Abstract
Mathematics for Hume is the exemplary field of demonstrative knowledge. Ideally, this knowledge is a priori as it arises only from the comparison of ideas without any further empirical input; it is certain because demonstration consist of steps that are intuitively evident and infallible; and it is also necessary because the possibility of its falsity is inconceivable as it would imply a contradiction. But this is only the ideal, because demonstrative sciences are human enterprises and as such they are just as fallible as their human practitioners. According to the reading suggested here, Hume develops a radical sceptical challenge for mathematics, and thereby he undermines the knowledge claims associated with demonstrative reasoning. But Hume does not stop there: he also offers resources for a sceptical solution to this challenge, one that appeals crucially to social practices, and sketches the social genealogy of a community-wide mathematical certainty. While explaining this process, he relies on the conceptual resources of his faculty psychology that helps him to distinguish between the metaphysics and practices of mathematical knowledge. His account explains why we have reasons to be dubious about our reasoning capacities, and also how human nature and sociability offers some remedy from these epistemic adversities.
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Notes
I discuss some aspects of this process in relation to Hume’s philosophy, and Hume’s explication of epistemic standards in Demeter (2016).
Owen (1999, p. 107), rightly I think, points out that Hume in the Treatise limits demonstrative knowledge to arithmetic and algebra. Recently Slavov (2017), rightly again I think, points out in the context of the Enquiry, syllogisms and definitional truths are also listed in this class. The present paper focuses exclusively on the Treatise.
This summary is consistent with the common wisdom that for Hume mathematical truths are analytic. There are others (most notably Coleman 1979) who maintain that mathematics for Hume is synthetic a priori.
Compare the following passage: “we must distinguish exactly betwixt the phaenomenon itself, and the causes, which I shall assign for it; and must not imagine from any uncertainty in the latter, that the former is also uncertain. The phaenomenon may be real, tho’ my explication be chimerical. The falsehood of the one is no consequence of that of the other” (T 1.2.5.19).
For the significance of analogy and comparison in Hume’s method see e.g. T I.10, 3.2.3.4n71 Enquiry 4.12, 8.13.
The term “inexact idea” might require some clarification because of Hume’s dictum that all perceptions are “determin’d in its degrees of both quantity and quality” (1.1.7.4). Several ways can be suggested as to how ideas could be inexact while sticking to this dictum. (1) Ideas can be inexact copies of impressions without violating the Copy Principle. Even if my impressions of Edinburgh are very detailed, my idea of Edinburgh may be inexact due to the liberty of the imagination to rearrange ideas. (2) Inexact ideas can be abstract ideas with vague borders for their “revival set” (for the term see Garrett (1997), p. 104]. For example, the abstract ideas “straight” and “curved” require a general “appearance” in spatial disposition, and it might be indeterminate whether some arrangements of individually unextended minima are “straight” or “curved.” (3) an idea might be inexact because, although we retain the same term and treat it as identical over time, it in fact varies somewhat in size and/or disposition of parts without our taking note of it. Such fluctuations would, of course, be a potential hindrance to precise reasoning. A more detailed discussion would exceed the scope of this paper. I am grateful to Don Garrett for helpfully pointing this out to me.
For example, the idea of a “necessary connection”, and our idea of “causation” along with it, does not derive purely from a corresponding impression. There is a contribution on the mind’s part, i.e. custom, that plays a crucial role in producing this idea. For an interpretation along these lines cf. Buckle (2002, esp. pp. 213–214).
For a useful background discussion see Badici (2011, pp. 462–463).
This may be one way to explain why the Enquiry (4.1, 7.1–2, 12.20) lists all of them as demonstrative.
For this reason the placement of Hume’s sceptical arguments should not pose a problem to the interpretation suggested here. It is true: Hume’s appeal to the significance of the approbation of epistemic peers occurs before one of Hume’s main arguments (T 1.4.1.3) to the conclusion that knowledge degenerates into probability and also the subsequent argument that probability reduces to nothing, and so they seem to already take into account the social factors that pointed out here. But given that Hume’s solution is a sceptical one, this does not undermine the present interpretation: even if the resources of a sceptical solution are at hand, the sceptical problem is still a legitimate problem. This is what Hume does elsewhere too. For example, his sceptical worries about causation in the Enquiry are preceded by the elements of its skeptical solution: the chapter on induction already contains what he needs for solving his sceptical problem about causation.
The distinction between straight and sceptical solutions to sceptical problems comes from Kripke (1982, p. 66).
Up to this point my reading is congenial to the one offered in Meeker (2007).
References
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Acknowledgements
I am indebted to Krisztián Pete for useful discussions leading to this paper. Its first version was presented at the Oxford Hume Forum in April 2017. I am grateful to Dan O’Brien and Lorenzo Greco for the invitation, and to the audience, especially Don Garrett, Peter Millican and Eric Schliesser for helpful comments and discussion. My work has been supported by the MTA BTK “Morals and Science” Lendület Project.
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Demeter, T. Hume on the social construction of mathematical knowledge. Synthese 196, 3615–3631 (2019). https://doi.org/10.1007/s11229-017-1655-x
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DOI: https://doi.org/10.1007/s11229-017-1655-x