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> That depends entirely on whether or not real numbers can be physically represented in our universe. That is unknown. At least some real numbers can be. Take

by btilly 2y ago

> That depends entirely on whether or not real numbers can be physically represented in our universe. That is unknown. 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.html 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_theorem 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?

lisper 2y ago
> you are showing a complete lack of awareness of literally decades of historical debate within mathematics 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?
- btilly 2y ago
  If you know more than you let on, can you summarize the positions of the Platonists versus the Formalists versus the Constructivists? And can you explain how the word "exists" means very different things in each of those schools of mathematical philosophy? 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?

by btilly 2y ago

lisper 2y ago

> you are showing a complete lack of awareness of literally decades of historical debate within mathematics 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?

btilly 2y ago
If you know more than you let on, can you summarize the positions of the Platonists versus the Formalists versus the Constructivists? And can you explain how the word "exists" means very different things in each of those schools of mathematical philosophy? 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?