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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? 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.

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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. The answer is 3, by placing them along the three edges emanating from a corner of your room.

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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?

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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. I was quite fascinated by all these mathematical clues back in grad school. My starting assumption, the external reality hypothesis, states that there exists an external physical reality completely independent of us.

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.

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:. 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. 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?

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.

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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. Above I described how we humans add baggage to our descriptions. 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.

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.

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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. Freeman San Francisco. Request this item to view in the Library's reading rooms using your library card.

Mathematics: The Man-Made Universe

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Mathematics: The Man-Made Universe

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