An Ineffable Theory?

by Chris Stones
TM_universe

It's not often that I have an idea that I can't stop thinking about.
In fact, this particular one kept my mind off of a host of personal
matters and for that I am thankful. You see, I've been applying
computational theory to our quest for finding the next physical theory
of the universe and seeing what I can logically say. Turns out I can
say at least one interesting thing. There's a chance that we may never
be able to describe our physical universe with any procedural physical
theory.

Just a possibility.

But let's consider the universe for a bit. Let's imagine writing down
the universe as one long string of characters. How might that look? I
can write down a list of particles together with their various
velocities, masses, dimensions and the fundamental forces at work. I
could write a huge string of text describing the entire observable
universe. And the amazing thing is that once I've represented the
universe as a pattern of symbols on one nearly infinite strip of tape
I can tell you some fun facts about what we can and can not say about
the language of the universe.

The term "language" here is what computer scientists dub any long
series of symbols on a hypothetical tape. Truthfully, there are a few
more conditions, such as only using a finite number of symbols etc.
but the idea is that whatever we can observe and measure we can write
down and if we specified "everything" we'd have a neat thing called a
language from the computational theoretical perspective.

For clarity let me write a snippet of what a version of this giant
tape of symbols might look like for the Newtonian Mechanical version
of our universe:

"timestamp:(1stParticleName,mass,radius,velocity_vector,position(X,Y,Z),forces)(2ndParticleName,mass,radius...)(3rdParticleName..)(... and on and on...

Given such a list of all the particles and a model of the universe a
machine could compute and record the future positions of all the
particles and thus predict how this Newtonian system evolves through
time. But in this case, let us consider a very special type of machine
It's called a Turing Machine.[0]

I'm not going to get into formal details here suffice it to say let's
imagine it to be a super powerful computer with infinite time and
memory able to run any program consisting of a finite number of
symbols. And that we can describe this machine in its entirety with a
finite number of rules. Essentially, we write down a rule book
telling this machine how to handle what it sees in its infinite memory
and what to do next. Like a normal computer, it has the ability to
read from memory, write to memory and move around in memory.

So what do we put into that memory?

Well let's say we put our description of the universe into its memory.
And tell this machine to compute all future positions of the
particles. When we have such a machine we say that it "recognizes" the
language we specified. Recognize is another technical term from
computational theory. The collection of strings that a Turing machine
accepts is the "language" of that machine.[1] And our fancy machine is
doing something even more wonderful. It's taking input and a list of
rules and outputting observational behavior. It's acting like a
physical theory!

And it gets better.

According to wonderful concept known as the Church-Turing Thesis[2]
we, here in computational land equate the definition of a Turing
machine with the concept of algorithm. That's right. Writing down a
model for a Turing Machine is equivalent to writing down an algorithm.
The catch being that the class of algorithms I'm pondering here are
all the theories of the universe we can write down.

This logic allows me to rephrase the current question in physics from
finding the next big theory of everything to what does the Turing
machine that at recognizes the language of the universe look like?
Could this representation of physics help us understand and
form new theories?

It's hard to describe the menagerie of thoughts without getting into
the definition of Turing Machines but basically I've fallen in love
with the notion that any complicated algorithm or theory can be
ultimately encoded as a series of points in a small 3D volume. That's
a strange and perhaps wonderful result. And there is nothing I want
more than a better way to apply computers to the human quest to
understand the universe so I'm delighted to be able to think about it.
But whether or not it actually becomes a useful tool for writing new
theories or even saying anything at all depends upon a few other as
yet unproven notions.

So let me say this.

On page 178 of my computational theory book, it states that there are
in fact more languages than we have Turing Machines for.[3] This means
that there are some languages out there that no Turing Machine, no
algorithm, no theory can state because there are just not enough
Turing machines to go around.[4] And so I wondered...

Could the language of our Universe require such an ineffable theory?

Human beings have been pretty lucky to be able to have explained
things this far. It's even more remarkable when you consider just how
many decimal places our current physical laws nail down for us. But
the next layer of the cosmic onion is still waiting. The theory
that makes Gravity and Quantum Mechanics play nice is out there.
Whether we'll ever be able to speak its language is a whole other
matter.



Originally Written: June 24, 2014
NOTES:

[0] Video of a Model Turing Machine

[1] Accepts - meaning here that a Turing Machine "Accepts" a language
if there exists a series of configurations leading to the accept
state.

[2] Church-Turing Thesis - it's unproven but widely believed. One has
to accept it a bit like an axiom but an important one at that.

[3] I left out countable vs uncountable because it complicates the
discussion. We can create a countably infinite number of Turing
Machines but we have an uncountably infinite number of languages.

[4] Because a Turing Machine is synonymous with algorithm according to
the Church-Turing Thesis. And physical theories are synonymous with
algorithms.

REFERENCES:

Introduction to the theory of computation
By: Sipser, Michael.
Thomson Course Technology
2006

DEDICATIONS:
This paper is dedicated to Rosie Records or as I call her these days,
Mehbob!



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