a perspective on mathematics, the pattern, and the abstract

Some background information:

- The number Pi is irrational, and when written as a decimal (in base 10) the sequence of digits continues without repetition: Pi = 3.141592…
- Numbers can be equivalently written in different ways using different counting systems or bases. For instance, in base 2 numbers are represented using just 0s and 1s. Pi expressed in this binary form may look something like this up to several digits:

Pi = 11.00100100001111110110

I am assuming*“calculating*" Pi means being able to calculate this sequence to any finite length of this infinite sequence. - Besides numbers themselves, anything
*reasonable*that permits a finite description of itself (like books, movies, source code, etc.) can be encoded as a string of 0s and 1s. - A
*normal number*has the property that its digit expansion in some counting base contains all possible finite sequence of digits. Consider the decimal that goes on forever formed by joining all the whole numbers in order: .123456789101112131415…

This number is normal by construction. - The conjecture that Pi is normal in binary is saying that any finite sequence of 0s and 1s is contained somewhere in the infinite sequence that represents Pi.
- I understand that the statements in the warning were probably intended as a joke.

This is an argument for why I think someone should not be considered guilty for calculating Pi in binary. This is because only knowing the sequence of digits in the binary expansion of Pi is not the same as knowing where any particular string of 1s and 0s is located in that infinite list. Nor is it the same as having knowledge that any arbitrary sequence is a proper encoding of somethings else.

Assuming Pi is normal and despite having the ability to calculate Pi in binary to an arbitrarily length, would this necessarily imply having the ability to efficiently find the position of any particular finite string in the sequence? It could be computed in principle but the efficiency of such a task is the important part. Algorithmically speaking, unless provided with some deeper structure of the binary expansion of Pi, the best way to locate a given string within that sequence may come down to a brute search.

Therefore, we should be careful in claiming someone to be guilty of things like those listed in the warning. Maybe if there was evidence suggesting that the person could locate a certain string in practice or had located one, then we could justify them being guilty.

Otherwise, having access to any part of the infinite expansion of Pi does not yield any information alone. How would someone even know where some string representing a certain book is? You could try to keep decoding random strings until you finally find the one you think you were looking for. Keep in mind that you could very well come across strings which seem to represent the book you’re looking for, but there would be typos or other variants of the original that you would need to sort through.

Even if you were provided with the proper encoding of something, which would defeat the point anyway, it would be difficult to locate or search for the position of that given string in the expansion of Pi. Imagine how long the bit strings would be for some of these things in any suitable encoding. There is no way of knowing which part of Pi’s expansion to even begin searching from. Trying to find a string of length N in some region of length M of Pi, could take 2^((M-N)N) steps in the worst case which is exponential in both N and M. For N greater than just 300, you would already be dealing with numbers exceeding estimates of the number of atoms in the universe (~10^80).

Calculating Pi, or any other normal number, doesn’t really help anyone trying to get a certain encoding of anything anymore then having a trivial list of all possible bit strings, which ought to be freely available to anyone interested.

Suppose there could be some elaborate way to use the positions of the stars in the sky to encode information about certain strings of bits within that framework. Even though we all have access to the stars we do not claim to have knowledge or ‘possession’ of information for any of the things that could be suitably encoded.

Essentially, calculating Pi should be a guilt-free experience because when it comes down to it encodings are arbitrary and there is no a priori meaning to a sequence of symbols.

To my defense, here is a nice site that lists the binary expansion of Pi to some number of digits which also happened to be ranked as #4 in a list of most useless pages on the WWW.

Source:
themathkid

High-res
Notes: 165
6/4/12 — 4:51pm
Filed under:
#Pi
#complexity theory
#mathematical ethics
#order of magnitude
#infinity
#reciprocity
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