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)

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 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.

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)

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)

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)

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)

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)

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)

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

In this article we consider some questions raised by F. Benoist, E. Bouscaren, and A. Pillay. We prove that infinitely $p$-divisible points on abelian varieties defined over function fields of transcendence degree one over a finite field are necessarily torsion points. We also prove that when the endomorphism ring of the abelian variety is $\mathbb{Z}$, then there are no infinitely $p$-divisible points of order a power of $p$.

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)

In this paper, we address an infamous argument against divisibility that dates back to Zeno. There has been an incredible amount of discussion on how to understand the critical notions of divisibility, extension, and infinite divisibility that are crucial for the very formulation of the argument. The paper provides new and rigorous definitions of those notions using the formal theories of parthood and location. Also, it provides a new solution to the paradox of divisibility which (...) does not face some threats that can possibly undermine the standard Lebesgue measure solution to such a paradox. (shrink)

In Physics, Aristotle starts his positive account of the infinite by raising a problem: “[I]f one supposes it not to exist, many impossible things result, and equally if one supposes it to exist.” His views on time, extended magnitudes, and number imply that there must be some sense in which the infinite exists, for he holds that time has no beginning or end, magnitudes are infinitely divisible, and there is no highest number. In Aristotle's view, a plurality cannot (...) escape having bounds if all of its members exist at once. Two interesting, and contrasting, interpretations of Aristotle's account can be found in the work of Jaako Hintikka and of Jonathan Lear. Hintikka tries to explain the sense in which the infinite is actually, and the sense in which its being is like the being of a day or a contest. Lear focuses on the sense in which the infinite is only potential, and emphasizes that an infinite, unlike a day or a contest, is always incomplete. (shrink)

In a fair, infinite lottery, it is possible to conclude that drawing a number divisible by four is strictly less likely than drawing an even number; and, with apparently equal cogency, that drawing a number divisible by four is equally as likely as drawing an even number.

Une preuve est pure si, en gros, elle ne réfère dans son développement qu’à ce qui est « proche » de, ou « intrinsèque » à l’énoncé à prouver. L’infinité des nombres premiers, un théorème classique de l’arithmétique, est un cas d’étude particulièrement riche pour les recherches philosophiques sur la pureté. Deux preuves différentes de ce résultat sont ici considérées, à savoir la preuve euclidienne classique et une preuve « topologique » plus récente proposée par Furstenberg. D’un point de vue (...) naïf, il semblerait que la première soit pure et la seconde impure. Des objections à cette vue naïve sont ici considérées et réfutées. Concernant la preuve euclidienne, la question relève de la logique, notamment de la définissabilité arithmétique de l’addition en termes de successeur et de divisibilité telle que démontrée par Julia Robinson, tandis qu’en ce qui concerne la preuve topologique, la question relève de la sémantique, notamment pour tout ce qui touche au problème de savoir ce qui est « inclus » dans le contenu d’énoncés particuliers.A proof is pure, roughly, if it draws only on what is « close » or « intrinsic » to the statement being proved. The infinitude of prime numbers, a classical theorem of arithmetic, is a rich case study for philosophical investigation of purity. Two different proofs of this result are considered, namely the classical Euclidean proof and a more recent « topological » proof by Furstenberg. Naively the former would seem to be pure and the latter to be impure. Objections to these naive views are considered and met. In the case of the former the issue rests on logical matters, specifically the arithmetic definability of addition in terms of successor and divisibility shown by Julia Robinson, while in the case of the latter the issue rests on semantic matters, specifically with respect to what is « contained » in the content of particular statements. (shrink)

We continue the exploration of various aspects of divisibility of ultrafilters, adding one more relation to the picture: multiplicative finite embeddability. We show that it lies between divisibility relations \ and \. The set of its minimal elements proves to be very rich, and the \-hierarchy is used to get a better intuition of this richness. We find the place of the set of \-maximal ultrafilters among some known families of ultrafilters. Finally, we introduce new notions of largeness (...) of subsets of \, and compare it to other such notions, important for infinite combinatorics and topological dynamics. (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)

Résumé — Cet article se propose de fournir une contribution au débat interprétatif sur le principe de la limitation de l’acte par la puissance dans la démonstration de l’infinité de Dieu de la Summa theologiae de Thomas d’Aquin à travers une enquête à caractère historique qui expose quelques-unes des étapes capitales de l’histoire de cette preuve. Le désaccord qui divise les interprètes contemporains à propos du rôle joué par le principe susdit hérite, en réalité, d’une opposition séculaire parmi les commentateurs (...) de Thomas qui remonte à la discussion qui suivit l’intervention de Tommaso de Vio et de Francesco Silvestri. D’autre part, l’incontestable présence chez certains commentateurs classiques de Thomas, Jean de Saint-Thomas in primis, d’une position qui sépare la preuve de la Summa du principe de la limitation de l’acte par la puissance réceptive, loin d’exprimer une orthodoxie par rapport à la tradition la plus ancienne, constitue un renversement par rapport à la lecture de Cajetan et de Silvestre de Ferrare, qui s’était affirmée comme l’interprétation canonique du texte de Thomas.— Questo articolo si propone di fornire un contributo al dibattito interpretativo sul principio della limitazione dell’atto attraverso la potenza nella dimostrazione dell’infinità di Dio nella Summa theologiae di Tommaso d’Aquino attraverso un’indagine di carattere storico che espone alcune delle tappe capitali della storia di questa prova. Il disaccordo che divide gli interpreti contemporanei sul ruolo giocato dal principio suddetto eredita, in realtà, un’opposizione secolare fra i commentatori di Tommaso che risale alla discussione che seguî gli interventi di Tommaso de Vio e di Francescio Silvestri. D’altra parte, la presenza, incontestabile, in alcuni commentatori classici di Tommaso, primo fra tutti Giovanni di san Tommaso, di una posizione che separa la prova della Summa dal principio della limitazione dell’atto attraverso la potenza recettiva, lungi dal configurarsi nei termini di una posizione ortodossa rispetto alla tradizione più antica, costituisce un ribaltamento della lettura di Cajetano e di Silvestro da Ferrare, che si era affermata come l’interpretazione canonica del testo di Tommaso. (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)

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)

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)

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)

This paper argues that, for Ockham, the parts of the continuum exist in act in the continuum: they are already there before any division of the continuum. Yet, they are infinitely many in that no division of the continuum will exhaust all the existing parts of the continuum taken conjointly. This reading of Ockham takes into account the crucial place of his new concept of the infinite in his analysis of the infinite divisibility of the continuum. Like (...) many of his fellow anti-atomists, Ockham stresses that the concept of a potential infinite seems to contradict Aristotle’s modal logic, in particular the central assumption that there is no potency that will never be realized. Ockham, like other fourteenth-century anti-atomists, tried not only to refute atomism, but also to propose an analysis of the infinite divisibility of the continuum that is not incompatible with their modal logic. (shrink)

This essay aims to show that the fourfold division theory of consciousness in the Cheng Weishi Lun 成唯識論 is the third way between phenomenology and the higher-order theories of consciousness. Regarding the problem of infinite regress, in particular, this theory represents an alternative between the reflexive model and the reflective model of self-consciousness. The main purpose of this essay is not to prove or to argue for the theory, but to clearly present its structure and the systematic or Abhidharmic (...) way of thinking that leads to the notion of awareness of self-awareness and provides a way out of the problem of infinite regress. It also points out some pertinent issues that need to be further addressed or explored, such as simultaneous causation, begging the question, reconciling the antireflexivity principle with the lamp simile for self-awareness, and aboutness. (shrink)

The topic of my dissertation is the treatment of the fallacies of composition and division during the scholastic period , the compounded/divided sense distinction which grew out of that treatment, and the philosophical use to which the distinction was put. For instance, a recognition of these fallacies during the twelfth and thirteenth centuries helped theologians deal with certain problems having to do with foreknowledge and human freedom. In addition, a recognition of the distinction between the compounded and divided senses of (...) propositions allowed medieval logicians after the first quarter of the fourteenth century to get clearer about what might broadly be described as the logic of propositional attitudes. ;In Chapter 1, I briefly summarize and compare theories of composition and division from Peter Abelard to Paul of Pergula . And in Chapters 2 and 3, I examine the treatments of composition and division presented by Lambert of Auxerra , Roger Bacon , and William Ockham . In these treatments, composition and division are viewed in terms of ambiguity. ;By the second quarter of the fourteenth century, one sees the discussion of composition and division move from a concern with ambiguous expressions to a concern with arguments comprised of unambiguous sentences having either the compounded sense or the divided sense in virtue of the sentence's word-order. "Infinitely fast does Socrates run; therefore, Socrates runs infinitely fast" is, according to William Heytesbury , invalid: the premise has the divided sense, and is possibly true; the conclusion has the compounded sense, and is impossible. With Heytesbury there is an increasing emphasis on the compounded/divided sense distinction in connection with verbs signifying mental acts or acts of will: 'know,' 'believe,' 'desire,' 'understand,' etc. In Chapter 4 I explain why these shifts may have occurred in the medieval treatment of composition and division. And in Chapter 5, I consider the effect of Heytesbury's treatment of composition and division on logicians and commentators who come after him. I focus in particular on the work of four fifteenth- through sixteenth-century Italian logicians and their views on epistemic logic. (shrink)

It is shown that the following three common understandings of Newton’s laws of motion do not hold for systems of infinitely many components. First, Newton’s third law, or the law of action and reaction, is universally believed to imply that the total sum of internal forces in a system is always zero. Several examples are presented to show that this belief fails to hold for infinite systems. Second, two of these examples are of an infinitely divisible continuous body with (...) finite mass and volume such that the sum of all the internal forces in the body is not zero and the body accelerates due to this non-null net internal force. So the two examples also demonstrate the breakdown of the common understanding that according to Newton’s laws a body under no external force does not accelerate. Finally, these examples also make it clear that the expression ‘impressed force’ in Newton’s formulations of his first and second laws should be understood not as ‘external force’ but as ‘exerted force’ which is the sum of all the internal and external forces acting on a given body, if the body is infinitely divisible. (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)

When "Finite and Infinite Goods" was published in 1999, it took its place as one of the few major statements of a broadly Augustinian ethical philosophy of the past century. By "broadly Augustinian" I refer to the disposition to combine a Platonic emphasis on a transcendent source of value with a traditionally theistic emphasis on the value-creating capacities of absolute will. In the form that this disposition takes with Robert Merrihew Adams, it is the resemblance between divine and a (...) finite excellence that makes the finite excellence objectively of value, and it is the correspondence of an obligation to a divine command that makes the obligation objectively obligatory. I look closely at the complexity of this ethical division of labor--between the good and the right--mainly as it appears in the context of "Finite and Infinite Goods", but also with attention to the broader corpus of Adams's writings, particularly his work on Leibniz and the essays of his that have been gathered together in "The Virtue of Faith". I argue that there is a creative tension in his work between his desire to secure an objective basis for ethics and his affirmation of the value of grace, a love that is not proportioned to the excellence of its object. This tension, I further argue, ought to be resolved in the direction of grace. (shrink)

I am so in favor of the actual infinite that instead of admitting that Nature abhors it, as is commonly said, I hold that Nature makes frequent use of it everywhere, in order to show more effectively the perfections of its Author. Thus I believe that there is no part of matter which is not, I do not say divisible, but actually divided; and consequently the least particle ought to be considered as a world full of an infinity of (...) different creatures. (shrink)

While the classical account of the linear continuum takes it to be a totality of points, which are its ultimate parts, Aristotle conceives of it as continuous and infinitely divisible, without ultimate parts. A formal account of this conception can be given employing a theory of quantification for nonatomic domains and a theory of region-based topology.

We give an answer to the question as to whether quantifier elimination is possible in some infinite algebraic extensions of Qp (‘infinite p-adic fields’) using a natural language extension. The present paper deals with those infinite p-adic fields which admit only tamely ramified algebraic extensions (so-called tame fields). In the case of tame fields whose residue fields satisfy Kaplansky’s condition of having no extension of p-divisible degree quantifier elimination is possible when the language of valued fields is (...) extended by the power predicates Pn introduced by Macintyre and, for the residue field, further predicates and constants. For tame infinite p-adic fields with algebraically closed residue fields an extension by Pn predicates is sufficient. (shrink)

In Alain Badiou’s most recent work, L’immanence des vérités ( The Immanence of Truths), psychoanalyst Jacques Lacan once again figures peripherally but saliently. What is their specific relation in this text, however? We argue that Badiou responds here to the problem raised precisely by the Lacanian subject, situated as it is between the radical subjectivity of the symptom and the possibility of formalization. In L’immanence, he introduces the term ‘absoluteness’ to secure truths against both relativism and transcendental construction. We show (...) that in drawing on Lacan to establish an understanding of the absolute, Badiou highlights the implicit tension between psychoanalysis and philosophy. We treat central cross-currents – truths, knowledge, the event and love – to help reveal the specific character of their confluence in this third book of Badiou’s trilogy. Although he stresses the unity of his and Lacan’s efforts, the impossible Real marking their divisions also invariably emerges the closer one investigates. (shrink)

In essays written between 1960 and 1963, Blanchot embarks on a new line of thought, beginning with fundamental philosophical division between language and vision. The contrast between the two domai...

Kant’s preoccupation with architectonics is a characteristic and noteworthy aspect of his thought. Various features of Kant’s argumentation and philosophical system are founded on the precise definitions of the various subdomains of human knowledge and the derivative borders among them. One science conspicuously absent from Kant’s routine discussions of the organization of knowledge is chemistry. Whereas sciences such as physics, psychology, and anthropology are all explicitly located in the architectonic, chemistry finds no such place. In this paper, I examine neglected (...) passages from Kant’s corpus as well as texts regarding chemistry that Kant himself read in order to unveil his views on the definition of chemistry and its relations with the other sciences. These considerations reveal chemistry to be the science that studies the changes of matter into new kinds. Yet Kant idiosyncratically believes that such a change requires an infinite division of matter, effected by chemical forces. Although this understanding of chemical change dovetails with Kant’s dynamical, continualist theory of matter, it implies that chemistry cannot be reduced to physics. Thus, although chemistry stands alongside empirical physics as an applied natural science in Kant’s architectonic, it remains a distinct, independent science. (shrink)