Number theory relates because his question suggests that a transcendental length might be especially problematic. I don't think it is, because of the hard limits on our ability to measure physical phenomena.

So it doesn't matter if we're talking about lengths that are integer, rational, algebraic, transcendental, computable, or otherwise. The end of the string is fuzzy at many different scales, so even defining its length at high precision becomes a problem, and at very high precisions, you actually change the length when you measure it.

If that sounds like me ducking the question it's because nature itself ducks the question.

> Number theory relates because his question suggests that a transcendental length

That isn't what the field of number theory concerns itself with.

I understand that. I'm just saying that the original post drags in the concept of transcendental numbers, a key aspect of real number theory, and attempts to mash it into the real world where it doesn't fit.

Recall what I said here: "In the real world, you always run into limits on how finely you can measure things before you run into quandaries about number theory."

In other words, when you measure entities and processes in the real world, it is unlikely that you'll ever have to think about the nature of the continuum, or transcendental numbers, or even for that matter precise integers.

For example, back in the '80s when I was using an Apple II computer to collect measurements from a microwave dish, it was obvious that we should measure amplitudes to four decimal places (or whatever, I forgot), and not concern ourselves with the impossible task of counting a precise integer number of energy quanta at each frequency.

Certainly when you're measuring energies in a particle accelerator you'll have different standards, but you will still always come face to face with the economics of "good enough" and even Heisenberg's hard limits on observability in principle.

So to reiterate: the constraints of the real physical world will always bind you long before you ever have to care about the vagaries of number theory.

Nevertheless, the abstract theory of real numbers is definitely useful even in the physical sciences, because that theory transcends all physical constraints and therefore imposes no a priori limits on observations. So it is not wise to fetter your mathematics with the chains of physical constraints.