(9/15/07) My "lite" version of this made the cover of New Scientist. The illustrated version is here (subscribers only); the text is freely available here.
(4/5/07) After only 11 years of procrastination, I've finished the sequel to my bananas mathematical multiverse paper.

Which mathematical structure is isomorphic to our Universe?

The new paper: The Mathematical Universe

(Download here)

Author: Max Tegmark

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 and initial conditions to broader issues like consciousness, parallel universes and Gödel incompleteness. I hypothesize that only computable and decidable (in Gödel'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.

Reference info: arXiv:0704.0646 [gr-qc]

Warning: Sections III, IV and the appendix of this paper are quite technical, so if you're among the 99.99% who don't have a Ph.D. in physics, perhaps skip those sections. The shorter and non-technical version I wrote for fNew Scientist is easier to read - it's here. The older paper below is slightly less technical. A much easier read covering related questions, certified 100% equation free, is the 3-way food fight here that I co-authored with Piet Hut and Mark Alford. Section IV of my parallel universe article here and the multiverse FAQ here may also be more accessible.
The arrows indicate the close relations between mathematical structures, formal systems, and computations. The question mark suggests that these are all aspects of the same transcendent structure (the Level IV multiverse, including our world), and that we still have not fully understood its nature.

Comments: I think of this paper as the sequel to one below that I wrote in 1996, clarifying and extending the ideas described therein, and including related ideas that I had fun thinking about in the interim but never got around to writing up.

The original 1996 paper: Is "the theory of everything'' merely the ultimate ensemble theory?

(Download here)

Author: Max Tegmark

Abstract: We discuss some physical consequences of what might be called "the ultimate ensemble theory'', where not only worlds corresponding to say different sets of initial data or different physical constants are considered equally real, but also worlds ruled by altogether different equations. The only postulate in this theory is that all structures that exist mathematically exist also physically, by which we mean that in those complex enough to contain self-aware substructures (SASs), these SASs will subjectively perceive themselves as existing in a physically "real'' world. We find that it is far from clear that this simple theory, which has no free parameters whatsoever, is observationally ruled out. The predictions of the theory take the form of probability distributions for the outcome of experiments, which makes it testable. In addition, it may be possible to rule it out by comparing its a priori predictions for the observable attributes of nature (the particle masses, the dimensionality of spacetime, etc) with what is observed.

Reference info: gr-qc/9704009. Annals of Physics 270, 1-51 (Received November 19, 1996; published November 20, 1998)

This paper was the cover story of New Scientist - you'll find it here. 

Comments:

The figure at the top of this page shows a small part of the "family tree'' of mathematical structures as described in the paper. The complete tree is probably infinitely large, so the figure way down below, at the bottom of this page where the arrows are explained, merely shows some of the most basic structures. Those complex enough to contain self-aware substructures (SASs), "observers'' such as us, are almost certainly not in this picture. Many of the highly complicated structures are likely to be dead as well, devoid of SASs. The figure below illustrates how little one needs to change some aspects of our world to make it hostile to life, and a similar figure for the dimensionality of space and time can be found here.
Figure 5. Anthropic constraints on the electromagnetic and strong coupling constants. The observed values around (1/137,0.1) are indicated with a filled square. In the purple region, no hydrogen survives the Big Bang. In the yellow region, there are almost no stable atomic elements.


Figure 1. Relationships between various basic mathematical structures. The arrows generally indicate addition of new symbols and/or axioms. Arrows that meet indicate the combination of structures - for instance, an algebra is a vector space that is also a ring, and a Lie group is a group that is also a manifold.
 

TOE links

Wei Dai's "everything'' mailing list

In case you've read this far, you may be interested to know that Wei Dai has set up a mailing list for discussing this sort of ideas. Here is his message with instructions for how to join:

Date: Thu, 15 Jan 1998
From: Wei Dai
Subject: ANNOUNCE: the "everything" mailing list

You are invited to join a mailing list for discussion of the idea that all possible universes exist. Some possible topics of discussion might include:

Here are some papers that can serve as a basis for the discussion: You can surf the postings to this list tying in with my paper here. To subscribe to the mailing list, please click here.
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This page was last modified September 25, 2007.
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