The idea of encoding a book as a string is incoherent once we look at it with intellectual rigour. How are images in illuminated manuscripts or graphic novels encoded? What does "all books" even mean? Are we talking about each individual artefact past present and future or some idealized representation of each: e.g. is my current copy of Catcher in the Rye encoded separately from the copy I was assigned in 10th grade English class?
It's all a matter of interpretation.We can just as easily choose a different arbitrary encoding and claim to have found all the books in π. There's no need to make things complicated. We are free to pick any interpretation we want once we are claiming that numbers represent books. Let:
B = {b1, b2,...bn}
Such that it contains the set of all books. And let:
def Find-books(num)
if 3.14 < num < 3.15
then return B
else return "all the books not found"
The article assumes that there is some natural way of encoding books. But digits of π are not Unicode or Ascii characters. Though we can interpret a digit or string of digits as such, that encoding is arbitrary not a property of the natural or mathematical world.
This objection is silly. Of course if you define an encoding that maps '3' to 'the contents of all books', then you've trivially found a way to encode it in pi. But I think you're aware that the intention of the question is to find an encoding that has comparable complexity to the text of all books.
Specifically, the property we want is that if some substring of the digits of pi encodes 'all books', then you should be able to extract certain books out of substrings of that block of pi. I don't know if there's a technical term for that, but it's clear that we're imagining encoding successive paragraphs of these books in successive blocks of digits in pi.
The thing about normality is that if a number is normal, it will contain all books not only for a given encoding of a book, but for any given encoding of any book. That is true only for normal sequences, of which pi is assumed to be an example.
Exactly. So we can choose an encoding that is computationally expensive or one which is cheap. The essence
of mathematical thinking is to use replace complexity with simplicity: a person who insists on calculating the sum of positive integers less than n with n - 1 operations is not to be admired for their mathematical insight [their dogged determination may be another matter].
Some encodings are more useful than others while being equivalent. All encoding requires an arbitrary interpretation. Each is equally valid...finding "books" in Pi is a flight of fantasy.
Comments
The idea of encoding a book as a string is incoherent once we look at it with intellectual rigour. How are images in illuminated manuscripts or graphic novels encoded? What does "all books" even mean? Are we talking about each individual artefact past present and future or some idealized representation of each: e.g. is my current copy of Catcher in the Rye encoded separately from the copy I was assigned in 10th grade English class?
It's all a matter of interpretation.We can just as easily choose a different arbitrary encoding and claim to have found all the books in π. There's no need to make things complicated. We are free to pick any interpretation we want once we are claiming that numbers represent books. Let:
Such that it contains the set of all books. And let: The article assumes that there is some natural way of encoding books. But digits of π are not Unicode or Ascii characters. Though we can interpret a digit or string of digits as such, that encoding is arbitrary not a property of the natural or mathematical world.This objection is silly. Of course if you define an encoding that maps '3' to 'the contents of all books', then you've trivially found a way to encode it in pi. But I think you're aware that the intention of the question is to find an encoding that has comparable complexity to the text of all books.
Specifically, the property we want is that if some substring of the digits of pi encodes 'all books', then you should be able to extract certain books out of substrings of that block of pi. I don't know if there's a technical term for that, but it's clear that we're imagining encoding successive paragraphs of these books in successive blocks of digits in pi.
The thing about normality is that if a number is normal, it will contain all books not only for a given encoding of a book, but for any given encoding of any book. That is true only for normal sequences, of which pi is assumed to be an example.
Nitpick: This is NOT true for only normal sequences. See disjunctive sequences: http://en.wikipedia.org/wiki/Disjunctive_sequence .
pi may contain all numbers as substrings and yet not be normal.
Exactly. So we can choose an encoding that is computationally expensive or one which is cheap. The essence of mathematical thinking is to use replace complexity with simplicity: a person who insists on calculating the sum of positive integers less than n with n - 1 operations is not to be admired for their mathematical insight [their dogged determination may be another matter].
Some encodings are more useful than others while being equivalent. All encoding requires an arbitrary interpretation. Each is equally valid...finding "books" in Pi is a flight of fantasy.