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.
Mathematical logicians sometimes say that PA is true. And that kind of talk puzzles me.
Of course, PA is true about everything within PA. However, that seems tautological, and not worth saying. If that is all that is meant by saying that PA is true, then why say it?
If saying PA is true means more than that trivial tautology, then I wonder about modular arithmetic. For if I do arithmetic mod 5, then 3+2 = 0. But that’s false in PA. So if the axioms of PA are true, does that mean that modular arithmetic is false? If the axioms of Euclidean geometry are true, does that mean that non-Euclidean geometry is false?
This is why I take axioms to be neither true nor false. I see them as defining principles for a particular area of mathematics. But it should mean more than that to say that they are true. Again, to put it in terms of PA, nothing seems to depend on whether the Peano axioms are true. Arithmetic depends on our taking PA as providing basic definitions and principles. And, of course, that makes the Peano axioms tautological truths within PA. However as best I can tell, it does not make the Peano axioms true in anything other than that tautological sense.
Perhaps this is why I am not a mathematical platonist.
