(In my previous post, I mentioned that the dialetheism proffered by Graham Priest should be of concern to the Meillassouxian position. I want to continue exploring this angle in a few posts, bear in mind that while I will attempt to draw some radical conclusions, they should be considered as transcissions rather than as finished products.)
Meillassoux does bring up “paraconsistent logics”, of which Priest’s dialetheism is a species, in After Finitude. To give a short summary of the background, the problem of paraconsistent logics surfaces in relation to one of the two things Meillassoux attempts to demonstrate on the basis of the “principle of factiality”: 1) That the thing-in-itself must exist, i.e., that there must be something rather than nothing, and 2) That the thing-in-itself cannot be contradictory. (See After Finitude, chapter 3, both for the principle of factiality itself, which states the absolute necessity of contingency, and the two demonstrations he proposes on the basis of this principle). The second demonstration proceeds by arguing that a thing must be non-contradictory in order to be contingent, and thus, since it must (per the principle of factiality) be contingent, it must also be non-contradictory.
After taking into account paraconsistent logics, however, Meillassoux is forced to admit that he has strictly speaking proven only the impossibility of an inconsistent being, i.e., a being of which all contradictions are true. For his demonstration depends upon the fact that such a being could not change, i.e., it could not be different, and thus it would no longer be contingent. But if only some, but not all, contradictions are true of it, then it could still change and thus still be contingent. Meillassoux is not content with this restricted version of his demonstration, however, and proposes that his demonstration could be enlarged to yield the originally intended conclusion (the impossibility of a contradictory being):
This time the question would be to know whether or not we could also use the principle of unreason to disqualify the possibility of real contradiction. We would need to point out that paraconsistent logics were not developed in order to account for actual contradictory facts, but only in order to prevent computers, such as expert medical systems, from deducing anything whatsoever from contradictory data (for instance, conflicting diagnoses about a single case) because of the principle of ex falso quodlibet. Thus, it would be a matter of ascertaining whether contradiction, which can be conceived in terms of incoherent data about the world, can still be conceived in terms of non-linguistic occurrences. We would then have to try to demonstrate that dialectics and paraconsistent logics are only ever dealing with contradictions inherent in statements about the world, never with real contradictions in the world – in other words, they deal with contradictory theses about a single reality, rather than with a contradictory reality. Dialectics and paraconsistent logics would then be shown to be studies of the ways in which the contradictions of thought produce effects in thought, rather than studies of the supposedly ontological contradictions which thought discovers in the surrounding world. Finally, our investigation would have to conclude by demonstrating that real contradiction and real inconsistency both violate the conditions for the conceivability of contingency. (After Finitude, p. 78-79)
It is in light of this statement that the quote I gave from Priest in the post below gives rise to some serious problems. In it, Priest imagines a scenario in which future science might come to rely on “inconsistent mathematics” in order to account for certain physical phenomena. Meillassoux’ problem here is of course that the conditions of his argument are so much stricter than most other’s. It is of no use to him to dismiss the possibility of real contradictions as extremely unlikely, like Ben Brugis over at Blog & Not Blog does. For Meillassoux’ argument is not pitched at the level of factual existence, but at the level of absolute possibility and necessity. So he must deduce the absolute impossibility of the scenario Priest imagines on the mere basis of the necessity of contingency. Of course, Meillassoux does not claim to offer such a demonstration in After Finitude, but he clearly suggests that it could be done. The thought experiment provided by Priest serves to underscore the difficulty of such a demonstration.
But the problem might go beyond mere difficulty, I believe. For as Priest says, “inconsistent mathematics” has already been developed and it seems likely that this area of mathematics will continue to grow (“there are already a number of inconsistent number theories, linear algebras, category theories; and it is clear that there is much more scope in this area”, as he says). And what are we, then, to make of the following ‘mission statement’ towards the very end of After Finitude?: “Thus, we must establish the following thesis, which we have already stated, by deriving it from the principle of factiality: what is mathematically conceivable is absolutely possible” (After Finitude, p. 126).
Prima facie, Meillassoux’ two projected demonstrations are at odds with each other. The crucial premise is that inconsistent mathematics is possible, which seems plausible given that such mathematics have already been developed. From this, one can deduce that if the mathematically conceivable is absolutely possible, then real contradiction is (absolutely) possible (since contradiction is mathematically conceivable). And conversely, if paraconsistent logics can be shown to “violate the conditions for the conceivability of contingency”, as Meillassoux projects, then not everything that is mathematically conceivable is absolutely possible (since inconsistent mathematics is mathematically conceivable but not possible). Thus, either one of these two projected demonstrations would seem to negate the other.
Perhaps one could deny the premise, on the basis that inconsistent mathematics is not really mathematics, or something along those lines. This appears dreadfully ad hoc to me at the moment, so though I cannot rule out there might be good independent reasons for doing so, I will not countenance this possibility further here. (This seems to summarize the present state of inconsistent mathematics)
My provisional conclusion is that the impossibility of real contradiction must give, since the absolute possibility of the mathematically conceivable seems such an integral part of Meillassoux’ position. We can thus glimpse the outline of a transconsistent speculative materialism.
In my next post I will consider an article by Frederick Kroon that argues against dialetheic realism, in favor of dialetheic fictionalism. I will argue that despite the fact that this conclusion is similar to the one Meillassoux wants to reach, the argument given in the article cannot be used to support Meillassoux’ projected demonstration that paraconsistent logics concern contradictions in thought, but not in reality. On the contrary, Meillassoux’ principle of factiality can be used to block Kroon’s argument against dialetheic realism, thus bolstering the conjunction of Priest’s dialetheism and Meillassoux’ speculative materialism: transconsistent speculative materialism.