This paper by Andreas Doering kicks off with an admirably clear overview of topos theory, and proceeds in a directioc of which I very much approve. Pity I don’t understand any of it, really…

UPDATE:

I do nevertheless want to try to say something about why, in spite of having at best a very unclear notion of what it involves, I think the work of Doering and Isham may be important. At bottom, it has to do with the construction of a mathematical context - a topos, or “world” in Badiou’s terminology - in which a physical system with the properties described by quantum theory can be modelled. The problem with non-realist quantum physics is that it forbids any such construction: it asserts, in effect, that there is no observer-independent “world” in which quantum events happen all by themselves. What I think Doering and Isham are trying to show is that this is true only for a narrow conception of “world”: the introduction of a privileged “observer” who mediates between subatomic-scale quantum weirdness and the definitive being-there of larger-scale physical entities is a way of resolving a problem thrown up by the inadequacies of the underlying model. It’s a very unsatisfactory solution, firstly because it introduces a rogue “continentalist” philosopheme into the language of science - with generally horrible and deleterious effects - and secondly because it makes the problem of quantum gravity unsolvable, since a non-realist solution would require a global observer to oversee the effects of quantum gravity throughout the entire universe. Doering and Isham aren’t quixotically attempting to reinstate “hidden state”, as far as I can see, but to provide a “transcendental” (or sub-object classifier) of sufficient complexity to capture the logic of quantum effects. The nature of this beast is totally beyond me - I haven’t the faintest idea what a spectral pre-sheaf is, or what it is that Doering and Isham’s “daseinisation operator” does - but what I take to be the shape of the argument seems very right to me, and I really hope to be able to approximate a better understanding of its contents over time.