Towards a Theory of Everything

This post may seem way off-topic, but for writing science fiction it helps to stay informed on the latest developments, as a sprinboard for our imagination. And a new approach to a "theory of everything" from Stephen Wolfram certainly provides fuel for the imagination. ("It's hypergraphs all the way down".)

I follow Joscha Bach on Twitter, and spent an hour reading a link from a tweet from him about an interesting blog post by Stephen Wolfram on Mathematics, Combinators and the Story of Computation. I was intrigued by its opening paragraph about a mathematical tool called 'combinators' (maybe discovered by a mathematician in 1920, Moses Schonfinkel). I think these combinators underlie what Wolfram believes may be a new framework he's created for thinking about physics, that will probably lead to a Theory of Everything.

Back in high school I read some excerpts of Newton's Principia Mathematica, in which he described some of the laws he'd discovered (like the inverse square law of gravity), in English.  It was a struggle to understand what he meant: expressed in English, the statements were really hard to understand, but easy to understand when expressed in modern mathematical notation.

Reading of the argument between Newton and Leibnitz regarding their (simultaneous and independent) discovery of calculus, I was struck similarly that Leibnitz's notation seemed more elegant and easier to work with than Newton's (which is presumably why Leibnitz's notation is the one we adopted).

And probably the capstone was reading the SF book Babel-17 by Samuel Delaney.

Those things made it clear to me that introducing excellent symbolism (notation) for the right concepts, coupled with rules for how to manipulate those symbols to make true statement, can produce really powerful tools for reasoning.  (That and asking the right question, or framing a problem the right way.)

But Wolfram's (long but fascinating) blog post offered me two more big ideas:

1) That there were and are people who think about symbolism and notation, and invent new ways to think about stuff. And that Moses Schonfinkel was one of those people, who also tried to distil mathematics down to a minimal framework - and through his invention of combinators reduced all maths down to just three combinators.

(A note for any computer scientists out there: combinators are equivalent to Turing machines which are equivalent to the Lambda calculus which is equivalent to cellular automata.)

2) Wolfram's claim that probably the single most important idea of this past century is that of universal computation: that in an absolutely real sense, the universe is an engine like a cellular automata, that's in operation.

His blog post also mentions his new physics project (Finally We May Have a Path to the Fundamental Theory of Physics…,and It’s Beautiful), announced and released several months ago, and today I've listened to the 4+ hours of his fascinating interview by Lex Fridman about it (Stephen Wolfram: Fundamental Theory of Physics, Life, and the Universe | Lex Fridman Podcast #124).  I highly recommend it.  Fridman conveniently breaks down the video with time coded links in the description (or you click on the "Chapters" link near the bottom of the video to call up a more graphical view of all those topic areas), if you want to just dip in.

(His explanation of the fundamental idea of his new framework, the hypergraph, is clearly described in that link above, in the section "How It Works".)  Wolfram is hoping to find "the right rule" (or small set of rules?) that would produce our observed universe and physical laws.

He says in the article:

"But in the early 1980s, when I started studying the computational universe of simple programs I made what was for me a very surprising and important discovery: that even when the underlying rules for a system are extremely simple, the behavior of the system as a whole can be essentially arbitrarily rich and complex.

"And this got me thinking: Could the universe work this way?"

To summarise some parts of the interview, I think he and his team have come up with a new mathematical symbolism for working with physics, and I think it's probably a major breakthrough.  I think it's important because it offers deep insights into quantum mechanics, as well as special and general relativity.

A few points that stood out for me:

- Space is quantised (I think he said thinks, at about the scale of 10^-100); and that there may be 10^400 or more points

- Everything is just space

- The key part of the symbolism is the idea of what he calls a hypergraph that captures the relations between points in space

- You could represent it as a graph (pairs of nodes connected by edges), but it's better to connect a node to multiple other nodes by a hyper edge (a surface?)

- Time is the sequence of applying cellular automata style rules. There may have been about 10^500 moments of time so far

- You can estimate how many physical spatial dimensions there are by how many dimensions you need to represent any specific hypergraph to avoid lots of crossings. For some hypergraphs that comes out as three.

- You can make statements about the curvature of space, and the expansion of the universe, in a hypergraph.

- Quantum mechanics, as formulated in the 20th century, falls out naturally from the representation.

- Ditto for general relativity, and also special relativity

- The new formalisation, the new mathematics, is relatively easy to learn, and there's plenty of low-hanging fruit (insights) from applying it.

"It’s always a test for scientific models to compare how much you put in with how much you get out. And I’ve never seen anything that comes close. What we put in is about as tiny as it could be. But what we’re getting out are huge chunks of the most sophisticated things that are known about physics. And what’s most amazing to me is that at least so far we’ve not run across a single thing where we’ve had to say “oh, to explain that we have to add something to our model”. Sometimes it’s not easy to see how things work, but so far it’s always just been a question of understanding what the model already says, not adding something new."

- One example is that fermions and bosons are fundamentally different because in his formulation the fermions are the particles that like to bifurcate in the hypergraph and the bosons like to join branches.

- I gather integer spin and half-integer spin particles have interesting explanations in the theory

- He has an estimate that in the hypergraph that represents our universe, there's 10^200 times more "activity" going on to "maintain the structure of space" itself, than into maintaining all the matter we know exists in the universe.

Wolfram Physics Project: https://www.wolframphysics.org/
Stephen Wolfram's Twitter: stephen_wolfram
Stephen's Blog: https://writings.stephenwolfram.com
His Books:
- A New Kind of Science
- A Project to Find the Fundamental Theory of Physics

Wolfram writes:

"Will we be able to bring together physics, computation and human understanding to deliver what we can reasonably consider to be a final, fundamental theory of physics? It is difficult to know how hard this will be. But I am extremely optimistic that we are finally on the right track, and may even have effectively already solved the fascinating problem of language design that this entails."

and

"For me, one of the most satisfying aspects of our discoveries over the past couple of months has been the extent to which they end up resonating with a huge range of existing—sometimes so far seemingly “just mathematical”—directions that have been taken in physics in recent years. It almost seems like everyone has been right all along, and it just takes adding a new substrate to see how it all fits together. There are hints of string theory, holographic principles, causal set theory, loop quantum gravity, twistor theory, and much more. And not only that, there are also modern mathematical ideas—geometric group theory, higher-order category theory, non-commutative geometry, geometric complexity theory, etc.—that seem so well aligned that one might almost think they must have been built to inform the analysis of our models.

"I have to say I didn’t expect this. The ideas and methods on which our models are based are very different from what’s ever been seriously pursued in physics, or really even in mathematics. But somehow—and I think it’s a good sign all around—what’s emerged is something that aligns wonderfully with lots of recent work in physics and mathematics."

He's also doing this all out in the open (publishing the software and papers), and inviting collaboration.

Exciting days (in a good way!) may lie ahead.