We don't even know if real numbers are "real."
— Leopold Kronecker
(“God made the integers; all else is the work of man.”)
That's not even clear, unless you also assume you can measure exactly a unit length, which is not physically possible. Assuming you can measure something as a perfect rational value implies infinite precision, which is not possible.
All physically measurable quantities have uncertainty is what I think you mean, but that doesn't say anything about possible cardinalities.
No, this most certainly isn't true.
For example, take a meter defined (as SI does) as a fraction of the speed of light.
Now, for every length in the universe to be a rational fraction of this length is most certainly not true, because lengths, under relativity, form a continuum. Any speed is possible, so any length (via contraction) is possible.
There'd be no physical reason all physical processes would be constrained to a subset of rational grid points - things can move freely.
Heck, if all physical quantities are rational, are you claiming velocities only occur as rational numbers? That time is only rational numbers?
It's far more likely that they'd be irrational, and our man made units are the weird things.
Suppose two diagonal corners of a square with rational sides are somehow physically exact.... What is the physical diagonal length?
Oh yeah, irrational.
So no, physics isn't formed from rational numbers.
I'm not making any kind of statement about whether space-time is continuous or not. You are modelling spacetime as continuous and seem to be saying this is the only possible option. That's not true; there are a number of different models of how space-time behave at the smallest length scales. Try reading about loop quantum gravity for example; in that model spacetime is taken to be a discrete lattice at the smallest scales. In your model with continuous space-time then it's true that a square of side length 1 would have irrational diagonal of length root 2. In reality we don't currently have experimental evidence to say either way, and it could be that your square of side length 1 has a rational diagonal which approximates root 2.
The cardinality of the natural number set is the same as the cardinality of the rational number set. So, in some sense, you are saying that we know that the natural counting numbers are "real," which is a self-evident truth. In another sense, you are saying that fractions of an inch/cm/etc are "real" which is another self-evident truth.
The question is whether uncountable infinity somehow exits in nature (in the case of real numbers, it is a question about the infinitesimally small scale). To say that it's a "polite fiction to make some proofs works" is too strong a statement, we do not know the answer to that question. At the same time, our measurement instrument will always be discrete and bounded, so the question is seemingly beyond science itself.
I think laws of conservation do not hold for information complexity. I don't believe you.
All physically measured numbers have uncertainty, meaning the value obtained is not an actual number, but is a range, perhaps with some associated probability spread.
This is not measure zero.
We are also very capable "directly" using various things that may be continuums, such as energy, or time, or velocities, or many other physical quantities.
That's not true :)
A nice example is Chaitin's constant [1], which I can use in proof and books and define and on and on....
And it's explicitly and most definitely NOT computable :)
There are lots of numbers in lots of areas of mathematics, even symbolic mathematics that are not computable in the sense you want them to be computable. Chaitin's constant is the tip of a very big iceberg.
You're using circular logic by claiming the only numbers I can are are the computable ones then claiming all numbers I can use are computable. That's not true. It's a circular argument.
The only thing they give you is the ability to declare R to be Cauchy complete in some proofs. They're a polite fiction.
>The subset of R that can be identified is countable, because language is countable
Define "identified" in mathematically precise terms without using circular reasoning - and there is the flaw in this line of claims. You will find such definitions miss common uses of real numbers in the same way no finite set of axioms catch all true statements about integers. You will maybe get a nice consistent math subset of the reals with measure zero, but it will not cover all the cases you want it to. Thanks Godel
For example, using your "proof" of "because language is countable" would imply there can be no infinity, yet we use it all the time. Time, for example, may be a physical continuum, so a finite interval of time contains infinitely many time steps, and if your math cannot even represent physical reality then it's a pretty weak system.
>You know they're out there, but they will never be usable the way numbers you can actually describe are
Plenty of formal computational systems are capable of using the same set of numbers I can describe.
You're conflating being able to list every number in use one at a time, and operating on sets of numbers, like computation has done almost since day 1 using interval arithmetic.
>they can't be used in computations
Interval arithmetic, formal proof system - both capable of using all the numbers as sets that I can use - well beyond (Lebesgue) measure zero.
And to be pedantic, even countable numbers have infinite measure for certain measures :)
Well, they contribute "this sequence has a limit" for all Cauchy sequences. That's it. Some proofs that depend on the existence of those limits work. But you never do anything else with those numbers. Their "existence" is about faith more than it is utility. That's a very different kind of "existence" than a number like 2, or ⅓, or π, or Chaitin's Ω, or anything else that be communicated uniquely in finite space.
If this is your definition, it falls to the barber paradox.
In finite space, if we're this sloppy, I can define the simplest real number not definable in finite space.... But I did it in finite space.... This this notion needs to be made more precise ... Then the next one falls... And so on.
Go ahead, you seem to know enough math to know what constitutes a solid definition and a proof. Try to define the set you're talking about, making sure it's not circular nor excludes any numbers you think it should contain. It's not trivial, nor even perhaps possible.
This is what I mean by you're going to have a very hard time making your definition precise enough to be a true statement.
People have tried your path many, many times. From my poking at the lit on it it never works. Chaitin has spent decades on these things, and the result always falls just outside precise enough to make a proof.
The fact that not every finite string of symbols in every notation identifies a number uniquely doesn't break anything.
This is in fact just like the barber paradox. "I cut the hair of everyone in town who does not cut their own hair" is also not a paradox unless you add in assumptions not present in the original formulation. (Hint: where does the barber say they don't also cut the hair of people who do cut their own hair?) All you're doing is constructing straw men to be smarter than.
But you haven't actually addressed my main point: the only thing those numbers provide is making the limits Cauchy sequences exist. And that isn't actually useful outside of writing proofs that assume they do. For any other purpose, those numbers might as well not exist.
No - for example, the barber paradox (in the form of Russell's paradox) is what led to a revolution in set theory around 1910 and Whitehead's Principia Mathematica. It is a real problem, and led to the notions of classes and sets being distinct (and universes? I forget..)
The sloppy "definable number" game runs afoul of exactly these issues. If you look on google scholar for "definable reals" or "non-constructive reals" you can get a taste of the research still being done in this area. It's not as simple as I think you think it is.
The short of it is that there are many, many notions of definable, and they are not equivalent. Thus I asked you to provide one you think is suitable for your claims.
> the only thing those numbers provide is making the limits Cauchy sequences exist.
This is far from true - depending on the (set-theoretic) model of the reals, you end up with all sorts of interesting pure math questions. If I recall, certain sequences of modules over the reals end up with different homology depending on crazy subtle questions of the model of reals.
There's also a lot of equivalent methods for defining reals, most require proof to show they are equivalent.
There are lot of interesting questions on the constructions of the reals (Conway called his surreal numbers his best discovery, and they are spectacular - they spit out the reals in a really interesting way from only two rules - no Cauchy sequences or Dedekind cuts or common constructions in sight. They were discovered analyzing perfect information two player games....). Here's a paper with many cool constructions of the reals [1]. Claiming the only thing they provide is Cauchy sequence limits is missing a lot of other useful properties.
So you may not find the reals useful, but they certainly are incredibly deep and fascinating. But they are useful far beyond the concept of Cauchy sequences.