In his account of necessity David Lewis proposes that given two worlds that are exactly alike at time_{1}, W and W*, and in which the same natural laws apply, then at any later time these two worlds will continue to be exactly alike. As a good Humean, however, Lewis encountered what he claimed to be a damning problem, the problem of undermining futures. On Lewis’s reading of Hume, any claims or truths we make regarding the world, including claims concerning necessary laws, supervene upon a given distribution of qualities. There cannot be a change in this distribution without there also being a change in the claims or truths that supervene upon them. Given the laws of probability, the chances of a dice coming up showing a six is one in six. Three or four sixes may show up in a row, but given a large enough number of throws the number of times it shows up sixes approaches one in six. These laws of probability therefore supervene upon a given distribution of qualities in the world up to and including time_{1}. If there is a non-zero chance, however, that after time_{1} sixes come up every time then that would effect the chance distribution at W at time_{1}—it would be something higher than one in six, but this contradicts Humean supervenience. Given the case of an undermining future we would assign a probability value x and non-x to the throw of the dice at time_{1.} In his response to this problem, Lewis proposes modifying the laws, but many have been unhappy with Lewis’s proposal. In his analysis of Hume in *After Finitude*, Meillassoux, following Badiou, would argue that the very laws themselves presuppose a totalized whole, an All, in order for there to be the regularities upon which these necessary laws supervene. If mathematics thinks the not-All, however, then there is no reason why Humean supervenience needs to stay the same or be different at a later time. The notion of an undermining future would be vacated of sense. We may continue to axiomatize and mathematize the distribution of qualities in the world, but there is no All which assures their necessity, and hence no necessity to be undermined. The only necessity for Meillassoux is the necessity of contingency. In his response to the problem of undermining futures, John Roberts argues that what gives rise to the problem is the idealization of our knowledge of chance at time_{1}. It is only under the assumption that we can specify a particular value to the chance of a particular event happening whereby we are led to a contradictory belief when the undermining future entails a different value and result. ‘But real evidence,’ Roberts claims, ‘never constrains these credences by specifying the objective chances of such events.’ (“Undermining Undermined,” p. 104) As Roberts clarifies, ‘if HS is correct, there could be such evidence only if there were no problem of induction,’ meaning that this evidence would have to ‘entail…contingent information about the future, something no evidence in principle available to creatures like us could ever do.’ (ibid.). By constraining evidence concerning chance and supervenience to ‘finite empirical cognizers,’ Roberts is able to block the reduction that results from undermining futures. In other words, people like us, finite cognizers, are simply unable to process all the evidence necessary to get the appropriate value, much less the undermining futures which we cannot even access, and thus we would be unable to generate the contradictory value. In doing this, however, Roberts fails to avoid the central critique of Meillassoux’s book. By calling upon the mathematical thinking of the not-All associated with Cantorian set-theory, Meillassoux sought to address the correlationist trap that has been in place since Kant—namely, we cannot know an object as it is in itself except for how it appears in its relationship to us as finite empirical cognizers. This is precisely what Roberts does, however, and the fact that Lewis himself was not attracted to the solution Roberts offers should give us pause. The reason for this is that Lewis sought, with the tools of modal logic, to do much what Meillassoux and Badiou would like to do—break free from the limiting cages of finite cognizers and arrive at truths about an autonomous reality that is not correlated to a finite cognizer. As a good Humean, however, Lewis would no doubt not accept Meillassoux’s rejection of the problem of undermining futures, the problem of induction, and similarly Meillassoux would reject Lewis’s approach since Humean supervenience continues the correlationsist legacy—knowledge of reality in itself is correlated to the various qualities of the world as they are related to a presupposed totality. This last claim I think is debatable since Lewis’s metaphysics may well involve a reality of possible worlds that are non-totalizable. Lewis nonetheless continued to believe in the necessity of natural law, thus even if there is a transfinite set of possible worlds (a not-All), the laws, whatever they are in each world, would hold as a consequence of the totality of that world; and hence Meillassoux’s argument would resurface.

At this point we can turn to Deleuze’s understanding of mathematics to clarify our take on Lewis’s position. As Daniel Smith has shown in his essay on the importance of mathematics in understanding Deleuze’s theory of multiplicities, and how this theory in turn differs from Badiou’s theory of the multiple-without-One, Smith shows that Deleuze’s theory is informed by a tradition of problematics in mathematics in contrast to the axiomatic approach favored by Badiou. (“Mathematics and the Theory of Multiplicities”). The difference becomes clear in Deleuze and Guattari’s very definition of the nondenumerable: ‘What characterizes the nondenumerable is neither the set nor its elements; rather, it is the connection, the “and” produced between elements, between sets, and which belongs to neither, which eludes them and constitutes a line of flight.’ (TP 518). The nondenumerable is problematic, for Deleuze, precisely because it constitutes problems that have, as Smith puts it, ‘an objectively determined structure, apart from its solution,’ and this objectively determined structure entails ‘a zone of objective indetermination’ that precludes being reduced to demonstrative and axiomatic methods in mathematics. The ‘genetic and problematic aspect of mathematics…remains inaccessible to set theoretical axiomatics,’ and yet, through continual movements and translations, the problematic in mathematics gives way to axiomatic innovations and recodings (Smith offers the example of the translation of infinitesimals and approaching the limit in calculus [an example of the problematics tradition] into the axiomatic epsilon-delta method as developed by Weierstrass). We have in short what you might call Deleuzian supervenience, whereby the discretization of the axiomatic maps or supervenes upon the continuity of the problematic, but the problematic forever exceeds the axiomatic, it is the ‘power of the continuum, tied to the axiomatic but exceeding it.’ (ibid. 466). Axiomatics, or what Deleuze will also call major or royal science, thus draws from problematics the necessity of inventing and innovating in response to the ‘objectively determined structure’ of the problem. Similarly problematics, minor or nomad science, calls upon axiomatics to actualize the solutions it lays out, if only indeterminately so. Deleuze and Guattari are clear on this point: ‘Major science has a perpetual need for the inspiration of the minor; but the minor would be nothing if it did not confront and conform to the highest scientific requirements.’ (ibid. 486). We can thus rethink Lewis’ problem of induction not as a problem intrinsic to the relationship between a finite cognizer and the distribution of qualities as they relate to this cognizer, but rather as an ‘objectively determined’ problem that exceeds the tools of modal realism and axiomatic logic. Lewis, in short, is encountering the necessity and insufficiency of invention. In the next post we’ll see that one interpretation of why Spinoza abandoned his *Treatise on the Emendation of the Intellect *was precisely because he, like Lewis, encountered the necessity and insufficiency of invention.

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