- Tegmark, Max.
Our Mathematical Universe.
New York: Alfred A. Knopf, 2014.
ISBN 978-0-307-59980-3.
-
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 four-level 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 fine-tuned 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 pre-existing
“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 fine-scale 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.
March 2014 