The Universe is Made of Mathematics

Max Tegmark

A couple of years ago I was reading Anathem by Neal Stephenson and a number of ideas started to click into place for me. Without going into the novel too much (spoilers), it prompted me to think about the nature of reality and why it might be that the universe exists.

Over the course of a few sleepless nights, it all came together and it seemed too make so much sense that I could entertain no doubt: the universe was what I thought of as a "Platonic algorithm".

To my knowledge at the time, this idea had never before been proposed, and a bit of half-hearted searching at the time turned nothing up. I became more convinced of my idea over time and started developing an interest in philosophy so as to learn how to communicate and argue for the idea.

As my research ramped up, I inevitably came across Max Tegmark's Mathematical Universe Hypothesis (MUH), and with a curious mixture of disappointment and vindication realised that it was essentially the same idea.

Nevertheless, Tegmark's idea was still relatively little known, so I decided I would devote some time to building up a rigorous argument with a view to perhaps writing a book on the topic some day. That is actually the single most important reason I started this blog, as many of the arguments I have outlined in my posts will serve to support the MUH.

I have now learned (courtesy of Massimo Pigliucci's blog Rationally Speaking) that Tegmark is in fact about to publish the popular science book that is so sorely needed. I guess I can put aside that ambition for now.

But it's also time for me to put this blog to the use it was originally intended for - enough beating around the bush!

Perhaps my hesitancy to address the subject arises out of the fact that it at first seems completely mad: I am utterly convinced that the universe is made of mathematics and that the concept of physical reality is incoherent.

Now let me try to explain why.

Why the Abstract Feels Physical

I will freely admit that this at first seems like a meaningless platitude to say "the universe is mathematics", akin to statements such as "God is Love". Massimo Pigliucci quite rightly stresses on his recent blog post that we are not claiming that reality is best described by mathematics -- the claim is literally that the fundamental substance of reality is mathematical structure.

As he pointed out, it's hard to see how this can be. How can mathematics possibly explain substance? Abstract concepts such as mathematics are surely entirely different from physical stuff, and to claim the two are the same seems to be perverse.

From Pigliucci's description, Tegmark's answer to this criticism seems rather weak. He insisted that electrons are mathematical objects having mathematical properties, but apparently failed to provide a convincing reason not to regard them as physical objects having physical properties which can be described mathematically. I can see why Pigliucci feared he may have been making a category mistake.

I would take another tack. I would argue that the intuition that mathematics cannot be the stuff of, well, stuff, arises from the false belief that this is a physical universe (and indeed that the physical universe is a concept that even makes sense).

But before I elaborate on this, let me state plainly that I believe the argument in favour of the mathematical universe rests on three crucial premises.

1) All mathematical objects exist abstractly and independently of minds (mathematical Platonism)
2) The mind is a computational process (The Computational Theory of Mind or CTM)
3) The universe behaves according to laws of physics which are expressible mathematically (metaphysical naturalism)

I have made arguments in favour of all of these premises previously on the blog (Platonism, CTM, Naturalism), so for now I am going to assume them to be true.

How can I seriously doubt that the universe is physical? Two powerful analogies help to explain this.

One hypothesis gaining currency in recent years is the idea that the universe is not physical but a computer simulation, as argued rather interestingly by philosopher Nick Bostrom. If Bostrom is right, the physical universe may not exist and the fundamental stuff of the universe could be information, i.e. the bits flowing through a computer program.

Another idea, explored in the book Sophie's World and elsewhere, is the idea that our world may not physically exist because we may be fictional characters living in a fictional world being described by some author. From the point of view of a fictional character, the world seems real, so none of us can really know for sure that we are not such a character.

In both of these ideas, the universe is not real (not physically at any rate), but this fact is forever hidden from us.

The MUH is quite similar in many ways. The crucial difference is that it removes the dependence on a greater reality. Unlike the computer simulation idea, we need no external hardware to support us. Unlike the characters in a novel, we are genuinely conscious. Unlike both, there is no creator, no programmer or author.

We have no need of a programmer or author, no cosmic computer or reader. The bedrock of our existence is mathematical Platonism, and unlike all other explanations this is sufficient as an ultimate cause. Mathematical objects are not created and need nothing to sustain them. They exist necessarily of their own nature.

Let's first consider the idea that the universe is a simulation. If the physics of the universe is computable (as it seems to be), then it could certainly be simulated by a computer of sufficient power, even if such a computer would be unfeasible to construct in this universe. It therefore follows that in principle it is entirely possible that it and we ourselves are simulated, virtual entities (although if you doubt the Computational Theory of Mind then this does not follow as we are evidently conscious).

However all computer programs are mathematical structures, and as such, Platonism holds that all computer programs exist in the abstract. It follows that even if no computer could ever be built to run the simulation of the universe, this program exists as a mathematical structure, and within this structure can be found perfect descriptions of all the objects in our universe, including our minds, our thoughts, etc.

But now, if we start thinking about an abstract mathematical object containing all of our thoughts and inner experiences and determining our life stories, it seems to me that we are getting close to the idea that we may all be characters in a work of fiction.

Unlike fiction, what happens in our universe is determined by mathematical laws and these have the regularities necessary to support life, brains and real intelligence. I don't believe characters in fiction are actually conscious because their actions are not determined by internal genuine thought processes. There is consciousness there, but it is not in the minds of the characters but in that of the writer. In contrast, our consciousness is our own and is the product of our own brains.

But like the lives of characters in a novel, our lives and our thoughts are all mapped out and there for the reading. We are implicit in the structure of this mathematical object, and in principle our stories could be discovered by simulating it on a powerful computer. This is just like the way we can explore mathematical objects such as the Mandelbrot set with computers.

We are the conscious characters in a cosmic narrative determined by no author but the laws of physics. Our lives exist even if no simulation is run, just as the stories of fictional characters continue to exist even while nobody is reading the novel and just as the Mandelbrot set has always existed, even before it was discovered.

So think of our universe and our life stories as being something like an enormous fractal structure arising from some simple mathematical rules. It's a beautiful, amazing, complex, surprising thing, but it needs no creator or sustainer.

The Incoherence of the Physical Universe Hypothesis

Far more than being merely a plausible account of reality, I view the Mathematical Universe Hypothesis as being necessarily true, because it reveals the incoherence of the concept of a physical universe.

Given that the universe obeys mathematical physical laws (naturalism), there must be a mathematical object (given Platonism) which perfectly describes the universe and which contains within it structures analogous to all objects within the universe, including ourselves.

What's more, given the computational theory of mind, those structures corresponding to our minds are necessarily conscious. As such, even if there is such thing as a real, physical universe, there must also be an isomorphic (having precisely the same form or structure) abstract non-physical universe. There is a physical you and an abstract you, and both have exactly the same experiences and neither has any way of knowing which universe they find themselves in.

In fact, there is no observer anywhere who can distinguish between the physical and the abstract universes. There is nothing we can say about the physical universe that is not also true about the abstract universe, except for the fact that it is physical.

So let's unpack that. What does physical mean? In everyday speech, it is used to distinguish between objects we can interact with directly (such as rocks) and objects we can only think about and discuss (such as numbers). For something to be physical it must be present at some time and place within the universe, and for something to be abstract it must exist outside of space and time.

So what do we mean when we claim that a universe is physical? It exists in time and space? But it doesn't -- it contains time and space within it.

Ok, so let's say that the universe is a special case, that it is physical not because it is inside spacetime but because it contains spacetime.

So now let's consider a hypothetical abstract universe other than our own and ask ourselves whether it is physical. (For illustrative purposes let's say that universe is Star Wars but let's assume that the universe is not a work of fiction but a mathematical object in much the same way that I'm claiming that our universe is a mathematical object).

This universe is not present within the spacetime of our universe, so from that point of view it is not physical. It does contain its own spacetime, so by our broader definition that would make it physical.

But it's just a made-up universe! Our instinct is that it ought not to be considered physical. So let's say that only universes which contain the spacetime of our own universe are physical.

So now, only our own universe can be physical, by definition. This seems to be a rather unsatisfactory result, because there are very good reasons (such as the anthropic principle) to believe there may be other universes. We want that option to be open to us, if only so we can discuss the possibility.

It's also a patently subjective definition of physical. From the point of view of an observer in this universe, the Star Wars universe is not physical. But from the point of view of Han Solo, our universe is not real.

It seems to me that the only way out of this mess is to realise that the application of the concept of physicality to a universe is a category mistake. Physicality as a concept only makes sense within the context of a given universe.

For example, you would no doubt regard yourself as physical but Luke Skywalker as abstract. However, Luke Skywalker is physical to Han Solo, while Han would consider you to be non-existent. There is no objective, universe-agnostic way to say that you are really physical but Luke Skywalker is not.

So given that physicality as applied to universes seems to be incoherent, and given that physicality is the only (completely undetectable) property that distinguishes the mathematical universe from the physical universe, it seems to me that the only sensible conclusion is that only the mathematical version of our universe exists. This accounts for the existence of the universe, fine-tuning and everything we observe.

The Physical Universe Hypothesis is therefore unnecessary, redundant and incoherent.

Reasons to Believe

I have explained why I think the Mathematical Universe Hypothesis follows necessarily from naturalism, Platonism and the computational theory of mind, however there are plenty of people who are skeptical of some or all of these propositions. Nevertheless, I think there are independent reasons to find the MUH plausible.

Firstly, and perhaps most importantly, it explains why the universe exists. It does not tell us what caused the Big Bang, or if the Big Bang had a cause at all, but it does explain why there is something rather than nothing, without appeal to a creator or any other unsatisfying ultimate cause. This echoes my rebuttal to the Kalam Cosmological argument.

For a creator God, we are left to ask who created the creator - but if the universe is a mathematical object, it needs no creator (on Platonism at least), so this is a very satisfying answer to that eternal question. It has always existed and will always exist outside of space-time as a mathematical construct.

It also provides a powerful explanation for fine-tuning. As discussed previously, this universe seems to be perfectly calibrated to support life. Much attention in discussions of fine tuning is focused on the physical constants such as the charge of the electron or the speed of light, etc, but very rarely is it asked why are the equations and the laws of physics themselves of the form they are. Why could the universe not be completely different, not just having different constants but having completely unrecognisable physical laws?

If all possible universes exist, then we have our answer. Our universe is fine-tuned because it is one which has the ability to support conscious thought selected from an infinite multitude of mathematical structures, most of which are lifeless.

Better than other multiverse hypotheses, where it is proposed that there might be a great number of universes, the MUH posits that all possible universes exist, not merely a great many. This is actually simpler because, as Tegmark says (and as explained previously on this blog), there are no free parameters. We have no reason to wonder why universe X exists but universe Y does not. If all universes exist, nothing is arbitrary. We have an ultimate explanation of everything.

There's something attractive to me about the idea that all universes exist. After all, what's to stop any given possible universe from existing? It is not subject to the laws of other universes. No law of this universe can prevent another entirely causally disconnected universe from existing. Even if there is a multiverse with its own meta-laws (e.g. the String Theory multiverse), what's to stop another multiverse with different meta-laws from existing? Even without asserting that the universe is made of mathematics, it seems to me to be perfectly sensible to propose that all possible universes exist. Why not?

Finally, the idea that the universe is literally a mathematical object explains what physicist Eugene Wigner called "the unreasonable effectiveness of mathematics" in modelling and predicting the natural world, often leading physicists to empirical discoveries they could not otherwise have made. If the universe really is a mathematical object, it is hardly surprising that mathematics should be effective in describing it.

There are objections to the Mathematical Universe Hypothesis of course, and I intend to return to them in time.