Leo Corry

Nicolas Bourbaki and the concept of mathematical structure

In the present article two possible meanings of the term “mathematical structure” are discussed: a formal and a nonformal one. It is claimed that contemporary mathematics is structural only in the nonformal sense of the term. Bourbakis definition of structure is presented as one among several attempts to elucidate the meaning of that nonformal idea by developing a formal theory which allegedly accounts for it. It is shown that Bourbakis concept of structure was, from a mathematical point of view, a superfluous undertaking. This is done by analyzing the role played by the concept, in the first place, within Bourbakis own mathematical output. Likewise, the interaction between Bourbakis work and the first stages of category theory is analyzed, on the basis of both published texts and personal documents.

The influence of David Hilbert and Hermann Minkowski on Einstein's views over the interrelation between physics and mathematics

In the early years of his scientific career, Albert Einstein considered mathematics to be a mere tool in the service of physical intuition. In later years, he came to consider mathematics as the very source of scientific creativity. A main motive behind this change was the influence of two prominent German mathematicians: David Hilbert and Hermann Minkowski.

The Origins of Eternal Truth in Modern Mathematics: Hilbert to Bourbaki and Beyond

The belief in the existence of eternal mathematical truth has been part of this science throughout history. Bourbaki, however, introduced an interesting, and rather innovative twist to it, beginning in the mid-1930s. This group of mathematicians advanced the view that mathematics is a science dealing with structures, and that it attains its results through a systematic application of the modern axiomatic method. Like many other mathematicians, past and contemporary, Bourbaki understood the historical development of mathematics as a series of necessary stages inexorably leading to its current state—meaning by this, the specific perspective that Bourbaki had adopted and were promoting. But unlike anyone else, Bourbaki actively put forward the view that their conception of mathematics was not only illuminating and useful to deal with the current concerns of mathematics, but in fact, that this was the ultimate stage in the evolution of mathematics, bound to remain unchanged by any future development of this science. In this way, they were extending in an unprecedented way the domain of validity of the belief in the eternal character of mathematical truths, from the body to the images of mathematical knowledge as well.

The Empiricist Roots of Hilbert’s Axiomatic Approach

Hilbert’s work on logic and proof theory—among the latest stages in his long and fruitful scientific career—appeared almost two decades after the publication of the epoch-making Grundlagen der Geometrie. In spite of the time span separating these two phases of his intellectual development, and given the centrality of the axiomatic approach to both, one might tend to consider the two as different manifestations of one and the same underlying conception. In particular, one runs the risk of examining the Grundlagen as an early expression of the ideas developed in Hilbert’s work on proof theory. A close historical examination of these works, however, brings to light important differences between them.

Introduction: The History of Modern Mathematics – Writing and Rewriting

The present issue of Science in Context comprises a collection of articles dealing with various, specific aspects of the history of mathematics during the last third of the nineteenth century and the first half of the twentieth. Like the September issue of 2003 of this journal (Vol. 16, no. 3), which was devoted to the history of ancient mathematics, this collection originated in the aftermath of a meeting held in Tel-Aviv and Jerusalem in May 2001, under the title: “History of Mathematics in the Last 25 Years: New Departures, New Questions, New Ideas.” Taken together, these two topical issues are meant as a token of appreciation for the work of Sabetai Unguru and his achievements in the history of mathematics. In the introduction to our first collection, guest editor Reviel Netz described the task of rewriting the history of early mathematics as “a necessary – but risky – enterprise.” This necessity, according to him, stems from reasons of both historiographical and philosophical import. To state it briefly, the historiographical reasons pertain to the scarcity of substantial evidence that implies the need for much interpretive additions from the historian. This being the case, shifts in the historical perspective may bring about significant changes in the historical accounts produced, thus sensibly enriching our historical understanding. The philosophical reasons pertain to the fact that a contextually sensitive account of the history of mathematics raises important questions about the nature of objectivity and rationality in science. In Netz’s words: “The task before us – Unguru’s historiographical descendants – is to show how a historicized mathematics need refine, but not destroy, our sense of rationality and truth.” Herein lies also, implicitly, the risky element of the enterprise: historicizing rationality has often been perceived as potentially calling into question its very essence. Although historicizing rationality has been undertaken in many ways and from many perspectives, it seems that when this is done in relation with one of the quintessential embodiments of rationality, mathematics, some particularly sensitive chords are sounded, and reactions are often colored with very sharp tones. It may sound strange to the standard reader of a journal like Science in Context that a guest editor feels compelled to provide, somewhat apologetically, a justification for the very need of his professional undertaking. It is true, as Netz stresses in his introduction, that the continued, and currently thriving, activity of historians of ancient mathematics is not based on the discovery and publication of new textual evidence, but rather on the

Zionist Internationalism through Number Theory: Edmund Landau at the Opening of the Hebrew University in 1925

This article gives the background to a public lecture delivered in Hebrew by Edmund Landau at the opening ceremony of the Hebrew University in Jerusalem in 1925. On the surface, the lecture appears to be a slightly awkward attempt by a distinguished German-Jewish mathematician to popularize a few number-theoretical tidbits. However, quite unexpectedly, what emerges here is Landaus personal blend of Zionism, German nationalism, and the proud ethos of pure, rigorous mathematics – against the backdrop of the situation of Germany after World War I. Landaus Jerusalem lecture thus shows how the Zionist cause was inextricably linked to, and determined by political agendas that were taking place in Europe at that time. The lecture stands in various historical contexts - Landaus biography, the history of Jewish scientists in the German Zionist movement, the founding of the Hebrew University in Jerusalem, and the creation of a modern Hebrew mathematical language. This article provides a broad historical introduction to the English translation, with commentary, of the original Hebrew text.

Fermat Meets SWAC: Vandiver, the Lehmers, Computers, and Number Theory

This article describes the work of Harry Schultz Vandiver, Derrick Henry Lehmer, and Emma Lehmer on calculations related with proofs of Fermats last theorem. This story sheds light on ideological and institutional aspects of activity in number theory in the US during the 20th century, and on the incursion of computer-assisted methods into pure fields of mathematical research.

Hilbert’s sixth problem: between the foundations of geometry and the axiomatization of physics

The sixth of Hilbert’s famous 1900 list of 23 problems was a programmatic call for the axiomatization of the physical sciences. It was naturally and organically rooted at the core of Hilbert’s conception of what axiomatization is all about. In fact, the axiomatic method which he applied at the turn of the twentieth century in his famous work on the foundations of geometry originated in a preoccupation with foundational questions related with empirical science in general. Indeed, far from a purely formal conception, Hilbert counted geometry among the sciences with strong empirical content, closely related to other branches of physics and deserving a treatment similar to that reserved for the latter. In this treatment, the axiomatization project was meant to play, in his view, a crucial role. Curiously, and contrary to a once-prevalent view, from all the problems in the list, the sixth is the only one that continually engaged Hilbet’s efforts over a very long period of time, at least between 1894 and 1932. This article is part of the theme issue ‘Hilbert’s sixth problem’.

Turing's pre-war analog computers: the fatherhood of the modern computer revisited

Turings machines of 1936 were a purely mathematical notion, not an exploration of possible blueprints for physical calculators.

Hermann Minkowski, Relativity and the Axiomatic Approach to Physics

This article surveys the general background to Minkowski’s incursion into relativity, of which Einstein’s work represented just one side. Special attention is paid to the idiosyncratic, rich, and complex interaction between mathematics and physics, that stood at the center of attention of the Gttingen mathematicians since the turn of the twentieth century. In particular the article explains Minkowski’s formulation of special relativity in terms of space-time against the background of David Hilbert’s program for the axiomatization of physics. In addition, the article sheds light on the changing attitudes of Einstein towards mathematics, in the wake of Minkowski’s work, and his increasing willingness to attribute significance to mathematical formalism in developing physical theories.