On universes

It’s become rather fashionable lately to talk about the existence of multiple universes.

Of course the talk can get fairly contentious, especially if you don’t first agree on terms. If, for example, by “universe” you mean “everything that exists”, then your definition automatically forbids the existence of other universes. Everything that exists is in the universe, therefore nothing outside the universe exists.

Another way to define a universe, in terms of general relativity, is, loosely, a complete topologically connected chunk of spacetime. That definition breaks down at the Planck scale, maybe, but it may be useful otherwise. Note that nothing in that definition explicitly forbids universes from interacting with each other — although interactions as we understand them (oversimplifying here) occur in an arena of spacetime; interactions between two disconnected chunks of spacetime would be something strange and entirely new.

Then again, one might define a universe in terms of interactions — if x is in the universe, then so are all things which interact with x. By that definition universes cannot interact.

But even if we agree on a definition of “universe”, there’s an even trickier word to define: “existence”. What does it mean to say such and such a universe exists or does not exist? If we’ve defined “universe” in such a way as to forbid universes from interacting, then other universes cannot be observed; how, then, do we distinguish existing universes from non-existing ones? And if we cannot, is “existence” a useful concept to apply to other universes?

Furthermore, most of us probably agree our universe exists. Indeed, “our universe does not exist” would seem to be a self contradictory assertion. But then “our universe exists” is a tautology, and again “existence” would not seem to be a useful concept even for our own universe.

We use “exists” to refer to thing for which we have observational evidence. This makes it subjective. We observe our universe, so we say it exists. Gilligan observes his universe, one in which the Skipper and Mr Howell and Mary Ann occur, so he says it exists; but we observe neither the Skipper nor Howell nor Mary Ann nor Gilligan himself, so we disallow his claim.

But is there some objective sense in which — to avoid the term “exists” — our universe has a special status that Gilligan’s universe doesn’t? This seems difficult to imagine, at least within the bounds of science.

Now, people in specialized fields often use a term to mean something different from its colloquial meaning. For example, mathematicians use “exist” in reference to things which, in everyday terms, do not exist. The number 2, for example, is not a thing that exists in the colloquial sense — it’s an abstraction. But in number theory, 2 is certainly a thing that exists — objectively. It’s not a matter of do you observe it or not. It’s a matter of does it emerge from the definitions and axioms of the mathematical system or not.

This might seem to have nothing to do with the existence of universes. But consider this: our universe is described by a mathematical system. Or at least, that’s the presumption underlying physics — that there are physical laws which are mathematical in nature and which describe the behavior of the universe. There are equations whose solutions are descriptions of quarks, gluons, leptons, photons, dark matter, and spacetime itself; put enough of those equations together, and a solution to them all corresponds to our universe. (Or if you’re Stephen Wolfram, there’s some other mathematical structure — like a cellular automaton in a particular state — that corresponds to our universe, rather than a solution to a set of equations. But in any case, there’s a mathematical structure of some sort.) So, we presume, within some mathematical system there exists a solution that corresponds to our universe.

Now, suppose this mathematical existence of an entity corresponding to our universe and the physical existence of our universe are in fact one and the same thing. In other words, within the mathematical object there are features that correspond to intelligent observers — us — and features corresponding to their observation of the world around them, and features corresponding to their synapses forming the concept that what they observe exists. Given the system, the solution, of which we and all we observe are a part, exists. It’s like the t shirt says: “And God said [Maxwell’s equations], And there was light.”

Or, having mentioned Wolfram, let’s go back to cellular automata. Conway’s Life is known to support configurations that behave as Turing machines. A Life Turing machine is a huge and complex object — at least by comparison to a glider or a puffer train — but in an infinite Life grid there’s room for far larger and more complex configurations. Could there be Life objects that behave a lot like organisms — growing, reproducing, responding to stimuli? Why not? Could some of these objects be capable of learning? Of signalling? Of forming abstractions? Of talking about themselves? Of, in other words, behaving like intelligent life? It may seem shocking to think so — yet is it really more shocking than to think quarks and gluons and electrons could do likewise? But if such configurations exist (in the mathematical sense) then surely they would believe they exist (in the colloquial sense). And note that such existence does not depend on our constructing these configurations and tracing their history in a supersized version of XLife. Life objects exist in Life whether anyone in our universe discovers and follows them or not. Turing machines existed in Life before anyone constructed one. Indeed, Turing machines existed in Life before Conway invented Life. Or rather, time in Life is independent of time in our universe, and what we do in our universe does not affect what happens in Life. Life is self contained.

We could be essentially the same as those intelligent Life creatures, existing within a mathematical system.

But why would that mathematical system have only one solution? Conway’s Life has an infinite number of initial states, so there’s an infinite number of Life universes, hypothetically including ones in which similar intelligent creatures observe, and do, different things. Likewise if our universe exists as a solution of a mathematical system, then so do all the other solutions — probably an infinite number of them. None has any objective special status.

We might seem to have replaced the question of why this universe exists with the question of why the mathematical system of which the universe is a solution exists. Why, in other words, are our many universes solutions to that mathematical system and not some other mathematical system? But that question practically answers itself: They aren’t. Our universe is a solution to one mathematical system. But just as it has no special status among solutions to that system, the system itself has no special status among all mathematical systems, and so our universe has no special status among solutions to all mathematical systems.

This idea, which Max Tegmark comes to from a somewhat different angle, that the universe is not described by mathematics, it is mathematics, is shocking and may seem crazy. Yet it answers some profound questions: Why are the laws of physics mathematical? (Because the universe is mathematics.) Why are the laws of physics what they are, and not otherwise? (They are otherwise, in other universes.) Why does the universe exist? (Same reason the number 2 exists in number theory, and gliders exist in Life.)

Still, the thought that Mozart (for instance) was a mathematical object on more or less equal footing with a sine function is rather mind bending.

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