### The Mathematical Universe

In the paper The Mathematical Universe (PDF,) cosmologist Max Tegmark takes the (epistemological) position that the External Reality Hypothesis (ERH) strongly implies the Mathematical Universe Hypothesis (MUH.) MUH holds that reality isn't just described (modeled) by mathematics, but that mathematics and reality are actually deeply equivalent.

The abstract of the paper says "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."

In another paper, Parallel Universes, Tegmark defines a taxonomy of different types of Universes (reminiscent of Georg Cantor's taxonomy of different types of infinities.) The existence of Type IV Universes would be a probable consequence of the Mathematical Universe Hypothesis.

In addition to making me think more deeply, and more outside the box, than anything I've read since encountering Eliezer Yudkowsky's essay Staring into the Singularity, these two papers by Tegmark strongly remind me of several things that may be deeply related:

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