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- I don’t understand constructivism at all.
No numbers are real. They are all an entirely abstract construction, like lines and planes and open sets and closed balls and metric spaces and everything else.
If I construct a number by saying it’s the limit of the sequence sqrt(2)-1/n as n->infinity that’s just as real as the number 1 or 1/2.
It just seems really arbitrary to privilege one kind of construction over another. We want a complete ordered field, so we built the reals. Saying they don’t exist or they aren’t real or whatever seems just to be completely beside the point. They are real enough to do the thing we want them to do.
They are just as real as anything else in maths.
by seanhunter - I think people overindex on the continuity problem with the reals. I'm personally a bit of real-number denier myself as a constructivist / intuitionalist.
But when we say things like "the rationals are discrete" or "the computable numbers are discrete" these are very specific claims in the domain of measure theory, a theory which yields almost nothing of value except endless paradoxes and naval-gazing nonsense. Similarly when people say "the rationals are countable" and "the computable numbers are countable" this is taking for granted the Cantor notion of measuring cardinality by bijective correspondence, once again, a theory that yields nothing of value except endless paradoxes and naval-gazing nonsense.
In the practical sense the rational numbers are quite continuous -- between any two rational numbers there are an infinite (unbounded) number of rational numbers -- there's no notion of a "leap" the way there is with the integers. And any useful number can be approximated arbitrarily closely by rationals.
And for computable numbers there's even less of a gap. With rationals you can only approximate. But you can have a computable number that is exactly equal to the square root of 2, because a computable number is the algorithm by which you form arbitrarily close approximations. The square of that computable number is itself computable and is exactly equal to 2.
What do "real" numbers buy you? That is, what do you get for the trouble of building your formalism around numbers that are not computable? Basically nothing.
And if you're worried that sticking to the rationals and the computable numbers is too much of a concession to "physical reality", rest assured -- the rationals are just as unphysical as the real numbers because they are continuous already, and physics does not give us the power to measure the difference between two sufficiently precise rational numbers just as it barfs when you throw "real" numbers at it.
by andrewla - We should also mention Nicolas Gisin.
One of his mantras is Time is real; Real numbers are not. He conceives of real numbers resulting from processes (approximations, relaxations, computable calculations) that unfold over time. So there is a Heisenbergish uncertainty principle of observable precision and elapsed time.
Selected papers:
Indeterminism in Physics, Classical Chaos and Bohmian Mechanics. Are Real Numbers Really Real?
https://arxiv.org/pdf/1803.06824
Real Numbers are the Hidden Variables of Classical Mechanics
https://philarchive.org/rec/GISRNA
Time Really Passes, Science Can’t Deny That
https://arxiv.org/pdf/1602.01497v1
Popular articles:
Real numbers don’t cut it in the real world, this physicist argues
https://www.sciencenews.org/article/real-numbers-physics-fre...
- Related. Others?
How real are real numbers? (2004) - https://news.ycombinator.com/item?id=24029791 - Aug 2020 (112 comments)
How real are real numbers? (2004) - https://news.ycombinator.com/item?id=14080024 - April 2017 (265 comments)
by dang - I’ve had this paper downloaded for about a decade and haven’t gotten around to reading it, but thinking about it, especially if space and time are quantized (an undetermined question last I checked and almost certainly still so), there would exist numbers in ℝ that cannot be expressed as physical quantities, even with an infinite universe. It’s possible that even the algebraic numbers include numbers that are non-physical (although it might be a larger subset of numbers than the constructible numbers depending on the structure of space-time’s quantization).by dhosek
- This doesn't seem like a very good point to make, sure reals are uncountable and any set of them with labels is countable and of measure zero. That doesn't say anything about physics at all. In QM things are only discrete in certain ways, like energy levels, not positions. A wave function over space can take on any real valued value in its range. Probabilities are real numbers(norms of the wave functions), and there is no reason to believe they would be discrete. Take cosine squared, given any angle it takes on all values between zero and one at some point.
- It is too bad we don't have a pithy familiar term for the computable / definable / "nameable" / constructible reals. The most general class of undisputed numbers, consistent with the forms we actually use to represent quantities and perform numerical operations.
I view "real" numbers, in the context of uncomputable, unnameable numbers, to be as unfortunately named as "imaginary" numbers.
by Nevermark - I've long admired Chaitin for his original thinking and especially his ability to clearly convey his ideas about foundations, complexity and information in concise and digestible short proofs.
I'm a little surprised, however, to see him here proselytizing for a particular side in the constructivism debate. I associate him more with what he has described as a "quasi empirical" approach to mathematics[0] where the adoption of new axioms (such as for example the axiom of choice) is justified by their resulting in new, interesting mathematics.
But here, it seems his goal is to arrive at a somewhat Platonic conclusion, that either the reals are valid numbers or (seemingly he prefers) not.
My lay and naive take would be: if you adopt these rules (this Formal Axiomatic System) then you can have Big Fun in the playground of ever more esoteric and complex infinite cardinals, or if you adopt this other FAS you get to discover which results can and can't be obtained under a strict constructivist regime, and whichever FAS you choose it's just the same process of choosing axioms and applying valid deductive steps to arrive at a result you find interesting, with no more "existence" implied than the thoroughly non-Platonic existence of a solution to a problem, which can be demonstrated by solving it: if you take such-and-such steps then such-and-such result will follow.
I suppose the point being made is that avoiding axioms which imply the "existence" of the reals is more useful for doing physics, but that seems non-obvious in a field which for the last 200 years has seemingly sought to progressively make more and more phenomena intelligible by means of differential equations from infinitesimal calculus!
0: see, for example: https://arxiv.org/pdf/math/0303352, 1.9 Is Mathematics Quasi-Empirical
by ttctciyf