Samuel R. Buss
"Nelson's Work on Logic and Foundations and Other Reflections on Foundations of Mathematics"
In Diffusion, Quantum Theory, and Radically Elementary Mathematics,
edited by W. Faris, Princeton University Press, Princeton and Oxford, 2006, pp. 183-208.
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Abstract: This paper starts by discussing
Nelson's philosophy of mathematics, which is a blend of mathematical formalism
and a radical constructivism. As such, it makes strong assertions about the
foundations of mathematic and the reality of mathematical objects. We then
offer our own suggestions for the definition of mathematics and the nature of
mathematical reality. We suggest a second characterization of mathematical
reasoning in terms of common sense reasoning and argue its relevance for
Nelson's philosophy is the foundation of his definition of predicative arithmetic. There are close connections between predicative arithmetic and the common theories of bounded arithmetic. We prove that polynomial space (PSPACE) predicates and exponential time (EXPTIME) predicates are predicative.
We discuss Nelson's formalist philosophies and his unpublished work in automatic theorem checking.
Slides from a presentation at Workshop on Analysis, Probability and Logic, Pacific Institute of Mathematical Studies, UBC, Vancouver, June 2004.
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