Hume on the Objects of Mathematics

Abstract

In this essay, I argue that Hume’s theory of Quantitative and Numerical Philosophical Relations can be interpreted in a way which allows mathematical knowledge to be about a body of objective and necessary truths, while preserving Hume’s nominalism and the basic principles of his theory of ideas. Attempts are made to clear up a number of obscure points about Hume’s claims concerning the abstract sciences of Arithmetic and Algebra by means of re-examining what he says and what he could comfortably have said about relations of qualitative resemblance.

Notes

1. This essay is a revised version of a paper presented at the 23^rd conference of the Hume Society in Park City, Utah, USA, in July of 1995. Thanks are due to Wade Robison and Tom Lennon for perceptive comments on the earlier version.Charles Echelbarger, “Hume on Deduction” Philosophy Research Archives 13 (1988): 351 – 365.

2. David Hume, A Treatise of Human Nature, ed. Peter. H. Nidditch and L. A. Selby-Bigge (Oxford: Oxford University Press, 1978), 79; hereinafter, Treatise.

3. An English translation of chapter 1, book. 2, vol. 1 of Brentano’s Psychologie Vom Empirischen Standpunkt may be found in Realism and the Background of Phenomenology, ed. and trans. Roderick Chisholm (Glencoe, IL: Ridgeview, 1960).

4. See Alexius Meinong on the “Theory of Objects,” in Chisholm, Realism and the Background of Phenomenology.

5. I have addressed this issue to some extent in “Hume on Deduction.”

6. In fact, Hume seems to have said precisely this on page 20 of the Treatise, where he says “to form an idea of an object and to form an idea simply is the same thing. The reference of an idea to an object being an extraneous denomination of which in itself it bears no mark or character.”

7. Ockham says, “It does not follow that if Socrates and Plato agree more than Socrates and the Donkey, there is some one thing with respect to which they agree more. But it is sufficient they agree more of and by themselves.” See Ockham’s Theory of Terms: Part I of the Summa Logicae, trans. Michael Loux (Notre Dame, MD: University of Notre Dame Press, 1974).

8. On universally quantified propositions, as Hume might have conceived them, see my, “Hume on Deduction.”

9. To take a highly relevant modern example, Locke in An Essay Concerning Human Understanding (1706), ed. Alexander C. Fraser (Mineola: Dover, 1959) said the objects of mathematics were “simple modes,” i.e. variations or combinations of the same simple idea without the mixture of any other, as a dozen or a score, which are nothing but the Ideas of so many distinct unites added together” (II.xii.5). Moreover, Locke said modes are “combinations not looked upon to be characteristical Marks of any real Beings that have a steady existence, but scattered and independent Ideas put together by the Mind” (II. xxii.1, 2). It is because modes are “voluntary collections” of ideas having “no particular foundation in Nature” that there is a coincidence between the real essence of a mode and its nominal essence. Locke says, “a figure including a space between three lines, is the real as well as nominal essence of a Triangle, it being not only the abstract idea to which the general Name is annexed but the very Essentia, or Being of the thing itself: that foundation from which all its properties flow” (III.iii.18).Locke’s and Hume’s views of the nature of mathematical objects seem to be much like that of Ockham. In Summa Logicae, Ockham maintained that ‘quantity’ is not predicable per se of any term signifying things outside the mind. Rather, it is only predicable of quantitative terms. But quantitative terms, Ockham says, only signify individual substances and their qualities through connoting divisions into parts (“intentio philosophi fuit assignare differentiam nominum et predicabilium intentionum quae no predicatur nisi de aliquo habente diversas partes, vel de rebus diversis et distinctis coniunctim sumptis”). Ockham classified all terms as written, spoken, or mental, assigning to mental terms the duty of signifying things by their nature rather than by convention. See William Ockham, Expositio Aurea (1496), ed. Marcus de Benevento, facsimile reprint (Ridgewood, NJ: Gregg, 1964).

10. Bertrand Russell, “Mathematical Logic as Based on the Theory of Types” (1908), reprinted in Logic and Knowledge, ed. Robert C. Marsh (London: Macmillan, 1964).

11. I suspect that Hume thought of the psychology of non-elementary demonstrative mathematical reasoning as a matter of mentally constructing numbers by means of various heuristic procedures, depending on the sort of problem requiring solution. These psychological processes may or may not resemble the tidy stepwise sequences exhibited in mathematics textbooks. The heuristic would be a rule for the construction of a proof-sequence. This suspicion of mine is based on what Hume says about the way that our abstract ideas enable us to call to mind ideas of individuals falling under them. At Treatise pages 22–23, he alludes to the fact that we form “customs” which are “analogous” to those involved in forming abstract ideas when we deal in quick and abbreviated fashion with large bodies of information or when we handle large numbers. We seldom or never form adequate ideas of large bodies of information or large numbers. In the case of some proof-heuristics, one may approach solutions to complex mathematical problems by breaking the problem up into parts, doing one part at a time, according to a certain order, until the solution is obtained. One thinks of the PEMDAS heuristic in elementary algebra, and the humorous associated memory devices often suggested to students as an aid to remembering the acronym. Perhaps there are natural mental strategies analogous to this artificial one.