Everything in the Universe Is Made of Math – Including You

In this excerpt from his new book, Max Tegmark proposes that our reality isn't just described by mathematics, it is mathematics.

By Max Tegmark|Monday, November 04, 2013
RELATED TAGS: MATH, PHYSICS
math-made-flesh
math-made-flesh
Rick Johnson/Discover; Akihiro Sugimoto/Getty Images

It’s a Friday morning in Princeton when I find this gem in my inbox from a senior professor I know: 

Subject: Not an easy e-mail to write ...

Dear Max, 

Your crackpot papers are not helping you. First, by submitting them to good journals and being unlucky so that they get published, you remove the "funny" side of them. ... I am the Editor of the leading journal...and your paper would have never passed. 

This might not be that important except that colleagues perceive this side of your personality as a bad omen on future development. ... You must realize that, if you do not fully separate these activities from your serious research, perhaps eliminating them altogether, and relegate them to the pub or similar places you may find your future in jeopardy.

I’ve had cold water poured on me before, but this was one of those great moments when I realized I’d set a new personal record, the new high score to try to top. When I forwarded this email to my dad, who’s greatly inspired my scientific pursuits, he referenced Dante: Segui il tuo corso et lascia dir le genti! “Follow your own path, and let people talk!” 

I’d fallen in love with physics precisely because I was fascinated with the biggest questions, yet it seemed clear that if I just followed my heart, then my next job would be at McDonald’s. I developed a secret strategy that I called my Dr. Jekyll/Mr. Hyde Strategy, and it exploited a sociological loophole: What you do after work is your own business and won’t be held against you as long as it doesn’t distract from your day job. 

So whenever authority figures asked what I worked on, I transformed into the respectable Dr. Jekyll and told them I worked on mainstream topics in cosmology. But secretly, when nobody was watching, I’d transform into the evil Mr. Hyde and do what I really wanted to do. 

This devious strategy worked beyond my wildest expectations, and I’m extremely grateful that I get to work without having to stop thinking about my greatest interests. But now, as a physics professor at MIT, I feel that I have a debt to pay to the science community. I have a moral obligation to more junior scientists to bring Mr. Hyde out of the academic closet and do my part to push the boundary a little. 

So what paper of mine triggered that “stop or you’ll ruin your career” email above? It was about the core idea that I’m about to discuss: that our physical world is a giant mathematical object.

Math, Math Everywhere! 

What’s the answer to the ultimate question of life, the universe and everything? In Douglas Adams’ science fiction spoof The Hitchhiker’s Guide to the Galaxy, the answer was 42; the hardest part turned out to be finding the real question. I find it very appropriate that Adams joked about 42 because mathematics has played a striking role in our growing understanding of the universe. 

The idea that everything is, in some sense, mathematical goes back at least to the Pythagoreans of ancient Greece and has spawned centuries of discussion among physicists and philosophers. In the 17th century, Galileo famously stated that our universe is a “grand book” written in the language of mathematics. More recently, the Nobel laureate Eugene Wigner argued in the 1960s that “the unreasonable effectiveness of mathematics in the natural sciences” demanded an explanation. 

Soon, we’ll explore a really extreme explanation. However, first we need to clear up exactly what we’re trying to explain. Please stop reading for a few moments and look around you. Where’s all this math that we’re going on about? Isn’t math all about numbers? You can probably spot a few numbers here and there — for example the page numbers of this magazine — but these are just symbols invented and printed by people, so they can hardly be said to reflect our universe being mathematical in any deep way. 

When you look around you, do you see any geometric patterns or shapes? Here again, human-made designs like the rectangular shape of this magazine don’t count. But try throwing a pebble, and watch the beautiful shape that nature makes for its trajectory! 

The trajectories of anything you throw have the same shape, called an upside-down parabola. When we observe how things move around in orbits in space, we discover another recurring shape: the ellipse. Moreover, these two shapes are related: The tip of a very elongated ellipse is shaped almost exactly like a parabola. So, in fact, all of these trajectories are simply parts of ellipses. 

We humans have gradually discovered many additional recurring shapes and patterns in nature, involving not only motion and gravity, but also electricity, magnetism, light, heat, chemistry, radioactivity and subatomic particles. These patterns are summarized by what we call our laws of physics. Just like the shape of an ellipse, all these laws can be described using mathematical equations. 

Equations aren’t the only hints of mathematics that are built into nature: There are also numbers. As opposed to human creations like the page numbers in this magazine, I’m now talking about numbers that are basic properties of our physical reality. 

For example, how many pencils can you arrange so that they’re all perpendicular (at 90 degrees) to each other? The answer is 3, by placing them along the three edges emanating from a corner of your room. Where did that number 3 come sailing in from? We call this number the dimensionality of our space, but why are there three dimensions rather than four or two or 42? 

There’s something very mathematical about our universe, and the more carefully we look, the more math we seem to find. So what do we make of all these hints of mathematics in our physical world? Most of my physics colleagues take it to mean that nature is for some reason described by mathematics, at least approximately, and leave it at that. But I’m convinced that there’s more to it, and let’s see if it makes more sense to you than to that professor who said it would ruin my career.

tegmark
tegmark

Max Tegmark says our reality isn't just described by mathematics — it is mathematics.

Winslow Townson

The Mathematical Universe Hypothesis

I was quite fascinated by all these mathematical clues back in grad school. One Berkeley evening in 1990, while my friend Bill Poirier and I were sitting around speculating about the ultimate nature of reality, I suddenly had an idea: Our reality isn’t just described by mathematics — it is mathematics, in a very specific sense. 

My starting assumption, the external reality hypothesis, states that there exists an external physical reality completely independent of us. When we derive the consequences of a theory, we introduce new concepts and words for them, such as “protons,” “cells” and “stars,” because they’re convenient. It’s important to remember, however, that it’s we humans who create these concepts; in principle, everything could be calculated without this baggage. 

But if we assume that reality exists independently of humans, then for a description to be complete, it must also be well-defined according to nonhuman entities — aliens or supercomputers, say — that lack any understanding of human concepts. That brings us to the Mathematical Universe Hypothesis, which states that our external physical reality is a mathematical structure. 

For example, suppose a tennis ball’s trajectory is that of a beautiful lob that wins you the match, and that you later want to describe what it looked like to a friend. Since the ball is made of elementary particles (quarks and electrons), you could in principle describe its motion without making any reference to tennis balls: 

• Particle 1 moves in a parabola.
• Particle 2 moves in a parabola.
• Particle 138,314,159,265,358,979,323,846,264 moves in a parabola. 

That would be slightly inconvenient, however, because it would take you longer than the age of our universe to say it. It would also be redundant, since all the particles are stuck together and move as a single unit. That’s why we humans have invented the word ball to refer to the entire unit, enabling us to save time by simply describing the motion of the whole unit. 

The ball was designed by humans, but it’s quite analogous for composite objects that aren’t man-made, such as molecules, rocks and stars: Inventing words for them is convenient both for saving time, and for providing concepts in terms to understand the world more intuitively. Although useful, such words are all optional baggage. 

All of this begs the question: Is it actually possible to find such a description of the external reality that involves no baggage? 

Mathematical Structures 

To answer this question, we need to take a closer look at mathematics. To a modern logician, a mathematical structure is precisely this: a set of abstract entities with relations between them. This is in stark contrast to the way most of us first perceive mathematics — either as a sadistic form of punishment or as a bag of tricks for manipulating numbers. 

Modern mathematics is the formal study of structures that can be defined in a purely abstract way, without any human baggage. Think of mathematical symbols as mere labels without intrinsic meaning. It doesn’t matter whether you write “two plus two equals four,” “2 + 2 = 4” or “dos mas dos igual a cuatro.” The notation used to denote the entities and the relations is irrelevant; the only properties of integers are those embodied by the relations between them. 

In summary, there are two key points to take away: The External Reality Hypothesis implies that a “theory of everything” (a complete description of our external physical reality) has no baggage, and something that has a complete baggage-free description is precisely a mathematical structure.

The bottom line is that if you believe in an external reality independent of humans, then you must also believe that our physical reality is a mathematical structure. Everything in our world is purely mathematical — including you. 

Life Without Baggage

Above I described how we humans add baggage to our descriptions. Now let’s look at the opposite: how mathematical abstraction can remove baggage and strip things down to their bare essence. 

Consider the sequence of chess moves that has become known as the Immortal Game, where white spectacularly sacrifices both rooks, a bishop and the queen to checkmate with the three remaining minor pieces. When chess aficionados call the Immortal Game beautiful, they’re not referring to the attractiveness of the players, but to a more abstract entity, which we might call the abstract game, or the sequence of moves.

chess-representations
chess-representations

A game of chess can be represented a number of ways (from left): a physical board, a drawn illustration, a computer simulation, and algebraic notation. Each is a form of baggage representing the mathematically pure game.

Chess involves abstract entities (different chess pieces, different squares on the board) and relations among them. For example, one relation that a piece may have to a square is that the former is standing on the latter. Another relation that a piece may have to a square is that it’s allowed to move there. Similarly, a description of a chess position given purely verbally in English is equivalent to a description given purely verbally in Spanish. 

What is it that’s left when you strip away all these equivalent descriptions, all this baggage? The Immortal Game itself, 100 percent pure, with no additives. There is only one unique mathematical structure that’s described by all these equivalent descriptions. 

The Mathematical Universe Hypothesis implies that we live in a relational reality, in the sense that the properties of the world around us stem not from properties of its ultimate building blocks, but from the relations among these building blocks. This crazy-sounding belief of mine that our physical world not only is described by mathematics, but that it is mathematics, makes us self-aware parts of a giant mathematical object. 

Ultimately, this demotes familiar notions such as randomness, complexity and even change to the status of illusions; it also implies a new and ultimate collection of parallel universes so vast and exotic that all the above-mentioned bizarreness pales in comparison, forcing us to relinquish many of our most deeply ingrained notions of reality. 

If my life as a physicist has taught me anything at all, it’s that Plato was right: Modern physics has made abundantly clear that the ultimate nature of reality isn’t what it seems.


This article, which originally appeared in print as "Math Made Flesh," is an excerpt from the book OUR MATHEMATICAL UNIVERSE: My Quest for the Ultimate Nature of Reality by Max Tegmark. Copyright © 2013 by Max Tegmark. Published by arrangement with Alfred A. Knopf, an imprint of The Knopf Doubleday Publishing Group, a division of Random House Inc. mathematicaluniverse.org

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