Mass Problems and Intuitionistic Higher-Order Logic
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Hubert Kennedy Eight Mathematical Biographies
Hubert Kennedy Eight Mathematical Biographies Peremptory Publications San Francisco 2002 © 2002 by Hubert Kennedy Eight Mathematical Biographies is a Peremptory Publications ebook. It may be freely distributed, but no changes may be made in it. Comments and suggestions are welcome. Please write to [email protected] . 2 Contents Introduction 4 Maria Gaetana Agnesi 5 Cesare Burali-Forti 13 Alessandro Padoa 17 Marc-Antoine Parseval des Chênes 19 Giuseppe Peano 22 Mario Pieri 32 Emil Leon Post 35 Giovanni Vailati 40 3 Introduction When a Dictionary of Scientific Biography was planned, my special research interest was Giuseppe Peano, so I volunteered to write five entries on Peano and his friends/colleagues, whose work I was investigating. (The DSB was published in 14 vol- umes in 1970–76, edited by C. C. Gillispie, New York: Charles Scribner's Sons.) I was later asked to write two more entries: for Parseval and Emil Leon Post. The entry for Post had to be done very quickly, and I could not have finished it without the generous help of one of his relatives. By the time the last of these articles was published in 1976, that for Giovanni Vailati, I had come out publicly as a homosexual and was involved in the gay liberation movement. But my article on Vailati was still discreet. If I had written it later, I would probably have included evidence of his homosexuality. The seven articles for the Dictionary of Scientific Biography have a uniform appear- ance. (The exception is the article on Burali-Forti, which I present here as I originally wrote it—with reference footnotes. -
Global Properties of the Turing Degrees and the Turing Jump
Global Properties of the Turing Degrees and the Turing Jump Theodore A. Slaman¤ Department of Mathematics University of California, Berkeley Berkeley, CA 94720-3840, USA [email protected] Abstract We present a summary of the lectures delivered to the Institute for Mathematical Sci- ences, Singapore, during the 2005 Summer School in Mathematical Logic. The lectures covered topics on the global structure of the Turing degrees D, the countability of its automorphism group, and the definability of the Turing jump within D. 1 Introduction This note summarizes the tutorial delivered to the Institute for Mathematical Sciences, Singapore, during the 2005 Summer School in Mathematical Logic on the structure of the Turing degrees. The tutorial gave a survey on the global structure of the Turing degrees D, the countability of its automorphism group, and the definability of the Turing jump within D. There is a glaring open problem in this area: Is there a nontrivial automorphism of the Turing degrees? Though such an automorphism was announced in Cooper (1999), the construction given in that paper is yet to be independently verified. In this pa- per, we regard the problem as not yet solved. The Slaman-Woodin Bi-interpretability Conjecture 5.10, which still seems plausible, is that there is no such automorphism. Interestingly, we can assemble a considerable amount of information about Aut(D), the automorphism group of D, without knowing whether it is trivial. For example, we can prove that it is countable and every element is arithmetically definable. Further, restrictions on Aut(D) lead us to interesting conclusions concerning definability in D. -
All That Math Portraits of Mathematicians As Young Researchers
Downloaded from orbit.dtu.dk on: Oct 06, 2021 All that Math Portraits of mathematicians as young researchers Hansen, Vagn Lundsgaard Published in: EMS Newsletter Publication date: 2012 Document Version Publisher's PDF, also known as Version of record Link back to DTU Orbit Citation (APA): Hansen, V. L. (2012). All that Math: Portraits of mathematicians as young researchers. EMS Newsletter, (85), 61-62. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain You may freely distribute the URL identifying the publication in the public portal If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. NEWSLETTER OF THE EUROPEAN MATHEMATICAL SOCIETY Editorial Obituary Feature Interview 6ecm Marco Brunella Alan Turing’s Centenary Endre Szemerédi p. 4 p. 29 p. 32 p. 39 September 2012 Issue 85 ISSN 1027-488X S E European M M Mathematical E S Society Applied Mathematics Journals from Cambridge journals.cambridge.org/pem journals.cambridge.org/ejm journals.cambridge.org/psp journals.cambridge.org/flm journals.cambridge.org/anz journals.cambridge.org/pes journals.cambridge.org/prm journals.cambridge.org/anu journals.cambridge.org/mtk Receive a free trial to the latest issue of each of our mathematics journals at journals.cambridge.org/maths Cambridge Press Applied Maths Advert_AW.indd 1 30/07/2012 12:11 Contents Editorial Team Editors-in-Chief Jorge Buescu (2009–2012) European (Book Reviews) Vicente Muñoz (2005–2012) Dep. -
Stephen Cole Kleene
STEPHEN COLE KLEENE American mathematician and logician Stephen Cole Kleene (January 5, 1909 – January 25, 1994) helped lay the foundations for modern computer science. In the 1930s, he and Alonzo Church, developed the lambda calculus, considered to be the mathematical basis for programming language. It was an effort to express all computable functions in a language with a simple syntax and few grammatical restrictions. Kleene was born in Hartford, Connecticut, but his real home was at his paternal grandfather’s farm in Maine. After graduating summa cum laude from Amherst (1930), Kleene went to Princeton University where he received his PhD in 1934. His thesis, A Theory of Positive Integers in Formal Logic, was supervised by Church. Kleene joined the faculty of the University of Wisconsin in 1935. During the next several years he spent time at the Institute for Advanced Study, taught at Amherst College and served in the U.S. Navy, attaining the rank of Lieutenant Commander. In 1946, he returned to Wisconsin, where he stayed until his retirement in 1979. In 1964 he was named the Cyrus C. MacDuffee professor of mathematics. Kleene was one of the pioneers of 20th century mathematics. His research was devoted to the theory of algorithms and recursive functions (that is, functions defined in a finite sequence of combinatorial steps). Together with Church, Kurt Gödel, Alan Turing, and others, Kleene developed the branch of mathematical logic known as recursion theory, which helped provide the foundations of theoretical computer science. The Kleene Star, Kleene recursion theorem and the Ascending Kleene Chain are named for him. -
Annals of Pure and Applied Logic Extending and Interpreting
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Annals of Pure and Applied Logic 161 (2010) 775–788 Contents lists available at ScienceDirect Annals of Pure and Applied Logic journal homepage: www.elsevier.com/locate/apal Extending and interpreting Post's programmeI S. Barry Cooper Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, UK article info a b s t r a c t Article history: Computability theory concerns information with a causal – typically algorithmic – Available online 3 July 2009 structure. As such, it provides a schematic analysis of many naturally occurring situations. Emil Post was the first to focus on the close relationship between information, coded as MSC: real numbers, and its algorithmic infrastructure. Having characterised the close connection 03-02 between the quantifier type of a real and the Turing jump operation, he looked for more 03A05 03D25 subtle ways in which information entails a particular causal context. Specifically, he wanted 03D80 to find simple relations on reals which produced richness of local computability-theoretic structure. To this extent, he was not just interested in causal structure as an abstraction, Keywords: but in the way in which this structure emerges in natural contexts. Post's programme was Computability the genesis of a more far reaching research project. Post's programme In this article we will firstly review the history of Post's programme, and look at two Ershov hierarchy Randomness interesting developments of Post's approach. The first of these developments concerns Turing invariance the extension of the core programme, initially restricted to the Turing structure of the computably enumerable sets of natural numbers, to the Ershov hierarchy of sets. -
THERE IS NO DEGREE INVARIANT HALF-JUMP a Striking
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 125, Number 10, October 1997, Pages 3033{3037 S 0002-9939(97)03915-4 THERE IS NO DEGREE INVARIANT HALF-JUMP RODNEY G. DOWNEY AND RICHARD A. SHORE (Communicated by Andreas R. Blass) Abstract. We prove that there is no degree invariant solution to Post's prob- lem that always gives an intermediate degree. In fact, assuming definable de- terminacy, if W is any definable operator on degrees such that a <W (a) < a0 on a cone then W is low2 or high2 on a cone of degrees, i.e., there is a degree c such that W (a)00 = a for every a c or W (a)00 = a for every a c. 00 ≥ 000 ≥ A striking phenomenon in the early days of computability theory was that every decision problem for axiomatizable theories turned out to be either decidable or of the same Turing degree as the halting problem (00,thecomplete computably enumerable set). Perhaps the most influential problem in computability theory over the past fifty years has been Post’s problem [10] of finding an exception to this rule, i.e. a noncomputable incomplete computably enumerable degree. The problem has been solved many times in various settings and disguises but the so- lutions always involve specific constructions of strange sets, usually by the priority method that was first developed (Friedberg [2] and Muchnik [9]) to solve this prob- lem. No natural decision problems or sets of any kind have been found that are neither computable nor complete. The question then becomes how to define what characterizes the natural computably enumerable degrees and show that none of them can supply a solution to Post’s problem. -
Uniform Martin's Conjecture, Locally
Uniform Martin's conjecture, locally Vittorio Bard July 23, 2019 Abstract We show that part I of uniform Martin's conjecture follows from a local phenomenon, namely that if a non-constant Turing invariant func- tion goes from the Turing degree x to the Turing degree y, then x ≤T y. Besides improving our knowledge about part I of uniform Martin's con- jecture (which turns out to be equivalent to Turing determinacy), the discovery of such local phenomenon also leads to new results that did not look strictly related to Martin's conjecture before. In particular, we get that computable reducibility ≤c on equivalence relations on N has a very complicated structure, as ≤T is Borel reducible to it. We conclude rais- ing the question Is part II of uniform Martin's conjecture implied by local phenomena, too? and briefly indicating a possible direction. 1 Introduction to Martin's conjecture Providing evidence for the intricacy of the structure (D; ≤T ) of Turing degrees has been arguably one of the main concerns of computability theory since the mid '50s, when the celebrated priority method was discovered. However, some have pointed out that if we restrict our attention to those Turing degrees that correspond to relevant problems occurring in mathematical practice, we see a much simpler structure: such \natural" Turing degrees appear to be well- ordered by ≤T , and there seems to be no \natural" Turing degree strictly be- tween a \natural" Turing degree and its Turing jump. Martin's conjecture is a long-standing open problem whose aim was to provide a precise mathemat- ical formalization of the previous insight. -
Addiction in Contemporary Mathematics FINAL 2020
Addiction in Contemporary Mathematics Newcomb Greenleaf Goddard College [email protected] Abstract 1. The Agony of Constructive Math 2. Reductio and Truth 3. Excluding the Middle 4. Brouwer Seen from Princeton 5. Errett Bishop 6. Brouwer Encodes Ignorance 7. Constructive Sets 8. Kicking the Classical Habit 9. Schizophrenia 10. Constructive Mathematics and Algorithms 11. The Intermediate Value Theorem 12. Real Numbers and Complexity 13. Natural Deduction 14. Mathematical Pluralism 15. Addiction in History of Math Abstract In 1973 Errett Bishop (1928-1983) gave the title Schizophrenia in Contemporary Mathematics to his Colloquium Lectures at the summer meetings of the AMS (American Mathematical Society). It marked the end of his seven-year campaign for Constructive mathematics, an attempt to introduce finer distinctions into mathematical thought. Seven years earlier Bishop had been a young math superstar, able to launch his revolution with an invited address at the International 1 Congress of Mathematicians, a quadrennial event held in Cold War Moscow in 1966. He built on the work of L. E. J. Brouwer (1881-1966), who had discovered in 1908 that Constructive mathematics requires a more subtle logic, in which truth has a positive quality stronger than the negation of falsity. By 1973 Bishop had accepted that most of his mathematical colleagues failed to understand even the simplest Constructive mathematics, and came up with a diagnosis of schizophrenia for their disability. We now know, from Andrej Bauer’s 2017 article, Five Stages of Accepting Constructive Mathematics that a much better diagnosis would have been addiction, whence our title. While habits can be kicked, schizophrenia tends to last a lifetime, which made Bishop’s misdiagnosis a very serious mistake. -
Independence Results on the Global Structure of the Turing Degrees by Marcia J
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume INDEPENDENCE RESULTS ON THE GLOBAL STRUCTURE OF THE TURING DEGREES BY MARCIA J. GROSZEK AND THEODORE A. SLAMAN1 " For nothing worthy proving can be proven, Nor yet disproven." _Tennyson Abstract. From CON(ZFC) we obtain: 1. CON(ZFC + 2" is arbitrarily large + there is a locally finite upper semilattice of size u2 which cannot be embedded into the Turing degrees as an upper semilattice). 2. CON(ZFC + 2" is arbitrarily large + there is a maximal independent set of Turing degrees of size w,). Introduction. Let ^ denote the set of Turing degrees ordered under the usual Turing reducibility, viewed as a partial order ( 6Î),< ) or an upper semilattice (fy, <,V> depending on context. A partial order (A,<) [upper semilattice (A, < , V>] is embeddable into fy (denoted A -* fy) if there is an embedding /: A -» <$so that a < ft if and only if/(a) </(ft) [and/(a) V /(ft) = /(a V ft)]. The structure of ty was first investigted in the germinal paper of Kleene and Post [2] where A -» ty for any countable partial order A was shown. Sacks [9] proved that A -» öDfor any partial order A which is locally finite (any point of A has only finitely many predecessors) and of size at most 2", or locally countable and of size at most u,. Say X C fy is an independent set of Turing degrees if, whenever x0, xx,...,xn E X and x0 < x, V x2 V • • • Vjcn, there is an / between 1 and n so that x0 = x¡; X is maximal if no proper extension of X is independent. -
Logique Combinatoire Et Physique Quantique
Eigenlogic Zeno Toffano [email protected] CentraleSupélec, Gif-sur-Yvette, France Laboratoire des Signaux et Systèmes, UMR8506-CNRS Université Paris-Saclay, France Zeno Toffano : “Eigenlogic” 1 UNILOG 2018 (6th World Congress and School on Universal Logic), Workshop Logic & Physics, Vichy (F), June 22, 2018 statement facts in quantum physics Quantum Physics started in 1900 with the quantification of radiation (Planck’s constant ℎ = 6.63 10−34 J.s) It is the most successful theory in physics and explains: nuclear energy, semiconductors, magnetism, lasers, quantum gravity… The theory has an increasing influence outside physics: computer science, AI, communications, biology, cognition… Nowadays there is a great quantum revival (second quantum revolution) which addresses principally: quantum entanglement (Bell inequalites, teleportation…), quantum superposition (Schrödinger cat), decoherence non-locality, non- separability, non-contextuality, non-classicality, non-… A more basic aspect is quantification: a measurement can only give one of the (real) eigenvalues of an observable. The associated eigenvector is the resulting quantum state. Measurements on non-eigenstates are indeterminate, the outcome probability is given by the Born rule. The eigenvalues are the spectrum (e.g. energies for the Hamiltonian). In physics it is natural to “work” in different representation eignespaces (of the considered observable) e.g: semiconductors “make sense” in the reciprocal lattice (momentum space 푝 , spatial Fourier transform of coordinate -
PDF (Dissertation.Pdf)
Kind Theory Thesis by Joseph R. Kiniry In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 2002 (Defended 10 May 2002) ii © 2002 Joseph R. Kiniry All Rights Reserved iii Preface This thesis describes a theory for representing, manipulating, and reasoning about structured pieces of knowledge in open collaborative systems. The theory's design is motivated by both its general model as well as its target user commu- nity. Its model is structured information, with emphasis on classification, relative structure, equivalence, and interpretation. Its user community is meant to be non-mathematicians and non-computer scientists that might use the theory via computational tool support once inte- grated with modern design and development tools. This thesis discusses a new logic called kind theory that meets these challenges. The core of the work is based in logic, type theory, and universal algebras. The theory is shown to be efficiently implementable, and several parts of a full realization have already been constructed and are reviewed. Additionally, several software engineering concepts, tools, and technologies have been con- structed that take advantage of this theoretical framework. These constructs are discussed as well, from the perspectives of general software engineering and applied formal methods. iv Acknowledgements I am grateful to my initial primary adviser, Prof. K. Mani Chandy, for bringing me to Caltech and his willingness to let me explore many unfamiliar research fields of my own choosing. I am also appreciative of my second adviser, Prof. Jason Hickey, for his support, encouragement, feedback, and patience through the later years of my work. -
Computable Functions A
http://dx.doi.org/10.1090/stml/019 STUDENT MATHEMATICAL LIBRARY Volume 19 Computable Functions A. Shen N. K. Vereshchagin Translated by V. N. Dubrovskii #AMS AMERICAN MATHEMATICAL SOCIETY Editorial Board Robert Devaney Carl Pomerance Daniel L. Goroff Hung-Hsi Wu David Bressoud, Chair H. K. BepemarHH, A. Illem* BMHHCJIHMME ^YHKUMM MIIHMO, 1999 Translated from the Russian by V. N. Dubrovskii 2000 Mathematics Subject Classification. Primary 03-01, 03Dxx. Library of Congress Cataloging-in-Publication Data Vereshchagin, Nikolai Konstantinovich, 1958- Computable functions / A. Shen, N.K. Vereshchagin ; translated by V.N. Dubrovskii. p. cm. — (Student mathematical library, ISSN 1520-9121 ; v. 19) Authors' names on t.p. of translation reversed from original. Includes bibliographical references and index. ISBN 0-8218-2732-4 (alk. paper) 1. Computable functions. I. Shen, A. (Alexander), 1958- II. Title. III. Se• ries. QA9.59 .V47 2003 511.3—dc21 2002038567 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904- 2294, USA. Requests can also be made by e-mail to reprint-permissionQams.org. © 2003 by the American Mathematical Society.