The Anthropic Principle and the Level IV Multiverse

A review of Our Mathematical Universe (Max Tegmark)

Our Mathematical Universe was my summer holiday reading this year.  But it turned out to be much more than just something to keep me occupied while lounging on the beach.  This book has changed my conception of reality.  The themes in the book are similar in nature to those in The Fabric of Reality by David Deutch - another one of my favourite books.  However, instead of restricting the argument to the parallel universes predicted by Everett's Universal Wavefunction, Tegmark takes us on a tour of four levels of multiverse.  More than this, he provides an overarching theoretical framework for understanding them based on what he calls "the A-word" (because using its full name is guaranteed to get your paper rejected).

The argument goes like this.  Whenever we find Nature appears finely tuned to make self-aware life possible, then there are 3 possible explanations

1. It's a fluke!
2. It's design! (by an intelligent entity who intended us as an outcome)
3. Other tunings also exist - we just haven't found them yet!

I crossed out number 2 because a) it is beyond arrogant; and b) it doesn't answer any questions.  (Or alternatively, it a) violates the Copernican principle, and b) leaves us needing to explain what created our creators.)

Number 1 may or may not be reasonable.  The deciding factor is the probability.  In medicine and most other scientific disciplines the cutoff is 5%.  If it's less than 5% likely for the observed outcome to have occurred were the null hypothesis true, then the null hypothesis can reasonably be rejected.

Tegmark claims that there are around 30 or so constants in modern physics - things like the gravitational constant G, the fine structure constant, the mass of the electron &c. - all of which seem to be fine tuned for our existence.  That is to say, if you change any one of them (within the bounds in which they make sense) you are likely to end up with a world without atoms, much less planets or humans.  If we let the null hypothesis be that these constants are random then we can work out the likelihood of them falling in the small range that allows for some sort of life.  The answer is around one in $10^{120}$, which is safely within the 5% we normally use.  This argument alone should rule out Number 1 and leave us with Number 3: There are other universes where the constants take different values$\dagger$.

Giordano Bruno -Rome

A similar argument can be made regarding the existence of other suns.  Our sun's mass falls within a small range of values that make life on Earth possible.  In fact there are quite a number of happy coincidences which make life on Earth possible.  Tegmark argues that this alone should have lead us to reject the "fluke" explanation and leave us to conclude Number 3: There are many Suns and many planets.  This is essentially what Giordano Bruno argued - that the stars are other suns - and his argument got him burned at the stake.

Obviously it is now accepted that there are other suns, but it wasn't this anthropic argument that led to it being accepted.  Instead we waited for a more complete theory to be developed which solved many other puzzles, but which happened to predict other suns too.  Surprisingly, it is also now largely accepted that there are other universes (at huge space-like intervals from the here-now) where the "constants" of nature take different values.  But, again, it wasn't the anthropic argument that led to it being accepted.  Instead we waited for a more complete theory (called inflation) to be developed which solved many other puzzles, but which happened to predict these other universes too.

Tegmark's clever point seems to be that we needn't have waited for the theories that predicted these other worlds, almost as a side effect.  The ridiculously small probabilities involved should have been enough for us to reject Number 1 (the fluke), and conclude Number 3 (all other options are explored - somewhere).

The book takes us on a tour of different multiverse levels.  For each of the first three Tegmark explains the theory that predicts it; the measurements which validated the theory; and also the bizarre coincidences that should have alerted us to the existence of that multiverse even before we had the theory.

• Level I - spacetime beyond our Hubble Horizon.  We haven't had time to be affected by it since the big bang.  We will be permanently separated from it if it turns out it's moving away from us faster than the speed of light.
• Level II - the L2 multiverse consists mostly of inflating material (material expanding by a factor of $10^{38}$ each second).  However, the inflation process creates ordinary (non-inflating) material as a by-product.  This condenses in bubbles that form L1 multiverses within which the 30 or so "constants" of nature really are constant.
• Level III - this multiverse consists of the "worlds" in the Many Worlds Interpretation.  Unlike the L1 and L2 multiverses its universes are not separated by spacetime, but by Hilbert space.
• Level IV - read on

One of the big outstanding tasks in modern physics is to combine our best two theories GR and QM.  Everyone (almost everyone) believes that this is possible.  If you don't then that's akin to saying either a) a human being is always required to judge when the equations of one or the equations of the other apply, or b) the boundary between the two is not defined by any rules, i.e. nature is fickle in a way she has never before demonstrated.  What if, argues Tegmark, you are the one to figure out the Grand Unified Theory - the fundamental equation that includes GR and QM as mere limiting cases.  So, you pick up your Nobel prize, you party, you go home and try to sleep.  That night, do you lie in bed and think "that's that, everything is now understood"?  Or - if you are not totally plastered - will you be thinking "why that equation?" Tegmark points out that we will be in exactly the same situation we have been many times before, because there are infinitely many possible equations and only a tiny measure of them make self-aware life possible.  The choice of equation is itself a fine tuning of reality that seems almost designed to allow us to exist.  The Level IV multiverse is Tegmark's answer to the question "why that equation".

The Level IV multiverse is nothing but the set of all mathematical structures, a tiny proportion of which are sufficiently complex to support the self-aware substructures we call life.  This is an amazing conclusion, whose dependence on the Anthropic principle would probably lead to it being rejected by most scientists $\dagger^1$.   But to me the argument holds water.  In fact I'll go further and say I believe it.  I did a pure maths degree and spent a lot of time wondering what the hell we were talking about.  I would ask myself, Is this a game in which we find out what marks on a piece of paper we can obtain by following some rules, or do these symbols describe the behaviour of an actual thing with some sort of real existence?  Tegmark answers that question resoundingly: mathematical structures are real, and in fact all features of reality are merely emergent features of those mathematical structures.

There is, it has to be admitted, one weak point.  Gödel pointed out that if you have a mathematical structure defined by some axioms, and the structure contains the integers as a substructure, then you can always find a statement such that either it or its negation can be added to the axioms without fear of contradiction.  Both of the new sets of axioms are consistent mathematical structures, but if one is physically realizable, the other ain't $\dagger^2$.  So, in the Level IV multiverse there must be some consistent mathematical structures missing.  To solve this Tegmark offers a couple of possible resolutions: i) only finite structures are allowed in the L4 multiverse (i.e. $\mathbb{Z}$ is never physically realized), or ii) only computable structures are allowed $\dagger^3$.  Both of these solutions feel like bodges $\dagger^4$.  There is a third option that he doesn't mention, which is that for each Gödel statement nature chooses one $\dagger^5$.

All in all, a fantastic book, full of detail but with a consistent theme running through it.

FOOTNOTES

• $\dagger$ But what if these constants are not arbitrary but defined by a law of physics we don't yet know?  Wouldn't that reduce the improbability?  As we will see later on, all that would do is replace the improbability of the arbitrary constants having values that allow for life with the improbability of the arbitrary laws having values that allow for life.
• $\dagger^1$  Although it also uses Occam's Razor - which has a better press.  Occam's Razor states that the theory with the lowest information content - i.e. the theory that can be compressed the most - and still predicts the facts is the right one.  And the L4 multiverse theory has zero information content!
• $\dagger^2$ For example, you can add to the axioms for the real numbers $\mathbb{R}$ the statement "there is a subset with cardinality strictly between $\mathbb{Z}$ and $\mathbb{R}$" or the statement "there are no subsets with cardinality strictly between $\mathbb{Z}$ and $\mathbb{R}$", and you get a self-consistent theory either way.
• $\dagger^3$ i.e. every member of the structure must be finitely addressable and every operation computable by a finite program that can be implemented on a Turing machine.
• $\dagger^4$ and both probably increase the information content of the theory, thereby reducing the support from Occam
• $\dagger^5$ He probably omitted that option because it increases the information content to infinity!