An Aristotelian Philosophy of Mathematics breaks the impasse between Platonist and nominalist views of mathematics. Neither a study of abstract objects nor a mere language or logic, mathematics is a science of real aspects of the world as much as biology is. For the first time, a philosophy of mathematics puts applied mathematics at the centre. Quantitative aspects of the world such as ratios of heights, and structural ones such as symmetry and continuity, are parts of the physical (...) world and are objects of mathematics. Though some mathematical structures such as infinities may be too big to be realized in fact, all of them are capable of being realized. Informed by the author's background in both philosophy and mathematics, but keeping to simple examples, the book shows how infant perception of patterns is extended by visualization and proof to the vast edifice of modern pure and applied mathematical knowledge. (shrink)

Quantity is the first category that Aristotle lists after substance. It has extraordinary epistemological clarity: "2+2=4" is the model of a self-evident and universally known truth. Continuous quantities such as the ratio of circumference to diameter of a circle are as clearly known as discrete ones. The theory that mathematics was "the science of quantity" was once the leading philosophy of mathematics. The article looks at puzzles in the classification and epistemology of quantity.

This paper deals with the metaphysics of the notion of quantity in the philosophy of Leibniz, and its aim is to defend the following bi-conditional: for any object x, x has a certain quantity if and only if x has a (metaphysical) limit or a bound. The direction from left to right is justified in §3, while in §4 I develop an argument to justify the direction from right to left. Since the bi-conditional links the metaphysical notion (...) of limit to the mathematical notion of quantity (and in this way it links Leibniz’s metaphysics with his conception of Mathesis Universalis), it allows the use of metaphysics to clarify the features of the mathematical notion of quantity. This task is accomplished in §5 and §6. Finally, §7 discusses a possible objection. (shrink)

Quantities are naturally viewed as functions, whose arguments may be construed as situations, events, objects, etc. We explore the question of the range of these functions: should it be construed as the real numbers (or some subset thereof)? This is Carnap's view. It has attractive features, specifically, what Carnap views as ontological economy. Or should the range of a quantity be a set of magnitudes? This may have been Helmholtz's view, and it, too, has attractive features. It reveals the (...) close connection between measurement and natural law, it makes dimensional analysis intelligible, and explains the concern of scientists and engineers with units in equations. It leaves the philosophical problem of the relation between the structure of magnitudes and the structure of the reals. What explains it? And is it always the same? We will argue that on the whole, construing the values of quantities as magnitudes has some advantages, and that (as Helmholtz seems to suggest in "Numbering and Measuring from an Epistemological Viewpoint") the relation between magnitudes and real numbers can be based on foundational similarities of structure. (shrink)

On which philosophical foundations is the attribution of numerical magnitudes to qualitative phenomena based? That is, what is the philosophical basis for attributing, through measurement operations, numbers to empirical qualities that our senses perceive in the outside world? This question, nowadays rarely addressed in such a way, actually refers to an old debate about the quantification of qualities. A historical analysis reveals that it was a major issue in the “context of discovery” of the first attempts to mathematize new fields (...) of knowledge, whereas the “context of justification” leading our contemporary perspective on such episodes often overlooks such problems. As a consequence, little attention has so far been given to tracing its emergence back to its original context, leaving in the shadows an important debate about measurement. In this respect, the work of William Stanley Jevons is worth analysis. As a preliminary step toward his ambitious project to mathematize economics, he first attempted to understand the mechanism underlying measurement in physics, through a very innovative approach. Its main merit consisted in trying to overcome the Aristotelian distinction between quantities and qualities. In this article, I will strive to highlight the significance of his work for the history of philosophy of science. (shrink)

Grice's Quantity maxims have been widely misinterpreted as enjoining a speaker to make the strongest claim that she can, while respecting the other conversational maxims. Although many writers on the topic of conversational implicature interpret the Quantity maxims as enjoining such volubility, so construed the Quantity maxims are unreasonable norms for conversation. Appreciating this calls for attending more closely to the notion of what a conversation requires. When we do so, we see that eschewing an injunction to (...) maximal informativeness need not deprive us of any ability to predict or explain genuine cases of implicature. Crucial to this explanation is an appreciation of how what a conversation, or a given stage of a conversation, requires, depends upon what kind of conversation is taking place. I close with an outline of this dependence relation that distinguishes among three importantly distinct types of conversation. (shrink)

Pt. I. Zeno and the metaphysics of quantity. Zeno's paradox of measure -- Tractarian nominalism -- Logical atoms and combinatorial possibility -- Strict coherence, sigma coherence, and the metaphysics of quantity -- pt. II. Coherent degrees of belief. Higher-order degrees of belief -- A mistake in dynamic coherence arguments? -- Dynamic coherence and probability kinematics -- Updating, supposing, and MAXENT -- The structure of radical probabilism -- Diachronic coherence and radical probabilism -- pt. III. Induction. Carnapian inductive logic (...) for Markov chains -- Carnapian inductive logic and Bayesian statistics -- Bayesian projectability. (shrink)

The paper interprets the concept “operator in the separable complex Hilbert space” (particalry, “Hermitian operator” as “quantity” is defined in the “classical” quantum mechanics) by that of “quantum information”. As far as wave function is the characteristic function of the probability (density) distribution for all possible values of a certain quantity to be measured, the definition of quantity in quantum mechanics means any unitary change of the probability (density) distribution. It can be represented as a particular case (...) of “unitary” qubits. The converse interpretation of any qubits as referring to a certain physical quantity implies its generalization to non-Hermitian operators, thus neither unitary, nor conserving energy. Their physical sense, speaking loosely, consists in exchanging temporal moments therefore being implemented out of the space-time “screen”. “Dark matter” and “dark energy” can be explained by the same generalization of “quantity” to non-Hermitian operators only secondarily projected on the pseudo-Riemannian space-time “screen” of general relativity according to Einstein's “Mach’s principle” and his field equation. (shrink)

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 (...) saying that something has zero mass (charge, spin, etc.) and saying that it has no mass (charge, spin, etc.)? Does the distinction cut any physical or philosophical ice? I argue that the answer to these questions is yes. Put briefly, the claim of this paper is that some zero-value physical quantities are not mere “privations”, “absences” or “holes in being”. They are respectable properties in the same sense in which their non-zero partners are. This, I will show, has implications for the debate between two rival views of the nature of property, dispositionalism and categoricalism. (shrink)

In this paper we explore the interpretation of quantity expressions in Yudja, an indigenous language spoken in the Amazonian basin, showing that while the language allows reference to exact cardinalities, it does not generally allow reference to exact measure values. It does, however, allow non-exact comparison along continuous dimensions. We use this data to argue that the grammar of exact measurement is distinct from a grammar allowing the expression of exact cardinalities, and that the grammar of counting and the (...) grammar of measurement may use numerals with different, though related interpretations. As Yudja shows, the language of measurement is not automatically acquired along with the knowledge of exact numeral expressions. We show that the ‘gap’ between prelinguistic intuitions about quantity in terms of numerosity and counting, which is bridged by the learning of language expressing exact cardinality, is paralleled by a similar gap between prelinguistic intuitions about quantity on a continuous dimension and measuring: this gap too must be bridged by language which expresses exact measure values. Our results suggest that the enculturation process by which we develop skills to perform abstract operations in the domain of measurement is (1) language dependent and (2) distinct from the process by which we learn to perform abstract calculations in the cardinal domain. (shrink)

What are physical quantities, and in particular, what makes them quantitative? This book presents an original answer to this question through the novel position of substantival structuralism, arguing that quantitativeness is an irreducible feature of attributes, and quantitative attributes are best understood as substantival structured spaces.

An unpublished manuscript on eucharistic metaphysics by Thomas White (1593–1676) supplies new information about his contributions to philosophy and theology—especially his irenic efforts to find middle ground between traditional Aristotelian views and challenges from the new mechanical philosophy. The work by White studied here, “A Discourse Concerning the Eucharist,” sheds light on his other writings and is illuminated by them. Substance, quantity, place, and accident were the main philosophical issues at stake in White's attempt to give a (...) reasoned account of controversial religious and theological issues, especially transubstantiation and the real presence. (shrink)

Since summer 2019 there is a new document that defines what in science should be regarded as being one second, one meter, and one kilogram, respectively. It is the ninth edition of the SI Brochure. Compared with older editions, a new definitional approach has been used. The seven base units are now defined by being directly related to a so-called defining constant. The paper discusses the second, the meter, and the kilogram. One odd salient, but nonetheless not discussed, feature of (...) the formulations of the definitions is that, from a purely linguistic point of view, they are circular. That is, symbols for what is to be defined appear also in the defining expressions. However, underneath the circularities there are, appearances notwithstanding, substantially non-circular definitions. The aim of the paper is to uncover these never before presented definitional structures. It is made by means of the two notions absolutely discrete quantities and metrical coordinative definitions. When the true definitional structures become visible, two important substantial metrological facts can be seen. First, the new definition of the kilogram is very much in need of further discussion. Second, contrary to what the Brochure hints at, there is no need to try to exchange the atomic parameter used in the definition of the second. (shrink)

Intermediate Quantitifiers presents and analyzes the logical and linguistic features of intermediate quantifiers, in a fashion typical of traditional logic. Intermediate quantifiers express logical quantities which fall between Aristotle's two quantities of categorical propositions - the universal and the particular. This book is the first to use traditional methods to integrate the logic and semantics of intermediate quantifiers with the two traditional quantities. Few, many and most express the most commonly referred to intermediate quantities, yet in this book Peterson argues (...) that an infinite number of additional quantities can be understood through a deeper examination of the logical nature of all intermediate quantifiers. Drawing on the work of Robert Carnes and taking a critical look at James McCawley's grammatical analysis, Peterson concludes that a new approach to foundations of mathematics based on the syllogistic logic of quantifiers is possible.Offering invaluable insights for those studying across areas of logic, linguistics, and the philosophy or semantics of natural language, this book presents the completion of an explanation for extending traditional syllogistic logic to handle intermediate quantifiers. (shrink)

Ever since David Lewis argued for the indispensibility of natural properties, they have become a staple of mainstream metaphysics. This dissertation is a critical examination of natural properties. What roles can natural properties play in metaphysics, and what structure do natural properties have? In the first half of the dissertation, I argue that natural properties cannot do all the work they are advertised to do. In the second half of the dissertation, I look at questions relating to the structure of (...) natural properties. I argue that the metric structure of fundamental quantitative properties cannot be reduced to mereological structure, and I argue that the simplistic picture of natural properties as monadic must be abandoned in light of theories of fundamental physics. (shrink)

I analyze the meaning of mass in Newtonian mechanics. First, I explain the notion of primitive ontology, which was originally introduced in the philosophy of quantum mechanics. Then I examine the two common interpretations of mass: mass as a measure of the quantity of matter and mass as a dynamical property. I claim that the former is ill-defined, and the latter is only plausible with respect to a metaphysical interpretation of laws of nature. I explore the following options (...) for the status of laws: Humeanism, primitivism about laws, dispositionalism, and ontic structural realism. (shrink)

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.

Thomists do not have an account of how modern mathematics relates to classical mathematics or more generally fits into the Aristotelian hierarchy of sciences. Rather than treat primarily of Aquinas’s theses on mathematical abstraction, I turn to considering what modern mathematics is in itself, seen from a broadly classical perspective. I argue that many modern quantities can be considered to be, not quantities as such or in themselves, but derived quantities, i.e., quantities that can be defined wholly in terms of (...) the principles of number or magnitude. I also interpret the parts of modern mathematics that study quantitative change as being properly-speaking parts of natural philosophy, for example, probability theory, statistics, calculus, etc. In conclusion, I consider the place that quantity as such has in the order of the world and why we should expect the world to be highly mathematical, as we have found it to be. (shrink)

Quantities like mass and temperature are properties that come in degrees. And those degrees (e.g. 5 kg) are properties that are called the magnitudes of the quantities. Some philosophers (e.g., Byrne 2003; Byrne & Hilbert 2003; Schroer 2010) talk about magnitudes of phenomenal qualities as if some of our phenomenal qualities are quantities. The goal of this essay is to explore the anti-physicalist implication of this apparently innocent way of conceptualizing phenomenal quantities. I will first argue for a metaphysical thesis (...) about the nature of magnitudes based on Yablo’s proportionality requirement of causation. Then, I will show that, if some phenomenal qualities are indeed quantities, there can be no demonstrative concepts about some of our phenomenal feelings. That presents a significant restriction on the way physicalists can account for the epistemic gap between the phenomenal and the physical. I’ll illustrate the restriction by showing how that rules out a popular physicalist response to the Knowledge Argument. (shrink)

Despite changeability of the world, the human mind also ponders on those quantities that remain constant over time. This was the case in ancient times, in the middle ages, and the same applies in modern physics. This paper discusses i.a. Zenon paradoxes, the principle of inertia, and the Emma Noether theorem, ending with the modern, so-called Zeno’s quantum effect. The foot-notes concern the ancient “Achilles” paradox, spot speed, as well as some of the facts taken out of the life-history of (...) Emma Noether, as well as the example of applying her theorem. (shrink)

This article introduces and motivates the notion of a “properly extensive” quantity by means of a puzzle about the reliability of certain canonical length measurements. An account of these measurements’ success, I argue, requires a modally robust connection between quantitative structure and mereology that is not mediated by the dynamics and is stronger than the constraints imposed by “mere additivity.” I outline what it means to say that length is not just extensive but properly so and then briefly sketch (...) an application of proper extensiveness to the project of providing a reductive ground for metric quantitative structure. (shrink)

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 paper proposes an extended version of the interventionist account for causal inference in the practical context of biological mechanism research. This paper studies the details of biological mechanism researchers’ practices of assessing the evidential legitimacy of experimental data, arguing why quantity and variety are two important criteria for this assessment. Because of the nature of biological mechanism research, the epistemic values of these two criteria result from the independence both between the causation of data generation and the causation (...) in question and between different interventions, not techniques. The former independence ensures that the interventions in the causation in question are not affected by the causation that is responsible for data generation. The latter independence ensures the reliability of the final mechanisms not only in the empirical but also the formal aspects. This paper first explores how the researchers use quantity to check the effectiveness of interventions, where they at the same time determine the validity of the difference-making revealed by the results of interventions. Then, this paper draws a distinction between experimental interventions and experimental techniques, so that the reliability of mechanisms, as supported by the variety of evidence, can be safely ensured in the probabilistic sense. The latter process is where the researchers establish evidence of the mechanisms connecting the events of interest. By using case studies, this paper proposes to use ‘intervention’ as the fruitful connecting point of literature between evidence and mechanisms. (shrink)

The Reign of Quantity gives a concise but comprehensive view of the present state of affairs in the world, as it appears from the point of view of the 'ancient wisdom', formerly common both to the East and to the West, but now almost entirely lost sight of. The author indicates with his fabled clarity and directness the precise nature of the modern deviation, and devotes special attention to the development of modern philosophy and science, and to the (...) part played by them, with their accompanying notions of progress and evolution, in the formation of the industrial and democratic society which we now regard as 'normal'. Guénon sees history as a descent from Form (or Quality) toward Matter (or Quantity); but after the Reign of Quantity-modern materialism and the 'rise of the masses'-Guénon predicts a reign of 'inverted quality' just before the end of the age: the triumph of the 'counter-initiation', the kingdom of Antichrist. This text is considered the magnum opus among Guénon's texts of civilizational criticism, as is Symbols of Sacred Science among his studies on symbols and cosmology, and Man and His Becoming according to the Vedanta among his more purely metaphysical works. (shrink)

In his Commentary on Ptolemy’s Harmonics, Porphyry debunks Ptolemy’s quantitative theory of pitches by demonstrating that pitches are qualitative attributes of sound. I argue that Porphyry’s main concern is to save the phenomenological dimension of sound while preserving the possibility of a quantitative analysis of music. I show how he draws on the Aristotelian tradition to develop a theory of pitches as emergent properties that covary with some underlying quantitative features without being reducible to them. Porphyry offers an original and (...) compelling solution to the classic problem of how to provide a quantitative analysis of a qualitative phenomenon. (shrink)

Kant's obscure essay entitled An Attempt to Introduce the Concept of Negative Quantities into Philosophy has received virtually no attention in the Kant literature. The essay has been in English translation for over twenty years, though not widely available. In his original 1983 translation, Gordon Treash argues that the Negative Quantities essay should be understood as part of an ongoing response to the philosophy of Christian Wolff. Like Hoffmann and Crusius before him, the Kant of 1763 is at (...) odds with the Leibnizian-Wolffian tradition of deductive metaphysics. He joins his predecessors in rejecting the assumption that the law of contradiction alone can provide proof of the principle of sufficient reason: -/- In his rejection of the possibility of deducing all philosophic truth from the law of contradiction, however, and in the clear recognition that this impossibility has immediate consequences for defense of the law of sufficient reason, Kant's work most definitely and positively constitutes a line of succession from Hoffmann and Crusius (Treash, 1983, p. 25). -/- The recognition that Kant's Negative Quantities essay is part of a response to the tradition of deductive metaphysics is, without a doubt, an important contribution to the Kant literature. However, there is still more to be said about this neglected essay. The full significance of the paper becomes known through its ties to a second, empiricist line of succession. Clues to this second line of succession can be found in Kant's prefatory remarks concerning Euler's 1748 Reflections on Space and Time and Crusius' 1749 Guidance in the Orderly and Careful Consideration of Natural Events. As I will show, these prefatory remarks suggest a reading of Kant's Negative Quantities paper that reaches beyond German deductive metaphysics to engage a debate regarding the application of mathematics in philosophy initiated by George Berkeley. (shrink)

In a recent paper (1994) Wesley Salmon has replied to criticisms (e.g., Dowe 1992c, Kitcher 1989) of his (1984) theory of causality, and has offered a revised theory which, he argues, is not open to those criticisms. The key change concerns the characterization of causal processes, where Salmon has traded "the capacity for mark transmission" for "the transmission of an invariant quantity." Salmon argues against the view presented in Dowe (1992c), namely that the concept of "possession of a conserved (...) quantity" is sufficient to account for the difference between causal and pseudo processes. Here that view is defended, and important questions are raised about the notion of transmission and about gerrymandered aggregates. (shrink)

A discussion of Suarez's views on continuous quantity in the context of his place in the history of philosophy. The paper raises issues about conceptual change in intellectual history. It advances original interpretations of Aristotle and Suarez on continuous quantity.

This rich book differs from much contemporary philosophy of mathematics in the author’s witty, down to earth style, and his extensive experience as a working mathematician. It accords with the field in focusing on whether mathematical entities are real. Franklin holds that recent discussion of this has oscillated between various forms of Platonism, and various forms of nominalism. He denies nominalism by holding that universals exist and denies Platonism by holding that they are concrete, not abstract - looking to (...) Aristotle for inspiration. (shrink)

In a recent paper Wesley Salmon has replied to criticisms of his theory of causality, and has offered a revised theory which, he argues, is not open to those criticisms. The key change concerns the characterization of causal processes, where Salmon has traded “the capacity for mark transmission” for “the transmission of an invariant quantity.” Salmon argues against the view presented in Dowe, namely that the concept of “possession of a conserved quantity” is sufficient to account for the (...) difference between causal and pseudo processes. Here that view is defended, and important questions are raised about the notion of transmission and about gerrymandered aggregates. (shrink)

I defend the conserved quantity theory of causation against two objections: firstly, that to tie the notion of “cause” to conservation laws is impossible, circular or metaphysically counterintuitive; and secondly, that the conserved quantity theory entails an undesired notion of identity through time. My defence makes use of an important meta-philosophical distinction between empirical analysis and conceptual analysis. My claim is that the conserved quantity theory of causation must be understood primarily as an empirical, not a conceptual, (...) analysis of causation. (shrink)

A description is given of the quantitative-qualitative distinction for terms in theories of measurable attributes, and, adjoined to that account, a suggestion is made concerning the sense in which empirical relational systems have an empirical attribute as their topic or focus. Since this characterization of quantitative terms, relative to a partition, makes no explicit reference to numbers, concatenation operations, or ordering relations, we show how our results are related to some standard theorems in the literature. Analogs of representation and uniqueness (...) theorems are proved, and the notions of exact quantitative term and the underlying attribute of a quantitative term, are described and studied. (shrink)

ABSTRACTThis paper presents a view of quantities as ‘adverbial’ entities of a certain kind—more specifically, determinate ways, or modes, of having length, mass, speed, and the like. In doing so, it will be argued that quantities as such should be distinguished from quantitative properties or relations, and are not universals but are particulars, although they are not objects, either. A main advantage of the adverbial view over its rivals will be found in its superior explanatory power with respect to both (...) certain fundamental principles of quantity and ordinary quantitative reasoning involving quantitative relations like three times as long as and 2 metres longer than. (shrink)

Immanuel Kant's Metaphysical Foundations of Natural Science (1786) provides metaphysical foundations for the application of mathematics to empirically given nature. The application that Kant primarily has in mind is that achieved in Isaac Newton's Principia (1687). Thus, Kant's first chapter, the Phoronomy, concerns the mathematization of speed or velocity, and his fourth chapter, the Phenomenology, concerns the empirical application of the Newtonian notions of true or absolute space, time, and motion. This paper concentrates on Kant's second and third chapters—the Dynamics (...) and the Mechanics, respectively—and argues that they are best read as providing a transcendental explanation of the conditions for the possibility of applying the (mathematical) concept of quantity of matter to experience. Kant again has in mind the empirical measures of this quantity that Newton fashions in the Principia, and he aims to make clear, in particular, how Newton achieves a universal measure for all bodies whatsoever by projecting the static quantity of terrestrial weight into the heavens by means of the theory of universal gravitation. Kant is not attempting to prove a priori what Newton has established empirically but, rather, to clarify the character of Newton's mathematization by building Newton's empirical measures into the very concept of matter that is articulated in the Metaphysical Foundations. (shrink)

The conserved quantity theory of causation aims to analyze causal processes and interactions in terms of conserved quantities. In order to be successful, the theory must correctly distinguish between causal processes and interactions, on the one hand, and pseudoprocesses and mere intersections on the other.Moreover, it must do this while satisfying two further criteria: it must avoid circularity; and the appeal to conserved quantities must not be redundant. I argue that the theory is not successful in meeting these criteria.

This paper is a book review of "An Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure" by James Franklin.

I defend the conserved quantity theory of causation against two objections: firstly, that to tie the notion of "cause"to conservation laws is impossible, circular or metaphysically counterintuitive ; and secondly, that the conser quantity theory entails an undesired notion of identity through time. My defence makes use of an important meta-philosophical distinction between empirical analysis and conceptual analysis. My claim is that the conserved quantity theory of causation must be understood primarily as an empirical, not a conceptual, (...) analysis of causation. (shrink)

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 considering quantity as an attribute and as divisible. There follows a brief consideration of the notions arrived at for their relevance to modern mathematics with particular reference to the real number system and the notion of infinity. (shrink)