In 2006, an eccentric Russian mathematician named Grigori Perelman solved one of the world's greatest intellectual puzzles. The Poincare conjecture is an extremely complex topological problem that had eluded the best minds for over a century. In 2000, the Clay Institute in Boston named it one of seven great unsolved mathematical problems, and promised a million dollars to anyone who could find a solution. Perelman was awarded the prize this year - and declined the money. Journalist Masha Gessen was determined to find out why. Drawing on interviews with Perelman's teachers, classmates, coaches, teammates, and colleagues in Russia and the US - and informed by her own background as a math whiz raised in Russia - she set out to uncover the nature of Perelman's astonishing abilities. In telling his story, Masha Gessen has constructed a gripping and tragic tale that sheds rare light on the unique burden of genius.

A gripping and tragic tale that sheds rare light on the unique burden of genius In 2006, an eccentric Russian mathematician named Grigori Perelman solved the Poincare Conjecture, an extremely complex topological problem that had eluded the best minds for over a century. A prize of one million dollars was offered to anyone who could unravel it, but Perelman declined the winnings, and in doing so inspired journalist Masha Gessen to tell his story. Drawing on interviews with Perelman’s teachers, classmates, coaches, teammates, and colleagues in Russia and the United States—and informed by her own background as a math whiz raised in Russia—Gessen uncovered a mind of unrivaled computational power, one that enabled Perelman to pursue mathematical concepts to their logical (sometimes distant) end. But she also discovered that this very strength turned out to be Perelman's undoing and the reason for his withdrawal, first from the world of mathematics and then, increasingly, from the world in general.

This Element looks at the contemporary debate on the nature of mathematical rigour and informal proofs as found in mathematical practice. The central argument is for rigour pluralism: that multiple different models of informal proof are good at accounting for different features and functions of the concept of rigour. To illustrate this pluralism, the Element surveys some of the main options in the literature: the 'standard view' that rigour is just formal, logical rigour; the models of proofs as arguments and dialogues; the recipe model of proofs as guiding actions and activities; and the idea of mathematical rigour as an intellectual virtue. The strengths and weaknesses of each are assessed, thereby providing an accessible and empirically-informed introduction to the key issues and ideas found in the current discussion.

More than any other major twentieth-century writer, Pierre Duhem has been the victim of ill-informed guesswork. For instance, many references to Duhem stress the importance of his Catholic faith, but nearly all of them draw the obvious-and entirely erroneous-conclusions about the role of Catholicism in Duhem's thinking. This book pays particular attention to the political and intellectual context of French Catholicism, wracked as it was by the tensions of Dreyfus affair and the so-called modernist crisis. Duhem took his inspiration, not from the papally-sponsored revival of the thought of St. Thomas Aquinas, but from Pascal, a fact that aroused suspicions of skepticism in the minds of conservative Catholics. The tensions between Duhem's work and authoritarian Catholic positions became more explicit as his historical work unfolded. Most famous for his denial of the possibility of a crucial experiment which could unambiguously decide between contending scientific theories, Duhem has often been interpreted as a mere instrumentalist or conventionalist, denying the meaningfulness of a reality behind the theory. Dr. Martin shows that Duhem was a Pascalian who argued for both logic and intuition as indispensable in approaching the truth. Duhem argues that physics could not legitimately be used to attack Christianity, but he held that physics was equally useless for the defense of Christianity, a position which made him unpopular with many Catholics.

Graciela De Pierris presents a novel interpretation of the relationship between skepticism and naturalism in Hume's epistemology, and a new appraisal of Hume's place within early modern thought. Whereas a dominant trend in recent Hume scholarship maintains that there are no skeptical arguments concerning causation and induction in Book I, Part III of the Treatise, Graciela De Pierris presents a detailed reading of the skeptical argument she finds there and how this argument initiates a train of skeptical reasoning that begins in Part III and culminates in Part IV. This reasoning is framed by Hume's version of the modern theory of ideas developed by Descartes and Locke. The skeptical implications of this theory, however, do not arise, as in traditional interpretations of Hume's skepticism, from the 'veil of perception.' They arise from Hume's elaboration of a presentational-phenomenological model of ultimate evidence, according to which there is always a justificatory gap between what is or has been immediately presented to the mind and any ideas that go beyond it. This happens, paradigmatically, in the causal-inductive inference, and, as De Pierris argues, in demonstrative inference as well. Yet, in spite of his firm commitment to radical skepticism, Hume also accepts the naturalistic standpoint of science and common life, and he does so, on the novel interpretation presented here, because of an equally firm commitment to Newtonian science in general and the Newtonian inductive method in particular. Hume defends the Newtonian method (against the mechanical philosophy) while simultaneously rejecting all attempts (including those of the Newtonians) to find a place for the supernatural within our understanding of nature.

This volume presents different conceptions of logic and mathematics and discuss their philosophical foundations and consequences. This concerns first of all topics of Wittgenstein's ideas on logic and mathematics; questions about the structural complexity of propositions; the more recent debate about Neo-Logicism and Neo-Fregeanism; the comparison and translatability of different logics; the foundations of mathematics: intuitionism, mathematical realism, and formalism. The contributing authors are Matthias Baaz, Francesco Berto, Jean-Yves Beziau, Elena Dragalina-Chernya, Günther Eder, Susan Edwards-McKie, Oliver Feldmann, Juliet Floyd, Norbert Gratzl, Richard Heinrich, Janusz Kaczmarek, Wolfgang Kienzler, Timm Lampert, Itala Maria Loffredo D'Ottaviano, Paolo Mancosu, Matthieu Marion, Felix Mühlhölzer, Charles Parsons, Edi Pavlovic, Christoph Pfisterer, Michael Potter, Richard Raatzsch, Esther Ramharter, Stefan Riegelnik, Gabriel Sandu, Georg Schiemer, Gerhard Schurz, Dana Scott, Stewart Shapiro, Karl Sigmund, William W. Tait, Mark van Atten, Maria van der Schaar, Vladimir Vasyukov, Jan von Plato, Jan Woleński and Richard Zach.

Proofs and Refutations is essential reading for all those interested in the methodology, the philosophy and the history of mathematics. Much of the book takes the form of a discussion between a teacher and his students. They propose various solutions to some mathematical problems and investigate the strengths and weaknesses of these solutions. Their discussion (which mirrors certain real developments in the history of mathematics) raises some philosophical problems and some problems about the nature of mathematical discovery or creativity. Imre Lakatos is concerned throughout to combat the classical picture of mathematical development as a steady accumulation of established truths. He shows that mathematics grows instead through a richer, more dramatic process of the successive improvement of creative hypotheses by attempts to 'prove' them and by criticism of these attempts: the logic of proofs and refutations.