Rticles the ORIGINS of the PROPOSITIONAL FUNCTIONS
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Discrete Mathematics
Discrete Mathematics1 http://lcs.ios.ac.cn/~znj/DM2017 Naijun Zhan March 19, 2017 1Special thanks to Profs Hanpin Wang (PKU) and Lijun Zhang (ISCAS) for their courtesy of the slides on this course. Contents 1. The Foundations: Logic and Proofs 2. Basic Structures: Sets, Functions, Sequences, Sum- s, and Matrices 3. Algorithms 4. Number Theory and Cryptography 5. Induction and Recursion 6. Counting 7. Discrete Probability 8. Advanced Counting Techniques 9. Relations 10. Graphs 11. Trees 12. Boolean Algebra 13. Modeling Computation 1 Chapter 1 The Foundations: Logic and Proofs Logic in Computer Science During the past fifty years there has been extensive, continuous, and growing interaction between logic and computer science. In many respects, logic provides computer science with both a u- nifying foundational framework and a tool for modeling compu- tational systems. In fact, logic has been called the calculus of computer science. The argument is that logic plays a fundamen- tal role in computer science, similar to that played by calculus in the physical sciences and traditional engineering disciplines. Indeed, logic plays an important role in areas of computer sci- ence as disparate as machine architecture, computer-aided de- sign, programming languages, databases, artificial intelligence, algorithms, and computability and complexity. Moshe Vardi 2 • The origins of logic can be dated back to Aristotle's time. • The birth of mathematical logic: { Leibnitz's idea { Russell paradox { Hilbert's plan { Three schools of modern logic: logicism (Frege, Russell, Whitehead) formalism (Hilbert) intuitionism (Brouwer) • One of the central problem for logicians is that: \why is this proof correct/incorrect?" • Boolean algebra owes to George Boole. -
Sets, Propositional Logic, Predicates, and Quantifiers
COMP 182 Algorithmic Thinking Sets, Propositional Logic, Luay Nakhleh Computer Science Predicates, and Quantifiers Rice University !1 Reading Material ❖ Chapter 1, Sections 1, 4, 5 ❖ Chapter 2, Sections 1, 2 !2 ❖ Mathematics is about statements that are either true or false. ❖ Such statements are called propositions. ❖ We use logic to describe them, and proof techniques to prove whether they are true or false. !3 Propositions ❖ 5>7 ❖ The square root of 2 is irrational. ❖ A graph is bipartite if and only if it doesn’t have a cycle of odd length. ❖ For n>1, the sum of the numbers 1,2,3,…,n is n2. !4 Propositions? ❖ E=mc2 ❖ The sun rises from the East every day. ❖ All species on Earth evolved from a common ancestor. ❖ God does not exist. ❖ Everyone eventually dies. !5 ❖ And some of you might already be wondering: “If I wanted to study mathematics, I would have majored in Math. I came here to study computer science.” !6 ❖ Computer Science is mathematics, but we almost exclusively focus on aspects of mathematics that relate to computation (that can be implemented in software and/or hardware). !7 ❖Logic is the language of computer science and, mathematics is the computer scientist’s most essential toolbox. !8 Examples of “CS-relevant” Math ❖ Algorithm A correctly solves problem P. ❖ Algorithm A has a worst-case running time of O(n3). ❖ Problem P has no solution. ❖ Using comparison between two elements as the basic operation, we cannot sort a list of n elements in less than O(n log n) time. ❖ Problem A is NP-Complete. -
The Development of Mathematical Logic from Russell to Tarski: 1900–1935
The Development of Mathematical Logic from Russell to Tarski: 1900–1935 Paolo Mancosu Richard Zach Calixto Badesa The Development of Mathematical Logic from Russell to Tarski: 1900–1935 Paolo Mancosu (University of California, Berkeley) Richard Zach (University of Calgary) Calixto Badesa (Universitat de Barcelona) Final Draft—May 2004 To appear in: Leila Haaparanta, ed., The Development of Modern Logic. New York and Oxford: Oxford University Press, 2004 Contents Contents i Introduction 1 1 Itinerary I: Metatheoretical Properties of Axiomatic Systems 3 1.1 Introduction . 3 1.2 Peano’s school on the logical structure of theories . 4 1.3 Hilbert on axiomatization . 8 1.4 Completeness and categoricity in the work of Veblen and Huntington . 10 1.5 Truth in a structure . 12 2 Itinerary II: Bertrand Russell’s Mathematical Logic 15 2.1 From the Paris congress to the Principles of Mathematics 1900–1903 . 15 2.2 Russell and Poincar´e on predicativity . 19 2.3 On Denoting . 21 2.4 Russell’s ramified type theory . 22 2.5 The logic of Principia ......................... 25 2.6 Further developments . 26 3 Itinerary III: Zermelo’s Axiomatization of Set Theory and Re- lated Foundational Issues 29 3.1 The debate on the axiom of choice . 29 3.2 Zermelo’s axiomatization of set theory . 32 3.3 The discussion on the notion of “definit” . 35 3.4 Metatheoretical studies of Zermelo’s axiomatization . 38 4 Itinerary IV: The Theory of Relatives and Lowenheim’s¨ Theorem 41 4.1 Theory of relatives and model theory . 41 4.2 The logic of relatives . -
Principle of Acquaintance and Axiom of Reducibility
Principle of acquaintance and axiom of reducibility Brice Halimi Université Paris Nanterre 15 mars 2018 Russell-the-epistemologist is the founding father of the concept of acquaintance. In this talk, I would like to show that Russell’s theory of knowledge is not simply the next step following his logic, but that his logic (especially the system of Principia Mathematica) can also be understood as the formal underpinning of a theory of knowledge. So there is a concept of acquaintance for Russell-the-logician as well. Principia Mathematica’s logical types In Principia, Russell gives the following examples of first-order propositional functions of individuals: φx; (x; y); (y) (x; y);::: Then, introducing φ!zb as a variable first-order propositional function of one individual, he gives examples of second-order propositional functions: f (φ!zb); g(φ!zb; !zb); F(φ!zb; x); (x) F(φ!zb; x); (φ) g(φ!zb; !zb); (φ) F(φ!zb; x);::: Then f !(φb!zb) is introduced as a variable second-order propositional function of one first-order propositional function. And so on. A possible value of f !(φb!zb) is . φ!a. (Example given by Russell.) This has to do with the fact that Principia’s schematic letters are variables: It will be seen that “φ!x” is itself a function of two variables, namely φ!zb and x. [. ] (Principia, p. 51) And variables are understood substitutionally (see Kevin Klement, “Russell on Ontological Fundamentality and Existence”, 2017). This explains that the language of Principia does not constitute an autonomous formal language. -
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Russell and Philip Jourdain a Study of Their Relationship
Russell and Philip Jourdain a study of their relationship One of the frustrations of Russellian scholarship is that there is often no trace among his papers of the letters he wrote Defore 1952 to his many correspondents. One of the most important missions of the Bertrand Russell Archives is to find as many of these letters as possible; for whenever some are found, new insights and detail s emerge. Without denying the central importance of the papers purchased from Russell in 1968, the most exciting development may be the growth of its section of recent acquisitions from 'other sources. A collection of letters of this type which is of unique importance is the surviving correspondence between Russell and the mathematician and historian of mathematics Philip Jourdain (1879-1919}. Jourdain suffered throughout his life from FriedriCh's ataxia, a crippling palsey which made writing extremely difficult for him. l Nevertheless he successfully pursued a career in research and scholarship. Indeed, all his brothers and sisters were remarkable in one way or another. For example, his eldest sister Eleanor was Vice-Principal of st. Hugh's College, Oxford (and Russell knew her quite we].l in the years when he lived in the city), and in 1911 she and her Principal Annie Moberley caused a sensation when they published pseudonymously a book claiming that in 1901 they had seen Marie Antoinette and members of her court during a visit to Versailles. 2 Philip had strong views, especially in favour of foundational studies in mathematics and against the church, and later in his life he shared with Russell strong pacifist inclinations. -
Philip Jourdain, Historiador De Las Matematicas
NOTAS PHILIP JOURDAIN, HISTORIADOR DE LAS MATEMATICAS ALEJANDRO R. GARCIADIEGO Facultad de Ciencias (México) RESUMEN ABSTRACT Philip Jourdain es conocido por Philip Jourdain is known by los historiadores de las matemáticas, historians of mathematics, among entre otras razones, a través de su other factors, by his correspondence correspondencia con algunos de los with some of the most famous matemáticos más famosos de su mathematicians of his days (e.g., tiempo (e.g., Georg Cantor (1845- Georg Cantor (1845-1918) and 1918) y Bertrand Russell (1872- Bertrand Russell (1872-1970) among 1970), entre otros); por sus artículos others); by his works on the history sobre la historia y filosofía de las and philosophy of mathematics; by his matemáticas; por sus artículos de publications on the theory of sets investigación en torno a la teoría de (especially, on the arithmetic of conjuntos (especialmente, en la transfinite cardinal and ordinal aritmética de los n ŭmeros cardinales y numbers); and, by his translation of ordinales transfinitos); y, por la Cantor's Beitráge... (1895-1897). The traducción del Beitráge... (1895-1897) goal of this essay is to discuss the de Cantor. En este ensayo discutiré la relation between his views of the relación entre sus puntos de vista nature of mathematics and his sobre la naturaleza de las matemáticas conception of the history of y su concepción de la historia de las mathematics which, apparently and at mismas que, aparentemente y a primera first sight, are inconsistent. vista, son contradictorias. 194 NOTAS LLULL 22 1. Introducción Philip Eduard Bertrand Jourdain, el más joven de los hijos del Reverendo F. -
Hugh Maccoll After One Hundred Years
Philosophia Scientiæ Travaux d'histoire et de philosophie des sciences 15-1 | 2011 Hugh MacColl after One Hundred Years Amirouche Moktefi and Stephen Read (dir.) Electronic version URL: http://journals.openedition.org/philosophiascientiae/351 DOI: 10.4000/philosophiascientiae.351 ISSN: 1775-4283 Publisher Éditions Kimé Printed version Date of publication: 1 April 2011 ISBN: 978-2-84174-551-7 ISSN: 1281-2463 Electronic reference Amirouche Moktefi and Stephen Read (dir.), Philosophia Scientiæ, 15-1 | 2011, “Hugh MacColl after One Hundred Years” [Online], Online since 01 April 2011, connection on 15 January 2021. URL: http:// journals.openedition.org/philosophiascientiae/351; DOI: https://doi.org/10.4000/philosophiascientiae. 351 This text was automatically generated on 15 January 2021. Tous droits réservés Preface Amirouche Moktefi LHSP, Archives H. Poincaré (UMR 7117), Nancy-Université & IRIST (EA 3424), Université de Strasbourg, France Stephen Read Department of Logic and Metaphysics, University of St. Andrews, Scotland “It all turns upon that little word if, Mr. Burton.” [MacColl 1891, 955] This volume is devoted to the life and work of Hugh MacColl (1837- 1909), the Scottish mathematician, philosopher and novelist. Although MacColl is nowadays remembered mainly for his contributions to the science of logic, his work covers many other domains and witnesses the richness and variety of his intellectual interests. MacColl’s early writings (from 1861 onwards) were devoted to mathematics, notably geometry, algebra and the theory of probabilities. In the late 1870s and early 1880s, he published several papers on logic that brought him to the attention of his contemporaries. In these writings, he expressed his preference for propositional calculus instead of the term/class approach, widespread in his day. -
The Foundations: Logic and Proof, Sets, and Functions
The Foundations: Logic and Proof, Sets, and Functions his chapter reviews the foundations of discrete mathematics. Three important topics are covered: logic, sets, and functions. The rules of logic specify the mean- ing of mathematical statements For instance, these rules help us understand and reason with statements such as "There exists an integer that is not the sum of two squares," and "For every positive integer n the sum of the positive integers not exceeding n is n(n + 1)/2." Logic is the basis of all mathematical reasoning, and it has practical applications to the design of computing machines, to system specifications, to artificial intelligence, to computer programming, to programming languages, and to other areas of computer science, as well as to many other fields of study. To understand mathematics, we must understand what makes up a correct mathemat- ical argument, that is, a proof. Moreover, to learn mathematics, a person needs to actively construct mathematical arguments and not just read exposition. In this chapter, we ex- plain what makes up a correct mathematical argument and introduce tools to construct these arguments. Proofs are important not only in mathematics, but also in many parts of computer science, including program verification, algorithm correctness, and system security. Furthermore, automated reasoning systems have been constructed that allow computers to construct their own proofs. Much of discrete mathematics is devoted to the study of discrete structures, which are used to represent discrete objects. Many important discrete structures are built using sets, which are collections of objects. Among the discrete structures built from sets are combinations, which are unordered collections of objects used extensively in counting; relations, which are sets of ordered pairs that represent relationships between objects; graphs, which consist of sets of vertices and of edges that connect vertices: and finite state machines, which are used to model computing machines. -
Some Views of Russell and Russell's Logic by His
SOME VIEWS OF RUSSELL AND RUSSELL’S LOGIC BY HIS CONTEMPORARIES, WITH PARTICULAR REFERENCE TO PEIRCE IRVING H. ANELLIS In the years prior to World War I, it was not yet definitively certain that the new logistic developed and espoused by Frege and Russell was the “wave of the future” of mathematical logic. Ernst Schröder’s Vorlesungen über die Algebra der Logik (1895– 1905) was still new and viewed as the cutting edge of (algebraic—as opposed to Aristote- lian) logic; Giuseppe Peano still proclaimed, in his earliest work on logic, his indebted- ness to George Boole, Charles Peirce, and Ernst Schröder, in the first footnote of his Arithmetices principia nova methodo exposita (1898), while, in a letter to Russell of 19 March 1901,1 Peano told Russell that his paper “Sur la logique des relations avec des ap- plications à la théorie des séries” on the logic of relations (1901) just “fills a gap between the work of Peirce and Schröder on the one hand and” his own “Formulaire on the other;” Alfred North Whitehead’s Treatise on Universal Algebra (1898) also still new, brought together the algebraic logic of Boole with the linear and multilinear algebras of Arthur Cayley, James Joseph Sylvester, William Rowan Hamilton, and Charles Peirce’s father Benjamin Peirce, and the calculus of extension of Hermann Grassmann. In his Treatise, Whitehead (p. x) not only named Hamilton and De Morgan as “the first to ex- press quite clearly the general possibilities of algebraic symbolism,” but continually ex- pressed his indebtedness to Boole, Grassmann, De Morgan, Schröder, and John Venn; and he described his Treatise (p. -
The Early Life of Russell's Notion of a Propositional Function
Baltic International Yearbook of Cognition, Logic and Communication Volume 4 200 YEARS OF ANALYTICAL PHILOSOPHY Article 1 2008 The Early Life Of Russell’s Notion Of A Propositional Function Michael Beaney University of York, UK Follow this and additional works at: https://newprairiepress.org/biyclc This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License. Recommended Citation Beaney, Michael (2008) "The Early Life Of Russell’s Notion Of A Propositional Function," Baltic International Yearbook of Cognition, Logic and Communication: Vol. 4. https://doi.org/10.4148/ biyclc.v4i0.125 This Proceeding of the Symposium for Cognition, Logic and Communication is brought to you for free and open access by the Conferences at New Prairie Press. It has been accepted for inclusion in Baltic International Yearbook of Cognition, Logic and Communication by an authorized administrator of New Prairie Press. For more information, please contact [email protected]. The Early Life of Russell’s Notion of a Propositional Function 2 The Baltic International Yearbook of weeks before the whole manuscript was submitted to Cambridge Uni- Cognition, Logic and Communication versity Press in late May 1902. The stage may have been set for its in- troduction, but given the importance of the notion, the precise nature of August 2009 Volume 4: 200 Years of Analytical Philosophy its appearance deserves elucidation. I offer an account of its emergence pages 1-25 DOI: 10.4148/biyclc.v4i0.125 in section 2. My interest in this is not merely antiquarian, however. The evolution of Russell’s conception of a propositional function plays a central role in the development of his thinking about analysis, and ex- MICHAEL BEANEY plaining this thinking is important not only in understanding Russell’s University of York, United Kingdom philosophy in all its changes but also in appreciating the complex na- ture of the practices of analysis in analytic philosophy today. -
Philosophy of Mathematics: Selected Writings
Philosophy Peirce The philosophy of mathematics plays a hilosophy of Peirce’s most vital role in the mature philosophy of Charles s. Peirce. Peirce received rigorous P importa-nt mathematical training from his father and Ma-thema-tic his philosophy carries on in decidedly philosophy Selected Writings writings on the mathematical and symbolic veins. For S Peirce, math was a philosophical tool and philosophy of many of his most productive ideas rest firmly on the foundation of mathematical cha-rles S. Peirce ma-thema-tics principles. This volume collects Peirce’s most important writings on the subject, many appearing in print for the first time. “Focuses on the major Peirce’s determination to understand writings Peirce produced matter, the cosmos, and “the grand design” of the universe remain relevant that are of greatest for contemporary students of science, significance for a correct technology, and symbolic logic. of appreciation of his larger Charles s. PeirCe (1839–1914) Ma-thema-ti was one of america’s most prolific philosophical agenda.” philosophers. he is noted for his —JosePh W. Dauben, contributions to logic, mathematics, City university oF neW york science, and semiotics. MattheW e. Moore is associate Professor of Philosophy at brooklyn College. he is editor of New Essays on Peirce’s Mathematical Philosophy. c Selections from the Writings of Cha-rles S. Peirce s PeirCe Edition ProJeCt INDIANA University Press Bloomington & Indianapolis www.iupress.indiana.edu INDIANA Edited by Ma-tthew E. Moore 1-800-842-6796 Phil of Mathematics MECH.indd 1 6/30/10 10:04 AM Beings of Reason.book Page i Wednesday, June 2, 2010 6:06 PM Philosophy of Mathematics Beings of Reason.book Page ii Wednesday, June 2, 2010 6:06 PM Beings of Reason.book Page iii Wednesday, June 2, 2010 6:06 PM Philosophy of Mathematics Selected Writings Charles S.