> But why think of any numbers as being "in nature"? And what does that really mean?

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.

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griffzhowl 2 days ago | parent [-]

> That there may be structures in nature that are 1-to-1 with any given (constructible) mathematical concept.

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...

	▲	Nevermark 2 days ago \| parent [-]
		> 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, 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!