That depends entirely on whether or not real numbers can be physically represented in our universe. That is unknown.
> And yet mathematics in the form of ZF says that the real numbers must include those uncountable sets.
Yes, and whether or not that is a model of something real is an empirical question that we do not yet know the answer to.
> The existence of those real numbers depends on philosophical debates about what "existence" means.
Yes, I address that here:
https://blog.rongarret.info/2015/02/31-flavors-of-ontology.h...
That is an earlier essay which is not a part of this series. The upshot is that the question is not whether or not the reals exist, but rather what ontological category the belong in. That is an open question.
> But you can't get there from physical systems.
Of course you can. Unless you are a dualist, we did get there from physical systems because we are physical systems. There is no other way to get anywhere.
At least some real numbers can be. Take the construction of reals from Cauchy sequences of rationals. We can easily write a computer program that produces a sequence of rationals that converges to e. To produce later terms it would have to run for a long time on a large computer, but the program still represents e.
The existence of numbers other than those which are representable as a computation on a computer is a question for philosophical debate. That is what I was referring to with:
> > The existence of those real numbers depends on philosophical debates about what "existence" means. > > Yes, I address that here: > > https://blog.rongarret.info/2015/02/31-flavors-of-ontology.h...
Except that you don't address that point there. You don't say anything relevant to it other than the trivial statement that existence is not a pithy value.
Again, you are showing a complete lack of awareness of literally decades of historical debate within mathematics.
> That is an earlier essay which is not a part of this series. The upshot is that the question is not whether or not the reals exist, but rather what ontological category the belong in. That is an open question.
Of course it is open. The answer depends, among other things, on your beliefs about God.
However the best that can be proven is that we have a collection of axioms which involve statements about existence, that we have good reason to believe will never contradict each other. What connection there is between this statement of existence, and normal notions, is very abstract and complex.
> > But you can't get there from physical systems. > > Of course you can. Unless you are a dualist, we did get there from physical systems because we are physical systems. There is no other way to get anywhere.
No. We proved formal statements about collections of axioms that made conclusions about existence, without attempting to connect the technical definition of existence in those formal definitions with any other notion of existence that people might have.
We certainly did not get to anything resembling a notion of actual existence. Merely the conclusion that if such and so axioms are true, then that statement is also true.
And there is no need to go to complicated stuff like "real numbers" to see all of the key issues.
The best example that I've seen is https://en.wikipedia.org/wiki/Robertson%E2%80%93Seymour_theo.... That theorem says that any category of finite graphs which is closed under taking graph minors, has a finite list of excluded minors that categorizes the category. The theorem's proof is nonconstructive, which means that it offers no way to find that finite list. It offers no way to put an upper bound on that finite list. If you were given a purported complete finite list, it offers no way to prove that this list is complete.
When I say "it offers no way", I'm being mild. In fact we can prove that there are categories with finite definitions such that there is no algorithm which can find the list. No algorithm which can verify an upper bound. And no algorithm to verify that an answer is correct. Each of these runs into Halting problem kind of liar's paradoxes.
Let's make this concrete. The set of planar graphs is closed under graph minors, and it has exactly two minimal forbidden minors. They are K5 and K3,3.
For the set of toroidal graphs - that's like planar except that you're drawing it on a torus - we have thousands of minimal forbidden minors. We currently have no reason to believe we have the complete list, no upper bound on how many there might be, no upper bound on the size of forbidden minors, and no reason to believe we could verify a correct list if it was given to us.
The Robinson-Seymour theorem leads to the statement that there must exist unverifiable finite things of unknowable size. The set of minimal forbidden minors for toroidal graphs may well be one of those things. But what does it mean to say that an unverifiable finite thing of unknowable size "really" exists?
I'm aware of more than I'm letting on in that article, though I do confess I'm not up on my graph theory. The Robertson–Seymour theorem went over my head, so I don't understand the point you intended to make with it.
> The Robinson-Seymour theorem leads to the statement that there must exist unverifiable finite things of unknowable size.
I have no idea what an "unverifiable finite thing" is.
But the existence of things that we cannot produce is not news. Chaitin's constant fits that description. Is the point you're trying to make substantively different from that?
Skipping on...
My point is substantively different from Chaitin's constant.
Chaitin's constant is unknowable because it is built off of an infinite series of answers to the halting problem. Which we can't solve. It's no surprise that a construction which starts with something we can't do, could result in a number that we can't determine.
What I'm talking about is a theorem that says that a certain completely finite thing must exist. But in general we have no way to find that thing, no way to put an upper bound on its size, and no way to verify that we were correctly given it if someone gave it to us. And I'm asking in what sense we can really say that this finite thing exists.
Now to the specific example. A graph is just a list of vertices, and a list of connecting edges between them. A graph is called planar if you can draw it in the plane without any edges crossing each other. Obviously if a graph is planar, it remains planar if you delete a vertex, delete and edge, or merge two vertices together that were connected by an edge. Those are the graph minor operations, and so we say that the set of planar graphs are closed under the graph minor operation.
Now as it happens, there are two important non-planar graphs. One has 5 vertices, all connected to each other. That's K5. The other is two groups of 3 vertices, all vertices in the first group connected to the second. That's K3,3. Every other graph that is non-planar will contain one of those two as a minor.
Planar graphs are not the only class of graphs closed under graph minors. Toroidal graphs, graphs you can draw on a torus, are also. As well as many other more exotic classes of graphs. The Robinson-Seymour says that each such class has a finite set of forbidden minors. Just like K5 and K3,3 for planar graphs.
But that set may be much larger. At last count, we have 17,523 forbidden minors for toroidal graphs. We don't know how many more there may be. We don't know if we have the full list. And we don't even know if the answers are provable in ZFC. They may not be.
So my question is this. We have proven the unique existence of a finite thing. But as far as we know, it is impossible for us to say anything useful about it other than that it exists, it is finite, and here are a bunch of things that are in it. In what sense does it really exist?
So first of all, the kinds of things you are talking about here are way beyond the scope of what I'm writing. This is a series about the scientific method, not about math per se, and the target audience is bright but non-technical high school-age kids. Math is relevant only insofar as it turns out to be extraordinarily useful for building scientific models.
Second, the topic of this post was not existence, it was truth. Existence is (mostly) a red herring. Yes, the thing you describe exists. It exists in the ontological category of mathematical objects, which is a sub-category of the ontological category of ideas. That's all I have to say about it. So yes, I could summarize Platonism vs Formalism vs Constructivism, but that is simply not what I'm writing about.
The reason I'm bringing up truth in math at all is to lay the foundation for a discussion of whether or not the quantum wave function describes something real. But that won't come until much, much later.
BTW, in case you're wondering, the answer is yes, the quantum wave function does describe something real. See:
https://blog.rongarret.info/2018/05/a-quantum-mechanics-puzz...
https://blog.rongarret.info/2018/05/a-quantum-mechanics-puzz...
In any case, the answer to your original question is obviously "no". Why do you think this matters?
Outside of that, maybe if you look somewhere like string theory?
Still, in general, the continuum is enough. And while it takes work, I'm pretty sure that nothing relevant to physics about the continuum cannot be said within a strictly Constructivist version of the continuum. So no, there is no physical theory from which we can justify the existence of more reals than rationals. The justification must be philosophical. And the answers that we accept will strongly depend on what our notions about what existence should mean.
Related to Hilbert spaces, consider the set of analytic functions with period 1. Under the L2 norm, this is an inner product space. It is also uncountable. However as Fourier showed, the Fourier series for the square wave does not converge within the set of analytic functions. In fact it is trying to converge to a function that not only fails to be analytic, it fails to be continuous!
First, I'm all for convincing people that the wave function is something real. If you believe that strongly enough, then you're pushed towards accepting the Everett Interpretation. Which makes far more sense to me than any other interpretation of QM.
But I would suggest that you get them there by the shortest reasoning that you can. With minimal detours. The more material you have, the more likely that someone with different preconceptions will find something to disagree about. Worse yet, people will lose track and abandon it.
I would also suggest that you simplify your writing. According to https://readabilityformulas.com/readability-scoring-system.p... your writing is college entry level. Readability is weird, we don't notice that it matters until we find ourselves unable to understand text. And then it is a wall. Meaningless phrasing choices can improve readability and broaden your audience.
Now for a more substantive criticism. But one of choice of topic, where you likely reasonably disagree with me. You are writing about the scientific method. But the style of your writing undermines the essential spirit behind the scientific method. What do I mean by that essential spirit? I mean the thing that Feynman was trying to explain in http://www.feynman.com/science/what-is-science/. It is the attitude that leads to statements like, "Science is the belief in the ignorance of experts."
Now you're telling them what to believe. You aren't telling them how they could think of it. You aren't encouraging them to disagree with you and question you. You're telling them what is true. I think that conveying the skeptical attitude is more important than convincing them of any particular fact.
Moving on, I see you've added a number of things to your article. They are significant improvements. But I would suggest an improvement to one in particular. Your arithmetic example.
Counting to negative numbers for debt makes sense. But you don't show how you get something that unexpectedly is useful. So I would make the chain that you lead people down to be counting, to positive integer arithmetic, to negative numbers for debt, to fractions, to the idea that we can put this on a line, to the question of whether we have everything on that line, to the reals. Just outline it. And then point out how we started with counting, and wound up most of the way to geometry...doesn't look like counting any more!
Finally, I think that high school students are capable of more philosophy than you do. In particular there are lots of youtube videos and articles that try to explain why uncountable means more. If they can see those explanations, why can't they encounter the idea that all such claims require additional assumptions? Thinking about how to connect reasoning and computation is something of interest. And leads to topics that are fundamental in such places as logic and CS.
Anyways, this is wrong and I don't know if you read it. But if you do, hopefully you found at least one thing that's worth thinking about.
> But I would suggest that you get them there by the shortest reasoning that you can. With minimal detours.
Yeah, well, that's the challenge, isn't it. The target audience is so far away from QM that it's hard to fathom. These are people who believe (or at least seriously consider the possibility) that Jesus rose from the dead, the earth is 6000 years old, demons are literally real...
> Now you're telling them what to believe.
Am I? That is actually something I was trying very hard to avoid. Can you point out where I'm doing that?
But it is also important that you make it easy to follow your explanation without having to challenge emotionally held beliefs. For example, consider those religious beliefs. A big part of why we hold them is that trying to give up those beliefs involves accepting a lot of pain. Both from external causes, like the condemnation of your religious community, and internal ones, like the gut level belief that even thinking about rejecting Jesus puts you at risk of going to Hell. It is very hard to engage in quiet contemplation while feeling that you're risking eternal damnation.
As for where you tell people what to believe:
And indeed it is a counter-example to the idea that mathematical truths are grounded in actual objective reality, but that is not news -- we already established that with the example of negative numbers and imaginary numbers.
You claim to have shown that the ideas were not grounded in actual objective reality. But that isn't true. You only showed that they aren't grounded in the first way that you tried to ground it. Both of those examples are things that can easily be grounded in objective reality in other ways.
But the Banach-Tarski construction is different. It depends on the axiom of choice. Which in turn is a statement about a process that cannot be actually done in objective reality. And without being able to do that process, we're unable to do Banach-Tarski in objective reality.
I think that's an important difference. And it is one that is missed when you tell them that this is like that other thing which you just saw.
It's pretty clear that you aren't actually familiar with the historical philosophical debates from a century ago about the foundations of mathematics.
See, I can return snark for snark.
If a model is a physical system, then a model can't include uncountable sets of numbers that can never actually be described. And yet mathematics in the form of ZF says that the real numbers must include those uncountable sets.
The existence of those real numbers depends on philosophical debates about what "existence" means. Debates which are well out of the realm of the physical.
To get to Banach-Tarski we have to go further. In addition to accepting the existence of uncountable hordes of numbers that cannot be specified in any way, we ALSO have to assume choice - essentially the existence of arbitrary kinds of infinite sets of coin flips. Which is also non-physical.
It is only after accepting both of these decidedly non-physical things as valid mathematically that we're in a position to prove Banach-Tarski. Deciding not to accept one or both leaves a system as consistent as classical mathematics. And denying either non-physical assumption leads to a system of mathematics where Banach-Tarski is false.
To believe Banach-Tarski, we have to accept one of two things. The first is that it is just a formal statement in an axiom manipulation game about something whose existence we're not worried about. That's Formalism. Or we have to accept the reality of the nonphysical existence of mathematics in a realm beyond us. That's Platonism, and it is no coincidence that Platonists like Gödel tend to also believe in God.
But you can't get there from physical systems. Or using any set of ideas modeled by starting with physical systems.