The key ideas

The Theory of Everything as the Ultimate Ensemble

Max Tegmark, 1998

Why does the world have these laws and not others? The physicist Max Tegmark puts forward a bold hypothesis: every mathematically possible world exists, and ours is just one of them. This retelling of his 1998 paper does without formulas and shows how such a hypothesis can be tested and why, by its logic, our space has three dimensions.

Who it is for

For anyone curious about why the world has exactly these laws and whether they could have been different. No formulas and no physics degree needed.

Cogito, ergo space is three-dimensional.

What breathes fire into the equations

Physicists have an old dream, a theory of everything: one elegant theory from which follows everything there is to know about the workings of the world. But even if it is found, a question will remain, one that Stephen Hawking once asked. What is it that breathes fire into the equations and makes a universe for them to describe? In other words, why do exactly these laws exist and not some others?

In 1998 the physicist Max Tegmark offered an unexpected answer: there is nothing to choose from at all. Every mathematically possible world exists. This is his hypothesis, and he himself writes that it is far from clear whether observations rule it out.

What you have in front of you is a retelling of a scientific paper, not a book. It is full of formulas, and there will be none here, only the line of thought. You will see how a theory without a single adjusted number can still make predictions, what the world looks like to a bird high above and through a frog's eyes, and why our space has three dimensions.

And at the end we will come back to Hawking's question. Tegmark has an answer to it, and that answer concerns you.

If the world is mathematics, why this mathematics

Tegmark begins with a question: is the world entirely mathematical, or is mathematics only a handy tool that describes it approximately? To put it more strictly: can we say that the physical world simply is a certain mathematical structure? By a structure he means a set of objects and the rules that connect them, like the natural numbers with addition and multiplication.

From this question grow four options. The world is not entirely mathematical, and then a theory of everything will in principle be unable to predict everything. The world is mathematical, but only some structures exist physically. None exists at all: this option is obviously ruled out and is there only for completeness. And the last option: they all exist.

Most physicists, by Tegmark's estimate, choose the first or the second. The second is vulnerable: it doesn't explain why this particular structure exists while its neighbours don't. Why is the proton about 1,836 times heavier than the electron, and not 1,996 times? Why does space have three dimensions and time one, rather than, say, seventeen and five?

A theory that doesn't answer such questions may turn out to be true. But in Tegmark's words, it looks arbitrary and may disappoint those who expect a theory of everything to answer every question.

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About the author

Max Tegmark is a physicist and cosmologist. He wrote this paper at the Institute for Advanced Study in Princeton, and in 1998 it was published in the journal Annals of Physics.

All books by this author: Max Tegmark

Terms from the book

  • Einstein
  • Hawking
  • Occam's razor
  • Theory of everything: The physicists' dream: one self-consistent theory that takes in both quantum physics and general relativity and describes the whole workings of the physical world.
  • Mathematical structure: A set of objects and the rules that connect them, for example the natural numbers with addition and multiplication. In the paper, a structure doesn't depend on the symbols you use to write it down.
  • Ultimate ensemble: Tegmark's term: the set of all mathematical structures at once, if each of them exists physically. It includes even worlds with altogether different equations.
  • Hilbert: David Hilbert, a German mathematician. By his definition, which Tegmark adopts, whatever is free from contradiction exists mathematically.
  • Bird perspective: An image from the paper: the way a mathematician sees a mathematical structure, from outside and as a whole.
  • Frog perspective: An image from the paper: the way a mathematical structure is experienced by someone living inside it.
  • Self-aware substructure: Tegmark's term: any part of a mathematical structure that is able to think and perceives itself as thinking. It could be a human being, a complex computer program or something else entirely.
  • Theory of relativity: Einstein's theory of space, time and gravity. In it, space and time are joined into a single four-dimensional world.
  • Eddington: Arthur Eddington, a British astrophysicist of the first half of the twentieth century. Tegmark cites his remark that what we see as an object is whatever keeps its form.
  • Ehrenfest: Paul Ehrenfest, a physicist who noticed in 1917 that in a space with more than three dimensions there are no stable atoms or planetary orbits.
  • Strong interaction: One of the four fundamental forces of nature. It holds the protons and neutrons in an atomic nucleus together.
  • Weak anthropic principle: The idea that we have no right to be surprised by the features of the world without which we would not exist: in a world without them, there would be nobody to be surprised.
  • Wigner: Eugene Wigner, a physicist and the author of a famous essay on the “unreasonable effectiveness” of mathematics in the natural sciences.

All terms

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