The mathematical form of measurement and the argument for Proposition I in Newton’s Principia

Newton characterizes the reasoning of Principia Mathematica as geometrical. He emulates classical geometry by displaying, in diagrams, the objects of his reasoning and comparisons between them. Examination of Newton??s unpublished texts (and the views of his mentor, Isaac Barrow) shows that Newton conceives geometry as the science of measurement. On this view, all measurement ultimately involves the literal juxtaposition??the putting-together in space??of the item to be measured with a measure, whose dimensions serve as the standard of reference, so that all quantity (which is what measurement makes known) is ultimately related to spatial extension. I use this conception of Newton??s project to explain the organization and proofs of the first theorems of mechanics to appear in the Principia (beginning in Sect. 2 of Book I). The placementof Kepler??s rule of areas as the first proposition, and the manner in which Newton proves it, appear natural on the supposition that Newton seeks a measure, in the sense of a moveable spatial quantity, of time. I argue that Newton proceeds in this way so that his reasoning can have the ostensive certainty of geometry.     

Naturalism,Experience, and Hume’s ‘Science of Human Nature’

Abstract

A standard interpretation of Hume’s naturalism is that it paved the way for a scientistic and ‘disenchanted’ conception of the world. My aim in this paper is to show that this is a restrictive reading of Hume, and it obscures a different and profitable interpretation of what Humean naturalism amounts to. The standard interpretation implies that Hume’s ‘science of human nature’ was a reductive investigation into our psychology. But, as Hume explains, the subject matter of this science is not restricted to introspectively accessible mental content and incorporates our social nature and interpersonal experience. Illuminating the science of human nature has implications for how we understand what Hume means by ‘experience’ and thus how we understand the context of his epistemological investigations. I examine these in turn and argue overall that Hume’s naturalism and his science of man do not simply anticipate a disenchanted conception of the world.     

Beyond Rigidity: The Unfinished Semantic Agenda of Naming and Necessity

To articulate their understanding of Hume’s discussion of ‘distinctions of reason’, commentators have often taken what I refer to as a ‘respect-first view’ on resemblance, in which they categorize resemblance as based on resembling respects. Holding this view, Donald Baxter argues that Hume’s view on the distinctions of reason leads to a contradiction. As an alternative, I offer ‘the resemblance-first view’, which is not dependent on resembling respects. I argue that this view is textually supported, and that it rescues Hume from the proposed contradiction.     

STANDARDS OF EQUALITY AND HUME'S VIEW OF GEOMETRY

It has been argued that there is a genuine conflict between the views of geometry defended by Hume in the Treatise and in the Enquiry: while the former work attributes to geometry a different status from that of arithmetic and algebra, the latter attempts to restore its status as an exact and certain science. A closer reading of Hume shows that, in fact, there is no conflict between the two works with respect to geometry. The key to understanding Hume's view of geometry is the distinction he draws between two standards of equality in extension.     

From Inexactness to Certainty: The Change in Hume's Conception of Geometry

Although Hume's analysis of geometry continues to serve as a reference point for many contemporary discussions in the philosophy of science, the fact that the first Enquiry presents a radical revision of Hume's conception of geometry in the Treatise has never been explained. The present essay closely examines Hume's early and late discussions of geometry and proposes a reconstruction of the reasons behind the change in his views on the subject. Hume's early conception of geometry as an inexact non-demonstrative science is argued to be a consequence of his attempt to discredit geometrical proofs of infinite divisibility of extension by anchoring the meaning of geometrical concepts in inherently inexact qualitative measurement procedures. This measurement-based attack on the exactness and certainty of geometry is analyzed and shown to be both self-refuting and inconsistent with the general epistemological framework of the Treatise. The revised conception of geometry as a demonstrative science in the first Enquiry is then interpreted as Hume's response to the failure of his earlier attempt to discredit geometrical proofs of infinite divisibility of extension. This revised version was published online in August 2006 with corrections to the Cover Date.     

The Place Of Geometry: Heidegger's Mathematical Excursus On Aristotle

'The Place of Geometry' discusses the excursus on mathematics from Heidegger's 1924–25 lecture course on Platonic dialogues, which has been published as Volume 19 of the Gesamtausgabe as Plato's Sophist , as a starting point for an examination of geometry in Euclid, Aristotle and Descartes. One of the crucial points Heidegger makes is that in Aristotle there is a fundamental difference between arithmetic and geometry, because the mode of their connection is different. The units of geometry are positioned, the units of arithmetic unpositioned. Following Heidegger's claim that the Greeks had no word for space, and David Lachterman's assertion that there is no term corresponding to or translatable as 'space' in Euclid's Elements , I examine when the term 'space' was introduced into Western thought. Descartes is central to understanding this shift, because his understanding of extension based in terms of mathematical co-ordinates is a radical break with Greek thought. Not only does this introduce this word 'space' but, by conceiving of geometrical lines and shapes in terms of numerical co-ordinates, which can be divided, it turns something that is positioned into unpositioned. Geometric problems can be reduced to equations, the length (i.e, quantity) of lines: a problem of number. The continuum of geometry is transformed into a form of arithmetic. Geometry loses position just as the Greek notion of 'place' is transformed into the modern notion of space.     

What Can Causal Claims Mean?

How can Hume account for the meaning of causal claims? The causal realist, I argue, is, on Hume's view, saying something nonsensical. I argue that both realist and agnostic interpretations of Hume are inconsistent with his view of language and intentionality. But what then accounts for this illusion of meaning? And even when we use causal terms in accordance with Hume’s definitions, we seem merely to be making disguised self-reports. I argue that Hume’s view is not as implausible as it sounds by exploring his conception of language.     

Hume's nominalism and the Copy Principle

I consider some ways in which the Copy Principle (CP) and Hume's nominalism impinge on one another, arguing for the following claims. First, Hume's argument against indeterminate ideas isn't cogent even if the CP is accepted. But this does not vindicate Locke: the imagistic conception of ideas, presupposed by the CP, will force Locke to accept something like Hume's view of the way general terms function, the availability of abstract ideas notwithstanding. Second, Hume's discussion of nominalism provides support for the “old Hume” interpretation, that which takes the CP to be a criterion of meaningfulness, as against the “new Hume” reading, according to which it constrains what we can know. Finally, nominalism forces Hume to adopt a more complicated theory of ideas.     

Hume on sympathy and agreeable qualities

Hume says that sympathy is the source of our moral feeling of approval for useful qualities. But does Hume give the same psychological explanation of our approval of immediately agreeable qualities as he does to our approval of useful qualities? Does he trace our moral approbation of immediately agreeable qualities to sympathy? Some commentators, including Rachel Cohon and Don Garrett, argue that he does not. Let us call this view the ‘narrow view’ of sympathy in contrast to the ‘wide view’ of sympathy, which holds that sympathy is required for every moral sentiment. There is indeed some apparent textual evidence in Hume’s work that seems to support the narrow view. My aim in this paper is to examine that evidence and show how it is merely apparent, in particular by showing how a number of passages can be and are misread. I thus want to argue indirectly for the wide view.     

Kant’s conception of proper science

Kant is well known for his restrictive conception of proper science. In the present paper I will try to explain why Kant adopted this conception. I will identify three core conditions which Kant thinks a proper science must satisfy: systematicity, objective grounding, and apodictic certainty. These conditions conform to conditions codified in the Classical Model of Science. Kant’s infamous claim that any proper natural science must be mathematical should be understood on the basis of these conditions. In order to substantiate this reading, I will show that only in this way it can be explained why Kant thought (1) that mathematics has a particular foundational function with respect to the natural sciences and (2) as such secures their scientific status.     

Kant on geometry and spatial intuition

I use recent work on Kant and diagrammatic reasoning to develop a reconsideration of central aspects of Kant??s philosophy of geometry and its relation to spatial intuition. In particular, I reconsider in this light the relations between geometrical concepts and their schemata, and the relationship between pure and empirical intuition. I argue that diagrammatic interpretations of Kant??s theory of geometrical intuition can, at best, capture only part of what Kant??s conception involves and that, for example, they cannot explain why Kant takes geometrical constructions in the style of Euclid to provide us with an a priori framework for physical space. I attempt, along the way, to shed new light on the relationship between Kant??s theory of space and the debate between Newton and Leibniz to which he was reacting, and also on the role of geometry and spatial intuition in the transcendental deduction of the categories.     

Self-knowledge, Externalism and Scepticism: David OwensScepticisms: Descartes and Hume

Contemporary discussion of scepticism focuses on the possibility that most or all of our beliefs might be false. I argue that the hypothesis of massive falsity and the associated 'problem of the external world' are inessential to the scepticisms of Descartes and Hume. What drives Cartesian and Humean scepticism is the demand for certainty: any possibility of error, however local, must be ruled out before we can claim either justified belief or knowledge. Contemporary philosophers have ignored this form of scepticism because they doubt that the demand for certainty can be motivated. But Descartes provides a sound motivation for this demand in the Meditations.     

Psychology as a Science of Subject and Comportment,beyond the Mind and Behavior

The turn of qualitative inquiry suggests a more open, plural conception of psychology than just the science of the mind and behavior as it is most commonly defined. Historical, ontological and epistemological binding of this conception of psychology to the positivist method of natural science may have exhausted its possibilities, and after having contributed to its prestige as a science, has now become an obstacle. It is proposed that psychology be reconceived as a science of subject and comportment in the framework of a contextual hermeneutic, social, human behavioral science. Thus, without rejecting quantitative inquiry, psychology recovers territory left aside like introspection and pre-reflective self-awareness, and reconnects with traditions marginalized from the main stream. From this perspective psychology might also recover its credibility as a human science in view of current skepticism.     

Hermann Weyl on Minkowskian Space–Time and Riemannian Geometry

Hermann Weyl as a founding father of field theory in relativistic physics and quantum theory always stressed the internal logic of mathematical and physical theories. In line with his stance in the foundations of mathematics, Weyl advocated a constructivist approach in physics and geometry. An attempt is made here to present a unified picture of Weyl’s conception of space–time theories from Riemann to Minkowski. The emphasis is on the mathematical foundations of physics and the foundational significance of a constructivist philosophical point of view. I conclude with some remarks on Weyl’s broader philosophical views.     

Kant's Criticisms of Hume's Dialogues Concerning Natural Religion

According to recent commentators like Paul Guyer, Kant agrees with Hume's Dialogues Concerning Natural Religion (1) that physico-theology can never provide knowledge of God and (2) that the concept of God, nevertheless, provides a useful heuristic principle for scientific enquiry. This paper argues that Kant, far from agreeing with Hume, criticizes Hume's Dialogues for failing to prove that physico-theology can never yield knowledge of God and that Kant correctly views Hume's Dialogues as a threat to, rather than an anticipation of, his own view that the concept of God provides a useful heuristic principle for science. The paper concludes that Kant's critique of physico-theology reflects Kant's deep dissatisfaction with Hume's manner of argumentation and suggests that Kant's attempt to provide a more successful critique of physico-theology merits continued philosophical attention.     

Erkenntnistheorie der Zahldefinition und philosophische Grundlegung der Arithmetik unter Bezugnahme auf einen Vergleich von Gottlob Freges Logizismus und platonischer Philosophie (Syrian, Theon von Smyrna u.a.)

The epistomology of the definition of number and the philosophical foundation of arithmetic based on a comparison between Gottlob Frege's logicism and Platonic philosophy (Syrianus, Theo Smyrnaeus, and others). The intention of this article is to provide arithmetic with a logically and methodologically valid definition of number for construing a consistent philosophical foundation of arithmetic. The – surely astonishing – main thesis is that instead of the modern and contemporary attempts, especially in Gottlob Frege's Foundations of Arithmetic, such a definition is found in the arithmetic in Euclid's Elements. To draw this conclusion a profound reflection on the role of epistemology for the foundation of mathematics, especially for the method of definition of number, is indispensable; a reflection not to be found in the contemporary debate (the predominate ‘pragmaticformalism’ in current mathematics just shirks from trying to solve the epistemological problems raised by the debate between logicism, intuitionism, and formalism). Frege's definition of number, ‘The number of the concept F is the extension of the concept ‘numerically equal to the concept F”, which is still substantial for contemporary mathematics, does not fulfil the requirements of logical and methodological correctness because the definiens in a double way (in the concepts ‘extension of a concept’ and ‘numerically equal’) implicitly presupposes the definiendum, i.e. number itself. Number itself, on the contrary, is defined adequately by Euclid as ‘multitude composed of units’, a definition which is even, though never mentioned, an implicit presupposition of the modern concept ofset. But Frege rejects this definition and construes his own - for epistemological reasons: Frege's definition exactly fits the needs of modern epistemology, namely that for to know something like the number of a concept one must become conscious of a multitude of acts of producing units of ‘given’ representations under the condition of a 1:1 relationship to obtain between the acts of counting and the counted ‘objects’. According to this view, which has existed at least since the Renaissance stoicism and is maintained not only by Frege but also by Descartes, Kant, Husserl, Dummett, and others, there is no such thing as a number of pure units itself because the intellect or pure reason, by itself empty, must become conscious of different units of representation in order to know a multitude, a condition not fulfilled by Euclid's conception. As this is Frege's main reason to reject Euclid's definition of number (others are discussed in detail), the paper shows that the epistemological reflection in Neoplatonic mathematical philosophy, which agrees with Euclid's definition of number, provides a consistent basement for it. Therefore it is not progress in the history of science which hasled to the a poretic contemporary state of affairs but an arbitrary change of epistemology in early modern times, which is of great influence even today. This revised version was published online in August 2006 with corrections to the Cover Date.     

Science and Life-World: Husserl, Schutz, Garfinkel

In this article I intend to explore the conception of science as it emerges from the work of Husserl, Schutz, and Garfinkel. By concentrating specifically on the issue of science, I attempt to show that Garfinkel’s views on the relationship between science and the everyday world are much closer to Husserl’s stance than to the Schutzian perspective. To this end, I explore Husserl’s notion of science especially as it emerges in the Crisis of European Sciences, where he describes the failure of European science and again preaches for a return to the “things themselves”. In this respect I interpret ethnomethodology’s most recent program as an answer to that call originating from a sociological domain. I then argue that the Husserlian turn within ethnomethodology marks the split between Garfinkel and Schutz. In fact I try to show that Schutz’s epistemological work is only partially inspired by phenomenology and that his conception of science retains a rationalist stance that ethnomethodology opposes. In the final section I briefly discuss Garfinkel’s most recent program as a way of closing the gap between theory and experience by linking the topics of science to the radical experiential phenomena.     

Reading Kant’s doctrine of schematism algebraically

Kant’s investigations into so-called a priori judgments of pure mathematics in the Critique of Pure Reason (KrV) are mainly confined to geometry and arithmetic both of which are grounded on our pure forms of intuition, space, and time. Nevertheless, as regards notions such as irrational numbers and continuous magnitudes, such a restricted account is crucially problematic. I argue that algebra can play a transcendental role with respect to the two pure intuitive sciences, arithmetic and geometry, as the condition of their possibility. It follows that Kant’s schematism of the concept of magnitude ought to be quantitatively represented in algebraic formulas in general, and also undergo several modifications in order to suit continuities.     

Kant and experimental philosophy

While Kant introduces his critical philosophy in continuity with the experimental tradition begun by Francis Bacon, it is widely accepted that his Copernican revolution places experimental physics outside the bounds of science. Yet scholars have recently contested this view. They argue that in Critique of the Power of Judgment Kant’s engagement with the growing influence of vitalism in the 1780s leads to an account of nature’s formative power that returns experimental physics within scientific parameters. Several critics are sceptical of this revised reading. They argue that Kant’s third Critique serves precisely to deflate the epistemological status of experimental physics, thereby protecting science from the threat of vitalism. In this paper I examine Kant’s account of science in the context of the experimental tradition of philosophy, particularly in relation to the generation dilemma of the eighteenth century. I argue that Kant does not deflate the epistemological status of experimental physics but rather introduces systematicity to the experimental tradition. By identifying the reflective use of reason to organize laws of experience into a systematic whole, Kant aims to ground experimental inquiry on the secure course of a science, opening a conception of science as a research programme.     

Mathematics and the Mystical in the Thought of Simone Weil

On Simone Weil’s “Pythagorean” view, mathematics has a mystical significance. In this paper, the nature of this significance and the coherence of Weil’s view are explored. To sharpen the discussion, consideration is given to both Rush Rhees’ criticism of Weil and Vance Morgan’s rebuttal of Rhees. It is argued here that while Morgan underestimates the force of Rhees’ criticism, Rhees’ take on Weil is, nevertheless, flawed for two reasons. First, Rhees fails to engage adequately with either the assumptions underlying Weil’s religious conception of philosophy or its dialectical method. Second, Rhees’ reading of Weil reflects an anti-Platonist conception of mathematics his justification of which is unsound and whose influence impedes recognition of the coherence of Weil’s position.