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 who have denied that space and time are infinitely divisible, the most notable are the ancient atomists, and Berkeley and Hume. All of these assert not only that space and time might be atomic, but that they must be. Infinite divisibility is, they say, impossible on purely conceptual grounds. (shrink)

Hume seems to argue unconvincingly against the infinite divisibility of finite regions of space. I show that his conclusion is entailed by respectable metaphysical principles which he held. One set of principles entails that there are partless (unextended) things. Another set entails that these cannot be ordered so that an infinite number of them compose a finite interval.

In lieu of an abstract, here is a brief excerpt of the content:Hume Studies Volume XX, Number 2, November 1994, pp. 219-240 Infinite Divisibility in Hume's First Enquiry DALE JACQUETTE The Limitations of Reason The arguments against infinite divisibility in the notes to Sections 124 and 125 of David Hume's Enquiry Concerning Human Understanding are presented as "sceptical" results about the limitations of reason. The metaphysics of infinite divisibility is introduced merely as a particular, (...) though especially representative problem, among several that Hume discusses. Hume first writes: The chief objection against all abstract reasonings is derived from the ideas of space and time; ideas, which, in common life and to a careless view, are very clear and intelligible, but when they pass through the scrutiny of the profound sciences (and they are the chief object of these sciences) afford principles, which seem full of absurdity and contradiction.1 There follows an intuitive appeal to the apparent incoherence of the consequences of infinite divisibility in the geometry of space and measurement of time. Hume enlists natural belief against the infinite divisibility thesis in the mathematics of extension: But what renders the matter more extraordinary, is, that these seemingly absurd opinions are supported by a chain of reasoning, the Dale Jacquette is at the Department of Philosophy, The Pennsylvania State University, 246 Sparks Building, University Park PA 16802 USA. e-mail: [email protected] 220 Dale Jacquette clearest and most natural; nor is it possible for us to allow the premises without admitting the consequences. Nothing can be more convincing and satisfactory than all the conclusions concerning the properties of circles and triangles; and yet, when these are once received, how can we deny, that the angle of contact between a circle and its tangent is infinitely less than any rectilinear angle, that as you may increase the diameter of the circle in infinitum, this angle of contact becomes still less, even in infinitum, and that the angle of contact between other curves and their tangents may be infinitely less than those between any circle and its tangent, and so on, in infinitum? The demonstration of these principles seems as unexceptionable as that which proves the three angles of a triangle to be equal to two right ones, though the latter opinion be natural and easy, and the former big with contradiction and absurdity. (EHU 156-157) The implied sense of contradiction is then extended by Hume to the problem of infinite divisibility in time. The absurdity of these bold determinations of the abstract sciences seems to become, if possible, still more palpable with regard to time than extension. An infinite number of real parts of time, passing in succession, and exhausted one after another, appears so evident a contradiction, that no man, one should think, whose judgment is not corrupted, instead of being improved, by the sciences, would ever be able to admit of it. (EHU 157) The contradictions involved in traditional concepts of extension and time demarcate the limits of which reason is capable. But Hume's answer is more complex than merely surrendering to skepticism over the prospects of reason achieving understanding in the metaphysics of space and time. He tries to show that what he calls "Pyrrhonian" skepticism is ultimately self-defeating in its application of reason against reason. In place of a negative conclusion, he offers another solution to the paradoxes of infinite divisibility. What is remarkable about Hume's pronouncements against the infinite divisibility of extension and time, that may signal a change in his methodology if not his overall position on divisibility in the Enquiry from that of A Treatise of Human Nature, is his straightforward claim that natural belief rebels against the concept of infinite divisibility as its consequences apply to the metric of space and time.2 As in the Treatise, Hume pits natural disbelief in infinite divisibility against abstract metaphysical reason. But in the Enquiry, he does not invoke reason merely to correct the excesses of natural belief for the skeptical deflation of natural belief, say, in the necessity of causal connection, or the existence of mind-independent continuants. On the contrary, Hume Studies Infinite Divisibility in Hume's First Enquiry 221 Hume here appears instead to invoke natural disbelief in... (shrink)

Hume on Infinite Divisibility and Sensible Extensionless Indivisibles DALE JACQUETTE 'Twere certainly to be wish'd, that some expedient were fallen upon to reconcile philosophy and common sense, which with regard to the question of infinite divisibility have wag'd most cruel wars with each other. David Hume, A Treatise of Human Nature 1. THE DIVISIBILITY ARGUMENTS David Hume's refutation of the infinite divisibility of space and time, and his doctrine of the sensible extensionless (...) indivisibles that constitute extension, are perhaps the least loved, and until recently, least examined aspects of his early philosophy. The fact that Hume reduced his lengthy discussion of the prob- lem from the Treatise to just two pages of the first Enquiry, together with his disavowal of the first work in the Enquiry's "Advertisement," have reinforced the tendency to ignore his arguments. ~ Opinions on the merits of Hume's analysis of space are widely divided. To give a sense of the disparity, Donald L. M. Baxter, in "Hume on Infinite Divisibility," maintains: "Far from begging the question, Hume has available a respectable argument against the infinite divisibility of finite spatial intervals."' C. D. Broad, on the contrary, in his Dawes Hicks Lecture on Philosophy to the British Academy, "Hume's Doctrine of Space," concludes: "I think, then, that Hume's whole account of spatial divisibility can David Hume, Essays and Treatises on... (shrink)

This essay examines David Hume's principal criticism of the idea of the infinite divisibility of extension in the ink-spot experiment of _Treatise<D>, Book I, Part II, and his arguments for his positive theory of finitely divisible space as composed of finitely many sensible extensionless indivisibles or _minima sensibilia<D>. The essay considers Hume's strict finitist metaphysics of space in the context of his reactions to a trilemma about the impossibility of the divisibility of extension on any (...) theory posed by Pierre Bayle in the article on "Zeno of Elea" in his _Dictionary Historical and Critical<D>. I evaluate Hume's arguments in light of complaints raised by contemporary commentators, and I compare the inference of the _Treatise<D> with Hume's final words on the subject, in the cryptic Berkeleyan 'hint' of the first _Enquiry<D>, to the effect that the idea of infinite divisibility is subject to general objections to 'all _abstract<D> reasonings'. (shrink)

Hume's Treatise arguments concerning space, time, and geometry, especially ones involving his denial of infinite divisibility; have suffered harsh criticism. I show that in the section "Of the ideas of space and time," Hume gives important characterizations of his skeptical approach, in some respects Pyrrhonian, that will be developed in the rest of the Treatise. When that approach is better understood, the force of Hume's arguments can be appreciated, and the influential criticisms of them can be (...) seen to miss the mark. (shrink)

§1. Introduction. Asymptotic cones of metric spaces were first invented by Gromov. They are metric spaces which capture the ‘large-scale structure’ of the underlying metric space. Later, van den Dries and Wilkie gave a more general construction of asymptotic cones using ultrapowers. Certain facts about asymptotic cones, like the completeness of the metric space, now follow rather easily from saturation properties of ultrapowers, and in this survey, we want to present two applications of the van den Dries-Wilkie approach. (...) Using ultrapowers we obtain an explicit description of the asymptotic cone of a semisimple Lie group. From this description, using semi-algebraic groups and non-standard methods, we can give a short proof of the Margulis Conjecture. In a second application, we use set theory to answer a question of Gromov.§2. Definitions. The intuitive idea behind Gromov's concept of an asymptotic cone was to look at a given metric space from an ‘infinite distance’, so that large-scale patterns should become visible. In his original definition this was done by gradually scaling down the metric by factors 1/nfornϵ ℕ. In the approach by van den Dries and Wilkie, this idea was captured by ultrapowers. Their construction is more general in the sense that the asymptotic cone exists for any metric space, whereas in Gromov's original definition, the asymptotic cone existed only for a rather restricted class of spaces. (shrink)

Comments on Fullerton's paper, 'The doctrine of space and time' (1901). Fullerton adapts both Kantian and Berkelian doctrine of space, but his view on space is dominated more by the Berkelian views. His views on time are the same as that on space. His comparative study on animals has been criticized since it is claimed that animals are living wholes and abstracting certain parts have little value for fundamental comparison. The concepts of infinity and infinite (...) divisibility are considered important. However, he misses the real import of infinity by recognizing the possibility of infinite divisibility and ignores the distinction between being infinitely divisible and infinitely divided. The world according to him is not given directly by intuition and consciousness can only represent it. (shrink)

Hume’s discussion of space, time, and mathematics at T 1.2 appeared to many earlier commentators as one of the weakest parts of his philosophy. From the point of view of pure mathematics, for example, Hume’s assumptions about the infinite may appear as crude misunderstandings of the continuum and infinite divisibility. I shall argue, on the contrary, that Hume’s views on this topic are deeply connected with his radically empiricist reliance on phenomenologically given sensory images. He insightfully (...) shows that, working within this epistemological model, we cannot attain complete certainty about the continuum but only at most about discrete quantity. Geometry, in contrast to arithmetic, cannot be a fully exact science. A number of more recent commentators have offered sympathetic interpretations of Hume’s discussion aiming to correct the older tendency to dismiss this part of the Treatise as weak and confused. Most of these commentators interpret Hume as anticipating the contemporary idea of a finite or discrete geometry. They view Hume’s conception that space is composed of simple indivisible minima as a forerunner of the conception that space is a discretely (rather than continuously) ordered set. This approach, in my view, is helpful as far as it goes, but there are several important features of Hume’s discussion that are not sufficiently appreciated. I go beyond these recent commentators by emphasizing three of Hume’s most original contributions. First, Hume’s epistemological model invokes the “confounding” of indivisible minima to explain the appearance of spatial continuity. Second, Hume’s sharp contrast between the perfect exactitude of arithmetic and the irremediable inexactitude of geometry reverses the more familiar conception of the early modern tradition in pure mathematics, according to which geometry (the science of continuous quantity) has its own standard of equality that is independent from and more exact than any corresponding standard supplied by algebra and arithmetic (the sciences of discrete quantity). Third, Hume has a developed explanation of how geometry (traditional Euclidean geometry) is nonetheless possible as an axiomatic demonstrative science possessing considerably more exactitude and certainty that the “loose judgements” of the vulgar. (shrink)

For infinite machines that are free from the classical Thomson’s lamp paradox, we show that they are not free from its inverted-in-time version. We provide a program for infinite machines and an infinite mechanism that demonstrate this paradox. While their finite analogs work predictably, the program and the infinite mechanism demonstrate an undefined behavior. As in the case of infinite Davies machines :671–682, 2001), our examples are free from infinite masses, infinite velocities, (...) class='Hi'>infinite forces, etc. Only infinite divisibility of space and time is assumed. Thus, the infinite devices considered are possible in a Newtonian Universe and they do not conflict with Newtonian mechanics. Note that the classical Thomson’s lamp paradox leads to infinite velocities which may not be producible in acceptable models of Newtonian mechanics. Finally, it is shown that the “paradox of predictability” is similar to the inverted Thomson’s lamp paradox. (shrink)

Mathematician and philosopher Hermann Weyl had our subject dead to rights.

Hume's discussion of the idea of space in his Treatise on Human Nature is fundamental to an understanding of his treatment of such central issues as the existence of external objects, the unity of the self, the relation between certainty and belief, and abstract ideas. Marina Frasca-Spada's rich and original study examines this difficult part of Hume's philosophical writings and connects it to eighteenth-century works in natural philosophy, mathematics and literature. Focusing on Hume's discussions of the infinite (...) class='Hi'>divisibility of extension, the origin of the idea of space, geometry, and the notion of a vacuum, she shows that the central questions of Hume's 'science of human nature' - what does the 'science of human nature' reveal about the mind and its operations? what is experience? - underlie all of these discussions. Her analysis points the way to a reassessment of the central current interpretative problems in Hume studies. (shrink)

This book provides an account of the unity of Immanuel Kant’s early metaphysics, including the moment he invents transcendental idealism. Matthew Rukgaber argues that a division between “two worlds”—the world of matter, force, and space on the one hand, and the world of metaphysical substances with inner states and principles preserved by God on the other—is what guides Kant’s thought. Until 1770 Kant consistently held a conception of space as a force-based material product of monads that are only (...) virtually present in nature. As Rukgaber explains, transcendental idealism emerges as a constructivist metaphysics, a view in which space and time are real relations outside of the mind, but those relations are metaphysically dependent on the subject. The subject creates the simple “now” and “here,” thus introducing into the intrinsically indeterminate and infinitely divisible continua of nature a metric with transformation rules that make possible all individuation and measurement. (shrink)

ABSTRACT This paper aims to show that intuitionistic Kripke models are a powerful tool for interpreting Kant’s ‘Critical Philosophy’. Part I reviews some old work of mine that applies these models to provide a reading of Kant’s second antinomy about the divisibility of matter and to answer several attacks on Kant’s antinomies. But it also points out three shortcomings of that original application. First, the reading fails to account for Kant’s second antinomy claim that matter is divisible ‘ad infinitum’ (...) and his first antinomy claim that empirical space extends only ‘ad indefinitum’. Secondly, in conformity with the ‘assertability’ heuristics for Kripke models, it attributes an assertability semantics to Kant. In fact, it attributes a pair of assertability semantics to Kant, but neither of these accords with modern assertabilism. And finally, one must wonder the propriety of using contemporary assertability and contemporary formal logic to save an eighteenth-century text. Part II goes historical to solve the problems about assertability. It outlines the Descartes–Leibniz dialectic about the infinite/indefinite distinction and demonstrates how each position reflects larger philosophical positions. It then demonstrates that Kant’s two versions of assertability and his version of the infinite/indefinite dichotomy emerge naturally from his attempts to resolve a tension in Leibniz’s philosophy. In light of this demonstration, Part III adapts the Kripke model interpretation so that it does reflect that dichotomy. It then refines the Kripke semantics in order to model some of Kant’s signature critical doctrines, and it shows how Kant’s critical version of the infinite/indefinite distinction differs from all those of his predecessors and from his own pre-critical version. It also addresses the question of using contemporary logical machinery to interpret Kant. Part IV turns the tables and uses what we have learned about Kant and modern semantics in order to address a pair of issues in the contemporary critical foundations of logic. (shrink)

According to a standard interpretation of Hume’s argument against infinite divisibility, Hume is raising a purely formal problem for mathematical constructions of infinite divisibility, divorced from all thought of the stuffing or filling of actual physical continua. I resist this. Hume’s argument must be understood in the context of a popular early modern account of the metaphysical status of the parts of physical quantities. This interpretation disarms the standard mathematical objections to Hume’s reasoning; I also defend (...) it on textual and contextual grounds. (shrink)

Let D be a division ring such that the number of conjugacy classes of the multiplicative group D ∗ is equal to the power of D ∗ . Suppose that H is the group GL or PGL, where V is a vector space of infinite dimension ϰ over D . We prove, in particular, that, uniformly in κ and D , the first-order theory of H is mutually syntactically interpretable with the theory of the two-sorted structure 〈κ,D〉 in (...) the second-order logic with quantification over arbitrary relations of power ⩽κ . A certain analogue of this results is proved for the groups ΓL and PΓL . These results imply criteria of elementary equivalence for infinite-dimensional classical groups of types H= ΓL , PΓL , GL, PGL over division rings, and solve, for these groups, a problem posed by U. Felgner. It follows from the criteria that if H≡H then κ 1 and κ 2 are second-order equivalent as sets. (shrink)

This paper analyzes two different proposals, one by Ellis and Brundrit, based on classical relativistic cosmology, the other by Garriga and Vilenkin, based on the DH interpretation of quantum mechanics, both concluding that, in an infinite universe, planets and beings must be repeated an infinite number of times. We point to possible shortcomings in these arguments. We conclude that the idea of an infinite repetition of histories in space cannot be considered strictly speaking a consequence of (...) current physics and cosmology. Such ideas should be seen rather as examples of «ironic science» in the terminology of John Horgan. (shrink)

This chapter contains section titled: Extension and Duration Hume's Reply to the Paradox of Composition Hume's Arguments for the Finite Divisibility of Perceptions (T 1.2.1) The Coherence of Hume's Account The Idea of Equality (T 1.2.4) The Infinite Divisibility of Objects (T 1.2.2) Manners of Disposition (T 1.2.3) The Simplicity of the Soul (T 1.4.5) The Idea of Vacuum (T 1.2.5) Hume's Account of Contiguity (T 1.1.5, 1.3.8, 2.3.7) Notes References Further reading.

This paper examines connections between concepts of space and extension on the one hand and immaterial spirits on the other, specifically the immanentist concept of spirits as present in rerum natura. Those holding an immanentist concept, such as Thomas Aquinas, typically understood spirits non-dimensionally as present by essence and power; and that concept was historically linked to holenmerism, the doctrine that the spirit is whole in every part. Yet as Aristotelian ideas about extension were challenged and an actual, (...) class='Hi'>infinite, dimensional space readmitted, a dimensionalist concept of spirit became possible—that asserted by the mature Henry More, as he repudiated holenmerism. Despite More’s intentions, his dimensionalist concept opens the door to materialism, for supposing that spirits have parts outside parts implies that those parts could in principle be mapped onto the parts of divisible bodies. The specter of materialism broadens our interest in More’s unconventional ideas, for the question of whether other early modern thinkers, including Isaac Newton, followed More becomes a question of whether they too unwittingly helped usher in materialism. This paper shows that More’s attack upon holenmerism fails. He illegitimately injects his dimensionalist concept of spirit into the doctrine, failing to recognize it as a consequence of the non-dimensionalist concept of spirit, which in itself secures indivisibility. The interpretive consequence for Newton is that there is no prima facie reason to suppose that the charitable interpretation takes him to deny holenmerism. (shrink)

Raphael Lataster and Herman Philipse present an argument which they think decisively demonstrates polytheism over monotheism, if theism is assumed. Far from being decisive, the argument depends on very controversial and likely false assumptions about how to treat infinities in probability. Moreover, these problems are well known. Here, we focus on three objections. First, the authors rely on both countable additivity and the Principle of Indifference, which contradict each other. Second, the authors rely on a particular way of dividing up (...) the possibility space, when there are equally as reasonable alternative divisions which give different answers to the one the authors arrived at. Third, the authors’ argument proves too much, giving us an argument against many positions we should not be able to argue against so easily. (shrink)

It has been argued that Hume's denial of infinite divisibility entails the falsity of most of the familiar theorems of Euclidean geometry, including the Pythagorean theorem and the bisection theorem. I argue that Hume's thesis that there are indivisibles is not incompatible with the Pythagorean theorem and other central theorems of Euclidean geometry, but only with those theorems that deal with matters of minuteness. The key to understanding Hume's view of geometry is the distinction he draws between a (...) precise and an imprecise standard of equality in extension. Hume's project is different from the attempt made by Berkeley in some of his later writings to save Euclidean geometry. Unlike Berkeley, who interprets the theorems of Euclidean geometry as false albeit useful approximations of geometrical facts, Hume is able to save most of the central theorems as true. (shrink)

Having been raised as a Catholic and educated in the West, then trained as a monk in India since the 1980s, Canadian author Swami Muktananda of Rishikesh is uniquely positioned to bring the Eastern tradition of Vedanta to Western spiritual seekers. In Awakening to the Infinite, he answers the eternal, fundamental question posed by philosophical seekers, "Who am I?" with straightforward simplicity. Knowing who you are and adopting a spiritual outlook, he counsels, can help solve problems in daily life (...) to do with relationships, work, children, sexuality, illness, and social injustice. In answering the question "Who am I?," Swami Muktananda of Rishikesh draws on the ancient teachings of advaita or nonduality, the truth that there is no division between ourselves and others at a fundamental level. The book is drawn from the many conversations he has throughout the year meeting with individuals from different backgrounds, cultures, and walks of life, as well as public talks and question and answer sessions. Covering subjects ranging from the spiritual path to everyday problems, Muktananda addresses questions with simplicity and compassion, from the standpoint of someone who is anchored in the infinite but fully in touch with the world. His teaching follows the words of Christ and the Old Testament as well as those of the sages of India, who all proclaim a universal truth that transcends all religions: 'You are not the body; you are not the mind; you are the immortal, divine Self. Realize this and be free.'. (shrink)

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. (shrink)

§1. Introduction.The motivation of this work comes from two different directions: infinite abelian groups, and ordered algebraic structures. A challenging problem in both cases is that of classification. In the first case, it is known for example (cf. [KA]) that the classification of abelian torsion groups amounts to that of reducedp-groups by numerical invariants called theUlm invariants(given by Ulm in [U]). Ulm's theorem was later generalized by P. Hill to the class of totally projective groups. As to the second (...) case, let us consider for instance the class of divisible ordered abelian groups. These may be viewed as ordered ℚ-vector spaces. Their theory being unstable, we cannot hope to classify them by numerical invariants. On the other hand, being o-minimal, the theory enjoys several good model theoretic properties (cf. [P-S]), so the search for some reasonable invariants is well motivated. The common denominator of the two cases, as well as of many others, is valuation theory. Indeed given an ordered vector space, one can consider it as a valued vector space, endowed with the natural valuation. Also, the socleG[p] of a reduced abelianp-groupG, endowed with the height functionhG, is a valued vector space over(the prime field of characteristicp) with values in the ordinals (cf. [F]). (shrink)

We study theorems from Functional Analysis with regard to their relationship with various weak choice principles and prove several results about them: “Every infinite‐dimensional Banach space has a well‐orderable Hamel basis” is equivalent to ; “ can be well‐ordered” implies “no infinite‐dimensional Banach space has a Hamel basis of cardinality ”, thus the latter statement is true in every Fraenkel‐Mostowski model of ; “No infinite‐dimensional Banach space has a Hamel basis of cardinality ” is (...) not provable in ; “No infinite‐dimensional Banach space has a well‐orderable Hamel basis of cardinality ” is provable in ; (the Axiom of Choice for denumerable families of non‐empty finite sets) is equivalent to “no infinite‐dimensional Banach space has a Hamel basis which can be written as a denumerable union of finite sets”; Mazur's Lemma (“If X is an infinite‐dimensional Banach space, Y is a finite‐dimensional vector subspace of X, and, then there is a unit vector such that for all and all scalars α”) is provable in ; “A real normed vector space X is finite‐dimensional if and only if its closed unit ball is compact” is provable in ; (Principle of Dependent Choices) + “ can be well‐ordered” does not imply the Hahn‐Banach Theorem () in ; and “no infinite‐dimensional Banach space has a Hamel basis of cardinality ” are independent from each other in ; “No infinite‐dimensional Banach space can be written as a denumerable union of finite‐dimensional subspaces” lies in strength between (the Axiom of Countable Choice) and ; implies “No infinite‐dimensional Banach space can be written as a denumerable union of closed proper subspaces” which in turn implies ; “Every infinite‐dimensional Banach space has a denumerable linearly independent subset” is a theorem of, but not a theorem of ; and “Every infinite‐dimensional Banach space has a linearly independent subset of cardinality ” implies “every Dedekind‐finite set is finite”. (shrink)

This paper analyzes two different proposals, one by Ellis and Brundrit, based on classical relativistic cosmology, the other by Garriga and Vilenkin, based on the DH interpretation of quantum mechanics, both concluding that, in an infinite universe, planets and beings must be repeated an infinite number of times. We point to possible shortcomings in these arguments. We conclude that the idea of an infinite repetition of histories in space cannot be considered strictly speaking a consequence of (...) current physics and cosmology. Such ideas should be seen rather as examples of «ironic science» in the terminology of John Horgan. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:205 TIME AND THE IDEA OF TIME Hume entitled Part II of Book I of the Treatise "Of the Ideas of Space and Time." Students of this most obscure Part of the Book are aware, however, that he spends little time in it on time. The main reason for his concentration on space. is polemical. In Part II his primary object is to exhibit the contradictions and (...) absurdities implicit in the science of his day with its postulation of empty space and of the mathematics with its notion of infinite divisibility, and then to show how these difficulties can be overcome by the adoption and implementation of the theory of ideas he had developed in Part I. As a result, with the exception of a discussion that occupies approximately three pages in Section III, most of his references to time in Part II are little more than brief appendages tacked on to his arguments concerning space. Hume commentators have generally taken their cue from him. Although the literature contains substantial analyses of his views about space, very little can be found on the subject of time. In this paper, I shall attempt to add something to that slim literature. As its title indicates, this paper will be concerned with two subjects, time and the idea of time. Hume accepts the ordinary distinction between thé two. Time and our idea of time are separate realities, the former being a part of the world independent of us and the latter being a part of the content of our consciousness. I shall observe this distinction, beginning with an account of what Hume means by time and following that with an explanation of his theory of the origin and nature of the idea of time. After finishing these two expository sections I shall devote 206 the remainder of the paper to a critical examination of Hume's views on both subjects, ending with a suggestion about a possible alternative route he might have followed in his argument about time and the idea of time in Part II. A final preliminary point: Although Hume devotes most of his attention to space in Part II, he makes it clear that, because space and time (as well as our ideas of them) are analogous to each other in certain vital respects, his views concerning the former can easily be transferred and applied to the latter. 1. Time Hume states or implies several things about the nature of time in Part II. (1)He equates time with duration, treating the two terms as equivalent throughout his discussion. (2)Time is an objective reality, or aspect of the external world. This interpretation may be questioned because Hume never asserts an objectivistic view of time unequivocally and in certain passages writes in a way that could be interpreted as implying that time is a characteristic of our conscious experience. "As 'tis from the disposition of visible and tangible objects we receive the idea of space, so from the succession of ideas and impressions we form the idea of time..." (T 35). But later he writes: "...time is nothing but the manner, in which some real objects exist..." (T 64). Also, in the titles of his first two sections he makes a clear distinction, labelling Section I "Of the infinite divisibility of our ideas of space and time" and Section II "Of the infinite divisibility of space and time." Since, in Section II, it is apparent that he is talking about space as a part 207 of the external world, the natural inference is that he is talking about time in the same way, as, for example, when he writes, "The infinite divisibility of space implies that of time, as is evident from the nature of motion" (T 31). (3)Time is not infinitely divisible but is made up of parts Hume calls 'moments' (T 31). These moments are the 'building blocks' of time and are ultimate and indivisible. They are analogous to the ultimate, indivisible parts of space, which Hume labels 'mathematical points' (T 40). (4)The moments of time are not co-existent but successive. In this respect time differs from space. "'Tis also evident, that these parts [of... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:76. A REFUTATION OF HUME'S THEORY OF CAUSALITY1 Given Hume's conceptions of space and time, which I take to be fundamental to his theory of causality, it is not always possible to meet all of those conditions definitive of the cause-effect relation, i.e., those "general rules, by which we may know when" objects really 2 are "causes or effects to each other" (T. 173). To show this, it (...) will be necessary, first, to give a very brief exposition of Hume's conceptions of space and time, with regard chiefly to their implications for the nature of motion. Then, after briefly summarizing his views on the nature of the cause-effect relation, I shall proceed immediately to my objection. It should be noted first, however, that, if one instance of a cause-effect relation can be found to which Hume's analysis will not apply, that, presumably, would be sufficient to refute his theory. I shall not here repeat all of Hume's arguments for the indivisibility of space and time. Suffice it to say, he takes it to be fairly obvious that each must consist of a finite number of indivisible points or moments, each of which is distinct, and therefore separate, from every other, and, in the case of time, none of which ever "co-exist." Only one of his arguments for the indivisibility of space need concern us here. The argument is brief: The infinite divisibility of space implies that of time, as is evident from the nature of motion. If the latter, therefore, be impossible, the former must be equally so. (T. 31) The argument here seems, roughly, to be this: the motion of an object can be described only by giving its positions at various times (the nature of motion). If, therefore, an object, such as a ball moving along a line AB, can be assigned an infinite number of positions taken between A and B (the infinite divisibility of space), 77. the time taken to traverse AB must also be infinite, i.e., must consist of an infinite number of instants. But the finite time taken to traverse the distance AB is not infinitely divisible; therefore, neither is the line AB. This argument, of course, relies upon the assumption that, in traversing the line AB, the ball will occupy each of the points (whether finite or infinite) of which it consists. The argument can also be turned around so as to derive the finite divisibility of time from that of space. Here, too, it must be assumed that the ball occupies each of those instants which constitute the time taken to traverse the line AB. It thus seems clear that, on Hume's view, the ball's motion can be neither more nor less divisible than the space and time it takes to move. x In its motion from A to B, then, the ball will occupy first one, then another of the points lying between A and B until finally it has occupied them all and has moved from A to B. But, in saying that it will occupy first one and then another, we are saying nothing more than that it successively occupies a series of points, and, insofar as its occupation of these different points is successive, it takes time. We are in a position, then, to define motion (at least motion along a straight line) as "the successive occupation of adjacent, linearly ordered points or places." Now, this is very different from the usual conception of motion. The ball's motion between any two adjacent points is not a moving from one to the other. It is simply its being in the one place at one instant and its being in the other place at the next. It is not a continuous motion, but the occupation of discrete places at different times. It is, as it were, a series of instantaneous leaps. This interpretation, if it is accurate, may also serve to account for Hume's ill-understood propensity for 78. referring to cause and effect as objects rather than as events. Events must, on Hume's view of time at least, be essentially static. Unlike those... (shrink)

Since both berkeley and hume are committed to the view that a line is composed of finitely many fundamental parts, They must find responses to the standard geometrical proofs of infinite divisibility. They both repeat traditional arguments intended to show that infinite divisibility leads to absurdities, E.G., That all lines would be infinite in length, That all lines would have the same length, Etc. In each case, Their arguments rest upon a misunderstanding of the concept (...) of a limit, And thus are not successful. Berkeley, However, Adds a further ingenious argument to the effect that the standard geometrical proofs of infinite divisibility misread the unlimited representational capacity of geometrical diagrams as a substantive feature of the objects that these diagrams represent. The article concludes that berkeley is right on this matter, And that the traditional proofs of infinite divisibility do not show what they are intended to show. (shrink)

This book celebrates the investigative power of phenomenology to explore the phenomenological sense of space and time in conjunction with the phenomenology of intentionality, the invisible, the sacred, and the mystical. It examines the course of life through its ontopoietic genesis, opening the cosmic sphere to logos. The work also explores, on the one hand, the intellectual drive to locate our cosmic position in the universe and, on the other, the pull toward the infinite. It intertwines science and (...) its grounding principles with imagination in order to make sense of the infinite. This work is the first of a two-part work that contains papers presented at the 62nd International Congress of Phenomenology, The Forces of the Cosmos and the Ontopoietic Genesis of Life, held in Paris, France, August 2012. It features the work of scholars in such diverse disciplines as biology, anthropology, pedagogy, and psychology who philosophically investigate the cosmic origins of beingness. Coverage in this first part includes: Toward a New Enlightenment: Metaphysics as Philosophy of Life, Transformation in Phenomenology: Husserl and Tymieniecka, Biologically Organized Quantum Vacuum and the Cosmic Origin of Cellular Life, Plotinus "Enneads" and Self-Creation, The Creative Potential of Humor, Transcendental Morphology - A Phenomenological Interpretation of Human and Non-Human Cosmos, and Cognition and Emotion: From Dichotomy to Ambiguity.. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:62. 'HUME ON SPACE AND GEOMETRY': ONE RESERVATION In so far as Rosemary Newman disagrees with any2 thing said in my 'Infinite Divisibility in Hume's Treatise ' - which seems, happily, not to be so very far - I hasten to report that I am now persuaded. Thus my suggested reason for refusing to allow that an impression of blackness could give rise to the idea (...) of extension was not, after all, 3 Hume's. And no doubt his conviction that whatever ^s capable of being divided in_ infinitum, must consist of an infinite number of parts (T26) was in part sustained by "the atomistic type of phenomenalism advanced in the first Part 4 of Book I of the Treatise". For to any objection that this nowadays so plainly unacceptable conviction had been accepted as obviously correct by such formidable predecessors as Berkeley and Hobbes, it could be replied that they too had their commitments to their own brands of atomism. In her further comments upon "Hume's phenomenalist atomism" Newman notices, as "a very curious claim", his statement that when - like the true philosopher he was! - he locked at a table:- My senses convey to me only the impressions of colour 'd points, dispos 'd in a certain manner (T34). So I was sorry that she seems to have missed the bold priority claim which, referring to the same passage, I made on Hume's behalf in that article: "Anyone familiar with the theories and the paintings of Seurat might also mischievously characterize the Hume of this Section as 'the Father of Pointillisme' "! But at the end of her paper, although I think only at the end, Newman begins to go very wrong indeed. She says: "Turning to the Enquiry there is little to indicate any shift in Hume's opinion of geometry as a body of synthetic but necessary knowledge." After remarking that Hume does not repeat "any of his earlier arguments" against infinite divisibility "nor those in support of a system of mathematical 63. points", she notices that, in the paragraph characterizing propositions stating only the relations of ideas, geometry "is classed as a science alongside arithmetic and algebra (E25) ".7 Next she quotes the end of that paragraph, and at once comments: "Flew thinks that in the Enquiry Hume is to be seen as 'restoring pure geometry to its place alongside Q the other two elements in the trinity',..." Newman does not, so far as I can see, offer any reason for dismissing this suggestion - apart, that is, from asserting that the paragraph from which she has just quoted "can be interpreted so as to remove any apparent inconsistency with the claim that the impossibility of conceiving the negation of a geometrical axiom has the same ground in the Enquiry as in the Treatise, namely, the invariant character of sensible space." She just tells us that "Where geometry is concerned I follow Atkinson's opinion in finding no reason to suppose that 9 Hume abandoned the Treatise stand." This I find, to say the least, surprising since the very sentence of Hume's Philosophy of Belief from which she quoted my contrary opinion does provide what I still consider to be a quite decisive reason. Since she does not discuss either my reason or the Hume passage to which I was referring, I can here do nothing else or better than repeat myself; adding only and by the way that Professor Atkinson's paper 'Hume on Mathematics' was published a year before that book:Besides restoring pure geometry to its place alongside the other two elements of the trinity, the Inquiry also sketches an account of applied mathematics. This is something the earlier book did not provide at all. 'Every part of mixed mathematics proceeds on the supposition that certain laws are established by nature in her operations, and abstract reasonings are employed, either to assist experience in the discovery of these laws or to determine their influence in particular instances... Thus it is a law of motion, discovered by experience, that the moment or force of any body in motion is in 64. the compound ratio or proportion of its solid contents and its velocity, and, consequently... (shrink)

This paper analyzes criteria of fair division of a set of indivisible items among people whose revealed preferences are limited to rankings of the items and for whom no side payments are allowed. The criteria include refinements of Pareto optimality and envy-freeness as well as dominance-freeness, evenness of shares, and two criteria based on equally-spaced surrogate utilities, referred to as maxsum and equimax. Maxsum maximizes a measure of aggregate utility or welfare, whereas equimax lexicographically maximizes persons' utilities from smallest to (...) largest. The paper analyzes conflicts among the criteria along with possibilities and pitfalls of achieving fair division in a variety of circumstances. (shrink)

This paper interprets the two pages devoted in the Critique of Pure Reason to a critique of Leibniz’s view of organisms as infinitely organized machines. It argues that this issue of organisms represents a crucial test-case for Kant in regard to the conflicting notions of space, continuity and divisibility held by classical metaphysics and by criticism. I first present Leibniz’s doctrine and its justification. In a second step, I explain the general reasoning by which Kant defines the problem (...) of the Antinomies, and then I specify the case of the Second Antinomy. Then, I ask why the organism raises specific issues for Kant’s solution of the Second Antinomy, and why it sheds light on the nature of Leibniz’s metaphysical assumptions about organisms. The last section considers this critique of Leibniz from the perspective of Kant’s future theory of organisms in the third Critique, specifying its role in the development of a Kantian view of organisms with regard to the changes occurring at the same time in the life sciences. (shrink)

We extend the hierarchy of finite-dimensional Ellentuck spaces to infinite dimensions. Using uniform barriers [Formula: see text] on [Formula: see text] as the prototype structures, we construct a class of continuum many topological Ramsey spaces [Formula: see text] which are Ellentuck-like in nature, and form a linearly ordered hierarchy under projections. We prove new Ramsey-classification theorems for equivalence relations on fronts, and hence also on barriers, on the spaces [Formula: see text], extending the Pudlák–Rödl theorem for barriers on the (...) Ellentuck space. The inspiration for these spaces comes from continuing the iterative construction of the forcings [Formula: see text] to the countable transfinite. The [Formula: see text]-closed partial order [Formula: see text] is forcing equivalent to [Formula: see text], which forces a non-p-point ultrafilter [Formula: see text]. This work forms the basis for further work classifying the Rudin–Keisler and Tukey structures for the hierarchy of the generic ultrafilters [Formula: see text]. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:62. 'HUME ON SPACE AND GEOMETRY': ONE RESERVATION In so far as Rosemary Newman disagrees with any2 thing said in my 'Infinite Divisibility in Hume's Treatise ' - which seems, happily, not to be so very far - I hasten to report that I am now persuaded. Thus my suggested reason for refusing to allow that an impression of blackness could give rise to the idea (...) of extension was not, after all, 3 Hume's. And no doubt his conviction that whatever ^s capable of being divided in_ infinitum, must consist of an infinite number of parts (T26) was in part sustained by "the atomistic type of phenomenalism advanced in the first Part 4 of Book I of the Treatise". For to any objection that this nowadays so plainly unacceptable conviction had been accepted as obviously correct by such formidable predecessors as Berkeley and Hobbes, it could be replied that they too had their commitments to their own brands of atomism. In her further comments upon "Hume's phenomenalist atomism" Newman notices, as "a very curious claim", his statement that when - like the true philosopher he was! - he locked at a table:- My senses convey to me only the impressions of colour 'd points, dispos 'd in a certain manner (T34). So I was sorry that she seems to have missed the bold priority claim which, referring to the same passage, I made on Hume's behalf in that article: "Anyone familiar with the theories and the paintings of Seurat might also mischievously characterize the Hume of this Section as 'the Father of Pointillisme' "! But at the end of her paper, although I think only at the end, Newman begins to go very wrong indeed. She says: "Turning to the Enquiry there is little to indicate any shift in Hume's opinion of geometry as a body of synthetic but necessary knowledge." After remarking that Hume does not repeat "any of his earlier arguments" against infinite divisibility "nor those in support of a system of mathematical 63. points", she notices that, in the paragraph characterizing propositions stating only the relations of ideas, geometry "is classed as a science alongside arithmetic and algebra (E25) ".7 Next she quotes the end of that paragraph, and at once comments: "Flew thinks that in the Enquiry Hume is to be seen as 'restoring pure geometry to its place alongside Q the other two elements in the trinity',..." Newman does not, so far as I can see, offer any reason for dismissing this suggestion - apart, that is, from asserting that the paragraph from which she has just quoted "can be interpreted so as to remove any apparent inconsistency with the claim that the impossibility of conceiving the negation of a geometrical axiom has the same ground in the Enquiry as in the Treatise, namely, the invariant character of sensible space." She just tells us that "Where geometry is concerned I follow Atkinson's opinion in finding no reason to suppose that 9 Hume abandoned the Treatise stand." This I find, to say the least, surprising since the very sentence of Hume's Philosophy of Belief from which she quoted my contrary opinion does provide what I still consider to be a quite decisive reason. Since she does not discuss either my reason or the Hume passage to which I was referring, I can here do nothing else or better than repeat myself; adding only and by the way that Professor Atkinson's paper 'Hume on Mathematics' was published a year before that book:Besides restoring pure geometry to its place alongside the other two elements of the trinity, the Inquiry also sketches an account of applied mathematics. This is something the earlier book did not provide at all. 'Every part of mixed mathematics proceeds on the supposition that certain laws are established by nature in her operations, and abstract reasonings are employed, either to assist experience in the discovery of these laws or to determine their influence in particular instances... Thus it is a law of motion, discovered by experience, that the moment or force of any body in motion is in 64. the compound ratio or proportion of its solid contents and its velocity, and, consequently... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Hume Studies Volume XXIII, Number 2, November 1997, pp. 227-244 Hume on Geometry and Infinite Divisibility in the Treatise H. MARK PRESSMAN Scholars have recognized that in the Treatise "Hume seeks to find a foundation for geometry in sense-experience."1 In this essay, I examine to what extent Hume succeeds in his attempt to ground geometry visually. I argue that the geometry Hume describes in the Treatise faces (...) a serious set of problems. Geometric Lines Hume maintains that ideas "are images" (T 6) which may be called up "when I shut my eyes" (T 3). "That we may fix the meaning of a word, figure," according to Hume, "we may revolve in our mind the ideas of circles, squares, parallelograms, triangles of different sizes and proportions, and may not rest on one image or idea" (T 22). As Arthur Pap notes, "'Idea' is in Hume's usage synonymous with 'mental image'." D.C.G. MacNabb also notes that "Hume thought of ideas as images, and primarily as visual images."2 When Hume argues, then, that "we have the idea of indivisible points, lines and surfaces conformable to the [geometer's] definition" (T 44), he is arguing that we have mental images of geometry's points, lines and planes. Consider geometric lines first. Geometers define a line "to be length without breadth or depth" (T 42). It is not apparent to everyone that we have mental images of such lines. According to "L'Art de penser" (T 43), published in 1662, it is impossible "to conceive a length without any breadth" (T 43). H. Mark Pressman is at the Department of Philosophy, University of California, Davis, 1238 Social Sciences and Humanities Building, Davis CA 95616-8673 USA. email: [email protected] 228 H. Mark Pressman We can only by an abstraction "consider the one without regarding the other" (T 43). Hume, however, offers "clear proof" (T 44), in the form of two arguments, that we actually possess ideas or mental images of geometry's lines. First, we have ideas or mental images of lengths with breadth, though these are supposed never to be without breadth and thus not geometric lines. (A mental image oflength is also a mental image with length, just as "to say the idea of extension agrees to any thing, is to say it is extended" [T 240].) Hume reduces to absurdity the view that our mental images of length are always with breadth and thus not images of geometric lines. If our length-images were always with breadth, then our mental images would be endlessly divisible in breadth, since they would always have some positive breadth (T 43). But no mental image is endlessly divisible, for the mind arrives "at an end in the division of its ideas, nor are there any possible means of evading the evidence of this conclusion" (T 27). We thus have an idea or image of length without (divisible) breadth conforming to the geometer's definition (T 44). Second, we can imagine or picture the termination of a geometric plane. But a geometric plane must terminate in a geometric line (T 44). (Proof: Assume L is a line with breadth which terminates plane P. Since L has breadth, L must contain at least two parts, w and y, exactly one of which actually terminates P. But whether it is w or y, L doesn't terminate P, which was the assumption.) Since we can picture the termination of a geometric plane, we can picture a geometric line, length without divisible breadth. It is one thing to have ideas or mental images of geometric lines which may be called up "when I shut my eyes" (T 3). It is another for such objects actually to exist "in nature." The conceptualist holds that though (a) geometry 's points, lines and planes are ideas, nevertheless (b) there are no such things in nature and that (c) there can't be such things in nature: [T]he objects of geometry... are mere ideas in the mind, and not only never did, but never can exist in nature. They never did exist; for no one will pretend to draw a line or make a... (shrink)

This work analyzes two perspectives, Atomism and Infinite Divisibility, in the light of modern mathematical knowledge and recent developments in computer graphics. A developmental perspective is taken which relates ideas leading to atomism and infinite divisibility. A detailed analysis of and a new resolution for Zeno's paradoxes are presented. Aristotle's arguments are analyzed. The arguments of some other philosophers are also presented and discussed. All arguments purporting to prove one position over the other are shown to (...) be faulty, mostly by question begging. Included is a sketch of the consistency of infinite divisibility and a development of the atomic perspective modeled on computer graphics screen displays. The Pythagorean theorem is shown to depend upon the assumption of infinite divisibility. The work concludes that Atomism and infinite divisibility are independantly consistent, though mutually incompatible, not unlike the wave/particle distinction in physics. (shrink)

Marina Frasca-Spada's Space and the Self in Hume's Treatise proposes a subjective idealist interpretation of Hume's account of space in part ii of Book I of the Treatise. The book is divided into four chapters. The first deals with Hume's position on infinite divisibility in I ii 1-2, the second with his position on the origin of the idea of space in I ii 3, the third with his account of geometrical knowledge in I ii (...) 4, and the final chapter with his position on vacuum in I ii 5. The subject matter of I ii 6 on the idea of existence is dealt with over the course of the third chapter. Between the second and third chapters, the exposition is interrupted by what Frasca-Spada calls an "Intermezzo." This piece begins by promising to explain why Hume's theory of space is "central to the philosophical substance of the Treatise", but turns to other topics before delivering on the promise. Its relation to the other parts of the book is unclear, and it would have been better included as a preface or appendix, or omitted altogether. (shrink)

This book gathers together for the first time an important body of texts written between 1672 and 1686 by the great German philosopher and polymath Gottfried Leibniz. These writings, most of them previously untranslated, represent Leibniz's sustained attempt on a problem whose solution was crucial to the development of his thought, that of the composition of the continuum. The volume begins with excerpts from Leibniz's Paris writings, in which he tackles such problems as whether the infinite division of matter (...) entails perfect points, whether matter and space can be regarded as true wholes, whether motion is truly continuous, and the nature of body and substance. Comprising the second section is Pacidius Philalethi, Leibniz's brilliant dialogue of late 1676 on the problem of the continuity of motion. In the selections of the final section, from his Hanover writings of 1677-1686, Leibniz abandons his earlier transcreationism and atomism in favor of the theory of corporeal substance, where the reality of body and motion is founded in substantial form or force.Leibniz's texts (one in French, the rest in Latin) are presented with facing-page English translations, together with an introduction, notes, appendixes containing related excerpts from earlier works by Leibniz and his predecessors, and a valuable glossary detailing important terms and their translations. (shrink)

We find the complete Euler characteristics for the categories of definable sets and functions in strongly minimal groups. Their images, which represent the Grothendieck semirings of those categories, are all isomorphic to the semiring of polynomials over the integers with nonnegative leading coefficient. As a consequence, injective definable endofunctions in those groups are surjective. For infinite vector spaces over arbitrary division rings, the same results hold, and more: We also establish the Fubini property for all Euler characteristics, and extend (...) the complete one to the eq-expansion of those spaces while preserving the Fubini property but not completeness. Then, surjective interpretable endofunctions in those spaces are injective, and conversely. Our presentation is made in the general setting of multi-sorted structures. (shrink)

Much has been written about the history of the work of men and women in the premodern past. It is now generally acknowledged that early modern ideological assumptions about a strict division of work and space between men and productive work outside the house on the one hand, and women and reproduction and consumption inside the house, on the other, bore little relation to reality. Household work strategies, out of necessity, were diverse. Yet what this spatial complexity meant in (...) particular households on a day-to-day basis and its consequences for gender relationships is less clear and has received relatively little historical attention. The aim of this paper is to add to our knowledge through a case study of the way that men and women used and organized space for work in the county of Essex during the “long seventeenth century”. Drawing on critiques of the concept of “separate spheres” and the models of economic change to which it relates, together with local/micro historical methods, it places evidence within an appropriate regional context to argue that spatial patterns were enormously varied in early modern England and a number of factors—time, place, occupation, and status, as well as gender—determined them. Understanding of the dynamic, complex, uneven purchase of patriarchy upon the organization, imagination, and experience of space has important implications for approaches to gender relations in early modern England. It raises additional doubts about the utility of the separate spheres analogy, and particularly the use of binary oppositions of male/female and public/private, to describe gender relations and their changes in this period and shows that a deeper understanding demands more research into the local contexts in which the gendered division and meaning of work was negotiated. (shrink)

We classify the measures on the lattice ℒ of all closed subspaces of infinite-dimensional orthomodular spaces (E, Ψ) over fields of generalized power series with coefficients in ℝ. We prove that every σ-additive measure on ℒ can be obtained by lifting measures from the residual spaces of (E, Ψ). The measures being lifted are known, for the residual spaces are Euclidean. From the classification we deduce, among other things, that the set of all measures on ℒ is not separating.

In lieu of an abstract, here is a brief excerpt of the content:Kant on the Method of MathematicsEmily Carson1. INTRODUCTIONThis paper will touch on three very general but closely related questions about Kant’s philosophy. First, on the role of mathematics as a paradigm of knowledge in the development of Kant’s Critical philosophy; second, on the nature of Kant’s opposition to his Leibnizean predecessors and its role in the development of the Critical philosophy; and finally, on the specific role of intuition (...) in Kant’s philosophy of mathematics. One of the points that I want to make is that recognizing the importance of these first two issues is essential to giving an adequate account of the third. That is, only by appreciating how Kant uses the example of mathematical knowledge, and in particular the objections he had to previous accounts of such knowledge, can we understand the philosophical role that he ascribes to intuition in his own account of mathematics.I don’t propose to explain the details of Kant’s philosophy of mathematics, but rather to compare his views on mathematics in the pre-Critical Prize Essay of 1764 to those of the Critique of Pure Reason, with the aim of showing how Kant’s doctrine of construction in pure intuition arises out of a combination of two factors: first, his attempt to ground the distinction between the respective methods of mathematics and philosophy, and second, his concern to defend the reality of mathematical knowledge against the views of those he called ‘the metaphysicians.’ The first factor demands an account of the certainty of mathematics, the second requires an account of its content. I shall argue that there is a possible conflict between Kant’s accounts of these features in the Prize Essay, a possibility which is finally ruled out only by means of the Critical doctrine of pure intuition. So I hope to show that there is a significant philosophical role for intuition in Kant’s account of mathematical knowledge (over and above the more logical role attributed to it on readings [End Page 629] such as those of Friedman, Beth, and Hintikka1) in reconciling the content of mathematics and its certainty.2. THE MATHEMATICIANS VS. THE METAPHYSICIANSThroughout his early writings, Kant appealed to the evidence and certainty of mathematics; his attitude towards metaphysics, however, underwent a dramatic shift. In the Physical Monadology of 1756, Kant asked how metaphysics and geometry can be united,[f] or the former peremptorily denies that space is infinitely divisible, while the latter, with its usual certainty, asserts that it is infinitely divisible. Geometry contends that empty space is necessary for free motion, while metaphysics hisses the idea off the stage. Geometry holds universal attraction or gravitation to be hardly explicable by mechanical causes but shows that it derives from the forces which are inherent in bodies at rest and which act at a distance, whereas metaphysics dismisses the notion as an empty delusion of the imagination [1:475–6].2He attempted in this work to reconcile the respective positions of metaphysics and geometry regarding infinite divisibility, thereby demonstrating that “it is neither the case that the geometer is mistaken nor that the opinion to be found among metaphysicians deviates from the truth” [1:480]. But in a series of works from 1762–3, Kant reveals a markedly different attitude towards metaphysics. For example, in “The only possible argument in support of a demonstration of the existence of God,” Kant says:the mania for method and the imitation of the mathematician, who advances with a sure step along a well-surfaced road, have occasioned a large number of such mishaps on the slippery ground of metaphysics. These mishaps are constantly before one’s eyes, but there is little hope that people will be warned by them, or that they will learn to be more circumspect as a result [2:71].In contrast, Kant describes the geometer as uncovering “with the greatest certainty” the most secret properties of that which is extended [2:70]. But the less conciliatory attitude to metaphysics comes out most clearly in Kant’s actual procedure in “The only possible argument.” In the Seventh Reflection, he presents an attempt to explain the origin of... (shrink)

This chapter rejects Equal Division, focusing on Hillel Steiner’s formulation of the view. First, further explanation of why one might take Equal Division to follow from Equal Original Claims is provided. Then, David Miller’s objection is introduced, according to which there is no defensible metric by which resource shares can be made commensurate, given the fact of reasonable value pluralism. The chapter argues that what the metric problem really shows, is that Equal Division possesses insufficient impartiality to satisfy the equal (...) original claims that motivate the view in the first place. This case is made by critiquing the three principal metrics proposed to amalgamate individual valuations of natural resources and thereby render Equal Division both coherent and defensible; namely, economic value, opportunity cost, and ecological space. The chapter concludes that to respect Equal Original Claims, the better approach will be to formulate a Common Ownership conception of justice for natural resources. (shrink)