Hume Society Meetings, Toronto, Canada, July 2005. Hume and Descartes

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Hume Society Meetings, Toronto, Canada, July 2005.

Hume and Descartes1 on skepticism with regard to demonstrative reasoning

Graciela De Pierris

At T 1.3.1 and T 1.3.2, Hume distinguishes between two kinds of philosophical relations and calls them, respectively, “Knowledge” and “Probability.” This distinction follows Locke’s terminology and Locke’s model of a phenomenological apprehension of image-like items before the mind. Nonetheless, Hume provides an original characterization of each of these two types of cognition. The dichotomy goes back at least to Descartes, who draws it against the background of an intended intellectualist conception of the apprehension of ideas. In Rule Two of the Rules for the Direction of the Mind (Regulae), Descartes claims that all “Scientia” (systematic knowledge based on indubitable foundations) is evident and certain. Descartes here contrasts Scientia with probable cognition, and regards arithmetic and geometry as the paradigm of the former.2 In the First Meditation, after entertaining the dream and the powerful deceiving God hypotheses, Descartes adopts the stringent policy of regarding all his habitual opinions as if they were false. Descartes justifies this policy by pointing out that although most of his previous beliefs are in need of a solid foundation, they are highly probable. This is one central reason why they are so powerful, so that, in particular, “hyperbolic” skeptical hypotheses are needed to counteract them: “I shall never get out of the habit of confidently assenting to these opinions, so long as I suppose them to be what in fact they are, namely highly probable opinions -- opinions which, despite the fact that they are in a sense doubtful, as has just been shown, it is still much more reasonable to believe than to deny. In view of this, I think it will be a good plan to turn my will in completely the opposite direction and deceive myself, by pretending for a time that these former opinions are utterly false and imaginary.”3

Hume develops his own distinction between knowledge and probability as two different ways in which we reflectively compare ideas, impressions or objects, and thereby establish that two kinds of philosophical relations obtain. By means of the first method of comparing items we can attain the “full certainty” (T of knowledge, precisely because this method relies solely on the direct inspection of features intrinsically contained in the items related.4 Thus, at T, in the context of arguing that the maxim “whatever begins to exist, must have a cause of existence” is not certain, Hume writes: “All certainty arises from the comparison of ideas, and from the discovery of such relations as are unalterable, so long as the ideas continue the same.” The first kind of philosophical relations are therefore certain and a priori. The intrinsic content of items so related might be presented in different events of apprehension, for example, in the sequence of steps of a mathematical proof. Yet, if the contents in the sequence are apprehended as qualitatively identical, or as one contained in the other, the relations among the contents are certain. In particular, if the intrinsic content of items are phenomenologically presented as resembling or contrary to each other, or as possessing different degrees of a quality, or as bearing a certain proportion of quantity or number, then these contents have these relations so long as they are apprehended as stable. Moreover, I can determine solely on the basis of an intrinsic content before my mind, before experiencing another particular content, which intrinsic features a new experience must have in order, for example, to resemble the one now before the mind. Particular phenomenological intrinsic features of the relata make the first kind of philosophical relations certain and a priori -- where the certainty and apriority in question are based on phenomenologically apprehended qualitative identity or containment.

By contrast, in order to determine whether the second kind of philosophical relations obtain, we are required to go beyond the intrinsic features of contents that are or have been apprehended. There are three relations of the second kind. One is the numerical identity (including temporal continuity) of objects presumed to be independent of the mental contents that refer to them. The other two are situations in time and place, and causation. Hume discusses these relations at T 1.3.2, under the title “Of probability; and of the idea of cause and effect.” Since we cannot determine whether the second kind of philosophical relations obtain solely by relying on the qualitative identity or containment of intrinsic features of the relata, these relations are extrinsic, contingent, and a posteriori. They depend on features of the relata that make the relata separable from, that is, independent of each other.

Hume marks a further distinction within each of the two kinds of philosophical relations. Within the first kind there is either intuitive or demonstrative knowledge. We can assess that resemblance, contrariety, and degrees in quality obtain by means of a single act of comparison. Hume writes (T “Three of these relations are discoverable at first sight, and fall more properly under the province of intuition than demonstration.” Hume continues in the same paragraph by pointing out that qualitative resemblance among contents “at first strike[s] the eye, or rather the mind; and seldom requires a second examination.” The comparison of ideas in simple cases of the fourth of the a priori relations -- proportions of quantity or number -- can likewise amount to intuitive knowledge (T “We might proceed, after the same manner, in fixing the proportions of quantity or number, and might at one view observe a superiority or inferiority betwixt any numbers, or figures; especially where the difference is very great and remarkable.” Thus, in the characterization of intuitive knowledge, Hume uses, besides the above phrases “at first strike the [mind],” “seldom requires a second examination,” “at one view,” other similar phrases such as “single consideration” and “comprehended in an instant” (T Most of mathematics, however, involves complex cases of relations of proportion, which require the use of inferential reasoning. Nevertheless, mathematical inferences are based solely on the intrinsic content of ideas, and can thus amount to demonstrative necessary knowledge and provide full certainty.

In the discussion of probable relations, Hume again refers to single acts of apprehension as opposed to inferential reasoning. The contrast is between, on the one hand, what Hume here calls “perception,” which consists in the apprehension by a single act of the mind of immediately given impressions and their immediately given extrinsic relations (such as the relations of time and place) and, on the other hand, reasoning based on sensory impressions and memories of them. Hume writes (T “When both the objects [which we compare] are present to the senses along with the relation, we call this perception rather than reasoning.” The causal inference to the unobserved is the most central of probable inferences and always provides less than full certainty.

Hume’s distinction between intuitive and demonstrative (a priori) knowledge, like the distinction between knowledge and probability, has an illustrious history. The most immediate antecedent is Locke’s distinction in An Essay concerning Human Understanding, Book IV, where Locke uses the same terminology of “intuition” and “demonstration” (although Locke does not use the same examples of intuitive knowledge as Hume uses later). Locke’s division has in turn immediate Cartesian roots. In the Regulae, Descartes gives a prominent place to the contrast between “mental intuition” (immediate mental apprehension) and deduction, starting with Rule Three.5 A more direct influence on Locke might be Descartes’s Fifth Meditation and the Second and Fourth Replies to the charge of circularity in the proof of the trustworthiness of clear and distinct ideas. In these later writings Descartes also suggests, as in the Regulae, that at least some cases of apprehension of clear and distinct ideas by a single act of an attentive mind (the Regulae’s mental intuition) is certain, whereas the apprehension of long demonstrations is not. (I return to this Cartesian thesis below.) However, Locke gives what I call a “sensible” version of Descartes’s intellectualist conception of a priori knowledge, since Locke recasts it in terms of the apprehension of particulars in the style of sensory perception and the imagination. Hume adopts and radicalizes this Lockean revision.6

The central target of Hume’s T 1.4.1, entitled “Of scepticism with regard to reason,” is, in my view, any “chain of reasoning,”7 which goes beyond a simple apprehension by a single act of the mind. Hume is not primarily concerned with either intuitive knowledge or with what in T 1.3.2 he calls “perception.” The first sentence of T 1.4.1 reveals that Hume’s key concern is inferential reasoning (T “In all demonstrative sciences the rules are certain and infallible.” In addition, Hume illustrates his initial points with the case of “discoveries” made by an “Algebraist” or a “Mathematician” by means of proofs, and also refers to “long numerations.” These illustrations also disclose that Hume follows Descartes in taking mathematics as the paradigm of a priori rational knowledge.8 This section offers first a radical skeptical argument concerning demonstrative knowledge and, on its basis, proceeds with a second radical skeptical argument concerning probable inference, thereby casting radical doubts regarding any idea whatsoever which is based on the employment of inferential reasoning. Thus, the word “reason” in the title refers both to demonstrative and probable inferential reasoning.

Both skeptical arguments, concerning demonstrative and probable reasoning respectively, proceed by taking into account the fallibility of our faculties. The appeal to the limitations and deceptive nature of human faculties in order to generate radical skeptical doubts also has Cartesian roots. Ordinary doubts, in contradistinction to radically skeptical doubts, always leave room for correction and improvement. For, in common life and science, we assume that we can continually approximate the exactitude and certainty of ideal rational inferences. In these same contexts, we take such inferences objectively to confer certainty on their conclusions (in the case of deductive inference), or to support them to a certain degree (in the case of probable inference). Radical skeptical doubts, by contrast, do not leave room for the idea of a positive progress towards accurately apprehending objective relationships of dependence between premises and conclusions. In T 1.4.1 Hume, like Descartes, raises radical skeptical rather than ordinary doubts about the employment of our faculties. Thus, despite Hume’s rejection of Descartes’s ideal of a pure intellect and of Descartes’s conception of belief as an act of the will,9 Descartes’s skeptical doubts about our faculties are a likely source of Hume’s approach. It might appear, therefore, that Hume is not proposing an original form of radical skepticism at T 1.4.1. However, as we shall see, Hume here uses the Cartesian legacy in a quite original way. Let us assess Hume’s contribution by considering first Descartes’s own radically skeptical argument regarding a priori demonstration.

In the First Meditation, Descartes raises the possibility that even the most trustworthy of our faculties might deceive us all the time. After the dream argument, which concerns our knowledge by means of the senses of the existence and character of corporeal nature, but before the Deceiving God hypothesis, Descartes writes: “So a reasonable conclusion from this might be that physics, astronomy, medicine, and all other disciplines which depend on the study of composite things, are doubtful; while arithmetic, geometry and other subjects of this kind, which deal only with the simplest and most general things, regardless of whether they really exist in nature or not, contain something certain and indubitable. For whether I am awake or sleep, two and three added together are five, and a square has no more than four sides. It seems impossible that such transparent truths should incur any suspicion of being false.”10 However, in the following paragraph, Descartes introduces the hypothesis of a deceiving God, which he later recasts as the hypothesis of a malicious demon. This hypothesis raises the possibility that we might be constantly deceived in the exercise of our faculties, even in acts of apprehension of the simplest and most transparent ideas of mathematics. Thus, Descartes writes: “What is more, since I sometimes believe that others go astray in cases where they think they have the most perfect knowledge, may I not similarly go wrong every time I add two and three or count the sides of a square, or in some even simpler matter, if that is imaginable?”11 At this point, therefore, in terms of Descartes’s distinction in Rule Two of the Regulae between Scientia and probable cognition, all cognition -- even the simplest such as adding two and three -- amounts only to highly probable opinion. For, there is a possibility of systematic deception in the exercise of even our most reliable faculty of the pure intellect. Neither claims about the external world nor mathematical claims have been shown to constitute Scientia, since no such claims have been shown to be indubitable. Hume also first reduces certainty, as we shall see, to probable opinion, but without using a device (like Descartes’s powerful deceiver) external to our own faculties.

In the Fifth Meditation, and in addressing the circularity objection in the Second and Fourth Replies,12 Descartes appears to correct the First Meditation’s suggestion that the skeptical doubt about purely intellectual ideas includes clear and distinct ideas apprehended in a single act of the mind. In these subsequent texts, Descartes directs the reader’s attention away from such a single act of apprehension and claims instead that the proof of the existence of a benevolent God is meant to secure the certainty of long demonstrative inferences or of demonstrative inferences to which we are not now attending. In the Regulae, Descartes had already raised the issue of the need to guarantee the certainty of long deductions. However, in that earlier work Descartes offers only ordinary doubts about demonstrative reasoning. There Descartes attributes the source of error in long deductions to memory, but he finds a remedy in a kind of training that, according to the Regulae, can improve our apprehension of long proofs. Rule Eleven, for example, shows that deduction can become certain when we can reduce its apprehension to a single intuitive act of the mind: “If, after intuiting a number of simple propositions, we deduce something else from them, it is useful to run through them in a continuous and completely uninterrupted train of thought, to reflect on their relations to one another, and to form a distinct and, as far as possible, simultaneous conception of several of them.”13 In the explanation of this Rule, Descartes says that the rule is needed in order to overcome the inherent fallibility of memory. Yet, to assume that running through a proof in a continuous and uninterrupted train of thought will solve the problem of the fallibility of memory is to grant that we cannot be deceived by our faculties all the time -- that we can improve their performance by applying tests, undergoing training, and so on -- contrary to the radical skeptical doubt of the Meditations.14 The Fifth Meditation and the Replies, unlike the Regulae, provide a guarantee for the certainty of long demonstrations which is external to the exercise of the faculties themselves. The doubt inevitably brought about by going beyond self-evident single acts of the mind (clear and distinct ideas) can only be eliminated when we attend to the proof that God exists and is not a deceiver. By appealing to this external device, Descartes reveals that he is still concerned, as in the First Meditation, with the possibility of being deceived all the time in the exercise of our faculties, even though now the doubt seems to be restricted to demonstrative inferences.

In addition to posing radical skeptical doubts about demonstrative reasoning, Descartes can be taken to be a source for Hume’s T 1.4.1 in another respect. Descartes invites the consideration of psychological, subjective, and empirical (causal) aspects of the human knower, which undermine the certainty of the pure intellect, and thus of a priori demonstrative knowledge. For Descartes calls attention to the fact that we are required to employ a faculty whose exercise, whether part of the intellect or not, can be fallible (even for the most attentive and well trained mind) precisely because we must go beyond a present mental intuition. The need to rely on the fallible performances of memory arises from the fact that an inference takes place in successive separate moments of time, through separate acts of apprehension. In calling attention to the time-bound aspect of inferences, Descartes moves away from focusing on the objectivity and non-empirical character of demonstrative proofs and the rules governing them -- contrary to what Leibniz, for example, will later very forcefully emphasize. In spite of this invitation to psychologism regarding demonstrative inference, however, Descartes also argues in the opposite direction. In the Fifth Meditation, for example, Descartes claims that we can know that there are immutable and eternal mathematical natures, which are independent of the human mind, precisely because we can demonstrate that certain properties necessarily belong to mathematical essences, whether we wish it or not.

Let us now consider the details of Hume’s first skeptical argument. In talking about fallible “faculties” in the plural throughout T 1.4.1, Hume remains neutral whether his argument about demonstrative reasoning concerns the pure intellect versus other faculties such as memory or the imagination. Hume confronts the reader with paradoxes about reasoning conceived either as a Cartesian pure intellect which apprehends eternal an immutable truths independent of the mind, or as a Lockean and Humean series of phenomenological acts of apprehension of particular items in the style of sensory perception or the imagination. Hume attempts to produce a challenge general enough to call into question the certainty of inference as such, in whatever way we conceive it.

Hume declares that “in all demonstrative sciences the rules are certain and infallible.” This claim does not mean that he endorses a rationalist conception of the objectivity of mathematical rules, according to which rules represent objectively valid relationships of dependence between premises and conclusions. Rather, Hume means that in common life and science -- in particular, in mathematical calculations -- we assume that the method of establishing claims by demonstrative reasoning provides certain knowledge. Hume initially refers to the fact that even in common life and science we can raise ordinary doubts about the certainty of our conclusions, given that we apply the rules only by means of “fallible and uncertain faculties.” The rules are assumed to be certain and infallible, “but when we apply them, our fallible and uncertain faculties are very apt to depart from them, and fall into error” (T No matter how experienced mathematicians (or merchants, or accountants) might be, they realize that they cannot be truly certain in their own calculations, because they are aware that they work with fallible faculties.

In common life and science ordinary reasoners and experts naturally attempt to assess and improve the reliability of the application of inferential rules. Hume observes that the standards for evaluating the application of inferential rules by fallible faculties are in fact those of causal reasoning. In common life and science we take into consideration past experience in estimating the degree of reliability of the inferential abilities of the reasoner. We also subject inferences to several tests, which include repeatedly running over the inferences and seeking intersubjective agreement. The natural and gradual increase in confidence after obtaining positive results (T “. . . is deriv’d from the constant union of causes and effects, according to past experience and observation.” Here, the constant union of causes and effects is the observation of a conjunction of performances of reasoners, on the one hand, and positive test results of the above kind associated with those performances, on the other. If we find a constant or near constant conjunction of past experiences of drawing conclusions and of having them certified by careful examination, then we can in fact increase our confidence in our inferential abilities, and thus in our present or next inferences. The performance of a priori demonstrations, as of any other inference, is thus subject to the control of causal reasoning. This control takes place because we need to evaluate the application of inferential rules by means of fallible human faculties. The event of application of rules is itself a causal process, as it consists in the application of what is taken to be a general rule in order to generate, at a particular time, a particular conclusion from particular premises. This is in part what Hume means when he writes (T “Our reason must be consider’d as a kind of cause, of which truth is the natural effect; but such-a-one as by the irruption of other causes, and by the inconstancy of our mental powers, may frequently be prevented.”

In this way, Hume initially records the ordinary doubts and solutions to these doubts that take expression in common life and science when an “Algebraist”, a mathematician, a merchant, or an “accomptant” evaluates his or someone else’s demonstrative inferences. This initial discussion can be regarded as a causal counterpart of Descartes’s ordinary doubts concerning human performances of generating deductions and Descartes’s non-skeptical prescription of exercising one’s ability to reduce a deduction to a mental intuition in the Regulae. And, like Descartes in the Meditations, Hume then turns these ordinary doubts into a radical skeptical doubt concerning demonstrative knowledge. This occurs when, contrary to the increase in confidence naturally acquired in common life and science, Hume points out that this increase is brought about by a causal reasoning concerning our fallible faculties, therefore, it amounts only to the addition of new probabilities, never to certainty (T “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.” In this way, Hume implicitly raises the possibility, as Descartes does explicitly, that our faculty of reasoning might deceive us all the time. More precisely, Hume dooms to failure the attempts to correct and improve the use of the faculty of reasoning in order to gain the certainty expected of mathematical demonstrations. Hume thereby reaches a radical skeptical doubt regarding the possibility of our ever attaining genuine certain demonstrative knowledge without the device of an all-powerful deceiver.

By contrast, mathematicians or accountants, for example, when engaged in evaluating their own results -- even if they take into consideration the fallibility of their faculties -- would not reach Hume’s skeptical conclusion. In common life and science, increased assurances gained by successful past performances, by careful checking of our proofs, by intersubjective agreement, and so on, point to the possibility of gradually approximating the certainty of demonstrative proofs. Hume reverses the force of such increased assurances: there is no consideration in Hume of a progressive adjustment of the exercise of our faculties to an assumed objective validity of the rules governing the inference. There is no gradual transition from probability to certain knowledge (T “But knowledge and probability are of such contrary and disagreeing natures, that they cannot well run insensibly into each other; and that because they will not divide, but must be either entirely present, or entirely absent.”

Hume is right that knowledge and probability “cannot well run insensibly into each other”, because probable inference, no matter how successful, can never establish the relation of containment of intrinsic features of ideas demanded by demonstrative knowledge. The state of our conviction in knowledge, by definition, must be entire or full certainty, whereas our conviction regarding conclusions of probable reasoning is always less that entire. Certainty is exclusively entire or full conviction; otherwise it is not certainty and not knowledge. Hume’s radical skeptical conclusion to the effect that what we initially took to be demonstrative knowledge degenerates into probable inference is deeply paradoxical. The paradox arises because, as previously characterized by Hume himself, Locke, Descartes, Leibniz, and others, “knowledge and probability are of such contrary and disagreeing natures,” knowledge cannot degenerate into probability -- yet Hume has shown that it does. With this paradox Hume intends to show, I believe, that despite the fact that we commonly regard the rules of mathematics as certain and infallible, such presumed certainty and infallibility signify nothing to us since we are prisoners of our own faculties. The exercise of fallible faculties constitutes the only actual realization of the rules: our inferences must always be tainted by the tools we use in applying the rules of inference, whatever the objective status of the rules. Hume also brings to the fore the unintended consequence of Descartes’s consideration of an inference as a series of successive events of apprehension, each of which takes place at a separate moment of time and is independent from the others. The separation of events of apprehension and the ideas apprehended reduces the certainty of demonstrations to the probability of causal reasoning. No amount of causal reasoning applied in order to improve our mathematical inferences can restore the certainty commonly attributed to demonstrative knowledge.

Thus, according to the first radical skeptical argument, when we acknowledge that we apply inferential rules using intrinsically fallible faculties, we must take the conclusions of our inferences to be at best merely probable opinion. These are the same conclusions that in common life and science we regard as amounting to demonstrative knowledge after we as reasoners undergo the process of increasing our confidence by causal reasoning -- but only because in those contexts we do not regard our faculties as capable of deceiving us all the time, and, more importantly, we do not fully acknowledge the fact that we increase our confidence only by relying on causal reasoning. Hume’s philosophical skeptical doubt about demonstrative knowledge is not the kind of doubt that a mathematician, a merchant, or an accountant, or even Hume himself when not immersed in philosophical skeptical reflection, would raise. For, in common life and science, we do not regard the application of rules of demonstrative reasoning, in the generation of proofs, as causal processes. We do not entertain the possibility that, in what we take to be demonstrative inferences, we might always be stuck with conclusions that merely have some probability or other. Instead, we retain the hope that an ever increasing improvement in the exercise of our reasoning faculty marks the path to full certainty.

Hume’s first skeptical argument at T1.4.1 shows that even the best case scenario of improving the exercise of our faculties in mathematical reasoning could at most approximate probability to 1, as it were. In this best possible state of cognition, we would only succeed in establishing external relations among ideas that still remain separable in terms of their intrinsic contents. We would not attain insight into relationships of containment of intrinsic features of the ideas as required by the definition of demonstrative knowledge. No matter how much we could progressively increase our mathematical reasoning abilities, we would never be entitled to reach the full certainty accorded to demonstrative knowledge, for our reasoning consists always in a causal process. At T Hume has distinguished, within the second kind of philosophical relations, between (inductive) proofs and (mere) probability. Hume here explicitly retains the notion of certain knowledge but points out that there are cases of “probable” reasoning (as they were called in T 1.3.2) in which we can be free from doubt and uncertainty (T “By knowledge, I mean the assurance arising from the comparison of ideas. By proofs, those arguments which are deriv’d from the relation of cause and effect, and which are entirely free from doubt and uncertainty.” However, in inductive proofs we can be free from doubt and uncertainty only to the degree, however high, that one can be in that state of conviction concerning external relations among ideas, impressions, or objects.15 The certainty of inductive proofs based on uniform experience is always open to revision, whereas full certainty based on the containment of ideas is not. In the end Hume retains the distinction between knowledge and inductive proofs only to show us, with the first skeptical argument of T 1.4.1, that the full and entire certainty of demonstrative knowledge is unattainable.

1 All English quotations of Descartes’s writings are from The Philosophical Writings of Descartes, John Cottingham, Robert Stoothoff, Dugald Murdoch, (editors), Volume I and II (Cambridge: Cambridge University Press, 1985), which I abbreviate as “Cottingham et al” followed by the volume and page numbers.

2 Rule Two prescribes: “We should attend only to those objects of which our minds seem capable of having certain and indubitable cognition.” At the beginning of the explanation of this rule, Descartes writes: “All knowledge is certain and evident cognition. Someone who has doubts about many things is no wiser than one who has never given them a thought; indeed, he appears less wise if he has formed a false opinion about any of them. Hence it is better never to study at all than occupy ourselves with objects which are so difficult that we are unable to distinguish what is true from what is false, and are forced to take the doubtful as certain; for in such matters the risk of diminishing our knowledge is greater than our hope of increasing it. So, in accordance with this Rule, we reject all such merely probable cognition and resolve to believe only what is perfectly known and incapable of being doubted.” At the end of the next paragraph, Descartes adds: “Accordingly, if my reckoning is correct, out of all the sciences so far devised, we are restricted to just arithmetic and geometry if we stick to this Rule.” Descartes, Rules for the Direction of the Mind, (composed around 1628), Cottingham et al, Vol. I, pp. 10-11.

3 Descartes, Meditations on First Philosophy, Cottingham et al, Vol. II, p. 15.

4 In my “Causation as a Philosophical Relation in Hume,” Philosophy and Phenomenological Research, Vol. LXIV, No. 3, May 2002, pp. 499-545, I make this thesis the central premise of my interpretation of Hume’s distinction between two kind of philosophical relations.

5 Descartes, Rules for the Direction of the Mind, Cottingham et al, Vol I, pp. 13f.

6 I argue for these theses in my book manuscript, Ideas, Evidence, and Method: Skepticism and Naturalism in Hume’s Epistemology.

7 For Hume’s use of this expression in talking about algebra and arithmetic, see T

8 Descartes takes mathematics to be such a paradigm to the detriment of logic. I discuss the extent of Descartes’s neglect of logic and the consequences for Descartes’s conception of a priori knowledge in my work cited in note Error: Reference source not found above.

9 These are two tenets of Cartesianism the rejection of which might also be guiding Hume’s discussion in T.1.4.1. For example, Hume emerges from the radical skeptical exercise of T 1.4.1 by reasserting his theory of belief, which is opposed not only to Descartes’s view of belief as an act of the will but also to views that make belief dependent on reasoning that is not mixed with impressions and natural propensities.

10 Descartes, Meditations on First Philosophy, Cottingham et al, Vol. II, p. 14.

11 Descartes, Ibid.

12Descartes, Author’s Replies to the Second Set of Objections, “Thirdly” and “Fourthly”, Cottingham et al, Vol. II, pp. 100f; Author’s Replies to the Fourth Set of Objections, op. cit., p. 171.

13 Descartes, Rules for the Direction of the Mind, Cottingham et al, Vol. I, p. 37.

14 An interpretative puzzle concerning Descartes’s conception of memory arises both in the Meditations and the Regulae. For, it is not obvious how Descartes could allow that a faculty of memory which is related to the body through the senses and the corporeal imagination can play a significant role in mathematical proofs. Thus, memory as used in demonstrations might be plausibly taken to be part or share attributes of the pure intellect. If a memory could in principle retrieve (although perhaps not always in a reliable way) the original immersion in an intellectual certain apprehension of clear and distinct ideas, such a memory must somehow be able to apprehend in the same way the same items the intellect apprehends.

15 In my paper “Hume and Locke on Scientific Methodology: The Newtonian Legacy” I argue that Hume’s distinction between (inductive) proof and (mere) probability plays a crucial role in Hume’s scientific methodology, and is Newtonian in character. I first pointed out that Hume’s distinction between proof and (mere) probability is a reflection of Hume’s Newtonianism, and, as such, marks a clear distinction with Locke, in my presentation at a symposium on Hume’s remark: “A wise man, therefore, proportions his belief to the evidence” (Enquiry, Section X), at the Twenty-eighth Annual Hume Society Conference, Victoria, Canada, 2001. I subsequently developed the parallel with Newton and the contrast with Locke in earlier versions of “Hume and Locke on Scientific Methodology” presented at the California Institute of Technology, April 2003; the Central Division Meetings of the American Philosophical Association, April 2004 (where, in particular, I first pointed out that Hume’s distinction between proof and mere probability is precisely modelled on Newton’s distinction between inductive proof and conjectures or hypotheses); and the Thirty-first Annual Hume Society Conference, Tokyo, Japan, August 2004.

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