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John Bell [53]John L. Bell [40]John F. Bell [6]John S. Bell [3]
John Fred Bell [1]
  1.  40
    John L. Bell, Logical Reflections On the Kochen-Specker Theorem.
    IN THEIR WELL-KNOWN PAPER, Kochen and Specker (1967) introduce the concept of partial Boolean algebra (pBa) and show that certain (finitely generated) partial Boolean algebras arising in quantum theory fail to possess morphisms to any Boolean algebra (we call such pBa's intractable in the sequel). In this note we begin by discussing partial..
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  2.  3
    L. Jonathan Cohen, Jean Nicod, John Bell & Michael Woods (1971). Geometry and Induction. Philosophical Quarterly 21 (85):376.
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  3.  36
    John L. Bell, Continuity and Infinitesimals.
    The usual meaning of the word continuous is “unbroken” or “uninterrupted”: thus a continuous entity —a continuum—has no “gaps.” We commonly suppose that space and time are continuous, and certain philosophers have maintained that all natural processes occur continuously: witness, for example, Leibniz's famous apothegm natura non facit saltus—“nature makes no jump.” In mathematics the word is used in the same general sense, but has had to be furnished with increasingly precise definitions. So, for instance, in the later 18th century (...)
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  4. John L. Bell (1999). Frege's Theorem in a Constructive Setting. Journal of Symbolic Logic 64 (2):486-488.
    then E has a subset which is the domain of a model of Peano's axioms for the natural numbers. (This result is proved explicitly, using classical reasoning, in section 3 of [1].) My purpose in this note is to strengthen this result in two directions: first, the premise will be weakened so as to require only that the map ν be defined on the family of (Kuratowski) finite subsets of the set E, and secondly, the argument will be constructive, i.e., (...)
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  5.  13
    John L. Bell, Hermann Weyl. Revue Philosophique de la France Et de l'Etranger.
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  6. John L. Bell (2007). The Continuous and the Infinitesimal in Mathematics and Philosophy. Bulletin of Symbolic Logic 13 (3):361-363.
     
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  7. John Bell & Michael Hallett (1982). Logic, Quantum Logic and Empiricism. Philosophy of Science 49 (3):355-379.
    This paper treats some of the issues raised by Putnam's discussion of, and claims for, quantum logic, specifically: that its proposal is a response to experimental difficulties; that it is a reasonable replacement for classical logic because its connectives retain their classical meanings, and because it can be derived as a logic of tests. We argue that the first claim is wrong (1), and that while conjunction and disjunction can be considered to retain their classical meanings, negation crucially does not. (...)
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  8.  95
    John L. Bell (2001). Observations on Category Theory. Axiomathes 12 (1-2):151-155.
    is a presentation of mathematics in terms of the fundamental concepts of transformation, and composition of transformations. While the importance of these concepts had long been recognized in algebra (for example, by Galois through the idea of a group of permutations) and in geometry (for example, by Klein in his Erlanger Programm), the truly universal role they play in mathematics did not really begin to be appreciated until the rise of abstract algebra in the 1930s. In abstract algebra the idea (...)
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  9.  3
    John S. Bell, J. Clauser, M. Horne & A. Shimony (1985). An Exchange on Local Beables. Dialectica 39 (2):85-96.
    Summarya) Bell tries to formulate more explicitly a notion of “local causality”: correlations between physical events in different space‐time regions should be explicable in terms of physical events in the overlap of the backward light cones. It is shown that ordinary relativistic quantum field theory is not locally causal in this sense, and cannot be embedded in a locally causal theory.b) Clauser, Home and Shimony criticize several steps in Bell's argument that any theory of local “beables” is (...)
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  10.  62
    John L. Bell (2004). Whole and Part in Mathematics. Axiomathes 14 (4):285-294.
    The centrality of the whole/part relation in mathematics is demonstrated through the presentation and analysis of examples from algebra, geometry, functional analysis,logic, topology and category theory.
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  11.  56
    John L. Bell, The Axiom of Choice. Stanford Encyclopedia of Philosophy.
    The principle of set theory known as the Axiom of Choice has been hailed as “probably the most interesting and, in spite of its late appearance, the most discussed axiom of mathematics, second only to Euclid's axiom of parallels which was introduced more than two thousand years ago” (Fraenkel, Bar-Hillel & Levy 1973, §II.4). The fulsomeness of this description might lead those unfamiliar with the axiom to expect it to be as startling as, say, the Principle of the Constancy of (...)
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  12.  44
    John L. Bell, Infinitary Logic. Stanford Encyclopedia of Philosophy.
    Traditionally, expressions in formal systems have been regarded as signifying finite inscriptions which are—at least in principle—capable of actually being written out in primitive notation. However, the fact that (first-order) formulas may be identified with natural numbers (via "Gödel numbering") and hence with finite sets makes it no longer necessary to regard formulas as inscriptions, and suggests the possibility of fashioning "languages" some of whose formulas would be naturally identified as infinite sets . A "language" of this kind is called (...)
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  13. John L. Bell (2011). Set Theory: Boolean-Valued Models and Independence Proofs. OUP Oxford.
    This third edition, now available in paperback, is a follow up to the author's classic Boolean-Valued Models and Independence Proofs in Set Theory. It provides an exposition of some of the most important results in set theory obtained in the 20th century: the independence of the continuum hypothesis and the axiom of choice.
     
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  14. John S. Bell (1976). How to Teach Special Relativity. Progress in Scientific Culture 1.
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  15.  14
    John L. Bell (2015). Reflections on Mathematics and Aesthetics. Aisthesis. Pratiche, Linguaggi E Saperi Dell’Estetico 8 (1):159-179.
    In this paper I reflect on the nature of mathematical beauty, and examine the connections between mathematics and the arts. I employ Plutarch’s distinction between the intelligible and the sensible, to compare the beauty of mathematics with the beauties of music, poetry and painting. While the beauty of mathematics is almost exclusively intelligible, and the beauties of these arts primarily sensible, it is pointed out that the latter share with mathematics a certain kind of intelligible beauty. The paper also contains (...)
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  16.  76
    John Bell, Hermann Weyl's Later Philosophical Views: His Divergence From Husserl.
    In what seems to have been his last paper, Insight and Reflection (1954), Hermann Weyl provides an illuminating sketch of his intellectual development, and describes the principal influences—scientific and philosophical—exerted on him in the course of his career as a mathematician. Of the latter the most important in the earlier stages was Husserl’s phenomenology. In Weyl’s work of 1918-22 we find much evidence of the great influence Husserl’s ideas had on Weyl’s philosophical outlook—one need merely glance through the pages of (...)
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  17. John Bell, Time and Causation in Gödel's Universe.
    In 1949 the great logician Kurt Gödel constructed the first mathematical models of the universe in which travel into the past is, in theory at least, possible. Within the framework of Einstein’s general theory of relativity Gödel produced cosmological solutions to Einstein’s field equations which contain closed time-like curves, that is, curves in spacetime which, despite being closed, still represent possible paths of bodies. An object moving along such a path would travel back into its own past, to the very (...)
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  18.  22
    John L. Bell, The Development of Categorical Logic.
    5.5. Every topos is linguistic: the equivalence theorem.
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  19.  89
    John L. Bell (2000). Hermann Weyl on Intuition and the Continuum. Philosophia Mathematica 8 (3):259-273.
    Hermann Weyl, one of the twentieth century's greatest mathematicians, was unusual in possessing acute literary and philosophical sensibilities—sensibilities to which he gave full expression in his writings. In this paper I use quotations from these writings to provide a sketch of Weyl's philosophical orientation, following which I attempt to elucidate his views on the mathematical continuum, bringing out the central role he assigned to intuition.
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  20.  6
    John L. Bell (1993). Hilbert's Ε‐Operator in Intuitionistic Type Theories. Mathematical Logic Quarterly 39 (1):323-337.
    We investigate Hilbert's ϵ-calculus in the context of intuitionistic type theories, that is, within certain systems of intuitionistic higher-order logic. We determine the additional deductive strength conferred on an intuitionistic type theory by the adjunction of closed ϵ-terms. We extend the usual topos semantics for type theories to the ϵ-operator and prove a completeness theorem. The paper also contains a discussion of the concept of “partially defined” ϵ-term. MSC: 03B15, 03B20, 03G30.
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  21.  98
    John Bell, The Axiom of Choice in the Foundations of Mathematics.
    The principle of set theory known as the Axiom of Choice (AC) has been hailed as “probably the most interesting and, in spite of its late appearance, the most discussed axiom of mathematics, second only to Euclid’s axiom of parallels which was introduced more than two thousand years ago”1 It has been employed in countless mathematical papers, a number of monographs have been exclusively devoted to it, and it has long played a prominently role in discussions on the foundations of (...)
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  22.  17
    John Bell, Cover Schemes, Frame-Valued Sets and Their Potential Uses in Spacetime Physics.
    In the present paper the concept of a covering is presented and developed. The relationship between cover schemes, frames (complete Heyting algebras), Kripke models, and frame-valued set theory is discussed. Finally cover schemes and framevalued set theory are applied in the context of Markopoulou’s account of discrete spacetime as sets “evolving” over a causal set. We observe that Markopoulou’s proposal may be effectively realized by working within an appropriate frame-valued model of set theory. We go on to show that, within (...)
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  23.  77
    John Bell (2008). Incompleteness in a General Setting (Vol 13, Pg 21, 2007). Bulletin of Symbolic Logic 14 (1):21 - 30.
    Full proofs of the Gödel incompleteness theorems are highly intricate affairs. Much of the intricacy lies in the details of setting up and checking the properties of a coding system representing the syntax of an object language (typically, that of arithmetic) within that same language. These details are seldom illuminating and tend to obscure the core of the argument. For this reason a number of efforts have been made to present the essentials of the proofs of Gödel’s theorems without getting (...)
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  24.  1
    John L. Bell (1994). Fregean Extensions of First‐Order Theories. Mathematical Logic Quarterly 40 (1):27-30.
    It is shown by Parsons [2] that the first-order fragment of Frege's logical system in the Grundgesetze der Arithmetic is consistent. In this note we formulate and prove a stronger version of this result for arbitrary first-order theories. We also show that a natural attempt to further strengthen our result runs afoul of Tarski's theorem on the undefinability of truth.
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  25.  33
    John L. Bell & William Demopoulos (1996). Elementary Propositions and Independence. Notre Dame Journal of Formal Logic 37 (1):112-124.
    This paper is concerned with Wittgenstein's early doctrine of the independence of elementary propositions. Using the notion of a free generator for a logical calculus–a concept we claim was anticipated by Wittgenstein–we show precisely why certain difficulties associated with his doctrine cannot be overcome. We then show that Russell's version of logical atomism–with independent particulars instead of elementary propositions–avoids the same difficulties.
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  26.  66
    John Bell (2008). The Axiom of Choice and the Law of Excluded Middle in Weak Set Theories. Mathematical Logic Quarterly 54 (2):194-201.
    A weak form of intuitionistic set theory WST lacking the axiom of extensionality is introduced. While WST is too weak to support the derivation of the law of excluded middle from the axiom of choice, we show that bee.ng up WST with moderate extensionality principles or quotient sets enables the derivation to go through.
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  27.  35
    John L. Bell (2000). Continuity and the Logic of Perception. Transcendent Philosophy 1 (2):1-7.
    If we imagine a chess-board with alternate blue and red squares, then this is something in which the individual red and blue areas allow themselves to be distinguished from each other in juxtaposition, and something similar holds also if we imagine each of the squares divided into four smaller squares also alternating between these two colours. If, however, we were to continue with such divisions until we had exceeded the boundary of noticeability for the individual small squares which result, then (...)
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  28.  68
    John Bell, The Philosophy of Mathematics.
    THE CLOSE CONNECTION BETWEEN mathematics and philosophy has long been recognized by practitioners of both disciplines. The apparent timelessness of mathematical truth, the exactness and objective nature of its concepts, its applicability to the phenomena of the empirical world—explicating such facts presents philosophy with some of its subtlest problems. We shall discuss some of the attempts made by philosophers and mathematicians to explain the nature of mathematics. We begin with a brief presentation of the views of four major classical philosophers: (...)
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  29.  54
    Molly Kao, Nicolas Fillion & John Bell (2010). J Ean -P Ierre M Arquis . From a Geometrical Point of View: A Study of the History and Philosophy of Category Theory. Philosophia Mathematica 18 (2):227-234.
    (No abstract is available for this citation).
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  30.  54
    John L. Bell (2000). Sets and Classes as Many. Journal of Philosophical Logic 29 (6):585-601.
    In this paper the view is developed that classes should not be understood as individuals, but, rather, as "classes as many" of individuals. To correlate classes with individuals "labelling" and "colabelling" functions are introduced and sets identified with a certain subdomain of the classes on which the labelling and colabelling functions are mutually inverse. A minimal axiomatization of the resulting system is formulated and some of its extensions are related to various systems of set theory, including nonwellfounded set theories.
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  31.  10
    John S. Bell (1973). Subject and Object. In Jagdish Mehra (ed.), The Physicist's Conception of Nature. Boston,Reidel 687--690.
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  32.  21
    John L. Bell (1999). Finite Sets and Frege Structures. Journal of Symbolic Logic 64 (4):1552-1556.
    Call a family F of subsets of a set E inductive if ∅ ∈ F and F is closed under unions with disjoint singletons, that is, if ∀X∈F ∀x∈E–X(X ∪ {x} ∈ F]. A Frege structure is a pair (E.
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  33.  50
    John Bell (2003). Some New Intuitionistic Equivalents of Zorn's Lemma. Archive for Mathematical Logic 42 (8):811-814.
    Two new intuitionistic equivalents to Zarn’s Lemma are stated and proved.
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  34.  37
    John Bell (1979). The Infinite Past Regained: A Reply to Whitrow. British Journal for the Philosophy of Science 30 (2):161-165.
    I show the inadequacy of whitrow's recent argument ("british journal for the philosophy of science", Volume 29, Pages 39-45) against the possibility of an infinite past. I argue that it is impossible to prove "a priori" the non-Existence of an infinite past or future.
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  35.  6
    John F. Bell (2001). Patterns of Subject Uptake and Examination Entry 1984–1997. Educational Studies 27 (2):201-219.
    In 1984, the APU science survey collected information on the courses followed by Year 11 pupils. In this paper, the APU survey will be compared with recent GCSE examination level data and will describe the impact of the National Curriculum on the sexes and on pupils of differing ability. In 1984, there were considerable differences in uptake by the sexes and by ability. In 1997, pupils were taking more examinations than were pupils in 1984. Also, girls were taking more GCSEs (...)
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  36.  53
    John Bell, Lectures on the Foundations of Mathematics.
    THE CLOSE CONNECTION BETWEEN mathematics and philosophy has long been recognized by practitioners of both disciplines. The apparent timelessness of mathematical truth, the exactness and objective nature of its concepts, its applicability to the phenomena of the empirical world— explicating such facts presents philosophy with some of its subtlest problems. Let me begin by reminding you of some celebrated past attempts made by philosophers and mathematicians to explicate the nature of mathematics.
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  37. John L. Bell, Algorithmicity and Consciousness.
    Why should one believe that conscious awareness is solely the result of organizational complexity? What is the connection between consciousness and combinatorics: transformation of quantity into quality? The claim that the former is reducible to the other seems unconvincing—as unlike as chalk and cheese! In his book1 Penrose is at least attempting to compare like with like: the enigma of consciousness with the progress of physics.
     
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  38.  25
    John L. Bell, Types, Sets and Categories.
    This essay is an attempt to sketch the evolution of type theory from its beginnings early in the last century to the present day. Central to the development of the type concept has been its close relationship with set theory to begin with and later its even more intimate relationship with category theory. Since it is effectively impossible to describe these relationships (especially in regard to the latter) with any pretensions to completeness within the space of a comparatively short article, (...)
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  39.  2
    John F. Bell (1989). A Comparison of Science Performance and Uptake by Fifteen‐Year‐Old Boys and Girls in Co‐Educational and Single‐Sex Schools—APU Survey Findings. Educational Studies 15 (2):193-203.
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  40.  42
    John Bell, The Continuum in Smooth Infinitesimal Analysis.
    The relation ≤ on R is defined by a ≤ b ⇔ ¬b < a. The open interval (a, b) and closed interval [a, b] are defined as usual, viz. (a, b) = {x: a < x < b} and [a, b] = {x: a ≤ x ≤ b}; similarly for half-open, half-closed, and unbounded intervals.
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  41.  42
    John L. Bell, Notes on Toposes and Local Set Theories.
    This book is written for those who are in sympathy with its spirit. This spirit is different from the one which informs the vast stream of European and American civilization in which all of us stand. That spirit expresses itself in an onwards movement, in building ever larger and more complicated structures; the other in striving in clarity and perspicuity in no matter what structure. The first tries to grasp the world by way of its periphery—in its variety; the second (...)
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  42.  42
    John L. Bell (2005). Oppositions and Paradoxes in Mathematics and Philosophy. Axiomathes 15 (2):165-180.
    In this paper a number of oppositions which have haunted mathematics and philosophy are described and analyzed. These include the Continuous and the Discrete, the One and the Many, the Finite and the Infinite, the Whole and the Part, and the Constant and the Variable.
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  43.  41
    John Bell, Infinitesimals and the Continuum.
    The opposed concepts of continuity and discreteness have figured prominently in the development of mathematics, and have also commanded the attention of philosophers. Continuous entities may be characterized by the fact that they can be divided indefinitely without altering their essential nature. So, for instance, the water in a bucket may be indefinitely halved and yet remain water. (For the purposes of illustration I ignore the atomic nature of matter which has been established by modern physics.) Discrete entities, on the (...)
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  44.  37
    John L. Bell (2005). Divergent Conceptions of the Continuum in 19th and Early 20th Century Mathematics and Philosophy. Axiomathes 15 (1):63-84.
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  45.  21
    John L. Bell, Cohesiveness.
    ABSTRACT: It is characteristic of a continuum that it be “all of one piece”, in the sense of being inseparable into two (or more) disjoint nonempty parts. By taking “part” to mean open (or closed) subset of the space, one obtains the usual topological concept of connectedness . Thus a space S is defined to be connected if it cannot be partitioned into two disjoint nonempty open (or closed) subsets – or equivalently, given any partition of S into two open (...)
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  46.  20
    John Bell (2004). The Perspective of the Ordinary Citizen on Law. Res Publica 10 (3):311-317.
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  47.  37
    John Bell (2007). Contribution to “Philosophy of Mathematics: 5 Questions”. In V. F. Hendricks & Hannes Leitgeb (eds.), Philosophy of Mathematics: Five Questions. Automatic Press/VIP
    V. Hendricks and H. Leitgeb, eds., Automatic Press, 2007.
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  48.  35
    John Bell, An Invitation to Smooth Infinitesimal Analysis.
    where f ′ (x) is the derivative of f(x) and A is a quantity whose value depends on both x and δx. Now if it were possible to take δx so small (but not demonstrably identical with 0) that (δx)2 = 0 then (1) would assume the simple form..
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  49.  35
    John Bell, Cosmological Theories and the Question of the Existence of a Creator.
    In a Vedic hymn, Reality or Being is proclaimed as having “arisen from Nothing”. By contrast, in Jaina cosmology time has no beginning; the universe, uncreated, has always existed.In Plato’s Timaeus the universe is conceived as not having existed eternally, but as having been created at some past time by a demiurge acting on pre-existing substance. We are all familiar with the arresting first line of Genesis.
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  50. John Bell, Basic Model Theory.
    A structure is a triple A = (A, {Ri: i ∈ I}, {ej: j ∈ J}), where A, the domain or universe of A, is a nonempty set, {Ri: i ∈ I} is an indexed family of relations on A and {ej: j ∈ J}) is an indexed set of elements —the designated elements of A. For each i ∈ I there is then a natural number λ(i) —the degree of Ri —such that Ri is a λ(i)-place relation on A, (...)
     
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