the view that inductive methods are at play when the hypotheses are established. The presence of hypotheses at the very heart of mathematics establishes an important similitude with physical theory and undermines the simple distinction between “formal” and “empirical” sciences. The chapter first elaborates on a hypothetical conception of mathematics before discussing the ideas (and ideals) of certainty and objectivity in mathematics. It then considers the modern problems of the continuum that exist in ancient Greek geometry, along with the so-called methodological platonism of modern mathematics and its focus on mathematical objects. Finally, it describes the Axiom of Completeness and the Riemann Hypothesis.

This book proposes a novel analysis of mathematical knowledge from a practice-oriented standpoint and within the context of the philosophy of mathematics. The approach it is advocating is a cognitive, pragmatist, historical one. It emphasizes a view of mathematics as knowledge produced by human agents, on the basis of their biological and cognitive abilities, the latter being mediated by culture. It also gives importance to the practical roots of mathematics—that is, its roots in everyday practices, technical practices, mathematical practices themselves, and scientific practices. Finally, the approach stresses the importance of analyzing mathematics' historical development, and of accepting the presence of hypothetical elements in advanced mathematics. The book's main thesis is that several different levels of knowledge and practice are coexistent, and that their links and interplay are crucial to mathematical knowledge. This chapter offers some remarks that may help readers locate the book's arguments within a general scheme.

This chapter considers two fundamental questions for detecting and correcting errors made while playing the card game SET®. The game is played with a special deck of eighty-one cards, and the objective is to find three cards that form a set. Over the course of a game, a player may make a mistake by taking three cards that do not form a set—a common occurrence which this chapter examines by first introducing coordinates for the cards and then uses these coordinates to define a Hamming weight for any subset of cards. The chapter then uses the facts about Hamming weight to describe a variant of the game, called the EndGame, which leads to error detection. Afterward, the chapter produces a perfect, single-error-correcting linear code solely from SET® cards. It concludes with additional topics that demonstrate the deep connections between the simple card game and advanced mathematics.

This chapter looks at how the application of error theory and of probability models to social statistics was pursued with growing success in Germany during the last third of the nineteenth century. The most successful and influential of those mathematical writers on statistics was the economist and statistician Wilhelm Lexis. The chapter then studies the index of dispersion that Lexis introduced in 1879. Lexis's writings on dispersion and the distribution of human physical attributes were influential within German anthropometry, which began to make interesting use of the analytical techniques associated with the error law during the last quarter of the nineteenth century. Meanwhile, Francis Edgeworth, the poet of statisticians, was led to probability in the context of his campaign to introduce advanced mathematics into the moral and social sciences. He hoped through analogies to bring the same rigor and elegance to economics and ethics.

This concluding chapter addresses how statistics has assumed the trappings of a modern academic discipline primarily during the last half century. The intellectual character of statistics had been thoroughly transformed by 1900. The period when statistical thinking was allied only to the simplest mathematics gave way to a period of statistical mathematics—which, to be sure, has not been divorced from thinking. In the twentieth century, statistics has at last assumed at least the appearance of conforming to that hierarchical structure of knowledge beloved by philosophers and sociologists in which theory governs practice and in which the “advanced” field of mathematics provides a solid foundation for the “less mature” biological and social sciences. The crystallization of a mathematical statistics out of the wealth of applications developed during the nineteenth century provides the natural culmination to this story.