Notes-4

Chapter 4: Hume’s Problem

  1. For Hume’s own formulation of the problem, see David Hume (1984), A

Treatise of Human Nature, C. Mossner (ed.) (Harmondsworth: Penguin), Book I, Part III, pp. 117–229; and David Hume (1957), Enquiry Concerning Human Understanding, C. W. Hendel (ed.) (New York: Liberal Arts Press), Sections 4 and 5, pp. 40–68.

  1. Popper clearly states his belief in the ‘principle of the uniformity of

nature’: ‘It never happens that old experiments one day yield new results. What happens is only that new experiments decide against an old theory.’ Karl Popper (2002), The Logic of Scientific Discovery (London and New York: Routledge), pp. 249–50. The reason we insist on distinguishing Hume’s problem from Popper’s problem is because Popper himself considerably muddied the waters by referring to the problem of the future validation of scientific theories as ‘Hume’s problem’. Thus, Popper believed himself to be dealing with the issue that preoccupied Hume when in fact he was only dealing with a problem that assumes that the Humean difficulty has already been resolved. For Popper’s question, which is to say the question of the future validity of our physical theories, assumes that in the future, even if those theories are refuted by new experiments, physics will still be possible, since such refutations will be carried out for the benefit of new physical theories. Thus Popper continues to assume that the principle of the uniformity of nature, which provides the condition for the possibility of physical experiments, will still be valid in the future, and it is by relying a priori on this supposedly necessary validity that he is able to elaborate the principles of his own epistemology.

  1. Hume (1957), p. 44.

  2. Hume (1957), p. 45.

  3. In the 1st edition of the Critique, the objective deduction of the categories

constitutes the third section of Chapter II in the ‘Analytic of Concepts’ (cf. Kant (1929), Critique of Pure Reason, tr. N. K. Smith (London: Macmillan), pp. 129–50); while in the 2nd edition of 1787, it takes up sections 15 to 24 of the same chapter – and even more specifically, sections 20 and 21 (cf. Kant (1929), pp. 151–75 and pp. 160–61). For a linear commentary of the objective deduction of 1781, cf. Jacques Rivelaygue (1992), Leçons de métaphysique allemande. Tome II: Kant, Heidegger, Habermas (Paris: Grasset), pp. 118–24.

  1. Jean-René Vernes (1982), Critique de la raison aléatoire, ou Descartes contre

Kant (Paris: Aubier), with a preface by Paul Ricoeur.

  1. Vernes (1982), p. 45.

  2. For a clear introduction to probability theory that covers both discrete

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and continuous distributions (or finite and infinite numbers of possible cases respectively), cf. J.L. Boursin (1989) Comprendre les probabilités (Paris: Armand Colin).

  1. Alain Badiou (2006), Being and Event, tr. O. Feltham (London:

Continuum).

  1. The reader should consult to the following ‘Meditations’ from Being

and Event: 1–5, 7, 12–14, and most importantly, Meditation 26, which concerns the pluralization of infinite multiplicities.

  1. Here is a simple example that may give non-mathematicians a better idea

of Cantor’s theorem. In the set-theoretical axiomatic, a set B is considered to be part of the set A if all the elements of B also belong to A. Consider the set A 5 (1, 2, 3), that is to say, a set comprising three elements. Now consider the set B, which comprises all the parts of A, and which can also be designated as p(A) – what are the elements of B? First of all, the set B contains the three singleton sets (1), (2) and (3) [a ‘singleton’ is a set comprising only one element – translator]. These parts of A are not identical with the elements of A (i.e. 1, 2 and 3), but with the grouping of these elements into sets containing them and them alone. These are the minimal parts of A – the parts comprising only one of the elements of A. Then we have parts in the usual sense of the term – those that comprise two of the three elements of A: (1, 2), (1, 3), (2, 3). Lastly, we have the maximal part of A, which is identical with A itself: (1, 2, 3). According to the set-theoretical definition of ‘part’, A is always part of itself, since all the elements of A do indeed belong to A. Finally, we would have to add the empty-set (whose existence standard set-theory posits, and whose uniqueness it demonstrates) to this list of the parts of A. The empty-set is part of every set, in the sense that the void, which has no elements, contains no elements that do not also belong to any set whatsoever (on this point, see Badiou’s Being and Event, Meditation 7, pp. 86–92). A simple count then suffices to show that p(A) contains more elements than A itself (i.e. eight, rather than three elements). The power of Cantor’s theorem resides in the way in which it generalizes this excess of the number of groupings of a set over the number of its elements to any set whatsoever, even infinite ones – whence the impossibility of halting the proliferation of infinities, since for any set whose existence we assume, we also assume the existence of its quantitative surpassing by the set of its parts.

  1. [Translation modified.]

  2. This surpassing of the proper limits of aleatory reason is just as evident in

the famous passage about cinnabar, where Kant, having just hypothetically assumed the absence of necessity in the laws of nature, infers what

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the consequences of this contingency would be for nature: ‘If cinnabar were sometimes red, sometimes black, sometimes light, sometimes heavy, if a man changed sometimes into this and sometimes into that animal form, if the country on the longest day were sometimes covered with fruit, sometimes with ice and snow, my empirical imagination would never find opportunity when representing red colour to bring to mind heavy cinnabar.’ (Kant 1929: 132, A 100). That it is indeed a frequent modification (‘the longest day’ sufficing to produce numerous chaotic occurrences) which is posited here as the consequence of the contingency of laws, is shown by the goal of the argument from cinnabar. For Kant, in this passage, it is a matter of demonstrating the absurdity of trying to explain, as Hume does, the necessity of natural laws solely in terms of our subjective habituation to the latter. For unless such objective necessity pre-existed the habituation which it induces in us, we would never have had occasion, Kant maintains, to become habituated to any event whatsoever, given the absence of sufficiently stable regularities in the empirical data.