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.

- 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).
- 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.

read more »

Advertisements