The Universe is Pure Math…or Is It?


Image from Hubble eXtreme Deep Field. Almost every spec of light in this image is a galaxy. Some of them are 13.2 billion years old – only 600 million years or so shy of the total age of the universe. 

What, at root, is the nature of reality? This is a thornier question than it might seem at first glance. Our sensory impressions might lead us to believe that the Universe is comprised of the things we can experience through sight, sound, touch, smell, and taste. This is the empiricist view. We might add to this list of elements our experience of a directional flow time and processes of change (social progress and upheaval, animal migration, seasonality, climatic variation, biological evolution, etc.).

None of this is entirely wrong. Nor – as has been known for some time – is it entirely right. The universe is filled with many things that leave distinct and reliable sensory impressions concerning the properties of reality that have proven pertinent to hairless, bipedal apes with lifespans measured in decades, over the course of our evolutionary history. Evolution has tuned us into phenomena that unfold on a certain macroscopic spatial scale, over the span of seconds, minutes, hours, days, weeks, years, and decades. Although these capacities are well suited to getting a decent living at those scales, they also have the effect of disguising deeper realities than unfold on vastly different scales. It is hard for us the fathom the very small or the massively large. Equal difficulties arise when contemplating processes that occur is vanishingly small fragments of a second or unfold over the slow march of eons.

Astrophysicist Arthur Eddington highlighted the dilemma inherent in the divergence between sensory experience and physical reality in his “two table” metaphor. Eddington drew a distinction between the table of sensory experience – the solid block of stained and polished wood, reflecting light in a spectrum of deep browns and reds, impenetrable to the human touch, supporting stacks of paper, human elbows, and writing instruments – and the scientific table – mostly empty space, filled with electrons whipping around atomic nuclei at frenetic speeds, held together by forces. Our sensory perceptions are not wrong, per se, but they fail to give us anything close to an adequate impression of the fundamental nature of reality.

Science is the best method available for peeling back the curtain of parochial sensory experience and navigating the errors that tend to flow from “common sense” thinking about the way the world works. It is a method for using our cognitive capacities – rigorously bound by an array of guidelines, fail-safes, and methods of recursive error detection and correction – to escape the myopia of folk induction and intuition. Science has, among other things, taught us that our very existence is predicated on processes that play out on scales that we can barely begin to comprehend. It has, to the lasting chagrin of those who cleave to literalist interpretations of various religions, revealed our existence is cosmically insignificant – that the universe is so old and so vast that all of human existence has been but a trivial dance on a tiny dust mote in the blink of a cosmic eye. And it has demonstrated that the nature of reality is, at root, incredibly weird.

Our Mathematical Universe, a recent book by theoretical physicist Max Tegmark, is at once an homage to and exploration of that fundamental weirdness. Tegmark himself seems to sit comfortably in the realm of bizarre and counterintuitive ideas, and scientific ideas really don’t get a lot weirder than they do when exploring some of potential consequences of quantum mechanics.

When Einstein first introduced his theory of general relativity, many physicists steeped in the well-established principles of Newtonian mechanics found the notion that space and time are mutable grossly unpalatable. It didn’t take long before the predictions deduced from the theory were borne out by experimental and observational evidence. As a result, general relativity has been enshrined as one of the most powerful and effective scientific theories of all time. However counter-intuitive it might seem, the idea that space and time are malleable, irrevocably dependent upon features like the mass and speed of objects, is an accurate representation of the basic nature of reality.

As strange as the consequences of general relativity might have seemed in the second decade of the 20th century, the understanding of quantum mechanics that emerged in succeeding decades is weirder still. At a subatomic level, the nature of reality is downright bizarre. The conception of reality spelled out by the likes of Werner Heisenberg, Erwin Schrödinger, and Neils Bohr runs so contrary to our common sense view of the way things work that its seems ridiculous. The world of the subatomic seems to be ruled by randomness, leaving us inevitably bound to a probabilistic understanding of the behavior of particles like quarks and electrons. Rather than describing the specific characteristics of a single particle – its speed, direction, position etc. – the equations of quantum mechanics describe a wave function, probabilistically defining the potential characteristics of a given particle. Not only does the behavior of these particles seem to be governed by certain amount of randomness, they seem to be influenced by observation. Prior to observation, a fundamental particle occupies a superposition of all possible positions. If this sounds outlandish, it’s because it is. But these ideas have been experimentally and observation corroborated.

If the recent history of science has taught us anything, it’s that things can always get weirder. Tegmark elucidates this expertly, diving into an exploration of an interpretation of quantum mechanics that is (at least to this subjective observer) more than a little unsettling. According to this view, wave functions never collapse and all potential outcomes of a given scenario are realized. In the traditional view, the wave function of a particle breaks down upon observation, cementing it into this or that position. Under the view outlined and favored by Tegmark, the wave function never collapses – a given observer merely has the subjective experience of one of the many potential results of an interaction, while every other potential outcome is realized in a parallel universe spawned at that instant. This leads to some truly peculiar outcomes, including the potential for subjective immortality (read the book if you want a full exploration of that particular bit of weirdness). Suffice it to say for now that this is a consequence of living in a world where every possible outcome of a given situation is realized through the endless propagation of parallel universes.

To the outside observer, this idea immediately confronts a rather troublesome epistemological speed-bump. How does one test the existence of these kinds of parallel universes? There seems to be an impenetrable partition between the universe we occupy and the others, meaning we can never really corroborate their existence. For a scientific enterprise, this is massively problematic. Without recourse to observation and experimental evidence, scientists have no way to evaluate the veracity of a hypothesis like the infinite proliferation of parallel universes. The idea might be consistent with established theory. It might even offer a more parsimonious explanation than competing interpretations, but absent some recourse to evidence, how are we to judge whether or not it is true?

For this, Tegmark hangs his hat on the following proposition: that for scientists to place confidence in a given theory, not all of its predictions need to be subjected to empirical evaluation – only some of them. To explain this, it takes a bit of digression into the world of cosmology. This is worthwhile for two reasons. First, Tegmark’s discussion of cosmology – though a retread of old ideas – was one of my favorite parts of his book. Second, it provides a good illustration of how the corroboration of certain features of theory can leads researchers to place confidence in others.

Under most conditions, general relativity predicts a dynamic universe, either expanding or contracting. Hubble’s observation of the spectral shifts of different galaxies confirmed this, their accelerating retreat pointing toward an earlier point at which they were much closer together. That is, the principles of relativity, together with Hubble’s observations, paint a picture of a universe that was at some point in the past much hotter and denser than it is now.

This is a situation in which we have a set of laws (i.e. general relativity) that have been highly corroborated – time slows at high speed and around massive objects, light bends around the sun, the orbit of Mercury is more accurately described and more adequately explained, etc. For all intents and purposes, general relativity has been confirmed to be true. Additionally, we have a number of particular facts about the universe derived from empirical observation. This universe is filled with galaxies and they are moving away from one another at an accelerating rate. Run the film in reverse and it is clear that at some point they must have been closer together. Add in estimates of the amount of matter in the universe and do some math well beyond my comprehension and you have a recipe for an axiomatic deduction about the nature of the early universe. If the laws of general relativity are true and the observations of Hubble and the estimates of mass are accurate, a certain picture of the early universe emerges as an inevitable consequences.

This is precisely what theoretical physicist George Gamow did, using the principles of general relativity, estimates of the amount of matter in the universe, and the rate of cosmological expansion to predict the consequences of a the universe’s backward contraction. Gamow predicted the temperature of the background of empty space that would result from a universe that was, at some point, a lot denser and hotter than it is today. When Arno Penzias and Robert Wilson discovered the microwave background radiation in 1964, they found a signature of the universe’s hot, dense origins that was within 2° of Gamow’s predictions.

Of course, science doesn’t work on pure logic alone, so the empirical measurements of Penzias and Wilson are really the lynchpins of the entire argument. But the whole story paints an awe-inspiring picture of the power of brilliant minds using rigorous methodology to engage in the process of scientific discovery. Returning to the quantum multiverse idea, Tegmark wants to position the idea that the wave function never collapses in the same ballpark as Gamow’s early prediction about our cosmic origins. The comparison holds insofar as we have a highly corroborated theory (quantum mechanics), a set of particular facts, and a prediction that flows as a logical consequence of those elements. Unfortunately, the similarities only extend so far. Gamow’s predictions are most remarkable in light of their eventual corroboration – this is what shifted the idea of a Big Bang from the status of a scientific hypothesis to that of a scientific fact. Until sympathetic quantum physicists find some way to experimentally evaluate the multiverse idea, it will remain an interesting – but unconfirmed – hypothesis.

In all this weirdness, I haven’t even touched the concept from which the book draws its title – Tegmark’s idea that the universe is, at bottom, a mathematical structure. This is a radical idea, and even digesting it takes some effort. Tegmark is arguing that the universe isn’t just remarkably well described by math. He is arguing that it is math. Despite his best efforts, Tegmark fell short of convincing this particular reader of his central thesis. When it comes to the astounding utility of mathematics, I prefer to take something of an instrumentalist view. This is the position that, for math to describe the universe, it merely needs to be useful in doing so – it doesn’t actually have to precisely mirror that nature of reality. As Jerry Coyne has pointed out, all we need to explain the usefulness of mathematics is a universe characterized by some amount of regularity. I don’t see the need to push beyond this and posit a more revolutionary perspective.

Tegmark also seems to inextricably link his mathematical universe hypothesis with the notion of a static four dimensional space-time, where the flow of time experienced by observers embedded in said space-time is illusory. This position does lend itself to thinking of everything in terms of mathematical structures, including Tegmark’s evocative metaphor that humans are complex, braided mathematical structures and that human consciousness is comprised of subjectively self-aware observer moments within said structures. Still, I can’t see past a set of troubles that emerge if one tries to situate the processes that lead to those subjectively self-aware observer moments within the static mathematical universe framework. The only process we know of capable of building thinking machines sophisticated enough to even contemplate the possibility of a mathematical universe is evolution. Humans – or the mathematical structures that view themselves as humans – are the product of evolutionary processes. Evolution, in turn, is a distinctly historical and highly contingent process. It is inherently about change, and therefore implicitly dependent on the existence of some temporal arrow. To say we are primates occupying a certain branch at a certain end of a 3.5 billion year chain of evolutionary change does more than partially specify our position on the time axis of a space-time coordinate system (though it does do that). It is a statement that demands not only the possibility of change, but the reality of a direction flow of time.

Of course, I may be taking Tegmark’s ideas too far here. Perhaps this is a semantic problem, arising from the use of terms like “appears” and “illusion” to describe processes of change and the flow of time. This confusion may well be an outworking of attempting to translate deeply mathematical concepts into a language that an archaeologist can comprehend. But absent some further clarification on the matter, the confusion stands. Though the equations of physics may work just as well backwards as they do forwards, obviating the need for a directional flow of time, the fact remains that the directional flow of time appears to be more than just an illusion. This is a phenomenon in need of explanation. In The Elegant Universe, Brian Greene attributes the directional flow of time to statistical mechanics – in particular, the subset of statistical mechanics commonly referred to as the second law of thermodynamics. Time’s arrow comes from a universe moving from a low entropy state (prior to the Big Bang) to a higher entropy state in the eons that followed. I would be very curious to read Tegmark’s view of this interpretation.

Most of the ideas presented in Our Mathematical Universe are intensely mathematical, and I have neither the skills nor the expertise to give them a fair or reliable evaluation. That I can even discuss them in the day to day language of an Internet blog is a testament to Tegmark’s skill as an explicator. Tegmark explores these ideas with an infectiously colloquial wit and clearly unbridled enthusiasm. This is clearly a man who absolutely loves science. Insofar as the book provided a window into that kind of mind, I consider it a resounding success. That it simultaneously offered up a mountain of grist for the cognitive mill is a pleasant bonus.
mathematical universe

Our Mathematical Universe, by Max Tegmark

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