Our mathematical universe. My quest for the ultimate nature of reality
In this book Max Tegmark defends his speculative and controversial theory of everything. His claim is the mathematical universe hypothesis (MUH) which he sees as a direct consequence of the external reality hypothesis (ERH). The latter is a kind of Platonic vision of mathematics: there is some external reality independent of human intervention and that we can only observe filtered by the limitations of our senses with subjective interpretations of our minds. In the MUH our universe and everything it contains is just a mathematical structure. Since any mathematical structure is a universe, there is a whole zoo of universes outside ours, and any self aware creature in such a universe will experience its physical environment as real as we do.
But the book starts in a much less provocative way. Part one (zooming out) begins explaining how people managed to measure the size of the moon, then the sun, planets, stars, galaxies etc. The farther we can see in distance the farther we see in the past, given the time needed by the light to travel through space. So we end up seeing young galaxies forming shortly after the Big Bang. Beyond that: only darkness. And yet there is some microwave background radiation. That is where Tegmark got involved in visualizing the picture of our baby universe in the WMAP project. It shows a bright plasma of a very hot free electron soup, cooling down and transforming hydrogen into helium in our infantile universe of only 400,000 years `young'. His story becomes very lively at this point him being a first hand witness. But if this was the state after the exponential inflation, where did mass and gravitation come from, that will finally form the galaxies? Tegmark gives clear answers to such fundamental questions and many others. If you define our universe as the sphere from where light can reach us since its origin some 14 billion year ago, then one might expect there is more beyond what we can observe, i.e., beyond the boundary of that sphere. So there may be more universes `out there'. This is what Tegmark calls the Level I multiverse. Our Big Bang is not the very beginning, but basically the end of the stage of exponential inflation. The Big Bang is caused by the inflation and not its origin. But the creation of a Big Bang is a very local phenomenon. In chaotic inflation theory there is a multitude of Big Bangs that will form bubbles in this for ever inflating multiverse and in each of these bubbles another universe will exist, some with different fundamental physical laws. This is the Level II multiverse.
Part two (zooming in) goes in the opposite direction and deals with particle physics and quantum mechanics. Much less details are given about the theory here, but it mainly serves to place the Copenhagen interpretation of a Schrödinger wave function collapse against the many worlds interpretation of Hugh Everett. All possible outcomes of the observation are possible, but they are alive in parallel worlds. Schrödinger's cat will be dead in one of the worlds, but it will be alive in a parallel one. This creates a Level III multiverse. This time the universes are not at a distance out of reach for us, but many versions of you will exist in as many parallel worlds that are separated in the Hilbert space in which the wave functions live. Since any possible outcome will be realized in one of these worlds, the Level III multiverse will include the Level I and Level II multiverses.
While all the multiverses defined so far have been considered also by others, the Level IV multiverse is Tegmark's idea. It is explained in the third part (stepping back) which fills almost completely the second half the book. Here he builds up his theory of the MUH and all the consequences that implies. For example he needs to explain how inhabitants of such a mathematical structure can be self-aware and how they experience the external reality. Since any mathematical structure is a universe, we are dealing with yet another kind of multiverse. This is what he calls the Level IV multiverse. However mathematics should rule. Thus Gödel's incompleteness or the Church-Turing undecidability should be avoided to form a consistent system. This rules out infinity. The `infinitely small' is related to continuity, but that can be removed because continuity is nothing but an approximation of reality that is only observed at a much higher scale. Zooming in at the details, everything there is just discrete particles, strings or branes or whatever, but always discrete. Real numbers are out too because they contain infinite information and thus are not computable in finite time. And so on and so further. Tegmark tackles one by one all possible objections and possible inconsistencies that may be raised by a critical opponent. Whether the reader will agree with all his arguments or not is of course up to the reader.