Truth in mathematics

I’m tentatively planning several posts on “truth”, and mathematics seems a good place to start — partly because I’m a mathematician, and partly because some of the distinctions seem clearer in mathematics than in other areas.

I can think of at least two different ways that we use “true” in mathematics. The most basic of these is with ordinary statements such as or the Pythagorean theorem. Both are generally recognized as true. The other way that some people use “true” is when they talk about whether the axioms are true. That could refer to the Peano axioms for arithmetic, the Zermelo-Fraenkel or about whether the axiom of choice is true or about whether the continuum hypothesis is true. As we shall discuss below, the way we think about the truth of axioms is different from the way that we think about the truth or ordinary mathematical statements.

Ordinary statements

When I talk of “ordinary statements” in mathematics, I am talking about statements such as in arithmetic or the Pythagorean theorem in geometry. We normally have a system of axioms that we use in our mathematics. For ordinary arithmetic, these are the Peano axioms. For geometric questions, we normally use Euclid’s axioms, supplemented by some version of the parallel postulate. For set theory, we most commonly use the Zermelo-Fraenkel axioms, possibly supplemented by the axiom of choice.

We normally consider an ordinary mathematical statement to be true, if it is a consequence of the axioms we are using. Note, however, that different axioms have different consequences. For example, the Pythagorean theorem is true in Euclidean geometry but is false in spherical geometry.

Because of this dependence on axioms, mathematics is sometimes described as a kind of if-then-ism. If we assume the Euclidean axioms, then the Pythagorean theorem is true. If we assume the Peano axioms, then is true.

In my experience, questions about the truth of ordinary statements is not controversial in mathematics. We generally use proofs to demonstrate truth when that isn’t trivially obvious. However, a statement could be a consequence of the axioms without us having discovered a proof. So truth is not quite the same as provability.

Questions could be raised about Gödel’s incompleteness theorem, and the undecidable statements that arise. I’m not inclined to think of those as ordinary statements.

Do we need axioms?

Could we manage without axioms? I presume that arithmetic started with simple counting. And we can count without depending on axioms. Notice, however, that counting is an imperfect procedure. We often have elections, where the outcome is close. And this leads to a recount. It is very common for the results of the recount to be different from the results of the original count. Usually, the difference is too small to affect the election outcome. That there is a difference illustrates that counting is not perfect.

These imperfections in counting are usually due to what we count. When counting votes, the counter has to decide whether the vote is valid. Mathematics idealizes this, so we avoid those imperfections. And the axioms we use are, in effect, the rules for idealization. So yes, we do need axioms.

Are axioms true?

Mathematicians don’t usually spend a lot of time thinking about whether their axioms are true. The mathematics itself does not depend on the truth of the axioms. It depends only on our assuming the axioms (or, roughly speaking, treating them as if they are true).

As a young mathematician, I did ponder whether the axiom of choice is true. But I was thinking of that in terms of whether it could be proved from other axioms of set theory. Once the axiom of choice was shown to be independent of other axioms, then there wasn’t a point in asking whether that axiom (often known as AC) is true. The only question was whether I should use it when doing set theory.

Some mathematicians do discuss the question of the truth of axioms. It is my impression that many mathematical Platonists are concerned with whether their axioms are true. Those of us who reject Platonism — and that includes me — are less likely to be concerned about the truth of axioms.

I once asked a Platonist what he means when he says that PA (the Peano axioms) are true. He hesitated before answering, and then suggested that the consistency of the axioms was part of it. However, as shown by Gödel, we cannot know whether PA is consistent. We take that as a matter of faith (no, not religious faith).

My personal view is that axioms don’t have a meaningful truth value.

If playing a game of baseball, we do not ask whether the rules are true. We only ask whether we are correctly following the rules. If we switched rules, and started following the rules of football instead of the rules of baseball, we would not be playing baseball with false rules. We would just be playing football. The rules are basically what defines the game.

Summary

I’ve discussed two types of question. For what I have called ordinary mathematical questions, the truth of a statement depends on whether we are correctly following the rules (the axioms).

The other type of statements are the rules themselves (i.e. the axioms). I don’t see those as having meaningful truth values. Yes, in some sense we agree to use the axioms. That agreement is something like a social convention among mathematicians. But agreeing to use axioms seems very different from concluding that they are true.