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> But pi is has infinite information It does not, according to any sane way of defining its information content. For example the Kolomogrov complexity of pi is

> But pi is has infinite information It does not, according to any sane way of defining its information content. For example the Kolomogrov complexity of pi is clearly finite - I can write down a program for a Turing machine which will run (forever) and keep writing down digits of pi as it does so.

pishpash 3y ago
Not true either. The original Kolmogorov complexity is for finite strings. Plus, the program would need to store that for whichever special strings you choose you will use this function but for other ones not. That will be a giant table that is part of your program. That's also a different point than the parent's. Seems they're saying if you were to specify pi as the limit of some expansion that describes the physical process of photons arriving in some area, then that specification's information increases with more terms added. Pi, being almost random by every statistical measure, has as much information as a random string, in fact, in any normal conception of information. You cannot wave that away by machine manipulation tricks or by defining a new constant, and this is borne out also by the parent's physical argument that in reality there are no low-complexity universal constants, but that there may be limits to information density (in space and time). Continuous physics can be a manipulation of limiting quantities without being literal.
- eigenket 3y ago
  > The original Kolmogorov complexity is for finite strings. I wrote "Kolmogorov complexity" not "original Kolmogorov complexity" so this isn't particularly relevant. The application of the concept to the infinite string which represents pi is essentially trivial. > Plus, the program would need to store that for whichever special strings you choose you will use this function but for other ones not. That will be a giant table that is part of your program. I honestly can't parse this. > Pi, being almost random by every statistical measure, has as much information as a random string This is wrong. You can consider something like a simple communication task. Alice and Bob share a phone line and she is attempting to tell him a number. Every second the line allows her to send a bit to Bob. For a truly random number she has to use the line infinitely many times to tell him the number. To send pi she can send a finite number of bits which amount to a program to compute pi and he can do the computation on his end.
  - pishpash 3y ago
    I urge you to go back and look at how Kolmogorov complexity is defined. It includes the notion that a program needs to decide whether to output the string directly or to generate it from some program. You're assuming Alice and Bob have already pre-synchronized what kind of computing machine is going to be used, one in which pi is the output of a relatively short program, as opposed to another type of machine where some other random-looking number has that property (random to you, pseudo-randomly generated via some machinery for all you know). You are assuming many things away. Also it is absolutely not trivial to extend Kolmogorov complexity to infinite strings. There are multiple formulations and they are a lot more difficult than for finite strings. Not the computation part but the complexity assignment part.
    - eigenket 3y ago
      I agree there is a bunch of complexity in generalising Kolmogorov complexity to general infinite strings. However I'm not really trying to do that here, all I want is enough to back-up the statement I made before, that the complexity of pi is finite. Doing that is much more trivial than what you're talking about. Theres a bunch of fine detail in getting it down to defining an actual number measuring complexity which I don't care about at all, all I care about (in the context of this discussion) is that the number is finite.

pishpash 3y ago

Not true either. The original Kolmogorov complexity is for finite strings. Plus, the program would need to store that for whichever special strings you choose you will use this function but for other ones not. That will be a giant table that is part of your program. That's also a different point than the parent's. Seems they're saying if you were to specify pi as the limit of some expansion that describes the physical process of photons arriving in some area, then that specification's information increases with more terms added. Pi, being almost random by every statistical measure, has as much information as a random string, in fact, in any normal conception of information. You cannot wave that away by machine manipulation tricks or by defining a new constant, and this is borne out also by the parent's physical argument that in reality there are no low-complexity universal constants, but that there may be limits to information density (in space and time). Continuous physics can be a manipulation of limiting quantities without being literal.

eigenket 3y ago
> The original Kolmogorov complexity is for finite strings. I wrote "Kolmogorov complexity" not "original Kolmogorov complexity" so this isn't particularly relevant. The application of the concept to the infinite string which represents pi is essentially trivial. > Plus, the program would need to store that for whichever special strings you choose you will use this function but for other ones not. That will be a giant table that is part of your program. I honestly can't parse this. > Pi, being almost random by every statistical measure, has as much information as a random string This is wrong. You can consider something like a simple communication task. Alice and Bob share a phone line and she is attempting to tell him a number. Every second the line allows her to send a bit to Bob. For a truly random number she has to use the line infinitely many times to tell him the number. To send pi she can send a finite number of bits which amount to a program to compute pi and he can do the computation on his end.
- pishpash 3y ago
  I urge you to go back and look at how Kolmogorov complexity is defined. It includes the notion that a program needs to decide whether to output the string directly or to generate it from some program. You're assuming Alice and Bob have already pre-synchronized what kind of computing machine is going to be used, one in which pi is the output of a relatively short program, as opposed to another type of machine where some other random-looking number has that property (random to you, pseudo-randomly generated via some machinery for all you know). You are assuming many things away. Also it is absolutely not trivial to extend Kolmogorov complexity to infinite strings. There are multiple formulations and they are a lot more difficult than for finite strings. Not the computation part but the complexity assignment part.
  - eigenket 3y ago
    I agree there is a bunch of complexity in generalising Kolmogorov complexity to general infinite strings. However I'm not really trying to do that here, all I want is enough to back-up the statement I made before, that the complexity of pi is finite. Doing that is much more trivial than what you're talking about. Theres a bunch of fine detail in getting it down to defining an actual number measuring complexity which I don't care about at all, all I care about (in the context of this discussion) is that the number is finite.