Talks by Sam Fletcher and Sam Sanders at MCMP (Oct. 15)

[Dardashti, Radin]

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.



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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.