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Tuesday December 09, 2008 16:13:21 -0600

Tegmark has posted a new paper that examines the deep mathematical nature of existence, the latest in a series of efforts that explore his personal interest in uncovering the ultimate Theory Of Everything. Tegmark has embarked on a personal quest to reveal the Holy Grail of Modern Physics.

The new paper titled, "The Mathematical Universe," explores a bold proposition: The physical world is nothing more nor less than an abstract mathematical structure.

Tegmark declares his goal to "push this idea to its extreme and argue that our universe is mathematics in a well-defined sense," and then proceeds to take the reader on a tour de force synthesis of physics, philosophy and the core problem of the unreasonable effectiveness of mathematics in describing the physical world.

Tegmark begins by examining two fundamental assumptions.

The first is reasonably non-controversial: The External Reality Hypothesis states there is a physical reality that exists independent of human experience.

The second assumption appears radical to the uninitiated, but is also relatively benign as it naturally falls out of the current understanding of quantum information theory: The Mathematical Universe Hypothesis states that our 'external reality' (presumably independent of out 'internal' perceptive state of mind) is a mathematical structure.

From these humble assumptions Tegmark unleashes a thorough examination of the intimate relationship between math, physics and the philosophy of existence.

In his paper Tegmark discusses the "baggage" of human language, a theme recently explored in another excellent paper from Jim Hartle.

Tegmark's personal "baggage" is presented in a "family tree" that organizes scientific theories by their relationship to other theories from which they may be derived. Not surprisingly, the top of Tegmark's tree features The Big Question Mark, the regime where it is generally assumed that Quantum Gravity plays a vital role on the road leading to the Theory of Everything.

Tegmark notes that physics "has come to focus on the way the external reality works rather than on the way it is."

Ideally, according to Tegmark, we would like to find a description of the entities that make up reality without invoking the baggage implied by human language and experience. He then argues that a mathematical structure is exactly what is required:

"Abstract entities with relations between them."

At this point Tegmark wisely leaves deeper mathematical details for the Appendix. Tegmark concludes that a reality external to human experience must be a theory of everything that has no baggage, thus the baggage free description is "precisely a mathematical structure."

Tegmark continues by introducing the "frog" observer within the structure of reality and the "bird perspective," the external point of view that studies mathematical structure.

As an example Tegmark invokes the concept of time, pointing out that mathematical structure is "an abstract, immutable entity existing outside of space and time. From the four-dimensional perspective of the bird, the motion of point particles appears as a tangle of spaghetti, thus the bird sees the frog as a thick bundle of pasta, whose highly complex intertwining corresponds to a cluster of particles that store and process information."

The core of Tegmark's argument invokes isomorphism, that is the presumed "one-to-one correspondence" between the external physical reality and the presently unknown mathematical structure that is our world. Tegmark wants to make certain the reader understands that although modern textbooks typically imply that the physical world is "described by mathematics," the isomorphism of the Theory of Everything literally means the external reality is in fact a mathematical structure.

The third section of Tegmark's paper examines the implications of physical reality as mathematical structure. Tegmark asks the question, "How do we compute the inside view [of the frog] from the outside view [of the bird]?"

Tegmark states, "In my opinion ... although understanding the detailed nature of human consciousness is an important challenge in its own right, it is not necessary for a fundamental theory of physics, which, in the case of us humans, corresponds to the mathematical description of our world found in physics textbooks."

Tegmark presents a clear explanation of the collective agreement between various subjective experiences of external reality, describing the consensus view from inside the physical world.

"Although the inside view looks black-and-white to a cat, iridescent to a bird seeing four primary colors, and still more different to a bee seeing polarized light, a bat using sonar, a blind person with keener touch and hearing, or the latest robotic vacuum cleaner, all agree on whether the door is open."

Following extensive examination of the relevant mathematics needed to create a "physics from scratch," Tegmark continues with a brief review of the history of what were once considered to be immutable fundamental laws. In Tegmark's vision, the laws underlying a Theory of Everything for a Mathematical Universe are "merely local bylaws," implying that what physicists once considered fundamental laws must vary across a vast landscape of parallel worlds.

A critical unknown that potentially might kill any Theory of Everything involves the complexity of the bird view outside the physical world versus the complexity of the frog observer on the inside. Tegmark raises the concern that "if describing the structure requires more bits than describing our observable universe, it is of course impossible to store the information about the structure in our universe."

On a positive note, and in spite of numerous struggles of string theory, Tegmark states that the cherished ideal of a simple and beautiful Theory of Everything has yet to be destroyed. If the frog, that is the observer within our universe, is found to have greater complexity than the bird perspective of the mathematician, our universe must be a member of a multiverse of parallel universes. In the multiverse the complexity we observe in our world encodes which of the parallel universes we inhabit. Tegmark explains that "an entire ensemble," in this case, the ensemble of parallel worlds, "is often much simpler than one of its members."

The reason our world appears complex from our frog perspective is that our attention is focused on only one universe in an ensemble that makes up a multiverse of parallel worlds.

Tegmark has categorized four types of parallel universes:

Level I universes include identical copies of our world physically separated from our observable universe by cosmic inflation. Tegmark muses that there is an "identical copy of you about ten raised to the power of ten to the power of twenty-nine meters away."

The Level II universes involve parallel worlds with different laws of physics existing on a landscape of possible vacuums, as has become popular in string theory.

Level III parallel worlds are implied by taking the unitary mathematics of quantum theory seriously, but as Tegmark comments, they "add nothing qualitatively new" to the predictions of the other levels.

The most extreme interpretation produces Level IV universes, where alternative mathematical structures produce universes with different laws of physics.

According to Tegmark, the existence of parallel worlds is not in question as they are already predicted by the standard cosmological model. Tegmark adds that "parallel universes are not a theory, but a prediction of certain theories."

Using the prediction of black holes coming from Einstein's General Theory of Relativity as an analogy, Tegmark states that the Theory of Inflation leads to Level I worlds, the addition of an energy landscape, as popularized in string theory leads to Level II, unitary quantum mechanics to Level III, and ultimately the realization that our physical world is isomorphic to a mathematical structure invokes Type IV.

In Tegmark's mind, mathematical existence and physical existence are one and the same, thus a mathematical structure does not describe a universe, but rather "it is a universe."

We found the section on "Implications for the Simulation Argument" of direct interest for our phenomenological investigations.

The Simulation Argument has been invoked to explain phenomena that appear to defy the laws of physics as they are presently understood. Our investigation of the U.S. Government records and personnel revealed serious interest in the existence of unexplained phenomena, as well as a disturbing effort by the government to cover and conceal this information from the public at large.

According to the simulation argument, our perception of reality is artificially created, presumably by advanced intelligence possessing far deeper understanding of the true nature of reality than can be accessed by the human mind. One need not consider a vast difference in evolutionary complexity to understand the potential gulf between the human mind and any 'caretakers' assumed to run such a simulation. Whereas the human mind can contemplate "life, the universe and everything," other species with which we share our world have not evolved brains capable of understanding Einstein's equations. Likewise, the human mind may have a limit in its ability to comprehend the complexity we find in the universe, although Tegmark continues to express hope for a simple and beautiful bird's eye point of view.

The idea that our existence might be simulated has given birth to serious papers on the subject; of the examples cited by Tegmark we are particularly fond of the work of Nick Bostrom. Tegmark also mentions an idea attributed to Seth Lloyd, that we live in an analog simulation created by a quantum fields forming a natural spatially distributed quantum computer.

Tegmark then proceeds to sidestep conventional concepts of the simulation argument. In the Mathematical Universe Hypothesis, computations do not evolve the universe step by step, since they define the bird's eye view of the mathematical structure from outside of time. Time itself is an emergent property coming from "clock subsystems." Tegmark also points out that even in the case of a classical computer simulation, there is no reason for the rate of time flow experienced within the simulation to correspond to the rate of computation of the computer that is running the simulation.

From the bird's point of view outside of space and time, nothing is computed at all. Instead the information required to encode all of the properties of our universe, even if it is a simulation, are stored in the mathematical structure that is our universe.

According to Tegmark, "Our universe could be simulated by quite a short computer program ... Every universe simulation corresponds to a mathematical structure ... eternal inflation predicts an infinite space with infinitely many planets, civilizations, and computers, and that the Level IV multiverse includes an infinite number of possible simulations."

The paper winds up in somewhat controversial territory by exploring the Computable Universe Hypothesis: The possibility that the mathematical structure we call the universe is defined by exactly computable functions. Tegmark reviews the undefined, undecidable
and uncomputable problems, noting "under certain circumstances, there are questions that can be posed but not answered."

In the Computable Universe there are only computable physical structures. Tegmark admits that the idea of a computable universe goes against the historical record of physics since "successful theories of physics violate" the Computable Universe Hypothesis. Perhaps, Tegmark suggests, there may ultimately exist a discrete computable structure hidden under layers of effective theory.

Difficult issues remain, but Tegmark expresses a personal hope for the Mathematical Universe Hypothesis, with a promise of an elegant and grand unification of physics, mathematics and information science.

We consider this paper essential reading for anyone concerned with the present-day state of the art convergence of what Oxford's David Deutsch has called the four strands of reality: Quantum physics, evolution, computation and knowledge.

Copyright (c) 2007 Gary S. Bekkum and Starstream Research. All rights reserved.