This is intended to explore a a thought I had, rather than making any particular argument about truth.
The canonical example of a thing which is true without any obvious physical referent is the statement 2+2=4. It is true about fingers, sheep, particles, and galaxies; but intuitively it does not seem that any of those truths encapsulates the full meaning of the statement. Moreover, it certainly seems that there is nothing anyone could do to make the statement untrue; it seems that it would have to hold in any universe whatsoever.
Now my thought: How do we know that the physical universe operates on this sort of arithmetic, and not arithmetic modulo some obscenely large number? Suppose we repeat the experiment that convinces us 2+2=4 (and let's note that babies are presumably not born knowing this; they learn it by counting on their fingers, even if they do so at too young an age to express it in words), but with much larger integers. Perhaps we might find that, when we take 3^^^^3 particles, and add 1, we are left with 3^^^^3 particles without any awareness that any particles have disappeared. And what is more, if we take three sets of 3^^^^3 particles, and measure their mass separately and then together, we find that we get the same mass. After some long sequence of such experiments, perhaps we might convince ourselves that physics actually operates on integer arithmetic modulo 3^^^^3. (Which would be unexpected in that the physics we know operates on complex numbers, not integers, but perhaps that's an approximation to some fantastically-finegrained two-dimensional integer grid.)
What would this mean, if anything, for the truth of such statements as 2+2=4? It seems that it would then be a contingent truth, not a universal one; that there could in principle exist a universe whose physics operated on arithmetic modulo 3, so that 2+2=1. (Presumably such a universe would not have any sentient beings in it.) What if 2+2=4 is an observed fact about our universe on the same order as the electromagnetic constant or the speed of light?
What does it mean that the universe operates on a certain sort of arithmetic? A lot of descriptions of the universe uses conventional arithmetics, some theories use rather SU(3) group or Z2 group or whatever. (Arithmetic is so general that the other mathematical constructions we use are usually somehow reducible to arithmetics, but that we have a fairly large formal system which unites all branches of useful models isn't particularly surprising.)
How do you in principle decide whether the strange behaviour of large number of particles is a fact about under what kind of arithmetic the universe operates or whether it is an additional physical law governing putting large number of particles together?
Anyway, the referents of "2+2=4" are all sets of balls, fingers, planets etc. on which we perform counting. It is a contingent truth, only abstracted a lot. Universal truths are either a philosophical confusion, or theorems of arbitrary formal systems. Either way, there is no need for that category.
Perhaps I am not phrasing my question very well. I am not asking about the existnece of universal truths, but about the human intuition that such truths exist. When someone says "2+2=4", it feels as though they are asserting a necessary truth, something that cannot possibly be otherwise. See, for example, Sniffnoy's comment above, where he asserts that even if fingers and balls and whatnot counted by integers mod 3, the plain unmodded integers would still exist. This seems to me like an assertion that unmodded arithmetic is a universal truth that... (read more)