‎BERGAMINI David‎
‎Mathematics.‎

‎Constructive formalism. Essays on the foundations of mathematics‎

‎Leicester, University College, 1951. 130 g In-8 broché, 91 pp.. . (Catégories : Mathématiques, Logique, )‎

‎[Bericht uber die 2. Tagung für Erkenntnislehre der exakten Wissenschaften Königsberg 1930]. Die Logizistische Grundlegung der Mathematik + Die intuitionistische Grundlegung der Mathematik + Die formalistische Grundlegung der Mathematik. [In: Erkenntn... - [ENDING THE FOUNDATIONAL CRISIS OF MATHEMATICS]‎

‎Leipzig, Felix Meiner, 1931. The entire volume present. 8vo. Orig. printed green wrappers. Sunning to spine, and a bit of soiling and minor wear to front wrapper w. minor loss of upper layer of paper at two pages, not gone through paper. A few leaves w. marginal markings, quite discreet. Library marking to inside of front wrapper, library stamp to title-page (Mathematical Institute of the University of Amsterdam). Overall a fine and nice copy. Pp. (91) - 105 + (106) - 115 + (116) - 121. The entire volume: (2) pp., Pp. (91) - 190.‎

‎First edition of the Erkenntnis-volume from the Königsberg congress of 1930, where Gödel introduced his incompleteness results and Carnap, Heyting and von Neumann held the seminal papers (here printed for the first time) that ended the ""Grundlagenkrise der Mathematik"" (foundational crisis of mathematics). It is also in this volume that the seminal discussions following Gödel's announcements of his results are printed for the first time (""Discussion on the Foundation of Mathematics"", between Gödel, von Neumann, Carnap, Hahn, Reidemeister, Heyting, and Scholz) (Gödel, Collected Works, 1931a) as well as the article which inaugurated the logicist foundation of mathematics, in which the modern sense of ""logicism"" is introduced (Carnap's contribution).In Königsberg in September 1930, Gödel presented his incompleteness results, a landmark in mathematical logic, at the second congress of scientific epistemology, -a congress which proved to be a turning point in the history of philosophical and mathematical logic. It is the papers presented at this congress which are printed in the present volume, apart from the contributions by Gödel and Scholtz (which were printed elsewhere) together with the seminal discussions that followed the presentation of the papers. The groundbreaking papers that are printed here include Carnap's ""Die Logizistische Grundlegung der Mathematik"", which furthermore introduced the modern sense of the term ""logicism"", Arend Heyting's ""Die intuitionistische Grundlegung der Mathematik"" and Johann von Neumann's ""Die formalistische Grundlegung der Mathematik"" as well as papers by Neugebauer, Reichenbach and Heisenberg. The present papers, as well as the following discussion, mark a turning point in the history of logic and a cornerstone in the future development of the field. The so-called ""Foundational Crisis of Mathematics"" was a phase within mathematics begun in the early 20th century due to the search for proper foundations of mathematics and the uncertainty of this quest, which was supported by the many difficulties that philosophy of mathematics faced at the end of the 19th and beginning of the 20th century . The crisis took its actual beginning with the publication of Russell's ""principles of Mathematics"" of 1903, culminated in the 1920'ies with the main advocates of Formalism and Intuitionism respectively, Hilbert and Brouwer, in what is called the ""foundational struggle of mathematics"", and ended with the present volume in 1931, following the congress of 1930.With the discovery of non-Euclidean geometry in the 18th century, it became evident that not only one sort of mathematics was possible, and even that some propositions could be true in one mathematical system, but false in another. This was the actual basis for the awareness of a mathematical foundation in the mathematical public, which again was the basis for the fact that the question of the foundation of mathematics could develop -and could develop into an actual crisis. During the first 30 years of the 20th century, almost all great mathematicians worked on their answer to the question of the correct foundation of mathematics, and thus it came to a crisis that developed into a struggle. It is this struggle and crisis that Carnap, Heyting and von Neumann break in 1930, where they present the three great positions of the struggling years: logicism (Carnap), intuitionism (Heyting) and formalism (von Neumann), and it is these three papars that pave the way for the discussion that follows, ""Diskussion zur Grundlegung der Mathematik"", between Gödel, Hahn, Carnap, Heyting, von Neumann, Reidemeister, and Scholz. They all presented their positions in the most conciliatory manner, out of the comprehension that all parties who had contributed to the crisis had also contributed because they wanted to solve it, and because they were also searching for the best possible foundation. It is also this comprehension that Hilbert takes over, when he, in his program, sets out to prove the contradiction-freedom of infinite mathematics on the basis of finite arithmetic.Thus, the seminal papers in the present volume once and for all ended the foundational crisis of mathematics and any fear of new antinomies.‎

‎The Principles of Mathematics. Vol I (all). - [""RUSSELL'S PARADOX""]‎

‎Cambridge, at the University Press, 1903. Royal 8vo. Original blue full cloth binding, all edges uncut. Capitals and upper front hinge with a bit of wear and corners a little bumped. But otherwise a very nice copy. Internally fresh and clean. XXIX, (1), 534 pp.‎

‎The uncommon first edition of Russell's landmark work in mathematical logic, in which theory of logicism is put forth and in which Russell introduces that which is now known as ""Russell's Paradox"". The work constitutes the forerunner of Russell and Whitehead's monumental ""Principia Mathematica"", and it seminally influenced logical thought and theories of the foundations of mathematics at this most crucial time for the development of modern mathematical and philosophical logic.""The present work has two main objects. One of these, the proof that all pure mathematics deals exclusively with concepts definable in terms of a very small number of fundamental logical concepts, and that its propositions are deducible from a very small number of fundamental logical principles, is undertaken in Parts II. - VI. Of this Volume, and will be established by strict symbolic reasoning in Volume II. ... The other object of the work, which occupies Part I., is the explanation of the fundamental concepts which mathematics accepts as indefinable. ..."" (Russell, Preface, p. (III)).At the age of 27, in 1898, Russell began working on the book that became ""The Principles of Mathematics"". He originally set out to investigate the contradiction that is inherent in the nature of number, and he originally imagined doing this from a Hegelian standpoint. However, after having read Whitehead's ""Universal Algebra"", Russell gave up his Hegelian approach and began working on a book that was to be entitled ""An Analysis of Mathematical Reasoning"". This book never appeared, as he gave it up in 1900, but much of it is what lies at the foundation of ""The Principles of Mathematics"". After having attended a congress in Paris in 1899, where Peano was present, Russell began rewriting large parts of the work, now with the aim of proving that all of mathematics could be reduced to a few logical concepts, that that which is called mathematics is in reality nothing but later deductions from logical premises. And thus he had developed his landmark thesis that mathematics and logic are identical"" a thesis that came to have a profound influence on logic and the foundations of mathematics throughout the 20th century.Since the congress, Russell had worked with the greatest of enthusiasm, and he finished the manuscript on the 31st of December 1900. However, in the spring of 1901, he discovered ""The Contradiction"", or as it is now called, ""Russell's Paradox"". Russell had been studying Cantor's proof, and in his own words, the paradox emerged thus: ""Before taking leave of fundamental questions, it is necessary to examine more in detail the singular contradiction, already mentioned, with regard to predicates not predictable of themselves. Before attempting to solve this puzzle, it will be well to make some deductions connected with it, and to state it in various different forms. I may mention that I was led to endeavour to reconcile Cantor's proof that there can be no greatest cardinal number with the very plausible supposition that the class of all termes (which we have seen to be essential to all formal propositions) has necessarily the greatest possible number of members."" (p. 101). The class of all classes that are not members of themselves, is this class a member of itself or not? The question was unanswerable (if it is, then it isn't, and if it isn't, then it is) and thus a paradox, and not just any paradox, this was a paradox of the greatest importance. Since, when using classical logic, all sentences are entailed by contradiction, this discovery naturally sparked a huge number of works within logic, set theory, foundations of mathematics, philosophy of mathematics, etc. Russell's own solution to the problem was his ""theory of types"", also developed in 1903.In December 1902 Russell had come to the point where he could write a preface, and the book finally appeared in May 1903. It was printed in merely 1.000 copies, and although it was well received, it was not a bestseller at its appearance. By 1909 the last copies of the first run were at the bookbinders. However, the book did play an enormous role in the development of mathematical and philosophical logic as well as the foundation of mathematics throughout the 20th century. Wittgentein's immense interest in the philosophy of logic stems from his reading of the present work and from Frege's ""Foundations of Arithmetic"", and no logician could neglect the impact of this seminal work, which still counts as one of the most important philosophical and logical works of the 20th century. The book also played an important part in spreading the works of Cantor and Frege to the English-speaking world. In 1903 the Spectator wrote ""we should say that Mr. Russell has an inherited place in literature or statesmanship waiting for him if he will condescend to come down to the common day."" Shearman's review in Mind hailed it as the most important work since Boole's ""Laws of Thought"". ""Bertrand Arthur William Russell (b.1872 - d.1970) was a British philosopher, logician, essayist, and social critic, best known for his work in mathematical logic and analytic philosophy. His most influential contributions include his defense of logicism (the view that mathematics is in some important sense reducible to logic), and his theories of definite descriptions and logical atomism. Along with G.E. Moore, Russell is generally recognized as one of the founders of analytic philosophy. Along with Kurt Gödel, he is also regularly credited with being one of the two most important logicians of the twentieth century."" (Stanford Encyclopedia of Philosophy).Russell had actually planned to write a second volume of the work, but as the contents of this further development would overlap considerably with the further research that Whitehead had undertaken after his ""Universal Algebra"", which he also planned two write a second volume of, the two great logicians decided to collaborate on that which became the ""Principia Mathematica"", which appeared 1910-13. ‎

‎Twentieth-Century Music and Mathematics‎

‎, Brepols, 2019 Hardback, xxxiii + 382 pages, Size:216 x 280 mm, Illustrations:Musical examples and figures, Languages: English, French, Italian. ISBN 9782503585703.‎

‎Summary Music and mathematics have been connected since ancient times. During the twentieth century, however, many composers consciously started using many mathematical concepts, algebraic operations and theorems as bases for their creative processes. The first part of this volume deals with the relationship between music and mathematics in the music of composers such as Olivier Messiaen, Iannis Xenakis, Franco Evangelisti, Pierre Boulez, Arvo P rt, Steve Reich and Philip Glass. A chapter is then dedicated to Spanish composers of the last forty years (Francisco Guerrero, Alberto Posadas, Jos Mar a S nchez-Verd , Iluminada P rez Frutos, Nur a Gimenez Comas, Helga Arias and Jos L pez Montes). The theme continues in the second part of the book through the examination of prominent theories (Neo-Riemannian theory, diatonic set theory, theory of musical kaleidocycles), the use of diagrams and charts in music, the algorithmic evolution of music, contemporary compositional practices inspired by mathematical concepts; it arrives at studies on double canons and trichords. We publish a philosophical study in order to ask what the future of the relationship between music and mathematics will be at the beginning of the 21st century. Includes an introduction by Massimiliano Locanto. TABLE OF CONTENTS Roberto Illiano Preface Massimiliano Locanto Music Composition, Mathematics, and the Modernist Legacy Twentieth-Century Composers and Mathematics Athanase Papadopoulos Messiaen et les math matiques Ronald Squibbs Xenakis's Method of Stochastic Composition: A Brief Introduction Peter Hoffmann My Music Makes no Revolution : Thoughts on the Role of Mathematics in the Work of Iannis Xenakis Alessandro Mastropietro Simboli matematici nei titoli delle composizioni di Franco Evangelisti C. Catherine Losada Boulez and Mathematics Andrew Shenton 1 + 1 = 1: The Elengant Mathematics of Arvo P rt's Tintinnabulation Joel Haack Mathematical Considerations of the Rhythmic Patterns in the Music of Steve Reich Richard Cohn Glass Graphs Pedro Ord ez Eslava Ars combinatoria: po ticas del n mero en la composici n espa ola contempor nea (1980-2018). Breve antolog a inacabada Music and Mathematics: Theories, Practices, Future Henry Klumpenhouwer Neo-Riemannian Theory Franck Jedrzejewski Graphes et diagrammes, l'en-de de la composition Angelo Orcalli L'evoluzione algoritmica della musica. Omaggio a Jean-Claude Risset Jos L. Besada Where Science-Based Music Comes From: Some Cognitive Remarks upon Contemporary Compositional Practices Inspired by Mathematical Concepts, Operations, and Objects Richard Hermann On Near Maximally Even Set-Classes Guerino Mazzola The Role of Mathematics for Music in Theory, Composition, and Performance Luigi Verdi Due esempi di Chim(us)ica Dodici canoni tricordali e loro combinazioni. Caleidocicli e canoni ritmici Fran ois Nicolas Du partage moderne/contemporain en math matiques et en musique la lumi re de Fernando Zalamea (Philosophie syst matique de la math matique contemporaine) Abstracts Biographies Index of Names‎

‎Great Breakthroughs in Mathematics: From Counting to Chaos Theory - How Numbers Changed the World‎

‎Arcturus Publishing Ltd 2020 192 pages 17 4x24 8x2 6cm. 2020. Relié. 192 pages.‎

‎Très bon état‎