Some features of Hume's conception of space

MARINA SOME FEATURES FRASCA SPADA’ OF HUME’S CONCEPTION OF SPACE zyxwvutsrqponmlkjihgfedcbaZYXWVUT intend to examine certain features of Hume’s conception of space, particularly as it is presented in the second part of Book 1 of the Trentise, my aim being to reconstruct the internal logic of his conception above and beyond its apparent contradictions. For various reasons, this is a task which is anything but straightforward. To begin then, it is the following features which strike us most forcibly on a first reading of Hume’s text: (I) It appears that certain fundamental passages of the argument can only with difficulty be reconciled with Hume’s philosophy taken as a whole, or at least with some of the more general features upon which the various and sometimes conflicting interpretations would, to some extent, seem to agree. Moreover, such features are usually justified by referring precisely to those pages which directly precede the discussion of the ideas of time and space. (2) To make matters worse, some of Hume’s arguments are decidedly “wrong”. No historian will be shocked by this, or at least they ought not to be. But the discussion of geometry, or, for instance, the way in which Hume claims to refute the so-called theory of infinite divisibility appear crudely empiricist, in places inconsistent, and based on inadequate information (this is true even with respect to other texts of the period - like Berkeley’s Analyst - where the thrust of the argument is in some sense similar to Hume’s). In short, l-lume’s conception is even somewhat irritating, to the point of trying the patience of the most refined of historiographical sensitivities. (3) Some passages are simply anything but easy to understand. (One needs only to look at the pages dedicated to the problem of a vacuum, for example.) The picture is already rather complex, and, moreover, further difficulties arise from the fact that in these pages, as I have already mentioned, we obviously find topics of fundamental importance to the discussion of mathematics. These contain ideas which are central both to Hume’s theory of relations and to the distinction between knowledge and probability, two arguments which are crucial to the philosophy of the Trearise as a whole. It should also be said that some passages seem to authorise, if they do not indeed encourage, the use of notions which are entirely alien to the overall argument, IN THIS ESSAYI *WolfsonCollege.Cambridge Studies in History Cl33 9BB, U.K. Correspondence Received 25 Rufus/ 1989. Stud. Hist. Phil. Sri.. Vol. 21, No. 3. pp. 371411, Printed in should be sent c/o the Editor. pnd Philosophy of Science. Great Britain 1990. c 371 0039-368 I/90 $3.00 + 0.00 1990. Pergamon Press plc. 372 Studies in History and Philosophy of Science and which therefore give rise to perverse interpretations revealing one anomaly after another. For example, many scholars employ the analytic/synthetic distinction understood in a sense which shows clear traces of Kantian analysis. Finally, and typically, the themes which seem most clearly epistemological are inextricably bound up with questions of a conspicuously metaphysical and theological origin. ’ In approaching these problems it is useful to begin by looking at considerations of an entirely general and external nature. In the first place, the ideas of space and time constitute one of the first subjects dealt with in the Treatise and occupy overall some 40 pages. It seems only natural then, although by no means entirely safe, to assume that Hume attaches a certain importance to the question2 At any rate, the position and length of this part of the book suggest that one of the main reasons why commentators should find it interesting lies in the fact that these pages represent one of the first instances of application of the fundamental principles set out in the previous part, and in this sense they constitute a kind of testing ground. They might, in other words, allow us to observe ‘what happens to the principle of the correspondence of impressions and ideas, for instance, at the point of transition from a general methodological position to an actual engagement with specific themes. In this transition from theory to application, we might expect to find certain adjustments. In some sense these adjustments can be read as contradictions. However, this is not necessarily so. Secondly, it is normal to distinguish three different, fundamental points zyxwvutsrqpon in Hume’s writing on space and time: (1) The discussion of infinite divisibility, evidently linked to the conception of mathematics; (2) The introduction of a relational theory of time and space, connected to the theory of abstract ideas expounded a few pages earlier, and similarly informed by a return to perception. This theory is also full of important consequences for geometry; (3) And finahy the examination and commentary on the notion of a vacuum. It is plain that the theoretical part, the actual theory of space and time, is neither apparent at the beginning, nor presented as a conclusion. It remains compressed, so to speak, between themes which are cIearly polemical . To a certain extent perhaps this might be ascribed to the crucial nature of the questions ‘As we shall see, these problems are well documented in the critical literature. This is contrary to what J. Laird, for instance, seems implicitly to state in Hutne’s zyxwvutsrqponmlkjih Philosophy of Human Nature (Archon Books: New York, 1967) (1st edn London 1932): according to Laud. “no philosopher can neglect these problems: and they permeated the thought of Hume’s time” (p. 64). He recognises, however. that “Hume long retained his interest in these subjects, although he was diffident of his capacity concernjng them” (p. 64). For a different opinion see R. Kuhns, ‘Hume’s Republic and the Universe of Newton’, in: P. Gay (ed.), Eighteenth Cenrury Studies presented to zyxwvutsrqponm Arthur M. Wilson (Russell & Russell: New York, 1972). pp~ 73-95. Kuhns considers the invcstigation of the ideas of space and time to be central in Hume’s moral philosophy. Htme’s Conception of Space 373 upon which Hume takes a stand. On the other hand, this kind of presentation can be considered a somewhat general characteristic of the asystematic way in which he shapes his discourse. It might also be linked to his explicit proposal to establish a new metaphysics, intending, above all, to rid himself of the “false and adulterated” metaphysics rather than to provide a complete and fully resolved picture. However, we obviously should not let ourselves be misled by the apparent ambiguity and heterogeneity of his discussion, and should as far as possible avoid considering particular passages in isolation. Such an approach would prevent a full grasp of Hume’s overall meaning and reduce the process of critical analysis to a mere enumeration of errors, inconsistencies and ambiguities.’ Thirdly, it may be useful to consider, though without laying any claim to completeness, which were Hume’s principal sources (or at least those of which one can be reasonably certain) for his conception of space, in order to relocate it in the context to which it must effectively belong. From the letter to Ramsey of 1737 it becomes clear above all that he considered the following texts as fundamental for an understanding of the Treatise in general: Malebranche’s Recherche de la WritP, Berkeley’s Principles, the ‘Zeno’ and ‘Spinoza’ entries in Bayle’s Dictionmire and Descartes’ Medirariunes. There are, indeed, various ways in which we can detect the presence of these works behind Hume’s thought, together with that of certain fundamental works of Newtonian theology like those of Samuel Clarke, and the writings of the Scottish moralists, especially Hutcheson. Moreover the pages in question also contain quotations from Locke’s Essay and L’Art de Penser. Isaac Barrow’s Usefulness zyxwvutsrq ofMathematical Learning and Nicolas Malezieu’s manual Ekmens de GPO& trie (on specific topics). All of this has been well documented.4 It is clear from The most striking example of this kind of critical approach is constituted by C. D. Broad’s essay, ‘Hume’s Doctrine of Space’, Proceedings zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPO ofrhe British Acudey,: 47 (1961). 161- 176, where one of the conclusions is that “Hume’s whole account of spatial dtvlsibility can be fairly zyxwvutsrqponm safely dismissed as rubbish” (p. 17l), as moreover, “Hume’s whole account of what is involved in having an idea of so-and-so is, and can be shown to be. rubbish” (p. 165). Indeed, Humc’s whole doctrine on space ought, according to Broad, to incline one to think that Hume - ” ‘sacred cow’ to many British philosophers . . he has never been one of my fetishes” (p. 176) - was merely a man of considerable ability rather than a truly great philosopher. C. W. Her&l, zyxwvutsrqponmlkjihgfed in Stuah in rhe Philosophy of David Hwne (Princeton University Press: Princeton, 1925). p. 167, places lroad amongst those who have taken up the suggestions made by Hume’s text on space and time. and maintains indeed that Broad’s arguments are the closest to those of Humc himself. ‘Hume’s letter to Ramsey is in R. H. Popkin, ‘So, Hume did read Berkeley’. Journal oj fhifosonhv 61(1964). 77475. The most comnlete study ofHume'ssources for the Trearise remains on thewhole &. K: Smith, The Phiiosophy ’of David- Hutne (Macmillan: London. 1964) (1st edn 1941). Chapter XIV, “Hume’s version of Hutcheson’s teaching that space and time are nonsensational”, is followed by five appendices in which. after having examined the discussion of infinite divisibility and vacuum (app. A). and the problem of “spatiaf location” connected to the immateriality of the soul (app. B), Smith considers the influence of Pierre Bayle (spp. C), and also quotes briefly the passages from Malezieu (app. D) and Barrow (app. E) referred to by Hume. 374 Studies in zyxwvutsrqponmlkjihgfedcbaZYXWVUTSR Hisror>nand Philosophy qf Science the above that any discussion of Hume’s idea of space must be read in the context of a system of references which we know must include major works of recognized philosophical importance, Scottish, English and Continental, of the seventeenth and eighteenth centuries. We must also include works (i.e. Barrow and Maletieu) of apparently lesser historical interest, but which nevertheless suggest a significant expansion of the philosophical context within which Hume worked. The latter two works mentioned above, each in its own way, represent different attitudes of mathematical enquiry. Furthermore, as they are quoted by Hume, they can reasonably be considered probable sources of his information regarding mathematics. Before looking at Hume’s writing directly then, it may be of some use to briefly consider these two texts and to try to reconstruct the picture which emerges from them. As we shall see, such an approach will help to prepare the ground by clearing away a basic ambiguity concerning the problematic context of Hume’s ideas about mathematics. This ambigirity has shown itself unusually persistent and misleading, and has led to the mathematical demonstrations which Hume mentions relative to the “theory of infinite divisibility”, being wholly and rather unquestioningly identifed with calculus, albeit in a very general way.5 Part I. The Elimination of an Ambiguity Barrow’s Lectures Barrow’s text is, in fact, quite familiar to historians of science and mathematics. It contains the first series of lectures given by the author as Lucasian frofessor of Mathematics in Cambridge in the three years from 1664 to 1666. Published in Latin in 1683, it appeared again in an English translation by John Kirkby in 1734.6 Barrow died in 1667, and these posthumous editions of his lectures are one indication of the persistence of his fame. This is also apparent from the way in which his name is commonly mentioned during this period. For instance, in his A View of Imnc Newton’s zyxwvutsrqponmlkjihgfedcbaZYXWVU Philosophy , of 1728, Henry Pemberton asserts that as a mathematician Barrow was second only to Newton 5cc. for example. the way in which the context of Hume’s discourse is treated by J. Laird, op. cit., pp. 64-65, where we read that ‘The theory of fluxions and of infinitesimals was as familiar to Hume’s age (and as little understood} as the Theory of Relativity is to ours”. See also R. Kuhns, op. cir., p. 79. el~aa~ Barrow, Lectiones Marhemaricae XXIII; In quibus Principia Marheseos generalia exponuntur (Habitat Cantabrigiac A. D. M.DC.LXIV, M.DC.LXV, M.DC.LXVI. Londini, 1683); reprinted in 1684 and in 1685, and republished in W. Whcwcll, The Marhemurical Works QJISII~C Barrow D.D. (Cambridge, 1860). The quotations here are from the facsimile edition (Frank Cass: London, 1970) of the U.fefulness of Mathemafical burning Explained and Demunsrrorcd Being Mathematical Lectures Read in the Public Schools at the University of Cambridge by lsaac Barrow, D.D. Professor of Mathematics, and Master of the Trinity College, etc. Transtatcd by the Revd Mr John Kirkby. of Egremont in Cumberland (London, 1734). Hwne’s 375 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Conception of space (who was until recently held to have been Barrow’s pupil and to have been strongly influenced by him).7 Barrow’s lectures typically reflect an intellectual personality in which mathematics, an interest in classical authors and theology combine to form a single field of interest. The lectures are more general and philosophical than technical, and present a view of mathematics which is by and large classical and conservative, set against an intellectual backcloth of a conception of space which belongs to the problematic Neoplatonism so characteristic of Cambridge at that time.* In the 1664 lectures, from T to VIII, he presents an introductory discussion on the nature of mathematics, its divisions, the kind of demonstrations it uses, and the principles upon which it is based. In the lectures of ‘65, from IX to XV, he deals with the properties of quantity, the nature of space, congruity (with respect to which he was later quoted by Hume) and measure. Finally, in the ‘66 lectures, from XVI to XXIII he discusses, primarily, the theory of proportions, dealing with the different “blemishes” identified by Euclid’s commentators. In relation to Hume’s text, what is especially interesting is the general conception presented in the lectures of 1664 and in some of those of 1665. According to Barrow, the object of mathematics is constituted by “magnitude”, or continued quantity: “this alone ought to be accounted the Object of Mathematics” [20], being the only given in Nature. It is, indeed, “the common Affection of all physical Things, it is interwoven in the Nature of Bodies, blended with all corporeal Accidents, and well nigh bears the principal Part in the Production of every natural Effect” [21]. The universal presence of quantity in nature is also reflected in the various sciences, all of which include the consideration of quantity, and this gives rise to the so-called mixed mathematics which, in fact, belong to the various natural sciences. This does not mean, however, that mathematics loses its object and thus its specificity. ‘On Barrow see P. H. Osmond, Isuoc Barrow. His life und Time (Society for Promoting Christian Knowledge: London, 1944); R. H. Kargon, Aromism in Englandfrom Hurior ro Newton (Clarendon Press: Oxford, 1966); id., ‘Newton, Barrow and the Hypothetical Physics’, Centaurus 11 (1965). 46-56; and, especially, D. T. Whiteside, S.V. in Dictionary ofScienrificBiography. Barrow has long teen remembered as Newton’s teacher (for instance. besides what one finds in Osmond and in Kargon, “Barrow, Lemaitre de Newton” is the title of an essay by H. G. Zeuthen published in Oversigt over det Kgl. Dunske Videnskabernes Selskabs Forhundlinger, 1897, pp. 565-606. This, at any rate, is the most common way in which Barrow’s name is mentioned). However, the relationship between Newton and Barrow has been entirely reappraised by D. T. Whiteside, ‘Isaac Newton: Birth of a Mathematician’, Niores and Records of the Royul Sociery, 19 (t964). 53-62, p. 61, n. 25. ‘for Barrow’s philosophical and Schule van Cambridge (Hcrdersche theological interests, see G. F. v. Hcrtling, 3ohn Locke und die Verlagshandlung: Frtiburg im Breisgau, l892), pp, 154-155, and the bibliography given there. Barrow’s name also appears in E. Cassircr, Die platonixhe RenaiFsonce in England und die Schule van Cambridge (Teubncr: Leibzig. 1932). p. 95. Some features of the Lecliones Mafhemnticue are analyscd in E. A. Burtt, The Meruphysicul Foundutionr oJModern Science (Routledge & Kegan Paul: London. 1980) (1st edn. 1924). pp. 150-161. 376 Studies in History nnd Philosoph.r of Science What characterises it is that it considers quantity as its subject universally and generally, working to this end according to that abstractive procedure which is, after all, proper to every science as such [2&22;13]. Mathematics then is in some way linked with experience, and consists in the universal consideration of continued quantity. Barrow excludes algebra, which in his opinion, given its instrumental nature, belongs rather to logic, and, in fact, is not strictly speaking a science at all [28, 441. The way to overcome the distinction between geometry and arithmetic, whose subject is discrete quantity or “multitude”, or number [10-l I], is to reduce the second to the first. After asserting that Number really differs nothing from what is called Continued.Quantity, but is only formed to express and declare it; and consequently that Arithmetic and Geometry are not conversant about different Matters, but do both equally demonstrate Properties common to one and the same Subject [30] Barrow iilustrates his own position by giving the most noteworthy example of correspondence between geometrical and arithmetical propositions to be found in the theory of proportions [3Off.]. Number, Barrow remarks, “has no Existence proper to itself, and really distinct from the Magnitude it denominates, but is only a kind of Nore or sign of Magnitude considered after a certain Manner” [41]. On the other hand, for the benefit of any follower of Aristotle who might consider Arithmetic more precise than Geometry because based on a notion of unity, simpler than the geometrical notion of point, Barrow denies the supposed correspondence between unity and point in the following way: Unity answers really to some Part of every Magnitude, but not to a Point: Thus if a Line be divided into six equal Parts, as the whole Line answers to the Number six, so every sixth Part answers to Unity, but not to a Point which is no Part of this Right Line. A Poinf [Barrow continues] is rightly termed Indivisible, not Unity. (For how ex. gr. can i+$ equal Unity. if Uniry be indivisible, and incomposed, and represent a Point) but rather only Unity is properly divisible. , , [48]. A point then is not a part of a line. It is unextended and corresponds to an Nothing” rather than to unity. In conclusion, “the Accuracy of Arithmetic and Geometry . . . is altogether the same, drawn from the same Principles, and employed about the same Things” [49]. Defined in this way, universal geometry is the most certain and the most rigorous of the sciences, No other “can afford Principles more evident, more certain, yea I will add, more simple, than Geometrical Axioms, or exercises a more strictly accurate Logic in drawing its Conclusions” [47-48]. Its certainty comes from being based upon self-evident principles, its rigour from the logic that underlies its demonstrations. As for the principles, they present themselves to the intellect in the same way in which particular events appear to the senses. “Arithmetical Hume’s Conception of Space 377 The mind of man “is by its native Faculty able to discern universcll Propositions, in the same manner, as the Sense does particulur ones”. These are the propositions which the M ind directly contemplates, and finds to be true by its native Force, without any previous Notion or applied Reasoning: Which Method of attaining Truth is by a peculiar Name stiled Intellection and the Faculty of attaining it the inreifecr. he continues, “what hinders but the Principles of Mathematical Demonstration may be perceived by such a Faculty?” [72-731. On the other hand, in the mathematical propositions, precisely as in the syllogisms. we find an essential connection, a rigid causality linking premises and conclusions. These can be directly inferred from the meaning of the terms of the first principles from which they derive. Definitions express the nature of things by means of one of the properties that determine them and are the formal causes of the conclusions, with respect to which “do all proceed from the Form and inward Constitution of the Thing expressed by the Definition*’ [90]. Furthermore, if the fundamental properties of a geometrical entity are linked by formal causality, they are interchangeable in the sense that each one can provide a basis for definition. Among them then Now, we usually seize upon that which is most obvious and ready. It is here as in the Progress of a Circle, from whatsoever Point of the Circumference- you take your Beginning, you will measure over the Whole. Such in Reality and no other is the mutual Causality and Dependence of the Terms of a M athematical Demonstmfion, viz. a most close and intimate Connection of them one with another [KS]. For instance, parallel straight lines can be properly defined according to the angles they form with an intersecting line and this definition thus becomes the formal cause of all their other properties, which follow on of necessity, “since they are connected together with such an essential, close and reciprocal Tie, that if any one be supposed, the rest must necessarily follow” 1861. Barrow, incidentally, does not fail to take a stand on this particular aspect of the thorny question of the theory of parallel straight lines. He shares the opinion of many interpreters who maintain Euclid “was not so right” to consider the negative property of not meeting even for an infinite distance as the easiest and clearest of the properties of parallel straight lines, and thus the most suitable for the definition [871. This does not mean, however, that the truth of a branch of geometry is brought into question. The adequacy of a definition might be questionable but in cases of this kind the question is not “about the Truth of Principles, but about the Order of some Propositions” [238]. In this respect there is more than one flaw in Euclid’s text, as we see again concerning the postulate of parallel lines and the IV postulate (which requires all right angles to be equal to one another): “. . this is unaccurateiy done by Euclid.. , For they are really Theorems consequent to the Definitions of a Right Angle and a 378 Studies in History and Philosophy of Science Parallel Line” [132]. However, both the identities of defined objects and their properties are unquestionable because we do clearly conceive, and readily obtain distinct Ideas of the Things which these Sciences contemplate; they being Things the most simple and common. such as lie exposed to Senses, capable to be represented by the most familiar Examples, and therefore most easy to be understood, as containing in them nothing abstruse. intricate or unusual 1531. Such is the case, for instance, with the straight line [93]. The strength of the foundations rests on the clarity with which entities are conceived. On the other hand, it would be senseless to object that in reahty entities defined by geometry are not, as such, given in Nature. They “agree with several particular Subjects occurring to the Sense” [19], that supply the occasion from which reason draws and understands them. By Sensation indeed . , . may be deduced the Possibility of Mathematical Hypotheses: Thus ex. gr. we know that a Right Line can be drawn between two assigned Points; because we perceive by the Sense, how a Progress may be made from one Point to another, wherein if there be any Unevenness or Deflection, it can be so far rectified by the K-land as to make a Line sensibly Right; from whence we infer by our Reason, there being no Repugnance on the Part of the Thing, that all other Roughness and Exorbitances may be pared off and corrected, and so the Line become perfectly Right [75]. The ideal of a rational standard is thus suggested by the manner in which sense works. From this point of view, indeed, the latter reveals geometrical figures only, even though they are mostly irregular. Barrow, in fact, maintains that “all imaginable Geometrical Figures are really inherent in every Particle of Matter”. His Neo-platonic version of the Cartesian geometer God might act like MichelangeIo, who asserted that he sculpted his figures by uncovering them from the material which concealed them: if the Hand of an Angel (at least the Power of God) should think fit to polish any Particle of Matter without Vacuity, a Spherical Superfice would appear to the Eyes of a Figure exactly round; not as created anew, but as unveiled and laid open from the Disguises and Covers of its circumjacent Matter [7677]. In short then, geometry is a privileged knowledge, and permits us to touch the inner secret of matter which only the angel or God Himself can reveal to the senses. Furthermore, it requires the omnipotence of God as a guarantee of its ontological basis, and so amounts to an argument for the existence of God. As Descartes had already asserted, every demonstration from this point of view “does in some sort suppose the Existence of God” [109]. On the other hand, it is precisely because of the divine omnipotence that it is also possible to reason about objects which do not actually exist. The intelligible world is vaster than the world of the senses, Hume’s Conceprion zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDC of Space 379 the Dominions of Reason do far exceed the Limits of Nature; the intelligible World is vastly farther extended and more diffusive than the zyxwvutsrqponmlkjihgfedcbaZY sensibleWorld, and the Understanding contemplates many more Things than the Sense fl 1I-1 121. This is also true, for instance, of the astronomers because “God has given us the Power of creating innumerable imaginary Worlds in out Thoughts, which himself, if he please, can cause to be real” [I I I]. On the grounds of this privileged knowledge afforded by geometry, Barrow forms his opinion on the problem of atomism or the infinite divisibility of quantity. If one holds that quantity is composed of indivisible particles, a huge and deplorable Ruin, Confusion, and Inconsistency is brought upon this most Divine Science; whose Principles notwithstanding, besides their Evidence and the Rigour of its Reasonings, are so firmly established, as well by their wonderful Consonance one with another which could not happen if they were false, as by their perpetual and exquisite Agreement with Experience [I%]. For instance, if lines were made up of points, a point would be the unit of measure for all magnitudes, which would obviously invalidate the theory of incommensurables. A point, however, as we have already seen, is not part of a line and, in general, the notions of point, line and surface are not derived by subdividing entities of a superior order. Experience presents us with bodies, and tells us they are bound by surfaces whose boundaries are constituted by lines which, in their turn, terminate in points, as in Euclid’s definitions [144]. However, the question is nevertheiess particularly complex and not so easily resolved. The properties of magnitude are in general elusive; the opposing and equally unsatisfactory positions of philosophers such as Descartes and Hobbes would suggest that it is perhaps better to consider such properties “after a meaner and poorer Manner, more accommodate to common Sense than to metaphysical Conceptions” [141]. But in this case in particular, as we find again with incommensurables, we are concerned with a “wonderful Property which almost exceeds human Capacity, and so puzzles such as are unaccustomed with these Things”. Barrow is thinking here of the methods of indivisibles and quotes Cavalieri [285]. It is worth remembering, in this respect, that it is characteristic of these lectures that the statements we have just considered co-exist with conceptions which are typical of the theory of indivisibles. According to this theory, for instance, a surface is equal to the sum of all its lines, or a line to all of its points. Furthermore, it is well-known that Barrow had direct knowledge of Cavalieri’s methods, and there are various examples which illustrate this. Besides some rather elementary instances [e.g. p. 311, Barrow also deals with Torricelli’s acute hyperbolic solid, in which “the infinite Diminution of one Dimension recompenses the infinite Augmentation 380 Studies in History and Philosophy qf Science of the other” [297].9 One example which is of interest here is the way Barrow treats the idea of congruity, the topic in relation to which he is quoted by Hume, as I have already mentioned. Congruity is “the chief Pillar and principal Bulwark of all the Mathematics” [185], and can be identified in five different ways. The first two are perfect congruity and possible congruity or, respectively, that which concerns entities of the same kind, straight lines with straight lines or plane surfaces with plane surfaces, and that which connects, for instance, straight lines and the perimeters of rectilinear polygons. Thirdly, one finds congruity when “all the Indivisible of each Magnitude may succeed in the same Place, and neither of the two vary the Position of their Parts” [193]. This is the case when we rotate a circle over a straight line. in which “all the Points of the circular Periphery are continually applied in a successive Order to all the Points of the Right Line”. Or when we rotate a cylinder over a plane surface, where “all the parallel Right Lines, which lie disposed one after another in the Superftce of the Cylinder, are applied, by a continued uninterrupted Series, to all the parallel Right Lines in the plane Superfice” 1194) Fourthly, there is congruity when all the lndivisibles of both Magnitudes do occupy the same Place in the same Time, the Situation of the Parts of one being in some measure unvaried, but yet their Order preserved, and the former Contiguity of the particular Parts retained [194]. Here Barrow talks amongst other things of one of the methods of indivisibles, maintaining that to this Mode also belongs, or is very near of kin to it, that Corlgruity ol which equal Figures included within the same parallel Lines or Planes are capable; of which all the Lines or Planes intercepted between the same Parallels are equal [ 1961. Barrow has already explained earlier, in mentioning the first propositions of Book II of Geomelria Indivisibilibus Confinuorum and the first of Cavalieri’s zyxwvutsrqpon Exercitafiunes, that the basis of the second method of indivisibles is, indeed, congruity [184-l 871. Barrow refers back to Cavalieri’s work for less generic explanations, and we find, finally, another reference to the latter’s second method expressed in the following terms: Way is, when the Cortgruity may be so performed, that the Position and Order of certain Parts is changed. This Way of Congruity [Barrow continues] of all the most imperfect and difficult to be understood, agrees with homogeneous Figures altogether unlike to one another. Ex. gr. A Triangle is no otherwise congruent with a Circle, a Cone with a Sphere, than by transposing the Parts of either. by applying a Part of one to a Part of the other, and a Part of the Residue in one to a part of the Thefifrh “Barrow mamtains that the methods of indivisibles are an apparent exception to the principle of homogeneity [302-3031. in his opinion, the acute hyperbolic solid is an example of the way in which the wisdom of the modern geometers has shattered the truth of the ancient axiom according to which there can be no proportion between the finite and infinity. The argument continues, mentioning the problem of the angle of contingence between the circumference and the tangent. Hwne ‘s Conception of Space 381 Remnant in the other; and so perpetually, till the Business be finished, and all the Parts of the one be at length applied all the Parts of the other [196-1971. It is plain that Hume would have found this argument interesting. Let us return though to the way Barrow presents the question of the composition of magnitude. The problem obviously has to do with the intervention of the “Conception of Infinity, which is not perfectly comprehensible to us” [152]. Yet, in conclusion he says, I deny not but it is difficult to be understood, how every single Part can be divided so as all not to be actually reduced by the Division to Indivisibles, or to Nothing or what is next to Nothing: Nor yet do I think, by reason of the Imperfection of the Mind of Man and the Smallness of our Capacities, that therefore the Truth is to bc deserted, when proved by so many evident Tokens, and supported by so many strong Arguments [162]. On the one hand, this final assertion of Barrow’s might in retrospect seem rather wise - more so than some of his ideas concerning even quite significant questions. It is clear, however, at this point that once again it is only by integrating his notions about mathematics into the framework of strong metaphysical and theological convictions that Barrow can speak of geometry as a very simple and evident science, both perfectly coherent and well-founded. Tt can be such, in his eyes, because it is a divine science. This last point appears all the more obvious when we note the central role played by theological and apologetic themes in the conception of space, taken into consideration because of its relation to magnitude. The problem concerns the independent existence of space, and Barrow argues, with Scripture and Aristotelian texts at his fingertips, both against Descartes’ identity of space and matter (he mentions [140; 175-1761 the well-known paradox of the immediate contact of the walls of an evacuated container given by Descartes in the Principia, II, 18; Barrow talks of the walls of a room), and against the theory of vacuum of Epicurean origin [e.g. 1791. First of all, “Space is a thing really distinct from Magnitude” [175]. In favour of this opinion Barrow points out the impiety of the theory of infinite matter, and the possibility which God has of creating or destroying any portion of matter [ 166, 1701. To distinguish space from magnitude signifies Clearly the limits of human reason cannot limit the truth. that something is designed by that Name, that a Conception answers it, that it is founded in the Nature of Things, that it is different from the Conception of Magnitude, and though Magnitude had no Existence at all, yet there would be space [I 751. Secondly, however, this does not mean that space actually exists: Space is not any thing actually existent, and actually diRerent from Quantity, much less it has any Dimensions proper to itself, and actually separate from the Dimensions of Magnitude [ 175-1761. 382 Srudies in History and Philosophy zyxwvutsrqponmlkjih qf Science Barrow has already reminded us that even to suppose the real existence of space independent of God would be impious. In saying this he has introduced the two sides of the dilemma. Indeed, space as it appears to common sense and as required by mathematics is “nothing else but the mew Power, Cupaci~,v. Ponibility, or . , . Interponibility oJM agnitu&” [ 1761. Its relation to Magnitude is that of power to act. It has the same kind of existence as its opposite, contiguity, in the sense that it is “the Mode of Magnitudes . , . which intimates that some other Magnitude may be interposed without moving them out of their Place” [178]. In other words, it serves to describe the way magnitude behaves in certain circumstances, and does not point to a being. The strength of Barrow’s position, which as we shal1 see contains parallels with certain passages of Hume’s, is guaranteed on theological grounds. Space as power is identified with the ubiquity of God - “which only signifies that God is present to all Space” [178] - and with “his unlimited Power of producing and disposing Bodies at his Pleasure” [178]. God can create new worlds and be present while remaining motionless, as his perfection requires, and this “can no otherwise be understood, than by conceiving him to have before been present in the Space, where they are now reposited” [l70-1711. Malezieu’s Manual The second text we shall consider has a somewhat unusual history. We will turn our attention to Nicolas de Malezieu, an intellectual who had been a pupil of Rohault. An accomplished courtier, after being principal tutor to the Duke of Maine, he became a loyal member of his circle. Our interest lies in the notes based upon Malezieu’s teaching made by another equally illustrious pupil, the Duke of Burgundy, entrusted to his care to study mathematics by Madame de Maintenon when he was I4 in 1696. The results of this work, which took four years, were kept by Maletieu and printed in 1705 by Boissitre, the iibrarian to the Duke of Maine. The text must have met with some success as a manual. Translated into Latin in Padua in 1713, it was reprinted in Paris in 1722, I729 and 1735. Even the Paduan edition was reprinted in 1734.j’ The work was still sufficiently well-known in 1763 to feature in Georg S. Kliigel’s dissertation concerning the attempts to prove Euclid’s Vth postulate.” “*The best source of information on Malezicu’s biography is to be found in Fontenelle’s obituary. The quotations here are from the 3rd edition. Hemens de Geonrerrie de Monseigneur le Due de Bourgopw. Troisicme edition, Revue, corrigee & augmentee d’un Trait6 des Logarithmes, Par M.De Malezieu. Avec I’lntroduction a I’Application de I’algebre a la geometrie (Paw. M.DCC.XXIX). The title of the Latin translation is SwenkFsimi Bwgundine Ducis Ekmen~tr Gaonwtrico Ex gallico Sermone in Latinum Translata ad usum Seminarii Patavini. Accessere Quatuor Propositiones ad Trigonometriam apprime utiles. Insuper Introductio ad Algebrae applicattonem ad Geometriam. Auctore Guisneo. Nunc primum Mine reddita (Patavii. M.DCC.XIII). “G. S. Kliigel, Conaruum praecipuorum rheoriam paraifelarum demon.ctrandi recensio (Goettigae, MDCCLXIII), part. pp. XI-XII. Mime’s CancepprionofzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDC Space 383 The character of this book is, of course, rather different from Barrow’s lectures in terms of its subject matter. It is, indeed, a’ proper manual of geometry, even if there are also some interesting considerations of a broadly philosophical nature. The themes explored in these considerations have to do with mathematical problems of infinity and infinitesimals, dealing with the nature of infinity, incommensurables (this is the part quoted by Hume) and the usefulness of indivisibles. The greater part of the text is. however, technical, although on the whole elementary. The real interest lies in seeing what kind of information this text supplied to its readers. bearing in mind that, of its kind. it was still probably considered to be not out-of-date during the 1730s and that it can therefore be considered typical. This is especially significant with regard to one reader in particular: it may provide a more concrete idea of Hume’s mathematical knowledge, if, as we are entitled to believe, he had read, or at least glanced at, Malezieu’s manual. At any rate, we have no indication as clear as this that Hume had any familiarity with less elementary or more recent, upto-date mathematical texts. The reference work chosen by Malezieu for his lessons to the Duke was the ElPmens attributed to Antoine Arnauld. This work, published in Paris in 1667, sets out explicitiy to reformulate geometry along simpler and clearer lines because, in the opinion of the author, the Elemenls of Euclid are “so confused and muddled, that far from giving to the mind the idea and the taste of the true order, they manage on the contrary only to accustom the mind to disorder and confusion”. I2 This attitude probably also reflects a certain uneasiness on the part of the author of the Arr de Penser about the “blemishes” which, rightly or wrongly, interpreters had found in Euclid’s books. It is an attitude also expressed in Malezieu’s preface, where the choice of Arnauld’s Elimens is explained by saying that it is “very much richer than the Elements of Euclid, more easily understood, and incomparably easier to remember”. There is a close thematic correspondence between the texts of Arnauld and Malezieu, the latter being essentially a simplified synthesis of the former. The expositions are cut down, abbreviated and rendered more intuitive. Furthermore, various propositions are omitted. The most apparent omission is certainly the absence at the beginning, apart from one very brief summary, of Arnauld’s first four books. These were dedicated to the formulation of the theory of proportions by means of algebra “in species”, that is, using letters; moreover, they were presented as being rather difficult and not strictly necessary to an understanding of the rest of the book. In other words, they were presumably inappro‘~Nour~eau.rekmens rie geomerrie, contenant. outre un ordre tout nouveau, & de nouvelles demonstrations des propositions les plus communes. De nouveaux moyens de faire voir quelles lignes sont hcommensurables, De nouvelles mesures des angles, dont on ne s’estoit point encore avisk. Et de nouveks manieres de trouver & de demontrer la proportion des lignes (A Paris, M.DC.LXVIl). Studies in Histq 384 and Philosophy of Science priate for a lCyear-old beginner. ” On the other hand the most conspicuous addition we find to Amauld’s text in the Duke’s notes is constituted by Book X, on solids, in which the calculation of volumes is set out with the method of indivisibles in analogy with the calculation of areas already also introduced by Arnauld with five simple propositions. In the opening pages we find definitions and axioms, following Euclid’s method of organization. The way in which these first principles are introduced is indicative of the author’s constant concern with the problems linked to infinitesimals. After stating that the subject of mathematics is quantity in general, that is to say extension, number and movement, and that the particular subject of geometry is extension, with its three dimensions, he introduces definitions of line - “lerrggthconsidered without hreadrh and without depth” of surface, and of solid. The straight line is defined in the zyxwvutsrqponmlkjihgfedcba usua l Archimedean fashion as “the shortest measure between two points” and the point as the extremity of a line, and considered as being without length, breadth or depth. In effect [expfains the author] it can have no breadth since the line itself has none; and it can have no length since it would itself become a line, and would no longer be merely the extremity of a line [il. There then follow definitions of various other geometrical entities, like the perpendicular to a given straight line (“a straight line is said to be perpendicular with regard to another straight line, when two of the points of the first line are placed directly above the same point of the line to which it is said to be perpendicular” [ii]); curved and mixed lines; plane, curved and mixed surfaces; circumference; diameter; radius; chord; arc; degrees and minutes, etc. Among the axioms we find that the whole is greater than the part and that it is equal to the sum of its parts. We also find the transitivity of equality and some properties of the four operations. There follows a presentation of “arithmetic by letter, which is usuaIly called in Species”, a few pages (seven in total in the 1729 edition), which does not go beyond the basic rules for the four operations with letters. Book I presents certain propositions concerning straight lines perpendicular or oblique to a straight line. Proposition IV is of some interest here. Deriving its proof from the definition of the perpendicular it states that the perpendicular to a straight line from a given point is unique and the shortest way between the line and the point. As a corollary it follows that two lines perpendicular to a straight line cannot meet even at an infinite distance. Otherwise “it would be true to say that two Iines perpendicular to the same line would begin from this meeting point” [4-51. In Book II this idea is developed into a reworking of the theory of parallel lines, based on the idea that if two straight lines have one “Only in the editions of 1722 and 1729 does Malezieu include, respectively, some geometrical problems solved by using algebra in species and, together with a brief treatise on logarithms, an “Introduction zi I’application de I’Algkbre d la zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLK Gbomttrie”. Hume’s Conception of Space 385 common perpendicular then every straight line which is perpendicular to one wilI also be perpendicular to the other, and that the segments of the perpendiculars will all be equal. From this one can derive the construction of the parallel to a given straight line on the basis of their equidistance - in which parallelism consists, according to a conception well known since antiquity and variously reformulated in modern times by Arnauld among others. Book III deals with internal and external secants, and tangents and chords of a circle, while the central theme of infinitesimals begins to feature more prominently. The tangent, defined like the perpendicular at the extremity of a radius, touches the circle at a single point (prop. XI), such that “it is impossible to make a single straight line pass between the tangent and the circle, although one can do so with an infinity of circular lines which will only meet at a single point of contingence” (prop. XII [26]), as already described by ITI, of Euclid’s Elements. As a commentary on this proposition we find the first of the philosophical digressions mentioned previously. There are echoes here of ancient reflections regarding this topic and traces too of the most recent and heated debate on the angle of contingence, in which Clavius and Peletier among others had intervened, as well as more general allusions to even more recent thinking. Proposition XII is most suited to humbling the human mind by convincing it that there are truths which are quite plain when examined each in turn, while it remains, however, impossible to conceive of any connection between them. And these truths are of such a nature that the one seems to destroy the other. It is, in fact, a question of “something infinitely small divided into an infinity of others. This can be demonstrated; but can it be clearly conceived?’ The example chosen to clarify the terms of the question is typical: To aid the imagination, imagine a perfect sphere resting on a surface. This sphere rests on a single point which has no extension. Otherwise the tangent and the circle would have more than one point in common. Now imagine another sphere much larger than the first, resting on the same surface. This large sphere rests like the first on a single point, and yet we can be quite certain that the curvature of the larger sphere is less than the curvature of the smaller; and consequently that from the point of contingence the circumference of the large sphere is at a lesser distance from the tangent, than the circumference of the small sphere, although it has been demonstrated that the space between the small sphere and its tangent is so small, that a quantity of infinitely small breadth such as a straight line would not pass between them. That is to say, [he concludes] that this space is infinitely small, and that it nevertheless encloses an infinite number of other spaces. This illustration of the question makes use of a classical example but it is also valid in other contexts, In fact, Everything which can be proved in the highest speculation of Geometry on asymptotes, asymptotic spaces and the infinitely small of Messieurs de Leibniz and de Studies in HistorJI and Philosophy qf zyxwvutsrqponmlkjihgfe Science 386 Wbpital whose principles are zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONML SO fruitful; in a word everything which can be proved about infinity, is of the same kind. He continues: The human mind is convinced of certain truths; but it is forced to admit its weakness when it wishes to understand, so to speak, how: in other words, how is it possible that these truths can co-exist? But as the human mind is limited, and as the Creator of our souls has not given them infinite light, it is up to us to remember our condition, Nothing could be more unreasonable than to want to deny the truths about which we are otherwise convinced, because we cannot understand the connection between them. We understand these truths because we possess a certain degree of reason; we do not understand the connection because we are not God and because our reason is not infinite. It is therefore quite wrong to wish to attack the Geometry of the Infinitely Small, and that of Indivisibles because there are certain things in the nature of infinity which cannot be understood. and which must indeed be incomprehensible. But it is one thing to .understand infinity and another to convince oneself that it exists. I sincerely confess that I am fully convinced of the truth of the twelfth proposition; but I admit that I do not understand it. For if I consider myself obliged to recognize incompatible truths in Geometry, where the human mind prides itself on being able to see more clearly than elsewhere, all the more should I submit to the truths of an order beyond my reason, and remind myself always that He who created my reason was not obliged to make it capable of everything [30-3 11. The paradoxes of the infinitely small give rise directly to the appeal to religious faith which we find in the conclusion. And the argument runs along lines which are common both to Malebranche and the authors of the Art de Penser. From the mathematical viewpoint we find, alongside the classical themes, mention of Leibniz’s and de l’H8pital’s infinitesimals, and of the indivisibles, which all seem, moreover, to be conffated. The themes discussed here recur, in more or less anatogous terms. at the end of Book IX, in the “RCflexion sur les Incommensurables” which follows the proof of the incommensurability of the diagonal and the side of the square. Here the author explains, picking up once again the thread of a fairly old argument, that this aiso proves the impossibility of indivisibles, and that to attempt to avoid the problem that then arises by saying “that there are no perfect squares and consequently neither sides nor diagonals, is to reason pitiably,” given that “it is not necessary that there are either squares, triangles or circles in the world in order to establish the truth of geometrical Proofs, the possibility of their existence is enough”, And this is guaranteed by God Himself since, “source of all truth, he would have known at least that a possibIe triangle was half points are thus impossible, of a possible parallelogram” [148]. If, however, the whole of geometry collapses. For example, returning again to the case of the tangent, from what we have already said it also follows that Hume’s Conception of Space 387 the circle is impossible. For if God makes a perfect sphere and pIaces it on a perfect surface, will the point of contingence have any extension? If it has, it is a surface or at least a line, so that the tangent and circle would have part of an extension in common. And here we arrive at the passage quoted by Hume: Moreover, when I consider carefully the existence of beings, I understand very clearly that existence belongs to units, and not to numbers. Let me explain. Twenty men only exist because each man exists. The number is merely an exterior name, or rather a repetition of the units to which alone existence can belong. There would be no numbers if not for units. There could not be twenty men. if there was not one man 11491. This typically nominalist argument, which incidentally could not in these terms fail to appeal to Hume,14 is then translated into the metaphysics of substances, with the conclusion that “matter is therefore composed of indivisible zyxwvutsrqponm sub stances. Thus our reason is reduced to strange extremities” [149-1501, from which - the author asserts again - a valuable lesson of humility may be learned. However, even if it is necessary to recognize the shortcomings of human reason, Malezieu also gives a way of using some of the partial truths which it is possible to attain. The first illustration is elementary indeed. Tn Book VTII, in the section given to measuring the area of rectilinear figures, Malezieu explains that the area of the rectangle is equal to the base multiplied by the height saying, if the line CD runs parallel to itself the length of the line CA, until it is coincident with the line AB; it will describe, or reveal if you like, the surface of the rectangle. Therefore the base CD is contained as many times in the height CA as there are points in this line CA (prop. XI [112]). In the following it is shown that “every parallelogram is equal to the Rectangle which has the same base and height” [113], and it is noted that, besides using classical methods, this proposition can also be proved in a way similar to the one before: proposition in both figures there is an equal sum of equal lines, because the perpendicular LH determines this sum, as far as it is possible, as perfectly equal. Thus the parallelogram is equal to the Rectangle . . . This method of proof is called the Geometry of lndivisibles. Its fruitfulness is admirable . . . [ 115). ‘7his is contrary to what we read in A. Flew, In_nile Divisibili~_rin zyxwvutsrqponmlkjihgfedcbaZYXW Hume3 Trtwise. in: D. W. Livingston, J. T. King (eds), Hume: R Re-evaluarion (Fordham University Press: New York, 1976), pp. 257-269: “This piece of o priori ontologizing is so incongruous with both the general temper and the stated object of this attempt to introduce the experimental method of reasoning into moral subjects. and so reminiscent of the sort of positive natural theology for which Hume had least respect. that he surely ought to have asked himself whether its aptness here is not an indication that something is going badly wrong; which it is” (p. 244). Studies in History and Philosophy qf Science 388 The same method is employed again for illustrative means in the second corollary of proposition XIII, according to which triangles contained by the same parallel lines and having the same base have equal areas. It is also in the corollary of proposition XXIII, in which the areas of a circle and of a rectangle are shown to be equal, where the height of the rectangle is equal to the radius and the base equal to half the circumference. Book X is, as I mentioned at the beginning, en&y innovative in comparison with Amauld’s text. Here the geometry of indivisible5 is introduced and employed in a more systematic way, although still on a rather elementary level. The reference model is probably constituted by the Geomerria Pracrico of C. F. Milliet de Challes, afterwards quoted by Malezieu in connection with the calculation of centres by gravity of Guldin’s method.15 Matezieu begins by explaining again and more broadly that the method consists in considering the surfaces as composed of parallel lines, such that a parallelogram is nothing other than a base running parallel to itself the length of the points of its perpendicular. It foliows from this that the base of a rectangle, square or parallelogram is contained as many times within its area as there are points in the perpendicular, and that to find this area one has only to multiply the base by the perpendicular. The same ratio (“analogie”) between the area and the sum of the lines applies also to the volume and the sum of the planes in solids. Thus, A Prism is nothing but an infinity of regular figures placed one on top of the other parallel to one another, or if you like, one can consider one of these figures running the length of the perpendicular of the Prism. Its volume is simply its base multiplied as many times as there are points in its perpendicular [I 531. Then, after having simply listed the most obvious results, he gives the simplest and commonest use for the indivisibles, in 12 propositions which consider the volume of pyramids, prisms and cones, the volume and surface of hemispheres, spheres and portions of spheres and the ratios between the volumes of similar solids. He then remarks in a note: We have just seen the usefulness of the Geometry of lndivisibles in the study of Solids. Those who are curious to further their speculation will not regret considering the following Propositions which open the way for an endless discovery of the most sublime truths of Geometry. To clearly understand what follows [he insists] it must be remembered that we consider surfaces as made up of parallel lines, and that we consider solids as composed of surfaces [ 174). Essentially the argument consists of two general propositions, panied by some illustrative corollaries. According to the first, If one has two figures, two solids, in a word, two homogenous each accommagnitudes to “The G~omerria Prac/ica (1st edn 1660) is included in volume II of Cur.rus. .scu mundu.r marhematicus (Lugduni. 1674). 389 Hurne’s zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Conceplion of Space compare with one another, and if these two figures, or solids, have the same height. the elements (Pkkmens) of one will not decrease while the elements of the other will always decrease according to height; the Figure or Solid whose elements do not decrease will be double the Figure or Solid whose elements decrease according to height [ 1761. (here a right-angle triangle) whose bases and heights are equal. Two elements of the triangle placed at an equal distance respectively from the base and the apex, taken together are equal to one element of the rectangle. And given that the heights are equal, they contain the same number of parallel lines (“there are as many parallel lines”), so that “all the lines of the triangle taken together must be equivalent to a half of all the lines of the Rectangle. And as there is no difference between all these lines taken togeiher and the surfaces, it follows that the surface of the Rectangle is double that of the other” 11781. Alternatively, according to the property of arithmetical progressions according to which the sums of two terms equidistant from the middle are always equal, two elements of the triangle equidistant from the mid-point of the height taken together are always equal. This is valid up to the extreme ends of the figure, in other zyxwvutsrqponmlkjihg wo rds up to the base and opposite apex - “point B, or rather zero” - taken together. They are therefore equal to the base itseif, or in other words to an element of the rectangle. “This proof’, the author concludes, “is general” [179], and to illustrate this, he proves in the first corollary that “the surface of a cylinder whose height is equal to the radius of the circle which forms its base is twice that of the circle” [179]. This is because the surface is made up of as many equal circumferences taken together as there are points in the radius, while contained in the surface of the circle there is the same number of concentric circumferences (“the same number of elements”) decreasing according to the radius itself. In the second corollary he shows that the volume of a rotated parabola is equal to half the volume of the cylinder which has the same base and height. The parabola is the curve defined by the fact that the squared segments of the perpendiculars which run from the curve to the axis are in the same ratio as the segments which they intersect on the axis itself. The same therefore applies to the circles of which they are the radii, in other wo rds the elements of the paraboIa, from which the result follows. According to the second proposition, The proof given is for rectangles and triangles if one has two Figures, two Solids, in a word, two homogenous magnitudes to compare, and if these two Figures or Solids have the same height, while the elements of one will not decrease, and the elements of the other will always decrease in duplicate ratio to the height, the Figure or Solid whose elements do not decrease will be thrice the Figure whose elements do decrease [1851. Since the geometric (or&&e) proof, according to the author, is rather confused, he gives another “through Arithmetic, which is more within every- 390 Studies in History and Philosoph?, qf Science one’s grasp” [185]. The height of two figures are divided corresponding to the description into a certain number of equal parts, which cut the elements in one figure so that they are all equal, and in the other so that they decrease geometrically. The ratio between the respective sums of elements slightly exceeds that required and approximates to it by increasing the number of divisions: so that by continually increasing the division of the height, I will reduce this difference to a quantity less than any given. from which fobws the perfect equality between the decreasing figure and the third of the total. supposing the number of these elements to be indefinite, as it in fact is [187]. From this, as a series of corollaries, it follows that: the volume of the cone is a third that of the cylinder of equal base and height; the volume of the conoid generated by rotating a mixed line triangle where the segments of parallels to the base decrease in duplicate ratio to the height, is equal to a fifth of the volume of the cyiinder; the volume of the hemisphere is equal to two thirds of the cylinder, and the “scodella” which constitutes the remainder is equal to a third, or, in other words, to the volume of the cone of equal base and height. The book concludes by giving an outline of Guldin’s method for calculating centres of gravity. It is a method of immense fruitfulness, not only for measuring all ordinary surfaces and zyxwvutsrqponmlkjihg so lids but for measuring an inIinite numbar of them where other methods are often lacking [191]. During the years when Hume was writing his Treatise, a large part of the debate amongst mathematicians was focused upon calculus. We have only to recall that Berkeley’s Analyst appeared in 1734 and Maclaurin’s Treatise on Fkcions in 1742. It goes without saying that the texts so far considered were, if only for reasons of chronology, outside the field of contemporary mathematical research. Despite this, however, the more or less latent problems concerning infinity, which are at least partly traditional, tend to reappear in the general thoughts of mathematicians (for example in the pages of Maclaurin or D’Alembert) in terms which are not altogether dissimilar, At any rate, the works of Barrow and Malezieu were still influential and widely read. Despite the differences between them it is possible to say that the instruction which they both afforded was presumably that of an average university course. Whatever else it was in the 173Os, the knowledge of mathematics was, to a certain extent, also reflected by these and similar texts. This consideration, as well as the fact that Hume apparently had these works in mind when he wrote about mathematics and infinite divisibility, would suggest that his work is better viewed in the context of this kind of background rather than in the light of more innovative work - in other words, in the context of the general level of information required by an intellectual who was notoriously omnivorous but not particularly interested in mathematics, let alone dedicated to it, even on an amateur level. From this viewpoint Hume’s “errors” might appear less 391 Hume’s zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Conception of Space conspicuous, teaching of judgements always clear for they can be seen as responses to crucial issues in the standard the period, issues on which, it is worth remembering, even the of the most dedicated and brilliant mathematicians were not or certain. Part II. The zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJI Pro b le m of Space in Hume’s Twatise The Pro b le m of Geometry: The Notion of Equality The tone and the very purpose of Hume’s writing on space are all to be found within the “science of human nature”, to use his own words. More specifically this means that the text can be seen as part of an attempt to reconstruct the origins of knowledge in perception and, in this way, to determine the limits within which it can be justified. This approach may explain the way he deais with geometry, which is one of the main and most complex subjects in this part of the Treatise. It is useful to recall in order the relevant points in the development of the argument, however familiar it may be. In the first two sections dedicated to space and time Hume argues in a decidedly metaphysical vein to show that ideas of space and time are not infinitely divisible. He then enlarges on this conclusion to cover space and time themselves, which would thus be composed of an infinite number of indivisible points (it is here that he uses Malezieu’s argument on unity). Then, after having shown the genesis of the ideas of space and time in perception, in the fourth section he examines the possible objections, setting out the most familiar arguments on the subject of the composition of extension, from the non-entity of mathematical points to the necessity that, being unextended, they penetrate one another through contact. He then goes on to discuss the fact that mathematics, and geometry in particular, turn up many objections to the theory of indivisible points. At this point the argument focuses on the contrast between this aspect of geometry and the way in which its entities are defined. He examines the classical Euclidean definitions of surface, line and point based on dimensions, and discounts those theories according to which geometrical entities do not exist or are the product of a distinctiu rationis. These definitions are only comprehensible if we accept the supposition of indivisible points. There follows the analysis of the notion of equality and its criteria, and a discussion of the way in which equality is accounted for by the theories of minimals and of infinite divisibility (Barrow is quoted here on the subject of congruity). Finally, he considers the definitions of the straight line and the plane surface, and the problem of the point of contingence between circumference and tangent. Hume concludes that the theory of indivisible points differs from that of infinite divisibility in allowing the fundamental notions of geometry, though only theoretically since these definitions are in reality based upon the way in which objects appear to us in common experience. The existence of proofs of infinite divisibility cannot WPS2134 392 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Studies in Hisrory and Philosophy of Science therefore constitute an argument capable of proving that extension is infinitely divisible, because this would be to ignore the fact that they are “built on ideas, which are not exact, and maxims, which are not precisely true” [45].16 The theme of time and space closes with two sections, the first on the idea of a vacuum and the second on the existence of external objects. Immediately afterwards Hume introduces the distinction between certain and probable knowledge, and discusses how geometry is a part of knowledge. Even from this short summary, we begin to see on the one hand how we may reasonably approach this text, and on the other hand which questions it is impossible, or at least not very reasonable, to address to it. On the whole, geometry is given an ambivalent role in these pages. On the one hand its place in the realm of certain knowledge is confirmed. And on the other, because it derives from perception, it tends towards probability, like the natural sciences, and therefore occupies a position quite different from arithmetic or algebra. It is worth noting that both aspects of the problem are examined in the Treatise, albeit with different emphases. (1) The fact that geometry belongs to certain knowIedge, along with arithmetic and algebra, is made clear on the very same page on which Hume describes certain knowledge as founded on relations which “depend entirely on the ideas, which we compare together” [69]. Typically this is the case with mathematics in general, and, the example which follows as an illustration of this kind of relation is precisely a geometrical one. It is concerned in particular with the relation of proportion in quantity or number upon which mathematics is based: ‘Tis from the idea of a triangle, that we discover the relation of equality, which its three angles bear to two right ones; and this relation is invariable, as long as our idea remains the same [69].” ‘b.4 Twmise of Human Nature, L. A. Selby-Bigge led.), 2nd edn revised by P. H. Niddich (The Clarendon Press: Oxford, 1978). All the quotations are from this edition. “Cp. Enquiries Concerning the Human Understanding. and Concerning the Principles of M orals, L. A. Selby-Bigge (cd.), 3rd edn revised by P. H. Niddich (The Clarendon Press: Oxford, 1975), p. 25, in which we find that certain propositions “are discoverable by the mere operation of thought, without dependence on what is anywhere existent in the universe. Though there never were a circle or triangle in nature, the truths demonstrated by Euclid would forever retain their certainty and evidence”. Some Hume scholars maintain that geometry is considered in the Trearise “as a factual science descriptive of a physical space”, while in the Enquiry all three branches of mathematics “do alike express analytic truths which rest upon the law of non-contradiction”: J. Noxon, Hume’s Philosophical Development (The Clarendon Press: Oxford, 1973), pp. 113K See also A. Flew, Hume’s Philosophy ~/Belief(Routledge & Kegan Paul: London, 1961), pp. 6243. For a reading which finds a g;e&r-continuity of empiricism I’nHume’s IWOworks, see 6..Zabeeh, Hume. Precursor ofM odern Emairicism fMartinus NiihoR The Hague. 1973) (1st edn 1961). where it is asserted that;‘Hume regaids geometry (in both the Twnrbeand En&&k) to bc a’kience which actually is conversant with properties of physical space” (p. 141). See also pp. 149ff.. and id.. ‘Hume on Pure and Applied Geometry’, Ratio 6 (19&l), 185-191, and Flew’s reply, ‘Did Hume distinguish pure from applied geometr;.’3, Rotio 8 (1966). 96- 100. lnteresting remarks also in R. Newman ‘Hume on &ace and Geometrv’. Hume Srudies 7 ! I98 I 1. I-3 1, PP.27ff.: A. Flew. ‘Hume on Space and Ceomeiry: One Reservation’. Hume Studies ‘8 ( 19k2). 62- &i. and Newman’s reply, ibid., pp. 6669. 393 Hume’s zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Space The issue here is not the extent to which mathematics considered in this light is analytical, as Kant thought. I8 The point that is at stake concerns a relation of equality with respect to the relationship between the idea of a geometrical entity and one of its properties. In order to change such a relation, Hume asserts, the idea ought also to change. Here geometry is made the example of certain knowledge. It is judged for its internal coherence, and thus the relation of equality does not constitute a problem but is taken for granted. A further confirmation of geometry’s full admission into the realm of certain knowledge occurs in the Abstract, published a few months after the Treatise, where it is maintained that all of our reasoning is probable, founded on causal inference, “excepting only geometry and arithmetic” 1650). It is worth noting, incidentally, that the certainty involved in defining this kind of knowledge does not correspond at all to the certainty of a psychological perspective. In other words, the viewpoint of the “science of human nature” cannot fail to influence the demonstrative sciences as a whole, so that “all knowledge resolves itself into probabihty” [181]. As Hume maintains in the introduction to the Treatise, “all the sciences have a relation, greater or less, to human nature; and . . . however wide any of them may seem to run from it, they stili return back by one passage or another” [xv], mathematics included. It too is in human hands, and there is no Algebraist nor Mathematician so expert in his science, as to place entire confidence in any truth immediately upon his discovery of it, or regard it as any thing, but a mere probability. Every time he runs over his proofs, his confidence encreases: but still more by the approbation of his friends: and is rais’d to its utmost perfection by the universal assent and applauses of the learned world [IBO]. So, even the most absolute necessity of any demonstrative reasoning is limited, on a psychoIogical level, by the repetition of proofs, and by the approbation, firstly of friends and then of the learned community. Even with regard to mathematics “our reason must be consider’d as a kind of cause, of which truth is the natural effect” [1X0]. The whole of mathematics, then, once contrasted with the concrete reality of knowledge, shows itself subject to doubt. This does not, however, mean that T-Iume intended to bring into question the certainty which depends upon internal coherence, or that this had escaped his attention. “The analytic/synthetic distinction is a recurrent temptation for Hume scholars. One of the many examples of this is in B. Rollin, ‘Hume’s Blue Patch’, Journal of rhe History OJ Idear 33 (1971), 119-128. where we find that in order to understand a particular question, “it is convenient to make use of some Kantian distinctions which are implicit in Hume. For Hume. all meaningful propositions are either u priori and analytic or a posreriori and synthetic” (p. 125).For a detailed analysis of this problem. approached in a varied, but generally well balanced way, see in particular R. F. Atkinson. ‘Hume on Mathematics’, Philosophical Quurtrrly 10 (19601, 127-137; W. A. Suchting. ‘Hume on Necessary Truth’, Diufogue 3 (l966-1967), 47-60; D. Gotterbarn, ‘Kant, Hume and Analyticity’. Kunl-S(udien 65 (1974) 274-283; L. W. Beck. Essays an Kunt and Humr (Yale University Press: New Haven. 1978). pp. 82tT. 394 Studies in History and Philosophy of Science (2) Let us return now to the geometrical notion of equality which Hume subjects to careful scrutiny a few pages before the passage quoted above. According to the theory of indivisible points, two entities would be equal when they contained the same number of points. However, although correct. “this standard of equality is entirely useless, and . . . it never is from such a comparison we determine objects to be equal or unequal with respect to each other” [45]. Even this criterion is obviously of no use to the infinite divisibility theorists. The congruity which Barrow speaks of is reduced to the same correct but useless criterion of the indivisible points, because congruity ought really to be applied even to the most minute parts. In other words, to the points: There are some, who pretend, that equality is best defin’d by congruity, and that any two figures are equal, when upon the placing of one upon the other. all their parts correspond to and touch each other. In order to judge of this definition let us consider, that since equality is a relation, it is not, strictly speaking, a property in the figures themselves, but arises merely from the comparison, which the mind makes betwixt them. If it consists, therefore, in this imaginary application and mutual contact of parts, we must at least have a distinct notion of these parts, and must conceive their contact. Now ‘tis plain, that in this conception we wou’d run up these parts to the greatest minuteness, which can possibly be conceiv’d; since the contact of large parts wou’d never render the figures equal. But the minutest parts we can conceive are mathematical points; and consequently this standard of equality is the same with that deriv’d from the equality of the number of points, which we have already determin’d to be a just but an useIess standard [46- 471. In fact, the criterion usually used to decide equality has more to do with the common sense of everyday life than the rationality of mathematics, and is based on the fact that the eye, or rather the mind is often able at one view to determine the proportions of bodies, and pronounce them equal to, or greater or less than each other, without examining or comparing the number of their minute parts [47]. This natural but imprecise judgement can then be corrected by various means ranging from juxtaposition to the use of a common measure, all of which increase the degree of accuracy according to the demands of the question in hand. In making judgements of this kind it is the nature of the problem which determines when the process of approximation can stop - when, that is to say, a satisfactory degree of precision has been attained. In conclusion, “we form a mix’d notion of equality deriv’d both from the looser and stricter methods of comparison” [48]. Clearly, the kind of equality required by geometry is generally a “standard corrected”, the most sophisticated kind. Yet, it always remains of equality, by which the appearances is indeed imaginary: problematic, and measuring and are exactIy as the very idea of equality is that of such a particular appearance corrected by juxta-position or a common measure, the notion of any correction beyond what we 395 Hvme’s Conception of Space have instruments and art to make, is a mere fiction of the mind, and useiess as well as incomprehensible [48]. It is not only incomprehensible, therefore, but also useless: an invention of the mind deriving from the fact that we are sensible, that the addition or removal of one of these minute parts. is not discernible either in the appearance or measuring; and as we imagine, that two figures, which were equal before, cannot be equal after this removal or addition. we therefore suppose some imaginary standard of equality, by which the appearances and measuring are exactly corrected, and the figures reduc’d entirely to that proportion [48]. Thus, the very meaning of geometry is betrayed, even if entirely naturally. (It should be noted in passing that, despite its naivety, Hume’s position does not in principle contradict some of the mathematical methods accepted by his age.) It is in this sense that geometry, given the way in which it is based upon perception, occupies as we said earlier. a special position In other words, with respect geometry, to arithmetic and algebra, that is the art, by which we fix the proportions of figures; tho’ it much excels both in universality and exactness, the loose judgements of the senses and imagination; yet never attains a perfect precision and exactness. So, given that we possess a precise standard of equality for algebra and arithmetic, they remain “the only sciences, in which we can carry on a chain of reasoning to any degree of intricacy, and yet preserve a perfect exactness and certainty” [7 I]. In some sense the reasoning here is clearly a reflection of the way in which the relationship between geometry and arithmetic was typically seen as a subject of mathematical thinking up until a few generations previous to Hum&s. (Let us recall that his point of reference could be said to be Barrow whose position was, as we have already seen, still well known and much respected during the first decades of the eighteenth century, however much it was already somewhat out-of-date during his own lifetime,) Hume’s reasoning often fails to distinguish clearly between precision and certainty, and couples assertions about the internal coherence of geometry with observations on the contradictions between its definitions and proofs. His discourse is most interesting, however, for the way in which geometry is considered to belong without any doubt to certain knowledge. In short, one might suggest that even here the distinction between certain and probable knowledge has a tendency to become less straightforward. The Notion of a Straight Line The same line of thought that we met on the subject of equality recurs in other parts of the section of geometry. What is brought into doubt is the 396 Studies in Hisroy and Philosophy of Science between geometrical metaphysical truths. that the of perception lost when go beyond limits. “how he [the prove to . . that two right lines cannot have one common segment? Or that ‘tis impossible to draw more than one right line betwixt any two points?” [51]. He cannot reply by appealing to the fact that “these opinions are obviously absurd, and repugnant to our clear ideas”; if geometry rests on what is evident, clear and obvious from perception, the limits of perception are somehow reflected in geometry: “supposing these two lines to approach at the rate of an inch in twenty leagues, I perceive no absurdity in asserting, that upon their contact they become one” [51]. Indeed, as in the case of equality, “the original standard of a right line is in reality nothing but a certain general appearance”. The definition adds Iittle,19 and presupposes that the reader has already formed an idea. Moreover, “‘tis evident right lines may be made to concur with each other, and yet cqrrespond to this standard” [52]. In other words, from the point of view of the study of perception or evidence, the straight lines in this example behave in the manner prescribed by the Euclidean theory of parallels only in so far as their behaviour can be perceived. The very fact that perception is limited means that any deviation from the requirements of such a theory is not incompatible with the standard of the straight line. In this respect, the truths of geometry are neither confirmed nor refuted by perception, and neither does perception have the power to force our assent. This is because the deviations from the dictates of geometry are not absurd, but conceivable and therefore also possible. Unlike Reid, with his geometry of visiblesZoa few years later, Hume does not systematically elaborate any geometrical theory for one of these other possibilities, and notes, anyway, that “its mistakes [of geometry] can never be of any consequence” 1721. It is, at any rate, certain that there are more possibilities contained within the information supplied by perception than there are in geometry. It is not surprising that Hans Reichenbach wrote that, with regard to geometry, presumably more due to good fortune than authentic genius, Hume “anticipates later results although he had no good argument for his conceptions”.” ‘Wurneis thinking of the Archimedean defmition according to which, to use his own words, the straight line “is the shortest way betwixt two points” [50]. The objection is that “this is more the discovery of one of the properties of a right Ime. than a just definition of it. For I ask anyone, if upon mention of a right line he thinks not immediately on such a particular appearance. and if ‘tis not by accident only that he considers that property?” [50]. As Suchting rightly observes, op. cir.. p. 54, “The justification for this statement is by no means clear.. These words suggest that he considered &tilinearity a primitive notion referring to an image or type of image”.-?“T. Reid. Inauirv of Common Sense (Edinburah. . _ inm the Human Mind on the Princi~ler . MDCCLXIV), chap. VI: “Of Seeing”. sect. 9: “Of the geometry of visibles”, pp. 237-261 (where the starting point is the distinction between tact& and visual space). On Reid’s geometry of I visibles. see N. Daniels. ‘Thomas Reid’s Discovery of a non-Euclidean Geometry’, Philosophy qf ScQnce 69 (1972). 219-234. ?-I. Reichcnbach, The Rise qf Scientific Philo.~ophv (University of California Press: Berkeley, 1964). pp~ 86-87. Hme’s Conceplion of Space 391 Returning then to Hume, geometry is a construction which is autonomous and independent from the reality of the external world. It is true and certain in as far as it is internally coherent and constituted at the same time on a basis which is perceptual. Its conclusions, however, can be converted into the language of a general description of space, or transferred to a metaphysical level only with the utmost caution. This seems the most probable reason for the polemical opening of the section about ideas of space and time. The theory of infinite divisibility is presented as the best example of the tendency to paradox typical of philosophers, who thus like to distinguish themselves from the common people, showing themselves at the same time to be subject to that general disposition of human nature according to which “any thing propos’d to us, which causes surprize and admiration, gives such a satisfaction to the mind, that it indulges itself in those agreeable emotions, and will never be perswaded that its pleasure is entirely without foundation” [26]. The Theory of Infinite Divisibility The reasoning Hume uses to refute the theory of the infinite divisibility of extension is simple and easily summarized. In the first place, the finite capacity of the mind means that it is impossible to attain “a full and adequate conception of infinity”. As others have noted, this assertion, presented as “universally allow’d,” closely recalls, for instance, the way in which infinity is presented by Locke (“a growing and fugitive idea, still in a boundless Progression that can stop no where”). **Its tone makes it perhaps more reminiscent of Locke’s statement than of Port Royal’s axiom, according to which “it is the nature of the finite mind to be incapable of comprehending infinity”, or indeed the Cartesian proposition condemned by the Jesuits: “Mens nostra, eo quod finita est, nihil certo scire potest de infinite”” (Our mind, because it is finite, can know nothing certain of the infinite). In fact, in Hume’s text, “comprehending” and “knowing something for certain” are transformed into “having a full and adequate conception”. Secondly, that which is infinitely divisible consists in an infinite number of parts. Enough has been written already about the many excellent reasons for saying that this principle is entirely mistaken, and so we need not dwell on it here.24 The conclusion follows directly from this principle: “the idea, which we form of any finite quality, is not infinitely divisible”, and by subdividing this idea we will at a certain point arrive at lesser ideas which will be perfectly simple and indivisible [27]. On the basis of this, “An E.mq Conrerning Human Undersmrding, Book II, chap. XVII. sect. 12, P. H. Niddich (ed.) (The Clarendon Press: Oxford, 1975). p, 216. “Arl de Penwr. P.W. chap. VII. ax. 9. The proposition condemned by the Jesuits and quoted in J. Laird. op. cir.. pi 67; A, Flew. “lntimte divisibility , .“, op. cd., p. 259. notes the same similarity discussed by Laird. R. Newman. op. cit.. p. 4, suggests the comparison with Locke. %e, for example. what is said in the texts mentioned in the previous note. 398 Studies in History and Philosophy qf Science zyxwvutsrqponmlkji Hume denies that there is any value at all in mathematical proofs concerning the division ad infinitum, without even going into the question or discussing how far these proofs can be said to be consistent. They remain in every case “mere scholastick quibbles, and unworthy of our attention” [32] The rigidity of this position finds expression also in a footnote where Hume flatly refuses to consider the distinction between aliquot and proportional parts (already known by Aristotle and, anyway, given by Bayle in the “Zeno” entry).25 Such a distinction “is entirely_frivolous”. This can be explained if we bear in mind that the problem which interests Hume is that of the composition of extension, in its metaphysical implications. If so, the fact that the theory of infinite divisibility (whatever Hume meant exactly by this term), in its mathematical aspects, rests on foundations that are at best uncertain, is conclusive for the rejection of its metaphysical implications. That this should have been the case must have been clear to him without going very deeply into the question. For geometry, as much as for metaphysics, he could well make use of the fact that, with regard to his students, “even the rabble without doors may judge from the noise and clamour, which they hear, that all goes not well within” [xiv]. If we exclude from Barrow’s and Malezieu’s argument every appeal to theology and religious faith, it is hard to see how one can maintain, in a situation bristling with paradoxes, that geometry is the clearest of the sciences because it deals in the clearest fashion with ideas which are perfectly evident, and that it therefore provides the key to understanding the reality of the natural world in its entirety. Furthermore, if this is Hume’s perspective in this case his (notorious)z6 habit of equating conceiving, imagining and having a mental image, is not illicit, (As an example of this one may refer to the above reference to the impossibility of a full and adequate conception of infinity, which is in itself a definitive argument,) Up to this point, at least, it is not a question of there being a flaw in his argument, or perhaps even in his forma zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONML mentis - which might possibly be attributed to his psychological atomism, whether true or only presumedZ7 but rather it is precisely a consequence of the specific aim of his argument. One has only to remember that if perception contradicts mathematical demonstrations of infinite divisibility because of the existence of perceptual minimas, the problems connected with the infinitesimal cannot be very different from those presented by anything which, because of its size, is not proportioned within the range of our perception, in other words the very small, for instance. We have already seen something similar on the subject of equality, where it was said that the criterion of points was correct but useless. Now, with regard “Here too the literature is extensive, See again the texts quoted in note 23. %ee particularly A. Flew, “Infinite Divisibility . .“, op. cd.. p, 259 and 262. T.ee, for example, R. Newman, op. cir.. pp. 7K 399 Hume’s Conceplion of zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Space to the very small, Hume admits that an idea might be conceived for which there was no corresponding image: When you tell me of the thousandth and ten thousandth part of a grain of sand, I have a distinct idea of these numbers and of their different proportions; but the images, which I form in my mind to represent the things themselves, are nothing different from each other, nor inferior to that image, by which I represent the grain of sand itself, which is suppos’d so vastly to exceed them [27]. first of all, for the image to enable him to talk about comprehension. But it implies equally well that he does not exclude other possibilities. He is talking precisely of a distinct idea existing despite the absence of a corresponding image. Therefore it is possible at least to deal with the very small when compatible with the limits of perception. when the ideas of the parts are substituted by ideas of the process of division and of the proportions between those parts. The traditional notion of potential infinity, as we find it expressed in LockeZs for instance, although it might seem to be generated in a fairly similar manner, has no part at all to play here, In contrast to the very small, not only does the infinitely small find no confirmation, or at least expression, at the level of perception, but it also directly contradicts it, in as much as it implies the denial of the points which are actually revealed by perception. So, the evidence of perception is directly opposed to the demonstrative force of geometry - and this, from the point of view of the “science of human nature”, that is to say of the problem in its metaphysical terms, simply means that it falsifies geometry: This implies that here Hume is looking, all the pretended demonstrations for the infinite divisibility of extension are equally sophistical; since ‘tis certain these demonstrations cannot be just without proving the impossibility of mathematical points; which ‘tis an evident absurdity to pretend to L331. One cannot therefore condone the attitude of all those, from Barrow and Malazieu to Malebranche and Amauld, who combine rationality and religious faith in order to assert that human reason, defeated by infinity even in the most minute parts of matter, must yield to the incomprehensible. If this is the case it would also clarify the immediate inference from perception to reality which Hume draws from the principle - which to a certain extent is a true metaphysical axiom, as has been rightly noted*’ - accor;ding to which “whatever appears impossible and contradictory upon the comparison of these ideas, must be zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA really impossible and contradictory, without any farther excuse or evasion” [29]. So, since our ideas of time and space are composed of an infinite number of simple and indivisible parts, “‘tis certain they actually do exist “Es.suy. Bk. II. chap. XXIX. sect. 16, pp. 37OK. *R. Newman. op. cit.. p. 4. Studies 400 in H&or! and Phifosoph>~ of Science conformable to it; since their infinite divisibility is utterly impossible and contradictory” [39]. In this way, the fact that geometry cannot be reconciIed with perception is transferred to the level of external reality, and in this sense the falsification of the theory of infinite divisibility. while still supported by geometry, is complete. In conclusion then, Geometry can tell us nothing of the ultimate reality which underIies perception. This is to say that it cannot be used with the aim of transcending reality in the pursuit of a higher knowledge: “and this is the nature and use of geometry, to run us up to such appearances, as, by reason of their simplicity, cannot lead us into any considerable error” [72]. This. Hume notes, is more than enough to guarantee the legitimacy of using geometry at a local level, even after having refuted the claim that geometry might provide the basis for a general description of the world. At any rate, when rationality reaches its extreme limits, the appeal is not to faith but to reasonableness. The Origin of the Idea of Space The section dedicated to the origin of the ideas of space and time opens with a reassertion of the fundamental principle underlying Hume’s investigation: No discovery cou’d have been made more happily for deciding all controversies concerning ideas, than that abovemention’d. that impressions always take the precedency of them, and that every idea, with which the imagination is furnish’d. first makes its appearance in a correspondent impression [33]. So, the verification of the ideas is the anaIysis of their origin. Upon opening my eyes, and turning them to the surrounding objects, I perceive many visible bodies; and upon shutting them again, and considering the distance betwixt these bodies, I acquire the idea of extension [33]. It is thus a question of identifying “the impressions similar to this idea of extension” [33]. The table before me is alone sufficient by its view to give me the idea of extension. This idea, then, is borrow’d from, and represents some impression, which this moment appears to the senses. But my senses convey to me only the impressions of colour’d points, dispos’d in a certain manner the idea of extension is nothing but a copy of these colour’d points, and of the manner of their appearance [34].W The idea of space - or extension (the two terms are used synonymously) the coIoured points, elementary “manner of appearance”, i.e. atoms and thus derives from two orders of impressions: units of perception, structure. and their ‘Tlew comments, “Infinite Divisibility .*.,op. cit.. p. 265: “Anyone familiar with the theories and paintings of Seurat might also mischievously characterize the Hume of this Section as ‘the Father of Pointitlisme’ “. 401 Hume’s zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Concephn zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA ofSpace Hume explains a little later what he means by perceptible points. They are “atoms or corpuscles endow’d with colour and solidity” [38], and only by virtue of these properties can they be conceived. Between the extension of physical points and the non-existence of mathematical points, the analysis of perception seems to reveal a psychological mean which is this sensible atom. This is the indivisible which Hume had in mind when he wrote concerning infinite divisibility: Put a spot of ink upon paper, fix your eye upon that spot. and retire to such a distance, that at last you lose sight of it; ‘tis plain, that the moment before it vanish’d the image or impression was perfectly indivisible [27]. which has provoked harsh criticism, was recently thought by a psychologist to merit comparison with Helmholtz’s study of sensation.3’ Hume continues: The passage, ‘Tis not for want of rays of light striking on our eyes, that the minute parts of distant bodies convey not any sensible impression; but because they are remov’d beyond that distance, at which their impressions were reduc’d to a minimum. and were incapable of any further diminution. A microscope or telescope, which renders them visible, produces not any new rays of light, but only spreads those, which always flow’d from them; and by that means both gives parts to impressions, which to the naked eye appear simple and uncompounded, and advances to a minimum, what was formerly imperceptible [27-281. His reasoning develops to show the origin of the belief in the existence of infinitely small objects. Nothing is smaller than certain impressions or ideas, however much what they represent might turn out to be composite in other situations. The indivisible that is perceived does not represent an indivisible object, and similarly, as we have seen, the idea of a grain of sand is no larger than the idea of its parts. Therefore: taking the impressions of those minute objects, which appear to the senses, to bc equal or nearly equal to the objects, and finding by reason, that there are other objects vastly more minute, we too hastily conclude, that these are inferior to any idea of our imagination or impression of our senses [28]. It has seemed appropriate to quote Hume at length here because some of the problems arising from this page are particularly subtle. It must, of course, be said that it is difficult to reconcile all of this with a rigidly phenomenal/St interpretation of Hume’s philosophy. In fact, he is clearly presupposing here the existence of external objects, corresponding essentially to collections of “See, for example. C. D. Broad, op. cit.. p. 165: “I am very doubtful whether the facts about the visual appearances of the ink-spot. on which the argument for punctiform visual sense-data is based. are correctly described”. For the comparison with Helmholtz. see P. BOA, Unitri fdenrirci Causaliki, Cappelli (Bologna, 1969). pp. 55Ff.and 64ff. 402 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Studies in Hisror_v and Philosoph!~ of Science several perceptions. ‘* When Hume talks about the inkspot which corresponds to the minimum sensibile when moved to an appropriate distance, he does not talk about an isolated perception, but rather he refers to the whole series of perceptions, decreasing in order of complexity as they occur during the distancing process. The series of successive perceptions is considered unified on an objective level. Nevertheless, this way of talking about external objects, even if it seems to take their existence for granted (in the same way, for instance, the discussion on the passions is based on the presupposed existence of the self and the force and liveliness of this idea), need not, in fact, contradict either Hume’s so-cahed phenomenalism or the fact that elsewhere in the Treatise he shows how our belief concerning external objects is without foundation. In fact, one might well suppose that Hume’s position is more sophisticated than it appears, and that he is not in any way concerned with the reality of external objects t’just as he is not concerned with the reality of the self in his discussion of the passions) - not even in order to call that reality into question. Or, in other words, we might think that the scepticism of his conclusions is not directed at the reasonableness of our beliefs and thus the legitimacy of their common use, but is intended rather to challenge any full theoretical justification of them and to serve as a warning against going too far in their metaphysical use. In this sense, the argument discussed above is entirely legitimate, if not indeed more coherently “phenomenalist”, in as far as nothing is said which might be attributed to the problem of the existence of external objects. The argument is concerned rather with the commonest conceptions. Anyway, to return to the issue at hand, the points which produce the impression of a table are conceivable solely on the basis of their colour and sohdity. In other words they are not extended. On the basis of the atomistic principle according to which “every idea, that is distinguishable, [is] also separable,” we can investigate the nature and the quality of one of these points considered on its own. And, in agreement with what we have seen so far, the conclusion must be that it does not coincide with the idea of extension, “for the idea of extension consists of parts; and this idea, according to the supposition, is perfectly simple and indivisible” [38]. Thus, the compound idea of extension has a quahty which does not appear in its minimal components. We have reached the question of the “manner of appearance” of the points taken together, or the way in which the overall impression is structured. We ‘C!k-e taird, op. cir.. pp. 48-69: “What is the spot if the ‘impressions’ form a series’? If the ‘image or impression’ were perfectly indivisible how could a pair of binoculars ‘spread’ it?“. Laird talks of the “halting and debased phenomenalism which characterized much of Hume’s argument” in these pages (p. 68). See also C Maund, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHG Humek Theory oJ Knawledge (Macmiffan: London, 1937) pp. 2OG2Of: “It is not sensible, in Hume’s philosophy, to say that the microscope spreads the rays of light from an impression which was formerly imperceptible and so makes it perceptible . this passage can only be made intelligible in terms of the object of the vulgar”. Hume’s Conception of Space 403 have already seen a somewhat similar idea in connection with the notion of a straight line which derives from ‘$a certain general appearance”; and this general appearance shows itself to consist, in principle, in the position of the points [52]. From the point of view of perception, then, it seems, at least as far as this example goes, that the disposition of parts in some way constitutes a primitive notion which causes “the entire impression of a curve or right line” [49],33And, in fact, correspondingly, in this more general part of the analysis the manner of appearance has nothing to do with any separate impression, but seems rather a characteristic peculiar to the actual act of perception. The impression of a table is not only made up of points. In the act of sensing, the punctiform impressions appear organized according to a structure which transcends them.” This is also, in Hume’s words, a common aspect of several perceptive contexts: suppose that in the extended object, or composition of colour’d points, from which we first receiv’d the idea of extension, the points were of a purple colour; it follows, that in every repetition of that idea we wou’d not only place the points in the same order with respect to each other, but also bestow on them that precise colour, with which alone we are acquainted. But afterwards having experience of the other colours of violet, green. red, white, black, and of all the dimerent compositions of these, and finding a resemblance in the disposition of colour’d points, of which they are compos’d, we omit the peculiarities of colour, as far as possible, and found an abstract idea merely on that disposition of points, or manner of appearance, in which they agree [34]. is the same as for abstract ideas, as Hume himself makes clear immediately afterwards, in the sense that The origin of the idea of space, therefore, all abstract ideas are really nothing but particular ones, consider’d in a certain light; but being annexed to general terms, they are able to represent a vast variety, and to comprehend objects, which, as they are alike zyxwvutsrqponmlkjihgfedcbaZYXWVUTSR in some particulars, are in others vastly wide of each other [34]. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDC the abstract idea of space refers also to tactile impressions, in as far as they are “found to be similar to those of sight in the disposition of their parts” [34]. However, this does not seem to mean that, in the end, space is merely an idea.” The abstract idea of space represents an aspect common to more than So, Yjce note 19 above. “See N. K. Smith, op. cit., pp. 273ff., whose basic thesis in this respect is that “the ideas of time and space are complex ideas which lie beyond the nature of each and all of our simple impressions”. For an interesting reading of the “manner of appearance”, see also R. W. Church, Humek fheorr of Understanding (George Allen & Unwin: London. 1968), pp, 6DK See also C. W. Hendel. Srudks‘in the Philosophy nf Duvid Hume (lndianapolis, 1963), pp. 418-419; A. Leroy, David Hume (Paris, 1953). p. 156. ‘?G.Tweyman rightly comes to this conclusion in Hume on Sepuruting the Inrepuruhle. in: W. B. Todd led.). Hume and the Enlightenment (The University Press: Edinburgh, f974), pp. 30-42. Tweyman considers the abstract ideas of space and time as an instance of the application of the distinctio rationis. In this sense they can be assimilated to the ideas of the colour and shape of an object, like the globe of white marble employed by Hume precisely to illustrate the distincrio rrrtionis. 404 Studies zyxwvutsrqponmlkjihgfedcbaZYXWVUTS in Histogs and Philosophjp of Science one sensorial context, a “manner of appearance” which cuts across every context and becomes, so to speak, isolated as a significant feature in the process which makes the term “space” available for general use. The underlying similarity is between the different contexts each considered in its entirety, between each “entire impression’+ (as in the case of the straight line [49]). In short, we are speaking about a relation between relations, for which, as such, there is no corresponding separate impression. In other words, even Hume’s psychological atomism, like his phenomenalism, if taken literally, appears ultimately to be little more than an historiographical myth. Intermezzo In his classical study of Hume’s philosophy, Norman Kemp Smith introduces his chapter on the ideas of space and time with a comment on these lines, which follows Hume’s assertion of the principle of the correspondence of impressions and ideas: These latter [simple] perceptions are all so clear and evident, that they admit of no controversy; tho’ many of our [complex] ideas are so obscure, that ‘tis almost impossible even for the mind, which forms them, to tell exactly their nature and composition [33]. According to Smith, this phrase of Hume’s “shows how far reaching are the reservations which he was prepared to introduce in the application of his principles.“36 Now, it has already been noted that the peculiarity of the position of geometry in knowledge seems in some way to complicate the distinction between certain and probable knowledge; in other words, it seems ultimately to entitle us to re-evaluate a doctrine which is decidedly central to Hume’s philosophy. Furthermore (and it is this presumabIy which Smith is referring to in particular), the investigation of the origin of the idea of space, because of the “manner of appearance”, seems to contradict the so-called atomistic principle, i.e. the fundamental principle of the correspondence between impressions and ideas. Now, all this might seem inconsistent, or at least rather confused.37 But the fact is that the fundamental philosophical principles of the Trearise, those which underpin the overall structure, are apparently held to be true only statistically, so to speak. Moreover, Hume does “6N.K. Smith. op. cit.. p. 273. See also the remarks on p. 267 where Smith quotes C. Maund, op. cit.. pp. 165ff. “See, for example. A. Seth. Scorrish Philosophy: A Comparison of the Scottish and German Answrs to Hume (William Blackwood: Edinburgh, 1890) (1st edn 1886), pp. 55ff,or more recently. A. Flew. “Jnfinite Divisibility zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA . ~ .“ , op. cir . pussim, and R. Kuhns. op. cit., p. 80, where we read that space and time “constitute (though Hume never explicitly recognises this) important exceptions to the principle of verification by reference. to impressions, and like the missing shade of blue mentioned earlier in the Treatise. they are ‘known’ without the mediation of impressions of themselves.. In offering this solution. Hume has denied both the mathematicians and his own epistemology”. For the shade of blue to which Kuhns refers. see below. 405 Hump’ s Conception zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA qf Space not even seem particularly reluctant to follow a general principle with an immediate counterexample. The most obvious illustration of this concerns precisely the principle of the precedence of impressions over ideas. It is a principle which Hume establishes by arguing inductively from the constant similarity we find between the two types of perception. He then gives the exception of a continuous range of shades of a colour broken by a single gap. The observer “will perceive a blank, where that shade is wanting”, and it would be possible for him, “from his own imagination, to supply this deficiency” despite the lack of a corresponding impression [6]. This exception, Hume asserts after having explained it, “does not merit that for it alone we should alter our general maxim”, which remains valid because the exceptions are very rare [6]. It has been said that this is a rhetorical artifice perpetrated by “the most astute of dialecticians” to put potential critics off the trail of more significant exceptions.38 But it is also a revealing symptom of Hume’s wellknown preference, already noted, for reasonableness over rationality.” This kind of elasticity in the fundamental principles of the new metaphysics and the freedom with which ‘its inventor makes use of them, provides a key to the meaning of Hume’s writing on space, besides perhaps also being a by no means inconsiderable lesson in common sense. On the other hand, it is enough perhaps to remember that, in any case, the “science of human nature” cannot but be included in probable knowledge. We need only look back at the introduction to the Treatise where Hume says that: “any hypothesis, that pretends to discover the ultimate original qualities of human nature, ought at first to be rejected as presumptuous and chimerical”. [xvii]. The “conjectures and hypotheses” on which the science of human nature is based must not be proposed as “the most certain principles” [xviii], because it is like any other science, based upon experience and analogy f22J. This is asserted first of all as a theoretical principle and then recurs in the actual investigation of the Treatise. The reason for this is that “the essence of the expression occurs in 8. E. Rolhn’r study of the shade of blue. op. cit., p. 120. ?‘On this last point, see for instance D. Deleule, Hume et la noissance du libkralisme Pconomipue (Paris, 1979). pp. 35ff. In his analysis of Hume’s notion of reason, focusing especially on the latter’s moral philosophy, Deleule concludes that “en un mot. au rutionel Hume substitute Ie rnisonnoble I I’intericur de la definition de la raison” , (p. 39). Interesting observations also in F. A. Hayck. ‘The Legal and Political Philosophy of David Hume’, II Politico 28 (1963), 691-721. It must be noted, however, that amongst Hume scholars there is no shortage of those who consider his primary interest to have been “in locating and describing those sciences which are exact and certain . for Hume, a priori evidence (depending ‘solely upon ideas’), in the form of noncontradiction. is the mark of genuine knowledge. hence of genuine science”: R. E. Butts, ‘Hume’s Scepticism’. Journal of fke Hisrory oJldeas 20 (1959), 413-419. Hume’s scepticism would thus be the product of the “contradiction, generated by trying to realise rationalist aims by empiricist means”(p.419). On a position of this kind. see also F. Zabeeh, op. cit., on p. 112 for example, “The where he maintains that mathematics **for him is the paradigm of all sciences”. 406 Sludies in HismrJ ond Philosopltj, cf Science mind [is] equally unknown to us with that of external bodies” [xvii]. With the “scjence of human nature”, on the other hand, we might hope to establish a system or set of opinions, which if not true (for that, perhaps, is too much to be hop’d for) might at least be satisfactory to the human mind, and might stand the test of the most critical examination [272]. So, it is only reasonably certain, and it appears further that it need not even be a system, but merely a simple set of opinions. One might ask, if so, what remains of the power of verification inherent in the principle of correspondence between impressions and ideas, if we recognize that in more than one case its validity is subject to limitations.40 One possible reply is that its basic inspiration is in no way altered. It is the means zyxwvutsrqponmlkjihgfedcb by which one exercises caution in passing from the words used to that which can with good reason be maintained concerning the world, and sets the limits within which such a description is reasonable. One confirmation of all this appears, for instance, in the analysis of the idea of a vacuum. The Idea of a Vacuum The way in which the origin of the notion of space has been examined in the previous pages is obviously a response to the demands of simplification posed by philosophical analysis. In everyday experience it is clearly not a matter of the table and the punctiform impressions going together in a certain way to make up that experience, but rather of a more complex series of concrete interactions. In this sense, it is the passage introducing the effects of contiguity or spatial distance on the passions which most clearly shows what is meant by “manner of appearance”.4’ ‘Tis obvious, that the imagination can never totally forget the points of space and time, in which we are existent; but receives such frequent advertisements of them from the passions and senses, that however it may turn its attention to foreign and remote objects, it is necessitated every moment to reflect on the present. ‘Tis also remarkable, that in the conception of those objects, which we regard as real and existent, we take them in their proper order and situation, and never leap from one object to another, which is distant from it, without running over, at least in a cursory manner, all those objects, which are interpos’d betwixt them. When we reflect, therefore, on any object distant from ourselves. we are oblig’d not only to reach it at first by passing thro’ all the intermediate space betwixt ourselves and the object, but also to renew our progress every moment; being every moment recall’d to the consideration of ourselves and our present situation [427428j. *‘Obviously one is thinking here especially of attempts to read Hume in neopositivistic terms. For example. F. Zabeeh. op. cir., pp. 5Off. This type of reading, however, is certainly not restricted to Zabeeh. See. roarinstance, D. F. Pears (ed.). David Hutne: A S.wnposiunr (Macmillan: London. 1963). “As J. Laird has rightly remarked. op. cir., p 70-71. 407 Hum’s zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Conception of zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFE Space The “manner of appearance” here is the arrangement of objects (that is to say of non-punctiform impressions) in relation to the perceiving subject. In the presentation of the activity of perceiving the emphasis falls wholly upon the continuity of space and the centrality of the body. Therefore, we are talking here of the perceptive organization of co-existent objects: “space or extension consists of a number of co-existent parts dispos’d in a certain order, and capable of being at once present to the sight or feeling” [429]. This emphasis is fundamental to the analysis and criticism of the notion of a vacuum. The conception of space, as a perceptive ordering or configuration of coexistents, is indissolubly linked to a perceptible set of objects, and therefore, in particular, “we can form no idea of a vacuum, or space, where there is nothing visible or tangible” [53]. The fact that a vacuum cannot be conceived is asserted by Hume in reference to three possible objections. Once again these are arguments which were widely and variously diffused throughout the texts of the period, usually in order to refute Descartes’ identification of extension with matter.42 More recently they had also been reformulated in a Newtonian context. The first objection argues that the mere fact of discussing whether or not a vacuum and its idea exist, implies that we have an idea of a vacuum just in order to make the discussion possible. The second points out the absurdity of the famous Cartesian paradox of the evacuated vessel whose walls ought to touch one another. The idea of a vacuum would thus be a matter of fact according to the first, or at least a possible idea on the basis of the second. The third objection takes up quite an old traditional argument, which is attributed here to the way in which Newtonian mechanics explains movement, and is in opposition to the Cartesian theory that movement in a vacuum is impossible. The necessity for a vacuum, and therefore also for an idea of it, is demonstrated precisely by the fact that movement wou!d b= impossible in the real world and inconceivable in the realm of ideas “without a vacuum, into which one body must move in order to make way for another” [55]. Typically, Hume classifies this third objection as radically heterogeneous with regard to the other two. The latter deal with the possibility or the existence of the idea of a vacuum, whereas the former departs from external reality and argues in terms of necessity, claiming to transcribe Newtonian mechanics directly into metaphysics. For instance, it is a position like that of Samuel Clarke, according to whom, “Space void of Body, is the Property of an INCORPOREAL SUBSTANCE . . . In All Void Space, God is CERTAINLY present”.43 Vacuum exists because space is divine. Hume, at any rate, declares his unwillingness to “For instance, they appear in Locke, Essay, 8.11, chap. IV. sect. 3 and XIII, sect. 2lR; and Bayle. Dichmnnire. under “&no”. “Corre~pondenc@ Leibniz-Clarke, present& d’apris les manumits originaux des Bibhothiques de Hanovre et de Londres par A. Robinet: Bibliothtque de philosophic contemporaine, Histoire de la philosophic et philosophie &&ale (Paris, 1957). p. 1IO. Studies in History and Philosoph_v~ of Science 408 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA go into the merits of this third objection, “because it principally belongs to natural philosophy, which lies without our present sphere” [55]. The reply to the third objection appears as a paradox in his conclusion to this discussion. Once again, the zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA first stage of the argument involves looking for the origin. If the idea of space is the idea of a composition of visible or tangible objects, it clearly cannot exist in their absence. The lack of visual or tactile impressions does not give rise to any conception of space: “darkness and motion, with the utter removal of every thing visible and tangible, can never give us the idea of extension without matter, or of a vacuum” [56]. Indeed, “a man, who enjoys his sight, receives no other perception from turning his eyes on every side, when entirely depriv’d of light, than what is common to him with one born blind” [5S-56].M Similarly, supposing “a man to be supported in the air, and to be softly convey’d along by some invisible power . . . even supposing he moves his limbs to and fro”, it is plain that he will not draw any notion of space from this [56]. The same goes for intervals of darkness and free movement between visible and tangible objects. The only difference between such intervals and the complete absence of visual and tactile impressions consists “in the object themselves, and in the manner they affect our senses” [58]. These intervals are the distance between two perceived objects, that is to say they belong to the spatial “manner of appearance” of the whole. It is precisely this latter situation, however, which explains the mistaken conviction of being able to conceive a vacuum. The invisible or intangible interval is confused with true extension, because these two kinds of distance have some characteristics in common. The reciprocal position of objects is indifferent with respect to the kind of distance interposed, as it is also to the conversion of one into the other, Both “have nearly the same effects on every natural phaenomenon”, that is to say “heat, cold, light, attraction &c. diminish in proportion to the distance” [59] leaving aside the question of its kind. The spatial separation is the same, regardless of whether the sequence of minimal sensibies which join the two bodies is there or not. This confusion between the two kinds of distance is a particular instance of the typical tendency of human nature to confuse narrowly related ideas, substituting one for the other in conversation and reasoning. In this way, one tends to use “space” to indicate both kinds of distance because of their similarity: “‘tis usual for men to use words for ideas, and to talk instead of thinking in their reasonings”. Combinatorial exercises with words takes the place of thought: “We use words for ideas, because they are commonly so closely connected, that the mind easily mistakes them” [61-621. Once again the “It has been frequently noted in the critical literature that darkness is not a good example absence of visual impressions. See N. K. Smith, op. cit.. p. 309. of the 409 Hume’s Conception of Space philosophers show themselves prone to a general shortcoming of human nature. The interchangeability of the two kinds of distance belongs precisely to this kind of phenomenon, and this is clearly the key to the problem. The first objection, according to which the very discussion of a vacuum presupposes that we have an idea of it, dissolves, given that these discussions obviously concern a word and not an idea; and, Hume asserts again, “nothing [is] more common, than to see men deceive themselves in this particular” [62]. On the other hand, considering the empirical information which has not been linguistically compromised, it is also clear that the annihilation of all the matter lying between two walls of a room would simply mean that the intervening distance becomes invisible and intangible: When every thing is annihilated in the chamber, and the walls continue immoveable, the chamber must be conceiv’d much in the same manner as at present, when the air that fills it, is not an object of the senses [62]. Obviously, Hume’s position on this issue is not particularly original. It recalls, for instance, the Cartesian argument as presented by Bayle or Locke, and its refutation proceeds along lines entirely analagous to Barrow’s As for the link between Newtonian mechanics and the ontology of a vacuum, the question is more complex. The movement can be explained as an instance of converting an invisible and intangible distance into a visible and tangible one. In this sense, the fact that the distant bodies are not affected “suffices to satisfy the imagination, and proves there is no repugnance in such a motion” [63]. Experience alone intervenes to demonstrate that such a conversion is not only conceivable, and thus possible, but also in fact occurs, and convinces us that two bodies, situated in the manner above-describ’d, have really such a capacity of receiving body betwixt them, and that there is no obstacle to the conversion of the invisible and intangible distance into one that is visible and tangible [63]. This is all that one can say, on a metaphysical level, about the way movement occurs, and the claim “to penetrate into the nature of bodies, or explain the secret causes of their operations”, besides failing outside the aims of the “science of human nature”, is anyway beyond what is humanly possible [64]. The attempt to translate Newtonian mechanics into the metaphysical terms of a description of reality is illicit, as further clarified in an addition to the second edition of the Treatise: AS long as we confine our speculations to the appemunces of objects to our senses, without entering into disquisitions concerning their real nature and operations, we are safe from all difficulties, and can never be embarass’d by any question [638], To the question about whether the invisible and intangible distance between 410 Studies in History and Philosophy of Science two objects is something or nothing, one can reply that it is “a property of the objects, which affect the genies after such a particular manner”. Once again, it is the same emphasis upon the notion of a “manner of appearance”. On the other hand, if we ask ourselves whether two objects separated by a distance of this kind are touching or not “it may be answer’d, that this depends upon the definition of the word, fouch” [638]. And the reply is different according to whether such a term indicates the situation of two objects between which nothing sensible is placed, or whether instead their images lie on continguous parts of the eye, and the sense of touch perceives them without any free movement being interposed. Apparently, the problem is that the common language, and here also that used by metaphysics, does not necessarily represent the phenomenon faithfully. The problem lies entirely in the language, because “the appearances of objects to our senses are all consistent; and no difficulties can ever arise, but from the obscurity of the terms we make use of” [639]. The originality of the sensation is here used to bring to light an ambiguity hidden in the words, from which an unfounded inference from science to metaphysics arises; “if rhe Newtonian philosophy be rightly understood, it will be found to mean no more”. In other words, it author&s a certain way of speaking in order somehow to represent the behaviour of bodies in movement, without, however, making any claim to deduce a comprehensive description of reality. In the absence of conclusive arguments, we could problematically consider that perhaps the invisible and intangible distance would remain such even with an improvement in our sensorial capacities. But this conclusion, far from being certain, is preferable only in as much as it is “more suitable to vulgar and popular notions” 16391(that is to say, precisely in its being as distant as possible from the taste for abstruse concepts which is characteristic of the most common kind of metaphysics and its students). This is also the meaning of the paradox which concludes the analysis of vacuum: if you are pleas’d to give the invisible and intangible distance, or in other words, to the capacity of becoming a visible and tangible distance, the name of a vacuum, extension and matter are the same, and yet there is a vacuum. If you will not give it that name, motion is possible in a plenum, without any impulse in infinitum, without returning in a circle, and without penetration [&I]. The “science of human nature” demonstrates with its analysis that the vacuum spoken of by metaphysics, even with the support of Newtonian science, is in reality a word whose usage is irremediably ambiguous. The methological lesson offered by that science would benefit from “a modest scepticism to a certain degree, and a fair confession of ignorance in subjects, that exceed all human capacity” [639]. In other words, only a thought willing to recognize that the capacities available to it are sufficient only for a knowledge which is definitively unsystematic, limited and local, can be said to be reasonable from Iiume~ Conceplion zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDC of Space 411 this point of view, in a metaphysics reasonableness. whose highest value lies precisely in Acknowledgentents - I am indebted to Mr D. Ruzicka for his help in preparing a translation from the Italian of this paper. I thank Julian Stargardt, Piers Bursill-Hall. Anna-Katherina Mayer and Nicholas Jardine for their generous advice and encouragement. This work was carried out during the tenure of a CNR-NATO Advanced Fellowship at the Department of History and Philosophy of Science, University of Cambridge.