This thinking has some superficial appeal, but I don’t think quantization in various domains frees you of the need for the continuum when those domains interact.

I don’t think there’s any reason to believe that, say, the fine structure constant, or the ratio of the mass of a neutrino to the mass of an electron, are rationals, or even algebraic.

Maybe there is though? That seems like very much a question for philosophy, since scientific measurement will never be able to tell us for certain.

In "reality", there is no such thing as an electron that has some mass. It's leaky abstractions and approximations all the way down. If we had a "complete" theory of physics, it would still be the same situation for all practical purposes, since you can never know "for sure" that is it "correct".

Hence, rationals are plenty sufficient for all of physics, assuming you bother to reformulate all the theories that are historically based on real numbers. However, there is no point in doing this, so only a few niche people try (who, of course, do it for philosophical/aesthetic reasons).

The tools of mathematics are actually just better suited to working on the continuum than on discrete numbers. So maybe that does suggest that when physicists use the tools of mathematics they will always wind up with real approximations to discrete reality.

No matter that physical reality insists that the number of people in a population or U-238 atoms in a lump of metal has to be an integer, mathematics will tell you that that integer nonetheless has a distinct, nonalgebraic natural logarithm, and that that number is useful in predicting how many people will be in that population or U-238 atoms will remain in that lump of metal at some later time - even if the raw result of any such calculation will be a nonsensical noninteger.

Indeed, the Avogadro number is an integer that is always given in the form of a floating point number. The reals is such a useful abstraction.

No, it is typically described using scientific notation - floating point is a fixed width representation of a number with fractional parts.

And floating point numbers - and scientific notation - can both only express rationals. They aren’t ‘reals’.

My point was that it seems that in calculations it is useful to think of very large integers that are known to some approximation as real numbers.

Just like integers, the real numbers are an abstraction, and so their reality as such is a philosophical question (which, despite being quite straightforward to answer, I have seen to confuse a lot of people).

Neither are most integers! As far as we know, everything is also finite.

Knowing that energy is quantized is different from knowing that it's discrete.

Hypothesizing that space & time are discrete is more difficult.

Reason and logic discretizes, but this doesn't mean reality is itself discrete.

If the universe is infinite in size (given that measurement is close to the topology being flat), or time is infinite in the future, then no. Or there is an infinite multiverse (take your pick of which one).

It is conceivable that the universe could be contiguous. Our current scientific thinking points to it being constituted from discrete pieces though.

> But at the lowest level, isn’t everything (time, space, energy etc) basically discrete, so the reals aren’t really real?

That literally doesn't matter one way or the other in the slightest, for two huge reasons:

1. No mathematics is exactly the same as reality itself, and cannot be, even in principle. Mathematics can be used to useful model certain limited aspects of reality, but of course models aren't the same as reality.

2. The mathematics of the reals and of continuity in general are essential for a vast number of kinds of practical computation that sometimes (not always!) turn out to be vastly harder to do without the assumption of continuity. We often want to bring to bear the whole apparatus of established mathematics to solve problems in order to get answers, and no one trying to get answers cares whether the mathematics used is "real", only whether the final answers are useful (accurate etc).

Actually there's a third reason that is even more important to some people: this is usually merely about word definitions. Words like "real" and "exist" have sharply defined unambiguous technical definitions in mathematics that are only indirectly related to the loosely defined vague non-technical definitions of the same words that we use in natural languages like English.

It is ultimately irrelevant whether e.g. the real numbers "exist" in the physical universe, because whether they do or not, they provably exist in (most) formal mathematics -- the word "exist" does not have the same meaning in the two realms.

It's still interesting to ask whether the physical universe has reals or continuity or infinities or infinitesimals etc., it just shouldn't be confused with the issue of their technical existence (or "reality") in mathematics.