Abstract

A decade ago, Isham and Butterfield proposed a topos-theoretic approach to quantum mechanics, which meanwhile has been extended by Doering and Isham so as to provide a new mathematical foundation for all of physics. Last year, the speaker with Heunen and Spitters redeveloped and refined these ideas by combining the C-star-algebraic approach to quantum theory with the so-called internal language of topos theory (see arXiv:0709.4364). The goal of the present talk (based on arXiv:0902.3201) is to illustrate our abstract setup through the concrete example of the C-star-algebra M_n© of complex n x n matrices. In our approach, the nondistributive lattice of projections in M_n© (which forms the basis of the traditional quantum logic of Birkhoff and von Neumann) is replaced by a specific distributive lattice, whose elements correspond to ‘Bohrified’ propositions, in the sense that to each classical context it associates a yes-no question (rather than being a single projection, as in standard quantum logic). Distributivity is recovered at the expense of the law of the excluded middle, whose demise is in our opinion to be welcomed, not just in intuitionistic logic in the spirit of Brouwer, but also in quantum logic in the spirit of von Neumann.