Oh Scarecrow, You’re The Strangest Of All

I mentioned earlier that Max Tegmark’s four levels of parallel universes, and described Level 1 in detail. As cool as it is, Tegmark’s Level 1 “parallel universe” is actually pretty obvious and not weird when you think about it — if the universe is large enough (or infinite), then there’s bound to be aliens out there very similar to us, just by chance. If you get a chance to read his article, you’ll find that Levels 2 and 3 bring in more weirdness — they describe ways that quantum mechanics allows for more parallel universes. But to me, the strangeness hits its peak at Tegmark’s Level 4 parallel universe, which touches on an idea that will be the subject of this last part of the article on the power of Infinite Monkeys.

So what is Tegmark’s Level 4 parallel universe? Well, it’s pretty strange, so let’s ease into it like you’d ease into a piping hot jacuzzi filled with abstract mathematical philosophy. Before revealing his weird idea, Tegmark asks us to ponder a particular question first: why is mathematics so central to our theories about how the universe works? While it sounds like a silly question, many have wondered about it, most notably the famous physicist Eugene Wigner, who questioned the “Unreasonable Effectiveness of Mathematics in the Natural Sciences“. Think about it for a sec — why exactly does math work so well to describe the laws of physics? Sure, counting numbers seem pretty obvious — makes sense why you’d want to count things in the real world. But non-euclidean geometry? Lie Algebra? Differential Equations? Some pretty complex math turns out to accurately model the real world. Wigner brings up a good example — why does the number Pi pop up in so many scientific equations that seemingly have nothing to do with a circle?

If you looked over the shoulder of a theoretical physicist while they worked, you’d probably think they were a mathematician instead. And you might startle them. Back away quietly, and let’s get out of her office before she notices us looking over her shoulder. In fact many areas in math came from physics, including calculus — invented because it was needed to solve physics problems. It’s not a stretch to say that when physicists are looking for a “Theory of Everything”, a unified theory of physics, they’re hunting for a mathematical object, a set of equations that closely mimic the real world. Those out on the frontiers searching for a Grand Unified Theory are often far beyond the possibility of doing experiments to test their theories, and therefore often have nothing except mathematical self-consistency and beauty to go on when judging among competing theories. You can’t blame the theoreticians from wondering whether the mathematical objects they find simply describe the universe, or if the relationship of math to our universe is much deeper. Which begs the question…

Do we discover math or do we invent it?

If humans weren’t around, would numbers still exist? Did we simply invent the concept of math, or is math somehow “real” outside our brains? It’s hard to believe that the integers (1, 2, 3…) exist solely in the minds of humans, but what about prime numbers? What about matrix multiplication? Would all that stuff still exist, somehow, if all humans suddenly disappeared? Is Fermat’s Last Theorem true without humans around to prove it? The jury is still out, though I’d wager most people fall into the Platonic camp — that math exists in and of itself, waiting to be discovered, and is not just a human invention. Seven is a prime number, and always was a prime number, regardless of whether we noticed or not.

Tegmark (and other “Radical Platonists”) subscribe to the following idea — that the underlying mathematical structure of the universe is the most fundamental thing about the universe. Rather than being simply a human invention that is handy for describing the universe, the mathematical theory in some sense creates the universe. Not only does mathematics exist independently from our minds, it’s quite important — the fundamental nature of the universe is mathematical.

Of course, mathematics isn’t a tangible substance — math doesn’t take up space, consume matter or energy, or bump into you in the sauna. If the Platonists are right and math does exist in and of itself, then it has some sort of abstract, ethereal “existence” different than what we mean when we say an apple exists. Or does it? (That’s what you call foreshadowing, right there…)

But wait…

Pondering whether something so abstract as math is “real” brings us back to Cliff Pickover’s dramatic view of how we’re encoded in the digits of Pi. Let’s ask ourselves whether Pickover is being metaphorical when he reassures us of our own immortality in the digits of Pi – after all, we’re not really immortal inside Pi, are we? Being represented as a string of digits somewhere in Pi (however cool that is) isn’t the same thing as physically existing in the real world. Right? Can we really take the solace that Cliff urges us to take, that our long lost loved ones are in there somewhere?

Ponder this — if you were somehow part of a very complex simulation instead of a physical object in the real world (say, you’re being simulated in a computer program on the mainframe of a very advanced alien civilization), there may be no way you’d know. The “Simulation Hypothesis” is the philosophical idea that what we perceive as a physical, concrete reality could simply be an elaborate virtual simulation, and we’re programmed not to notice. After all, if we’re little self-aware subroutines in someone else’s computer, whoever programmed us could simply build us to perceive the world as real – all our sensations tell us the world is real because we’re designed to feel that way. Do you dispute the unreality of the world by kicking your foot on a rock to show its solidity? Good, that means the “make rocks hard” subroutine is working properly.

So the first mind-bending idea here is that we could be a computer simulation and not realize it — all of reality could be “virtual”, and we’d never know the difference.

Furthermore, this insane-sounding idea isn’t just for the bongwater-stained dormroom — it has actually been discussed about in modern physics and philosophy. It’s not just that our little planet is (unbeknownst to us) a computer (as in the Hitchhiker’s Guide books1), but the entire universe could be seen as one giant computer program. There has always been a strong relationship between information theory and modern physics, for example John Wheeler’s “It from Bit” idea. Stephen Wolfram (and others) propose the entire universe is a sort of cellular automaton along the lines of Conway’s Game of Life.

With a shift in perspective, you could see the entire universe as a gigantic computation, with the rules of physics serving as the rules (i.e. the source code). Instead of lines of code operating on symbols in memory, this particular “computation” operates on physical objects using the laws of physics. You could see this as an analogy, but many have taken it at face value and pondered whether the universe could be seen as essentially a giant computer program running a vast “simulation”. Perhaps we really are “virtual” and simply are programmed to perceive the world as physically real, or perhaps it really doesn’t matter what the “medium” is of the computation (be it computer code or physical electrons) — as long as there’s a mathematical rule set (in the form of the laws of physics) underneath it all, our universe is mathematically equivalent to and might as well be called a “computation”.

We’ve reached maximum weirdage, right?

Wrong. This brings us to the next concept — hold onto your butt, because it’s going to get loopier. There’s a related philosophical idea that any conceivable world actually exists in its own right. Modal Realism is one version of this idea in which any possible world which can be dreamed up ipso facto exists, somehow, and is just as real as our own. For example, if it’s theoretically possible to simulate a self-consistent world with laws of physics and self-aware little people, then that world therefore exists. I’m by no means giving this idea justice, but the conclusion would be that any conceivable simulation that could give rise to self-aware conscious creatures does exist, and those creatures therefore feel alive and existing in their own world.

So the bottom line is, this hypothetical computer simulation of an entire universe doesn’t actually have to be implemented on a computer somewhere. If such a complex simulation is possible, then *poof* it hereby exists. At least to the people inside such a simulation. So we don’t have to rely on some alien civilization to power up their Windows 486 machines to run the computer program that simulates our universe, and likewise we don’t have to worry about them tripping over the power cord and snuffing our universe out of existence. Just by the fact that such a simulation could exist, could be self-contained and hold together without falling apart, it does exist, and that little universe feels just as physical and tangible to the people inside it as our world does to us. We could be inside one of these “theoretical” simulations now, and we’d never know. This dovetails with Tegmark’s idea — a “computable simulation” is after all, a mathematical idea.

So the second mind-bending idea is that our world as a simulation doesn’t have to actually be running on a computer somewhere — a theoretically-possible simulation, that isn’t actually coded up, compiled, and executed, is good enough.

Now of course if we’re inside one of these simulations, we’re trapped inside — there’s no way to peek outside the simulation to see if we’re actually running on an alien computer, or are just a theoretical concept sprung to life. We could no less escape from our simulation to see what’s outside than a character in a book could perceive the person reading the book. Dont’ think of it as a “theoretically possible simulation” / mathematical equation sitting in a textbook that suddenly spits out our universe in a conflagration of light and energy. We exist within the computer program / mathematical concept, and this existence is just as valid and real as any other kind of physical existence we can conceive of. In fact Tegmark (and others) would argue, this is the only kind of existence there is.

So… that brings us to Pi. Presumably a full numeric encoding of ourselves is embedded in Pi somewhere, and further there must be a full numeric embedding of the entire universe, somewhere in Pi. Putting aside the numerical concerns I mentioned earlier (about whether the integers in the digits of Pi can really do the job of storing all the real numbers we need to encode the world around us), let’s presume that it is theoretically possible to encode everything about you in a long string of numbers, which is then found somewhere in the decimal expansion of Pi. If these philosophical ideas are right, that means the “you” embedded in the numeric encoding does indeed exist, does feel as fully alive and real as you do, reading this webpage. So in a philosophical sense, we really are immortal in the digits of Pi.

Back to Level 4

Which brings us back to Tegmark’s Level 4 parallel universe. We’ve heard that Tegmark suggests that the foundation of the universe is mathematical — that under it all, we are ultimately made of mathematics. Tegmark further suggests that not only does the mathematical structure somehow imbue the universe with existence, any possible mathematical structure that can create a universe does create that universe. If there are other possible mathematical theories that could create a universe with conscious observers, then those observers do exist, somewhere. Somewhere there is another universe just like ours, except somewhere in its underlying equations it has a minus sign where we have a plus sign. And those folks over there are pondering whether we exist, just as we ponder whether they exist. According to Tegmark, every possible mathematical structure creates its own “universe”, just by its very existence. Instead of envisioning math as a human invention, existing solely inside our minds, we’re just visions inside math’s head, so to speak.

Tegmark calls this the “Mathematical Universe Hypothesis” — our universe is just one of infinitely many possible universes, one for every possible conceivable mathematical structure. Ours is characterized by the fundamental equations of string theory or loop quantum gravity or whatever the final “Theory of Everything” turns out to be (we don’t know it yet). Others correspond to completely different mathematical concepts. Some of these mathematical “universes” will be so simple to be essentially trivial, while others (including ours) will be complex enough for life to evolve. If the mathematical structure is just right to support the appearance of conscious life, then that “universe” will wind up containing conscious creatures who are pondering their existence. Others that aren’t sufficient for the evolution of complex life will remain “empty”, but exist nonetheless. Our universe is just one of infinitely many variations of universes, all of which exist just as much as our universe exists.

Even though this sounds crazy, Tegmark points out it’s kind of attractive, from a philosophical point of view — it helps explain why we have the universe we have, rather than some other universe. It’s a compelling answer to that old philosophical question, “why do we have something rather than nothing?” In the grand scheme of things, anything that is *possible* mathematically does exist, so we are just one of an infinitely many possible universes. Our universe appears to be describable mathematically, and therefore it has sprung into existence. If you were a ruthless applier of Occam’s razor, you might conclude this view of the universe makes more sense than a theory where one particular mathematical structure was chosen arbitrarily to form the laws of physics.

Many folks have pondered these questions and have pushed them even further into ever weirder directions. I won’t go into them here, as they’re pretty much beyond the realm of science (if you even grant me that anything I’ve talked about here still is science). If you’re curious, there’s the absolutely bizarre Omega Point concept by Frank Tipler, which takes these concepts to a breathtaking extreme at the end of the universe. I mentioned Stephen Wolfram’s Theory of Everything based on cellular automata — check out his book for the mind-numbing details or my review for the mind-numbing review. Read the works of David Deutsch, a pioneer of quantum computing and world-class strange-idea-generator, especially his book The Fabric of Reality. And finally, read Paul Davies‘ book The Mind of God for a whole lot more nuttiness in this area. Once you’re done, come visit me somewhere in the digits of Pi, and we’ll talk it all over.

Next Up: Experimental Verification >>

Footnotes:

1. Spoiler Alert