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Quantity and number

In Daniel D. Novotný & Lukáš Novák (eds.), Neo-Aristotelian Perspectives in Metaphysics. New York, USA: Routledge. pp. 221-244 (2014)

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  1. Motivating Reductionism About Sets.Alexander Paseau - 2008 - Australasian Journal of Philosophy 86 (2):295 – 307.
    The paper raises some difficulties for the typical motivations behind set reductionism, the view that sets are reducible to entities identified independently of set theory.
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  • Quantities in Quantum Mechanics.John Forge - 2000 - International Studies in the Philosophy of Science 14 (1):43 – 56.
    The problem of the failure of value definiteness (VD) for the idea of quantity in quantum mechanics is stated, and what VD is and how it fails is explained. An account of quantity, called BP, is outlined and used as a basis for discussing the problem. Several proposals are canvassed in view of, respectively, Forrest's indeterminate particle speculation, the "standard" interpretation of quantum mechanics and Bub's modal interpretation.
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  • Classes Are States of Affairs.D. M. Armstrong - 1991 - Mind 100 (2):189-200.
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  • The Indispensability of Mathematics.Mark Colyvan - 2001 - Oxford University Press.
    This book not only outlines the indispensability argument in considerable detail but also defends it against various challenges.
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  • Philosophy of Mathematics: Structure and Ontology.Stewart Shapiro - 1997 - 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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  • The Reality of Numbers: A Physicalist's Philosophy of Mathematics.John Bigelow - 1988 - Oxford University Press.
    Challenging the myth that mathematical objects can be defined into existence, Bigelow here employs Armstrong's metaphysical materialism to cast new light on mathematics. He identifies natural, real, and imaginary numbers and sets with specified physical properties and relations and, by so doing, draws mathematics back from its sterile, abstract exile into the midst of the physical world.
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  • Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century.Paolo Mancosu - 1996 - Oxford University Press.
    The seventeenth century saw dramatic advances in mathematical theory and practice. With the recovery of many of the classical Greek mathematical texts, new techniques were introduced, and within 100 years, the rules of analytic geometry, geometry of indivisibles, arithmatic of infinites, and calculus were developed. Although many technical studies have been devoted to these innovations, Mancosu provides the first comprehensive account of the relationship between mathematical advances of the seventeenth century and the philosophy of mathematics of the period. Starting with (...)
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  • Mathematical Truth.Paul Benacerraf - 1973 - Journal of Philosophy 70 (19):661-679.
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  • Achievements and Fallacies in Hume's Account of Infinite Divisibility.James Franklin - 1994 - Hume Studies 20 (1):85-101.
    Throughout history, almost all mathematicians, physicists and philosophers have been of the opinion that space and time are infinitely divisible. That is, it is usually believed that space and time do not consist of atoms, but that any piece of space and time of non-zero size, however small, can itself be divided into still smaller parts. This assumption is included in geometry, as in Euclid, and also in the Euclidean and non- Euclidean geometries used in modern physics. Of the few (...)
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  • Aristotelianism in the Philosophy of Mathematics.James Franklin - 2011 - Studia Neoaristotelica 8 (1):3-15.
    Modern philosophy of mathematics has been dominated by Platonism and nominalism, to the neglect of the Aristotelian realist option. Aristotelianism holds that mathematics studies certain real properties of the world – mathematics is neither about a disembodied world of “abstract objects”, as Platonism holds, nor it is merely a language of science, as nominalism holds. Aristotle’s theory that mathematics is the “science of quantity” is a good account of at least elementary mathematics: the ratio of two heights, for example, is (...)
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  • Space, Time, and Spacetime.Lawrence Sklar - 1974 - University of California Press.
    In this book, Lawrence Sklar demonstrates the interdependence of science and philosophy by examining a number of crucial problems on the nature of space and ...
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  • Philosophy of Mathematics: An Introduction to a World of Proofs and Pictures.James Robert Brown - 1999 - Routledge.
    _Philosophy of Mathematics_ is an excellent introductory text. This student friendly book discusses the great philosophers and the importance of mathematics to their thought. It includes the following topics: * the mathematical image * platonism * picture-proofs * applied mathematics * Hilbert and Godel * knots and nations * definitions * picture-proofs and Wittgenstein * computation, proof and conjecture. The book is ideal for courses on philosophy of mathematics and logic.
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  • Principles and Proofs: Aristotle's Theory of Demonstrative Science.Richard D. McKirahan (ed.) - 1992 - Princeton University Press.
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  • A Combinatorial Theory of Possibility.D. M. Armstrong - 1989 - Cambridge University Press.
    David Armstrong's book is a contribution to the philosophical discussion about possible worlds. Taking Wittgenstein's Tractatus as his point of departure, Professor Armstrong argues that nonactual possibilities and possible worlds are recombinations of actually existing elements, and as such are useful fictions. There is an extended criticism of the alternative-possible-worlds approach championed by the American philosopher David Lewis. This major work will be read with interest by a wide range of philosophers.
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  • Science and Necessity.John Bigelow & Robert Pargetter - 1990 - Cambridge University Press.
    This book espouses an innovative theory of scientific realism in which due weight is given to mathematics and logic. The authors argue that mathematics can be understood realistically if it is seen to be the study of universals, of properties and relations, of patterns and structures, the kinds of things which can be in several places at once. Taking this kind of scientific platonism as their point of departure, they show how the theory of universals can account for probability, laws (...)
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  • Armstrong on Quantities and Resemblance.Maya Eddon - 2007 - Philosophical Studies 136 (3):385-404.
    Resemblances obtain not only between objects but between properties. Resemblances of the latter sort - in particular resemblances between quantitative properties - prove to be the downfall of a well-known theory of universals, namely the one presented by David Armstrong. This paper examines Armstrong's efforts to account for such resemblances within the framework of his theory and also explores several extensions of that theory. All of them fail.
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  • Zero-Value Physical Quantities.Yuri Balashov - 1999 - Synthese 119 (3):253-286.
    To state an important fact about the photon, physicists use such expressions as (1) “the photon has zero (null, vanishing) mass” and (2) “the photon is (a) massless (particle)” interchangeably. Both (1) and (2) express the fact that the photon has no non-zero mass. However, statements (1) and (2) disagree about a further fact: (1) attributes to the photon the property of zero-masshood whereas (2) denies that the photon has any mass at all. But is there really a difference between (...)
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  • Quantities.John Bigelow, Robert Pargetter & D. M. Armstrong - 1988 - Philosophical Studies 54 (3):287 - 304.
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  • The Metaphysics of Quantity.Brent Mundy - 1987 - Philosophical Studies 51 (1):29 - 54.
    A formal theory of quantity T Q is presented which is realist, Platonist, and syntactically second-order (while logically elementary), in contrast with the existing formal theories of quantity developed within the theory of measurement, which are empiricist, nominalist, and syntactically first-order (while logically non-elementary). T Q is shown to be formally and empirically adequate as a theory of quantity, and is argued to be scientifically superior to the existing first-order theories of quantity in that it does not depend upon empirically (...)
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  • What Science Knows: And How It Knows It.James Franklin - 2009 - Encounter Books.
    In What Science Knows, the Australian philosopher and mathematician James Franklin explains in captivating and straightforward prose how science works its magic. It offers a semipopular introduction to an objective Bayesian/logical probabilist account of scientific reasoning, arguing that inductive reasoning is logically justified (though actually existing science sometimes falls short). Its account of mathematics is Aristotelian realist.
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  • Deflating Existential Commitment: A Case for Nominalism.Jody Azzouni - 2004 - Oup Usa.
    If we take mathematical statements to be true, then must we also believe in the existence of invisible mathematical objects, accessible only by the power of thought? Jody Azzouni says we do not have to, and claims that the way to escape such a commitment is to accept - as an essential part of scientific doctrine - true statements which are 'about' objects which don't exist in any real sense.
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  • Are Quantities Relations? A Reply to Bigelow and Pargetter.D. M. Armstrong - 1988 - Philosophical Studies 54 (3):305 - 316.
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  • Aquinas and Suarez on the Essence of Continuous Physical Quantity.David P. Lang - 2002 - Laval Théologique et Philosophique 58 (3):565-595.
    The development of the notion of continuous physical quantity is traced from Aristotle to Aquinas to Suarez. It is concluded that Aristotle’s divisibility definition fails to excavate the ontological core of material quantification. Although the basic germ of the solution to the problem is discovered in Aquinas, it is Suarez who fully articulates the essence of continuous physical quantity with his explicit concept of aptitudinal extension — which has crucial theological implications. Résumé Nous considérons ici le développement de la notion (...)
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  • Sketch for a Systematic Metaphysics.D. M. Armstrong - 2010 - Oxford, UK: Oxford University Press UK.
    In his last book, David Armstrong sets out his metaphysical system in a set of concise and lively chapters each dealing with one aspect of the world. He begins with the assumption that all that exists is the physical world of space-time. On this foundation he constructs a coherent metaphysical scheme that gives plausible answers to many of the great problems of metaphysics. He gives accounts of properties, relations, and particulars; laws of nature; modality; abstract objects such as numbers; and (...)
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  • Aristotelian Realism.James Franklin - 2009 - In A. Irvine (ed.), The Philosophy of Mathematics (Handbook of the Philosophy of Science series). North-Holland Elsevier.
    Aristotelian, or non-Platonist, realism holds that mathematics is a science of the real world, just as much as biology or sociology are. Where biology studies living things and sociology studies human social relations, mathematics studies the quantitative or structural aspects of things, such as ratios, or patterns, or complexity, or numerosity, or symmetry. Let us start with an example, as Aristotelians always prefer, an example that introduces the essential themes of the Aristotelian view of mathematics. A typical mathematical truth is (...)
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  • The Applicability of Mathematics as a Philosophical Problem.Mark Steiner - 1998 - Harvard University Press.
    This book analyzes the different ways mathematics is applicable in the physical sciences, and presents a startling thesis--the success of mathematical physics ...
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  • The Philosophy of Mathematics: An Introductory Essay.Stephan Körner - 1960 - Dover Publications.
    Lucid and comprehensive essay surveys the views of Plato, Aristotle, Leibniz and Kant on the nature of mathematics; examines the propositions and theories of the schools these philosophers inspired; and concludes with a discussion on the relation between mathematical theories, empirical data and philosophical presuppositions.
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  • Berkeley's Philosophy of Mathematics.Douglas M. Jesseph - 1993 - University of Chicago Press.
    In this first modern, critical assessment of the place of mathematics in Berkeley's philosophy and Berkeley's place in the history of mathematics, Douglas M. Jesseph provides a bold reinterpretation of Berkeley's work.
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  • Intellectual Abstraction as Incompatible with Materilism.David McGraw - 1995 - Southwest Philosophy Review 11 (2):23-30.
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  • Aristotelian-Thomistic Philosophy of Measure and the International System of Units (Si).Charles Bonaventure Crowley - 1996
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  • Foundations of Measurement, Vol. I: Additive and Polynomial Representations.David Krantz, Duncan Luce, Patrick Suppes & Amos Tversky (eds.) - 1971 - New York Academic Press.
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  • Realism in Mathematics.Penelope MADDY - 1990 - Oxford University Prress.
    Mathematicians tend to think of themselves as scientists investigating the features of real mathematical things, and the wildly successful application of mathematics in the physical sciences reinforces this picture of mathematics as an objective study. For philosophers, however, this realism about mathematics raises serious questions: What are mathematical things? Where are they? How do we know about them? Offering a scrupulously fair treatment of both mathematical and philosophical concerns, Penelope Maddy here delineates and defends a novel version of mathematical realism. (...)
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  • On the Origin of Objects.Brian Cantwell Smith - 1996 - MIT Press.
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  • The Foundations of Arithmetic.Gottlob Frege - 1953 - Evanston: Ill., Northwestern University Press.
    In arithmetic, if only because many of its methods and concepts originated in India, it has been the tradition to reason less strictly than in geometry, ...
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  • Aristotle's Philosophy of Mathematics.David Bostock - 2012 - In Christopher Shields (ed.), The Oxford Handbook of Aristotle. Oup Usa. pp. 465.
    Much of Aristotle's thought developed in reaction to Plato's views, and this is certainly true of his philosophy of mathematics. To judge from his dialogue, the Meno, the first thing that struck Plato as an interesting and important feature of mathematics was its epistemology: in this subject we can apparently just “draw knowledge out of ourselves.” Aristotle certainly thinks that Plato was wrong to “separate” the objects of mathematics from the familiar objects that we experience in this world. His main (...)
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  • St. Thomas on the Object of Geometry Under the Auspices of the Aristotelian Society of Marquette University.Vincent Edward Smith - 1954 - Marquette University Press.
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  • Aristotle's Philosophy of Mathematics.Hippocrates George Apostle - 1952 - University of Chicago Press.
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  • Frege, Mill, and the Foundations of Arithmetic.Glenn Kessler - 1980 - Journal of Philosophy 77 (2):65-79.
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  • The Nature of Number.Peter Forrest & D. M. Armstrong - 1987 - Philosophical Papers 16 (3):165-186.
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  • The Shape of Space.Graham Nerlich - 1994 - Cambridge University Press.
    This is a revised and updated edition of Graham Nerlich's classic book The Shape of Space. It develops a metaphysical account of space which treats it as a real and concrete entity. In particular, it shows that the shape of space plays a key explanatory role in space and spacetime theories. Arguing that geometrical explanation is very like causal explanation, Professor Nerlich prepares the ground for philosophical argument, and, using a number of novel examples, investigates how different spaces would affect (...)
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  • Universals: An Opinionated Introduction.D. M. ARMSTRONG - 1989 - Westview Press.
    In this short text, a distinguished philosopher turns his attention to one of the oldest and most fundamental philosophical problems of all: How it is that we are able to sort and classify different things as being of the same natural class? Professor Armstrong carefully sets out six major theories—ancient, modern, and contemporary—and assesses the strengths and weaknesses of each. Recognizing that there are no final victories or defeats in metaphysics, Armstrong nonetheless defends a traditional account of universals as the (...)
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  • Introduction: The Philosophy of Vectors.Philipp Keller Stephan Leuenberger - 2009 - Dialectica 63 (4):369-380.
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  • Bigelow and Pargetter on Quantities.John Forge - 1995 - Australasian Journal of Philosophy 73 (4):594 – 605.
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  • Universal Mathematics in Aristotelian-Thomistic Philosophy the Hermeneutics of Aristotelian Texts Relative to Universal Mathematics.Charles B. Crowley - 1980
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  • Aristotle’s Notion of Quantity and Modern Mathematics.Seamus Hegarty - 1969 - Philosophical Studies 18:25-35.
    THE notion of quantity is basic and it is no surprise that Aristotle refers to it in many places. There are two main discussions, that in the Categories—a part of the Organon which is of great interest to modern logicians and that spread over the physical treatises. Naturally the two treatments overlap, but modern logic is at a far remove from classical cosmology and it is fairly easy to separate them at their sources. This I have attempted to do by (...)
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  • Aristotle and Modern Mathematical Theories of the Continuum.Anne Newstead - 2001 - In Demetra Sfendoni-Mentzou & James Brown (eds.), Aristotle and Contemporary Philosophy of Science. Peter Lang.
    This paper is on Aristotle's conception of the continuum. It is argued that although Aristotle did not have the modern conception of real numbers, his account of the continuum does mirror the topology of the real number continuum in modern mathematics especially as seen in the work of Georg Cantor. Some differences are noted, particularly as regards Aristotle's conception of number and the modern conception of real numbers. The issue of whether Aristotle had the notion of open versus closed intervals (...)
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  • Aristotle on Species Variation.James Franklin - 1986 - Philosophy 61 (236):245 - 252.
    Explains Aristotle's views on the possibility of continuous variation between biological species. While the Porphyrean/Linnean classification of species by a tree suggests species are distributed discretely, Aristotle admitted continuous variation between species among lower life forms.
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  • Measurement in Psychology: A Critical History of a Methodological Concept.Joel Michell - 1999 - Cambridge University Press.
    This book traces how such a seemingly immutable idea as measurement proved so malleable when it collided with the subject matter of psychology. It locates philosophical and social influences reshaping the concept and, at the core of this reshaping, identifies a fundamental problem: the issue of whether psychological attributes really are quantitative. It argues that the idea of measurement now endorsed within psychology actually subverts attempts to establish a genuinely quantitative science and it urges a new direction. It relates views (...)
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  • The Question of Intensive Magnitudes According to Some Jesuits in the Sixteenth and Seventeenth Centuries.Jean-Luc Solère - 2001 - The Monist 84 (4):582-616.
    The problem of the intensification and remission of qualities was a crux for philosophical, theological, and scientific thought in the Middle Ages. It was raised in Antiquity with this remark of Aristotle: some qualities, as accidental beings, admit the more and the less. Admitting more and less is not a trivial property, since it belongs neither to every category of being, nor to every quality. Rather it applies only to states and dispositions such as virtue, to affections of bodies such (...)
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  • An Introduction to Logic and Scientific Method.Morris Raphael Cohen - 1934 - [Madison, Wis.]Pub. For the United States Armed Forces Institute by Harcourt, Brace and Company.
    A text that would find a place for the realistic formalism of Aristotle, the scientific penetration of Peirce, the pedagogical soundness of Dewey, and the ...
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