I think the argument is mostly that there are universal truths than math and physics describe not that our current level of math and physics are universal truths. So finding an example that we do not understand fully doesn't mean that there aren't truths in other aspects of math and physics. As for things like complex numbers there is an underlying debate that has been around in philosophy of science and math that distinguishes between discovery and invention. Our representation may have been invented but we discovered some thing that works historically and has predictive power.

> because they are useful?

But the usefulness is objective, that is, it is not an arbitrary product of the mind but rather it is dictated by the logic of things once the goal is set, so invention (or discovery) of useful things is more or less unavoidable.

As to quantum mechanics, you are talking about the variety of interpretations which from the practical standpoint are simply different ways of looking at quantum behavior, which, in turn, sometimes leads to different methods of calculation.

> But the usefulness is objective, that is, it is not an arbitrary product of the mind but rather it is dictated by the logic of things once the goal is set, so invention (or discovery) of useful things is more or less unavoidable.

It certainly seems like it is objective, and often it probably is, but in a more general sense, any instance of "x 'is' y" very often turns out to be subjective very quickly. Even with "is useful", things get complicated if one explicitly injects the dimension of Time into the question (it is there in the first place implicitly, but is easily overlooked).

I think you have the right idea. Most human truths are contingencies of our evolution and the evolution of life on earth. It's very hard to extrapolate from this to universal and alien truth.

> but didn't we just invent complex numbers because they are useful?

Not any more than we invented natural numbers because they are useful.

There are several ways to naturally derive complex numbers, either from mathematics or from physics.

For one, complex numbers are probably the simplest possible extension of the real numbers in which all real-valued polynomials have roots (for example, x^2 + 1 doesn't have a root if x has to be real). This is the same reason why the non-transcendental irrational numbers were invented (such as sqrt(2) ).

(Incidentally, the transcendental numbers (pi, e) are less justified than the complex numbers from this point of view - any polynomial of any rank whose coefficients are non-transcendental real numbers has roots that are either real non-transcendental numbers, or a complex number whose real and imaginary parts are real non-transcendental numbers )

For a physical explanation, complex numbers are the best way we know of describing wave mechanics (either classical or quantum), and in general periodic phenomena and how they compose.