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.
Numbers and other mathematical concepts are used to describe and reason about physical systems. More or less everyone agrees on that much. Why make the further claim that some of these mathematical concepts or objects are "real"?
There isn't any one-to-one mapping between numerical concepts and physical systems. Even for a finite collection of physical objects, we could associate with it a number, which is the number of items in the collection, but we could also associate with it another number, which is the number of possible combinations of items of the collection. It depends on our interests and what we want to do. Even grouping items into particluar collections is dependent on the goals we might have in some situation. We might choose to group items differently, or just measure their total mass. More generally, any physical magni9tude can be associated with an arbitrary number, just by a choice of unit.
We use numbers, and mathematical concepts more generally, in many different ways to reason about physical systems and in our technical constructions. I don't see why we need any more than that, and to say that some mathematical concepts are "real" while some are not.
> 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.
Can you explain what it would mean to "find a constructible real in nature"? Maybe we just have different ideas about how this would be spelt out
That there may be structures in nature that are 1-to-1 with any given (constructible) mathematical concept.
Anywhere there is conservation of quantity we get addition and subtraction. Anywhere quantity can be looked at from two directions we get reversibility, i.e. positive and negative perspectives of the quantity. Multiplication happens anywhere two scalar values operate on each other, or orthogonal quantities create a commutative space between each other.
We find reversibility, associativity, commutativity, and many more basic algebraic structures appearing with corresponding structures in physics. And more complex algebra where simpler structures interact.
Wherever there is a dependency between constraints applying, we have logical relationships.
So that is what I mean about mathematical structures appearing in nature. Numbers/quantities just being subset of those structures.
> Can you explain what it would mean to "find a constructible real in nature"? Maybe we just have different ideas about how this would be spelt out
My emphasis is really that we don't/won't find un-constructible reals.
I WEAKLY claim (given that reality increasingly looks likely to extend beyond our universe, and more conjecturally, may be infinite), that any given constructible math structure has a possibility of appearing somewhere. Perhaps all constructible math appears somewhere.
However, that is the weaker claim I would make.
I more STRONGLY claim that un-constructible mathematical structures are highly unlikely to have counterparts. Which includes un-constructible reals.
The un-constructible real invented by Cantor, was a value r, which has infinite decimal digits, but with no finite description. No algorithm to even generate.
It is an interesting concept, but an un-instantiatable (even in theory) one. Infinite information structures immediately present difficult problems just for abstract reasoning. How corresponding structures might exist and relate in a physical analogue isn't something anyone has even attempted, as far as I know.
--
I don't think I am saying something controversial.
If there is anything surprising in what I am saying, it is that I am addressing the fact that reals got defined in a way that includes un-constructible reals.
The only implication most people know about that, is that the cardinality of reals is greater than the cardinality of integers.
But what might be very surprising for many, is that the cardinality of constructible reals, every possible scalar number we could ever calculate, measure or apply, is in fact the same as the integers. A distinction/insight that seems highly relevant and useful when dealing with instantiatable math, physics and computation.
There may be, or there may not be. I don't think we can make a definitive argument either way without perfect knowledge of the structure of the physical world, or at least some part of it. What we have are mathematical models of physical systems that are valid to within some error margin within some range of parameters. The ultimate structure of the physical world is so far unknown, and may be forever unknown. Any actual physical situation is too complex for us to fully analyze, and we can only make our mathematical models work (to within some error) when we can simplify a physical system sufficiently.
I think I understand better though what your main point is: that whatever physical theories or models we might have, the unconstructible reals won't be an essential part of it, i.e. even if we have some physical theory or model whose standard formulation might be committed to unconstructable reals, we could always reformulate it into a predictively equivalent model which doesn't have this commitment. Is that fair?
That might be true for all I know. I'm not sure how to evaluate it (IANA mathematical physicist). It seems plausible though. There's the example of synthetic differential geometry, which has a different conceptual basis to the standard formulation of differential geometry, and at least suggests that you can't a priori rule out the possibility of alternate formalizations of any mathematical structure. I don't know enough about it to say whether or not it postulates something equivalent to unconstructible reals, it's just something that came to mind as maybe being along the lines of your point of view
https://ncatlab.org/nlab/show/synthetic+differential+geometr...
Yes, that is a good way to put it.
Unconstructible reals are a major and interesting "what if". What if there were numbers that had no finite relation to other numbers?
It is a great idea, from a mathematical boundary pushing way. So abstract we can never do anything not abstract with it!
> even if we have some physical theory or model whose standard formulation might be committed to unconstructable reals, we could always reformulate it into a predictively equivalent model which doesn't have this commitment. Is that fair?
But unconstructible structures can never be reformulated as constructible by definition. That would mean they were constructible.
We can never define a specific unconstructible real.
But anyone who manages to create an interesting systems theory that uses them, with dynamics that constructible math can't match, would have created a major work of mathematical art!