Archive for May 3rd, 2013

May 3, 2013

Truth and axioms in mathematics

by Neil Rickert

There’s been some discussion of truth in mathematics in the comments to my previous post.  Here, I want to expand a little on my view and express puzzlement at the idea that axioms are themselves true or false.

In response to a question, said “Actually, I take axioms to be neither true nor false, and I take the truth of mathematical theorems to be relative to the assumed axioms.”  Let me restate that in terms of the Peano axioms for ordinary arithmetic.

  1. The Peano axioms are neither true nor false.  Rather, they are definitional statements.  They define that part of mathematics known as Peano Arithmetic (or PA, or simply arithmetic).
  2. Theorems proved in PA are true in a relative sense.  Their truth is relative to the PA axioms.  They are true as used within PA, but perhaps not even meaningful outside of PA.

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