DOCSLIB.ORG

Does Mathematics Need Something Other Than Logic?

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Does Mathematics Need Something Other Than Logic? Does Mathematics Need Something other than Logic? Katuzi ONO Mathematical Institute, Nagoya University Allow me to reason freely in the primitive logic. Then, I will show you how to develop mathematical theories without assuming anything in it. Introduction "Does mathematics really need something other than logic?" This is a question I have asked myself over and over again. Something in my mind has tempted me very strongly to answer "No. Mathematics needs nothing other than logic. If mathematics at present needs something other than logic, mathematics in the ideal form should be established purely logically." Most people would agree to the answer idealistically. Anyone would like to establish mathematics purely logically, if he could. But, how can he know that we can establish mathematics purely logically without being able to do so? I have always believed mathematics is absolutely applicable. If something is concluded mathematically, I believe it absolutely. If there arises something which contradicts a mathematical conclusion, I would only seek after illusional perceptions of the matter or mis-reasonings leading to the mathematical conclusion. I have to admit that illusional perceptions or mis-reasonings are really possible, but I can not even imagine such a situation that some correct perceptions contradict correct mathematical conclusions. Why can we believe in mathematics so absolutely? The reason seems to be that mathematics can be established purely logically. I do not know whether all the theories of mathematics are really established purely logically, but some thing in my mind forces me to say that all mathematical theories should be es tablished purely logically. In reality, however, we live in the tradition of mathematics, in the long lived tradition from the era of Euclid, which is to develop all mathematical theories starting from respective systems of axioms. I do not assert that we have really developed every mathematical theory starting from a certain system of its axioms , but I must admit that, if not, we are strongly tempted to try so . In the good old days, mathematicians could believe that axioms are self-evi dent truths. Nowadays, we can not agree even that axioms should be self-evident , nor that they could be so. However, if axioms are not self-evident , how can we believe so confidently in the absolute applicability of mathematics? -93- 8 K. ONO Vol. 3 Anyway, axioms are something which are not purely logical. If axioms are really necessary for developing mathematical theories, can we still say that mathematics needs nothing other than logic? If we stick to the axiomatic formali zation of mathematics, I think, we can not assert that mathematics needs nothing other than logic. The axiomatic approach has been really effective for clearly disclosing the logical structure of mathematical theories. However, it has not been the sole approach in the logical study of mathematics. In fact, there has been once an epoch when mathematicians seem to have been trying (successfully or not) to construct fundamental mathematical notions such as "real numbers", exclusively on a logical basis. I am thinking that set theory in the beginning was going to be developed in such an atmosphere. Someone seemed to believe that we needed natural numbers even at the beginning of mathematical theories but we can con struct all the others purely logically. Some mathematicians, however, seem to have been trying to construct even the natural numbers "set-theoretically", this perhaps being synonymous with "logically". I think that the category-theoretical approach of these days must be on the same line. A typical example lying along this line in traditional mathematics must be the definition of real numbers by cuts of all rational numbers. We need not regard real numbers always as cuts of all rational numbers, but cuts of all rational numbers really behave just as real numbers. This is why real numbers can be represented nicely by cuts of all rational numbers. A similar situation arises when we introduce all logical constants in terms of "implication" (notation:"•¨","_??_•¨_??_ is read "_??_ implies _??_") , "universal quanti fication" (notation: "( )", "(ƒÔ)_??_(ƒÔ)" is read "For every ƒÔ,_??_(ƒÔ) holds"), and "nega tion" (notation:"_??_","_??__??_is read "Not _??_"). In classical logic, for instance, "_??_ or _??_" (notation' "_??__??__??_") can be paraphrased as "the denial of W implies 3" (in notation:"_??__??__??__??_") and "There exists an x which satisfies _??_(ƒÔ)" (notation: "(_??_ƒÔ)_??_(ƒÔ)")can be paraphrased as "It does not hold that _??_(ƒÔ) does not hold for every ƒÔ" (in notation: "- _??_(ƒÔ)_??___??_(ƒÔ)"). We need not regard existence notion always as being defined in terms of universal quantification and negation, but the existence notion thus defined behaves just as the real existence notion in classical logic. Therefore, the existence notion can be defined nicely in this way in classical logic. These two examples are really similar in some respects. They differ, however, in the respect that, in the former example, mathematical notions only are paraphrased in a system without the notions but, in the latter example, logical constants are paraphrased in a system without the notions. In constructive theories of mathematics, some mathematical notions are introduced, but we keep usually logical constants unchanged. In our study of logical structure of mathematical theories, we are also in -94- No. 3 Does Mathematics Need Something other than Logic ? 9 terested in mathematical theories standing on the intuitionistic logic, the positive logic, or furthermore, other various logics. In these cases, it seems impossible to keep every logical constant unchanged. In fact, I have been able to develop every finitely axiomatizable formal theory in a very simple logic called primitive logic, which has been introduced in my former papers [2] and [3]. Technical treatment of the matter has been described in details in my former papers [5] and [6]. In the present paper, I will describe our study on the same subject, disclosing my schedules, my failures, and my successes frankly. (1) Reduction of a Formal Theory to Another by Paraphrasing Every Word of the Former in the Latter To speak of the development of one formal theory in another formal theory or to speak of reduction of one formal theory to another formal theory clearly, I would like to study at first how to define these notions. In mathematics, a typical example of reduction of a formal theory to another would be the reduction of geometry to analysis by introducing coordinates. For reducing plane geometry to analysis, any point is paraphrased by its coordinates, i.e. a pair of real numbers, a straight line is paraphrased as a condition expressed by a linear equation with respect to two variables, and "a straight line passes through a point" is paraphrased as "the coordinates of the point satisfies the equation expressing the straight line" . For plane geometry, "point", "straight line", and "a straight line passes through a point" are basic notions in the sense that other plane geometrical notions can be defined in terms of these basic notions. More generally, every formal theory has a system of its basic notions, in terms of which all the other notions of the theory can be defined. Accordingly, if we give the translations of all the basic notions of a formal theory into another, we would have a complete translation of the former theory into the latter. When every proposition holds in the former theory if and only if its translation holds in the latter theory, the former theory can be duly called "reduced to the latter theory." Reductions in mathematics are usually reductions of this kind. Such a kind of reductions rely on the paraphrasing of their basic notions. These basic notions of a theory are usually called the primitive notions of the formal theory. Any primitive notion of a formal theory is usually denoted by a technical term (maybe a word or a phrase) of the theory, and it is denoted by a special predicate symbol when the theory is formalized. If it is denoted by a phrase, the phrase is translated as a whole as if it were a single word. So, the paraphrasing is essenti ally something like a word-for-word translation. Going more into details, any formal theory has a limited number of its primitive notions. Any other notion in the formal theory is paraphrased in terms of these primitive notions. The paraphrasing is usually called the definition of the - 95- 10 K. ONO Vol. 3 notion in the theory. Anyway, any sentence in a formal theory can be expressed in principle in terms of its primitive notions (practicable or not). Accordingly, any sentence in a formal theory is completely interpreted in another theory, if every primitive notion of the former theory is paraphrased in terms of admitted notions in the latter theory. Even if this can not be called "word-for-word" in the strict sense, we can easily modulate it as if it is a word-for-word translation. It is enough, 1) to introduce a word in the latter theory corresponding to each primitive notion in the former theory, and 2) instead of paraphrasing every primitive notion of the former theory in the latter, to translate the primitive notion by the corresponding notion just introduced and regard the paraphrasing of the primitive notion of the former in the latter as a paraphrasing of the word in terms of already admitted notions exclusively in the latter theory. If every sentence in a language can be translated literally into another language without any change of meaning and nuance, both languages could be called equivalent in the most strict sense of the word.
Load more

Recommended publications
  • Basic Concepts of Set Theory, Functions and Relations 1. Basic
    Ling 310, adapted from UMass Ling 409, Partee lecture notes March 1, 2006 p. 1 Basic Concepts of Set Theory, Functions and Relations 1. Basic Concepts of Set Theory........................................................................................................................1 1.1. Sets and elements ...................................................................................................................................1 1.2. Specification of sets ...............................................................................................................................2 1.3. Identity and cardinality ..........................................................................................................................3 1.4. Subsets ...................................................................................................................................................4 1.5. Power sets .............................................................................................................................................4 1.6. Operations on sets: union, intersection...................................................................................................4 1.7 More operations on sets: difference, complement...................................................................................5 1.8. Set-theoretic equalities ...........................................................................................................................5 Chapter 2. Relations and Functions ..................................................................................................................6
    [Show 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 .
    [Show full text]
  • Definitions and Nondefinability in Geometry 475 2
    Definitions and Nondefinability in Geometry1 James T. Smith Abstract. Around 1900 some noted mathematicians published works developing geometry from its very beginning. They wanted to supplant approaches, based on Euclid’s, which han- dled some basic concepts awkwardly and imprecisely. They would introduce precision re- quired for generalization and application to new, delicate problems in higher mathematics. Their work was controversial: they departed from tradition, criticized standards of rigor, and addressed fundamental questions in philosophy. This paper follows the problem, Which geo- metric concepts are most elementary? It describes a false start, some successful solutions, and an argument that one of those is optimal. It’s about axioms, definitions, and definability, and emphasizes contributions of Mario Pieri (1860–1913) and Alfred Tarski (1901–1983). By fol- lowing this thread of ideas and personalities to the present, the author hopes to kindle interest in a fascinating research area and an exciting era in the history of mathematics. 1. INTRODUCTION. Around 1900 several noted mathematicians published major works on a subject familiar to us from school: developing geometry from the very beginning. They wanted to supplant the established approaches, which were based on Euclid’s, but which handled awkwardly and imprecisely some concepts that Euclid did not treat fully. They would present geometry with the precision required for general- ization and applications to new, delicate problems in higher mathematics—precision beyond the norm for most elementary classes. Work in this area was controversial: these mathematicians departed from tradition, criticized previous standards of rigor, and addressed fundamental questions in logic and philosophy of mathematics.2 After establishing background, this paper tells a story about research into the ques- tion, Which geometric concepts are most elementary? It describes a false start, some successful solutions, and a demonstration that one of those is in a sense optimal.
    [Show full text]
  • Taboo Versus Axiom
    TABOO VERSUS AXIOM KATUZI ONO Some important formal systems are really developable from a finite number of axioms in the lower classical predicate logic LK or in the intuitionistic predicate logic LJ. Any system of this kind can be developed in LK (or in LJ) from the single conjunction of all the axioms of the system. I have been intending to develop usual formal systems starting from TABOOS and standing on the primitive logic LO at first, since the logic too could be brought up to the usual logics by means of TABOOS in LOυ. In developing any formal system from TABOOS, we are forced to assume unwillingly that all the TABOOS are mutually equivalent, as far as we adopt more than one TABOOS for the system. If we could develop formal systems from single- TABOO TABOO-systems, we could get rid of this unwilling assumption. It would be well expected that any formal theory developable from a finite number of axioms in LK or in LJ would be developable from a single TABOO.2) In the present paper, I will prove that this is really the case. Before stating the theorem, let us define the δ-TRANSFORM Spy] of any sentence ® with respect to an h-ary relation £y (λ>0). SΓ&3 is introduced by a structural recursive definition as follows: @C8Qs (r)((©->g(r))->#(r)) for any elementary sentence ©, Received January 20, 1966. *> As for this plan, see Ono [1]. LO is called PRIMITIVE SYSTEM OF POSITIVE LOGIC in Ono [1]. The terminology PRIMITIVE LOGIC together with its reference notation LO has been introduced in Ono [2], 2) In my work [1], I had to assume this to bring up logics by means of TABOO systems.
    [Show full text]
  • Gödel's Unpublished Papers on Foundations of Mathematics
    G¨odel’s unpublished papers on foundations of mathematics W. W. Tait∗ Kurt G¨odel: Collected Works Volume III [G¨odel, 1995] contains a selec- tion from G¨odel’s Nachlass; it consists of texts of lectures, notes for lectures and manuscripts of papers that for one reason or another G¨odel chose not to publish. I will discuss those papers in it that are concerned with the foun- dations/philosophy of mathematics in relation to his well-known published papers on this topic.1 1 Cumulative Type Theory [*1933o]2 is the handwritten text of a lecture G¨odel delivered two years after the publication of his proof of the incompleteness theorems. The problem of giving a foundation for mathematics (i.e. for “the totality of methods actually used by mathematicians”), he said, falls into two parts. The first consists in reducing these proofs to a minimum number of axioms and primitive rules of inference; the second in justifying “in some sense” these axioms and rules. The first part he takes to have been solved “in a perfectly satisfactory way” ∗I thank Charles Parsons for an extensive list of comments on the penultimate version of this paper. Almost all of them have led to what I hope are improvements in the text. 1This paper is in response to an invitation by the editor of this journal to write a critical review of Volume III; but, with his indulgence and the author’s self-indulgence, something else entirely has been produced. 2I will adopt the device of the editors of the Collected Works of distinguishing the unpublished papers from those published by means of a preceding asterisk, as in “ [G¨odel, *1933o]”.
    [Show full text]
  • Formal System Having Just One Primitive Notion
    FORMAL SYSTEM HAVING JUST ONE PRIMITIVE NOTION KATUZI ONO Introduction In this paper, I would like to point out some characteristic features of formal systems having just one primitive notion. Most remarkable systems of this kind may be Zermelo's and FraenkeΓs set-theories, both having just one primitive notion e. In my former work [1], I have proved that J-series logics (intuitionistic predicate logic L J, Johansson's minimal predicate logic LM, and positive predicate logic LP) as well as K-series logics (fortified logics LK, LN, and LQ of LJ, LM, and LP by Peirce's rule that (A->B)->A implies Λy respectively; LK being the lower classical predicate logic) can be faithfully interpreted in the primitive logic LO. However, this does not mean that CONJUNCTION, DISJUNCTION, NEGATION (except for LP and LQ), and EXISTENTIAL QUANTIFICATION of these logics can be defined in LO in such ways that propositions behave with respect to the defined logical constants together with the original logical constants of LO just as propositions of J- and K-series logics concerning their provability. In reality, these logical constants are defined so only for a certain class of propositions in LO. If we restrict ourselves only to formal systems having just one primitive notion, however, we can really define CONJUNCTION, DISJUNCTION, and EXISTENTIAL QUANTIFICATION together with one NEGATION for LJ, a series of NEGATIONS for LM, and naturally no NEGATION for LP in such way that propositions behave with respect to the defined logical constants together with the original logical constants of LO just as propositions of these J-series logics, respectively.
    [Show full text]
  • Neo-Logicism? an Ontological Reduction of Mathematics to Metaphysics
    Edward N. Zalta 2 certain axiomatic, mathematics-free metaphysical theory. This thesis ap- pears to be a version of mathematical platonism, for if correct, it would make a certain simple and intuitive philosophical position about mathe- Neo-Logicism? An Ontological Reduction of matics much more rigorous, namely, that mathematics describes a realm of abstract objects. Nevertheless, there are two ways in which the present Mathematics to Metaphysics∗ view might constitute a kind of neo-logicism. The first is that the compre- hension principle for abstract objects that forms part of the metaphysical theory can be reformulated as a principle that `looks and sounds' like an Edward N. Zaltay analytic, if not logical, truth. Although we shall not argue here that the Center for the Study of Language and Information reformulated principle is analytic, other philosophers have argued that Stanford University principles analogous to it are. The second is that the abstract objects systematized by the metaphysical theory are, in some sense, logical ob- jects. By offering a reduction of mathematical objects to logical objects, the present view may thereby present us with a new kind of logicism. Die nat¨urlichen Zahlen hat der liebe Gott gemacht, der Rest ist Menschenwerk. To establish our thesis, we need two elements, the first of which is Leopold Kronecker already in place. The first element is the axiomatic, metaphysical theory of abstract objects. We shall employ the background ontology described It is now well accepted that logicism is false. The primitive notions by the axiomatic theory of abstract objects developed in Zalta [1983] and proper axioms of mathematical theories are not reducible to primi- and [1988].1 The axioms of this theory can be stated without appealing tive logical notions and logical axioms.
    [Show full text]
  • Bertrand Russell – Principles of Mathematics
    Principles of Mathematics “Unless we are very much mistaken, its lucid application and develop- ment of the great discoveries of Peano and Cantor mark the opening of a new epoch in both philosophical and mathematical thought” – The Spectator Bertrand Russell Principles of Mathematics London and New York First published in 1903 First published in the Routledge Classics in 2010 by Routledge 2 Park Square, Milton Park, Abingdon, Oxon OX14 4RN Routledge is an imprint of the Taylor & Francis Group, an informa business This edition published in the Taylor & Francis e-Library, 2009. To purchase your own copy of this or any of Taylor & Francis or Routledge’s collection of thousands of eBooks please go to www.eBookstore.tandf.co.uk. © 2010 The Bertrand Russell Peace Foundation Ltd Introduction © 1992 John G. Slater All rights reserved. No part of this book may be reprinted or reproduced or utilized in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging in Publication Data A catalog record for this book has been requested ISBN 0-203-86476-X Master e-book ISBN ISBN 10: 0-415-48741-2 ISBN 10: 0-203-86476-X (ebk) ISBN 13: 978-0-415-48741-2 ISBN 13: 978-0-203-86476-0 (ebk) CONTENTS introduction to the 1992 edition xxv introduction to the second edition xxxi preface xliii PART I THE INDEFINABLES OF MATHEMATICS 1 1 Definition of Pure Mathematics 3 1.
    [Show full text]
  • A Set Theory Formalization
    Research Practice Final Report A Set Theory Formalization Alejandro Calle-Saldarriaga acalles@eafit.edu.co Mathematical Engineering, Universidad EAFIT Tutor: Andrés Sicard-Ramírez June 3, 2017 1 Problem definition A little preliminary definition: a set is any kind of mental collection of things. Each of the things in a set is called a member of the set. Set theory is the area of mathematics that study sets, and we can construct almost all known mathematics from it [Enderton, 1977]. Sets in set theory are a primitive notion so they are not defined in terms of previously defined concepts (we just appeal to intuition, akin to points and lines in axiomatic geometry), so the general approach is to describe what is it that one can do with sets via some axioms and prove some properties about sets using those axioms. In this work we will only use pure sets. A set is pure when all of its members are sets, and the members of their members are sets, and so on. Let ? be the set which has no members, called the empty set. Then the sets ?; f?g; f?; f?gg and ff?gg are pure sets. We will restrict our attention to the von Neumann universe [Singh & Singh, 2007] of pure sets, since pure sets gives us many advantages and little generality lost, as most mathematical concepts can be modeled using these sets. You can see that the von Neumann universe in figure (1) has a very particular hierarchy of sets: V0 = ? is the atom, and Vα+1 = PVα, where PVα is the power set of Vα, meaning the set of all subsets of Vα.
    [Show full text]
  • Tarski and Primitivism About Truth
    Philosophers’ volume 13, no. 17 1. Introduction Imprint august 2013 The view that truth is a primitive, indefinable notion was of central importance to the originators of analytic philosophy. Moore and Rus- sell adopted the view after abandoning their idealism (though they soon turned to correspondence accounts), and Frege subscribed to it until the end of his life.1 But save for some attention given to the view by Davidson (1990, 1996), primitivism about truth has laid low for the last century. During that time, by far the dominant force in the theory Tarski and Primitivism of truth has been Tarski, and much subsequent discussion has been focused around the question of whether Tarski’s work better motivates a robust, correspondence-style theory of truth,2 or a more deflationary approach.3 I reject this dichotomy, and argue in this paper that Tarski’s About Truth work on truth is actually most consonant with a primitivist perspec- tive on truth; hence, his views should not be thought to lend support to either correspondence or deflationary theories. Given that Tarski shows how to offer a definition of truth, the congeniality between his views and primitivism may not be immediately obvious, and my aim is to draw the appropriate connections. I do not argue that Tarski himself subscribed to a primitivist conception of truth, though I shall show how the view is open to him, and is more amenable to his views on truth than are the more familiar theories of truth. 1. See, inter alia, Moore 1899, Russell 1904, and Frege 1956.
    [Show full text]
  • The Axiom of Choice and the Class of Hyperarithmetic Functions
    MATHEMATICS THE AXIOM OF CHOICE AND THE CLASS OF HYPER­ ARITHMETIC FUNCTIONS BY G. KREISEL*) (Communicated by Prof. A. HEYTING at the meeting of January 27, 1962) In sections 1 and 2 we give a simple new characterization of the class of hyperarithmetic functions (Theorem 2), and then we use the cor­ responding proof theoretic form to give a convenient formulation of elementary predicative analysis (EPA). The position with regard to predicativity is summarized in section 8. l. Let HYP mean hyperarithmetic, let R and S denote binary primitive recursive relations, a a number theoretic function variable of one argument, a 1 of two arguments, <i(n) (the number of) the sequence (a(O), ... , a(n-1)), <i1(m, ii) the sequence (a1(m, 0), ... , a1(m, n-1)). Theorem 1. ( /\m)( V a)HYP( 1\n)R[<i(n), m] _,.. ( V al)HYP( 1\m)( l\n)R[&1(m, ii), m]. Lemma 1. If P(x, y) is II~ then ( 1\m)( V n)P(m, n) -i>- ( V 'I])HYP( 1\m)P[m, 'l](m)]. Standard proof. Since P(m, n) is IIL there is a primitive recursive R(a, b, c) such that P(m, n) +--+ ( 1\fJ)( Vy)R[p(y), m, n]. For each pair m and n, consider the set O'm,n of sequences of R(a, m, n) which are not past­ secured, i.e. {(ao, ... ,as) : ( 1\r)[r<s -7-, R((ao, ... , ar), m, n)J}. Define n(x) as the n such that ( 1\{J)( V y)R[p(y), x, n ], and, for p < n, ax,p cannot be mapped into ax,n, and no O'x,p can be mapped into a proper section of ax,n (by an order-preserving mapping).
    [Show full text]
  • Bertrand Russell's Logic
    BERTRAND RUSSELL’S LOGIC Andrew D. Irvine Bertrand Russell is generally recognized as one of the most important English- speaking philosophers, logicians and essayists of the twentieth century. Often cited along with G.E. Moore as one of the founders of modern analytic philosophy and along with Kurt G¨odel as one of the most influential logicians of his time, Russell is also widely recognized for his sustained public contributions to many of the most controversial social, political and educational issues of his day. Even so, more than anything else, it is Russell’s work in logic and the foundations of mathematics that serves as his core contribution to intellectual history and that makes Russell the seminal thinker he is. His most significant achievements include 1. his refining and popularizing of Giuseppe Peano’s and Gottlob Frege’s first attempts at developing a modern mathematical logic, 2. his discovery of the paradox that bears his name, 3. his introduction of the theory of types (his way of avoiding the paradox), 4. his defense of logicism, the view that mathematics is in some important sense reducible to logic, and his many detailed derivations supporting this view, 5. his ground-breaking advances in technical philosophy, including both his theory of definite descriptions and his theory of logical constructions, 6. his theory of logical relations, including his impressively general theory of relation arithmetic, 7. his formalization of the reals, 8. his theory of logical atomism, and 9. his championing of the many connections between modern logic, mathemat- ics, science, and knowledge in general.
    [Show full text]