Abstract
I have argued elsewhere that Hume ’s views on space and geometry in Book One, Part Two of A Treatise of Human Nature are rooted in his radically empiricist model of apprehension and ultimate evidence This epistemological model ultimately relies on phenomenologically given particular sensory images. Diagrams in geometry are always regarded as themselves particular sensory images, thus they cannot be taken as representatives of ideal geometrical objects. Guided by this conception, Hume offers a radically skeptical view of geometry (the study of continuous quantity) according to which it cannot be a fully exact and certain science. Only arithmetic and algebra (the sciences of discrete quantity) attain complete exactitude and certainty.
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Notes
- 1.
- 2.
Hume claims that his arguments against infinite divisibility (including his “experiments” discussed below) are demonstrative (T 1.2.2.6/SBN 31): “I doubt not but it will readily be allow’d by the most obstinate defender of the doctrine of infinite divisibility, that these arguments are difficulties … But here we may observe, that nothing can be more absurd, than this custom of calling a difficulty what pretends to be a demonstration, and endeavouring by that means to elude its force and evidence. ’Tis not in demonstrations as in probabilities, that difficulties can take place, and one argument counter-ballance another, and diminish its authority. A demonstration, if just, admits of no opposite difficulty; and if not just, ’tis a mere sophism, and consequently can never be a difficulty. ’Tis either irresistible, or has no manner of force.” All citations of A Treatise of Human Nature (abbreviated T), are from Hume (2000), and thus include the book, part, section, and paragraph numbers; this is followed by a reference to the corresponding page number in Hume (1978), abbreviated SBN. All citations of An Enquiry concerning Human Understanding (abbreviated EHU) are from Hume (1999), which includes section and paragraph numbers; this is followed by a reference to the page number in Hume (1975), abbreviated SBN.
- 3.
See again note 1 above.
- 4.
For the background to my brief discussion in the present essay of Hume ’s relationship to Newton ’s scientific methodology, see De Pierris (2006).
- 5.
Notice that this and subsequent experiments of reaching a simple indivisible minimum shed light on Hume ’s earlier distinction between simple and complex impressions and ideas (introduced at T 1.1.1.2/SBN 2) and on the closely related separability principle, which he renders here as “what consists of parts is distinguishable into them, and what is distinguishable is separable”. Hume first introduces the separability principle at T 1.1.3.4 (SBN 10): “Where-ever the imagination perceives a difference among ideas, it can easily produce a separation.” At T 1.1.7.3 (SBN 18) he adds the condition of distinguishability: “[W]hatever objects are different are distinguishable, and whatever objects are distinguishable are separable [and vice versa] by the thought and imagination.” Separability is also said to be a necessary property of “distinct” ideas (T 1.3.3.3/SBN 79): “[A]ll distinct ideas are separable from each other.” The above quoted text from T 1.1.3.4 (SBN 10) occurs in the context of pointing out that the imagination (unlike memory) has perfect “liberty … to transpose and change its ideas,” Here Hume makes explicit the close relationship between the simple/complex distinction and the separability principle (T 1.1.3.4/SBN 10): “Nor will this liberty of the fancy appear strange, when we consider, that all our ideas are copy’d from our impressions, and that there are not any two impressions which are perfectly inseparable. Not to mention that this is an evident consequence of the division of ideas into simple and complex. Where-ever the imagination perceives a difference among ideas, it can easily produce a separation.”
- 6.
Given these two notions of perceptible part, there are two different senses in which space can appear to have parts of equal or different “sizes.” On the one hand, there are the extended sub-wholes of extension that are perceivable as such in a single given whole appearance. Hume accepts a rough and approximate geometrical notion of “size” for such sub-wholes, in so far as they may phenomenologically appear (roughly) as either of the same magnitude or of different magnitudes. (These are only rough appearances, for Hume , since the exact magnitude—the exact number of minima in each of these sub-wholes—cannot be determined). On the other hand, with respect to the simple and indivisible minimum parts whose “confounding” results in all the appearances of extension in question, the situation is completely different. For, despite the fact that a minimum can be determined to be such only after a (finite) temporal process of diminutions or divisions, there is nonetheless a sense in which Hume takes the minima, as I argued in the first article cited in note 1 above, to be all of the same arithmetical “size”: they are ultimate simple units, which sensibly appear as discrete at the phenomenological threshold immediately before the appearance vanishes.
- 7.
Hume puts this point very clearly for the case of time (T 1.2.3.10/SBN 36–37): “In order to know whether any objects, which are join’d in impression, be separable in idea, we need only consider, if they be different from each other; in which case, ’tis plain they may be conceiv’d apart. Every thing, that is different, is distinguishable; and every thing, that is distinguishable, may be separated, according to the maxims above-explain’d. If on the contrary they be not different, they are not distinguishable; and if they be not distinguishable, they cannot be separated. But this is precisely the case with respect to time, compar’d with our successive perceptions. The idea of time is not deriv’d from a particular impression mix’d up with others, and plainly distinguishable from them, but arises altogether from the manner, in which impressions appear to the mind, without making one of the number. Five notes play’d on a flute give us the impression and idea of time; tho’ time be not a sixth impression, which presents itself to the hearing or any other of the senses. Nor is it a sixth impression, which the mind by reflection finds in itself. These five sounds making their appearance in this particular manner, excite no emotion in the mind, nor produce an affection of any kind, which being observ’d by it can give rise to a new idea. For that is necessary to produce a new idea of reflection, nor can the mind, by revolving over a thousand times all its ideas of sensation, ever extract from them any new original idea, unless nature has so fram’d its faculties, that it feels some new original impression arise from such a contemplation. But here it only takes notice of the manner, in which the different sounds make their appearance; and that it may afterwards consider without considering these particular sounds, but may conjoin it with any other objects. The ideas of some objects it certainly must have, nor is it possible for it without these ideas ever to arrive at any conception of time; which since it appears not as any primary distinct impression, can plainly be nothing but different ideas, or impressions, or objects dispos’d in a certain manner, that is, succeeding each other.” Note the importance of the explicit appeal to the separability principle at the very beginning of this passage.
- 8.
See note 6 above and the first article cited in note 1 above.
- 9.
See the Editors’ Annotations to T 1.2.1.5 in Hume (2000, p. 435).
- 10.
Compare the discussion in the section “Of skepticism with regard to the senses” of how the “fiction” of numerical identity of external objects arises from resemblance (e.g., at T 1.4.2.34–35).
- 11.
Hume cannot here be referring to physical atoms, for he has just introduced sensible minima as “perfectly simple and indivisible” parts of extension at T 1.2.1.2–4 (SBN 27–28)—he is explicitly considering “impressions of the senses” and “ideas of the imagination” (T 1.2.1.4/SBN 27–28). (The word “atom” simply means indivisible).
- 12.
Hume points out at T 1.2.1.4 (SBN 28) that “[a] microscope or telescope, which renders [previously unseen objects] visible, produces not any new rays of light, but only spreads those, which always flow’d from them.” The reference to “spreading” rays of light suggests Newton ’s famous experiments on the refraction of light rays by a prism. In my article on Hume , Locke and Newton cited in note 4 above, I develop Hume ’s understanding of Newton ’s prism experiments, and I argue, in addition, that the way both Hume and Newton accept corpuscularianism depends on a purely inductive advance into smaller and smaller observable parts of matter rather than on a postulation beforehand of an unobservable hidden microstructure—a postulation characteristic of the mechanical philosophy of Descartes , Leibniz , Locke and Boyle .
- 13.
I introduced for the first time what I take to be Hume ’s two standpoints—the philosophical standpoint of radical skepticism and the standpoint of common like and science—in De Pierris (2001, 2002). I further develop this distinction in De Pierris (2015). In particular, I argue that Hume ’s radical skepticism concerning our causal reasoning and its presupposed principle of the uniformity of nature is directed against his own and Newton ’s employment of this reasoning, which is naturally unavoidable and normatively required by our best method of inquiry in common life and science.
- 14.
A virtually identical passage occurs at T 1.3.12.5 (SBN 132). As I discuss in detail in my article on Hume , Locke and Newton cited in note 4 above, Hume goes on to say that “this possibility is converted into certainty by farther observation,” where the “certainty” in question is that of (inductive) proof as opposed to (mere) probability. The methodological principle that is most relevant here is Hume ’s sixth rule (T 1.3.15.8/SBN 174): “The difference in the effects of two resembling objects must proceed from that particular, in which they differ. For as like causes always produce like effects, when in any instance we find our expectation to be disappointed, we must conclude that this irregularity proceeds from some difference in the causes.” As I explain in the cited article, this and the other rules are grounded in the desideratum of attempting to achieve, in the Newtonian style, full (inductive) proof or perfect uniformity (under the guidance, therefore, of the principle of the uniformity of nature).
- 15.
The Annotations to EHU 8.14 in Hume (1999, p. 240), explain that “Chambers defined animal œconomy as ‘the first branch of the theory of medicine; or that which explains the parts of the human body, their structure and use; the nature and causes of life and health, and the effects or phænomena arising from them’.” The Editors’ Annotations to T 1.2.1.5 in Hume (2000, p. 435), refer us, for “animal spirits,” to their later annotation to 1.2.5.20 (p. 443): “Chambers describes animal spirits as ‘an exceedingly thin, subtile, moveable fluid juice or humour separated from the blood in the cortex of the brain, hence received into the minute fibres of the medulla, and by them discharged into the nerves, by which it is conveyed through every part of the body, to be the instrument of sensation, muscular motion, &c’. Although he grants that the ‘existence of the animal spirits’ is controversial, Chambers contends that they provide the best explanation of bodily motion and function: ‘the infinite use they are of in the animal œconomy, and the exceedingly lame account we should have of any of the animal functions without them, will still keep the greatest part of the world on their side’ (Cyclopædia, ‘Spirit’).”
- 16.
The one clear exception to this conception of the fundamental principles or axioms is the parallel postulate (Euclid , Postulate 5), which cannot be grasped intuitively in a limited region of extension.
- 17.
See my discussion of Hume ’s distinction between what we would now call “a priori” and “a posteriori” methods of reasoning in De Pierris (2005). I further developed the discussion of this distinction in my 2015 book cited in note 13 above.
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De Pierris, G. (2015). Hume ’s Skepticism and Inductivism Concerning Space and Geometry. In: De Risi, V. (eds) Mathematizing Space. Trends in the History of Science. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-12102-4_11
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