The Mathematical Universe

[neutralino]

I found this rather interesting paper today that had been reposted to the arxiv. It's by a guy called Max Tegmark, and is called, as the thread title suggests, The Mathematical Universe. Here is the abstract:

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

It's quite a mathematically complex paper, but does have some nice diagrams throughout. There is also a shorter version of the paper available that was written for New Scientist entitled: Shut up and calculate. This non-technical, so may be better for the lay-reader. Here is the abstract:

I advocate an extreme "shut-up-and-calculate" approach to physics, where our external physical reality is assumed to be purely mathematical. This brief essay motivates this "it's all just equations" assumption and discusses its implications.

These papers are both very interesting. I like Tegmark, as he's a little different to most scientists; he's more philosophical.

I'll post more when I've read the paper completely, but please feel free to comment if you've read either paper, or are interested.

[Profpat]

I agree with you Neutralino, about Max Tagmark. He is one of my favorite physicists. It was his article on Parallel universes that I found so interesting and informative. He is now at MIT, prior to that he was at UPenn. Some of his "stuff" may now be the intellectual property at UPenn. However he has a very interesting home page filled with "stuff" at:
( http://space.mit.edu/home/tegmark/no...ontlinkme.html )

Best to all,

Pat

[neutralino]

Thanks, Pat. The first time I heard of him was when I stumbled upon a paper of his entitled On the dimensionality of spacetime (warning: that links straight to a .pdf document) which I found to be a very interesting read.

[neutralino]

I've reopened this thread, for the time being, in case anyone wishes to discuss the topic.

[phoenixthoth]

Hi All.

In the time since I last posted, I have been looking for that mathematical structure that Tegmark posits is isomorphic to our universe. I may or may not be on the right track. Right now, a colleague of sorts is looking over my work in this area and I'd like to put a link to it when I've polished it further.

I will post the abstract and some results.

In this document, the author presents a structure U with the property that all structures are elementarily embeddable within it. One essential tool is a version of New Foundations set theory, first developed by Quine, as presented by Holmes in [4]. The motivation is given by The Mathematical Universe article by Max Tegmark, [6], in which it is hypothesized that physical existence is mathematical existence. Consequently, it is hypothesized that the structure with the aforementioned property could be central in the Mathematical Universe Hypothesis. Finally, some basic facts about U are explored.
The author assumes some knowledge of mathematical logic such as, for example, the inductive definition of a first-order well-formed formula and the definition of the satisfaction relation, |=. A survey of relevant concepts of mathematical logic is available online in [1], and offline in [2] and [3].

A couple of results...

I have a mathematical structure U that is ultimate in that, basically, it contains all other structures. Part IV is the part that is unfinished but it will be a list of several properties of "the theory of U," which the set of true statements about U. So far, the theory of U has the following properties that I know of:

1. the theory of U is complete. This means that, given a statement, that statement or its negation is true for U.
2. the theory of U is consistent. This means you will never be able to prove a statement as both true and false.
3. the theory of U is finitely axiomatizable, meaning that there is a finite list of assumptions such that the entirety of the consequences of those axioms is identical to the theory of U
4. the theory of U is undecidable. This means there is no "effective procedure" (eg, a mathematical proof or, eg, a program on a computer) for determining if a statement is true or false. Another way to put it is that some statements that although they are true (or false) by completeness, a proof of that fact is impossible (no matter how clever you are).

Cheers

[damccut]

The one thing that all of you, including Max, have overlooked is that it would take too much mathematics to fully describe a universe that is not exceedingly simple, which ours isn't. The best we can hope for is to generalize aspects of it into equations that are merely approximations of the true behavior of its component parts.

Don't get me wrong, I like the way Tegmark gets "out there" on a lot of topics that the mainstream guys stay away from, but it means that a lot of it ends up being worthless in the end.