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)

Every set can been thought of as a directed graph whose edge relation is ∈. We show that many natural examples of directed graphs of this kind are indivisible: for every infinite κ, for every indecomposable λ, and every countable model of set theory. All of the countable digraphs we consider are orientations of the countable random graph. In this way we find indivisible well‐founded orientations of the random graph that are distinct up to isomorphism, and ℵ1 (...) that are distinct up to siblinghood. (shrink)

In this paper I reconstruct and discuss Antonio Rubio (1546-1615)’s theory of the composition of the continuum, as set out in his Tractatus de compositione continui, a part of his influential commentary on Aristotle’s Physics, published in 1605 but rewritten in 1606. Here I attempt especially to show that Rubio’s is a significant case of Scholastic overlapping between Aristotle’s theory of infinitely divisible parts and indivisibilism or ‘Zenonism’, i.e. the theory that allows for indivisibles, extensionless points, lines, and surfaces, which (...) are supposed to take part in the composition of the continuum. Even if such a syncretic tendency was, in many different ways, already developing in the medieval period and then at the end of the sixteenth century, Rubio’s position is indeed peculiar. He maintains that indivisibles are real and actual, infinite in act, really distinct from each other, and that, although they indwell in substance, indivisibles do not contribute directly to the constitution of the continuum. In this reconstruction I emphasize notably Rubio’s usage of mereological notions like those of part, whole, completeness and incompleteness. (shrink)

The pascalian use of indivisibles is here considered in the context of the theological and mathematical debates of the time, by distinguishing it clearly from this of Cavalieri. The combinatory and geometrical approaches are closely linked in Pascal’s work. His use of indivisibles has a heuristic, inventive character and not only a demonstrative one. Ontologically speaking, it stems out from the acceptance of actual infinite. The use of the symmetry axiom of Archimedes is the basis of the pascalian use (...) of the infinitesimals, which has, in other respects, some close connexions with the Leibnizian conception of infinitesimals. (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)

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)

As Aristotle classically defined it, continuity is the property of being infinitely divisible into ever-divisible parts. How has this conception been affected by the process of mathematization of motion during the 14th century? This paper focuses on Nicole Oresme, who extensively commented on Aristotle’s Physics, but also made decisive contributions to the mathematics of motion. Oresme’s attitude about continuity seems ambivalent: on the one hand, he never really departs from Aristotle’s conception, but on the other hand, he uses it in (...) a completely new way in his mathematics, particularly in his Questions on Euclidean geometry, a tantamount way to an atomization of motion. If the fluxus theory of natural motion involves that continuity is an essential property of real motion, defined as a res successiva, the ontological and mathematical structure of this continuity implies that continuum is in some way “composed” of an infinite number of indivisibles. In fact, Oresme’s analysis opened the path to a completely new kind of mathematical continuity. (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)

This article analyzes Galileo's mathematization of motion, focusing in particular on his use of geometrical diagrams. It argues that Galileo regarded his diagrams of acceleration not just as a complement to his mathematical demonstrations, but as a powerful heuristic tool. Galileo probably abandoned the wrong assumption of the proportionality between the degree of velocity and the space traversed in accelerated motion when he realized that it was impossible, on the basis of that hypothesis, to build a diagram of the law (...) of fall. The article also shows how Galileo's discussion of the paradoxes of infinity in the First Day of the Two New Sciences is meant to provide a visual solution to problems linked to the theory of acceleration presented in Day Three of the work. Finally, it explores the reasons why Cavalieri and Gassendi, although endorsing Galileo's law of free fall, replaced Galileo's diagrams of acceleration with alternative ones. (shrink)

The seventeenth century saw dramatic advances in mathematical theory and practice. With the recovery of many of the classical Greek mathematical texts, new techniques were introduced, and within 100 years, the rules of analytic geometry, geometry of indivisibles, arithmatic of infinites, and calculus were developed. Although many technical studies have been devoted to these innovations, Mancosu provides the first comprehensive account of the relationship between mathematical advances of the seventeenth century and the philosophy of mathematics of the period. Starting with (...) the Renaissance debates on the certainty of mathematics, Mancosu leads the reader through the foundational issues raised by the emergence of these new mathematical techniques, including the influence of the Aristotelian conception of science in Cavalieri and Guldin, the foundational relevance of Descartes' Geometrie, the relation between geometrical and epistemological theories of the infinite, and the Leibnizian calculus and the opposition to infinitesimalist procedures. In the process Mancosu draws a sophisticated picture of the subtle dependencies between technical development and philosophical reflection in seventeenth century mathematics. (shrink)

Prime matter plays an indispensable role in Aristotle’s philosophy, enabling him to avoid the pitfalls of both naïve Platonism and nominalism. Prime matter is best thought of as a kind of infinitely divisible and atomless bare particularity, grounding the distinctness of distinct members of the same species. Such bare particularity is needed in symmetrical situations, like a world consisting of indistinguishable Max Black spheres. Bare particularity is especially important in modern physics, given the homogeneity and isotropy of space. With the (...) importance of fields in classical, relativistic, and quantum physics, we have good reason to prefer something like Aristotle’s continuous, infinitely divisible matter over indivisible particles. Mass and energy in relativistic physics also points in the direction of prime matter as the enduring substrate of these quantities. Recent work on Aristotelian interpretations of quantum mechanics, further underscores the contemporary relevance of prime matter. (shrink)

Whenever the membranes of the egg in which the foetus emerges on its way to becoming a new-born are broken, imagine for a moment that something flies off, and that one can do it with an egg as easily as with a man, namely the hommelette, or the lamella. The lamella is something extra-flat, which moves like the amoeba. It is just a little more complicated. But it goes everywhere. And as it is something - I will tell you shortly (...) why - that is related to what the sexed being loses in sexuality, it is, like the amoeba in relation to sexed beings, immortal - because it survives any division, and scissiparous intervention. And it can turn around. Well! This is not very reassuring. But suppose it comes and envelopes your face while you are quietly asleep... I can't see how we would not join battle with a being capable of these properties. But it would not be a very convenient battle. This lamella, this organ, whose characteristic is not to exist, but which is nevertheless an organ - I can give you more details as to its zoological place - is the libido. It is the libido, qua pure life instinct, that is to say, immortal life, irrepressible life, life that has need of no organ, simplified, indestructible life. It is precisely what is subtracted from the living being by virtue of the fact that it is subject to the cycle of sexed reproduction. And it is of this that all the forms of the objet a that can be enumerated are the representatives, the equivalents. Every word has a weight here, in this deceivingly poetic description of the mythic creature called by Lacan "lamella". Lacan imagines lamella as a version of what Freud called "partial object": a weird organ which is magically autonomized, surviving without a body whose organ it should have been, like a hand that wonders around alone in early Surrealist films, or like the smile in Alice in Wonderland that persists alone, even when the Cheshire cat's body is no longer present: "'All right', said the Cat; and this time it vanished quite slowly, beginning with the end of the tail, and ending with the grin, which remained some time after the rest of it had gone. 'Well! I've often seen a cat without a grin', thought Alice; 'but a grin without a cat! It's the most curious thing I ever saw in my life!'" The lamella is an entity of pure surface, without the density of a substance, an infinitely plastic object that can not only incessantly change its form, but can even transpose itself from one to another medium: imagine a "something" that is first heard as a shrilling sound, and then pops up as a monstrously distorted body. A lamella is indivisible, indestructible, and immortal - more precisely, undead in the sense this term has in horror fiction: not the sublime spiritual immortality, but the obscene immortality of the "living dead" which, after every annihilation, re-composes themselves and clumsily goes on. As Lacan puts it in his terms, lamella does not exist, it insists: it is unreal, an entity of pure semblance, a multiplicity of appearances which seem to envelop a central void - its status is purely fantasmatic. This blind indestructible insistence of the libido is what Freud called "death drive," and one should bear in mind that "death drive" is, paradoxically, the Freudian name for its very opposite, for the way immortality appears within psychoanalysis: for an uncanny excess of life, for an "undead" urge which persist beyond the cycle of life and death, of generation and corruption. This is why Freud equates death drive with the so-called "compulsion-to-repeat," an uncanny urge to repeat painful past experiences which seems to outgrow the natural limitations of the organism affected by it and to insist even beyond the organism's death - again, like the living dead in a horror film who just go on. This excess inscribes itself into the human body in the guise of a wound which makes the subject "undead," depriving him of the capacity to die : when this wound is healed, the hero can die in peace. (shrink)

Do you believe it to be impossible that God is infinite, without parts?-Yes. I wish therefore to show you an infinite and indivisible thing. It is a point moving everywhere with an infinite velocity; for it is one in all places, and is all totality in every place. Let this effect of nature, which previously seemed to you impossible, make you know that there may be others of which you are still ignorant. Do not draw this (...) conclusion from your experiment, that there remains nothing for you to know; but rather that there remains an infinity for you to know. (shrink)

El corpuscularismo sirvió a los físicos del XVII para matematizar la naturaleza al considerarla un conjunto de sistemas mecánicos. Pero la discontinuidad del atomismo chocaba con la continuidad de las magnitudes básicas, espacio y el tiempo, y derivadas. En su madurez, Galileo fundió física y matemáticas propo-niendo componer tanto los cuerpos como las magnitudes continuas a base de átomos inextensos (indivisibles). En el proceso inició el análisis de las propiedades de los conjuntos infinitos, pero no logró elaborar un cálculo que (...) le permitiese computar diferentes movimientos acelerados, mientras que en física no resolvió en problema fundamental de la condensación y la rarefacción.Seventeenth century atomism envisioned Nature as a set of mechanical systems to be treated mathematically. But the basic discontinuity of atomic theory of matter appeared inconsistent with the essential continuous character of geometrical magnitudes. In his old age Galileo devised a way to unify mathematics and physics via composing matter and continuous magnitudes out of an infinity of indivisible (atomic) units. Even though he forwarded the analysis of infinite sets, he couldn’t establish a calculus to compute and compare different accelerated motions. In physics he never solved the basic problem of condensation and rarefaction of substances. But the side results were interesting and even fascinating. (shrink)

Since the Middle Ages, Augustine and the wealth of his writings have had an enormous impact on Western philosophical thinking. His approach to time and memory, which he sets out in his eleventh book of the Confessions, is one of the most important sources for research about the philosophy of time. Augustine describes time as a permanent movement in which the future passes unceasingly through an unrelated present into the past. Only the very present moment exists, but this present moment (...) is infinitely short. If the past is no more, the future is not yet, and the present is indivisibly brief, memoria becomes the determining factor of human consciousness. Augustine’s theory of memory and time reveals an epistemological connection with the functioning of human consciousness. Since his understanding of time and memory is already of great relevance in many interdisciplinary approaches, it can also become important to the philosophy of music education. Revisiting Augustine’s concept of memoria can give us new impetus on questions about aesthetic education, cancel culture, and utopian thinking. (shrink)

There are many good reasons for seeing Aristotelian hylemorphism and atomism as diametrically opposed theories of matter. Aristotle himself had forcefully combatted the physical model of Leucippus and Democritus, whose ontology consisted of indivisible material bodies moving in an immaterial void, presenting his own model as an alternative. This alternative excluded both indivisibles and the void and postulated instead a plenist world made up of substances all of which were infinitely divisible continua composed of universal matter and specific substantial (...) forms. Unlike elementary atoms, Aristotle’s elementary substances could transmute into one another: water, when evaporating, was said to turn into air. (shrink)

Spinoza seems to argue both that “God or Nature” is mereologically simple, and that this being is mereologically complex insofar as it is composed of parts. This chapter proposes on Spinoza's behalf a resolution of this antinomy. This resolution focuses on Spinoza's mereology of the material world. It offers an alternative interpretation according to which Spinoza adheres both to the indivisibility of extended substance and to the reality of the finite modal parts that compose an infinite modal whole. In (...) the Oldenburg Letter, Spinoza claims that “concerning whole and parts, the chapter considers things as parts of some whole to the extent that the nature of the one so adapts itself to the nature of the other so that they agree with one another as far as possible”. The chapter argues that a similar assumption underlies Peterson's argument against the reality in Spinoza of divisible spatial quantity. (shrink)

In this paper, we will consider the theoretical aspects of Leibniz’s thought on infinitely small and infinite quantities in the context of the natural philosophy developed by him in the Parisian period. We will hold that in the texts of this period an attempt of problematizing concepts of infinitary mathematics is found, which is not in the strictly mathematical texts. In this perspective, we also propose that there is in Leibniz a “double methodological record” concerning the question of the (...) infinity and of the infinitesimal. As a mathematician, Leibniz is concerned about technical questions related to the construction of the infinitesimal calculus, in its different aspects, without be concerned about the philosophical challenges its adoption implies. However, as a philosopher and a metaphysician, Leibniz is obliged to deal with epistemological and ontological problems that emerge from the adoption of technical resources in matters of infinitesimal mathematics. (shrink)

Definitions Axioms Prop. I. Substance is by nature prior to its modifications Prop. II. Two substances, whose attributes are different, have nothing in common Prop III. Things, which have nothing in common, cannot be one the cause of the other Prop. IV. Two or more distinct things are distinguished one from the other either by the difference of the attributes of the substance, or by the differences of their modifications Prop. V. There cannot exist in the universe two or more (...) substances having the same nature or attribute Prop. VI. One substance cannot be produced by another substance Prop. VII. Existence belongs to the nature of substance Prop. VIII. Every substance is necessarily infinite Prop. IX. The more reality or being a thing has, the greater the number of its attributes Prop. X. Each particular attribute of the one substance must be conceived through itself Prop. XI. God, or substance consisting of infinite attributes, of which each expresses eternal and infinite essentiality, necessarily exists Prop. XII. No attribute of substance can be conceived, from which it would follow that substance can be divided Prop. XIII. Substance absolutely infinite is indivisible Prop. XIV. Besides God no substance can be granted or conceived Prop. XV. Whatsoever is, is in God, and without God nothing can be, or be conceived Prop. XVI. From the necessity of the divine nature must follow an infinite number of things in infinite ways--that is, all things which fall within the sphere of infinite intellect Prop. XVII. God acts solely by the laws of his own nature and is not constrained by anyone Prop. XVIII. God is the indwelling and not the transient cause of all things Prop. XIX. God and all the attributes of God are eternal Prop. XX. The existence of God and his essence are one and the same Prop. XXI. All things, which follow from the absolute nature of any attribute of God, must always exist and be infinite, or, in other words, are eternal and infinite through the said attribute Prop.. (shrink)

In this paper I reconstruct and discuss Antonio Rubio (1546-1615)’s theory of the composition of the continuum, as set out in his Tractatus de compositione continui, a part of his influential commentary on Aristotle’s Physics, published in 1605 but rewritten in 1606. Here I attempt especially to show that Rubio’s is a significant case of Scholastic overlapping between Aristotle’s theory of infinitely divisible parts and indivisibilism or ‘Zenonism’, i.e. the theory that allows for indivisibles, extensionless points, lines, and surfaces, which (...) are supposed to take part in the composition of the continuum. Even if such a syncretic tendency was, in many different ways, already developing in the medieval period and then at the end of the sixteenth century, Rubio’s position is indeed peculiar. He maintains that indivisibles are real and actual, infinite in act, really distinct from each other, and that, although they indwell in substance, indivisibles do not contribute directly to the constitution of the continuum. In this reconstruction, I emphasize notably Rubio’s usage of mereological notions like those of part, whole, completeness and incompleteness. (shrink)

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, 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)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Ideas 62.2 (2001) 193-210 [Access article in PDF] Zeno Against Mathematical Physics Trish Glazebrook Galileo wrote in The Assayer that the universe "is written in the language of mathematics," and therein both established and articulated a foundational belief for the modern physicist. 1 That physical reality can be interpreted mathematically is an assumption so fundamental to modern physics that chaos and super-strings are examples (...) of physical theories developed on the basis of success in mathematics. The mathematics came first, then the physical theory. Quantum theory is an awkward and similar case in point. Despite the fact that there is much agreement among physicists about the mathematical theory, its physical interpretation remains a matter of controversy. There are several interpretations, all of which challenge our everyday assumptions about reality. This led Bohr in 1935 to call for "a radical revision of our attitude towards the problem of physical reality." 2 Arthur Fine, originally a staunch defender of realist interpretations of quantum theory, saw the success of Bohr's Copenhagen interpretation as the end of Einsteinian realism, and subsequently Fine declared that "realism is well and truly dead." 3 Mathematical operators simply do not correspond with real entities in any conceivable way.Ilkka Niiniluoto has argued convincingly, however, that realism is alive and well in quantum theory. 4 The problem is that Bohr is right: quantum theory can be considered to concern the real only upon radical revision of ordinary conceptions of reality. Quantum theory does not describe the real in any meaningful sense of "the real." Rather, it is mathematical projection: ideal rather than real. [End Page 193]Aristotle did not hold that the physicist deciphers nature mathematically. He distinguished the physicist from the mathematician at Physics 2.2 by analogy to the definitions of "snub nose" and "curved." 5 The definition of "snub nose" requires the mention of a nose, that is, of matter which is inseparable from motion, whereas "curved" and other mathematical concepts like " 'odd' and 'even', 'straight'... 'number,' 'line,' and 'figure,' " which are separable from matter and from motion, are investigated by the mathematician. 6 Aristotle claims that physics is different from mathematics because the mathematician investigates physical lines, but not qua physical, and the physicist investigates mathematical lines "but qua physical, not qua mathematical." 7A peculiar consequence of my argument is that it establishes a common principle between two thinkers whose beliefs are otherwise so fundamentally in opposition. Zeno's paradoxes are commonly read as a denial of motion. For Aristotle, that move is axiomatic to the physicist. At 185a15 he considers arguments against this assumption outside the realm of objections to which the physicist must reply. If I am right that Zeno's paradoxes of motion challenge the very notion of mathematical physics, then he and Aristotle agree, contra Galileo, that the universe is not written in the language of mathematics. They simply come to this point from opposite directions, Zeno from an Eleatic thesis that physical reality is illusion, Aristotle from the natural scientist's pragmatic commitment to a truth about nature.I intend to use Zeno's paradoxes of motion to argue that the application of mathematical concepts to the physical world results in paradox. Zeno's arguments are limited within mathematics to geometry. This paper is accordingly a beginning from which I hope more work can be done later. Furthermore, Zeno's paradoxes are standardly read as directed against anyone who disagrees with Parmenides' thesis that there is but one thing which is motionless and indivisible. The paradoxes achieve this end by means of a reductio against the possibility of dividing space and time. That there are four paradoxes is accordingly explicable by the fact that there are four possibilities for the divisibility of space and time. The Achilles argues against the possibility that space and time are both infinitely divisible; the Dichotomy, against the infinite divisibility of space and finite divisibility of time; the Arrow, against the infinite divisibility of time and the finite divisibility of space; and finally, the Stadium shows that time and space cannot both... (shrink)

In this paper I intend to present the Dilemma of Continuity of Matter and a possible solution to it. This dilemma consists in choosing between two misfortunes in explaining the continuity of matter: or to say that material objects are infinitely divisible and not explain what constitutes the continuity of some kind of object, or to say that there is a certain kind of indivisible object and not explain what constitutes the continuity of such an object. The solution we (...) provide is precisely the thesis that material objects consists of points, a thesis we try to make clearer, although we have not developed it much. (shrink)

Once we realize that the indivisible and infinitely repeatable One of the arithmetic lesson in Republic7 is generated by διάνοια at Parmenides 143a6-9, it becomes possible to revisit the Divided Line’s Second Part and see that Aristotle’s error was not to claim that Plato placed Intermediates between the Ideas and sensible things but to restrict that class to the mathematical objects Socrates used to explain it. All of the One-Over-Many Forms of Republic10 that Aristotle, following Plato, attacked with the (...) Third Man, are equally dependent on Images and above all on the Hypothesis of the One. (shrink)

In 1675, Leibniz elaborated his longest mathematical treatise he everwrote, the treatise ``On the arithmetical quadrature of the circle, theellipse, and the hyperbola. A corollary is a trigonometry withouttables''. It was unpublished until 1993, and represents a comprehensive discussion of infinitesimalgeometry. In this treatise, Leibniz laid the rigorous foundation of thetheory of infinitely small and infinite quantities or, in other words,of the theory of quantified indivisibles. In modern terms Leibnizintroduced `Riemannian sums' in order to demonstrate the integrabilityof continuous functions. (...) The article deals with this demonstration,with Leibniz's handling of infinitely small and infinite quantities,and with a general theorem regarding hyperboloids. (shrink)

On our ordinary representations of space, space is composed of indivisible, dimensionless points; extended regions are understood as infinite sets of points. Region-based theories of space reverse this atomistic picture, by taking as primitive several relations on extended regions, and recovering points as higher-order abstractions from regions. Over the years, such theories have focused almost exclusively on the topological and geometric structure of space. We introduce to region-based theories of space a new primitive binary relation (‘qualitative probability’) that (...) is tied to _measure_. It expresses that one region is _smaller than or equal in size_ to another. Algebraic models of our theory are _separation_ _σ_-_algebras with qualitative probability_: \((\mathbb {B}, \ll, \preceq )\), where \(\mathbb {B}\) is a Boolean _σ_-algebra, ≪ is a separation relation on \(\mathbb {B}\), and ≼ is a qualitative probability on \(\mathbb {B}\). We show that from algebraic models of this kind we can, in an interesting class of cases, recover a compact Hausdorff topology _X_, together with a countably additive measure _μ_ on a _σ_-field of Borel subsets of that topology, and that \((\mathbb {B}, \ll, \preceq )\) is isomorphic to a ‘standard model’ arising out of the pair (_X_, _μ_). It follows from one of our main results that any closed ball in Euclidean space, \(\mathbb {R}^{n}\), together with Lebesgue measure arises in this way from a separation _σ_-algebra with qualitative probability. (shrink)

In the preface to his Theodicy, Leibniz describes the whole of his philosophical work as an attempt to follow Ariadne’s thread through “the two famous labyrinths in which our reason goes astray.” The first and best known of these—the labyrinth of freedom—concerns the relation between contingency and necessity in history. The second—and the one I want to discuss—is what Leibniz calls the labyrinth of the composition of the continuum. The problem itself is relatively simple: how can indivisible and distinct (...) elements constitute a continuum? How, that is, can singularity possibly be thought if the real is everywhere continuous, without interruption or break? Leibniz, as is well known, affirms both extremes of these questions: what really exists are simple unities or wholes, and yet the totality of these unities constitutes a single, organically complete world. While one mode of approach would be to turn immediately to Leibniz’s elaboration of monadic expression and intrinsic predication, I want to examine the effects the problem generates for the specifically Leibnizian conception of the real and the ideal, as well as the transition this entails from the metaphorics of the labyrinth to the figure of the infinite library and the problem of repetition. (shrink)

The first half of the 17th century was a time of intellectual ferment when wars of natural philosophy were echoes of religious wars, as we illustrate by a case study of an apparently innocuous mathematical technique called adequality pioneered by the honorable judge Pierre de Fermat, its relation to indivisibles, as well as to other hocus-pocus. André Weil noted that simple applications of adequality involving polynomials can be treated purely algebraically but more general problems like the cycloid curve cannot be (...) so treated and involve additional tools–leading the mathematician Fermat potentially into troubled waters. Breger attacks Tannery for tampering with Fermat’s manuscript but it is Breger who tampers with Fermat’s procedure by moving all terms to the left-hand side so as to accord better with Breger’s own interpretation emphasizing the double root idea. We provide modern proxies for Fermat’s procedures in terms of relations of infinite proximity as well as the standard part function. (shrink)

El corpuscularismo sirvió a los físicos del XVII para matematizar la naturaleza al considerarla un conjunto de sistemas mecánicos. Pero la discontinuidad del atomismo chocaba con la continuidad de las magnitudes básicas, espacio y el tiempo, y derivadas. En su madurez, Galileo fundió física y matemáticas propo-niendo componer tanto los cuerpos como las magnitudes continuas a base de átomos inextensos (indivisibles). En el proceso inició el análisis de las propiedades de los conjuntos infinitos, pero no logró elaborar un cálculo que (...) le permitiese computar diferentes movimientos acelerados, mientras que en física no resolvió en problema fundamental de la condensación y la rarefacción.Seventeenth century atomism envisioned Nature as a set of mechanical systems to be treated mathematically. But the basic discontinuity of atomic theory of matter appeared inconsistent with the essential continuous character of geometrical magnitudes. In his old age Galileo devised a way to unify mathematics and physics via composing matter and continuous magnitudes out of an infinity of indivisible (atomic) units. Even though he forwarded the analysis of infinite sets, he couldn’t establish a calculus to compute and compare different accelerated motions. In physics he never solved the basic problem of condensation and rarefaction of substances. But the side results were interesting and even fascinating. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Atoms, Gunk, and God:Natural Theology and the Debate over the Fundamental Composition of MatterTravis DumsdayLET US SAY we take a rock and divide it in two. We then divide each of the halves again. We repeat. We keep repeating, over and over and over again, until we have reached down to the level of molecules and then to atoms and then to subatomic particles and beyond. What, eventually, will (...) we end up with? (A) Do we eventually reach “rock bottom,” a layer of fundamental objects that can be divided no further? Or (B) could the dividing process be carried on forever?Let us take (A), the idea that there are fundamental, indivisible objects. We may call these “atoms,” and the theory that affirms their existence atomism. (I use the label for historical reasons; the “atoms” discussed in the history of metaphysics are of course not equivalent to the atoms of contemporary physics, which are divisible into constituent subatomic particles.) What must be the nature of these objects? Here we can identify two distinct versions of atomism.According to the first, these entities cannot be spatially extended, since extension entails divisibility. After all, if something has discernible length, then its conceptually distinguishable left and right halves, for instance, could be separated from each other, at least in theory. The existence of the matter constituting the left half seems not to entail the existence of the matter constituting the right half or vice versa, so one could persist without the other (e.g., the left half could be annihilated while the right half continues in existence), and [End Page 227] could be divided from the other. Whether they could actually be separated by us, using current or future technology, is of course an entirely different question from the philosophically significant issue of divisibility in principle. Atoms are, on this view, extensionless “point particles” with no actual or potential proper parts. This may be called atomism version 1.According to the second version of atomism, these indivisible objects are nevertheless spatially extended. The reality of atoms is affirmed, but the relation obtaining between extension and divisibility, which is supposedly entailed, is rejected. Despite being spatially extended, these particles cannot possibly be divided. Consequently, although they are spatially extended, they do not possess actual proper parts in the sense that “proper part” is typically understood (i.e., as involving smaller constituent objects, which could in theory exist apart from the whole). The claim is not merely that they cannot be divided by us using current or future technology, nor that their division would be inconsistent with contingent physical laws. The impossibility referenced here is not mere physical impossibility. Rather, the claim is that it is metaphysically impossible (and logically impossible, for those who equate metaphysical and logical impossibility) for them to be divided. There is no possible world in which these entities are in any way divided, either by splitting in two or by destroying one section while leaving the rest intact. This may be called atomism version 2.Atomism’s major competitors may be grouped under the affirmation of (B), according to which division could go on forever. Matter is infinitely divisible. But what does infinite divisibility imply? Here there is a further divergence of views. One camp affirms the gunky view of matter. On this view, a material object is infinitely divisible because it is composed of an actual infinity of proper parts: any material object is composed of smaller proper parts that are themselves real objects composed of smaller proper parts that are themselves real objects composed of smaller proper parts, and so on, ad infinitum. Another camp denies the reality of actual infinities (at least in the material realm), and affirms instead the reality of extended simples. On this view matter is infinitely divisible, as [End Page 228] on the gunky view, but not because material objects are composed of an actual infinity of smaller objects. Instead, material nature bottoms out (in a sense) at objects that are spatially extended and potentially subject to division, but that lack actual proper parts. They are not composed of real, smaller objects. Rather, what makes it true that the object... (shrink)

Philosophers no longer argue whether Hume ever read Berkeley, yet some remain puzzled as to why so little of Berkeley appears in Hume's works. Professor Popkin has remarked that even “where Hume and Berkeley come closest to discussing the same subject or holding the same view, Hume neither uses Berkeley's terms nor refers to him.” An apparent exception to this generalization is Berkeley's doctrine ofminima sensibilia, for both philosophers use this term to denote indivisible sensible points, and both invoke (...) such points in order to show that sensible extension is not infinitely divisible. Yet it has been suggested that, although Berkeley and Hume employ the same terms, they nevertheless maintain different doctrines: Hume'sminimaare unextended, whereas Berkeley's are extended. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Parmenides: The Road to Reality A New Verse Translation RICHARD MCKIM introduction i. In the history of Presocratic Greek philosophy, the poetry of Parmenides seems to loom up suddenly out of the blue like a spectral mountain peak. Depicting a vision of ultimate reality that transcends the sensory world, his towering verse manifesto revolutionized both how philosophers thought and what they thought about, with profound repercussions that still reverberate (...) today. Parmenides has not, however, been well served by his English translators. He wrote poetry and yet is almost always translated into prose. His poem describes a divinely inspired revelation and yet is persistently translated as if it were an exercise in deductive logic. His Greek can be strange and difficult but is never unintelligible, which is more than can be said for the Anglo-gibberish his translators too often force him to speak. Too many subscribe to the misguided notion that a “literal” translation, as close as possible to word-forword, best represents the original. In fact, the painful English that results, so far from being faithful to the Greek, actually betrays it, creating the obscurities it purports to reflect. Parmenides ’ reputation for being hard to understand is largely his translators’ fault, not his. I’ve undertaken to make amends by translating his poem as a poem, in a loose English approximation of the same meter. My goal is to capture some of what gets lost in prose—to mirror, however dimly, the vital role of poetic form in shaping Parmenides’ vision. The demands of meter make literal translation impossible—not at all a bad thing— while paradoxically freeing the translator to be more faitharion 27.2 fall 2019 ful. I’ve tried to use this freedom to demonstrate that Parmenides is not the obscurantist would-be logician of so many other translations but a philosopher who thinks in poetry, and whose thought is as clear and accessible as it is astonishing. ii. parmenides was a citizen of Elea, a Greek colonial town on the west coast of the ankle of Italy’s boot. Born around 515 BCE, he was an older contemporary of Socrates, who was born in 470 (according to Plato, the two men met when Parmenides was sixty-five). Several other so-called Presocratics were active even later—Democritus for one was Socrates’ junior by a decade. But the label is still useful: while Socrates reoriented philosophy toward moral values and politics, the later Presocratics maintained their predecessors’ focus on physics and cosmology. Their project was to account for the physical workings of the universe—its origins, its elements, and the forces or laws that govern its behavior— without falling back on the gods of traditional myth. But Parmenides splits their story in two. Before him, Presocratics took the reality of the sensory world for granted— a world of multiplicity and variety, motion and change, becoming and passing away. Parmenides radically rejected this assumption. The sensory world, he declared, is an illusion. What really exists is what he calls “being,” whose true nature can be perceived only by the mind. And by clearly perceiving what being must be like, the mind perceives that it can’t be anything like the world as our senses perceive it. Being is the exact opposite of the sensory world in every respect. It is timeless, motionless, changeless and imperishable, an eternal and indivisible One. After Parmenides, the reality of the sensory world needed defending. Subsequent Presocratics had to explain, at a minimum, how it could circumvent his ban on multiplicity and motion. Most notably, he provoked Leucippus and Democri106 parmenides: the road to reality tus into developing their theory of atomism—that the universe consists entirely of atoms in a void. Atoms are microscopic Parmenidean beings insofar as they’re indivisible, indestructible and immutable, but they can be infinite in number, they can move, collide and combine, thanks to the roominess of the infinite void, defined as the absence of being. Never mind that Parmenides ruled out the existence of such an absence—by inspiring atomism, he became, in his otherworldly way, the unwitting godfather of modern physics. As for metaphysics, it may be debatable whether he arrived at the idea... (shrink)

Evangelista Torricelli is perhaps best known for being the most gifted of Galileo’s pupils, and for his works based on indivisibles, especially his stunning cubature of an infinite hyperboloid. Scattered among Torricelli’s writings, we find numerous traces of the philosophy of mathematics underlying his mathematical practice. Though virtually neglected by historians and philosophers alike, these traces reveal that Torricelli’s mathematical practice was informed by an original philosophy of mathematics. The latter was dashed with strains of Thomistic metaphysics and theology. (...) Torricelli’s philosophy of mathematics emphasized mathematical constructs as human-made beings of reason, yet mathematical truths as divine decrees, which upon being discovered by the mathematician ‘appropriate eternity’. In this paper, I reconstruct Torricelli’s philosophy of mathematics—which I label radical mathematical Thomism—placing it in the context of Thomistic patterns of thought.Keywords: Evangelista Torricelli; Thomism; Eternity; Indivisibles; Proportions. (shrink)

Before establishing his mature interpretation of infinitesimals as fictions, Gottfried Leibniz had advocated their existence as actually existing entities in the continuum. In this paper I trace the development of these early attempts, distinguishing three distinct phases in his interpretation of infinitesimals prior to his adopting a fictionalist interpretation: (i) (1669) the continuum consists of assignable points separated by unassignable gaps; (ii) (1670-71) the continuum is composed of an infinity of indivisible points, or parts smaller than any assignable, with (...) no gaps between them; (iii) (1672- 75) a continuous line is composed not of points but of infinitely many infinitesimal lines, each of which is divisible and proportional to a generating motion at an instant (conatus). In 1676, finally, Leibniz ceased to regard infinitesimals as actual, opting instead for an interpretation of them as fictitious entities which may be used as compendia loquendi to abbreviate mathematical reasonings. (shrink)

The article presents Leibniz's preoccupation (in 1675?6) with the difference between the notion of infinite number, which he regards as impossible, and that of the infinite being, which he regards as possible. I call this issue ?Leibniz's Problem? and examine Spinoza's solution to a similar problem that arises in the context of his philosophy. ?Spinoza's solution? is expounded in his letter on the infinite (Ep.12), which Leibniz read and annotated in April 1676. The gist of Spinoza's solution (...) is to distinguish between three kinds of infinity and, in particular, between one that applies to substance, and one that applies to numbers, seen as auxiliaries of the imagination. The rest of the paper examines the extent to which Spinoza's solution solves Leibniz's problem. The main thesis I advance is that, when Spinoza and Leibniz say that the divine substance is infinite, in most contexts it is to be understood in non-numerical and non-quantitative terms. Instead, for Spinoza and Leibniz, a substance is said to be infinite in a qualitative sense stressing that it is complete, perfect and indivisible. I argue that this approach solves one strand of Leibniz's problem and leaves another unsolved. (shrink)

The relationship between Ars and Natura undergoes a manifest evolution from Aristotle to the medieval period and the time of Walter Burley. While Aristotle views all the possible complementarities between natural things and artificial things with regard to their perfection, the medieval commentators from Doctor planus and perspicuus on define artificialia in natural philosophy increasingly as perfect figures in actuality, divisible and with infinitely arrangeable parts. In Burley's works in particular, the products of art are likened to geometrical figures within (...) the structure of the continuous, such as lines, points and surfaces, and used as arguments against Ockham in their dispute over the ontology of indivisibles. (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)

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)

Druga knjiga Petrićeve Pancosmije potpuno nam otkriva što je Petrić mislio de continuo ili de divisibilitate quantitatis te nam ujedno nudi mnoge detalje Petrićeve neuspješne strategije pri osporavanju Aristotelovih pojmova neprekidnine i potencijalne beskonačnine. Prigovarajući Aristotelu, Petrić i ne htijući upozorava na glavne domete Aristotelova nauka o neprekidnini, ali nudi svoja, drugačija rješenja, poput zamisli o najmanjoj nedjeljivoj crti. Iako svojim rješenjima ne uspijeva postići ono što je Aristotel blistavo postigao pojmom neprekidnine u tumačenju prirode i matematike, Petrić unutar polemike (...) s Aristotelom postavlja nova pitanja kad se bavi računom s beskonačninama i genealogijom znanosti.The second book of Pancosmia provides full insight into Petrić’s view of the continuum or de divisibilitate quantitatis, offering us many details on Petrić’s futile strategy in refuting Aristotle’s notions of the continuum and the potential infinite. In his objections to Aristotle, Petrić implicitly points the major contributions of Aristotle’s doctrine of continuum, but also submits his own solutions such as the idea of the minimal indivisible line. Although his solutions failed to bring him the results as brilliant as those of Aristotle with the notion of the continuum in the interpretation of nature and mathematics, within his polemic with Aristotle, Petrić comes forward with new questions dealing with the calculus of infinite quantities and genealogy of science. (shrink)

The essence of the theory of minima naturalia is the contention that a physical body is not infinitely divisible qua that specific body. A drop of water cannot be divided again and again and still maintain its “wateriness”. There are several statements in Aristotle's Physics which suggest such an interpretation, and the theory of minima naturalia is commonly considered to have originated in the thirteenth century as an interpretation of these statements. The present paper is a preliminary presentation of the (...) role of Ibn Rushd in the evolution of the theory, hitherto neglected. His theory developed not only as an elaboration on the “suitable” statements of Aristotle, but mainly as an attempt to solve the difficulties raised by Aristotle's thesis that body and motion are continuous, infinitely divisible entities and are associated qua such. According to Ibn Rushd's interpretation, body and motion are associated not qua being continuous but qua having indivisible minimal parts. It seems that Epicurus' and Ibn Rushd's theories of minima developed as responses to Physics VI and offer modifications of classical atomism and of classical Aristotelianism , which to a certain extent reduce the gap between these two systems. (shrink)

Translated by Drew S. Burk and Anthony Paul Smith. Excerpted from Struggle and Utopia at the End Times of Philosophy , (Minneapolis: Univocal Publishing, 2012). THE END TIMES OF PHILOSOPHY The phrase “end times of philosophy” is not a new version of the “end of philosophy” or the “end of history,” themes which have become quite vulgar and nourish all hopes of revenge and powerlessness. Moreover, philosophy itself does not stop proclaiming its own death, admitting itself to be half dead (...) and doing nothing but providing ammunition for its adversaries. With our sights set on clearing up this nuance, we differentiate philosophy as an institutional entity, and the philosophizability of the World and History, of “thought-world,” which universalizes the narrow concept of philosophy and that of “capital.” We also give an eschatological and apocalyptical cause to this end, of “times” or “ages” rather than those of philosophical practice. Last but not least, it is the Future itself in the performativity of its ultimatum that determines this end times, reversing these times from the Identity the Future accorded to it, withdrawing the thought-world from the lie of its death. Why this style of axioms and oracles, these more or less subtle distinctions, old and debased, with an appeal to the ultimata , to end times and last words? We fight to give, parallel to the concept of Hell, its new theoretical position, for its philosophical return and its non-philosophical transformation. No more so than any other word, Hell is not a metaphor here, just the Principle of Sufficient World. Every man, no doubt, has his “hell” readily available, connivance, control, conformism, domestication, schooling, alienation, extermination, exploitation, oppression, anxiety, etc. We have our little list that the Contemporaries established in the previous century in the same way one used to construct lists of virtues and vices or honors and wealth. They invented it for us without knowing it, for us-the-Futures who have as our responsibility to invent its use. In the Christian and Gnostic tradition, the struggle of the End Times takes place “on earth.” The most sophisticated of believers have it taking place in Heaven as well, above all in Heaven. The various kinds of gnosis imagine infinite falls and vertiginous highs, the vertigo of salvation. On Earth as in Heaven, a hell is available. The Marxists have the law of profit and the control of production, the class struggle Capital imposes on man. The Nietzscheans, the dull grumble of the struggle in the foundations of World and History, the domestication of man, the society of control. The phenomenologists, the capture of being, the most superficial amongst them, the age of suspicion. But all of these “hells” are taken from World, History, Society, and Religion. What we call Hell is no longer of the order of these specific and total intra-worldly generalities, it is both more singular and more universal, no longer being at all of the same order, it is the determinant Identity of these small hells strung out through history but unified here in the name of the last Humaneity. It is even found within the French idiom for hell [ enfer ], en-fer literally means “in-irons.” We are as much “in-irons” as we are “alive” [ en-Vie ]. We believe in Hell but as non-philosophers and it is even because we are non-philosophers that we can believe in it outside of any sort of religion. Hell is less mythological than ideological, it combines philosophizability with universal capital. Under what form? A single term could work for them without being a metaphor or something they would participate in by analogy, a more innovative and conjectural term than control, more universal than profit. It would denote the growing and permanent extortion of a surplus-value of communication, of speed and of urgency in change, in productivity and in work, in the pressure of images and slogans. It would be worse than solicitation, more tenacious than capture, more active and persecutory than control, softer and more insidious than a frontal attack, just as perverse as questioning and accusation, less brutal and offensive than extermination, less ritualized than inquisition, it would be soft and dispersed, instantaneous and vicious, it would be a crime without declared violence. Collusion and conformism, it would be afraid to show itself. Related to rumor, from which it borrows its infinite and tortuous ways. It is harassment. As a modernized form of Hell, perhaps harassment has a long future in front of it, of innocent torture, slow assassination, in short the fall, but radical with no way of recovering and which tolerates only salvation. THE PHILOSOPHICAL PAST OF NON-PHILOSOPHY Non-philosophy is thus Man as the utopian identity of the philosophical form of the World, a utopia destined to transform it. We still have to understand these equations, in particular that of the being-uni-versed from Man, and this book adheres to this by re-exposing non-philosophy in a different way via one of its new possibilities. It uses this opportunity to once again take up formulations that lead to objections answering certain external critics, as well as revisiting diverging interpretations specific to non-philosophers. A portion of this book is devoted to going through these theories via a strict or “lengthy” presentation of non-philosophy, and its defense against more expeditious solutions. This work of rectification is the occasion, merely the occasion, for refocusing non-philosophy on Man (the “Man-in-person,” “Humaneity”), and in a more innovative manner, on its utopian vocation established since the book Future Christ . As for this “occasion,” it is quite obvious. A school of posture, not to say a school of thought, supposes a minimum of closure from the most liberating of knowledge, a heritage, its utilization, and its no less certain dispersal. Within its development, a variety of interpretations will no doubt appear, deviations that are as much normalizations, and a struggle against this multiplication of divergent tendencies. These are perhaps not inevitable evils, especially here, merely a normal development according to the twisting paths of history. But the problem is made worse by the fact that this school of non-philosophers is that of utopia. Not the former attempts devoted to commenting on the worst authoritarian and criminal forms of the past and the present, but utopia as the determinant principle for human life, or to put it another way, of the Future as an irreducible presupposition of (for) thinking the World and History. Non-philosophers are engaged in an aleatory navigation between the respect for the most rigorous utopia, whose rules are not that of the reproductive imagination but those from the Future determining imagination itself, as well as the temptations, diversions, and remorse of history. Little by little, we will begin to understand that the Future as we understand it no longer has any temporal consistency or positive content, without being an empty form or a nothingness, but that it is foreclosed to past and present History, just as it is foreclosed to the place of places, the World, and that it is the only method for establishing the practice of thinking in a non-imaginary instance. Because it is the World and History that are imaginary and have a terrible materiality, it is not necessarily utopia. We will overlap two objectives here: the defense of non-philosophy against the (non-) philosophers that we are, only occasionally, and the introduction of philosophy to a rigorous future. Together they set out to definitively render, without any possible return to philosophical conformism or towards the facilities of the past and present, the non-philosophical enterprise understood as utopia or uchronia. Imagination and speculation, left to themselves and thus undistinguished, are quite good for participating in the grand game of History but have little value or worse for the Future which is unimaginable and unintelligible and must be maintained as such. MAN-IN-PERSON AS SUSPENSION OF THE PHILOSOPHICAL CHORA The point where philosophical resistance is concentrated is without a doubt the invention of the Name-of-Man, first name, oracular as much as axiomatic, of the determining cause for the non-philosophical posture. And that which concentrates the differends is the style of non-philosophy as identity that possesses the dual aspect, of discipline and of the oeuvre, of the theorem and of the oracle. But the real difficulty in understanding the simplicity of non-philosophy is profoundly hidden in the depths of philosophy itself. Because philosophy, from Parmenides to Derrida, even Levinas, continues to be a divided gesture, without a veritable immanence, transforming its thematic contents of transcendence in also forgetting to transform the operative transcendence in the element from which the ontology of surface is established which we will call, in memory of Plato, the chorismos. The general effect of the chora literally gives place to philosophy, demanding binding and sutures to which we will once and for all “oppose” the Man-in-person, his power of cloning, and his future being. All philosophy contains a hinter-philosophy in which it deploys its operations and weaves its tradition like an understudy of a topographical nature and in the best of cases being itself topological. Philosophy as well consists of two levels, its pre-ontological operative conditions on the one hand, and its superficial theme on the other, it too has its presupposed, but is not aware of it or erases it within the unity of appearance named Logos. Rightly, the Logos, and its flash or lightning nature, possesses a “dark precursor,” the chora, which is as much a virtual image, and philosophy, dazzled by its own lightning flash, seems to completely forget about it at the same time it sets itself up within it. Non-philosophy risks taking this same path, of confusing what it believes to be the real with its phantom double, contenting itself to working on the thematic level of philosophy, not its surface objects and its idle chatter (we stopped talking about this a long time ago and in any case they are merely simple materials for inducing a work of transformation), but the transcendence-form of its objects. In the end it risks, through precipitation, taking back up the heritage of philosophy, a heritage of a misunderstood presupposed, even more profound than the play of transcendences. This is what the imperative of the radicality of immanence meant, to treat immanence in an immanent manner, not to make a new object out of it. And from here we get non- (philosophy) and its refusal of the Platonic chorismos , symbol of all abstraction, and thus all transcendental appearance. There are no illusions. The message will leave a heritage in tattered pieces and interpretations. But it was difficult not to dispute the differend to its core. There will be complete confusion of the multiple, possible, and necessary effectuations of non-philosophy with its interpretations. The non-philosophical or human freedom of philosophical effectuation and the philosophical freedom of interpretation. Effectuations demand non-philosophy to return to zero from the point of view of its philosophical material and thus also but within these limits the formulation of its axioms , but in no way providing from the outset divergent interpretations of the aforementioned axioms. They are divergent because they do not take into account the material from which these axioms are derived within non-philosophy, and because they do not see themselves as symptoms of another vision of the World. The utterances of non-philosophy are not mathematical theorems and pure axioms, they merely have a mathematical aspect . They are, by their extraction or origin, mathematical and transcendental. And by their determined function in-Real, within non-philosophy, they are identically in-the-last-Humaneity entities which have an aspect of an axiom and an aspect of interpretation (or an oracular aspect as we say) that attempts (sometimes it is ourselves who provide the occasion) to isolate and transform, in complete freedom of interpretation. There will be an opportunity to complain about the complex character of the language of non-philosophy, an idiom saturated with classical references, sophisticated in a contemporary way. Its freedom of decision up against the whole of philosophy demands these effects of “complication” and “privatization,” as the saying goes. But it also demands fighting against the drift [ dérive ] of the pedagogical-all and the mediatic-all that leads philosophy into the shallow depths of opinion, which is the site of its impossible death. The noble idealism of “pop-philosophy” has been consumed into a “philo-reality;” against this we propose philo-fiction. Parricide, which is at the bottom of these interpretations and which we can judge as being quite fertile, although it has informed tradition, only takes place once or within one lone meaning. In regards to Parmenides, it was possible; Plato introduced the Other as non-being and language, bringing into existence the philosophical system of the World, but is it possible to repeat it again with the same fecundity in regards to non-philosophy, this time in introducing (non) religion or (non) art, still mixing them without taking into account this mixture, alternatively as a philosophical or religious resentment? If philosophy begins via a crime, it is no doubt obliged to continue down similar pathways, to the effect that the crimes of philosophy, once the founding crime has been committed, are a reaction of self-defense. It is undoubtedly from this that we get Marx’s declaration that history begins by tragedy and repeats itself or ends in farce. The preservation of rigor and fecundity is, in every respect, a psychologically difficult task within a theory such as non-philosophy. Having posited an essential objective of liberation in regards to philosophy and its services, one has often understood this objective as an authorization of providing particular interpretations of its axioms and ends up obliterating their scope. This ends up confusing, on one hand, two kinds of freedoms in regards to non-philosophy, the freedom of its interpretations and the freedom of its effectuations. On the other hand, any defense of “principles” against precipitated interpretations is immediately taxed with a will to orthodoxy, a prohibitive objection when we are dealing with, as is the case here, a heretical theory of thought. Nevertheless, it is time to stop confusing heresy as the cause of thought with an ideology of heresies, which is certainly not at all our object, but rather a form of normalization. As for the “disciplinary” aspect, which is not the only aspect, it demands something other than philosophical “answers to objections,” a precision in the definition and use of its procedures in the formation of utterances, since non-philosophy is neither a supplementary doctrine interior to philosophy nor a vision of the world but one whose priority is a “vision of Man,” or rather Man as “vision” that implies a theory and a practice of philosophy. In the end, struggle is only one aspect of non-philosophy, not its whole or telos, struggle coming only from its materiality. In particular, if the discipline of non-philosophy is inseparable from struggle, it is not a question of reducing the monomaniacal obsession of its “marching orders.” This would reduce its complexity and kill its indivisibility, deploying it in a “long march” and a form of Maoization whose philosophical presuppositions no longer have any pertinence here, a case of the One and the Two, which are now cloned and no longer tied together. More generally, non-philosophy is a complex thought composed of a multitude of aspects, which is to say, unilateral interpretations, of a philosophical origin but reduced by their determination in-the-last-instance . The “liberalism” of non-philosophy is merely one of the aspects of which it is capable, not an essence. Similarly, it is only capable of having a “Maoist” aspect. Let us generalize. The weakness of non-philosophy is due to a specific cause, the determination-in-the-last-Humaneity of a subject for the World. Everything that has a right to the philosophical city can be said about it in turn and in a retaliatory mode since Man contributes nothing of himself that Man takes from the World. We can consider non-philosophy as being pretentious, absurd, idealistic, empty, materialistic, formalistic, contradictory, modern, post-modern, Zen, Buddhist, Marxist; it endures or tolerates, perhaps “appeals” to, or at least renders possible, sarcasms, ironies, and insults without even talking about the misunderstandings, partly for the same reason as psychoanalysis. All of this goes beyond simple “deviations.” They are its aspects, which is to say, its “unilateral” philosophical interpretations in both senses of the word, being either sufficient coming from the mouth of philosophers, or reduced to their absolute dimension of sufficiency and totality in the mouths of non-philosophers, and both times due to the weakness and strength of Man-in-person as their determination only in-the-last-instance. The non-philosopher is certainly not a Saint Paul fantasizing about a new Church. The non-philosopher is either a (Saint) Sebastian whose flesh is pierced with as many arrows as there are Churches, or a Christ persecuted by a Saint Paul. What is engaged in here is the practice of retaliation. A negative rule of the non-philosophical ethics of outlawed discussion by way of argumentation (the sufficient is you, the orthodox one is always you, you are the fashionable one, and when a master you are someone else) that is founded on the confusion of effectuations of non-philosophy and of its overall interpretations. Retaliation is the law but as with any too-human law, it must acquire a dimension that displaces it, or rather emplaces it and takes away its authority but not all of its effectiveness. If the non-philosopher is only authorized by himself, which is to say by philosophy but limited by the Real-of-the-last-instance, its critique of other non-philosophers can merely be retaliatory under the same conditions, only by the Real limited in-the last-instance. THE TREE OF PHILOSOPHICAL SAINTLINESS The thematic horizon or material of these debates is in the relationships between philosophy, religion as gnosis, and non-philosophy. It is inevitable, regarding non-philosophers in general (whether they are non-philosophers by name or simply its neighbors) that we often end up evoking Marx’s Holy Family and imagining, arranged on the neighboring branches of the tree of philosophy and annexed, sometimes abusively, to non-philosophy, authors who would quite evidently and quite rightly refuse this label. So it is that we find, for example, a Saint Michel, a Saint Alain, a Saint Gilles 1 without even mentioning the youngest who aspire as well to the freedom of “saintliness” and who make their muted voices heard here. If there is a Holy Family of non-philosophers, it extends completely beyond these three, provided that the sectarian spirit can save us. This book is organized in the following manner: To begin, in order to recall the essential part of the problematic, we have organized a Summary of Non-Philosophy , a vade-mecum of notions and basic problems, in a classical style. Secondly, there is Clarifications On the Three Axioms of Non-Philosophy , designed to posit their proper use as much as to elucidate their meaning. Thirdly, an analysis of Philosophizability and Practicity , both being extreme constituents of philosophical material or the contents of the third axiom. Fourthly, the heart of this work: Let us Make a Tabula Rasa of the Future or of Utopia as Method . Fifthly, we have a theoretical outline of a non-institutional utopia, The International Organization of Non-Philosophy, L’Organisation Non-Philosophique Internationale, (ONPHI) already created in practice but under the conditions of possibility and functioning from which here we put into question “de jure,” thus not without a perplexity concerning “facts,” in any case, without the capability of “getting to the bottom of things.” Sixthly, an essay characterizing The Right and the Left of Non-Philosophy , a brief topology of several philosophizing and normalizing positions of proponents or tenants of this problematic. Seventhly, Rebel in the Soul: A Theory of Future Struggle , a systematic discussion starting from a confrontation of non-philosophical gnosis and non-religious gnosis to the extent that they pose, posed or perhaps still will pose themselves as rivals to non-philosophy in a mixture of fidelity and infidelity. Despite the fact that it can also be read as putting non-philosophy into perspective: it pits against a standard Platonism two contemporary appropriations of Gnosticism. On the basis of the paradigm of Man who never ceases to come as the Future-in-person, each one of these moments strives to reestablish not the “true” non-philosophy and its orthodoxy, but the minimal conditions to respect in order to allow for its maximum fecundity. And in order to bring about one of the last possibilities of its development, making explicit Humaneity as a utopia-for-the-World. In introducing these considerations in the form of a “testament” and “ultimatum,” we want to indicate two things: First, that this is the last time we will intervene in order to caution non-philosophers against the temptation of returning and looking backwards towards philosophy. Only a disillusioned nostalgia for the former World and its traditions barely remain permissible to us.… Secondly, that non-philosophy is also a sort of ultimatum for considering one’s life and transforming one’s thought from the perspective of a uni-version rather than a conversion. Man as future is this ultimatum in action, not an impatient self-proclaimed genius, and philosophy is his testament. It is obviously the ultimatum that determines this testament as “old” with a view towards a life that is, itself, non-testamentary. In and of itself, the “old” can never bear a veritable eschaton. Thus, this book intersects according to the logic of this paradigm, under the sign of the ultimate or “last” as future, philosophy as testament and cautionary note for maintaining the non-philosophical oeuvre as “future” or “utopian.” We will see that between these two dimensions it cradles a theory of struggle. In the end, this book envisions non-philosophers in multiple ways. It inevitably sees them as subjects of knowledge, most often academics insofar as life in the world demands, but above all as close relatives of three great human types. The analyst and political militant are quite obvious, for non-philosophy is close to psychoanalysis and Marxism insofar as it transforms the subject in transforming philosophy. Here again, one must have a sense not of certain nuances but of aspects (of the interpretations, albeit unilateralized) and not in order to construct a simple proletarization or militarization of thought as theory. To be rigorous, rather than authoritarian or spiteful, is its task. And lastly, non-philosophy is a close relative of the spiritual but definitely not the spiritualist. Those who are spiritual are not at all spiritualists, for the spiritual oscillate between fury and tranquil rage, they are great destroyers of the forces of Philosophy and the State, which are united under the name of Conformism. They haunt the margins of philosophy, gnosis, mysticism, science fiction and even religions. Spiritual types are not only abstract mystics and quietists; they are heretics for the World. The task is to bring their heresy to the capacity of utopia, and their utopia to the capacity of the paradigm. NOTES We will recognize allusions, and sometimes references, to closely related or distant themes, but which are related, in the work of Michel Henry, Alain Badiou and via the representative of “non-religious” gnosis of a Platonic origin, in Gilles Grelet. It goes without saying that these discussions are current and local, neither concerning the ensemble of doctrines nor prejudging the eventual evolution of certain amongst them. This concerns defining certain proximities with non-philosophy (rather than adversaries which in some sense they are) and typological and emblematic differends (rather than conflicts with a certain author). (shrink)

Non-classical science gives a very specific answer to the question of scientific errors and their epistemological value. But for all the specificity of this answer, it casts light on a problem that remains with us century after century, the historically constant problem of truth and error—one of the most fundamental problems of knowledge. At first sight, these two poles have always stood opposite each other, like good and evil, beauty and ugliness. But moral and aesthetic theories have long since left (...) behind this initial conception, and shown that the two polar concepts are in fact inseparable. As regards truth and error, their indivisibility has only become apparent in the context of non-classical science. Truth has ceased to be the absolute contradiction of error. Contemporary science finds it to be something relative, inseparable from its opposite pole. But the non-classical, retrospective approach, a re-evaluation of the past of science in the light of its present, and indeed even more in the light of its prognosis, its future, allows one to see that scientific truths that are adequate for the object of knowledge always reflect only one side of the objective world, a world which has an infinite number of sides. The irreversible process that leads from relative error to relative truth has an absolute character: an irreversible process of generalization, concretization and complication of a Weltanschauung that more accurately reflects the infinite complexity of life. (shrink)

In this paper I attempt to trace the development of Gottfried Leibniz’s early thought on the status of the actually infinitely small in relation to the continuum. I argue that before he arrived at his mature interpretation of infinitesimals as fictions, he had advocated their existence as actually existing entities in the continuum. From among his early attempts on the continuum problem I distinguish four distinct phases in his interpretation of infinitesimals: (i) (1669) the continuum consists of assignable points separated (...) by unassignable gaps; (ii) (1670-71) the continuum is composed of an infinity of indivisible points, or parts smaller than any assignable, with no gaps between them; (iii) (1672-75) a continuous line is composed not of points but of infinitely many infinitesimal lines, each of which is divisible and proportional to a generating motion at an instant (conatus); (iv) (1676 onward) infinitesimals are fictitious entities, which may be used as compendia loquendi to abbreviate mathematical reasonings; they are justifiable in terms of finite quantities taken as arbitrarily small, in such a way that the resulting error is smaller than any pre-assigned margin. Thus according to this analysis Leibniz arrived at his interpretation of infinitesimals as fictions already in 1676, and not in the 1700's in response to the controversy between Nieuwentijt and Varignon, as is often believed. (shrink)

The seventeenth century saw dramatic advances in mathematical theory and practice. With the recovery of many of the classical Greek mathematical texts, new techniques were introduced, and within 100 years, the rules of analytic geometry, geometry of indivisibles, arithmetic of infinites, and calculus were developed. Although many technical studies have been devoted to these innovations, Mancosu provides the first comprehensive account of the relationship between mathematical advances of the seventeenth century and the philosophy of mathematics of the period. Starting with (...) the Renaissance debates on the certainty of mathematics, Mancosu leads the reader through the foundational issues raised by the emergence of these new mathematical techniques, including the influence of the Aristotelian conception of science in Cavalieri and Guldin, the foundational relevance of Descartes' Geometrie, the relation between geometrical and epistemological theories of the infinite, and the Leibnizian calculus and the opposition to infinitesimalist procedures. In the process Mancosu draws a sophisticated picture of the subtle dependencies between technical development and philosophical reflection in seventeenth century mathematics. (shrink)