A forgotten episode of mathematical resistance reveals the rise of modern mathematics and its cornerstone, mathematical purity, as political phenomena. The nineteenth century opened with a major shift in European mathematics, and in the Kingdom of Naples, this occurred earlier than elsewhere. Between 1790 and 1830 its leading scientific institutions rejected as untrustworthy the “very modern mathematics” of French analysis and in its place consolidated, legitimated, and put to work a different mathematical culture. The Neapolitan mathematical resistance was a complete reorientation of mathematical practice. Over the unrestricted manipulation and application of algebraic algorithms, Neapolitan mathematicians called for a return to Greek-style geometry and the preeminence of pure mathematics. For all their apparent backwardness, Massimo Mazzotti explains, they were arguing for what would become crucial features of modern mathematics: its voluntary restriction through a new kind of rigor and discipline, and the complete disconnection of mathematical truth from the empirical world—in other words, its purity. The Neapolitans, Mazzotti argues, were reacting to the widespread use of mathematical analysis in social and political arguments: theirs was a reactionary mathematics that aimed to technically refute the revolutionary mathematics of the Jacobins. Reactionaries targeted the modern administrative monarchy and its technocratic ambitions, and their mathematical critique questioned the legitimacy of analysis as deployed by expert groups, such as engineers and statisticians. What Mazzotti’s penetrating history shows us in vivid detail is that producing mathematical knowledge was equally about producing certain forms of social, political, and economic order.

Paul Erdős published more papers during his lifetime than any other mathematician, especially in discrete mathematics. He had a nose for beautiful, simply-stated problems with solutions that have far-reaching consequences across mathematics. This captivating book, written for students, provides an easy-to-understand introduction to discrete mathematics by presenting questions that intrigued Erdős, along with his brilliant ways of working toward their answers. It includes young Erdős's proof of Bertrand's postulate, the Erdős-Szekeres Happy End Theorem, De Bruijn-Erdős theorem, Erdős-Rado delta-systems, Erdős-Ko-Rado theorem, Erdős-Stone theorem, the Erdős-Rényi-Sós Friendship Theorem, Erdős-Rényi random graphs, the Chvátal-Erdős theorem on Hamilton cycles, and other results of Erdős, as well as results related to his work, such as Ramsey's theorem or Deza's theorem on weak delta-systems. Its appendix covers topics normally missing from introductory courses. Filled with personal anecdotes about Erdős, this book offers a behind-the-scenes look at interactions with the legendary collaborator.

In Russia at the turn of the twentieth century, mysticism, anti-Semitism, and mathematical theory fused into a distinctive intellectual movement. Through analyses of such seemingly disparate subjects as Moscow mathematical circles and the 1913 novel Petersburg, this book illuminates a forgotten aspect of Russian cultural and intellectual history.

With his groundbreaking and controversial DIM hypothesis, Dr. Leonard Peikoff casts a penetrating new light on the process of human thought, and thereby on Western culture and history. In this far-reaching study, Peikoff identifies the three methods people use to integrate concrete data into a whole, as when connecting diverse experiments by a scientific theory, or separate laws into a Constitution, or single events into a story. The first method, in which data is integrated through rational means, he calls Integration. The second, which employs non-rational means, he calls Misintegration. The third is Disintegration—which is nihilism, the desire to tear things apart. In The DIM Hypothesis Peikoff demonstrates the power of these three methods in shaping the West, by using the categories to examine the culturally representative fields of literature, physics, education, and politics. His analysis illustrates how the historical trends in each field have been dominated by one of these three categories, not only today but during the whole progression of Western culture from its beginning in Ancient Greece. Extrapolating from the historical pattern he identifies, Peikoff concludes by explaining why the lights of the West are going out—and predicts the most likely future for the United States.

Computation is the process of applying a procedure or algorithm to the solution of a mathematical problem. Mathematicians and physicists have been occupied for many decades pondering which problems can be solved by which procedures, and, for those that can be solved, how this can most efficiently be done. In recent years, quantum mechanics has augmented our understanding of the process of computation and of its limitations. Perspectives in Computation covers three broad topics: the computation process and its limitations, the search for computational efficiency, and the role of quantum mechanics in computation. The emphasis is theoretical; Robert Geroch asks what can be done, and what, in principle, are the limitations on what can be done? Geroch guides readers through these topics by combining general discussions of broader issues with precise mathematical formulations—as well as through examples of how computation works. Requiring little technical knowledge of mathematics or physics, Perspectives in Computation will serve both advanced undergraduates and graduate students in mathematics and physics, as well as other scientists working in adjacent fields.