There has been a recent uprise in philosophy (and it seems to me to be a very popular topic there), which is called Speculative Realism or Object-Oriented Ontology. One of the founding texts there is Quentin Meillassoux's "After Finitude" (including "Iteration, Reiteration, Repetition: A Speculative Analysis of the Meaningless Sign"), which discusses mathematics very prominently, e. g.:
one of the main points, very roughly, is that correlationism, " every philosophy that maintains the impossibility of acceding through thought to a being independent of thought. We never, according to this type of philosophy, have access to any intended thing (understood in the most general sense) that is not always already correlated to an act of thinking (understood, again, in the most general sense). Consequently, correlationism posits, against all materialism, that thought cannot escape from itself so as to accede to a world not yet affected by the modes of apprehension of our subjectivity.", could be overcomed by mathematics and mathematized physics: "My plan is thus as follows: I will try to exhibit a minimal condition, modest yet fundamental, of various contemporary formal languages – logical as well as mathematical. This minimal condition, as we shall see, has to do with our capacity to think a meaningless sign. I will then derive this capacity to think a meaningless sign from the principle of factiality, by showing that there is an essential link between this sort of sign and absolutized contingency. I therefore will have to show in what way this factial derivation of the meaningless sign allows us to argue that physics (or any other science of nature) must be based upon this absoluteness of the void sign in order to produce hypothetical (revisable) descriptions of the present world, capable, in turn, of being true in an absolute sense – that is to say, independently of our existence"
he makes logical proofs about things-in-itself. For references, you could read modal logic intepretation of some of the arguments in Jon Cogburn's blog: here, here, here and here.
he also discusses possible applications of paraconsistent logic, but maybe Graham Priest would be more appropriate here (in terms of not-purely-analytical philosophers, who use or mention mathematical logic in their works).
In overall, I have a somewhat strange impression, that this work in theory should/could be read by (addressed to) serious mathematicians, but I haven't found any commentary from mathematicians about recent developments in philosophy. (As a related question on hsm, you could read comments from physicists about continental philosophy pre-2005).
So my question is, has any natural scientists have commented on speculative realism, object-oriented ontology, or more specifically, on the works of Quentin Meillassoux?
I have also found more understandable review of his work by philosopher Fabio Gironi : These two opening passages summarise Quentin Meillassoux‘s two-fold ambition: to denounce the restrained nature (and inconsistency)of the premises of post-Kantian continental philosophy and to propose anew, speculative way of philosophising, based on direct access to theexternal world obtained through mathematical reasoning. These two moments in Meillassoux‘s project are both a negative critique of contemporary philosophy and a positive demonstration of a new principle on which philosophical speculation can be founded: the absolute necessity of contingency and the recognition of mathematical discourse as its expression.
