References in:
Quantity and number
In Daniel D. Novotný & Lukáš Novák (eds.), NeoAristotelian Perspectives in Metaphysics. New York, USA: Routledge. pp. 221244 (2014)
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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. 



This book not only outlines the indispensability argument in considerable detail but also defends it against various challenges. 

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 (...) 

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. 

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 (...) 



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 nonzero 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 (...) 

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 (...) 

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 ... 

_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 * pictureproofs * applied mathematics * Hilbert and Godel * knots and nations * definitions * pictureproofs and Wittgenstein * computation, proof and conjecture. The book is ideal for courses on philosophy of mathematics and logic. 



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 alternativepossibleworlds approach championed by the American philosopher David Lewis. This major work will be read with interest by a wide range of philosophers. 

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 (...) 

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 wellknown 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. 

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 nonzero mass. However, statements (1) and (2) disagree about a further fact: (1) attributes to the photon the property of zeromasshood whereas (2) denies that the photon has any mass at all. But is there really a difference between (...) 



A formal theory of quantity T Q is presented which is realist, Platonist, and syntactically secondorder (while logically elementary), in contrast with the existing formal theories of quantity developed within the theory of measurement, which are empiricist, nominalist, and syntactically firstorder (while logically nonelementary). 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 firstorder theories of quantity in that it does not depend upon empirically (...) 

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. 

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. 



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 (...) 

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 spacetime. 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 (...) 

Aristotelian, or nonPlatonist, 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 (...) 

This book analyzes the different ways mathematics is applicable in the physical sciences, and presents a startling thesisthe success of mathematical physics ... 

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. 

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. 







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. (...) 



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, ... 

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 (...) 









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 (...) 

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 (...) 







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 (...) 

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 (...) 

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. 

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 (...) 

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 (...) 

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 ... 