Mandelbrot set (Source: Wikipedia)
I recently read Max Tegmark’s latest book, ‘Our Mathematical Universe‘, about his views on multiverses and the ultimate nature of reality. This is the fourth and final post in a series on the concepts and views he covers in the book.
The previous entries were:
Tegmark’s Level I Multiverse: infinite space
Tegmark’s Level II Multiverse: bubble universes
Tegmark’s Level III Multiverse: The many worlds interpretation of quantum mechanics
This final post in the series is a commentary on the overall book. Tegmark spends the early parts reviewing the current state of cosmology and physics. As described in the previous entries, he covers three increasingly diverse and grander definitions of the multiverse. These are fairly standard multiverse conceptions, and they aren’t all the one in currently circulation, but they are the ones most relevant to his main thesis.
The Mathematical Universe Hypothesis
Philosophy is written in this grand book, the universe … It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures –Galileo
Galileo wasn’t the first to say this of course. The ancient Greeks were also well aware of it. Mathematics is at the heart of science. Isaac Newton is credited with explaining the universal role of gravity, not because he was the first to come up with the idea (others had already contemplated it), but because he was the first to demonstrate the mathematics that described its dynamics.
The uncanny usefulness of mathematics in describing the world has often been a source of puzzlement for many philosophers. Indeed, there is a philosophy of mathematics field where a number of theories about this are discussed and debated, such as empiricism, platonism, nominalism, and many others.
So, the idea that mathematics describes the universe is well accepted. Tegmark, however, goes further by asserting that the universe is not just described by mathematics, but that it is mathematics, characterizing this as a radical form of platonism.
Now, immediately we have to do an important semantic clarification. When Tegmark refers to mathematics, he isn’t referring to the notation, the nomenclature, or the techniques that we use to express or explore mathematics. The ancient Greeks worked in math with a different notation than we use today, and no doubt an alien from Andromeda would have a radically different notation and process than anything humans have conceived of. But all these notations and processes should refer to the same underlying structures, the same underlying realities.
Tegmark points out that these mathematical structures are often identical to the underlying structures in nature. We have a tendency to view mathematical structures as abstract and separate from physical reality. But if those abstract structures match the physical ones, if we have two descriptions that are equivalent, then it makes sense to regard them as describing the same thing.
Many properties in science, such as empty space, the quantum wave function, or the spin property of elementary particles, are really only known only by their numeric properties. (“Spin” was originally thought to be descriptive of particles rotating in some classical manner. Subsequent developments showed that to be naive, but the name stuck.)
Most scientific theories are mathematical at their core, but require a qualitative explanation of one or more of the variables. In physics, this is often referred to as “baggage”. For example, the equation E=mc2 is fairly meaningless if you don’t know that E is energy, m is mass, and c is the speed of light.
Tegmark speculates that, if the Mathematical Universe Hypothesis is true, then the much sought after Theory of Everything should be a purely mathematical theory. It shouldn’t need any baggage. It’s entities should merely serve as points in relationships that should be enough to explain all of reality.
Addressing commons criticisms of the MUH, Tegmark spends a chapter on time. Mathematical structures are timeless structures, so how does that relate to a universe that evolves with time? Thinking in terms of spacetime, with time as one of the dimensions, the universe, including all of its history, could be viewed as a static structure. Tegmark uses the example of a DVD movie that appears to change when watching it, but is actually a static unchanging construct. He describes this concept in fascinating detail, in a manner that I can’t do justice to here.
Tegmark has an interesting discussion on time, infinity, and strange predictions that may call into question whether infinity is a valid concept. I found this section interesting because infinity seems to be an important assumption for the Level I and II multiverses. This discussion also included an excellent description of problems such as Boltzmann brains.
Finally, Tegmark addresses the most glaring criticism, that many mathematical structures do indeed match real world patterns, but not all of them. Many, such as the Mandelbrot set, exist only abstractly. Here is where all the earlier discussion of multiverses come to fruition. Tegmark’s answer is that all mathematical structures correspond with actual physical patterns, just not all in this universe.
The Level IV multiverse is one of mathematical structures. If our universe is a mathematical structure, then it is only one of an infinite variety of structures. All mathematical structures have physical reality in this multiverse. Exploring this multiverse is a matter of computation and ideas.
Before reading this book, I was agnostic about the MUH, and I’m forced to say that I remain largely agnostic, albeit now in a much more informed fashion. Tegmark does an excellent job of describing the concept, along with the many required supporting ideas. But I often found him to exude a level of certainty that felt unwarranted.
His certitude is often related to what he sees as the inevitable mathematical consequences of well accepted theories. I don’t understand the mathematics of most of those theories well enough to judge first hand whether or not that certitude is warranted. But I’m aware that many physicists, who do understand those theories at the mathematical level, don’t necessarily concur.
I’m also aware that just because the mathematics lead to a certain conclusion doesn’t make that conclusion inevitable. The mathematical consequences of Newtonian mechanics allowed astronomers to predict the existence of Neptune because of Uranus’s orbit, but it also led them to predict the existence of Vulcan because of Mercury’s orbit. One was right, but the other was wrong, and a new theory (general relativity) was necessary to understand why.
I do strongly believe that mathematics rest on empirical foundations, foundations found in the patterns of nature. As a result, many mathematical constructs have real world correlates, and many others approximate real world patterns. This, to me, is sufficient to explain the powerful utility of mathematics in science, without necessarily having to adopt an absolutist position about all mathematical structures having physical existence.
Of course, many abstract mathematical structures have no known physical correlates. Here Tegmark’s extensive descriptions of multiverses serve an important purpose, since multiverses are necessary to explain how these abstract structures could exist physically. Interestingly, Tegmark himself does speculate that some mathematical structures might actually not exist. His focus is on infinite ones, but it doesn’t seem like much of a cognitive leap to conclude that many other types might not as well.
But if those abstract structures don’t have a physical existence, then where do they come from? I’m tempted to say that they come from the same place as Vulcan, that is a tautological conclusion with no real world correlate. But this implies that they’re not valid, and I don’t think that, particularly since abstract structures sometimes turn out to correspond to something physical that we just weren’t aware of when they were formulated.
To be clear, I do think the MUH is a valid candidate for reality. It might be true. In the first post on this blog, I discussed the possibility that reality might be structure all the way down, and the MUH is definitely compatible with that. Even if reality does have a brute physical layer, everything above it are patterns, most of which, if not all, are describable in mathematical terms.
I tend to think that whether or not the MUH is true is a philosophical matter. Tegmark asserts that the idea is falsifiable since if it isn’t true, physics will eventually hit a brick wall where mathematics is no longer useful. The problem is that if we hit such a wall, MUH proponents can always claim that we simply don’t know enough yet to apply mathematics to that wall.
Indeed, a case could be made that this is exactly what the indeterminancy of a single quantum particle is, and that quantum interpretations that rescue determinism are just saving appearances. Now, I’m agnostic on the major quantum interpretations, and I certainly don’t think it’s productive to assume we’ll never know more than we do about it, but it does seem that the MUH needs one of the deterministic interpretations of quantum mechanics to be true.
All that said, Tegmark is an excellent writer, and if you’ve found the ideas in this series interesting, then I highly recommend his book. It’s an excellent introduction to many ideas and I’ve only lightly scratched the surface in this and the previous posts.
Fellow blogger, Disagreeable Me, a advocate of the MUH, has written an excellent blog post on it, which I know some of you have already read. DM approaches the issue from a philosophical angle, and I found myself returning to his post after I had completed the book. A highly recommended read.