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  1. William M. Farmer (1995). Reasoning About Partial Functions with the Aid of a Computer. Erkenntnis 43 (3):279 - 294.
    Partial functions are ubiquitous in both mathematics and computer science. Therefore, it is imperative that the underlying logical formalism for a general-purpose mechanized mathematics system provide strong support for reasoning about partial functions. Unfortunately, the common logical formalisms — first-order logic, type theory, and set theory — are usually only adequate for reasoning about partial functionsin theory. However, the approach to partial functions traditionally employed by mathematicians is quite adequatein practice. This paper shows how the traditional approach to partial functions (...)
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Algebra
  1. Jeremy Avigad, Notes on a Formalization of the Prime Number Theorem.
    On September 6, 2004, using the Isabelle proof assistant, I verified the following statement: (%x. pi x * ln (real x) / (real x)) ----> 1 The system thereby confirmed that the prime number theorem is a consequence of the axioms of higher-order logic together with an axiom asserting the existence of an infinite set. All told, our number theory session, including the proof of the prime number theorem and supporting libraries, constitutes 673 pages of proof scripts, or roughly 30,000 (...)
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  2. Jeremy Avigad (2006). Methodology and Metaphysics in the Development of Dedekind's Theory of Ideals. In Jose Ferreiros Jeremy Gray (ed.), The architecture of modern mathematics.
    Philosophical concerns rarely force their way into the average mathematician’s workday. But, in extreme circumstances, fundamental questions can arise as to the legitimacy of a certain manner of proceeding, say, as to whether a particular object should be granted ontological status, or whether a certain conclusion is epistemologically warranted. There are then two distinct views as to the role that philosophy should play in such a situation. On the first view, the mathematician is called upon to turn to the counsel (...)
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  3. Michael Beeson (1976). The Unprovability in Intuitionistic Formal Systems of the Continuity of Effective Operations on the Reals. Journal of Symbolic Logic 41 (1):18-24.
  4. Alexander Berenstein (2004). Dividing in the Algebra of Compact Operators. Journal of Symbolic Logic 69 (3):817-829.
    We interpret the algebra of finite rank operators as imaginaries inside a Hilbert space. We prove that the Hilbert space enlarged with these imaginaries has built-in canonical bases.
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  5. Xavier Caicedo & Roberto Cignoli (2001). An Algebraic Approach to Intuitionistic Connectives. Journal of Symbolic Logic 66 (4):1620-1636.
    It is shown that axiomatic extensions of intuitionistic propositional calculus defining univocally new connectives, including those proposed by Gabbay, are strongly complete with respect to valuations in Heyting algebras with additional operations. In all cases, the double negation of such a connective is equivalent to a formula of intuitionistic calculus. Thus, under the excluded third law it collapses to a classical formula, showing that this condition in Gabbay's definition is redundant. Moreover, such connectives can not be interpreted in all Heyting (...)
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  6. Sergio A. Celani & Hernán J. San Martín (2012). Frontal Operators in Weak Heyting Algebras. Studia Logica 100 (1-2):91-114.
    In this paper we shall introduce the variety FWHA of frontal weak Heyting algebras as a generalization of the frontal Heyting algebras introduced by Leo Esakia in [ 10 ]. A frontal operator in a weak Heyting algebra A is an expansive operator τ preserving finite meets which also satisfies the equation $${\tau(a) \leq b \vee (b \rightarrow a)}$$, for all $${a, b \in A}$$. These operators were studied from an algebraic, logical and topological point of view by Leo Esakia (...)
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  7. Roger M. Cooke & Michiel Lambalgen (1983). The Representation of Takeuti's *20c ||_ -Operator. Studia Logica 42 (4):407 - 415.
    Gaisi Takeuti has recently proposed a new operation on orthomodular lattices L, ⫫: $\scr{P}(L)\rightarrow L$ . The properties of ⫫ suggest that the value of ⫫ $(A)(A\subseteq L)$ corresponds to the degree in which the elements of A behave classically. To make this idea precise, we investigate the connection between structural properties of orthomodular lattices L and the existence of two-valued homomorphisms on L.
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  8. Maarten de Rijke & Yde Venema (1995). Sahlqvist's Theorem for Boolean Algebras with Operators with an Application to Cylindric Algebras. Studia Logica 54 (1):61-78.
    For an arbitrary similarity type of Boolean Algebras with Operators we define a class ofSahlqvist identities. Sahlqvist identities have two important properties. First, a Sahlqvist identity is valid in a complex algebra if and only if the underlying relational atom structure satisfies a first-order condition which can be effectively read off from the syntactic form of the identity. Second, and as a consequence of the first property, Sahlqvist identities arecanonical, that is, their validity is preserved under taking canonical embedding algebras. (...)
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  9. Janet Folina (2012). Newton and Hamilton: In Defense of Truth in Algebra. Southern Journal of Philosophy 50 (3):504-527.
    Although it is clear that Sir William Rowan Hamilton supported a Kantian account of algebra, I argue that there is an important sense in which Hamilton's philosophy of mathematics can be situated in the Newtonian tradition. Drawing from both Niccolo Guicciardini's (2009) and Stephen Gaukroger's (2010) readings of the Newton–Leibniz controversy over the calculus, I aim to show that the very epistemic ideals that underpin Newton's argument for the superiority of geometry over algebra also motivate Hamilton's philosophy of algebra. Namely, (...)
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  10. Josep M. Font & Ventura Verdú (1993). The Lattice of Distributive Closure Operators Over an Algebra. Studia Logica 52 (1):1 - 13.
    In our previous paper Algebraic Logic for Classical Conjunction and Disjunction we studied some relations between the fragmentL of classical logic having just conjunction and disjunction and the varietyD of distributive lattices, within the context of Algebraic Logic. The central tool in that study was a class of closure operators which we calleddistributive, and one of its main results was that for any algebraA of type (2,2) there is an isomorphism between the lattices of allD-congruences ofA and of all distributive (...)
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  11. Han Geurdes, The Construction of Transfinite Equivalence Algorithms.
    Context: Consistency of mathematical constructions in numerical analysis and the application of computerized proofs in the light of the occurrence of numerical chaos in simple systems. Purpose: To show that a computer in general and a numerical analysis in particular can add its own peculiarities to the subject under study. Hence the need of thorough theoretical studies on chaos in numerical simulation. Hence, a questioning of what e.g. a numerical disproof of a theorem in physics or a prediction in numerical (...)
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  12. Burghard Herrmann (1997). Characterizing Equivalential and Algebraizable Logics by the Leibniz Operator. Studia Logica 58 (2):305-323.
    In [14] we used the term finitely algebraizable for algebraizable logics in the sense of Blok and Pigozzi [2] and we introduced possibly infinitely algebraizable, for short, p.i.-algebraizable logics. In the present paper, we characterize the hierarchy of protoalgebraic, equivalential, finitely equivalential, p.i.-algebraizable, and finitely algebraizable logics by properties of the Leibniz operator. A Beth-style definability result yields that finitely equivalential and finitely algebraizable as well as equivalential and p.i.-algebraizable logics can be distinguished by injectivity of the Leibniz operator. Thus, (...)
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  13. Jose Ferreiros Jeremy Gray (ed.) (2006). The Architecture of Modern Mathematics.
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  14. Ladislav Kvasz (2006). The History of Algebra and the Development of the Form of its Language. Philosophia Mathematica 14 (3):287-317.
    This paper offers an epistemological reconstruction of the historical development of algebra from al-Khwrizm, Cardano, and Descartes to <span class='Hi'>Euler</span>, Lagrange, and Galois. In the reconstruction it interprets the algebraic formulas as a symbolic language and analyzes the changes of this language in the course of history. It turns out that the most fundamental epistemological changes in the development of algebra can be interpreted as changes of the pictorial form (in the sense of Wittgenstein's Tractatus) of the symbolic language of (...)
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  15. Jean-Pierre Marquis (2013). Mathematical Forms and Forms of Mathematics: Leaving the Shores of Extensional Mathematics. Synthese 190 (12):2141-2164.
    In this paper, I introduce the idea that some important parts of contemporary pure mathematics are moving away from what I call the extensional point of view. More specifically, these fields are based on criteria of identity that are not extensional. After presenting a few cases, I concentrate on homotopy theory where the situation is particularly clear. Moreover, homotopy types are arguably fundamental entities of geometry, thus of a large portion of mathematics, and potentially to all mathematics, at least according (...)
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  16. Jean-pierre Marquis (1997). Abstract Mathematical Tools and Machines for Mathematics. Philosophia Mathematica 5 (3):250-272.
    In this paper, we try to establish that some mathematical theories, like K-theory, homology, cohomology, homotopy theories, spectral sequences, modern Galois theory (in its various applications), representation theory and character theory, etc., should be thought of as (abstract) machines in the same way that there are (concrete) machines in the natural sciences. If this is correct, then many epistemological and ontological issues in the philosophy of mathematics are seen in a different light. We concentrate on one problem which immediately follows (...)
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  17. Robert K. Meyer (2008). Ai, Me and Lewis (Abelian Implication, Material Equivalence and C I Lewis 1920). Journal of Philosophical Logic 37 (2):169 - 181.
    C I Lewis showed up Down Under in 2005, in e-mails initiated by Allen Hazen of Melbourne. Their topic was the system Hazen called FL (a Funny Logic), axiomatized in passing in Lewis 1921. I show that FL is the system MEN of material equivalence with negation. But negation plays no special role in MEN. Symbolizing equivalence with → and defining ∼A inferentially as A→f, the theorems of MEN are just those of the underlying theory ME of pure material equivalence. (...)
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  18. Alejandro Petrovich (1996). Distributive Lattices with an Operator. Studia Logica 56 (1-2):205 - 224.
    It was shown in [3] (see also [5]) that there is a duality between the category of bounded distributive lattices endowed with a join-homomorphism and the category of Priestley spaces endowed with a Priestley relation. In this paper, bounded distributive lattices endowed with a join-homomorphism, are considered as algebras and we characterize the congruences of these algebras in terms of the mentioned duality and certain closed subsets of Priestley spaces. This enable us to characterize the simple and subdirectly irreducible algebras. (...)
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  19. Charles C. Pinter (1975). Algebraic Logic with Generalized Quantifiers. Notre Dame Journal of Formal Logic 16 (4):511-516.
  20. G. Spencer-Brown (1972). Laws of Form. New York,Julian Press.
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  21. Alasdair Urquhart (1979). Distributive Lattices with a Dual Homomorphic Operation. Studia Logica 38 (2):201 - 209.
    The lattices of the title generalize the concept of a De Morgan lattice. A representation in terms of ordered topological spaces is described. This topological duality is applied to describe homomorphisms, congruences, and subdirectly irreducible and free lattices in the category. In addition, certain equational subclasses are described in detail.
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  22. Andrzej Wroński (2004). The Distance Function in Commutative ℓ-Semigroups and the Equivalence in Łukasiewicz Logic. Studia Logica 77 (2):241 - 253.
    The equivalence connective in ukasiewicz logic has its algebraic counterpart which is the distance function d(x,y) =|x–y| of a positive cone of a commutative -group. We make some observations on logically motivated algebraic structures involving the distance function.
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Analysis
  1. J. L. Bell (1994). Introduction. Philosophia Mathematica 2 (1):4-4.
    Continuous as the stars that shine And twinkle on the milky way, They stretched in never-ending line Along the margin of a bay: Ten thousand saw I at a glance, Tossing their heads in sprightly dance.
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  2. John Bell, Chapter.
    Despite the great success of Weierstrass, Dedekind and Cantor in constructing the continuum from arithmetical materials, a number of thinkers of the late 19th and early 20th centuries remained opposed, in varying degrees, to the idea of explicating the continuum concept entirely in discrete terms. These include the mathematicians du Bois-Reymond, Veronese, Poincaré, Brouwer and Weyl, and the philosophers Brentano..
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  3. 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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  4. 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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  5. John P. Burgess (2000). Critical Studies / Book Reviews. Philosophia Mathematica 8 (1):84-91.
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  6. Piotr Błaszczyk, Mikhail G. Katz & David Sherry (2013). Ten Misconceptions From the History of Analysis and Their Debunking. Foundations of Science 18 (1):43-74.
    The widespread idea that infinitesimals were “eliminated” by the “great triumvirate” of Cantor, Dedekind, and Weierstrass is refuted by an uninterrupted chain of work on infinitesimal-enriched number systems. The elimination claim is an oversimplification created by triumvirate followers, who tend to view the history of analysis as a pre-ordained march toward the radiant future of Weierstrassian epsilontics. In the present text, we document distortions of the history of analysis stemming from the triumvirate ideology of ontological minimalism, which identified the continuum (...)
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  7. Martin Cooke, To Continue with Continuity.
    The metaphysical concept of continuity is important, not least because physical continua are not known to be impossible. While it is standard to model them with a mathematical continuum based upon set-theoretical intuitions, this essay considers, as a contribution to the debate about the adequacy of those intuitions, the neglected intuition that dividing the length of a line by the length of an individual point should yield the line’s cardinality. The algebraic properties of that cardinal number are derived pre-theoretically from (...)
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  8. S. S. Demidov (1988). On an Early History of the Moscow School of Theory of Functions. Philosophia Mathematica (1):29-35.
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  9. Fausto di Biase (2009). True or False? A Case in the Study of Harmonic Functions. Topoi 28 (2):143-160.
    Recent mathematical results, obtained by the author, in collaboration with Alexander Stokolos, Olof Svensson, and Tomasz Weiss, in the study of harmonic functions, have prompted the following reflections, intertwined with views on some turning points in the history of mathematics and accompanied by an interpretive key that could perhaps shed some light on other aspects of (the development of) mathematics.
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  10. Bradley H. Dowden (1991). A Linear Continuum of Time. Philosophia Mathematica (1):53-64.
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  11. Jens Erik Fenstad (1985). Is Nonstandard Analysis Relevant for the Philosophy of Mathematics? Synthese 62 (2):289 - 301.
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  12. Fernando Ferreira (2008). A Most Artistic Package of a Jumble of Ideas. Dialectica 62 (2: Table of Contents"/> Select):205–222.
    In the course of ten short sections, we comment on Gödel's seminal dialectica paper of fifty years ago and its aftermath. We start by suggesting that Gödel's use of functionals of finite type is yet another instance of the realistic attitude of Gödel towards mathematics, in tune with his defense of the postulation of ever increasing higher types in foundational studies. We also make some observations concerning Gödel's recasting of intuitionistic arithmetic via the dialectica interpretation, discuss the extra principles that (...)
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  13. Han Geurdes, The Construction of Transfinite Equivalence Algorithms.
    Context: Consistency of mathematical constructions in numerical analysis and the application of computerized proofs in the light of the occurrence of numerical chaos in simple systems. Purpose: To show that a computer in general and a numerical analysis in particular can add its own peculiarities to the subject under study. Hence the need of thorough theoretical studies on chaos in numerical simulation. Hence, a questioning of what e.g. a numerical disproof of a theorem in physics or a prediction in numerical (...)
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  14. Jeremy Gwiazda (2013). Throwing Darts, Time, and the Infinite. Erkenntnis 78 (5):971-975.
    In this paper, I present a puzzle involving special relativity and the random selection of real numbers. In a manner to be specified, darts thrown later hit reals further into a fixed well-ordering than darts thrown earlier. Special relativity is then invoked to create a puzzle. I consider four ways of responding to this puzzle which, I suggest, fail. I then propose a resolution to the puzzle, which relies on the distinction between the potential infinite and the actual infinite. I (...)
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  15. Bob Hale (2002). Real Numbers, Quantities, and Measurement. Philosophia Mathematica 10 (3):304-323.
    Defining the real numbers by abstraction as ratios of quantities gives prominence to then- applications in just the way that Frege thought we should. But if all the reals are to be obtained in this way, it is necessary to presuppose a rich domain of quantities of a land we cannot reasonably assume to be exemplified by any physical or other empirically measurable quantities. In consequence, an explanation of the applications of the reals, defined in this way, must proceed indirectly. (...)
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  16. Bob Hale (2000). Reals by Abstractiont. Philosophia Mathematica 8 (2):100--123.
    On the neo-Fregean approach to the foundations of mathematics, elementary arithmetic is analytic in the sense that the addition of a principle wliich may be held to IMJ explanatory of the concept of cardinal number to a suitable second-order logical basis suffices for the derivation of its basic laws. This principle, now commonly called Hume's principle, is an example of a Fregean abstraction principle. In this paper, I assume the correctness of the neo-Fregean position on elementary aritlunetic and seek to (...)
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  17. Deirdre Haskell (2012). Model Theory of Analytic Functions: Some Historical Comments. Bulletin of Symbolic Logic 18 (3):368-381.
    Model theorists have been studying analytic functions since the late 1970s. Highlights include the seminal work of Denef and van den Dries on the theory of the p-adics with restricted analytic functions, Wilkie's proof of o-minimality of the theory of the reals with the exponential function, and the formulation of Zilber's conjecture for the complex exponential. My goal in this talk is to survey these main developments and to reflect on today's open problems, in particular for theories of valued fields.
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  18. Geoffrey Hellman (1994). Real Analysis Without Classes. Philosophia Mathematica 2 (3):228-250.
    This paper explores strengths and limitations of both predicativism and nominalism, especially in connection with the problem of characterizing the continuum. Although the natural number structure can be recovered predicatively (despite appearances), no predicative system can characterize even the full predicative continuum which the classicist can recognize. It is shown, however, that the classical second-order theory of continua (third-order number theory) can be recovered nominalistically, by synthesizing mereology, plural quantification, and a modal-structured approach with essentially just the assumption that an (...)
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  19. Geoffrey Hellman & Stewart Shapiro (2013). The Classical Continuum Without Points. Review of Symbolic Logic 6 (3):488-512.
    We develop a point-free construction of the classical one-dimensional continuum, with an interval structure based on mereology and either a weak set theory or a logic of plural quantification. In some respects, this realizes ideas going back to Aristotle, although, unlike Aristotle, we make free use of contemporary . Also, in contrast to intuitionistic analysis, smooth infinitesimal analysis, and Eret Bishopgunky lineindecomposabilityCantor structure of ℝ as a complete, separable, ordered field. We also present some simple topological models of our system, (...)
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  20. Friedrich Kaulbach (1967). Philosophisches Und Mathematisches Kontinuum. Philosophia Mathematica (1-2):47-69.
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  21. Michael Kohlhase, A Foundational View on Integration Problems.
    The integration of reasoning and computation services across system and language boundaries has been mostly treated from an engineering perspective. In this paper we take a foundational point of view. We identify the following form of integration problems: an informal (mathematical; i.e, logically underspecified) specification has multiple concrete formal implementations between which queries and results have to be transported. The integration challenge consists in dealing with the implementation-specific details such as additional constants and properties. We pinpoint their role in safe (...)
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  22. Vojtěch Kolman (2010). Continuum, Name and Paradox. Synthese 175 (3):351 - 367.
    The article deals with Cantor's argument for the non-denumerability of reals somewhat in the spirit of Lakatos' logic of mathematical discovery. At the outset Cantor's proof is compared with some other famous proofs such as Dedekind's recursion theorem, showing that rather than usual proofs they are resolutions to do things differently. Based on this I argue that there are "ontologically" safer ways of developing the diagonal argument into a full-fledged theory of continuum, concluding eventually that famous semantic paradoxes based on (...)
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  23. O. B. Lupanov (ed.) (2005). Stochastic Algorithms: Foundations and Applications: Third International Symposium, Saga 2005, Moscow, Russia, October 20-22, 2005: Proceedings. [REVIEW] Springer.
    This book constitutes the refereed proceedings of the Third International Symposium on Stochastic Algorithms: Foundations and Applications, SAGA 2005, held in Moscow, Russia in October 2005. The 14 revised full papers presented together with 5 invited papers were carefully reviewed and selected for inclusion in the book. The contributed papers included in this volume cover both theoretical as well as applied aspects of stochastic computations whith a special focus on new algorithmic ideas involving stochastic decisions and the design and evaluation (...)
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  24. Moshé Machover (1993). The Place of Nonstandard Analysis in Mathematics and in Mathematics Teaching. British Journal for the Philosophy of Science 44 (2):205-212.
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  25. Jean-Pierre Marquis (2006). John L. BELL. The Continuous and the Infinitesimal in Mathematics and Philosophy. Monza: Polimetrica, 2005. Pp. 349. ISBN 88-7699-015-. [REVIEW] Philosophia Mathematica 14 (3):394-400.
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  26. Matthew W. Parker (2003). Three Concepts of Decidability for General Subsets of Uncountable Spaces. Theoretical Computer Science 351 (1):2-13.
    There is no uniquely standard concept of an effectively decidable set of real numbers or real n-tuples. Here we consider three notions: decidability up to measure zero [M.W. Parker, Undecidability in Rn: Riddled basins, the KAM tori, and the stability of the solar system, Phil. Sci. 70(2) (2003) 359–382], which we abbreviate d.m.z.; recursive approximability [or r.a.; K.-I. Ko, Complexity Theory of Real Functions, Birkhäuser, Boston, 1991]; and decidability ignoring boundaries [d.i.b.; W.C. Myrvold, The decision problem for entanglement, in: R.S. (...)
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Category Theory
  1. Jeremy Avigad & Jeffrey Helzner (2002). Transfer Principles in Nonstandard Intuitionistic Arithmetic. Archive for Mathematical Logic 41 (6):581-602.
    Using a slight generalization, due to Palmgren, of sheaf semantics, we present a term-model construction that assigns a model to any first-order intuitionistic theory. A modification of this construction then assigns a nonstandard model to any theory of arithmetic, enabling us to reproduce conservation results of Moerdijk and Palmgren for nonstandard Heyting arithmetic. Internalizing the construction allows us to strengthen these results with additional transfer rules; we then show that even trivial transfer axioms or minor strengthenings of these rules destroy (...)
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