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I'm not sure I'm comfortable with equating "these numbers would eventually be found in the digits of pi" with "this is 'stored' in pi." Do the digits of pi exi

I'm not sure I'm comfortable with equating "these numbers would eventually be found in the digits of pi" with "this is 'stored' in pi." Do the digits of pi exist independent of doing a computation process to turn pi into a digit-by-digit representation? In which case it gets into messy stuff about infinities, right, e.g. it could take an infinitely long amount of time to find the sequence you're looking for. And is it valid to say "any circle contains an infinite amount of information" just because there's this ratio, given that just from looking at the circle, you couldn't compute the ratio to infinite precision? Whereas if we're talking a theoretical circle, and the equations for finding all the digits of pi, how much do those concepts truly "exist" in a way that they could "store" information?

paskozdilar 5y ago
I've thought a lot about this. My conclusion is that there is no difference between a number and an algorithm. Pi is not a "number" in a sense that integers and rationals are - it is not a finite string of symbols that represents a quantity for us. But it does represent some quantity, though in an indirect way - all irrational numbers can only be represented by an algorithm that approaches the infinitely-precise value. So, integers and rationals only seem "like numbers" to us because we don't consider them "algorithms" - they seem intuitive to us. But looking from the perspective of Peano axioms, even integers are just algorithms for computing numbers - 3 is just s(s(s(1))), which is an algorithm that states that the successor function has to be applied 3 times to the number 1. So we can't really draw any kind of objective line between a number and an algorithm. Every number is an algorithm, it's just that some of them are trivial for us. Same thing with circle - circle is just an algorithm for drawing a specific shape. All the properties of circles are just algorithms for approximating real-world properties of our drawings of circles. So basically, pi (the algorithm) does in fact "contain" all sequences of digits when computed - but it does it in the same way that an infinite grassfield contains a grassleaf of any size - the size of the grassfield only makes it harder for us to find the grassleaf.
titzer 5y ago
A circle doesn't contain infinite information, because there is a finite string of symbols that we can write that completely describe it. The question is, what symbols are we working with, and how can we combine them? That leads to the definition of a language, and the Chomsky hierarchy of languages comes into play, with Turing-complete languages as the sine-qua-non. Then Kolmogorov comes along and says "The ultimate measure of information in a string is the size of the smallest Turing machine that can compute the string".

paskozdilar 5y ago

I've thought a lot about this. My conclusion is that there is no difference between a number and an algorithm. Pi is not a "number" in a sense that integers and rationals are - it is not a finite string of symbols that represents a quantity for us. But it does represent some quantity, though in an indirect way - all irrational numbers can only be represented by an algorithm that approaches the infinitely-precise value. So, integers and rationals only seem "like numbers" to us because we don't consider them "algorithms" - they seem intuitive to us. But looking from the perspective of Peano axioms, even integers are just algorithms for computing numbers - 3 is just s(s(s(1))), which is an algorithm that states that the successor function has to be applied 3 times to the number 1. So we can't really draw any kind of objective line between a number and an algorithm. Every number is an algorithm, it's just that some of them are trivial for us. Same thing with circle - circle is just an algorithm for drawing a specific shape. All the properties of circles are just algorithms for approximating real-world properties of our drawings of circles. So basically, pi (the algorithm) does in fact "contain" all sequences of digits when computed - but it does it in the same way that an infinite grassfield contains a grassleaf of any size - the size of the grassfield only makes it harder for us to find the grassleaf.

titzer 5y ago

A circle doesn't contain infinite information, because there is a finite string of symbols that we can write that completely describe it. The question is, what symbols are we working with, and how can we combine them? That leads to the definition of a language, and the Chomsky hierarchy of languages comes into play, with Turing-complete languages as the sine-qua-non. Then Kolmogorov comes along and says "The ultimate measure of information in a string is the size of the smallest Turing machine that can compute the string".