§1. The origins of recursion theory. In dedicating a book to Steve Kleene, I referred to him as the person who made recursion theory into a theory. Recursion theory was begun by Kleene's teacher at Princeton, Alonzo Church, who first defined the class of recursive functions; first maintained that this class was the class of computable functions ; and first used this fact to solve negatively some classical problems on the existence of algorithms. However, it was Kleene who, in his (...) thesis and in his subsequent attempts to convince himself of Church's Thesis, developed a general theory of the behavior of the recursive functions. He continued to develop this theory and extend it to new situations throughout his mathematical career. Indeed, all of the research which he did had a close relationship to recursive functions.Church's Thesis arose in an accidental way. In his investigations of a system of logic which he had invented, Church became interested in a class of functions which he called the λ-definable functions. Initially, Church knew that the successor function and the addition function were λ-definable, but not much else. During 1932, Kleene gradually showed1 that this class of functions was quite extensive; and these results became an important part of his thesis 1935a. (shrink)

Cet article étudie l’édition des œuvres de mathématiciens au xixe siècle Je me concentre sur une étude de cas : l’édition des œuvres du mathématicien allemand B. Riemann, par R. Dedekind et H. Weber, publiées pour la première fois en 1876, puis republiées en 1892 et en 1902, par Teubner, et partiellement traduites en français en 1898 chez Gauthier-Villars. Pour l’édition des textes de mathématiciens au xixe siècle, les éditeurs ne sont plus historiens ou philologues, mais eux-mêmes des mathématiciens de (...) premier plan. Le mathématicien éditeur devient celui qui à la fois lit et fabrique le texte à publier. Le cas des œuvres de Riemann est particulièrement intéressant, car une large majorité du travail éditorial s’est effectué par lettres. Ces lettres, ainsi que les Nachlässe de Riemann et Dedekind fournissent une documentation exceptionnellement riche. Il est possible d’obtenir une vision détaillée du processus d’édition. Après avoir replacé l’édition des œuvres de Riemann et ses rééditions dans leur contexte, j’étudierai certains choix faits par les éditeurs dans la sélection des textes à publier. Enfin, je considérerai la question des modifications et adaptations des textes de Riemann par les éditeurs. (shrink)

Kurt Gdel was the most outstanding logician of the 20th century and a giant in the field. This book is part of a five volume set that makes available all of Gdel's writings. The first three volumes, already published, consist of the papers and essays of Gdel. The final two volumes of the set deal with Gdel's correspondence with his contemporary mathematicians, this fourth volume consists of material from correspondents from A-G.

In his book on The Mathematics of Great Amateurs Coolidge starts the chapter on Bolzano saying that he included Bolzano because it seemed interesting to him ‘that a man who was a remarkable pulpit orator, only removed from his chair for his political opinions, should have thought so far into the deepest problems of a science which he never taught in a professional capacity’ [Coolidge, 1990, p. 195]. In fact, considering Bolzano's poor health and his enormous productivity in his ‘professional (...) areas’, the results, the breadth and the extent of this ‘great amateur's’ thoughts on mathematics, and its proper scientific presentation are simply astonishing. Even this new selection of more than 650 printed pages of text, well edited and translated by Steve Russ, represents only a part of Bolzano's legacy on mathematical topics: the complete German edition of Bolzano's writings, the Bernard Bolzano-Gesamtausgabe edited by J. Berg, E. Morscher, B. van Rootselaar, and others is projected to include over 120 volumes of which more than 36 volumes will contain his mathematical works. About 25 volumes have been published so far in this mathematical subseries including Bolzano's extensive Miscellanea Mathematica .To most readers of this journal, Bolzano will be well-known for his contributions to logic—his ‘logic of variation’ is viewed as a precursor of the modern semantic conception of logical consequence—and his contributions to the foundations of calculus, especially his ‘purely analytic proof’ of the intermediate-value theorem and the Bolzano-Weierstrass theorem, or his early discovery of a counterintuitive function which is continuous in its domain, but nowhere-differentiable.Despite his lifelong interest in the foundations of mathematics, Bolzano studied theology and taught philosophy of religion in his native city, Prague, from 1805 until his dismissal in 1820. Part of Bolzano's duties …. (shrink)

Hermann Grassmann's Ausdehnungslehre of 1844 and his Lehrbuch der Arithmetik of 1861 are landmark works in mathematics; the former not only developed new mathematical fields but also both contributed to the setting of modern standards of rigor. Their very modernity, however, may obscure features of Grassmann's view of the foundations of mathematics that were not adopted since. Grassmann gave a key role to the learning of mathematics that affected his method of presentation, including his emphasis on making initial assumptions (...) explicit. In order to better understand this less well-known aspect of his work it will help to examine why some commentators have overlooked his theme of unifying logic, pedagogy and foundations, while others have recognised it. (shrink)

ArgumentThe purpose of this paper is to provide an analysis of Giordano Bruno’s conception of mathematics. Specifically, it intends to highlight two aspects of this conception that have been neglected in previous studies. First, Bruno’s conception of mathematics changed over time and in parallel with another concept that was central to his thought: the concept of infinity. Specifically, Bruno undertook a reform of mathematics in order to accommodate the concept of the infinitely small or “minimum,” which was introduced at a (...) later stage. Second, contrary to what Héléne Védrine claimed, Bruno believed that mathematical objects were mind-dependent. To chart the parallel development of the conceptions of mathematics and infinity, a seven-year time span is considered, from the publication of Bruno’s first Italian dialogue (La cena de le ceneri,1584) to the publication of one of his last Latin works (De minimo, 1591). (shrink)

Foundational work in mathematics by some of the other participants in the symposium helps towards answering the question whether a heterodox mathematics could in principle be used as successfully as is orthodox mathematics in scientific applications. This question is turn, it will be argued, is relevant to the question how far current science is the way it is because the world is the way it is, and how far because we are the way we are, which is a central (...) question, if not the central question, of philosophy of science. (shrink)

The main claim of this talk is that mathematical physics and philosophy of physics are not different. This claim, so formulated, is obviously false because it is overstated; however, since no non-tautological statement is likely to be completely true, it is a meaningful question whether the overstated claim expresses some truth. I hope it does, or so I’ll argue. The argument consists of two parts: First I’ll recall some characteristic features of von Neumann’s work on mathematical foundations (...) of quantum mechanics and will claim that von Neumann’s motivation and results are essentially philosophical in their nature; hence, to the extent von Neumann’s work exemplifies what is considered to be mathematical physics, mathematical physics appears as formally explicit philosophy of physics. The second argument is based on a rather trivial interpretation of what mathematical physics is. That interpretation implies that mathematical physics shares some key characteristic features with philosophy of physics which make the two almost indistinguishable. (shrink)

Review by: Achim Blumensath The Bulletin of Symbolic Logic, Volume 19, Issue 3, Page 394-396, September 2013.

We examine the contrast between the “Newtonian style” and the Cartesian, hypothetico-deductive method in order to expand on George Smith's account of working hypotheses and theory-mediation. We stress the pivotal role of theory-mediation in turning experience into well-defined phenomena and introduce the complementary notions of conditional and independent evidence. Conditional evidence is evidence in favor of an hypothesis that is based on phenomena that would not be constituted as phenomena without the hypothesis in question. Direct evidence is evidence based on (...) phenomena that are independently available. We use the distinction to interrogate the role of mathematical representation as the most basic constitutive posit of the Newtonian style, as the working hypothesis that enables all further theory-mediation. Our primary examples are Newton's struggles with motion in fluids and his defense of the Laws of Motion. (shrink)

Many studies have indicated that mathematics anxiety, and other negative attitudes and emotions toward mathematics, are pervasive and are associated with lower mathematical performance. Some previous research has suggested that working memory is related to both mathematics anxiety and mathematics. Moreover, both gender and chosen course of study appeared likely to influence students’ attitudes to mathematics. In the present study, 40 university undergraduates completed a battery of assessments investigating working memory, attitude to mathematics, test anxiety. and mental and written (...) arithmetic. Attitudes to mathematics were significantly associated with the other variables: working memory, test anxiety, and both measures of mathematical performance. The other variables were not strongly associated with one another. There were no gender differences in mathematical performance, but females exhibited more negative attitudes to mathematics and higher test anxiety than males. After controlling for test anxiety, there ceased to be significant gender differences in attitudes to mathematics. Science students had more positive attitudes to mathematics than humanities students, but the groups did not differ in test anxiety, Science students were better at written but not mental arithmetic. They were also better at working memory, but this was not a significant covariate when the groups were compared on mathematical performance and attitudes to mathematics The results are discussed, with particular focus on implications for future research on influences on mathematics anxiety. (shrink)

Review by: Richard Pettigrew The Bulletin of Symbolic Logic, Volume 19, Issue 3, Page 396-397, September 2013.

The main claim of this talk is that mathematical physics and philosophy of physics are not different. This claim, so formulated, is obviously false because it is overstated; however, since no non-tautological statement is likely to be completely true, it is a meaningful question whether the overstated claim expresses some truth. I hope it does, or so I’ll argue. The argument consists of two parts: First I’ll recall some characteristic features of von Neumann’s work on mathematical foundations (...) of quantum mechanics and will claim that von Neumann’s motivation and results are essentially philosophical in their nature; hence, to the extent von Neumann’s work exemplifies what is considered to be mathematical physics, mathematical physics appears as formally explicit philosophy of physics. The second argument is based on a rather trivial interpretation of what mathematical physics is. That interpretation implies that mathematical physics shares some key characteristic features with philosophy of physics which make the two almost indistinguishable. (shrink)

We give an overview of Lakatos’ life, his philosophy of mathematics and science, as well as of this issue. Firstly, we briefly delineate Lakatos’ key contributions to philosophy: his anti-formalist philosophy of mathematics, and his methodology of scientific research programmes in the philosophy of science. Secondly, we outline the themes and structure of the masterclass Lakatos’ Undone Work – The Practical Turn and the Division of Philosophy of Mathematics and Philosophy of Science, which gave rise to this special issue. (...) Lastly, we provide a summary of the contributions to this issue. (shrink)

The main claim of this talk is that mathematical physics and philosophy of physics are not different. This claim, so formulated, is obviously false because it is overstated; however, since no non-tautological statement is likely to be completely true, it is a meaningful question whether the overstated claim expresses some truth. I hope it does, or so I’ll argue. The argument consists of two parts: First I’ll recall some characteristic features of von Neumann’s work on mathematical foundations (...) of quantum mechanics and will claim that von Neumann’s motivation and results are essentially philosophical in their nature; hence, to the extent von Neumann’s work exemplifies what is considered to be mathematical physics, mathematical physics appears as formally explicit philosophy of physics. The second argument is based on a rather trivial interpretation of what mathematical physics is. That interpretation implies that mathematical physics shares some key characteristic features with philosophy of physics which make the two almost indistinguishable. (shrink)

ArgumentRecent studies in Sinology have shown that Qing dynasty editors acted as philologists. This paper argues that the identification of their philological methods and editorial choices suggests that their choices were not totally neutral and may have significantly shaped the way modern historians interpreted specific works edited by mathematicians of that dynasty. A case study of the re-edition in 1798 of a Song dynasty treatise, theYigu yanduan(1259), by a Qing dynasty mathematician will illustrate this point. At the end of the (...) eighteenth century, Li Rui (1773–1817) was asked to prepare an edition of the mathematical works written by Li Ye (1192–1279) for a private collection. Li Rui was a talented mathematician, but he was also a meticulous editor and trained philologist. He adopted his editorial model from the preparation of the imperial encyclopaedia, theSiku quanshu, but Li Rui also made some corrections to the text in an effort to restore an older version of Li Ye's treatises that had been lost. Convinced of the Chinese origin of algebra, Li Rui used philological techniques to recover the lost materials and to restore the roots of “Chinese mathematics.” TheYigu yanduancontains two algebraic procedures to set up quadratic equations, one from the procedure ofCelestial Source(tian yuan shu) and the other from theSection of Pieces[of Areas] (tiao duan). Curiously, the second procedure has not yet attracted the attention of scholars so far, although Li Rui's edition is the one typically used by twentieth-century historians of mathematics. Today, theCelestial Sourcecharacterizes “Chinese algebra.” However, the specific concerns of Li Rui about the procedure ofCelestial Source, combined with his editorial methods, contributed to this perspective. (shrink)

A common analysis of Edmund Husserl’s early works on the philosophy of logic and mathematics presents these writings as the result of a combination of two distinct strands of influence: on the one hand a mathematical influence due to his teachers is Berlin, such as Karl Weierstrass, and on the other hand a philosophical influence due to his later studies in Vienna with Franz Brentano. However, the formative influences on Husserl’s early philosophy cannot be so cleanly separated into a (...) philosophical and a mathematical pathway. Growing evidence indicates that a Brentanist philosophy of mathematics was already in place before Husserl. Rather than an original combination at the confluence of two different streams, his early writings represent an elaboration of topics and problems that were already being discussed in the School of Brentano within a pre-existing framework. The traditional account understandably neglects Brentano’s own work on the philosophy of mathematics and logic, which can be found mostly in his unpublished manuscripts and lectures, and various works by Brentano’s students on the philosophy of mathematics which have only recently emerged from obscurity. Husserl’s early works must be correctly placed in this preceding context in order to be fully understood and correctly assessed. (shrink)

This book provides a reading of Kant's theory of the construction of mathematical concepts through a fully contextualised analysis. In this work the author argues that it is only through an understanding of the relevant eighteenth century mathematics textbooks, and the related mathematical practice, that the material and context necessary for a successful interpretation of Kant's philosophy can be provided.

This paper is concerned with the status of mathematical fictions in Leibniz’s work and especially with infinitary quantities as fictions. Thus, it is maintained that mathematical fictions constitute a kind of symbolic notion that implies various degrees of impossibility. With this framework, different kinds of notions of possibility and impossibility are proposed, reviewing the usual interpretation of both modal concepts, which appeals to the consistency property. Thus, three concepts of the possibility/impossibility pair are distinguished; they give rise, (...) in turn, to three concepts of mathematical fictions. Moreover, such a distinction is the base for the claim that infinitesimal quantities, as mathematical fictions, do not imply an absolute impossibility, resulting from self-contradiction, but a relative impossibility, founded on irrepresentability and on the fact that it does not conform to architectural principles. In conclusion, this “soft” impossibility of infinitesimals yields them, in Leibniz view, a presumptive or “conjectural” status. (shrink)

Mathematical argument in the elementary grades : what and why? -- Elementary students as mathematicians -- The teaching model -- Using the lesson sequences : what the teacher does -- Mathematical argument in the elementary classroom : impact on students and teachers.

Traces the development of British philosopher Russell's (1872-1970) ideas on mathematics from the 1890s to the publication of his Principles of mathematics in 1903. Draws from Russell's unpublished manuscripts, correspondence, and published works to point out the influence of Hegel, Cantor, Whitehead, Peano, and others. No index. Annotation copyrighted by Book News, Inc., Portland, OR.

It is well known that Felix Klein took a decisive step in investigating the invariants of transformation groups. However, less attention has been given to Klein’s considerations on the epistemological implications of his work on geometry. This paper proposes an interpretation of Klein’s view as a form of mathematical structuralism, according to which the study of mathematical structures provides the basis for a better understanding of how mathematical research and practice develop.