Friday, July 01, 2016

Finding yourself in the mathematical universe of pi

I’m reading Tegmark’s Our Mathematical Universe. It’s the layperson version of a 2007 paper (downloadable)

I explore physics implications of the External Reality Hypothesis (ERH) that there exists an external physical reality completely independent of us humans. I argue that with a sufficiently broad definition of mathematics, it implies the Mathematical Universe Hypothesis (MUH) that our physical world is an abstract mathematical structure. I discuss various implications of the ERH and MUH, ranging from standard physics topics like symmetries, irreducible representations, units, free parameters, randomness and initial conditions to broader issues like consciousness, parallel universes and Godel incompleteness. I hypothesize that only computable and decidable (in Godel's sense) structures exist, which alleviates the cosmological measure problem and help explain why our physical laws appear so simple. I also comment on the intimate relation between mathematical structures, computations, simulations and physical systems.

As best as I can tell, Tegmark, more or less like Einstein, thinks of spacetime as a static thing; we relate to spacetime as a tune relates to a CD. It is a bit of an illusion that one thing follows another, or that we seem to have a past and a future, or that things are created and end. Rather as a block of granite contains all sculpture, or the alphabet all knowledge, meaning is just an arrangement that fits. “I think therefore I am” indeed.

Which made me wonder if I could find “I am John” in pi. Seems a bit similar. If pi is random then that string should show up somewhere.

Happily the “National Energy Research Scientific Computing Center” will search pi for strings based on a 5bit English character set encoding.

Alas, “I am John” is not in the span they search…

search string = "i_am_john"
45-bit binary equivalent = 010010000000001011010000001010011110100001110
string does not occur in first 4 billion binary digits of pi

My daughter’s name does turn up:

search string = "brinna"
30-bit binary equivalent = 000101001001001011100111000001
search string found at binary index = 872397609 

binary pi : 0010101000010100100100101110011100000101111010011010101101000011
binary string: 000101001001001011100111000001
character pi : aqyfuc_nj_--yjbrinnaoiuma.fvfkbv.bcfgz
character string: brinna

This essay should be in there too, assuming pi digits really are deterministically random. Somewhere. Which I think has something to do with the Mathematical Universe. Somehow.

PS. From the arXiv paper: “… quantum field theory states such as the Hawking-Hartle wave function or the inflationary Bunch-Davies vacuum have very low algorithmic complexity (since they can be de- fined in quite brief physics papers), yet simulating their time evolution would simulate not merely one universe like ours, but a vast decohering ensemble corresponding to the above-mentioned Level III multiverse. It is therefore plausible that our universe could be simulated by quite a short computer program.”

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