Which would mean integers are baked in, rationals too, but non-constructible reals (essentially all reals, given any degree of approximation) are a useful abstraction but don't actually exist in any way.
Reals are not real.
(Roughly) Equivalently: There may be no perfect circles in nature.
For one thing, they're speculative: the current theories that give the most precise and accurate predictions within their respective domains of applicability are general relativity and quantum field theory. These theories are based on continuous space and time, and no attempt to base them on discretized space and time has been successful (AFAIU both QFT and GR rely on Lorentz invariance, which means there's no absolute rest frame, hence no absolute unit of time and space, but a discretized spacetime would require absolute unit values for space and time, hence an absolute rest frame).
Should we conclude then that the reals are real, because they're components of our best current physical theories? Maybe, maybe not: these are features of our current best models, but we don't know, and possibly will never know, the ultimate nature of physical space and time.
For another thing, even if space and time are fundamentally discrete, there's still no doubt that the mathematical theories based on real numbers are effective in making predictions, and we would still like to use them. That's means they should have some logical foundation which can guarantee that reasoning using them is correct.
But this claim is nowhere made in the comment? Like clearly the transfinite ordinals aren't real, but no one would say that implies they aren't a very useful mathematical idea (and also just interesting in and of themselves).
"It seems wrong to me to base mathematical ontology on speculative theories about the nature of physical space and time."
Given the theme and title of the article.
Given one of the primary (but often not emphasized) properties of the class of reals, is that it contains and is actually dominated by the un-constructible reals, an argument that the total class of reals isn't represented in nature seems unremarkable to me.
Un-constructible reals are a highly exotic abstract concept. You can never actually identify or operate on one.
I do believe all the constructible reals (what most people think of when they think of real numbers), are likely to be found in nature/reality.
It is a mishap of terminology, that the larger class, which is unnecessary for middleschool math, or any math most people will ever encounter, has a pithy name "real".
While the practical concept, that directly correlates with arithmetic, algebra, calculus, diff eq, etc., has the unwieldily title of "unconstructible reals".
So in order to avoid having to talk about constructible vs. unconstructible numbers to kids who shouldn't need to care, we use the pithy term and actually throw unconstructible reals into the mix, where it wasn't necessary at all.
We tell kinds how the reals have a higher-order uncountable infinite cardinality, relative to integers. Which is true, but gives the impression that cardinality is somehow a concept necessarily linked to algebraic and other practical numbers that we introduce at the same time, which it is not.
The set of constructible reals has the same countably infinite cardinality as the integers.
I'll use this definition: a constructible real is a real that you can describe uniquely in a purely mathematical way (and prove that your description identifies a unique real). IE the positive solution to x^2=2 identifies a unique real. Also, the first positive solution to sin x.
Now if you accept that the universe is infinite (and has infinite matter in every direction), you could get representations of un-constructible reals in your universe. One pretty contrived way to get an un-constructible real from this infinite universe is this: start at earth with some velocity, lets say 0.01c in some direction. Start with r = "0." Every second, find the closest particle. Take the amount of meters (rounded down) that this particle is away from you and append it to r.
So if after one second the closest particle is 145m away, r becomes 0.145. If after another second the closest particle is 0.14m away, r becomes 0.1450
The value this process converges to could be un-constructible.
I would argue that is still a constructible real. Only practical issues make calculating that value difficult.
Since we are instances of physical constraints ourselves, just because we can't do a particular measurement, directly or indirectly, doesn't make a value un-constructible in the mathematical sense.
(Also side noting, that we handle superposition/quantum collapse explicitly, by actually generating many alternate counts, or an expression that covers all the counts.)
Note that your "algorithm" was finitely statable, and that its "data", consists of a finite number of particles (in any given superposition).
But if I were going to argue for an un-constructible number with a physical counterpart, your thought experiment is a good starting point!
Well if the universe is actually infinite, the amount of data in the number this process approaches is infinite.
> I would argue that is still a constructible real.
That is what I was going for. I was trying to think up a construction that leads to uncountably many reals, but the construction I gave doesn't really work.
Consider a different situation:
Start with r = 0. (a number in binary)
Look for an unstable radioactive isotope. Wait for its half time. If it decays within its half time, concatenate 1 to r. Else concatenate 0. Look for another radioactive isotope and repeat.
The number this process approaches could be real number between 0 and 1 (including both bounds). Is the resulting number constructible?
That's a good one.
I am going to say it absolutely is. Then acknowledge why others may feel very strongly that it isn't.
So that's quantum mechanics, which from a field theory standpoint is completely deterministic. It just appears non-deterministic to us, because we are also superpositions. We are quantum structures too. And our field would keep splitting in two, at each measurement/decision point, so our total quantum field would remain completely predictable.
But, it is true that each of our superpositions would have the experience of a completely random set of digits, going off to infinity.
But, despite it adding additional physics and not explaining any more, some physicists seem to still think that there is a real collapse, not just an already explainable experience of collapse, of quantum fields.
So, I think it is fair to say that if that was true, then truly unconstructible events would be happening. There would be no way to form an expression or algorithm to ever predict the flow of digits, even in principle.
So you nailed the best possibility for it that I can think of.
And this is a little circular, but between collapses adding a new phenomenon with no additional explanatory power (Occam's Razor be damned!), and the magic event decisions, are why I don't believe collapses happen.
Collapses don't just imply that a magical event decision is made whenever we set up some careful experiment with one particle, but that all possible event situations in space-time, even in us, are constantly being magically decided as we are exposed to information about them, all the time.
Given virtual particles are constantly frothing around even in empty space, this means that all of space-time is constantly flooding us with an unimaginable amount of magically created information. The magic bandwidth would be insane.
One magical fundamental physical constant seems implausible to me. But 10^(very very big number) of magical decisions animating all of our universe and us every pico-second? Well, that would just be ... unconstructible!
This interpretation of reality, at least how I stand it, seems like the correct one to me. (At least that's how it feels to me.)
But this combined with:
> I am going to say it absolutely is.
Means that all reals in [0, 1] are constructible, and as a result of that, all reals by modifying the starting value of r to ie 1. instead of 0., or 10. (2. in decimal).
Constructible reals are continuous over [0, 1] in that there are no gaps between any interval between constructible reals [r1 r2] that are not filled by more constructible reals (and in fact, the cardinality of the constructible reals within any interval is the same, i.e. countably infinite, in a fractal way).
So there is no obvious motivation for anything but constructible reals from that standpoint.
Unconstructible reals were invented (or at least used) by the mathematician Cantor to explore ideas about different infinities. A "real" number with infinite decimal digits but not any finite description lets him create a number space larger than the constructible reals, a larger infinite cardinality.
So there is nothing missing in a [0 1] constructible interval. Or to put it another way, constructible numbers are closed. There is no sqrt(-1) situation requiring unconstrucible numbers to fill, like the square root of -1 required imaginary numbers (or geometric algebra dimensions) to fill.
But [0 1] contains the higher infinity of unconstructible reals in it, if you want. But I am unaware of any claim that they solve any problems by being included, other than exploring interesting puzzles related to unconstructible numbers as interesting ideas in themselves.