Search results for 'Induction (Mathematics' (try it on Scholar)

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  1.  3
    Thomas Glaß, Michael Rathjen & Andreas Schlüter (1997). On the Proof-Theoretic Strength of Monotone Induction in Explicit Mathematics. Annals of Pure and Applied Logic 85 (1):1-46.
    We characterize the proof-theoretic strength of systems of explicit mathematics with a general principle asserting the existence of least fixed points for monotone inductive definitions, in terms of certain systems of analysis and set theory. In the case of analysis, these are systems which contain the Σ12-axiom of choice and Π12-comprehension for formulas without set parameters. In the case of set theory, these are systems containing the Kripke-Platek axioms for a recursively inaccessible universe together with the existence of a stable (...)
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  2.  10
    Jeffry L. Hirst (1999). Ordinal Inequalities, Transfinite Induction, and Reverse Mathematics. Journal of Symbolic Logic 64 (2):769-774.
    If α and β are ordinals, α ≤ β, and $\beta \nleq \alpha$ , then α + 1 ≤ β. The first result of this paper shows that the restriction of this statement to countable well orderings is provably equivalent to ACA 0 , a subsystem of second order arithmetic introduced by Friedman. The proof of the equivalence is reminiscent of Dekker's construction of a hypersimple set. An application of the theorem yields the equivalence of the set comprehension scheme ACA (...)
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  3. Alan Baker (2007). Is There a Problem of Induction for Mathematics? In M. Potter (ed.), Mathematical Knowledge. Oxford University Press 57-71.
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  4.  5
    Alan L. T. Paterson (2000). The Successor Function and Induction Principle in a Hegelian Philosophy of Mathematics. Idealistic Studies 30 (1):25-60.
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  5.  15
    Alan L. T. Paterson (2000). The Successor Function and Induction Principle in a Hegelian Philosophy of Mathematics. Idealistic Studies 30 (1):25-60.
  6.  1
    Thomas Glass, Michael Rathjen & Andreas Schlüter (1997). On the Proof-Theoretic Strength of Monotone Induction in Explicit Mathematics. Annals of Pure and Applied Logic 85 (1):1-46.
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  7.  20
    Cassiano Terra Rodrigues (2011). The Method of Scientific Discovery in Peirce's Philosophy: Deduction, Induction, and Abduction. [REVIEW] Logica Universalis 5 (1):127-164.
    In this paper we will show Peirce’s distinction between deduction, induction and abduction. The aim of the paper is to show how Peirce changed his views on the subject, from an understanding of deduction, induction and hypotheses as types of reasoning to understanding them as stages of inquiry very tightly connected. In order to get a better understanding of Peirce’s originality on this, we show Peirce’s distinctions between qualitative and quantitative induction and between theorematical and corollarial deduction, (...)
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  8. Itay Neeman (2011). Necessary Use of [Image] Induction in a Reversal. Journal of Symbolic Logic 76 (2):561 - 574.
    Jullien's indecomposability theorem (INDEC) states that if a scattered countable linear order is indecomposable, then it is either indecomposable to the left, or indecomposable to the right. The theorem was shown by Montalbán to be a theorem of hyperarithmetic analysis, and then, in the base system RCA₀ plus ${\mathrm{\Sigma }}_{1}^{1}\text{\hspace{0.17em}}$ induction, it was shown by Neeman to have strength strictly between weak ${\mathrm{\Sigma }}_{1}^{1}$ choice and ${\mathrm{\Delta }}_{1}^{1}$ comprehension. We prove in this paper that ${\mathrm{\Sigma }}_{1}^{1}$ induction is (...)
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  9.  1
    Yiannis N. Moschovakis (1974/2008). Elementary Induction on Abstract Structures. Dover Publications.
    Hailed by the Bulletin of the American Mathematical Society as "easy to use and a pleasure to read," this research monograph is recommended for students and professionals interested in model theory and definability theory. The sole prerequisite is a familiarity with the basics of logic, model theory, and set theory. 1974 edition.
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  10. David S. Gunderson (2010). Handbook of Mathematical Induction: Theory and Applications. Chapman & Hall/Crc.
  11. Ryszard Stanisław Michalski (1977). Toward Computer-Aided Induction: A Brief Review of Currently Implemented Aqval Programs. Dept. Of Computer Science, University of Illinois at Urbana-Champaign.
  12. I. S. Sominskiĭ (1963). The Method of Mathematical Induction. Boston, Heath.
     
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  13.  67
    Stewart Shapiro (1997). Philosophy of Mathematics: Structure and Ontology. Oxford University Press.
    Do numbers, sets, and so forth, exist? What do mathematical statements mean? Are they literally true or false, or do they lack truth values altogether? Addressing questions that have attracted lively debate in recent years, Stewart Shapiro contends that standard realist and antirealist accounts of mathematics are both problematic. As Benacerraf first noted, we are confronted with the following powerful dilemma. The desired continuity between mathematical and, say, scientific language suggests realism, but realism in this context suggests seemingly intractable epistemic (...)
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  14.  41
    Alexander Paseau (2011). Mathematical Instrumentalism, Gödel's Theorem, and Inductive Evidence. Studies in History and Philosophy of Science Part A 42 (1):140-149.
    Mathematical instrumentalism construes some parts of mathematics, typically the abstract ones, as an instrument for establishing statements in other parts of mathematics, typically the elementary ones. Gödel’s second incompleteness theorem seems to show that one cannot prove the consistency of all of mathematics from within elementary mathematics. It is therefore generally thought to defeat instrumentalisms that insist on a proof of the consistency of abstract mathematics from within the elementary portion. This article argues that though some versions of mathematical instrumentalism (...)
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  15.  62
    Miguel Hoeltje, Benjamin Schnieder & Alex Steinberg (2013). Explanation by Induction? Synthese 190 (3):509-524.
    Philosophers of mathematics commonly distinguish between explanatory and non-explanatory proofs. An important subclass of mathematical proofs are proofs by induction. Are they explanatory? This paper addresses the question, based on general principles about explanation. First, a recent argument for a negative answer is discussed and rebutted. Second, a case is made for a qualified positive take on the issue.
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  16.  5
    Michael Rathjen (2006). A Note on Bar Induction in Constructive Set Theory. Mathematical Logic Quarterly 52 (3):253-258.
    Bar Induction occupies a central place in Brouwerian mathematics. This note is concerned with the strength of Bar Induction on the basis of Constructive Zermelo-Fraenkel Set Theory, CZF. It is shown that CZF augmented by decidable Bar Induction proves the 1-consistency of CZF. This answers a question of P. Aczel who used Bar Induction to give a proof of the Lusin Separation Theorem in the constructive set theory CZF.
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  17.  2
    Mohammad Ardeshir & Rasoul Ramezanian (2009). Decidability and Specker Sequences in Intuitionistic Mathematics. Mathematical Logic Quarterly 55 (6):637-648.
    A bounded monotone sequence of reals without a limit is called a Specker sequence. In Russian constructive analysis, Church's Thesis permits the existence of a Specker sequence. In intuitionistic mathematics, Brouwer's Continuity Principle implies it is false that every bounded monotone sequence of real numbers has a limit. We claim that the existence of Specker sequences crucially depends on the properties of intuitionistic decidable sets. We propose a schema about intuitionistic decidability that asserts “there exists an intuitionistic enumerable set that (...)
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  18. Abraham Adolf Fraenkel & Yehoshua Bar-Hillel (eds.) (1966). Essays on the Foundations of Mathematics. Jerusalem, Magnes Press Hebrew University.
    Bibliography of A. A. Fraenkel (p. ix-x)--Axiomatic set theory. Zur Frage der Unendlichkeitsschemata in der axiomatischen Mengenlehre, von P. Bernays.--On some problems involving inaccessible cardinals, by P. Erdös and A. Tarski.--Comparing the axioms of local and universal choice, by A. Lévy.--Frankel's addition to the axioms of Zermelo, by R. Mantague.--More on the axiom of extensionality, by D. Scott.--The problem of predicativity, by J. R. Shoenfield.--Mathematical logic. Grundgedanken einer typenfreien Logik, von W. Ackermann.--On the use of Hilbert's [epsilon]-operator in scientific theories, (...)
     
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  19. David Harriman (2010). The Logical Leap: Induction in Physics. New American Library.
    The nature of concepts -- Generalizations as hierarchical -- Perceiving first-level causal connections -- Conceptualizing first-level causal connections -- The structure of inductive reasoning -- Galileo's kinematics -- Newton's optics -- The methods of difference and agreement -- Induction as inherent in conceptualization -- The birth of celestial physics -- Mathematics and causality -- The power of mathematics -- Proof of Kepler's theory -- The development of dynamics -- The discovery of universal gravitation -- Discovery is proof -- Chemical (...)
     
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  20.  39
    Friedrich Waismann (1951/2003). Introduction to Mathematical Thinking: The Formation of Concepts in Modern Mathematics. Dover Publications.
    "With exceptional clarity, but with no evasion of essential ideas, the author outlines the fundamental structure of mathematics."--Carl B. Boyer, Brooklyn College. This enlightening survey of mathematical concept formation holds a natural appeal to philosophically minded readers, and no formal training in mathematics is necessary to appreciate its clear exposition. Contents include examinations of arithmetic and geometry; the rigorous construction of the theory of integers; the rational numbers and their foundation in arithmetic; and the rigorous construction of elementary arithmetic. Advanced (...)
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  21.  5
    James Franklin (2013). Non-Deductive Logic in Mathematics: The Probability of Conjectures. In Andrew Aberdein & Ian J. Dove (eds.), The Argument of Mathematics. Springer 11--29.
    Mathematicians often speak of conjectures, yet unproved, as probable or well-confirmed by evidence. The Riemann Hypothesis, for example, is widely believed to be almost certainly true. There seems no initial reason to distinguish such probability from the same notion in empirical science. Yet it is hard to see how there could be probabilistic relations between the necessary truths of pure mathematics. The existence of such logical relations, short of certainty, is defended using the theory of logical probability (or objective Bayesianism (...)
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  22. Ronald Christensen (1964/1980). Foundations of Inductive Reasoning. Entropy,Ltd.].
     
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  23. Harvey Friedman, The Inevitability of Logical Strength: Strict Reverse Mathematics.
    An extreme kind of logic skeptic claims that "the present formal systems used for the foundations of mathematics are artificially strong, thereby causing unnecessary headaches such as the Gödel incompleteness phenomena". The skeptic continues by claiming that "logician's systems always contain overly general assertions, and/or assertions about overly general notions, that are not used in any significant way in normal mathematics. For example, induction for all statements, or even all statements of certain restricted forms, is far too general - (...)
     
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  24.  38
    James Ladyman & Stuart Presnell (2015). Identity in Homotopy Type Theory, Part I: The Justification of Path Induction. Philosophia Mathematica 23 (3):386-406.
    Homotopy Type Theory is a proposed new language and foundation for mathematics, combining algebraic topology with logic. An important rule for the treatment of identity in HoTT is path induction, which is commonly explained by appeal to the homotopy interpretation of the theory's types, tokens, and identities as spaces, points, and paths. However, if HoTT is to be an autonomous foundation then such an interpretation cannot play a fundamental role. In this paper we give a derivation of path (...), motivated from pre-mathematical considerations, without recourse to homotopy theory. (shrink)
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  25.  1
    Solomon Feferman & Gerhard Jäger (1996). Systems of Explicit Mathematics with Non-Constructive Μ-Operator. Part II. Annals of Pure and Applied Logic 79 (1):37-52.
    This paper is mainly concerned with proof-theoretic analysis of some second-order systems of explicit mathematics with a non-constructive minimum operator. By introducing axioms for variable types we extend our first-order theory BON to the elementary explicit type theory EET and add several forms of induction as well as axioms for μ. The principal results then state: EET plus set induction is proof-theoretically equivalent to Peano arithmetic PA <0).
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  26.  22
    G. von Wright (1951). A Treatise on Induction and Probability. Routledge and Kegan Paul.
    ... and Induction Nicod V The Foundations of Mathematics Braithwaite VI Logical Studies von Wright VII A Treatise on Induction and Probability von Wright..
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  27.  1
    Solomon Feferman & Gerhard Jäger (1993). Systems of Explicit Mathematics with Non-Constructive Μ-Operator. Part I. Annals of Pure and Applied Logic 65 (3):243-263.
    Feferman, S. and G. Jäger, Systems of explicit mathematics with non-constructive μ-operator. Part I, Annals of Pure and Applied Logic 65 243-263. This paper is mainly concerned with the proof-theoretic analysis of systems of explicit mathematics with a non-constructive minimum operator. We start off from a basic theory BON of operators and numbers and add some principles of set and formula induction on the natural numbers as well as axioms for μ. The principal results then state: BON plus set (...)
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  28. Harvey M. Friedman, Strict Reverse Mathematics.
    An extreme kind of logic skeptic claims that "the present formal systems used for the foundations of mathematics are artificially strong, thereby causing unnecessary headaches such as the Gödel incompleteness phenomena". The skeptic continues by claiming that "logician's systems always contain overly general assertions, and/or assertions about overly general notions, that are not used in any significant way in normal mathematics. For example, induction for all statements, or even all statements of certain restricted forms, is far too general - (...)
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  29.  2
    Gerhard Jäger, Reinhard Kahle & Thomas Studer (2001). Universes in Explicit Mathematics. Annals of Pure and Applied Logic 109 (3):141-162.
    This paper deals with universes in explicit mathematics. After introducing some basic definitions, the limit axiom and possible ordering principles for universes are discussed. Later, we turn to least universes, strictness and name induction. Special emphasis is put on theories for explicit mathematics with universes which are proof-theoretically equivalent to Feferman's.
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  30. Stewart Shapiro (2000). Philosophy of Mathematics: Structure and Ontology. Oxford University Press Usa.
    Moving beyond both realist and anti-realist accounts of mathematics, Shapiro articulates a "structuralist" approach, arguing that the subject matter of a mathematical theory is not a fixed domain of numbers that exist independent of each other, but rather is the natural structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle.
     
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  31.  11
    Thomas Glass (1996). On Power Set in Explicit Mathematics. Journal of Symbolic Logic 61 (2):468-489.
    This paper is concerned with the determination of the proof-strength of the power set axiom relative to axiom systems for Feferman's explicit mathematics. As conjectured by Feferman, we obtain that the presence of the power set axiom does not increase proof-strength. Results are achieved by reducing the systems including the power set axiom to subsystems of classical analysis. In those cases where only the induction axiom is available, we make use of the technique of asymmetrical interpretations.
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  32.  31
    Frank Waaldijk (2005). On the Foundations of Constructive Mathematics – Especially in Relation to the Theory of Continuous Functions. Foundations of Science 10 (3):249-324.
    We discuss the foundations of constructive mathematics, including recursive mathematics and intuitionism, in relation to classical mathematics. There are connections with the foundations of physics, due to the way in which the different branches of mathematics reflect reality. Many different axioms and their interrelationship are discussed. We show that there is a fundamental problem in BISH (Bishop’s school of constructive mathematics) with regard to its current definition of ‘continuous function’. This problem is closely related to the definition in BISH of (...)
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  33.  3
    Francesco Ciraulo (2008). A Constructive Semantics for Non-Deducibility. Mathematical Logic Quarterly 54 (1):35-48.
    This paper provides a constructive topological semantics for non-deducibility of a first order intuitionistic formula. Formal topology theory, in particular the recently introduced notion of a binary positivity predicate, and co-induction are two needful tools.
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  34.  18
    Mohammad Ardeshir & Rasoul Ramezanian (2012). A Solution to the Surprise Exam Paradox in Constructive Mathematics. Review of Symbolic Logic 5 (4):679-686.
    We represent the well-known surprise exam paradox in constructive and computable mathematics and offer solutions. One solution is based on Brouwer’s continuity principle in constructive mathematics, and the other involves type 2 Turing computability in classical mathematics. We also discuss the backward induction paradox for extensive form games in constructive logic.
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  35. Stewart Shapiro (2003). Philosophy of Mathematics. In Peter Clark & Katherine Hawley (eds.), Philosophy of Science Today. Clarendon Press
    Moving beyond both realist and anti-realist accounts of mathematics, Shapiro articulates a "structuralist" approach, arguing that the subject matter of a mathematical theory is not a fixed domain of numbers that exist independent of each other, but rather is the natural structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle.
     
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  36.  9
    Yvon Gauthier (2009). La Descente Infinie, l'Induction Transfinie Et le Tiers Exclu. Dialogue 48 (1):1.
    ABSTRACT: It is argued that the equivalence, which is usually postulated to hold between infinite descent and transfinite induction in the foundations of arithmetic uses the law of excluded middle through the use of a double negation on the infinite set of natural numbers and therefore cannot be admitted in intuitionistic logic and mathematics, and a fortiori in more radical constructivist foundational schemes. Moreover it is shown that the infinite descent used in Dedekind-Peano arithmetic does not correspond to the (...)
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  37.  1
    Jacqueline Boniface (2004). Poincaré et le principe d'induction. Philosophiques 31 (1):131-149.
    Le principe d’induction est lié à la définition des nombres entiers d’une façon à la fois essentielle et sujette à controverse. Fonde-t-il ces nombres, ou bien trouve-t-il en eux son fondement ? Son statut lui-même peut être conçu de diverses manières. Est-il donné par l’expérience, par l’intuition, par la logique, par convention ? Ces questions furent l’objet d’une âpre discussion, autour des années 1905-1906, dans le cadre plus large d’un débat sur les fondements des mathématiques qui opposa Poincaré aux (...)
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  38.  17
    James Franklin (1996). Proof in Mathematics. Quakers Hill Press.
    A textbook on proof in mathematics, inspired by an Aristotelian point of view on mathematics and proof. It explains how to prove statements in mathematics, from evident premises. It deals with the proof of "all" statements, "some" statements, multiple quantifiers and mathematical induction.
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  39.  4
    Bahareh Afshari & Michael Rathjen (2010). A Note on the Theory of Positive Induction, {{Rm ID}^*_1}. Archive for Mathematical Logic 49 (2):275-281.
    The article shows a simple way of calibrating the strength of the theory of positive induction, ${{\rm ID}^{*}_{1}}$ . Crucially the proof exploits the equivalence of ${\Sigma^{1}_{1}}$ dependent choice and ω-model reflection for ${\Pi^{1}_{2}}$ formulae over ACA 0. Unbeknown to the authors, D. Probst had already determined the proof-theoretic strength of ${{\rm ID}^{*}_{1}}$ in Probst, J Symb Log, 71, 721–746, 2006.
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  40.  22
    Shunsuke Yatabe (2009). Comprehension Contradicts to the Induction Within Łukasiewicz Predicate Logic. Archive for Mathematical Logic 48 (3-4):265-268.
    We introduce the simpler and shorter proof of Hajek’s theorem that the mathematical induction on ω implies a contradiction in the set theory with the comprehension principle within Łukasiewicz predicate logic Ł ${\forall}$ (Hajek Arch Math Logic 44(6):763–782, 2005) by extending the proof in (Yatabe Arch Math Logic, accepted) so as to be effective in any linearly ordered MV-algebra.
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  41.  5
    Roman Murawski (2007). Troubles with (the Concept of) Truth in Mathematics. Logic and Logical Philosophy 15 (4):285-303.
    In the paper the problem of definability and undefinability of the concept of satisfaction and truth is considered. Connections between satisfaction and truth on the one hand and consistency of certain systems of omega-logic and transfinite induction on the other are indicated.
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  42.  24
    Juliette Kennedy & Roman Kossak (eds.) (2012). Set Theory, Arithmetic, and Foundations of Mathematics: Theorems, Philosophies. Cambridge University Press.
    Machine generated contents note: 1. Introduction Juliette Kennedy and Roman Kossak; 2. Historical remarks on Suslin's problem Akihiro Kanamori; 3. The continuum hypothesis, the generic-multiverse of sets, and the [OMEGA] conjecture W. Hugh Woodin; 4. [omega]-Models of finite set theory Ali Enayat, James H. Schmerl and Albert Visser; 5. Tennenbaum's theorem for models of arithmetic Richard Kaye; 6. Hierarchies of subsystems of weak arithmetic Shahram Mohsenipour; 7. Diophantine correct open induction Sidney Raffer; 8. Tennenbaum's theorem and recursive reducts James (...)
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  43. Imre Lakatos, British Society for the Philosophy of Science, London School of Economics and Political Science & International Union of the History and Philosophy of Science (1967). Proceedings of the International Colloquium in the Philosophy of Science, London, 1965. Monograph Collection (Matt - Pseudo).
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  44. Richard Heck (1995). Definition by Induction in Frege's Grundgesetze der Arithmetik. In W. Demopoulos (ed.), Frege's Philosophy of Mathematics. OUP
    This paper discusses Frege's account of definition by induction in Grundgesetze and the two key theorems Frege proves using it.
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  45.  20
    Clark Glymour (1991). The Hierarchies of Knowledge and the Mathematics of Discovery. Minds and Machines 1 (1):75-95.
    Rather than attempting to characterize a relation of confirmation between evidence and theory, epistemology might better consider which methods of forming conjectures from evidence, or of altering beliefs in the light of evidence, are most reliable for getting to the truth. A logical framework for such a study was constructed in the early 1960s by E. Mark Gold and Hilary Putnam. This essay describes some of the results that have been obtained in that framework and their significance for philosophy of (...)
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  46.  3
    Keita Yokoyama (2013). On the Strength of Ramsey's Theorem Without S1-Induction. Mathematical Logic Quarterly 59 (1):108-111.
    In this paper, we show that equation image is a equation image-conservative extension of BΣ1 + exp, thus it does not imply IΣ1.
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  47.  35
    Lance J. Rips, Amber Bloomfield & Jennifer Asmuth (2008). From Numerical Concepts to Concepts of Number. Behavioral and Brain Sciences 31 (6):623-642.
    Many experiments with infants suggest that they possess quantitative abilities, and many experimentalists believe that these abilities set the stage for later mathematics: natural numbers and arithmetic. However, the connection between these early and later skills is far from obvious. We evaluate two possible routes to mathematics and argue that neither is sufficient: (1) We first sketch what we think is the most likely model for infant abilities in this domain, and we examine proposals for extrapolating the natural number concept (...)
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  48.  98
    Mary Leng, Alexander Paseau & Michael D. Potter (eds.) (2007). Mathematical Knowledge. Oxford University Press.
    What is the nature of mathematical knowledge? Is it anything like scientific knowledge or is it sui generis? How do we acquire it? Should we believe what mathematicians themselves tell us about it? Are mathematical concepts innate or acquired? Eight new essays offer answers to these and many other questions. Written by some of the world's leading philosophers of mathematics, psychologists, and mathematicians, Mathematical Knowledge gives a lively sense of the current state of debate in this fascinating field. Contents 1. (...)
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  49.  33
    R. T. Brady & P. A. Rush (2008). What is Wrong with Cantor's Diagonal Argument? Logique Et Analyse 51 (1):185-219..
    We first consider the entailment logic MC, based on meaning containment, which contains neither the Law of Excluded Middle (LEM) nor the Disjunctive Syllogism (DS). We then argue that the DS may be assumed at least on a similar basis as the assumption of the LEM, which is then justified over a finite domain or for a recursive property over an infinite domain. In the latter case, use is made of Mathematical Induction. We then show that an instance of (...)
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  50.  71
    S. Shapiro (1998). Induction and Indefinite Extensibility: The Gödel Sentence is True, but Did Someone Change the Subject? Mind 107 (427):597-624.
    Over the last few decades Michael Dummett developed a rich program for assessing logic and the meaning of the terms of a language. He is also a major exponent of Frege's version of logicism in the philosophy of mathematics. Over the last decade, Neil Tennant developed an extensive version of logicism in Dummettian terms, and Dummett influences other contemporary logicists such as Crispin Wright and Bob Hale. The purpose of this paper is to explore the prospects for Fregean logicism within (...)
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