3 ms·

This is good. I really liked it. But just to continue the mind-games, yes there are an infinity more indescribable numbers than there are describable ones, but

This is good. I really liked it. But just to continue the mind-games, yes there are an infinity more indescribable numbers than there are describable ones, but should there be? Given the balloon example, he makes the case that as an imaginary god: "...stare imploringly into the siren call of the final ellipsis, for you know that no matter how often you expand it, I will always smile when giving you more and more digits, because you and I will both know that the final wisdom of The Number will always be mine and never yours..." But we know that, given any exact point in time, only a certain number of atoms are inside the balloon. Therefore, whatever the number is to answer the equation, it is one with a finite number of digits. (One imagines the experiment being done in such a manner that quantum tunneling is minimized) So while yes, using symbols you can certainly construct numbers which are indescribable (given either an infinite series of symbols or a self-recursive way of generating more), does the real world actually work that way? In other words, is mathematics truly isomorphic with reality as we observe it? Or is it somehow a superset of logic sitting above an infinite number of possible realities? In a universe full of discrete things (perhaps at the sub-quark level), do we need a system of mathematics that works over continuities? Is such a model always helpful? If not, where might it be tripping us up?

danbruc 13y ago
I would find it very beautiful if we could get away with only integers, for example a large number of interacting state machines. This way we could get rid of the complex ideas like space and time (if you manage to convince your brain that there is no need to embed a net of state machines in space and that they have to change state over time).
tripzilch 13y ago
What you describe has to do with mathematical constructivism[0], and also with the Axiom of Choice, and it's really complicated (to me, I'm not a mathematician, and I only sort-of get it). You may have heard about the Banach-Tarski paradox[1], which tells you that if you assume "Real numbers" are actually reality, and the Axiom of Choice, you can divide a sphere into five pieces (one of which is just a point) and rearrange them into two spheres of the same volume. Obviously you can't do that in reality. This is the point where it gets really complex and I won't fill in the details as I understand them, because they are probably wrong. R. Buckminster Fuller also wrote some ideas on this constructivism (or something a lot like it), basically advocating against the set of "Real Numbers" as an accurate depiction of reality. For instance if you make a 1m x 1m square out of a real material. The sides of that square will have N atoms, and the number of atoms (and how far they are apart, which is a property of the material used) is really what describes the length. This is a whole number. But according to geometry math, the diagonal of this square has a length of sqrt(2) metres, or sqrt(2) * N atoms. That can't be because you can't have an irrational number of atoms. Then what is really going on suddenly depends on how these atoms are packed inside the square, and again, it gets really complex and that's where my understanding (of both atomic physics and mathematical constructivism) stops. [0] http://en.wikipedia.org/wiki/Constructivism_(mathematics) http://en.wikipedia.org/wiki/Constructivism_(mathematics) [1] http://en.wikipedia.org/wiki/Banach-Tarski_paradox http://en.wikipedia.org/wiki/Banach-Tarski_paradox
- graycat 13y ago
  > Banach-Tarski Yes, but in that construction, not all the sets are Lebesgue measurable. It's a curious fact, already true just in one dimension, i.e., the reals, that we can't have a nice way to assign a length, area, or measure to all the subsets. Instead there have to be some subsets to which we can't assign a length, etc. Here the usual proof on the reals does use the axiom of choice! With measurable sets, which already can be wildly bizarre, can't do anything like Banach-Tarski. So, suddenly in this thread, instead of hackers, we are all turning into undergraduate pure math majors!
- tel 13y ago
  Constructivism is significantly more fundamental than AC. AC is notable for being famously perplexing, less foundationally perplexing. For a more graspable investigation of the perplexities and arbitrariness of our choice of foundational set theory, take a look at Vicious Cycles where the author investigates what happens when set theory does not assume a terminating world (and thus introduces a lot of things that we use in CS often, try to model on sets, and are left dissatisfied because we really wanted costa, not data). From there please take a deep look into topos theory and you'll discover again that AC is just one choice of properties that make for a meaningful "foundation of mathematics (of a kind)" and a not particularly remarkable one at that.
  - tripzilch 13y ago
    "Costa, not data", what does that mean? And Vicious Cycles is a book, I presume? Who is the author?
- deleted 13y ago
  [deleted]
StevenXC 13y ago
When I tell many people that I do my research in infinite combinatorial games, they immediately assume that my research is more impressive[0] than my officemate's research dealing with finite combinatorics. I always move quickly to correct them: dealing with infinity often has the effect of making mathematics easier, and unfairly "trivializes" distinctly difficult problems. After all, in my work, 2^100 is "just finite", even though that many milliseconds would compose over 40 quintillion years. To your point, I don't know the answer of whether the universe is finitely describable or not. But I certainly know it's easier to abstract many small aspects of the universe with an approximation by an "infinite model" than to deal with the ridiculously large finite numbers involved. [0] if more useless
yk 13y ago
I think you have your argument backwards, we use math to describe reality. So whatever axiom system you choose to build your physical theory, there is then some correspondence between the objects of your axiom system and reality. And if you somehow manage to construct a physical theory with integers alone, then there will only be integers in your description. Interestingly: Indescribable numbers do not appear in physical theories. Proof: The number would need to be described by the theory in order to have a correspondence to nature. ( None of the above should be constructed as a statement about the existence of Platonic ideals.)

danbruc 13y ago

I would find it very beautiful if we could get away with only integers, for example a large number of interacting state machines. This way we could get rid of the complex ideas like space and time (if you manage to convince your brain that there is no need to embed a net of state machines in space and that they have to change state over time).

tripzilch 13y ago

What you describe has to do with mathematical constructivism[0], and also with the Axiom of Choice, and it's really complicated (to me, I'm not a mathematician, and I only sort-of get it). You may have heard about the Banach-Tarski paradox[1], which tells you that if you assume "Real numbers" are actually reality, and the Axiom of Choice, you can divide a sphere into five pieces (one of which is just a point) and rearrange them into two spheres of the same volume. Obviously you can't do that in reality. This is the point where it gets really complex and I won't fill in the details as I understand them, because they are probably wrong. R. Buckminster Fuller also wrote some ideas on this constructivism (or something a lot like it), basically advocating against the set of "Real Numbers" as an accurate depiction of reality. For instance if you make a 1m x 1m square out of a real material. The sides of that square will have N atoms, and the number of atoms (and how far they are apart, which is a property of the material used) is really what describes the length. This is a whole number. But according to geometry math, the diagonal of this square has a length of sqrt(2) metres, or sqrt(2) * N atoms. That can't be because you can't have an irrational number of atoms. Then what is really going on suddenly depends on how these atoms are packed inside the square, and again, it gets really complex and that's where my understanding (of both atomic physics and mathematical constructivism) stops. [0] http://en.wikipedia.org/wiki/Constructivism_(mathematics) http://en.wikipedia.org/wiki/Constructivism_(mathematics) [1] http://en.wikipedia.org/wiki/Banach-Tarski_paradox http://en.wikipedia.org/wiki/Banach-Tarski_paradox

graycat 13y ago
> Banach-Tarski Yes, but in that construction, not all the sets are Lebesgue measurable. It's a curious fact, already true just in one dimension, i.e., the reals, that we can't have a nice way to assign a length, area, or measure to all the subsets. Instead there have to be some subsets to which we can't assign a length, etc. Here the usual proof on the reals does use the axiom of choice! With measurable sets, which already can be wildly bizarre, can't do anything like Banach-Tarski. So, suddenly in this thread, instead of hackers, we are all turning into undergraduate pure math majors!
tel 13y ago
Constructivism is significantly more fundamental than AC. AC is notable for being famously perplexing, less foundationally perplexing. For a more graspable investigation of the perplexities and arbitrariness of our choice of foundational set theory, take a look at Vicious Cycles where the author investigates what happens when set theory does not assume a terminating world (and thus introduces a lot of things that we use in CS often, try to model on sets, and are left dissatisfied because we really wanted costa, not data). From there please take a deep look into topos theory and you'll discover again that AC is just one choice of properties that make for a meaningful "foundation of mathematics (of a kind)" and a not particularly remarkable one at that.
- tripzilch 13y ago
  "Costa, not data", what does that mean? And Vicious Cycles is a book, I presume? Who is the author?
deleted 13y ago
[deleted]

StevenXC 13y ago

When I tell many people that I do my research in infinite combinatorial games, they immediately assume that my research is more impressive[0] than my officemate's research dealing with finite combinatorics. I always move quickly to correct them: dealing with infinity often has the effect of making mathematics easier, and unfairly "trivializes" distinctly difficult problems. After all, in my work, 2^100 is "just finite", even though that many milliseconds would compose over 40 quintillion years. To your point, I don't know the answer of whether the universe is finitely describable or not. But I certainly know it's easier to abstract many small aspects of the universe with an approximation by an "infinite model" than to deal with the ridiculously large finite numbers involved. [0] if more useless

yk 13y ago

I think you have your argument backwards, we use math to describe reality. So whatever axiom system you choose to build your physical theory, there is then some correspondence between the objects of your axiom system and reality. And if you somehow manage to construct a physical theory with integers alone, then there will only be integers in your description. Interestingly: Indescribable numbers do not appear in physical theories. Proof: The number would need to be described by the theory in order to have a correspondence to nature. ( None of the above should be constructed as a statement about the existence of Platonic ideals.)