I've been looking at one of Max Tegmark's articles about his 'Mathematical Universe' hypothesis, here on arXiv.(adsbygoogle = window.adsbygoogle || []).push({});

As a preliminary, note that Tegmark's framework has four 'levels' of multiverses, with each level being an infinite collection of multiverses at the level below it. The second or third level is to do with quantum superpositions and has strong similarities to Everett's Many-Worlds framework. A Level 1 'multiverse' is just a single spacetime, but Tegmark assumes each Level 1 spacetime is spatially infinite.

One of the attention-grabbing aspects of Tegmark's hypothesis is that it says that everything that can happen, does happen.

That is not surprising when we are talking about quantum superpositions, as Everett says more or less the same thing.

What is surprising is that Tegmark claims this to be the case for every single Level 1 spacetime as well. The claim relies on an assumption that a single Level 2 multiverse, which is an infinite collection of Level 1 spacetimes, considered as a probability space, is ergodic, and that each Level 1 spacetime is spatially infinite. He writes:

That last paragraph (' Max Tegmark said:The physics description of the world is traditionally split into two parts: initial conditions and laws of physics specifying how the initial conditions evolve. Observers living in parallel universes at Level I observe the exact same laws of physics as we do, but with different initial conditions than those in our Hubble volume. The currently favored theory is that the initial conditions (the densities and motions of different types of matter early on) were created by quantum fluctuations during the inflation epoch (see section 3). This quantum mechanism generates initial conditions that are for all practical purposes random, producing density fluctuations described by what mathematicians call an ergodic random field.

Ergodic means that if you imagine generating an ensemble of universes, each with its own random initial conditions, then the probability distribution of outcomes in a given volume is identical to the distribution that you get by sampling different volumes in a single universe.

In other words, it means that everything that could in principle have happened here did in fact happen somewhere else.In other words....') does not seem to me to follow from what goes before it. Ergodicity is about expected values, and ergodic properties such as the one I linked above are carefully constrained with uses of the technical terms 'almost surely' or 'almost everywhere'. In an ergodic ensemble of infinite spacetimes, for a given spacetime S, the probability is 1 that an event E that occurs anywhere in the ensemble will occur somewhere in S. But that means that E occursalmost surelysomewhere in S, which is not the same as saying that it does in fact occur in S.

I am trying to work out why Tegmark might have written this. Possibilities that occur to me are:

I would greatly appreciate people's thoughts on this.

- Tegmark is trained as a physicist, not a mathematician, and has never studied formal probability theory, and does not understand that, in an infinite sample space, probability one does not imply certainty.
- Tegmark has learned that, but has forgotten it.
- Tegmark knows the distinction, but in the course of simplifying his conclusion in order to reach a wider audience, 'simplified' it to the point of making it misleading (which reminds me of Einstein's dictum: '
we should simplify as much as possible, but not more than that').- There is some other reason, beyond mere ergodicity, why every possible thing must happen somewhere in an infinite Level 1 spacetime; or
- I have misunderstood ergodicity.

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# I MUST every possible thing happen in a Tegmark L1 Multiverse?

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