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Wednesday, March 12, 2014
Reading List: Our Mathematical Universe
 Tegmark, Max. Our Mathematical Universe. New York: Alfred A. Knopf, 2014. ISBN 9780307599803.

In 1960, physicist Eugene Wigner wrote an essay titled
“The
Unreasonable Effectiveness of Mathematics in the Natural
Sciences”
in which he observed that “the enormous usefulness of
mathematics in the natural sciences is something bordering
on the mysterious and that there is no rational
explanation for it”. Indeed, each time physics has
expanded the horizon of its knowledge from the human
scale, whether outward to the planets, stars, and galaxies; or
inward to molecules, atoms, nucleons, and quarks it has been
found that mathematical theories which precisely model these
levels of structure can be found, and that these theories
almost always predict new phenomena which are subsequently
observed when experiments are performed to look for them. And yet
it all seems very odd. The universe seems to obey laws written
in the language of mathematics, but when we look at the universe
we don't see anything which itself looks like mathematics. The
mystery then, as posed by Stephen Hawking, is “What is it
that breathes fire into the equations and makes a universe for
them to describe?”
This book describes the author's personal journey to answer these deep questions. Max Tegmark, born in Stockholm, is a professor of physics at MIT who, by his own description, leads a double life. He has been a pioneer in developing techniques to tease out data about the early structure of the universe from maps of the cosmic background radiation obtained by satellite and balloon experiments and, in doing so, has been an important contributor to the emergence of precision cosmology: providing precise information on the age of the universe, its composition, and the seeding of large scale structure. This he calls his Dr. Jekyll work, and it is described in detail in the first part of the book. In the balance, his Mr. Hyde persona asserts itself and he delves deeply into the ultimate structure of reality.
He argues that just as science has in the past shown our universe to be far larger and more complicated than previously imagined, our contemporary theories suggest that everything we observe is part of an enormously greater fourlevel hierarchy of multiverses, arranged as follows.
The level I multiverse consists of all the regions of space outside our cosmic horizon from which light has not yet had time to reach us. If, as precision cosmology suggests, the universe is, if not infinite, so close as to be enormously larger than what we can observe, there will be a multitude of volumes of space as large as the one we can observe in which the laws of physics will be identical but the randomly specified initial conditions will vary. Because there is a finite number of possible quantum states within each observable radius and the number of such regions is likely to be much larger, there will be a multitude of observers just like you, and even more which will differ in various ways. This sounds completely crazy, but it is a straightforward prediction from our understanding of the Big Bang and the measurements of precision cosmology.
The level II multiverse follows directly from the theory of eternal inflation, which explains many otherwise mysterious aspects of the universe, such as why its curvature is so close to flat, why the cosmic background radiation has such a uniform temperature over the entire sky, and why the constants of physics appear to be exquisitely finetuned to permit the development of complex structures including life. Eternal (or chaotic) inflation argues that our level I multiverse (of which everything we can observe is a tiny bit) is a single “bubble” which nucleated when a preexisting “false vacuum” phase decayed to a lower energy state. It is this decay which ultimately set off the enormous expansion after the Big Bang and provided the energy to create all of the content of the universe. But eternal inflation seems to require that there be an infinite series of bubbles created, all causally disconnected from one another. Because the process which causes a bubble to begin to inflate is affected by quantum fluctuations, although the fundamental physical laws in all of the bubbles will be the same, the initial conditions, including physical constants, will vary from bubble to bubble. Some bubbles will almost immediately recollapse into a black hole, others will expand so rapidly stars and galaxies never form, and in still others primordial nucleosynthesis may result in a universe filled only with helium. We find ourselves in a bubble which is hospitable to our form of life because we can only exist in such a bubble.
The level III multiverse is implied by the unitary evolution of the wave function in quantum mechanics and the multiple worlds interpretation which replaces collapse of the wave function with continually splitting universes in which every possible outcome occurs. In this view of quantum mechanics there is no randomness—the evolution of the wave function is completely deterministic. The results of our experiments appear to contain randomness because in the level III multiverse there are copies of each of us which experience every possible outcome of the experiment and we don't know which copy we are. In the author's words, “…causal physics will produce the illusion of randomness from your subjective viewpoint in any circumstance where you're being cloned. … So how does it feel when you get cloned? It feels random! And every time something fundamentally random appears to happen to you, which couldn't have been predicted even in principle, it's a sign that you've been cloned.”
In the level IV multiverse, not only do the initial conditions, physical constants, and the results of measuring an evolving quantum wave function vary, but the fundamental equations—the mathematical structure—of physics differ. There might be a different number of spatial dimensions, or two or more time dimensions, for example. The author argues that the ultimate ensemble theory is to assume that every mathematical structure exists as a physical structure in the level IV multiverse (perhaps with some constraints: for example, only computable structures may have physical representations). Most of these structures would not permit the existence of observers like ourselves, but once again we shouldn't be surprised to find ourselves living in a structure which allows us to exist. Thus, finally, the reason mathematics is so unreasonably effective in describing the laws of physics is just that mathematics and the laws of physics are one and the same thing. Any observer, regardless of how bizarre the universe it inhabits, will discover mathematical laws underlying the phenomena within that universe and conclude they make perfect sense.
Tegmark contends that when we try to discover the mathematical structure of the laws of physics, the outcome of quantum measurements, the physical constants which appear to be free parameters in our models, or the detailed properties of the visible part of our universe, we are simply trying to find our address in the respective levels of these multiverses. We will never find a reason from first principles for these things we measure: we observe what we do because that's the way they are where we happen to find ourselves. Observers elsewhere will see other things.
The principal opposition to multiverse arguments is that they are unscientific because they posit phenomena which are unobservable, perhaps even in principle, and hence cannot be falsified by experiment. Tegmark takes a different tack. He says that if you have a theory (for example, eternal inflation) which explains observations which otherwise do not make any sense and has made falsifiable predictions (the finescale structure of the cosmic background radiation) which have subsequently been confirmed by experiment, then if it predicts other inevitable consequences (the existence of a multitude of other Hubble volume universes outside our horizon and other bubbles with different physical constants) we should take these predictions seriously, even if we cannot think of any way at present to confirm them. Consider gravitational radiation: Einstein predicted it in 1916 as a consequence of general relativity. While general relativity has passed every experimental test in subsequent years, at the time of Einstein's prediction almost nobody thought a gravitational wave could be detected, and yet the consistency of the theory, validated by other tests, persuaded almost all physicists that gravitational waves must exist. It was not until the 1980s that indirect evidence for this phenomenon was detected, and to this date, despite the construction of elaborate apparatus and the efforts of hundreds of researchers over decades, no direct detection of gravitational radiation has been achieved.
There is a great deal more in this enlightening book. You will learn about the academic politics of doing highly speculative research, gaming the arXiv to get your paper listed as the first in the day's publications, the nature of consciousness and perception and its complex relation to consensus and external reality, the measure problem as an unappreciated deep mystery of cosmology, whether humans are alone in our observable universe, the continuum versus an underlying discrete structure, and the ultimate fate of our observable part of the multiverses.
In the Kindle edition, everything is properly linked, including the comprehensive index. Citations of documents on the Web are live links which may be clicked to display them.
Posted at March 12, 2014 23:53