So I finally got around to reading Max Tegmark’s book Our Mathematical Universe, and while the book answered the question that had led me to read it, namely, how one might reconcile Plato’s idea of eternal mathematical forms with the concept of multiple universes, it also threw up a whole host of new questions. This beautifully written and thought provoking book made me wonder about the future of science and the scientific method, the limits to human knowledge, and the scientific, philosophical and moral meaning of various ideas of the multiverse.

I should start though with my initial question of how Tegmark manages to fit something very much like Plato’s Theory of the Forms into the seemingly chaotic landscape of multiverse theories. If you remember back to your college philosophy classes, you might recall something of Plato’s idea of forms, which in its very basics boils down to this: Plato thought there was a world of perfect, eternally existing ideas of which our own supposedly real world was little more than a shadow. The idea sounds out there until you realize that Plato was thinking like a mathematician. We should remember that over the walls of Plato’s Academy was written “Let no man ignorant of geometry enter here”, and for the Greeks geometry was the essence of mathematics. Plato aimed to create a school of philosophical mathematicians much more than he hoped to turn philosophers into a sect of moral geometers.

Probably almost all mathematicians and physicists hold to some version of platonism, which means that they think mathematical structures are something discovered rather than a form of language invented by human beings. Non- mathematicians, myself very much included, often have trouble understanding this, but a simple example from Plato himself might help clarify.

When the Greeks played around with shapes for long enough they discovered things. And here we really should say discover because they had no idea shapes had these properties until they stumbled upon them through play.Plato’s dialogue Meno gave us the most famous demonstration of the discovery rather than invention of mathematical structures. Socrates asks a “slave boy” (we should take this to be the modern day equivalent of the man off the street) to figure out the area of a square which is double that of a square with a length of 2. The key, as Socrates leads the boy to see, is that one should turn the square with the side of 2 into a right triangle the length of whose hypotenuse is then seen as equal to one of the lengths of the doubled square allowing you easily calculate its area. The slave boy explains his measurement epiphany as the “recovery of knowledge from a past life.”

The big gap between Plato and modern platonists is that the ancient philosopher thought the natural world was a flawed copy of the crystalline purity of the mathematics of thought. Contrast that with Newton who saw the hand of God himself in nature’s calculable regularities. The deeper the scientists of the modern age probed with their new mathematical tools the more nature appeared as Galileo said “ a book written in the language of mathematics”. For the moderns mathematical structures and natural structures became almost one and the same. The Spanish filmmaker and graphic designer Cristóbal Vila has a beautiful short over at AEON reflecting precisely this view.

It’s that “almost” that Tegmark has lept over with his Mathematical Universe Hypothesis (MUH). The essence of the MUH is not only that mathematical structures have an independent identity, or that nature is a book written in mathematics, but that the nature is a mathematical structure and just as all mathematical structures exist independent of whether we have discovered them or not, all logically coherent universes exists whether or not we have discovered their structures. This is platonism with a capital P, the latter half explaining how the MUH intersects with the idea of the multiverse.

One of the beneficial things Tegmark does with his book is to provide a simple to understand set of levels for different ideas that there is more than one universe.

Level I: Beyond our cosmological horizon

A Level I multiverse is the easiest for me to understand. It is within the lifetime of people still alive that our universe was held to be no bigger than our galaxy. Before that people thought the entirety of what was consisted of nothing but our solar system, so it is no wonder that people thought humanity was the center of creation’s story. As of right now the observable universe is around 46 billion light years across, actually older than the age of the universe due to its expansion. Yet, why should we think this observable horizon constitutes everything when such assumption has never proved true in the past? The Level I multiverse holds that there are entire other universes outside the limit of what we can observe.

Level II: Universes with different physical constants

The Level II multiverse again makes intuitive sense to me. If one assumes that the Big Bang was not the first or the last of its kind, and if one assumes there are whole other, potentially an infinite number of universes, why assume that our is the only way a universe should be organized? Indeed, having a variety of physical constants to choose from would make the fine tuning of our own universe make more sense.

Level III: Many-worlds interpretation of quantum mechanics

This is where I start to get lost, or at least this particular doppelganger of me starts to get lost. Here we find Hugh Everett’s interpretation of quantum unpredictability. Rather than Schrodinger’s Cat being pushed from a superposition of states between alive and dead when you open the box, exposing the feline causes the universe to split- in one universe you have an alive cat, and in another a dead one. It gets me dizzy just thinking about it, just imagine the poor cat- wait, I am the cat!

Level IV: Ultimate ensemble

Here we have Tegmark’s model itself where every universe that can represented as a logically consistent mathematical structure is said to actually exist. In such a multiverse when you roll a six-sided die, there end up being six universes corresponding to each of the universes, but there is no universe where you have rolled a “1 not 1” , and so on. If a universe’s mathematical structure can be described, then that universe can be said to exist there being, in Tegmark’s view, no difference between such a mathematical structure and a universe.

I had previously thought the idea of the multiverse was a way to give scale to the shadow of our ignorance and expand our horizon in space and time. As mentioned, we had once thought all that is was only as big as our solar system and merely thousands of years old. By the 19th century the universe had expanded to the size of our galaxy and the past had grown to as much as 400 million years. By the end of the 20th century we knew there were at least 100 billion galaxies in the universe and that its age was 13.7 billion. There is no reason to believe that we have grasped the full totality of existence, that the universe, beyond our observable horizon isn’t even bigger, and the past deeper. There is “no sign on the Big Bang saying ‘this happened only once’” as someone once said cleverly whose attribution I cannot find.

Ideas of the multiverse seemed to explain the odd fact that the universe seems fine-tuned to provide the conditions for life, Martin Rees “six numbers” such as Epsilon (ε)- the strength of the force binding nucleons to nuclei. If you have a large enough sample of universes then the fact that some universes are friendly for life starts to make more sense. The problem, I think, comes in when you realize just how large this sample size has to be to get you to fine tuning- somewhere on the order of 10 ^200. What this means is that you’ve proposed the existence of a very very large or even infinite number of values, as far as we know which are unobservable to explain essentially six. If this is science, it is radically different from the science we’ve known since Galileo dropped cannon balls off of the Leaning Tower of Pisa.

For whatever reason, rather than solidify my belief in the possibility of the multiverse, or convert me to platonism, Tegmark’s book left me with a whole host of new questions, which is what good books do. The problem is my damned doppelgangers who can be found not only at the crazy quantum Level III, but at the levels I thought were a preserve of Copernican Mediocrity – Levels I and II, or as Tegmark says.

The only difference between Level I and Level III is where your doppelgängers reside.

Yet, to my non-physicist eyes, the different levels of multiverse sure seems distinct. Level III seems to violate Copernican Mediocrity with observers and actors being able to call into being whole new timelines with even the most minutea laden of their choices, whereas Levels I and II simply posit that a universe sufficiently large enough and sufficiently extended enough in time would allow for repeat performances down to the smallest detail- perhaps the universe is just smaller than that, or less extended in time, or there is some sort of kink whereby when what the late Stephen J Gould called the “life tape” is replayed you can never get the same results twice.

Still, our intuitions about reality have often been proven wrong, so no theory can be discounted on the basis of intuitive doubts. There are other reasons, however, why we might use caution when it comes to multiverse theories, namely, their potential risk to the scientific endeavor itself. The fact that we can never directly observe parts of the multiverse that are not our own means that we would have to step away from falsifiability as the criteria for scientific truth. The physicist Sean Carroll argues that falsifiability is a weak criteria, what makes a theory scientific is that it is “direct” (says something definite about how reality works) and “empirical”, by which he no longer means the Popperian notion of falsifiability, but its ability to explain the world. He writes:

Consider the multiverse. If the universe we see around us is the only one there is, the vacuum energy is a unique constant of nature, and we are faced with the problem of explaining it. If, on the other hand, we live in a multiverse, the vacuum energy could be completely different in different regions, and an explanation suggests itself immediately: in regions where the vacuum energy is much larger, conditions are inhospitable to the existence of life. There is therefore a selection effect, and we should predict a small value of the vacuum energy. Indeed, using this precise reasoning, Steven Weinberg did predict the value of the vacuum energy, long before the acceleration of the universe was discovered. We can’t (as far as we know) observe other parts of the multiverse directly. But their existence has a dramatic effect on how we account for the data in the part of the multiverse we do observe.

One could look at Tegmark’s MUH and Carroll’s comments as a broadening of our scientific and imaginative horizons and the continuation of our powers to explain into realms beyond what human beings will ever observe. The idea of a 22nd version of Plato’s Academy using amazingly powerful computers to explore all the potential universes ala Tegmark’s MUH is an attractive future to me. Yet, given how reliant we are on science and the technology that grows from it, and given the role of science in our society in establishing the consensus view of what our shared physical reality actually is, we need to be cognizant and careful of what such a changed understanding of science actually might mean.

The physicist, George Ellis, for one, thinks the multiverse hypothesis, and not just Tegmark’s version of it, opens the door to all sorts of pseudoscience such as Intelligent Design. After all, the explanation that the laws and structure of our universe can be understood only by reference to something “outside” is the essence of explanations from design as well, and just like the multiverse, cannot be falsified.

One might think that the multiverse was a victory of theorizing over real world science, but I think Sean Carroll is essentially right when he defends the multiverse theory by saying:

Science is not merely armchair theorizing; it’s about explaining the world we see, developing models that fit the data.

It’s the use of the word “model” here rather than “theory” that is telling. For a model is a type of representation of something whereas a theory constitutes an attempt at a coherent self-contained explanation. If the move from theories to models was only happening in physics then we might say that this had something to do merely with physics as a science rather than science in general. But we see this move all over the place.

Among, neuroscientists, for example, there is no widely agreed upon theory of how SSRIs work, even though they’ve been around for a generation, and there’s more. In a widely debated speech Noam Chomsky argued that current statistical models in AI were bringing us no closer to the goal of AGI or the understanding of human intelligence because they lacked any coherent theory of how intelligence works. As Yaden Katz wrote for The Atlantic:

Chomsky critiqued the field of AI for adopting an approach reminiscent of behaviorism, except in more modern, computationally sophisticated form. Chomsky argued that the field’s heavy use of statistical techniques to pick regularities in masses of data is unlikely to yield the explanatory insight that science ought to offer. For Chomsky, the “new AI” — focused on using statistical learning techniques to better mine and predict data — is unlikely to yield general principles about the nature of intelligent beings or about cognition.

Likewise, the field of systems biology and especially genomic science is built not on theory but on our ability to scan enormous databases of genetic information looking for meaningful correlations. The new field of social physics is based on the idea that correlations of human behavior can be used as governance and management tools, and business already believes that statistical correlation is worth enough to spend billions on and build an economy around.

Will this work as well as the science we’ve had for the last five centuries? It’s too early to tell, but it certainly constitutes a big change for science and the rest of us who depend upon it. This shouldn’t be taken as an unqualified defense of theory- for if theory was working then we wouldn’t be pursuing this new route of data correlation whatever the powers of our computers. Yet, those who are pushing this new model of science should be aware of its uncertain success, and its dangers.

The primary danger I can see from these new sorts of science, and this includes the MUH, is that it challenges the role of science in establishing the consensus reality which we all must agree upon. Anyone who remembers their Thomas Kuhn can recall that what makes science distinct from almost any system of knowledge we’ve had before, is that it both enforces a consensus view of physical reality beyond which an individual’s view of the world can be considered “unreal”, and provides a mechanism by which this consensus reality can be challenged and where the challenge is successful overturned.

With multiverse theories we are in approaching what David Engelman calls Possibilism the exploration of every range of ways existence can be structured that is compatible with the findings of science and is rationally coherent. I find this interesting as a philosophical and even spiritual project, but it isn’t science, at least as we’ve understood science since the beginning of the modern world. Declaring the project to be scientific blurs the lines between science and speculation and might allow people to claim the kind of understanding over uncertainty that makes politics and consensus decisions regarding acute needs of the present, such a global warming, or projected needs of the future impossible.

Let me try to clarify this. I found it very important that in Our Mathematical Universe Tegmark tried to tackle the problem of existential risks facing the human future. He touches upon everything from climate change, to asteroid impacts, to pandemics to rogue AI. Yet, the very idea that there are multiple versions of us out there, and that our own future is determined seems to rob these issues of their urgency. In an “infinity” of predetermined worlds we destroy ourselves, just as in an “infinity” of predetermined worlds we do what needs to be done. There is no need to urge us forward because, puppet-like, we are destined to do one thing or the other on this particular timeline.

Morally and emotionally, how is what happens in this version of the universe in the future all that different from what happens in other universe? Persons in those parallel universes are even closer to us, our children, parents, spouses, and even ourselves than the people of the future on our own timeline. According to the deterministic models of the multiverse, the world of these others are outside of our influence and both the expansion or contraction of our ethical horizon leave us in the same state of moral paralysis. Given this, I will hold off on believing in the multiverse, at least on the doppelganger scale of Level I and II, and especially Levels III and IV until it actually becomes established as a scientific fact,which it is not at the moment, and given our limitations, perhaps never will be, even if it is ultimately true.

All that said, I greatly enjoyed Tegmark’s book, it was nothing if not thought provoking. Nor would I say it left me with little but despair, for in one section he imagined a Spinoza-like version of eternity that will last me a lifetime, or perhaps I should say beyond. I am aware that I will contradict myself here: his image that gripped me was of an individual life seen as a braid of space-time. For Tegmark, human beings have the most complex space-time braids we know of. The idea vastly oversimplified by the image above.

About which Tegmark explains:

At both ends of your spacetime braid, corresponding to your birth and death, all the threads gradually separate, corresponding to all your particles joining, interacting and finally going their own separate ways. This makes the spacetime structure of your entire life resemble a tree: At the bottom, corresponding to early times, is an elaborate system of roots corresponding to the spacetime trajectories of many particles, which gradually merge into thicker strands and culminate in a single tube-like trunk corresponding to your current body (with a remarkable braid-like pattern inside as we described above). At the top, corresponding to late times, the trunk splits into ever finer branches, corresponding to your particles going their own separate ways once your life is over. In other words, the pattern of life has only a finite extent along the time dimension, with the braid coming apart into frizz at both ends.

Because mathematical structures always exist whether or not anyone has discovered them, our life braid can be said to have always existed and will always exist. I have never been able to wrap my head around the religious idea of eternity, but this eternity I understand. Someday I may even do a post on how the notion of time found in the MUH resembles the medieval idea of eternity as nunc stans, the standing-now, but for now I’ll use it to address more down to earth concerns.

My youngest daughter, philosopher that she is, has often asked me “where was I before I was born?”. To which my lame response has been “you were an egg” which for a while made big breakfasts difficult. Now I can just tell her to get out her crayons to scribble, and we’ll color our way to something profound.