Mathematical truth: seven cannot be factored into integers other than itself and one. Mathematical formalism: multiplication is commutative.
That is quite a leap of faith, isn't it? Not that I have anything against faith, you understand. But how can we know that there are infinitely many potential realities but only one mathematical truth?
No, I'm not going all wobbly on you. No Zen here. If you flip a coin, it's either going to be heads or tails. Both outcomes are possible (potential realities) but only one occurs. Avoiding the topic of quantum superposition, there's only one reality. But just about anything you can point to in "our" reality--including, yes, much of what we call the "laws of physics"--could consistently have been otherwise.
Al Gore might have won the election. The Greeks might have been defeated at Thermopylae. The Permo-Triassic impactor might have missed the Earth. The electroweak symmetry might have broken in such a way that photons were as massive as the Z boson. But seven would be prime regardless.
Positive integers, I mean! :^)
Ah, now I see what you mean. Multiplication could have been defined so that it is not commutative. And indeed, matrix multiplication is not.
On the other hand, primality appears to be an intrinsic property of the integer seven. I cannot imagine how one could "redefine" seven to make it composite. (Unless one wanted to redefine multiplication to make that true.)
Still, I have my doubts. As Kronecker said, "God created the whole numbers; everything else was man's handiwork." Integers suffice if all we are going to do is count things. However, if we also want to measure things, we need more sophisticated mathematics. And the further we get from the integers, the more formalism is found in our mathematics. Thus we decide that multiplication of scalars is to be commutative, because it is more useful that way.
When I said that reality is more fundamental than mathematics, I was thinking that nature is always richer in detail than our mathematics. Mathematical modeling is difficult; invariably, we are forced to omit things from our models. Even the "laws" of physics turn out to be, on closer examination, merely good approximations.
This is very much an intuitionist statement, i.e., it's obvious that 7 is prime.
A formalist, however, goes to great pains to prove that 7 is prime using no more than his axiom set. On the way he incidently defines 7, defines multiplication, and in the process also finds that multiplication commutes. The 7's primeness and multiplication's commutivity have equal standing.
Kronecker might have been the consummate intuitionist in stating that only integers are obvious. He also argued that infinity must be excluded from mathematics. But since he couldn't do much useful mathematics from those positions he blithely used all manner of "non-intuitionist" notions in his serious work. It was all a matter of not allowing his philosophy to interfere with his industry.