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The Universe Is Made Of Mathematics

Sam Woolfe recounts the mathematical metaphysics of physicist Max Tegmark.

Max Tegmark is a Swedish-American cosmologist currently teaching at MIT. He has made important contributions to physics, such as measuring dark matter and understanding how light from the early universe informs the Big Bang model of the universe’s origins. He has also proposed his own Theory of Everything. His Theory of Everything is known as the Ultimate Ensemble or by the more attention-grabbing name, the Mathematical Universe Hypothesis. This hypothesis can be summed up in one phrase: “Our external physical reality is a mathematical structure.” In this case, a ‘mathematical structure’ means a set of abstract entities, such as numbers, and the mathematical relations between them. So the Mathematical Universe Hypothesis states that mathematics is not just a useful tool we have invented to describe the universe. Rather, mathematics itself defines and structures the universe. In other words, the physical universe is mathematics. This is a very strange and bold statement, and at first glance it’s not easy to wrap your head around it, but let’s try.

Tegmark & Plato

The Mathematical Universe Hypothesis has a very philosophical nature to it. It can be considered a form of Platonism, the philosophy of Plato, who argued that certain abstract ideas have a real independent existence beyond our minds. Similarily, Tegmark’s hypothesis argues that mathematical entities such as numbers exist independently of us – these abstract entities are not merely imaginary; they exist as part of mind-independent reality. In a sense, Tegmark’s hypothesis goes well beyond Platonism, since Tegmark claims that ultimately only mathematical objects exist and nothing else does! In his own words, “there is only mathematics; that is all that exists” (Discover magazine, July 2008). This position is known as mathematical monism.

Some may view Tegmark’s mathematical monism as an extreme and nonsensical position, due to the fact that we never perceive these mathematical objects, whereas we do perceive a physical world, full of physical objects. Based on our experience, it would seem that there is no evidence for the existence of mathematical objects, whereas there is unavoidable evidence for a physical world. However, in his paper ‘The Mathematical Universe’ in Foundations of Physics (2007), Tegmark argues that, “in those [worlds] complex enough to contain self-aware substructures [they] will subjectively perceive themselves as existing in a physically ‘real’ world.” So we shouldn’t be surprised to find that we perceive a physical world, because this perception is the inevitable result of a mathematical universe which is sufficiently complex. Ultimately, then, our perception of a physical world is due to the nature of our consciousness r, and not due to the true nature of the universe itself.

In a way this is similar to Plato’s belief that ordinary minds cannot perceive or even understand the true nature of things. The true nature of things, Plato claims, can be traced to what he calls Forms or Ideas, which are abstract, timeless, archetypal, non-physical entities. In order to go beyond the illusory appearance of things, we need to use reason to uncover their true nature, not visual or other perception. This, he argued, only those trained in philosophy could do.

Similarily, Tegmark argues that there are two possible ways to view reality; from inside the mathematical structure, and from outside it. We view it from within it, and so see a physical reality which exists in time. From the (purely hypothetical) external point of view, however, Tegmark thinks that there is only a mathematical structure which exists outside of time. Some might respond to this by saying that the idea of ‘outside of time’ and ‘timelessness’ is verging on the mystical.

World of Mathematics © Ken Laidlaw 2016. Please visit www.kenlaidlaw.com to see more of Ken’s art

Mathematical Reasoning & Science

Indeed, Tegmark admits that he is in a minority of scientists who believe his Mathematical Universe Hypothesis. It took a while before he got his ideas published in a scientific journal, and he was warned that his MUH would damage his reputation and career. But there are some reasons why one might believe it. The physicist Eugene Wigner wrote an essay called ‘The Unreasonable Effectiveness of Mathematics in the Natural Sciences’ (Communications in Pure and Applied Mathematics, vol. 13, No.1, 1960), asking why nature is so accurately described by mathematics. Tegmark answers that the unreasonable effectiveness of mathematics in describing reality implies that mathematics is at the very foundation of reality.

The ancient Greek thinker Pythagoras and his followers also believed that the universe was built on or from mathematics; whilst Galileo said that nature is a “grand book” written in “the language of mathematics.” But it is also worth reminding ourselves that there are those who think mathematics is purely a human invention, albeit one which is extremely useful. For instance, in their book Where Mathematics Comes From (2001), George Lakoff and Rafael Nunez maintain that mathematics arises from our brains, our everyday experiences, and from the needs of human societies, and that mathematics is simply the result of normal human cognitive abilities, especially the capacity for conceptual metaphor – understanding one idea in terms of another. Mathematics is effective because it is the result of evolution, not because it has its basis in an objective reality: numbers or mathematical principles are not independent truths. (However, these authors do praise the invention of mathematics as one of the greatest and most ingenious inventions ever made.) An extreme version of this evolutionary idea is the mathematical fictionalism put forward by Hartry Field in his book, Science Without Numbers (1980). Field said that mathematics does not correspond to anything real. Instead he believes that mathematics is a kind of useful fiction: that statements such as ‘2+2=4’ are just as fictional as statements such as ‘Harry Potter lives at Hogwarts’. We know what they mean, but their assertions do not correspond to anything real.

Tegmark In The Multiverse

Interestingly, Tegmark’s Mathematical Universe Hypothesis also relates to the multiverse hypothesis, in that he maintains that all structures that exist mathematically also exist physically. This means that anything that can be described by mathematics actually exists. It follows, then, that there are other universes in which I don’t exist, whereas there are an infinite number of me in still other universes.

Tegmark also writes in his paper ‘Parallel Universes’ in Science and Ultimate Reality (J.D. Barrow, P.C.W. Davies, & C.L. Harper, eds, 2003), that his Ultimate Ensemble/Mathematical Universe Hypothesis encompasses all levels of multiverse, of which he says that there are four types or levels. The first type of multiverse is a universe which is infinite in space in which there are regions which we cannot observe, but which may be similar (or even identical) to our observable region. For this type of multiverse, the physical constants and laws are the same everywhere.

The second type is a multiverse in which some regions of space form distinct non-interacting bubble universes, like gas pockets in a loaf of rising bread. Different bubbles may have different fundamental physical constants, such as the strength of gravity, the weight of an electron, and so on.

The third type or level of multiverse, is one in which all possible courses of action actually take place in separate or parallel universes. If, for example, I decide to take the bus to work instead of the train, reality will split at the point of my decision such that there will be another universe, which is just as real, where I take the train to work and not the bus. This idea was originally Hugh Everett’s many-worlds interpretation of quantum mechanics, and it is quite mainstream in the physics community. The Level III multiverse can be thought of as a tree with an infinite number of branches, where every possible quantum event creates a new universe and so signifies the growth of a new branch.

Tegmark writes, “The only difference between Level I and Level III is where your doppelgängers reside.” In a Level I concept of the multiverse, my doppelgängers (copies) live exist somewhere else in the same universe as me; whereas in Level III they exist in a different universe altogether.

The Level IV type of multiverse is the Ultimate Ensemble, and it contains all the other levels of multiverse, or describes all the other levels. This is why the Ultimate Ensemble is considered a Theory of Everything – because it can supposedly explain every single universe that possibly exists. To Tegmark, every different universe is ultimately a different mathematical structure.

Sam Woolfe is a Philosophy graduate from Durham University who currently lives in London and blogs at www.samwoolfe.com.