DOCSLIB.ORG

The Development of Mathematical Logic from Russell to Tarski: 1900–1935

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
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 . 44 4.3 Lowenheim’s¨ theorem . 46 4.4 Skolem’s first versions of Lowenheim’s¨ theorem . 56 i Contents 5 Itinerary V: Logic in the Hilbert School 59 5.1 Early lectures on logic . 59 5.2 The completeness of propositional logic . 60 5.3 Consistency and completeness . 61 5.4 Axioms and inference rules . 66 5.5 Grundzuge¨ der theoretischen Logik . 70 5.6 The decision problem . 71 5.6.1 The decision problem in the tradition of algebra of logic 72 5.6.2 Work on the decision problem after 1920 . 73 5.7 Combinatory logic and λ-calculus . 74 5.8 Structural inference: Hertz and Gentzen . 76 6 Itinerary VI: Proof Theory and Arithmetic 81 6.1 Hilbert’s Program for consistency proofs . 81 6.2 Consistency proofs for weak fragments of arithmetic . 82 6.3 Ackermann and von Neumann on epsilon substitution . 87 6.4 Herbrand’s Theorem . 92 6.5 Kurt Godel¨ and the incompleteness theorems . 94 7 Itinerary VII: Intuitionism and Many-valued Logics 99 7.1 Intuitionistic logic . 99 7.1.1 Brouwer’s philosophy of mathematics . 99 7.1.2 Brouwer on the excluded middle . 101 7.1.3 The logic of negation . 102 7.1.4 Kolmogorov . 103 7.1.5 The debate on intuitionist logic . 106 7.1.6 The formalization and interpretation of intuitionistic logic . 108 7.1.7 Godel’s¨ contributions to the metatheory of intuition- istic logic . 110 7.2 Many-valued logics . 111 8 Itinerary VIII: Semantics and Model-theoretic Notions 117 8.1 Background . 117 8.1.1 The algebra of logic tradition . 117 8.1.2 Terminological variations (systems of objects, mod- els, and structures) . 118 8.1.3 Interpretations for propositional logic . 119 8.2 Consistency and independence for propositional logic . 120 8.3 Post’s contributions to the metatheory of the propositional calculus . 123 8.4 Semantical completeness of first-order logic . 124 8.5 Models of first order logic . 129 8.6 Completeness and categoricity . 130 8.7 Tarski’s definition of truth . 134 ii Notes 141 Bibliography 149 Index of Citations 175 iii Introduction The following nine itineraries in the history of mathematical logic do not aim at a complete account of the history of mathematical logic during the period 1900–1935. For one thing, we had to limit our ambition to the technical developments without attempting a detailed discussion of issues such as what conceptions of logic were being held during the period. This also means that we have not engaged in detail with historiographical de- bates which are quite lively today, such as those on the universality of logic, conceptions of truth, the nature of logic itself etc. While of extreme interest these themes cannot be properly dealt with in a short space, as they often require extensive exegetical work. We therefore merely point out in the text or in appropriate notes how the reader can pursue the connection between the material we treat and the secondary literature on these debates. Second, we have not treated some important developments. While we have not aimed at completeness our hope has been that by fo- cusing on a narrower range of topics our treatment will improve on the existing literature on the history of logic. There are excellent accounts of the history of mathematical logic available, such as, to name a few, Kneale and Kneale (1962), Dimitriu (1977), and Mangione and Bozzi (1993). We have kept the secondary literature quite present in that we also wanted to write an essay that would strike a balance between covering material that was adequately discussed in the secondary literature and presenting new lines of investigation. This explains, for instance, why the reader will find a long and precise exposition of Lowenheim’s¨ ( 1915) theorem but only a short one on Godel’s¨ incompleteness theorem: Whereas there is hitherto no precise presentation of the first result, accounts of the second result abound. Finally, the treatment of the foundations of mathematics is quite restricted and it is ancillary to the exposition of the history of mathemati- cal logic. Thus, it is not meant to be the main focus of our exposition.1 Page references in citations are to the English translations, if available; or to the reprint edition, if listed in the bibliography. All translations are the authors’, unless an English translation is listed in the references. 1 1 Itinerary I. Metatheoretical Properties of Axiomatic Systems 1.1 Introduction The two most important meetings in philosophy and mathematics in 1900 took place in Paris. The First International Congress of Philosophy met in August and so did, soon after, the Second International Congress of Mathe- maticians. As symbolic, or mathematical, logic has traditionally been part both of mathematics and philosophy, a glimpse at the contributions in mathematical logic at these two events will give us a representative se- lection of the state of mathematical logic at the beginning of the twenti- eth century. At the International Congress of Mathematicians Hilbert pre- sented his famous list of problems ( Hilbert 1900a), some of which became central to mathematical logic, such as the continuum problem, the consis- tency proof for the system of real numbers, and the decision problem for Diophantine equations (Hilbert’s tenth problem). However, despite the at- tendance of remarkable logicians like Schroder,¨ Peano, and Whitehead in the audience, the only other talk that could be classified as pertaining to mathematical logic were two talks given by Alessandro Padoa on the ax- iomatizations of the integers and of geometry, respectively. The third section of the International Congress of Philosophy was de- voted to logic and history of the sciences ( Lovett 1900–01). Among the contributors of papers in logic we find Russell, MacColl, Peano, Burali- Forti, Padoa, Pieri, Poretsky, Schroder,¨ and Johnson. Of these, MacColl, Poretsky, Schroder,¨ and Johnson read papers that belong squarely to the algebra of logic tradition. Russell read a paper on the application of the theory of relations to the problem of order and absolute position in space and time. Finally, the Italian school of Peano and his disciples—Burali- Forti, Padoa and Pieri—contributed papers on the logical analysis of math- ematics. Peano and Burali-Forti spoke on definitions, Padoa read his fa- mous essay containing the “logical introduction to any theory whatever,” and Pieri spoke on geometry considered as a purely logical system. Al- though there are certainly points of contact between the first group of 3 1. Metatheoretical Properties of Axiomatic Systems logicians and the second group, already at that time it was obvious that two different approaches to mathematical logic were at play. Whereas the algebra of logic tradition was considered to be mainly an application of mathematics to logic, the other tradition was concerned more with an analysis of mathematics by logical means. In a course given in 1908 in Gottingen,¨ Zermelo captured the double meaning of mathemat- ical logic in the period by reference to the two schools: The word “mathematical logic” can be used with two different meanings. On the one hand one can treat logic mathematically, as it was done for instance by Schroder¨ in his Algebra of Logic; on the other hand, one can also investigate scientifically the logical components of mathematics. ( Zermelo 1908a, 1)2 The first approach is tied to the names of Boole and Schroder,¨ the second was represented by Frege, Peano and Russell.3 We will begin by focusing on mathematical logic as the logical analysis of mathematical theories but we will return later (see itinerary IV) to the other tradition. 1.2 Peano’s school on the logical structure of theories We have mentioned the importance of the logical analysis of mathemat- ics as one of the central motivating factors in the work of Peano and his school on mathematical logic. First of all, Peano was instrumental in em- phasizing the importance of mathematical logic as an artificial language that would remove the ambiguities of natural language thereby allowing a precise analysis of mathematics. In the words of Pieri, an appropriate ideographical algorithm is useful as “an instrument appropriate to guide and discipline thought, to exclude ambiguities, implicit assumptions, men- tal restrictions, insinuations and other shortcomings, almost inseparable from ordinary language, written as well as spoken, which are so damag- ing to speculative research.” ( Pieri 1901, 381).
Load more

Recommended publications
  • Poznań Studies in the Philosophy of the Sciences and the Humanities), 21Pp
    Forthcoming in: Uncovering Facts and Values, ed. A. Kuzniar and J. Odrowąż-Sypniewska (Poznań Studies in the Philosophy of the Sciences and the Humanities), 21pp. Abstract Alfred Tarski seems to endorse a partial conception of truth, the T-schema, which he believes might be clarified by the application of empirical methods, specifically citing the experimental results of Arne Næss (1938a). The aim of this paper is to argue that Næss’ empirical work confirmed Tarski’s semantic conception of truth, among others. In the first part, I lay out the case for believing that Tarski’s T-schema, while not the formal and generalizable Convention-T, provides a partial account of truth that may be buttressed by an examination of the ordinary person’s views of truth. Then, I address a concern raised by Tarski’s contemporaries who saw Næss’ results as refuting Tarski’s semantic conception. Following that, I summarize Næss’ results. Finally, I will contend with a few objections that suggest a strict interpretation of Næss’ results might recommend an overturning of Tarski’s theory. Keywords: truth, Alfred Tarski, Arne Næss, Vienna Circle, experimental philosophy Joseph Ulatowski ORDINARY TRUTH IN TARSKI AND NÆSS 1. Introduction Many of Alfred Tarski's better known papers on truth (e.g. 1944; 1983b), logical consequence (1983c), semantic concepts in general (1983a), or definability (1948) identify two conditions that successful definitions of “truth,” “logical consequence,” or “definition” require: formal correctness and material (or intuitive) adequacy.1 The first condition Tarski calls “formal correctness” because a definition of truth (for a given formal language) is formally correct when it is constructed in a manner that allows us to avoid both circular definition and semantic paradoxes.
    [Show full text]
  • “The Church-Turing “Thesis” As a Special Corollary of Gödel's
    “The Church-Turing “Thesis” as a Special Corollary of Gödel’s Completeness Theorem,” in Computability: Turing, Gödel, Church, and Beyond, B. J. Copeland, C. Posy, and O. Shagrir (eds.), MIT Press (Cambridge), 2013, pp. 77-104. Saul A. Kripke This is the published version of the book chapter indicated above, which can be obtained from the publisher at https://mitpress.mit.edu/books/computability. It is reproduced here by permission of the publisher who holds the copyright. © The MIT Press The Church-Turing “ Thesis ” as a Special Corollary of G ö del ’ s 4 Completeness Theorem 1 Saul A. Kripke Traditionally, many writers, following Kleene (1952) , thought of the Church-Turing thesis as unprovable by its nature but having various strong arguments in its favor, including Turing ’ s analysis of human computation. More recently, the beauty, power, and obvious fundamental importance of this analysis — what Turing (1936) calls “ argument I ” — has led some writers to give an almost exclusive emphasis on this argument as the unique justification for the Church-Turing thesis. In this chapter I advocate an alternative justification, essentially presupposed by Turing himself in what he calls “ argument II. ” The idea is that computation is a special form of math- ematical deduction. Assuming the steps of the deduction can be stated in a first- order language, the Church-Turing thesis follows as a special case of G ö del ’ s completeness theorem (first-order algorithm theorem). I propose this idea as an alternative foundation for the Church-Turing thesis, both for human and machine computation. Clearly the relevant assumptions are justified for computations pres- ently known.
    [Show full text]
  • Set Theory, by Thomas Jech, Academic Press, New York, 1978, Xii + 621 Pp., '$53.00
    BOOK REVIEWS 775 BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 3, Number 1, July 1980 © 1980 American Mathematical Society 0002-9904/80/0000-0 319/$01.75 Set theory, by Thomas Jech, Academic Press, New York, 1978, xii + 621 pp., '$53.00. "General set theory is pretty trivial stuff really" (Halmos; see [H, p. vi]). At least, with the hindsight afforded by Cantor, Zermelo, and others, it is pretty trivial to do the following. First, write down a list of axioms about sets and membership, enunciating some "obviously true" set-theoretic principles; the most popular Hst today is called ZFC (the Zermelo-Fraenkel axioms with the axiom of Choice). Next, explain how, from ZFC, one may derive all of conventional mathematics, including the general theory of transfinite cardi­ nals and ordinals. This "trivial" part of set theory is well covered in standard texts, such as [E] or [H]. Jech's book is an introduction to the "nontrivial" part. Now, nontrivial set theory may be roughly divided into two general areas. The first area, classical set theory, is a direct outgrowth of Cantor's work. Cantor set down the basic properties of cardinal numbers. In particular, he showed that if K is a cardinal number, then 2", or exp(/c), is a cardinal strictly larger than K (if A is a set of size K, 2* is the cardinality of the family of all subsets of A). Now starting with a cardinal K, we may form larger cardinals exp(ic), exp2(ic) = exp(exp(fc)), exp3(ic) = exp(exp2(ic)), and in fact this may be continued through the transfinite to form expa(»c) for every ordinal number a.
    [Show full text]
  • Logical Truth, Contradictions, Inconsistency, and Logical Equivalence
    Logical Truth, Contradictions, Inconsistency, and Logical Equivalence Logical Truth • The semantical concepts of logical truth, contradiction, inconsistency, and logi- cal equivalence in Predicate Logic are straightforward adaptations of the corre- sponding concepts in Sentence Logic. • A closed sentence X of Predicate Logic is logically true, X, if and only if X is true in all interpretations. • The logical truth of a sentence is proved directly using general reasoning in se- mantics. • Given soundness, one can also prove the logical truth of a sentence X by provid- ing a derivation with no premises. • The result of the derivation is that X is theorem, ` X. An Example • (∀x)Fx ⊃ (∃x)Fx. – Suppose d satisfies ‘(∀x)Fx’. – Then all x-variants of d satisfy ‘Fx’. – Since the domain D is non-empty, some x-variant of d satisfies ‘Fx’. – So d satisfies ‘(∃x)Fx’ – Therefore d satisfies ‘(∀x)Fx ⊃ (∃x)Fx’, QED. • ` (∀x)Fx ⊃ (∃x)Fx. 1 (∀x)Fx P 2 Fa 1 ∀ E 3 (∃x)Fx 1 ∃ I Contradictions • A closed sentence X of Predicate Logic is a contradiction if and only if X is false in all interpretations. • A sentence X is false in all interpretations if and only if its negation ∼X is true on all interpretations. • Therefore, one may directly demonstrate that a sentence is a contradiction by proving that its negation is a logical truth. • If the ∼X of a sentence is a logical truth, then given completeness, it is a theorem, and hence ∼X can be derived from no premises. 1 • If a sentence X is such that if it is true in any interpretation, both Y and ∼Y are true in that interpretation, then X cannot be true on any interpretation.
    [Show full text]
  • Chapter 2 Introduction to Classical Propositional
    CHAPTER 2 INTRODUCTION TO CLASSICAL PROPOSITIONAL LOGIC 1 Motivation and History The origins of the classical propositional logic, classical propositional calculus, as it was, and still often is called, go back to antiquity and are due to Stoic school of philosophy (3rd century B.C.), whose most eminent representative was Chryssipus. But the real development of this calculus began only in the mid-19th century and was initiated by the research done by the English math- ematician G. Boole, who is sometimes regarded as the founder of mathematical logic. The classical propositional calculus was ¯rst formulated as a formal ax- iomatic system by the eminent German logician G. Frege in 1879. The assumption underlying the formalization of classical propositional calculus are the following. Logical sentences We deal only with sentences that can always be evaluated as true or false. Such sentences are called logical sentences or proposi- tions. Hence the name propositional logic. A statement: 2 + 2 = 4 is a proposition as we assume that it is a well known and agreed upon truth. A statement: 2 + 2 = 5 is also a proposition (false. A statement:] I am pretty is modeled, if needed as a logical sentence (proposi- tion). We assume that it is false, or true. A statement: 2 + n = 5 is not a proposition; it might be true for some n, for example n=3, false for other n, for example n= 2, and moreover, we don't know what n is. Sentences of this kind are called propositional functions. We model propositional functions within propositional logic by treating propositional functions as propositions.
    [Show full text]
  • Automating Free Logic in HOL, with an Experimental Application in Category Theory
    Noname manuscript No. (will be inserted by the editor) Automating Free Logic in HOL, with an Experimental Application in Category Theory Christoph Benzm¨uller and Dana S. Scott Received: date / Accepted: date Abstract A shallow semantical embedding of free logic in classical higher- order logic is presented, which enables the off-the-shelf application of higher- order interactive and automated theorem provers for the formalisation and verification of free logic theories. Subsequently, this approach is applied to a selected domain of mathematics: starting from a generalization of the standard axioms for a monoid we present a stepwise development of various, mutually equivalent foundational axiom systems for category theory. As a side-effect of this work some (minor) issues in a prominent category theory textbook have been revealed. The purpose of this article is not to claim any novel results in category the- ory, but to demonstrate an elegant way to “implement” and utilize interactive and automated reasoning in free logic, and to present illustrative experiments. Keywords Free Logic · Classical Higher-Order Logic · Category Theory · Interactive and Automated Theorem Proving 1 Introduction Partiality and undefinedness are prominent challenges in various areas of math- ematics and computer science. Unfortunately, however, modern proof assistant systems and automated theorem provers based on traditional classical or intu- itionistic logics provide rather inadequate support for these challenge concepts. Free logic [24,25,30,32] offers a theoretically appealing solution, but it has been considered as rather unsuited towards practical utilization. Christoph Benzm¨uller Freie Universit¨at Berlin, Berlin, Germany & University of Luxembourg, Luxembourg E-mail: [email protected] Dana S.
    [Show full text]
  • Mathematical Logic. Introduction. by Vilnis Detlovs And
    1 Version released: May 24, 2017 Introduction to Mathematical Logic Hyper-textbook for students by Vilnis Detlovs, Dr. math., and Karlis Podnieks, Dr. math. University of Latvia This work is licensed under a Creative Commons License and is copyrighted © 2000-2017 by us, Vilnis Detlovs and Karlis Podnieks. Sections 1, 2, 3 of this book represent an extended translation of the corresponding chapters of the book: V. Detlovs, Elements of Mathematical Logic, Riga, University of Latvia, 1964, 252 pp. (in Latvian). With kind permission of Dr. Detlovs. Vilnis Detlovs. Memorial Page In preparation – forever (however, since 2000, used successfully in a real logic course for computer science students). This hyper-textbook contains links to: Wikipedia, the free encyclopedia; MacTutor History of Mathematics archive of the University of St Andrews; MathWorld of Wolfram Research. 2 Table of Contents References..........................................................................................................3 1. Introduction. What Is Logic, Really?.............................................................4 1.1. Total Formalization is Possible!..............................................................5 1.2. Predicate Languages.............................................................................10 1.3. Axioms of Logic: Minimal System, Constructive System and Classical System..........................................................................................................27 1.4. The Flavor of Proving Directly.............................................................40
    [Show full text]
  • Pluralisms About Truth and Logic Nathan Kellen University of Connecticut - Storrs, [email protected]
    University of Connecticut OpenCommons@UConn Doctoral Dissertations University of Connecticut Graduate School 8-9-2019 Pluralisms about Truth and Logic Nathan Kellen University of Connecticut - Storrs, [email protected] Follow this and additional works at: https://opencommons.uconn.edu/dissertations Recommended Citation Kellen, Nathan, "Pluralisms about Truth and Logic" (2019). Doctoral Dissertations. 2263. https://opencommons.uconn.edu/dissertations/2263 Pluralisms about Truth and Logic Nathan Kellen, PhD University of Connecticut, 2019 Abstract: In this dissertation I analyze two theories, truth pluralism and logical pluralism, as well as the theoretical connections between them, including whether they can be combined into a single, coherent framework. I begin by arguing that truth pluralism is a combination of realist and anti-realist intuitions, and that we should recognize these motivations when categorizing and formulating truth pluralist views. I then introduce logical functionalism, which analyzes logical consequence as a functional concept. I show how one can both build theories from the ground up and analyze existing views within the functionalist framework. One upshot of logical functionalism is a unified account of logical monism, pluralism and nihilism. I conclude with two negative arguments. First, I argue that the most prominent form of logical pluralism faces a serious dilemma: it either must give up on one of the core principles of logical consequence, and thus fail to be a theory of logic at all, or it must give up on pluralism itself. I call this \The Normative Problem for Logical Pluralism", and argue that it is unsolvable for the most prominent form of logical pluralism. Second, I examine an argument given by multiple truth pluralists that purports to show that truth pluralists must also be logical pluralists.
    [Show full text]
  • The Logic of Brouwer and Heyting
    THE LOGIC OF BROUWER AND HEYTING Joan Rand Moschovakis Intuitionistic logic consists of the principles of reasoning which were used in- formally by L. E. J. Brouwer, formalized by A. Heyting (also partially by V. Glivenko), interpreted by A. Kolmogorov, and studied by G. Gentzen and K. G¨odel during the first third of the twentieth century. Formally, intuitionistic first- order predicate logic is a proper subsystem of classical logic, obtained by replacing the law of excluded middle A ∨¬A by ¬A ⊃ (A ⊃ B); it has infinitely many dis- tinct consistent axiomatic extensions, each necessarily contained in classical logic (which is axiomatically complete). However, intuitionistic second-order logic is inconsistent with classical second-order logic. In terms of expressibility, the intu- itionistic logic and language are richer than the classical; as G¨odel and Gentzen showed, classical first-order arithmetic can be faithfully translated into the nega- tive fragment of intuitionistic arithmetic, establishing proof-theoretical equivalence and clarifying the distinction between the classical and constructive consequences of mathematical axioms. The logic of Brouwer and Heyting is effective. The conclusion of an intuitionistic derivation holds with the same degree of constructivity as the premises. Any proof of a disjunction of two statements can be effectively transformed into a proof of one of the disjuncts, while any proof of an existential statement contains an effec- tive prescription for finding a witness. The negation of a statement is interpreted as asserting that the statement is not merely false but absurd, i.e., leads to a contradiction. Brouwer objected to the general law of excluded middle as claim- ing a priori that every mathematical problem has a solution, and to the general law ¬¬A ⊃ A of double negation as asserting that every consistent mathematical statement holds.
    [Show full text]
  • Aristotle's Assertoric Syllogistic and Modern Relevance Logic*
    Aristotle’s assertoric syllogistic and modern relevance logic* Abstract: This paper sets out to evaluate the claim that Aristotle’s Assertoric Syllogistic is a relevance logic or shows significant similarities with it. I prepare the grounds for a meaningful comparison by extracting the notion of relevance employed in the most influential work on modern relevance logic, Anderson and Belnap’s Entailment. This notion is characterized by two conditions imposed on the concept of validity: first, that some meaning content is shared between the premises and the conclusion, and second, that the premises of a proof are actually used to derive the conclusion. Turning to Aristotle’s Prior Analytics, I argue that there is evidence that Aristotle’s Assertoric Syllogistic satisfies both conditions. Moreover, Aristotle at one point explicitly addresses the potential harmfulness of syllogisms with unused premises. Here, I argue that Aristotle’s analysis allows for a rejection of such syllogisms on formal grounds established in the foregoing parts of the Prior Analytics. In a final section I consider the view that Aristotle distinguished between validity on the one hand and syllogistic validity on the other. Following this line of reasoning, Aristotle’s logic might not be a relevance logic, since relevance is part of syllogistic validity and not, as modern relevance logic demands, of general validity. I argue that the reasons to reject this view are more compelling than the reasons to accept it and that we can, cautiously, uphold the result that Aristotle’s logic is a relevance logic. Keywords: Aristotle, Syllogism, Relevance Logic, History of Logic, Logic, Relevance 1.
    [Show full text]
  • 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.
    [Show full text]
  • Alfred Tarski His:Bio:Tar: Sec Alfred Tarski Was Born on January 14, 1901 in Warsaw, Poland (Then Part of the Russian Empire)
    bio.1 Alfred Tarski his:bio:tar: sec Alfred Tarski was born on January 14, 1901 in Warsaw, Poland (then part of the Russian Empire). Described as \Napoleonic," Tarski was boisterous, talkative, and intense. His energy was often reflected in his lectures|he once set fire to a wastebasket while disposing of a cigarette during a lecture, and was forbidden from lecturing in that building again. Tarski had a thirst for knowledge from a young age. Although later in life he would tell students that he stud- ied logic because it was the only class in which he got a B, his high school records show that he got A's across the board| even in logic. He studied at the Univer- sity of Warsaw from 1918 to 1924. Tarski Figure 1: Alfred Tarski first intended to study biology, but be- came interested in mathematics, philosophy, and logic, as the university was the center of the Warsaw School of Logic and Philosophy. Tarski earned his doctorate in 1924 under the supervision of Stanislaw Le´sniewski. Before emigrating to the United States in 1939, Tarski completed some of his most important work while working as a secondary school teacher in Warsaw. His work on logical consequence and logical truth were written during this time. In 1939, Tarski was visiting the United States for a lecture tour. During his visit, Germany invaded Poland, and because of his Jewish heritage, Tarski could not return. His wife and children remained in Poland until the end of the war, but were then able to emigrate to the United States as well.
    [Show full text]