The Hubble eXtreme Deep Field, recording the "deepest-ever view of the universe." Image via NASA Goddard Space Flight Center on Flickr.
Max Tegmark has a theory about reality. According to Max, who is a cosmologist and professor of physics at MIT, all that exists, all this familiar stuff—that ergonomic chair you are sitting on, your body and your brain, even the space surrounding you— is math and we are merely “self-aware parts of a giant mathematical object.”
It’s a heady concept, but what does it even mean? In his new book Our Mathematical Universe, Max calls this idea the Mathematical Universe Hypothesis, wherein the universe is envisaged as a mathematical structure. A mathematical structure is “an abstract set of entities with relations between them,” expounds Max in his book, and these relations do not just describe all that is, but actually are all that is.
Reading Our Mathematical Universe, which is part mind-bending scientific treatise and part autobiography, is no casual jaunt. While the book offers a lot to readers, it also asks a lot in return. When I had the opportunity to chat with Max just before the holidays, I felt obligated to preface our conversation with the fact that although I had read the entirety of the book, I experienced difficulties in understanding chunks of it. To him, this presented no problem at all.
“You have to remember, Lex, that if you don’t feel you understand 100 percent about our Universe, nobody else does either!”
Fortunately for me, and anyone else interested in the possible realities of reality, Max is open to having his brain probed, which is what I hoped to achieve in our conversation. What I got from him was not just a deeper understanding of the details and implications of the contentious Mathematical Universe Hypothesis, but also how thinking about such grand ideas could assist you next time you find a parking ticket wedged under your windshield wiper.
MOTHERBOARD: Your excitement for your subject is really contagious in the book. I’ve sort of noticed a similar excitement in other physicists—Richard Feynman, Carl Sagan, Neil deGrasse Tyson. Why do you think physics provokes that sort of wonderment with the world?
Tegmark: I can only speak for myself. I have always been fascinated by mysteries. To me, physics is the greatest mystery of them all, the mystery of understanding reality.
We find ourselves self-aware parts of this Universe. What’s going on here? Where does everything come from? What is all this stuff? What happened to it? I find it incredibly exciting that we little humans, through serious reasoning, have actually been able to understand so much of it already, that we’ve been able, to a certain extent, to take charge of our destinies and change the Universe to make it more humane for us.
What are the most elegant aspects of physics or cosmology in your opinion?
I think the most elegant aspect of physics and cosmology is that it turns out we can describe absolutely everything with math. For example, we have a list of 32 numbers from which we mathematically can calculate, in principle, every single number that we’ve ever measured in the history of science. It’s so elegant that it really begs the question: “why is this?”
I’m not the first person to marvel over this elegance. Galileo exclaimed famously that our Universe is a book written in the language of mathematics. He thought it was so elegant that mathematics and geometric shapes could describe the motions of the planets and so on.
Where it gets controversial is, what does this all mean? Is this just some kind of fluke? Or is our Universe really mathematical in some sense? I’m on the most radical fringe of this spectrum of views: I think it’s completely mathematical and that’s why we keep finding all these elegant things.
Many of the physics discoveries you trace in the book seem to require more than just looking at equations. How big of a role do you think things like imagination and philosophy play in subjects, like physics, that are otherwise considered hard sciences?
I feel that philosophy, creativity, and imagination are enormously important in the history of physics because often the hardest part has been having creativity and imagination to question assumptions that everybody else has bought into. Einstein, for example, was the first person to really question the idea that time was just this boring thing that ticks at the same rate for everybody. That led to relativity theory. And I suspect that also today there are a lot of assumptions that we’re making about reality which just aren’t true.
Imagination is crucial for imagining how things could be different. You’re never going to be the first person to discover something if you’re always following everybody else.
In your book, you talk about the importance of finding the “right questions” versus the “right answers.” So are there any “right questions” that you’re looking forward to tackling in the future?
Absolutely! To me, this idea that our Universe is completely mathematical is actually most interesting not as a final answer, but rather as a way of generating awesome questions to follow up with research. If my starting point is that it can all be ultimately understood with math, it leads to questions which we still can’t answer.
I’m very optimistic because if I’m wrong and there is something fundamentally non-mathematical about reality, then physics is ultimately doomed.
For instance, if you’re at a party and you look around yourself, as a physicist, what you see is enormous numbers of elementary particles bouncing around which can all be described by mathematical equations. And if it’s all math, then there should be some way of just taking the motions of all those particles and figuring out how they are grouped into objects, like people. If it’s all math, it should be possible to understand that there’s a very, very special pattern of particle motions that somehow correspond to information processing in very complex ways that might be what consciousness is. You can then set out to understand it one day.
I think there’s huge discoveries out there waiting to be made. In other words, I’m very optimistic because if I’m wrong and there is something fundamentally non-mathematical about reality, then physics is ultimately doomed. We’re going to hit a roadblock beyond which we can’t proceed, beyond which we just can’t understand things. Whereas if I’m right and it’s all math, then there is no roadblock! Our ultimate ability to understand our world is only going to be limited by our own imagination.
How long do you think it would take to figure out an equation for consciousness? Would that be something that would happen in the next century or would that be a thousand years down the line?
I think this could easily happen in our lifetime. It raises all sorts of fascinating questions.
Like if you build super intelligent computers, are they conscious or not? If you actually understand what kinds of motions of particles are conscious and what aren’t, you can actually figure this out, right? If you have a patient in a vegetative state and you want to know are they conscious or not, some treat this as something you can never know by definition. But I don’t think that’s true. If consciousness is fundamentally information being processed in a certain way and we can look from the outside and see how these particles are moving around in the patient, we should ultimately be able to understand this.
It has huge implications for ethics, medical issues, cruelty to animals… If someone says, oh, this cow cannot feel pain when it’s being treated in this way, there is no fundamental reason why we shouldn’t be able to ultimately answer and understand whether that’s true or not.
"Just as art and poetry can capture a lot in just a few symbols, so can the equations of physics. From left to right, top to bottom, these masterpieces describe electromagnetism, near-lightspeed motion, gravity, quantum mechanics, and our expanding universe. We still haven't found equations for a unified theory of everything." Image and caption courtesy of Max Tegmark.
I gather it’s a pretty divisive subject among physicists, but what do mathematicians think of the mathematical universe idea?
I’ve mostly gotten positive feedback, like from Ed Witten who got the Fields Medal and from a colleague of mine, David Vogan at MIT. He has a big poster on his wall of this super cool mathematical structure called E8 that he has studied and discovered parts of, so for him, it’s natural. He doesn’t feel he invented it, this mathematical structure. He feels that he discovered its properties and that he is studying it, so that it’s out there in some sense.
My dad, who is a retired mathematician, has this attitude, which I think we all have as kids, that ultimately reality is made of stuff. End of story. If you look around yourself in the world, you don’t see anything that’s mathematical, right? But physicists have discovered that all this stuff is made out of elementary particles, like quarks and electrons. What properties does an electron actually have? It has the properties -1, ½, and 1. These are properties that we physicists have made up geeky names for, like electric charge and spin and lepton number. But they’re just numbers! They’re just mathematical properties!
So all these building blocks of nature, these particles, actually have no properties at all, except for mathematical properties. So in that sense, they are purely mathematical objects. Classic materialism is dead: “stuff” isn’t the end of the story.
And the same deal with the fabric of what’s around us, space itself. It has the property three, the number of dimensions. That’s a number. That’s a mathematical property. All this space and the stuff in space is purely mathematical with only mathematical properties.
So overall, it’s been positive from mathematicians?
I would say yes, absolutely. So far, I think everybody agrees that there is something kind of mathematical about nature. Every year that goes by, that becomes even more obvious. Like, with what tool was the Higgs-Boson predicted? A pencil. Through pure math, Peter Higgs predicted that if you built this awesome machine, you’d find a new particle there. We built it. Boom! There it was.
I challenge people to come up with any single property of our world which is actually not mathematical.
Where it gets controversial is what to make of this. So some people will say it’s just a fluke, it means nothing, or mathematics is just something we humans have made up or maybe something to do with the brain, but it’s nothing fundamental. Then there are people who say, well, nature has some properties that are really mathematical, but others are not mathematical. And then at the opposite extreme, you have me saying that nature has only mathematical properties.
I challenge people to come up with any single property of our world which is actually not mathematical. We thought there were a lot of things that couldn’t be described by math, like light and magnetism, but then we discovered equations for those, too!
One of my biggest problems in understanding your hypothesis was the idea you presented that math has no human baggage. I just can’t wrap my head around it.
So can you not wrap your head around the idea that something could exist if there were no humans?
No. I can think about that… but I can’t imagine what the mathematical structure of the Universe would look like without trying to put it in human terms. I guess that’s what I’m saying.
Could you imagine that the Andromeda galaxy could still exist even if Earth somehow blew up?
No, I understand that! Now you’re pushing me…
I’m just raising that because if the Andromeda galaxy still would exist, with nobody able to see it, then it becomes very interesting to ask, what does it mean that it exists? What properties does it have? How could you describe it without reference to any human language, now that there are no humans?
It would seem like it would exist. It still has hundreds of billions of stars, for example, so there’s that number associated with it. It has a lot of other properties, too. And in what language would you be able to describe that?
The only language that seems to be up to the task is actually the language of mathematics. In English, you say two plus two is four. In Swedish, you say två plus två är fyra. We have different words for the same thing, but if you want to strip away all human baggage, these are equivalent descriptions of the same thing. So the thing itself, the numbers, are actually independent of human language. They don’t depend on what you call them. They don’t depend on what shapes the symbols have that you write them with. None of that baggage matters.
The numbers have these properties—two plus two is four anyway, regardless of what we call them, much like the Andromeda galaxy has hundreds of billions of stars whether we call them stars or stjärnot or estrellas or whatever.
So math is a sort of universal language in that it just is?
Exactly. And I think that’s what Galileo meant. That is the true nature of our Universe. Then we humans come on the scene and we have different words depending on which country we grew up in and stuff like that. But the Universe doesn’t care.
Max Tegmark on BBC's Horizon answering the question, "what is reality?"
Can you summarize your feelings on infinity?
On one hand, I got seduced by it at an early age because it’s so elegant, this infinite hierarchy of infinites. On the other hand, I think that it’s not real in the sense that I don’t think that there’s anything truly infinite in our physical world.
Moreover, assuming the infinite has caused many of the worst problems in physics right now. One of the biggest problems we have in cosmology is called the measure problem, which prevents us from predicting anything rigorously. It comes from the assumption that space can be stretched out ad infinitum. Space is truly continuous. If that were really true, to even measure the distance between two points, you would need to write out a real number like 5.732… with infinitely many decimals, yet we’ve never measured any number better than 17 decimals. That’s a pretty far cry from infinity!
Most of the stuff I teach at MIT which assumes infinity we even know is wrong. When I calculate why it is that you can hear me by using the equation for sound waves, it assumes that air is continuous, that at infinitely many points in the air there is this pressure you can measure there. But we know it’s wrong because air is made of atoms. It’s much more convenient to make that continuum approximation than to deal with all those pesky atoms.
My guess is that we will ultimately discover some other mathematical description of the Universe which is infinity-free and find that all this infinite math that we’re using today is just a really convenient approximation.
Right. So beyond pure curiosity, do you see any practical benefit in the average reader learning about the mathematical structure of the Universe? Is there any way it would affect our daily lives?
I think seeing the biggest picture of what we humans are part of has and really should affect what we do here on this planet. We tend to be so focused on petty little things that we lose sight of all the enormous potential we have. For example, I find it utterly depressing that there are fewer people who have heard of Vasili Arkhipov than Justin Bieber, even though it was not Justin Bieber who single-handedly prevented a Soviet nuclear attack during the Cuban Missile Crisis.
In other words, we humans have already come very close to wreaking enormous havoc with our civilization, and yet, we’re not even paying that much attention to many of these problems. That makes me really worried that we’re not paying much attention to the future of our species either. I talk a lot about this in the book, how there are bad things the Universe is going to do to us. Like, our sun is going to boil off the oceans in a billion years and we might get hit by asteroids. But we actually have the technology to solve all those problems if we get our act together reasonably soon.
The biggest threat is actually our own stupidity—things like, you know, we might start an accidental nuclear war, we might wipe ourselves out by synthetic biology getting out of control, we might wipe ourselves out as a result of super intelligent computers falling into the wrong hands, etcetera. I would guess there’s about a 60 percent chance that I’m not going to die of old age, but from some kind of human-caused calamity. Which would suggest that I should spend a significant portion of my time actually worrying about this. We should in society, too.
I would guess there’s about a 60 percent chance that I’m not going to die of old age, but from some kind of human-caused calamity.
I think there’s a tendency to say, well, you know, nobody knows for sure that any of these disasters are going to happen, so let’s not think about them until someone can prove that it’s really a problem. But that’s silly! Imagine that somebody just had a baby and they go to buy a stroller. And the guy in the store says “Well, I have this really, really solid stroller here for a hundred bucks. It’s been sold for over ten years and there have never been any reports of problems with it. But I also have this other one for a special bargain price. There have been some unsubstantiated reports that it sometimes collapses and crushes the baby, but nobody’s been able to prove that it is caused by any design flaw. I’d suggest you don’t worry about it. And it’s ten percent off! So which one are you going to buy?”
If that’s not how we would behave with one child, then that’s not the way we should behave when we talk about not just our child, but everybody’s children, all future generations.
The big theme in this book is that it was again and again realized that everything we thought existed was just a small part of something even grander. A planet is part of solar system, which is part of a galaxy, which is part of this amazing Universe, which is all part of the grandest structure of all, the Level IV multiverse of all mathematical structures. And that means we have enormous potential. We have 1057 times more volume that we can in principle spread out into. We also have not just hundreds of years at our disposal, but billions upon billions of years during which life could do wonderful things in our Universe.
So I’m hoping that the more aware we become as a species of the wonderful potential that we have, the harder we will work to not squander that potential.
And even on a very personal level, last time I got a parking ticket for not noticing a fire hydrant that had been buried under a pile of snow, instead of getting really worked up about it, I just thought, well, in this wonderful galaxy with hundreds of billions of stars, it’s really not such a big deal. And then I got my smile back.