What does it mean to assume an axiom is “true” in mathematics?

Question

1. I honestly don’t even know what “true” means anymore :(.

2. Is mathematics typographical?

3. Are we just saying a string of symbols is “true” to kickstart our theory?

Answer 1

The appropriate phrasing is to say that an axiom is satisfied in a model. Thus, the group axioms are satisfied by the group $\mathbb Z_2$. The axiom of separation is satisfied by any model of set theory. The Peano Axioms are satisfied by the natural numbers. Here truth would be a separate issue.

Answer 2

Mathematical 'truth' is different from the way we normally talk about 'truth'. For example, an everyday claim like 'grass is green' is considered 'true' because it accurately describes our world .. the very physical world we live in. Put simply, 'grass is green' is true because the grass around us is in fact green. And 'grass is pink' is false, because in our world grass is not pink.

Mathematicians, however, are not necessarily concerned with what our world looks like. Like logicians, they are happy to consider imaginary and abstract worlds where there are 'flubbers' that are 'brillig'. But note: in such a world, 'flubbers are brillig' would be true. In other words, 'truth' is relative to a world: in one world a claim would be true, but in some other world, that same claim would be false. In Euclidian worlds, Euclid's 'parallel postulate' would be true, but in non-Euclidian worlds, that same postulate would be false.

But this is exactly why in mathematics we have axioms. Using axioms, a mathematician is restricting their considerations to certain types of worlds: those worlds in which the axioms are true. And by deriving theorems from those axioms, we can find out other things about those kinds of worlds, i.e. other statements that are 'true-in-those-worlds'. As such, mathematicians can prove all kinds of things without having to worry about what things are true in our world.

The same goes for 'existence' claims that you often encounter in mathematics. For example: 'there exists an infinite number of primes'. By this, mathematics are not necessarily talking about our world .. or are even asserting that in our world numbers exist ... what they really say is that 'in any world where the axioms for basic numbers (e.g. the Peano axioms) are true, there are infinitely many prime numbers'.

Having said all that though, it is a bit of straw man or exaggeration to say that mathematicians and logicians are not interested in our world. Of course it would be nice if these axioms would be true in our world, i.e. if they describe something (approximately) accurately about our world. And this is why we don't have branches of mathematics that talk about flubbers being brillig, but instead talk about numbers, geometrical shapes, etc.: things that do seem to apply to our world pretty well ... and sometimes really well ... as scientists are finding out.

Answer 3

We can think of axioms as the rules of a game. So by analogy, consider the question, “Can I move my pawn three squares to the left?” When we are playing chess, the answer is that no, you cannot move your pawn three squares to the left. But can the sentence

You cannot move your pawn three squares to the left.

be called true? I mean even a mere toddler might very well come up to the board, pick up that pawn, and place it down on the very square in question. So in some contexts the sentence isn’t true, the answer is that, yes, it’s quite possible.

That’s the very nature of axioms. That’s how mathematicians build things. Now, a physicist say, might be more likely to build some math in order to encapsulate and model empirical observations about the world around us. And that’s a different beast entirely. For physicists, the ultimate consideration is conformity to empirical observations, whereas for mathematicians it’s conformity to our definitions and the laws of logic. In that sense, axioms are one flavor of mathematicians’ definitions.

For a more mathematical example, consider whether one can divide by zero, or to put it another way, whether there can be any two nonzero “things” whose product is zero. In the real numbers, the answer is, no, the definition of division on $\bf R$ precludes $0$ as a divisor. But in a different setting, say the ring of $2\times 2$ matrices on $\bf R$, things are different. Here, zero means $\left(\begin{array}{cc} 0&0 \\ 0&0\end{array}\right)$, and we have facts like $$ \left(\begin{array}{cc} x&x\\y&y \end{array}\right) \left(\begin{array}{cc} 1&1\\-1&-1 \end{array}\right) =\left(\begin{array}{cc} 0&0\\0&0 \end{array}\right) $$ for any $x,y\in \bf R$.

If you want a mathematical example where the “things” are more like everyday numbers, consider the 10-adics, a number system in which there exists a number $$x\approx \ldots92256259918212890625$$ with the property that $x^2=x$, which can be checked by performing the first $k$ steps of the multiplication for any $k\geq 1$. But this means that $x^2-x=0$, and thus that $x(x-1)=0$. So we have a pair of 10-addic numbers, $x$ and $x-1$, neither of which is zero but whose product is zero.