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Nelson’s Work on Logic and Foundations and Other Reflections on Foundations of Mathematics
, 2006
"... 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 defi ..."
Abstract

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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 mathematics education. 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 We discuss Nelson’s formalist philosophies and his unpublished work in automatic theorem checking. This paper was begun with the plan of discussing Nelson’s work in logic and foundations and his philosophy on mathematics. In particular, it is based on our talk at the Nelson meeting in Vancouver in June 2004. The main topics of this talk were Nelson’s predicative arithmetic and his unpublished work on automatic theorem proving. However, it proved impossible to stay within this plan. In writing the paper, we were prompted to think carefully about the nature of mathematics and more fully formulate our own philosophy of mathematics. We present this below, along with some discussion about mathematics education. Much of the paper focuses on Nelson’s philosophy of mathematics, on how his philosophy motivates his development of predicative arithmetic, and on his unpublished work on computer assisted theorem proving. We also discuss the