5 ms·

> It's pretty clear you didn't actually read all the way to the end. It's pretty clear that you aren't actually familiar with the historical philosophical deba

by btilly 2y ago

> It's pretty clear you didn't actually read all the way to the end. 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.

lisper 2y ago
> If a model is a physical system, then a model can't include uncountable sets of numbers that can never actually be described. 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.html 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.
- 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?

by btilly 2y ago

lisper 2y ago

> If a model is a physical system, then a model can't include uncountable sets of numbers that can never actually be described. 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.html 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.

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?