A monthly thread for posting rationality-related quotes you've seen recently (or had stored in your quotesfile for ages).
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ETA: It would seem that rationality quotes are no longer desired. After several days this thread stands voted into the negatives. Wolud whoever chose to to downvote this below 0 would care to express their disapproval of the regular quotes tradition more explicitly? Or perhaps they may like to browse around for some alternative posts that they could downvote instead of this one? Or, since we're in the business of quotation, they could "come on if they think they're hard enough!"
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:
is_number(0)
,is_number(1)
, ...zero(0)
next(0,1)
,next(1,2)
, ...add_up_to(0,0,0)
,add_up_to(0,1,1)
,add_up_to(1,0,1)
...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" describi... (read more)