Results for 'irrational numbers'

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  1. 3. The Monotone Series and Multiplier and Divisor Relative Numbers.Divisor Relative Numbers - 1987 - International Logic Review: Rassegna Internazionale di Logica 15 (1):26.
     
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  2.  4
    How the Abstract Becomes Concrete: Irrational Numbers Are Understood Relative to Natural Numbers and Perfect Squares.Purav Patel & Sashank Varma - 2018 - Cognitive Science 42 (5):1642-1676.
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  3.  24
    Frege Gottlob. Der Gedanke. Beiträge Zur Philosophie des Deutschen Idealismus, Vol. 1 No. 2 , Pp. 58–77.Frege Gottlob. Die Verneinung. Beiträge Zur Philosophie des Deutschen Idealismus, Vol. 1 No. 3–4 , Pp. 143–157.Black Max. Frege Against the Formalists. A Translation of Part of Grundgesetze der Arithmetik. Introductory Note. The Philosophical Review, Vol. 59 , Pp. 77–78.Frege Gottlob. Frege Against the Formalists. E. Heine's and J. Thomae's Theories of Irrational Numbers. The Philosophical Review, Vol. 59 , Pp. 79–93, 202–220, 332–345. Frege Gottlob. On Concept and Object. Mind, N.S. Vol. 60 , Pp. 168–180. Gromska Daniela. L'Abbé Stanisław Kobyłecki. Studia Philosophica , Vol. 3 , Pp. 40–41. [4631-2; V 43.]Gromska Daniela. Edward Stamm. Studia Philosophica , Vol. 3 , Pp. 43–45. [1851–12.3.]Gromska Daniela. Stanisław Leśniewski. Studia Philosophica , Vol. 3 , Pp. 46–51. [2021-13; V 83, 84.]Gromska Daniela. Leon Chwistek. Studia Philosophica , Vol. 3 , Pp. 51–54. [2201-15, 4911-2; II 1. [REVIEW]Alonzo Church - 1953 - Journal of Symbolic Logic 18 (1):93-94.
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  4.  54
    Concerning Professor Sawyer's Reflections on Irrational Numbers.George Goe - 1965 - Philosophia Mathematica (1):38-43.
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  5.  6
    Logic and Arithmetic. Vol. 2: Rational and Irrational Numbers.Mary Tiles & David Bostock - 1981 - Philosophical Quarterly 31 (124):277.
  6.  19
    E. Heine's and J. Thomae's Theories of Irrational Numbers.Gottlob Frege - 1950 - Philosophical Review 59 (1):79-93.
  7.  6
    Review: David Bostock, Logic and Arithmetic. Volume 1. Natural Numbers; David Bostock, Logic and Arithmetic. Volume 2. Rational and Irrational Numbers[REVIEW]Michael D. Resnik - 1982 - Journal of Symbolic Logic 47 (3):708-713.
  8.  2
    Concerning Professor Sawyer's Reflections On Irrational Numbers.George Goe - 1965 - Philosophia Mathematica (1):38-43.
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  9.  1
    Bostock David. Logic and Arithmetic. Volume 1. Natural Numbers. The Clarendon Press, Oxford University Press, Oxford 1974, X + 219 Pp.Bostock David. Logic and Arithmetic. Volume 2. Rational and Irrational Numbers. The Clarendon Press, Oxford University Press, Oxford 1979, Ix + 307 Pp. [REVIEW]Michael D. Resnik - 1982 - Journal of Symbolic Logic 47 (3):708-713.
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  10. Logic and Arithmetic, Vol. II--Rational and Irrational Numbers.David Bostock - 1981 - Mind 90 (359):473-475.
     
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  11. Logic and Arithmetic. Vol. 2: Rational and Irrational Numbers.D. Bostock - 1981 - Tijdschrift Voor Filosofie 43 (4):763-764.
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  12. The Contents of the Fifth and Sixth Books of Euclid: With a Note on Irrational Numbers.M. J. M. Hill - 2014 - Cambridge University Press.
    First published in 1908 as the second edition of a 1900 original, this book explains the content of the fifth and sixth books of Euclid's Elements, which are primarily concerned with ratio and magnitudes. Hill furnishes the text with copious diagrams to illustrate key points of Euclidian reasoning. This book will be of value to anyone with an interest in the history of education.
     
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  13. BOSTOCK, D. "Logic and Arithmetic, Vol. II-Rational and Irrational Numbers". [REVIEW]N. Tennant - 1981 - Mind 90:473.
  14.  50
    Did the Greeks Discover the Irrationals?Philip Hugly & Charles Sayward - 1999 - Philosophy 74 (2):169-176.
    A popular view is that the great discovery of Pythagoras was that there are irrational numbers, e.g., the positive square root of two. Against this it is argued that mathematics and geometry, together with their applications, do not show that there are irrational numbers or compel assent to that proposition.
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  15.  41
    Wittgenstein on Pseudo-Irrationals, Diagonal Numbers and Decidability.Timm Lampert - 2008 - In The Logica Yearbook 2008. London: pp. 95-111.
    In his early philosophy as well as in his middle period, Wittgenstein holds a purely syntactic view of logic and mathematics. However, his syntactic foundation of logic and mathematics is opposed to the axiomatic approach of modern mathematical logic. The object of Wittgenstein’s approach is not the representation of mathematical properties within a logical axiomatic system, but their representation by a symbolism that identifies the properties in question by its syntactic features. It rests on his distinction of descriptions and operations; (...)
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  16. Continuity and Number.B. Goussinsky - 1959 - Tel Aviv, Israel.
  17.  7
    Étude Constructive de Problèmes de Topologie Pour les Réels Irrationnels.Mohamed Khalouani, Salah Labhalla & Et Henri Lombardi - 1999 - Mathematical Logic Quarterly 45 (2):257-288.
    We study in a constructive manner some problems of topology related to the set Irr of irrational reals. The constructive approach requires a strong notion of an irrational number; constructively, a real number is irrational if it is clearly different from any rational number. We show that the set Irr is one-to-one with the set Dfc of infinite developments in continued fraction . We define two extensions of Irr, one, called Dfc1, is the set of dfc of (...)
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  18. Undefinability Results in o-Minimal Expansions of the Real Numbers.Ricardo Bianconi - 2005 - Annals of Pure and Applied Logic 134 (1):43-51.
    We show that if is not in the field generated by α1,…,αn, then no restriction of the function xβ to an interval is definable in . We also prove that if the real and imaginary parts of a complex analytic function are definable in Rexp or in the expansion of by functions xα, for irrational α, then they are already definable in . We conclude with some conjectures and open questions.
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  19.  21
    Deleuze's Third Synthesis of Time.Daniela Voss - 2013 - Deleuze and Guatarri Studies 7 (2):194-216.
    Deleuze's theory of time set out in Difference and Repetition is a complex structure of three different syntheses of time – the passive synthesis of the living present, the passive synthesis of the pure past and the static synthesis of the future. This article focuses on Deleuze's third synthesis of time, which seems to be the most obscure part of his tripartite theory, as Deleuze mixes different theoretical concepts drawn from philosophy, Greek drama theory and mathematics. Of central importance is (...)
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  20.  15
    Frege's Approach to the Foundations of Analysis (1874–1903).Matthias Schirn - 2013 - History and Philosophy of Logic 34 (3):266-292.
    The concept of quantity (Größe) plays a key role in Frege's theory of real numbers. Typically enough, he refers to this theory as ?theory of quantity? (?Größenlehre?) in the second volume of his opus magnum Grundgesetze der Arithmetik (Frege 1903). In this essay, I deal, in a critical way, with Frege's treatment of the concept of quantity and his approach to analysis from the beginning of his academic career until Frege 1903. I begin with a few introductory remarks. In (...)
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  21.  65
    Fractal Geometry is Not the Geometry of Nature.Orly Shenker - 1994 - Studies in History and Philosophy of Science Part A 25 (6):967-981.
    In recent years the magnificent world of fractals has been revealed. Some of the fractal images resemble natural forms so closely that Benoit Mandelbrot's hypothesis, that the fractal geometry is the geometry of natural objects, has been accepted by scientists and non-scientists alike. The present paper critically examines Mandelbrot's hypothesis. It first analyzes the concept of a fractal. The analysis reveals that fractals are endless geometrical processes, and not geometrical forms. A comparison between fractals and irrational numbers shows (...)
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  22.  40
    Computable Chaos.John A. Winnie - 1992 - Philosophy of Science 59 (2):263-275.
    Some irrational numbers are "random" in a sense which implies that no algorithm can compute their decimal expansions to an arbitrarily high degree of accuracy. This feature of (most) irrational numbers has been claimed to be at the heart of the deterministic, but chaotic, behavior exhibited by many nonlinear dynamical systems. In this paper, a number of now classical chaotic systems are shown to remain chaotic when their domains are restricted to the computable real numbers, (...)
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  23. Eudoxos Versus Dedekind.Piotr Błaszczyk - 2007 - Filozofia Nauki 2.
    All through the XXth century it has been repeated that "there is an exact correspondence, almost coincidence between Euclid's definition of equal ratios and the modern theory of irrational numbers due to Dedekind". Since the idea was presented as early as in 1908 in Thomas Heath's translation of Euclid's Elements as a comment to Book V, def. 5, we call it in the paper Heath's thesis. Heath's thesis finds different justifications so it is accepted yet in different versions. (...)
     
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  24.  39
    The Significance of a Non-Reductionist Ontology for the Discipline of Mathematics: A Historical and Systematic Analysis. [REVIEW]D. F. M. Strauss - 2010 - Axiomathes 20 (1):19-52.
    A Christian approach to scholarship, directed by the central biblical motive of creation, fall and redemption and guided by the theoretical idea that God subjected all of creation to His Law-Word, delimiting and determining the cohering diversity we experience within reality, in principle safe-guards those in the grip of this ultimate commitment and theoretical orientation from absolutizing or deifying anything within creation. In this article my over-all approach is focused on the one-sided legacy of mathematics, starting with Pythagorean arithmeticism (“everything (...)
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  25.  10
    Les Commentaires d'Al-Māhānī Et d'Un Anonyme du Livre X des Éléments d'Euclide.Marouane Ben Miled - 1999 - Arabic Sciences and Philosophy 9 (1):89.
    This paper presents the first edition, translation and analyse of al-Mns commentary of the Book X of Euclid one. For the first time, irrational numbers are defined and classified. The algebraisation of Elementsrizms Algebra, shows irrational numbers as solution to algebraic quadratic equations. The algebraic calculus makes here the first steps. On this occasion, negative numbers and their calculation rules appears. Simplifications imposed by the algebraic writings are sometimes in opposition with the conclusions of propositions (...)
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  26.  3
    Analyse de complexité pour un théorème de Hall sur les fractions continues.Salah Labhalla & Henri Lombardi - 1996 - Mathematical Logic Quarterly 42 (1):134-144.
    We give a polynomial time controlled version of a theorem of M. Hall: every real number can be written as the sum of two irrational numbers whose developments into a continued fraction contain only 1, 2, 3 or 4.
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  27. Reference to Numbers in Natural Language.Friederike Moltmann - 2013 - Philosophical Studies 162 (3):499 - 536.
    A common view is that natural language treats numbers as abstract objects, with expressions like the number of planets, eight, as well as the number eight acting as referential terms referring to numbers. In this paper I will argue that this view about reference to numbers in natural language is fundamentally mistaken. A more thorough look at natural language reveals a very different view of the ontological status of natural numbers. On this view, numbers are (...)
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  28. Of Numbers and Electrons.Cian Dorr - 2010 - Proceedings of the Aristotelian Society 110 (2pt2):133-181.
    According to a tradition stemming from Quine and Putnam, we have the same broadly inductive reason for believing in numbers as we have for believing in electrons: certain theories that entail that there are numbers are better, qua explanations of our evidence, than any theories that do not. This paper investigates how modal theories of the form ‘Possibly, the concrete world is just as it in fact is and T’ and ‘Necessarily, if standard mathematics is true and the (...)
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  29. Taurek, Numbers and Probabilities.Rob Lawlor - 2006 - Ethical Theory and Moral Practice 9 (2):149 - 166.
    In his paper, “Should the Numbers Count?" John Taurek imagines that we are in a position such that we can either save a group of five people, or we can save one individual, David. We cannot save David and the five. This is because they each require a life-saving drug. However, David needs all of the drug if he is to survive, while the other five need only a fifth each.Typically, people have argued as if there was a choice (...)
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  30.  20
    Sustainability Ratings and the Disciplinary Power of the Ideology of Numbers.Mohamed Chelli & Yves Gendron - 2013 - Journal of Business Ethics 112 (2):187-203.
    The main purpose of this paper is to better understand how sustainability rating agencies, through discourse, promote an “ideology of numbers” that ultimately aims to establish a regime of normalization governing social and environmental performance. Drawing on Thompson’s (Ideology and modern culture: Critical social theory in the era of mass communication, 1990 ) modes of operation of ideology, we examine the extent to which, and how, the ideology of numbers is reflected on websites and public documents published by (...)
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  31.  27
    Arbitrary Reference, Numbers, and Propositions.Michele Palmira - 2018 - European Journal of Philosophy 26 (3):1069-1085.
    Reductionist realist accounts of certain entities, such as the natural numbers and propositions, have been taken to be fatally undermined by what we may call the problem of arbitrary identification. The problem is that there are multiple and equally adequate reductions of the natural numbers to sets (see Benacerraf, 1965), as well as of propositions to unstructured or structured entities (see, e.g., Bealer, 1998; King, Soames, & Speaks, 2014; Melia, 1992). This paper sets out to solve the problem (...)
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  32.  86
    Defusing Easy Arguments for Numbers.Brendan Balcerak Jackson - 2013 - Linguistics and Philosophy 36 (6):447-461.
    Pairs of sentences like the following pose a problem for ontology: (1) Jupiter has four moons. (2) The number of moons of Jupiter is four. (2) is intuitively a trivial paraphrase of (1). And yet while (1) seems ontologically innocent, (2) appears to imply the existence of numbers. Thomas Hofweber proposes that we can resolve the puzzle by recognizing that sentence (2) is syntactically derived from, and has the same meaning as, sentence (1). Despite appearances, the expressions ‘the number (...)
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  33.  9
    Relational Priming Based on a Multiplicative Schema for Whole Numbers and Fractions.Melissa DeWolf, Ji Y. Son, Miriam Bassok & Keith J. Holyoak - 2017 - Cognitive Science 41 (8):2053-2088.
    Why might it be beneficial for adults to process fractions componentially? Recent research has shown that college-educated adults can capitalize on the bipartite structure of the fraction notation, performing more successfully with fractions than with decimals in relational tasks, notably analogical reasoning. This study examined patterns of relational priming for problems with fractions in a task that required arithmetic computations. College students were asked to judge whether or not multiplication equations involving fractions were correct. Some equations served as structurally inverse (...)
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  34.  4
    Relational Priming Based on a Multiplicative Schema for Whole Numbers and Fractions.Melissa DeWolf, Ji Y. Son, Miriam Bassok & Keith J. Holyoak - 2017 - Cognitive Science 41 (8):2053-2088.
    Why might it be beneficial for adults to process fractions componentially? Recent research has shown that college-educated adults can capitalize on the bipartite structure of the fraction notation, performing more successfully with fractions than with decimals in relational tasks, notably analogical reasoning. This study examined patterns of relational priming for problems with fractions in a task that required arithmetic computations. College students were asked to judge whether or not multiplication equations involving fractions were correct. Some equations served as structurally inverse (...)
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  35. Holistic Biology: Back on Stage? Comments on Post-Genomics in Historical Perspective.Alfred Gierer - 2002 - Philosophia Naturalis 39 (1):25-44.
    A strong motivation for the human genome project was to relate biological features to the structure and function of small sets of genes, and ideally to individual genes. However, it is now increasingly realized that many problems require a "systems" approach emphasizing the interplay of large numbers of genes, and the involvement of complex networks of gene regulation. This implies a new emphasis on integrative, systems theoretical approaches. It may be called 'holistic' if the term is used without (...) overtones, in the general sense of directing attention to integrated features of organs and organisms. In the history of biology, seemingly conflicting reductionist and holistic notions have alternated, with bottom-up as well as top-down approaches eventually contributing to the solutions of basic problems. By now, there is no doubt that biological features and phenomena are rooted in physico-chemical processes of the molecules involved; and yet, integrated systems aspects are becoming more and more relevant in developmental biology, brain and behavioural science, and socio-biology. -/- . (shrink)
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  36.  23
    Learning to Represent Exact Numbers.Barbara W. Sarnecka - forthcoming - Synthese:1-18.
    This article focuses on how young children acquire concepts for exact, cardinal numbers. I believe that exact numbers are a conceptual structure that was invented by people, and that most children acquire gradually, over a period of months or years during early childhood. This article reviews studies that explore children’s number knowledge at various points during this acquisition process. Most of these studies were done in my own lab, and assume the theoretical framework proposed by Carey. In this (...)
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  37.  10
    More on D-Logics of Subspaces of the Rational Numbers.Guram Bezhanishvili & Joel Lucero-Bryan - 2012 - Notre Dame Journal of Formal Logic 53 (3):319-345.
    We prove that each countable rooted K4 -frame is a d-morphic image of a subspace of the space $\mathbb{Q}$ of rational numbers. From this we derive that each modal logic over K4 axiomatizable by variable-free formulas is the d-logic of a subspace of $\mathbb{Q}$ . It follows that subspaces of $\mathbb{Q}$ give rise to continuum many d-logics over K4 , continuum many of which are neither finitely axiomatizable nor decidable. In addition, we exhibit several families of modal logics finitely (...)
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  38.  58
    Differential Calculus Based on the Double Contradiction.Kazuhiko Kotani - 2016 - Open Journal of Philosophy 6 (4):420-427.
    The derivative is a basic concept of differential calculus. However, if we calculate the derivative as change in distance over change in time, the result at any instant is 0/0, which seems meaningless. Hence, Newton and Leibniz used the limit to determine the derivative. Their method is valid in practice, but it is not easy to intuitively accept. Thus, this article describes the novel method of differential calculus based on the double contradiction, which is easier to accept intuitively. Next, the (...)
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  39. Numbers and Propositions: Reply to Melia.Tim Crane - 1992 - Analysis 52 (4):253-256.
    Is the way we use propositions to individuate beliefs and other intentional states analogous to the way we use numbers to measure weights and other physical magnitudes? In an earlier paper [2], I argued that there is an important disanalogy. One and the same weight can be 'related to' different numbers under different units of measurement. Moreover, the choice of a unit of measurement is arbitrary,in the sense that which way we choose doesn't affect the weight attributed to (...)
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  40.  43
    Prospect-Theory's Diminishing Sensitivity Versus Economics' Intrinsic Utility of Money: How the Introduction of the Euro Can Be Used to Disentangle the Two Empirically. [REVIEW]Peter P. Wakker, Veronika Köbberling & Christiane Schwieren - 2007 - Theory and Decision 63 (3):205-231.
    The introduction of the euro gave a unique opportunity to empirically disentangle two components of utility: intrinsic value, a rational component central in economics, and the numerosity effect (going by numbers while ignoring units), a descriptive and irrational component central in prospect theory and underlying the money illusion. We measured relative risk aversion in Belgium before and after the introduction of the euro, and could consider changes in intrinsic value while keeping numbers constant, and changes in (...) while keeping intrinsic value constant. Intrinsic value significantly affected risk aversion, and the numerosity effect did not. Our study is the first to confirm the classical hypothesis of increasing relative risk aversion while avoiding irrational distortions due to the numerosity effect. (shrink)
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  41.  24
    Michael Friedmans Behandlung des Unterschiedes zwischen Arithmetik und Algebra bei Kant in Kant and the Exact Sciences.Peter Ospald - 2010 - Kant-Studien 101 (1):75-88.
    In the second chapter of his book Kant and the Exact Sciences Michael Friedman deals with two different interpretations of the relation or the difference between algebra and arithmetic in Kant's thought. According to the first interpretation algebra can be described as general arithmetic because it generalizes over all numbers by the use of variables, whereas arithmetic only deals with particular numbers. The alternative suggestion is that algebra is more general than arithmetic because it considers a more general (...)
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  42.  59
    The Abductive Loop: Tracking Irrational Sets. [REVIEW]Addis Tom, Addis Jan Townsend, Billinge Dave, Gooding David & Visscher Bart-Floris - 2008 - Foundations of Science 13 (1):5-16.
    We argue from the Church-Turing thesis (Kleene Mathematical logic. New York: Wiley 1967) that a program can be considered as equivalent to a formal language similar to predicate calculus where predicates can be taken as functions. We can relate such a calculus to Wittgenstein’s first major work, the Tractatus, and use the Tractatus and its theses as a model of the formal classical definition of a computer program. However, Wittgenstein found flaws in his initial great work and he explored these (...)
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  43.  39
    The Psychology and Philosophy of Natural Numbers.Oliver R. Marshall - 2017 - Philosophia Mathematica (1):nkx002.
    ABSTRACT I argue against both neuropsychological and cognitive accounts of our grasp of numbers. I show that despite the points of divergence between these two accounts, they face analogous problems. Both presuppose too much about what they purport to explain to be informative, and also characterize our grasp of numbers in a way that is absurd in the light of what we already know from the point of view of mathematical practice. Then I offer a positive methodological proposal (...)
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  44.  49
    Numbers and Propositions Versus Nominalists: Yellow Cards for Salmon & Soames. [REVIEW]Rafal Urbaniak - 2012 - Erkenntnis 77 (3):381-397.
    Salmon and Soames argue against nominalism about numbers and sentence types. They employ (respectively) higher-order and first-order logic to model certain natural language inferences and claim that the natural language conclusions carry commitment to abstract objects, partially because their renderings in those formal systems seem to do that. I argue that this strategy fails because the nominalist can accept those natural language consequences, provide them with plausible and non-committing truth conditions and account for the inferences made without committing themselves (...)
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  45. Don’T Count on Taurek: Vindicating the Case for the Numbers Counting.Yishai Cohen - 2014 - Res Publica 20 (3):245-261.
    Suppose you can save only one of two groups of people from harm, with one person in one group, and five persons in the other group. Are you obligated to save the greater number? While common sense seems to say ‘yes’, the numbers skeptic says ‘no’. Numbers Skepticism has been partly motivated by the anti-consequentialist thought that the goods, harms and well-being of individual people do not aggregate in any morally significant way. However, even many non-consequentialists think that (...)
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  46.  45
    A Representation Theorem for Frequently Irrational Agents.Edward Elliott - 2017 - Journal of Philosophical Logic 46 (5):467-506.
    The standard representation theorem for expected utility theory tells us that if a subject’s preferences conform to certain axioms, then she can be represented as maximising her expected utility given a particular set of credences and utilities—and, moreover, that having those credences and utilities is the only way that she could be maximising her expected utility. However, the kinds of agents these theorems seem apt to tell us anything about are highly idealised, being always probabilistically coherent with infinitely precise degrees (...)
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  47.  38
    On Irrational Guilt.Juha Räikkä - 2005 - Ethical Theory and Moral Practice 7 (5):473 - 485.
    A person raised in a religious family may have been taught that going to the theater is not allowed, and even if he has rejected this taboo years ago, he still feels guilty when attending theater. These kinds of cases may not be rare, but they are strange. Indeed, one may wonder how they are even possible. This is why an explanation is needed, and in my paper I aim to give such an explanation. In particular, I will first provide (...)
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  48.  35
    Frege, the Complex Numbers, and the Identity of Indiscernibles.Wenzel Christian Helmut - 2010 - Logique Et Analyse 53 (209):51-60.
    There are mathematical structures with elements that cannot be distinguished by the properties they have within that structure. For instance within the field of complex numbers the two square roots of −1, i and −i, have the same algebraic properties in that field. So how do we distinguish between them? Imbedding the complex numbers in a bigger structure, the quaternions, allows us to algebraically tell them apart. But a similar problem appears for this larger structure. There seems to (...)
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  49.  15
    Explaining Irrational Actions.Jesse S. Summers - 2017 - Ideas Y Valores 66 (S3):81-96.
    We sometimes want to understand irrational action, or actions a person undertakes given that their acting that way conflicts with their beliefs, their desires, or their goals. What is puzzling about all explanations of such irrational actions is this: if we explain the action by offering the agent’s reasons for the action, the action no longer seems irrational, but only a bad decision. If we explain the action mechanistically, without offering the agent’s reasons for it, then the (...)
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  50.  23
    The Numbers Count.Peter A. Graham - 2017 - Thought: A Journal of Philosophy 6 (2):129-134.
    Numbers Skeptics deny that when faced with a choice between saving some innocent people from harm and saving a larger number of different, though equally innocent, people from suffering a similar harm you ought to save the larger number. In this article, I aim to put pressure on Numbers Skepticism.
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