There, I said it.
Last week Elon Musk told us that there is a 1 in billions chances that we're not living in a simulation. Musk is a smart dude and I don't think he actually believes the argument. It seems more like the kind of trolling argument you have with your contrarian college roommate about Zeno's paradox or .9 repeating. You can see what I mean when Musk asks "is there a flaw in that argument?"
A little back-of-the-napkin math shows the flaw right away.
Before we get to the math, I have to disagree about games being photorealistic. They're often breathtakingly beautiful, but stare at any part of them and the gap between the game and reality is painfully obvious.
Musk's argument makes me wonder how much he understands videogames. Most objects in games are not simulated in any interesting way. They're hollow shells, incredibly detailed, but hollow nonetheless. Like us, videogame characters have skin but they don't typically have muscles or organs or blood or nervous systems or trillions of bacteria.
Imagine you walk into a VR game today and pick up a simple orange. You can't slice the orange in half. You can't rip up the rind into a thousand pieces. You can't squeeze the orange and watch the juice drip out. It's just an orange ball (maybe with some interesting tricks to make it look OK as long as you're looking at it from several feet away).
That's why Minecraft is so interesting. You actually can poke into the world and slice things up and explore forever. But remember that the atomic particle in Minecraft is a 1 meter block. And at the most extreme graphics settings, it simulates less than a square kilometer of those blocks at a time. That's the resolution we're at now. One meter.
That doesn't impress me. Not in the way I need to be impressed for this argument.
Musk makes a handwavy arugment that we'll have billions of set-top boxes capable of games/simulations indistinguishable from reality. And we'll get them in 10,000 years EVEN IF our current rate of technological advancement slows by 1000 fold. It's an off the cuff remark, but a specific claim and a demonstrably silly one.
What's our current rate of advancement? Let's assume Moore's Law is still going strong (I don't necessarily believe that, but let's assume it). Then our processing power increases at a rate of 2t/2 where t is measured in years.
You can interpret "a thousandth" our current rate in one of two ways: reduce the base or reduce the exponent. That leaves you with either
1.00110,000/2 = 148
210,000/2,000 = 32
Let's take the higher number. A 148x improvement is about 15 years of steady Moore's Law progress. I don't think that's what Musk had in mind.
You can think about it like this: what did videogames look like in 2001, what do they look like today, and what will they look like in 2031 (assuming Moore's law doesn't slow down).
Who knows. Maybe 2031 graphics will look photorealistic, but I still don't think they'll be realistic. I have no reason to think in 2031 I'll be able to cut up an orange in a realistic way. And I'm only talking about a macroscopic object!
Maybe I'm being unfair, so let's break down what it takes to simulate everything. We'll ask the question "how fast do computers have to be to simulate the universe?"
This is very rough and I'll address objections later, but here goes. The number of calculations per second required for a universe simulation is something like:
The numerator is squared because each particle affects each other particle in an N-body simulation (and, well, in reality too). There are ways to optimize a simulation to reduce this dramatically, but only at the cost of reduced accuracy. Even without the square, the numbers are ridiculous. More on this later.
How many particles?
What are we simulating here? The universe right? Well, there are about 1080 particles in the observable universe.
One obvious guess would be the Planck time (5.3x10-44s), but not being a physicist, I can't say if that number has any particular significance to our simulation.
At least it's got to be smaller than the smallest time we can currently measure: 1.2x10-17s.
The final calculation:
So all we need is a computer that can run about this many calculations per second:
That's a big number, innit?
Our most powerful supercomputer, Tianhe-2, can do this many calculations per second:
And that computer cost $400 million! In Musk's scenario we're running universe simulators in every home. Oh and assuming we did get that 148x improvement described earlier, the fastest supercomputer in the world could now do this many calculations per second:
One more time for emphasis. This is what we need instead:
Of course I hear you screaming "exponential growth will get us there eventually!" Moore's law indeed gives you another "zero" every 7 years. There's only 177 zeros up there, so we're really only a millenia of solid growth away. No problem, right?
Here's the fly in the ointment though: there's no proof we're going to have exponential growth in processing power forever. There's no proof that any sort of advancement over the long run is guaranteed. Five decades of Moore's law is surely impressive, but there's no reason to believe the party will continue forever. Last year Gordon Moore himself said "I see Moore's law dying here in the next decade or so."
For a moment, let's ignore the fact that Moore's Law has already seen some slowdown in recent years.
Let's ignore the fact that we're already making transistors that are only 100 times bigger than single atoms.
The one thing we can't ignore is the speed of light. Bremermann's limit is a speed limit on how fast computers can ever be made. Turn the entire Earth into a computer running at that limit and you're only able to achieve 1075 calculations per second.
Notice that's over a hundred orders of magnitude (a googol) below what we calculated above.
So even if each of our post-human descendants has their own set-top box (each as massive as Earth) and they've overclocked them to Bremermann's limit (no reason to think we'd ever get anywhere close) and billions of them work together, they still won't be able to simulate a single timestep of the simulation after trillions of years.
On to the objections....
Our descendants only have to simulate human brains. Much easier.
Sorry, no. You're still missing an entire world. That leaves everyone in a sensory deprivation tank. If instead you neglect to do a proper simulation and poorly approximate the physical world, the simulation will fall apart (i.e. stop making sense) as soon as the beings inside started doing science.
Our descendants could alter the simulation on the fly to meet our expectations.
Maybe you're thinking of a puppet show instead of a simulation.
That's not how videogames work anyway. Dungeons and Dragons maybe.
Per Musk's arguments, there are billions of simulated beings per real being. It doesn't make sense in the slightest that the base reality would have the time to manage each of us individually.
Well, our descendants don't have to simulate the entire universe. Only Earth.
Really? What happens when humans land on the moon or we send a probe to Mars? Or humans detect gravitational waves from two motherfucking black holes colliding? Or when our spacecraft exit the solar system? Or when we look anywhere at the night sky?
Even if this argument was granted, the Earth has 1050 particles of its own. Run the calculation again and you still wind up with absurd numbers.
Ok, but our descendants don't have to simulate in real time.
True, but as I said earlier the math doesn't seem to allow for even a single timestep to be processed in trillions of years. Anyway, who wants to run their videogame in anything less than real time?!?!
Even if you have some massive optimization that makes the thing run in real time, you have more problems....
Are you going to patiently simulate 13.8 billion years of solar and biological evolution before humans arrive? If you do, then your futuristic society has to wait until your real sun dies out before you can see the simulated Earth being formed.
And if you don't wait, how are you going to get the initial conditions? Are you going to scan the entire universe as it stands, all its matter, and its electromagnetic waves? If not, how do you guarantee every piece of the physical universe has internal consistency and a plausible origin?
Come on, at least the timestep can be longer. No faster than human perception!
How is your simulation going to handle simulated computers that run faster than your timestep?
Funny thing that.
The entire simulation argument rests on computers getting faster, but the faster they get, the harder it will be to simulate them!
Well, the outer reality doesn't have Bremermann's limit. The laws of physics there are completely different and alien.
And now you've really jumped the shark. Instead of the argument being an extrapolation of current technological trends, you're just positing new fantasy universes while having absolutely zero evidence of them existing.
I mean, it's not impossible. No more so than solipsism being true. Or a tiny teapot floating between Earth and Mars.
Elon Musk got many of these ideas from Nick Bostrom, one of the popularizers of the idea that the universe might be a simulation. While all very interesting, I don't find Bostrom's argument totally coherent. Granted, I haven't read Bostrom's books, so perhaps I'm missing some nuance there. But the summary certainly doesn't seem airtight. In a nutshell, Bostrom argues that one of the following three possibilities is true:
It's a bit like saying if our civilization can go to Mars (we can), and some people are interested in going to Mars (they are), then almost everyone now lives on Mars.
There's a lot wrong with this reasoning. It misses the point that the enthusiasts of any given subject (whether that's manned space travel, computer simulations, or civil war reenactment) don't usually control the entire world's economy. Maybe our descendants, most of them anyway, will have more interesting shit to do.
Beyond that, it ignores a fourth possibility: that a civilization might be able to just barely run a single simulation, but not necessarily a plethora of them.
Interestingly, even Bostrom says it's less than 50% likely that we're actually in a simulation.
Odd how that turns into a greater than 99.9999999% likelihood for Musk.