armhold / infinity

literally everything digital appears in the set of natural numbers

Infinity

Literally everything digital appears in the set of natural numbers.

Let's find a famous painting in the set of Natural Numbers

There's a famous painting hidden among the integers in the set of Natural Numbers. Michelangelo's Creation of Adam happens to reside at this number:

25519637684623283551604138549800901322256540519555396335820617899064263322699621
95191687493347723876911325592757091527584399836110327600568809057692273948571948
08265352774295543927463930227303989499812656204789274349264352246002620646195049
80743234655174227362545608569251825909819314291568940243800729262655548989815338
54946158978193172722003989771610404779019733491375454052733009346842568422009696
74942993143476565213005563390876806416489284574517449719256327217688989948909567
[ ... ]
66075392419937785208072076024225421796258819260095866487851661074765988817906435
95488615051785336967485705800344694295344605450895377295574423161887490912578460
55236362647031265463447554894035823105547224316844121735586980345834721507867209
39588739309981549612372297190758970535136257561299604029885537909193046058541165
07892131487938814207003552600673252186494827588209195808322307452037885128631910
361

_{NB: I've elided most of the lines for brevity; you can download all digits of the full number here.)}

That is a number with 160,165 digits. It's a simple (though long) integer on the number line, and it happens to contain Michelangelo's Creation of Adam. It's always been there, even before Michelangelo created his famous fresco in 1511.

Background

Anything that can be described as a sequence of bytes can also be described by a single, very large integer. Just take the bytes and stack them end to end. You can interpret these bytes as a single integer. By definition, that integer appears in the set of Natural Numbers.

So in a sense, everything that can be expressed digitally (software programs, books, music, pictures, etc) already exists- we just have to find it on the number line.

This means that, for example, the Bible appears in the set of numbers- all possible versions of it. Same for the Koran, and every other book that's ever been written (or will be written). Tomorrow's newspaper is in there. Your photographs and your favorite movie are all in there too.

This program explores how we can discover the numbers associated with digital files.

Really Big Numbers

Any file contains a sequence of bytes- we string these bytes together to create a single number out of the many smaller ones. Each byte has eight bits of storage. Every added bit doubles the potential "range" of numbers. Since a byte has eight bits, each time you add a byte, the range is doubled eight times.

This quickly leads to some pretty big numbers.

A kilobyte of data (1024 bytes- not at all a big file) is 8,192 bits. To get a ballpark idea of where that data might lie on the number line we need to double "1" 8,192 times. That's 2⁸¹⁹², or about 1.09 × 10²⁴⁶⁶.

That's more than the number of atoms in the universe. A big number for sure, but still a number.

Demonstration

Consider the string "ABC". This can be written as three bytes. In decimal, the bytes are (assuming ASCII/Unicode) 65, 66, and 67. In binary, this gives us:

A = 65 = 01000001
B = 66 = 01000010
C = 67 = 01000011

If we concatenate the binary, we get:

01000001 01000010 01000011  

Or 4,276,803 in decimal. So the number for the string "ABC" is 4,276,803. Therefore "ABC" exists on the number line near the 4.2 millionth number.

This project includes a program to print the number line "coordinate" for any file. Install it like this:

$ go install github.com/armhold/infinity/...

Now you can use it to get the number coordinate for any given file like this:

$ find_number some_file.jpg
[ ... pages of numbers omitted...]

Or if you just want to look for a particular word or phrase, you can do:

$ find_number -t "meaning of life"
568016600252838567935837243227858533

And if you already have a number, and you'd like to see what it holds:

$ decode_number -n 568016600252838567935837243227858533
meaning of life

Wait, aren't you just encoding this data?

In a sense, yes. You can check the source code (it's very short) and see how it works- we take a sequence of letters, turn them into bytes, and then print the bytes as a long number. It doesn't change the fact that at the end, you get a single number back; a number that maps to your data, which has always been there in the number. This program merely identifies where it resides.

A somewhat spooky corollary to this is the fact that all your data is out there on the number line, and you cannot ever edit or delete it. Like some kind of cosmic blockchain, the data exists in the number line- it's always been there, and there is no way for you to remove it. Take comfort in the fact that infinite variations and permutations of the numbers exist, and from our perspective, it's mostly just noise.

Notes

The image "Creation of Adam" by Michelangelo is in the Public Domain.