There is a fascinating series of posts over at Blog & Not Blog concerning Russell’s paradox and Graham Priest’s dialetheism, i.e., the view that there are true contradictions. Though the posts are intended to defend monoaletheism against Priest (based on a Quinean view of existence that I am at least thoroughly non-committal to), the most immediate eye-opener is the following quote from Priest:

“In modern science, the inferentially sophisticated part is nearly always mathematical. An appropriate mathematical theory is found, and its theorems are applied. Hence, a likely way for an inconsistent theory to arise now in science is via the application of an inconsistent mathematical theory. Though the construction of inconsistent mathematical theories (based on adjunctive paraconsistent logics) is relatively new, 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. The theories have not been developed with an eye to their applicability in science—just as classical group theory was not. But once the paraconsistent revolution has been digested, it is by no means implausible to suggest that these theories, or ones like them, may find physical application—just as classical group theory did. For example, we might determine that certain physical magnitudes appear to be governed by the laws of some inconsistent arithmetic, where, for example, if n and m are magnitudes no smaller than some physical constant k, n + m = k (as well as its being the case that n+ m ≠k). There are, after all, plenty of episodes in the history of science where we came to accept that certain physical magnitudes had somewhat surprising mathematical properties (being imaginary, non-commuting, etc.). Why not inconsistency?”

Some immediate ramifications that stand out:

– The scenario sketched here seems to me, considered commonsensically, neither more or less plausible than the fact that physical magnitudes have the mathematical properties of being imaginary or non-commutative. Or p-adic, for that matter. These other possibilites (that are in fact actualities) are already so completely *baffling* to me that I have no reason why inconsistent arithmetic should not be usable in physics that would not apply to these other examples as well.

– I have thought for some time already that Meillassoux needs to come to terms with paraconsistent logic in a much more serious way than he has done so far. But this quote from Priest really brings it out, as it wreaks havoc upon any remaining way I see of justifying Meillassoux’ claim concerning the law of non-contradiction in *After Finitude*.

– It also sets Badiou’s claim concerning the necessity of physics according with ontology (as he sees it) in a strange light. What would such a discovery as the indispensability of inconsistent arithmetic for explaining some physical phenomenon do to Badiou’s account of ontology?

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