I’m responding here to your invitation in the parent, since this post provides some good examples of what you’re not getting.
I didn’t say that. Read it again. I said that there is some finite axiom list that can describe squirrels, but it’s not just the axioms that suffice to let you use arithmetic.
Simulating squirrels and using arithmetic require information, but that information is not supplied in the form of axioms. The best way to imagine this in the case of arithmetic is in terms of a structure.
Starting from the definition in that wikipedia page, you can imagine giving the graphs of the universe and functions and relations as Datalog terms. (Using terms instead of tuples keeps the graphs disjoint, which will be important later.) Like so:
Then you use some simple recursive coding of datalog terms as binary. What you’re left with is just a big (infinite) set of binary strings. The Kolmogorov complexity of the structure N, then (the thing you need to use arithmetic) is the size of the shortest program that enumerates the set, which is actually very small.
Note that this is actually the same arithmetic that Steve is talking about! It is just a different level of description, one that is much simpler but entirely sufficient to conduct simulations with. It is only in understanding the long-term behavior of simulations without running them that one requires any of the extra complexity embodied in T(N) (the theory). To actually run them you just need N (the structure).
The fact that you don’t seem to understand this point yet leads me to believe you were being a little unfair when you said:
By the way, I really hope your remark about Splat’s comment being “enlightening” was just politeness, and that you didn’t actually mean it. Because if you did, that would mean you’re just now learning the distinction between N and T(N), the equivocation between which undermines your claims about arithmetic’s relation to the universe.
Now, if you want to complete the comparison, imagine you’re creating a structure that includes a universe with squirrel-states and times, and a function from time to squirrel state. This would look something like:
The squirrel states, though, will not be described by a couple of words like that, but by incredibly detailed descriptions of the squirrel’s internal state—what shape all its cells are, where all the mRNAs are on their way to the ribosomes, etc. The structure you come up with will take a much bigger program to enumerate than N will. (And I know you already agree with the conclusion here, but making the correct parallel matters.)
Simulating squirrels and using arithmetic require information, but that information is not supplied in the form of axioms.
I wasn’t careful to distinguish axioms from other kinds of information in the model, and I think it’s a distraction to do so because it’s just an issue of labels (which as you probably saw from the discussion is a major source of confusion). My focus was on tabulating the total complexity of whatever-is-being-claimed-is-significant. For that, you only need to count up how much information goes into your “message” describing the data (in the “Minimum Message Length criterion” sense of “message”). Anything in such a message can be described without loss of generality as an axiom.
If I want to describe squirrels, I will find, like most scientists find, that the job is much easier of I can express things using arithmetic. Arithmetic is so helpful that, even after accounting for the cost of telling you how to use it (the axioms-or-whatever of math), I still save in total message length. Whether you call the squirrel info I gathered from nature, or the specification of math, the “axioms” doesn’t matter.
...which is actually very small. Note that this is actually the same arithmetic that Steve is talking about! It is just a different level of description, one that is much simpler but entirely sufficient to conduct simulations with. It is only in understanding the long-term behavior of simulations without running them that one requires any of the extra complexity embodied in T(N) (the theory). …
But it’s not the same arithmetic SteveLandsburg is talking about, if you follow through to the implications he claims fall out from it. He claims arithmetic—the infinitely complex one—runs the universe. It doesn’t. The universe only requires the short message specifying N, plus the (finite) particulars of the universe. Whatever infinitely-complex thing he’s talking about from a “different level of description” isn’t the same thing, and can’t be the same thing.
What’s more, the universe can’t contain that thing because there is no (computable) isomorphism between it and the universe. As we derive the results of longer and longer chains of reasoning, our universe starts to contain more and more complex pieces of that thing, but it still wouldn’t be somehow fundamental to the universe’s operation—not if we’re just now getting to contain pieces of it.
The fact that you don’t seem to understand this point yet leads me to believe you were being a little unfair when you said: … Now, if you want to complete the comparison, imagine you’re creating a structure that includes a universe with squirrel-states and times, and a function from time to squirrel state. This would look something like: … The squirrel states, though, will not be described by a couple of words like that,
I’m sorry, I don’t see how that contradicts what I said or shows a different parallel. Now, I certainly didn’t use the N vs. T(N) terminology you did, but I clearly explained how there have to be two separate “arithmetics” in play here, as best summarized in my comment here. Whatever infinitely complex arithmetic SteveLandsburg is talking about, isn’t the one that runs the universe. The insights on one don’t apply to the other.
I’m responding here to your invitation in the parent, since this post provides some good examples of what you’re not getting.
Simulating squirrels and using arithmetic require information, but that information is not supplied in the form of axioms. The best way to imagine this in the case of arithmetic is in terms of a structure.
Starting from the definition in that wikipedia page, you can imagine giving the graphs of the universe and functions and relations as Datalog terms. (Using terms instead of tuples keeps the graphs disjoint, which will be important later.) Like so:
Universe:
is_number(0),is_number(1), …0:
zero(0)S:
next(0,1),next(1,2), …+:
add_up_to(0,0,0),add_up_to(0,1,1),add_up_to(1,0,1)…and so on.
Then you use some simple recursive coding of datalog terms as binary. What you’re left with is just a big (infinite) set of binary strings. The Kolmogorov complexity of the structure N, then (the thing you need to use arithmetic) is the size of the shortest program that enumerates the set, which is actually very small.
Note that this is actually the same arithmetic that Steve is talking about! It is just a different level of description, one that is much simpler but entirely sufficient to conduct simulations with. It is only in understanding the long-term behavior of simulations without running them that one requires any of the extra complexity embodied in T(N) (the theory). To actually run them you just need N (the structure).
The fact that you don’t seem to understand this point yet leads me to believe you were being a little unfair when you said:
Now, if you want to complete the comparison, imagine you’re creating a structure that includes a universe with squirrel-states and times, and a function from time to squirrel state. This would look something like:
is_time(1:00:00),is_time(1:00:01), …is_squirrel_state(<eating nut>),is_squirrel_state(<rippling tail>),is_squirrel_state(<road pizza>)squirrel_does(1:00:00, <rippling tail>), …The squirrel states, though, will not be described by a couple of words like that, but by incredibly detailed descriptions of the squirrel’s internal state—what shape all its cells are, where all the mRNAs are on their way to the ribosomes, etc. The structure you come up with will take a much bigger program to enumerate than N will. (And I know you already agree with the conclusion here, but making the correct parallel matters.)
(Edit: fixed markup.)
I wasn’t careful to distinguish axioms from other kinds of information in the model, and I think it’s a distraction to do so because it’s just an issue of labels (which as you probably saw from the discussion is a major source of confusion). My focus was on tabulating the total complexity of whatever-is-being-claimed-is-significant. For that, you only need to count up how much information goes into your “message” describing the data (in the “Minimum Message Length criterion” sense of “message”). Anything in such a message can be described without loss of generality as an axiom.
If I want to describe squirrels, I will find, like most scientists find, that the job is much easier of I can express things using arithmetic. Arithmetic is so helpful that, even after accounting for the cost of telling you how to use it (the axioms-or-whatever of math), I still save in total message length. Whether you call the squirrel info I gathered from nature, or the specification of math, the “axioms” doesn’t matter.
But it’s not the same arithmetic SteveLandsburg is talking about, if you follow through to the implications he claims fall out from it. He claims arithmetic—the infinitely complex one—runs the universe. It doesn’t. The universe only requires the short message specifying N, plus the (finite) particulars of the universe. Whatever infinitely-complex thing he’s talking about from a “different level of description” isn’t the same thing, and can’t be the same thing.
What’s more, the universe can’t contain that thing because there is no (computable) isomorphism between it and the universe. As we derive the results of longer and longer chains of reasoning, our universe starts to contain more and more complex pieces of that thing, but it still wouldn’t be somehow fundamental to the universe’s operation—not if we’re just now getting to contain pieces of it.
I’m sorry, I don’t see how that contradicts what I said or shows a different parallel. Now, I certainly didn’t use the N vs. T(N) terminology you did, but I clearly explained how there have to be two separate “arithmetics” in play here, as best summarized in my comment here. Whatever infinitely complex arithmetic SteveLandsburg is talking about, isn’t the one that runs the universe. The insights on one don’t apply to the other.