1 Hume, D. A Treatise of Human Nature, Norton, D.F. and Norton, M.J. eds. (Oxford: Oxford University Press 2000).Google Scholar Citations from the Treatise will be designated by ‘T’ followed by the book, part, section, and paragraph.

2 The ‘magnitude’ that is under investigation in this paper is one that abstracts from the more general concerns over the distance between an object and the observer. In the Early Modern period, questions were raised about whether this distance, and consequently the magnitude of objects insofar as it varies with this distance, is an immediate object of perception. Hume was not particularly concerned with this issue, however. His interest centered upon the more fundamental question of how it is that we perceive objects as having any magnitude at all, and with what leads us to perceive one object as having a greater magnitude than another. These are the questions that constitute the subject of this investigation.

3 Broad, CD. ‘Hume's Theory of Space,’ Proceedings of the British Academy 47 (1961) 161-76;Google Scholar Falkenstein, L. ‘Hume on Manners of Dispositions and the Ideas of Space and Time,’ Archiv fur Geschichte der Philosophie 79 (1997) 179–201;CrossRefGoogle Scholar Jacquette, D. ‘Hume on Infinite Divisibility and Sensible Extensionless Indivisibles,’ Journal of the History of Philosophy 34 (1996) 61–78CrossRefGoogle Scholar

4 Waxman, W. ‘The Psychologistic Foundation of Hume's Critique of Mathematical Philosophy,’ Hume Studies 22 (1996) 123-67.CrossRefGoogle Scholar This view of Hume's theory of extended visible figure has not gathered as much support as the opposite (summation of indivisibles) approach.

5 Some commentators have attempted to make sense of Hume's theory of space (and time) by assigning a fundamental role to these higher mental faculties (imagination and/or distinctions of reason) in the very perception of the impressions and ideas of extension. For example, David Fate Norton concludes that ‘the particular impressions [of extension] are by the imagination transformed into a “compound impression, which represents extension, ” or the abstract idea of space itself ('An Introduction to Hume's Thought,’ in The Cambridge Companion to Hume, Norton, David Fate ed. (Cambridge: University of Cambridge Press 1993), 8;CrossRefGoogle Scholar author's emphasis). Yet this suggestion, that imagination actively constructs the compound impressions of extension, is in no way substantiated by the texts. A complex impression of extension, as a species of sensible impression, ‘arises in the soul originally, from unknown causes [presumably external causes]’ (T 1.1.2.1), with no known role assigned to any higher mental faculty. Hume does allow imagination to play a crucial role in the formulation of the ‘relations’ classified under space and time (e.g., ‘distance,’ ‘contiguous,’ ‘above,’ ‘below,’ etc.; T 1.1.3-5), but these ideas are more properly classified under the mind's more active capacity to construct abstract ideas (since it involves the ‘comparing of objects’; T 1.1.5.1), and thus it fails to account for the ideas of extension copied directly from our complex impressions of extension (as will be discussed below). Since it not the subject of this essay to explore all of the postulated reconstructions of Hume's theory of space, the reader would profit from exploring Falkenstein's analysis of the competing approaches (179-201).

6 A number of commentators have striven to interpret Hume's theory of space (and time) as a phenomenalist rendition of Leibniz's relationalist theory, the latter being (possibly) an ontological theory: see, Mijuskovic, B. ‘Hume on Space (and Time),’ Journal of the History of Philosophy 6 (1977) 387-94.Google Scholar It is unclear if Hume was, in fact, inspired by Leibniz, but a much more direct and obvious source for his relationalism/nominalism probably stems from the similar views of Berkeley, especially given the influence that Berkeley's theory of perception and nominalist approach to conceptual abstraction is known to have had on Hume's Overall philosophy. See Berkeley, G. An Essay Towards a New Theory of Vision, in The Works of George Berkeley, Bishop of Cloyne, vol. 1,Google Scholar A.A. Luce and T.E. Jessop, eds. (Edinburgh: Thomas Nelson 1948-1957). In fact, Berkeley describes (tangible) extension, in Leibnizian fashion, as ‘being made up of several distinct co-existent parts’ (sec. 145; he then makes the same claim for visible extension).

7 See Bayle, P. ‘Zeno of Elea,’ in The Dictionary Historical and Critical of Mr. Peter Bayle, 2nd ed., Maizeaux, P. Des ed. (New York: Garland Publishing 1984 [1738]), 605-19;Google Scholar Berkeley, G. A Treatise Concerning the Principles of Human Knowledge in ibid., vol. 2, sec. 124.Google Scholar The literature on Hume's rejection of infinite divisibility is vast, and largely accounts for the recent upsurge in interest in Hume's theory of space and time. For earlier treatments of this topic (in addition to Broad), see, for example Flew, A. 'Infinite Divisibility in Hume's Treatise,’ in Hume: A Re-evaluation, Livingston, D.W. and King, J.T. eds. (New York: Fordham University Press 1976) 257-69;Google Scholar Smith, N. Kemp The Philosophy of David Hume (London: Macmillan 1941), 287.CrossRefGoogle Scholar For a few recent, and important, surveys of these issues (in addition to those cited above), see Frasca-Spada, M. ‘Some Features of Hume's Conception of Space,’ Studies in History and Philosophy of Science 21 (1990) 371–411;CrossRefGoogle Scholar Frankin, J. ‘Achievements and Fallacies in Hume's Account of Infinite Divisibility,’ Hume Studies 20 (1994) 85–101;Google Scholar Baxter, D. 'Hume on Infinite Divisibility,’ History of Philosophy Quarterly 5 (1988) 133-40;Google Scholar and Holden, T. ‘Infinite Divisibility and Actual Parts in Hume's Treatise,’ Hume Studies 28 (2002) 3–25.CrossRefGoogle Scholar It should be noted at this point that the debate on infinite divisibility of extension does not have a serious impact on the conclusions reached in this essay.

8 A fairly clear Statement of the composite hypothesis can be found in Berkeley, as well as the appeal to indivisible minimals of visible and tangible figure: ‘Each of these magnitudes [visible and tangible] are greater or lesser, according as they contain in them more or fewer points, they being made up of points or minimums’ (New Theory of Vision), sec. 54. However, like Hume (as will be argued below), Berkeley does not discuss how his individual minimals can combine to provide a magnitude for the whole extended figure.

9 Owen, D. Hume's Reason (Oxford: Oxford University Press 1999), 84Google Scholar

10 Owen has provided a careful analysis of Hume's use of the faculty of imagination, and while judgments of demonstrative and probable reasoning are at times associated with imagination, intuition does not appear to fall under this category (see T 1.4.7.7, yet, in other contexts, Hume strives to separate demonstrative and probable reasoning from the imagination, especially T 1.3.9.19, fn.). Owen's overall judgment of this issue would also seem to support our conclusion, since Hume's explanations of imagination and intuition are kept separate: ‘I take ail this to be further evidence of Hume's lack of commitment to faculty talk: the important point is not which faculty an activity occurs in, but the general principles which explain that activity’ (66).

11 If one returns to the distinction between impressions and ideas, then there exists an intriguing possibility for resurrecting the composite hypothesis in the wake of the problems just discussed. Bearing in mind that Hume specifically declares that a complex impression can contain information not copied in our corresponding ideas (T 1.1.1.4), one can thus claim that our impressions of spatial magnitude uphold the composite hypothesis (such that magnitude is directly correlated to the summation of the aggregate points), but that our ideas of magnitude copied from those impressions do not. That is, somehow the composite hypothesis, or our mind's ability to recall using it, is lost in the act of copying the information embodied in the impressions to the corresponding ideas. Consequently, the lost information would explain why the mind cannot compute the aggregate points needed to ascertain the magnitude of the ideas of sensible figure. Despite the ingenuity of this stratagem for rescuing the composite hypothesis, it still fails to account for the mind's actual ability to determine the magnitude of spatial ideas, which apparently are grasped in an intuitive fashion.

12 For a nice discussion of these foundational debates in mathematics, especially as they pertain to the Early Modem empiricist tradition, see Jesseph, D.M. Berkeley's Philosophy of Mathematics (Chicago: University of Chicago Press 1993), ch. 3.CrossRefGoogle Scholar It should be noted, moreover, that Hume cited both Barrow and Nicolas de Malezieu in presenting his theory of space and geometry, although how well acquainted he was with their work is uncertain. See, Frasca-Spada, M. (ibid.) on the historical background to these issues, as well as in Space and the Self in Hume's Treatise (Cambridge: Cambridge University Press 1998).Google Scholar

13 In this context, the trend among recent scholars to categorize Hume's approach to geometry as ‘synthetic a priori,’ as opposed to ‘analytic a priori,’ becomes all the more clear. Hume does not regard all geometric truths as analytic, as is evident in both the Enquiry (An Enquiry Concerning Human Understanding, Selby-Bigge, L.A. ed. [Oxford: Clarendon Press 1973], 163Google Scholar) and the Treatise. In the latter work, with respect to the claim that ‘a right line is the shortest distance between two points,’ he comments (in Kantian fashion) that ‘this is more properly the discovery of one of the properties of a right line, than just a definition of it,’ which suggests a synthetic reading (T 1.2.4.26). Moreover, although all ideas stem from sense experience, it is nevertheless the case that, once we obtain various sense ideas, our mind is able to ascertain invariant and irrefutably true relationships among these ideas. Algebra and arithmetic (and some geometry, as will be discussed below) fall within this classification, hence these certain truths could be classified as ‘a priori.’ Of course, this type of a priori is quite different from, say, the Cartesian notion (which is prior to, and independent of, sense perception). Nevertheless, Hume's claim of a ‘discovery’ seems to move in a Kantian direction, for once we obtain the knowledge of the invariant (arithmetical/geometric) relationships among certain sense ideas (such as ‘the shortest distance between two points is a straight line’), that knowledge will be true of all of our possible, future sense experience (pertaining to points and lines, etc.) — and this is at least one possible way to interpret Kant's understanding of the a priori. The synthetic a priori reading of Hume is convincingly argued for by Coleman, D.P. ‘Is mathematics for Hume Synthetic a Priori?’ Southwestern Journal of Philosophy 10 (1979) 113-26,CrossRefGoogle Scholar and Newman, R. ‘Hume on Space and Geometry,’ Hume Studies 7 (1981) 1–31.Google Scholar

14 If this rendering of the various discussions and arguments in the Treatise is at all correct (and it is uncertain given the paltry references to mathematics), then it leaves a somewhat ironic ambiguity at the heart of Hume's theory of the foundations of mathematics: first, since the indivisible sensible points provide the nominalist basis for our concepts of number, arithmetic cannot thus provide the foundations for geometry (since arithmetic is thus no more basic than geometry); but, second, since the indivisibles are extensionless (i.e., have no shape or surface, as argued above), they cannot really be classified as more geometric in nature than arithmetic, and thus geometry cannot be regarded as the indisputable grounds of arithmetic either. Accordingly, the peculiar attributes of Hume's indivisible minimal points would seem to elude the entire arithmetic-geometry priority dispute!

15 I would like to thank members of the Mid-Atlantic Seminar in Early Modern Philosophy (held at Johns Hopkins University, May 2003) for their helpful comments on an earlier version of this paper presented at the seminar. I would especially like to thank Lorne Falkenstein for his insightful comments and generous help in the development of this project.