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. Now ’tis evident, that this gradual encrease of assurance is nothing but the addition of new probabilities, and is deriv’d from the constant union of causes and effects, according to past experience and observation.--Hume Treatise, 184.108.40.206
I have been revisiting a terrific paper by Kevin Meeker (recall this post). Like a lot of Kevin's other work the paper argues for interpreting Hume as an old-fashioned skeptic (even about purported mathematical knowledge). Because of it I re-read Hume's Treatise 1.4.1 ("Of scepticism with regard to reason"). In historical 18th century context the significance of 1.4.1 is two-fold:
1. Together with much of Treatise 1.2.4, Treatise 220.127.116.11 reinforces Hume's attack on the epistemic status of applying mathematical concepts to nature: "all [purported] knowledge degenerates into probability." (18.104.22.168) In particular, the epistemic status of mathematical, natural philosophy "becomes at last of the same nature with that evidence, which we employ in common life," (22.214.171.124) That is, Hume is here part of a much larger trend of what I call, 18th century "anti-mathematics." This is much indebted to Spinoza's "Letter on the Infinite" and even if we allow that Hume had not read Spinoza (which goes against a lot of circumstantial evidence), Hume almost certainly would have been familiar with species of such anti-mathematics either in Berkeley or in Mandeville (recall here; also here and here).
2.A: I had never noticed before that Hume's "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" commits Hume to (use anachronistic jargon) a kind of epistemic externalism, even social epistemology (note that "still more"). This feature here goes against the general (and probably correct) tendency to treat Hume as a kind of epistemic internalist.But when it comes to purported knowledge claims that belong to the sciences, Hume appears to me to be an externalist. 2.B: In fact, "reason" in this section, seems to mean something like "the fruits of socially disciplined enquiry."
As a final aside, at least one author may have noticed this second (A) feature about Hume. Adam Smith treats the history of mathematical astronomy (and self-consciously by implication, the whole history of philosophy) in such social epistemological terms, namely as successive systems that "have successively been adopted by the learned and ingenious; and, without regarding their absurdity or probability, their agreement or inconsistency with truth and reality, let us consider them only in that particular point of view which belongs to our subject." ("History of Astronomy," 2.12) There is a further after-life to this aside because as I learned from Laura Snyder, Whewell (who coined "scientist" and did much to create science as a distinctive enterprise) was a careful reader of this essay by Smith, but about these matters some other time more.