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To reduce math to models of objective reality, you need to say what a model is. To get Banach-Tarski you need to either accept Formalism - it is just a formal

by btilly 2y ago

To reduce math to models of objective reality, you need to say what a model is. To get Banach-Tarski you need to either accept Formalism - it is just a formal game whose concepts like uncountable sets do not "really" exist, or you need to accept that Platonic reality with uncountable things in it. If you try to model math in a way that can actually be rooted in objective reality, then you wind up with some form of Constructivism. And now Banach-Tarski goes away.

lisper 2y ago
> To reduce math to models of objective reality, you need to say what a model is. I did. See the paragraph that starts, "A model is any physical system whose behavior correlates in some way with another physical system." > If you try to model math in a way that can actually be rooted in objective reality... It's pretty clear you didn't actually read all the way to the end.
- layer8 2y ago
  > "A model is any physical system whose behavior correlates in some way with another physical system." So when X is a model of Y, then Y is always also a model of X? (Since correlation is generally a symmetric relationship.) That seems like a strange definition of “model”.
  - gavagai691 2y ago
    I agree that this is a problem with the above definition of model, analogous to the (IMO fatal) problems with https://en.wikipedia.org/wiki/Deductive-nomological_model#Weaknesses https://en.wikipedia.org/wiki/Deductive-nomological_model#We....
    - lisper 2y ago
      That has to do with causality, which is a complicated topic that I was trying very hard to avoid at this point in the exposition. But maybe that was a mistake.
  - lisper 2y ago
    > So when X is a model of Y, then Y is always also a model of X? Yes. > That seems like a strange definition of “model”. Yes, it does, but I can defend it. It's analogous to the definition of "information". If a system X contains information about a system Y then it is always the case that Y also contains the same amount of information about X. The distinction between "model" and "modeled" is purely artificial in the most literal sense: the thing we generally label "model" is the thing we humans made.
- gavagai691 2y ago
  You don't think that a model is something abstract? Abstract doesn't have to imply nonphysical in the sense that people think souls or God are nonphysical. I mean abstract in the sense that language, mathematics, or a sketch are abstract. To expand on this: I think models are representations, and whether or not something is a model depends in some way on human minds. (In particular, it depends on whether a something would be understood by a human mind to be a representation.) I don't think that any correlation between physical systems qualifies one as a model for the other. Your definition as written would include any two things that are connected causally, or have a common cause, as models for one another. One problem (though not the only one) that I have is that your definition removes any mention of human minds. In particular, I think "representation" is, broadly speaking, some kind of correspondence relationship between linguistic or pictorial things (where I include mathematics as "linguistic") and physical reality, and "a representation" is some linguistic or pictorial thing that corresponds to reality. I think that a model is a kind of representation. A model is a kind of representation where for convenience and tractability, certain aspects of reality are left out or "abstracted away" (deliberately), with the goal of understanding the real world by understanding the simpler representation of the real world.
  - lisper 2y ago
    > You don't think that a model is something abstract? They certainly can be, but they don't have to be. An orrery, for example, is not abstract. > whether or not something is a model depends in some way on human minds Well, yeah, but what exactly is that "some way"? That is exactly the complication I was trying to avoid. Maybe the word "model" is too loaded and I need to pick a different word. I'm open to suggestions.
  - seanhunter 2y ago
    > You don't think that a model is something abstract? Models definitely don't have to be abstract. For example, researchers will talk about studying a disease or the effectiveness of a treatment in a "mouse model"[1]. That model is an actual concrete mouse that is being used as a model of a human. It's not abstract in the sense of language, mathematics, or a sketch, and they do the research by looking for the physical effects on the mouse model and drawing an analogy to what would correspondingly happen in a human. [1] eg https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7121329/ https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7121329/
- 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.
gavagai691 2y ago
Well, it is an empirical question whether or not matter is continuous and infinitely divisible, or discrete. Our best theories tentatively suggest that matter is discrete (though it is hard to say how much confidence to put in that, or how we could really know either way).
- btilly 2y ago
  While interesting, that doesn't matter for what I said. An example of a key question is whether mathematical constructions can use the absolute truth of statements in mathematical constructions. Even if those absolute truths are not themselves something that can be settled by any algorithm. If they can, then we get all of the weirdness of infinite set theory, such as that there must be more reals than rationals. If they can't, then all of mathematics could fit within a countable universe, and things like Cantor's diagonalization proof just demonstrates a Halting problem kind of self-reference in the definitions of the real numbers. And this brings us to my point. It doesn't matter whether the universe is continuous and measured to finite precision, or discrete. The set of measurements that we could potentially ever make within this universe is finite (though large). The set of measurements that could be made in principle from the principles that we can discover within this universe is countable. And we have no way to produce an oracle that can always decide truth or falsity. And therefore reality cannot encompass the actual existence of the uncountable infinites that ZFC claim must exist.

by btilly 2y ago

lisper 2y ago

> To reduce math to models of objective reality, you need to say what a model is. I did. See the paragraph that starts, "A model is any physical system whose behavior correlates in some way with another physical system." > If you try to model math in a way that can actually be rooted in objective reality... It's pretty clear you didn't actually read all the way to the end.

layer8 2y ago
> "A model is any physical system whose behavior correlates in some way with another physical system." So when X is a model of Y, then Y is always also a model of X? (Since correlation is generally a symmetric relationship.) That seems like a strange definition of “model”.
- gavagai691 2y ago
  I agree that this is a problem with the above definition of model, analogous to the (IMO fatal) problems with https://en.wikipedia.org/wiki/Deductive-nomological_model#Weaknesses https://en.wikipedia.org/wiki/Deductive-nomological_model#We....
  - lisper 2y ago
    That has to do with causality, which is a complicated topic that I was trying very hard to avoid at this point in the exposition. But maybe that was a mistake.
- lisper 2y ago
  > So when X is a model of Y, then Y is always also a model of X? Yes. > That seems like a strange definition of “model”. Yes, it does, but I can defend it. It's analogous to the definition of "information". If a system X contains information about a system Y then it is always the case that Y also contains the same amount of information about X. The distinction between "model" and "modeled" is purely artificial in the most literal sense: the thing we generally label "model" is the thing we humans made.
gavagai691 2y ago
You don't think that a model is something abstract? Abstract doesn't have to imply nonphysical in the sense that people think souls or God are nonphysical. I mean abstract in the sense that language, mathematics, or a sketch are abstract. To expand on this: I think models are representations, and whether or not something is a model depends in some way on human minds. (In particular, it depends on whether a something would be understood by a human mind to be a representation.) I don't think that any correlation between physical systems qualifies one as a model for the other. Your definition as written would include any two things that are connected causally, or have a common cause, as models for one another. One problem (though not the only one) that I have is that your definition removes any mention of human minds. In particular, I think "representation" is, broadly speaking, some kind of correspondence relationship between linguistic or pictorial things (where I include mathematics as "linguistic") and physical reality, and "a representation" is some linguistic or pictorial thing that corresponds to reality. I think that a model is a kind of representation. A model is a kind of representation where for convenience and tractability, certain aspects of reality are left out or "abstracted away" (deliberately), with the goal of understanding the real world by understanding the simpler representation of the real world.
- lisper 2y ago
  > You don't think that a model is something abstract? They certainly can be, but they don't have to be. An orrery, for example, is not abstract. > whether or not something is a model depends in some way on human minds Well, yeah, but what exactly is that "some way"? That is exactly the complication I was trying to avoid. Maybe the word "model" is too loaded and I need to pick a different word. I'm open to suggestions.
- seanhunter 2y ago
  > You don't think that a model is something abstract? Models definitely don't have to be abstract. For example, researchers will talk about studying a disease or the effectiveness of a treatment in a "mouse model"[1]. That model is an actual concrete mouse that is being used as a model of a human. It's not abstract in the sense of language, mathematics, or a sketch, and they do the research by looking for the physical effects on the mouse model and drawing an analogy to what would correspondingly happen in a human. [1] eg https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7121329/ https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7121329/
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.

gavagai691 2y ago

Well, it is an empirical question whether or not matter is continuous and infinitely divisible, or discrete. Our best theories tentatively suggest that matter is discrete (though it is hard to say how much confidence to put in that, or how we could really know either way).

btilly 2y ago
While interesting, that doesn't matter for what I said. An example of a key question is whether mathematical constructions can use the absolute truth of statements in mathematical constructions. Even if those absolute truths are not themselves something that can be settled by any algorithm. If they can, then we get all of the weirdness of infinite set theory, such as that there must be more reals than rationals. If they can't, then all of mathematics could fit within a countable universe, and things like Cantor's diagonalization proof just demonstrates a Halting problem kind of self-reference in the definitions of the real numbers. And this brings us to my point. It doesn't matter whether the universe is continuous and measured to finite precision, or discrete. The set of measurements that we could potentially ever make within this universe is finite (though large). The set of measurements that could be made in principle from the principles that we can discover within this universe is countable. And we have no way to produce an oracle that can always decide truth or falsity. And therefore reality cannot encompass the actual existence of the uncountable infinites that ZFC claim must exist.