Speaker: Samuel Fletcher (MCMP) Wednesday 15th Oct. 2014 Location: Ludwigstr. 31 room 021 Time: 16:15 - 17:45 Title: On the Local Flatness of Spacetime Abstract: Many discussions of the foundations of general relativity put a special emphasis on describing every relativistic spacetime as “locally flat”, or as “locally Minkowskian”. Such claims are prima facie puzzling: after all, curvature is itself a local property, being described by a tensor field on spacetime. In general, relativistic spacetimes have non-vanishing curvature, so there is a straightforward sense in which they are not locally flat. Still, there is a natural intuition behind claims of “local flatness” arising from analogy with a sufficiently small region of a curved surface, like that of the Earth, which can to a good approximation be described as planar. But like many “principles” of general relativity, there does not seem to be much consensus regarding how to make this intuition more precise. Without attempting a comprehensive survey, we note three common articulations of what it could mean for spacetime to be “locally flat” or “locally Minkowskian,” arguing that each of them is unsatisfactory. We then explore a different, but precise and coordinate-independent sense in which relativistic spacetimes might be described as (approximately) locally flat. --------------------------------------- Speaker: Sam Sanders (MCMP) Wednesday 15th Oct. 2014 Location: Ludwigstr. 31 room 021 Time: 18:15 - 19:45 Title: Constructivism in Physics and Platonism in Mathematics Abstract: First of all, I argue that constructive mathematics (in the sense of Bishop, going back to Brouwer) is an essential part of the mathematical practice of physics. My motivation is the observation that to test a physical theory against experiments, we have to compute or approximate the mathematical objects in that theory in order to compare them to our experimental data. Such approximation procedure is generally impossible for non-constructive objects. In particular, I show that the intuitive “calculus with infinitesimals” from physics naturally gives rise to constructive mathematics. Secondly, I argue that predicativist mathematics (in the sense of Russell, Weyl, and Feferman) is incoherent from the point of view of Nonstandard Analysis. In particular, to remove the paradoxes in naive set theory, Russell proposed accepting the set of natural numbers as a finished totality, but banning subsets defined using the “vicious circle principle”, nowadays called “impredicative definition”. The textbook example of the latter is the Suslin functional, while arithmetical comprehension is the perfect example of predicativist mathematics. I show that the Suslin functional is equivalent to a nonstandard principle of arithmetical comprehension over a predicativist system of Nonstandard Analysis.