A Post about Stubbornness1
Let’s assume for the moment that the universe is completely, totally, 100% deterministic.
If this is the case, then given a complete understanding of the laws of nature and a total snapshot of the universe at some instant, a very dedicated individual (or machine, more likely) would be able to roll out the calculations and predict future states of the universe. In principal, anyway. I admit the details would probably get a little gnarly.
So let’s say modern science has figured out more or less how to do this: a means of predicting the future with 100% certainty. It’s a neat trick. Something to impress people at parties, anyway.
Now imagine a scientist—the stereotypical labcoat-clad clipboard-wielding boffin will do nicely. In fact, let’s pretend it’s Jen, circa strip 1. She has access to the skyscraper-sized supercomputer that Science has constructed for the purpose of brute-forcing its way through the mind-boggling infinitude of calculations necessary to predict the future. It’s pretty cool. She’s pretty happy about it.
One day the gang is at Razor Burger for lunch and they get to talking. Jen recently bought a new cat, and she’s rapturously describing it to Roy. In passing, she mentions that the universe is deterministic and that science has developed a method with which it can perfectly predict the future. But god is it a beautiful cat. Her name is Agnes and she’s a calico Japanese Bobtail. You really need to come over and see it one day.
In the meantime, Roy doesn’t really care about cats. I mean, they’re okay, but in his opinion Jen seems a smidge overenthusiastic. Then again, at least she’s not talking about her tuba again. So it could be worse. But wait, what was this whole thing about the universe being deterministic and science predicting the future? That sounds bad for Roy’s free will. Roy likes his free will. Quite a bit, in fact. If he has any say in the matter, he’d rather not part with it.
Roy says as much, and Jen replies, “Oh, free will? No, that’s a myth. A farce! Here, let me show you.” She whips out a handheld device from her coat pocket. It looks like a miniature Etch-A-Sketch with a retractable keyboard. “Have you decided what you’re ordering?”
“What? No. I’m trying to decide between the Shank-wich and the Shuriken Salad. I’ve been trying to watch my figure, but the Shank-wich is undeniably delectable…”
“Perfect! See, this unit is a portable interface to The Computer. I’ll just ask it to predict what you eventually decide to order. It’ll spit out an answer in a minute or two; you won’t even have to bother deliberating!” Thinking she’s doing him a small favor, Jen begins tapping away at the device’s keyboard. She’s hits SUBMIT before Roy can retaliate:
“No? What? That’s preposterous! I refuse to listen to your device. In fact, whatever it tells me to do, I shall do the exact opposite! Let’s see how this Compute-O-Whiz of yours likes that!”
And now we’ve come to the real crux of the story. If the machine can predict exactly how the future will unravel, then by definition whatever answer it yields must be the thing that Roy orders. But Roy is predicating his whole decision on the basis of doing the exact opposite of what the machine tells him to do. The machine cannot be incorrect, but Roy is infinitely stubborn. He would sooner be stricken dead than follow the whims of a capricious mechanical deity. The unstoppable object has run into its immovable wall. In short: a paradox. Therefore, it is logically impossible for Jen to tell Roy what he will be ordering.
Well, okay, fine. Just don’t tell Roy. Easy enough. Unfortunately, this doesn’t really fix anything. See, Milo’s been remarkably quiet this whole conversation. Actually, he’s been distracted. Daydreaming about the scientific consequences regarding the putative existence of leprechauns, mostly. But he catches the tail end of Jen and Roy’s exchange, and he finds Roy’s stubbornness amusing. He decides to try it out with his own portable interface to The Computer. If it tells Milo that he’s going to order the Cutlass Supreme Combo, he’ll get the Masamune Masher, and vice versa.
Now that the predictor and the stubborn person are one and the same individual, The Computer isn’t even allowed to generate a result. Think about it: how can The Computer calculate a correct answer if that answer is necessarily the one that will not come to pass? Back when it was Jen and Roy, Jen could feasibly make the query, secretly record the result somewhere, wait for Roy to place his order, and then reveal the (correct) prediction after the fact. There’s nothing logically inconsistent about that. In Milo’s case, however, knowledge of the prediction directly results in a paradox.
The weak conclusion of our little mental exercise is that it’s impossible for a stubborn individual to know their future. A stronger conclusion might be that it is generally impossible for a person to have 100% accurate information about the future. I can envision an interesting universe where the weak conclusion holds but the strong one does not: in this case, The Computer implicitly knows who is stubborn (and at what times, if we make the reasonable assumption that folks can be transiently stubborn) and fudges its responses accordingly. That is to say, The Computer occasionally lies. In point of fact, The Computer really does “know” the accurate answer—but it also “knows” that revealing this answer will self-defeat the prophecy. (Of course, we should be careful about personifying The Computer with words like know: although it is in a sense “omniscient”, it also necessarily lacks a certain degree of agency. It should, for example, be impossible for The Computer itself to be stubborn!)
The point is: in this theoretical world, stubborn individuals run the risk of getting a false prediction. Thus, in order to ensure that they’re always getting a true and correct prediction, they need to promise to themselves that they’ll follow whatever advice The Computer gives them. Talk about a self-fulfilling prophecy!
I’ll leave you with one last interesting corollary of the weak conclusion. It’s succinctly expressed as follows: If God exists and is omniscient, then it follows that God can’t be stubborn!
1 This post is largely based on an argument presented in Raymond Smullyan’s excellent book This Book Needs No Title.
No one is infinity stubborn.
All the computer must do is present an order to Roy or Milo that they can not turn down, and it does not have to be anything on the menu at the restaurant.
Given that the universe is deterministic Roy and Milo are computers that if given the right inputs will produce one output.
“He”(Milo)” decides to try it out with his own portable interface to The Computer. If it tells Milo that he’s going to order the Cutlass Supreme Combo, he’ll get the Masamune Masher, and vice versa.”
Only when Milo can predict the future better then the computer can Milo go against the computer’s stated outcome.
Now, Milo very well may be able to predict the future better then the computer in this particular case, if say the predictive simulation is numerical in origin or has any possibility for error in fact.
The proposed paradox may be a more numerically intensive problem to solve correctly, but still solvable in a deterministic universe.
Answer: Jen receives the output and tells Roy the opposite.
@ PJW: Perhaps we could simplify this by making it a simply binary decision instead of selecting from a whole menu of items. Roy submits his query—”Will I order the Masamune Masher or not?”—and The Computer returns a single boolean value as its output. This removes the computer’s ability to present a third option that wouldn’t necessarily cause a paradox, and therefore sidesteps the workaround you mention in the first paragraph of your response.
In this modified scenario, Roy has made a promise to himself before asking the computer: if it replies “yes”, he won’t order the item; if it replies “no”, he WILL order it. It sounds like you’re questioning whether Roy would be able to actually follow through with this promise to himself. After all, if it’s logically impossible for him to contradict the machine, he shouldn’t be able to!
But it’s not clear to me via what mechanism Roy would break his promise. We don’t have to assume he’s infinitely stubborn—just give him the unexceptional characteristic that he’ll do something he intends to do. (Unless, of course, he’s prevented by some external occurrence. Or if he changes his mind.)
Anyway, the point is that if he intends to oppose the machine’s prediction, there should be something to keep him from following through on the action. Sure, it should logically be impossible if the machine is correct, but that doesn’t change the fact that Roy is going to contradict the machine unless something stops him. What is that something?
@ Nick: But that doesn’t solve the scenario with only Milo!
@Greg
There is no something, there is no seemingly external force that stops Roy. A remarkable coincidence that forces Roy to renege on his promise to do the opposite of the computer is possible, however interacting with the computer does not bring this about, so it is unlikely.
The answer is mundane, there are some hidden assumptions in the universe where this is scenario is transpiring and therefore in the proposed paradox.
A brute force calculation does suggest a numerical simulation, and with any numerical simulation there comes a certain of fine grain size. Jen and Milo are asking the computer to preform an extremely demanding prediction. It must correctly simulate the interaction of itself with the rest of the universe and it must able to simulation this interaction with infinite recursion to predicate the future with 100% accuracy. Can this be done in a timely manner with available computational resources? If not the computer spins it’s wheels trying to come up with an answer.
To really run the proposed paradox to the ground I assume this computation, including infinite recursion, is possible with a bare brute force solution. If so what does this say about the computer and the universe it lives in?
The computer simulates the universe, in this simulated universe is the computer running a simulation of the universe continuing down into the depths of infinity. In order for this computation to happen in a timely fashion each sub-universe must be simulated more quickly then the last.
This is the hidden assumption. That you can simulated the infinite with the finite. Or if you somehow allow the computer to also be infinite that you can simulate an infinite number of infinite with the infinite.
This is the bad assumption leads to what seems like a paradoxical conclusion. Connected the dots between the bad assumption and the proposed paradox is not immediately clear to me, I will get back to you on that. It is not surprising that some contradiction is found when the base assumption include some sort of infinity.
I agree! To perform the required calculations, the computer would necessarily have to be at least as powerful as the universe in which it’s situated. Your point about the computer needing to be able to simulate itself is an excellent insight.
Now, presumably you wouldn’t need to actually brute force every molecular interaction in the entire universe to produce an accurate result. So instead, let’s say you have a planet-sized computer off-world that’s entirely devoted to the problem of the closed system of Roy, Jen, and the restaurant. Could this work?
Unfortunately, no. As soon as they submit their query to The Computer, the system is no longer closed. The Computer becomes a part of the system, and thus the machine must simulate itself—which as you’ve pointed out doesn’t work so well.
But really, the paradox arises even without the determinism and computation aspects of the thought experiment. Let’s just assume Roy has a time machine and decides to visit the future to find out what he orders. He then plans to return to the present and order something else. This is standard movie time travel stuff, and generally the resolution to the paradox is that the future has changed to fit the new reality. But then we’re still stuck with the conclusion that it’s impossible to have a 100% accurate prediction of the future—even with a fully-functional time machine!
I guess it’s not a particularly profound point, but it’s something fun (at least for me) to ponder through
I do not know if you do not calculate everything error will build in the simulation and given some amount of time the simulation will be wrong. Constant input of new more accurate conditions could counter act this.
The time machine adds a new twist I think. How can Roy visit the future to see what he orders. If Roy is there in the future to order then it is predetermined that he will go back in time so that he can order and see himself do so.
In a non deterministic universe if I traveled to the future, my future not some alternative or possible future, I would not expect to find a copy of myself since it would not yet be determined that I would travel back in time.
Good point about the computer needing to include itself (and its prediction being given to Roy) in its prediction. One solution given in some time-travel stories has been that things happen just as they said it would, but through a complicated or unforseen set events that bring it about.
So, for example, after Roy says he will do the opposite of what the computer predicts, Milo runs over to the trash bin, turns it upside down and starts sorting through it, looking for leprechauns. Roy is so embarrassed/unnerved that he makes his order without waiting for Jen to tell him what was predicted, or even if he was told, the ruckus causes him to forget what he was told, and he ends up ordering what the computer predicted he would order, after all. Humans may be stubborn, but they are still fallible beings in a complex universe, deterministic or not.
wow, that was deliciously geeky
I love it!
@macsnafu: You’re absolutely right. It’s a really interesting idea, and one that I think is cool from a storytelling perspective. It basically requires the universe to have some sort of hidden machinery that self-rights itself in these situations where causality has gone all funky. In some sense, the universe would somehow have to produce some improbable scenario to keep the paradoxical event from happening. If things worked like this, then you’d be able to produce an improbable event with certainty simply by visiting the future! (Does that make any sense? I think it might, but I could be way wrong about that.)
This talk of generating improbability with great certainty kind of reminds me of Douglas Adams’s famed improbability drive.
@Gary: It’s good to hear that I’m not the only one interested in pondering these geeky things
Right. In a 100% deterministic universe, knowledge of the future should not allow changes to occur. But suppose, instead, that it’s only *mostly* deterministic, but not completely, say 75-90% deterministic. That is, the larger structure, context, or setting is determined, but smaller details are not. In that case, which item Roy orders may not be predictable, or only a certain probability of his order can be predicted, but without 100% certainty.
I suggest this because I do believe in free will, but nonetheless, there is a certain order to the universe, and not random chaotic-ness.
Looking at the structure of the atom, as a basis, they tell us that the exact location/velocity of an electron in that atom cannot be determined. However, it certainly can be determined that that electron is within a certain orbit at a certain distance around the nucleus of the atom, even if its exact location is not determinable. We know more generally where it is, even if it’s not exact.
The problem, as already said, is that the computer can’t be part of the universe it is simulating. Or, to be more precise, it can only make 100% accurate predictions for points outside it’s future light cone. Anything placed in the computer’s future light cone could be causally linked to the computer itself and would require the computer to simulate itself.
This shows that computer CAN make 100% accurate predictions about a part of the universe, but anyone influenced by the prediction cannot find that answer before the event actually happens, because the event is outside the computer’s future light cone, so outside any possible means of communication with the computer (the two light cones will eventually intersect, so the answer can be verifiable after the event).
@daniel: Yeah, it’s stuff like this that makes me think that faster-than-light travel is really impossible, or at the least effectively an equivalent to time travel. If you can break the cosmic speed limit, you’ve pretty much tossed causality out the window.
Between this light cone issue and macsnafu’s point about the Heisenberg uncertainty principle, I think we’ve pretty firmly established the impossibility of actually realizing this thought experiment.
Well, if the Computer knows everything, probably knows even that Roy will do the opposite of what it says, so the answer to jen question could be “Roy will order “A” but only if you say him I said “B”.
In true reality, the Heisemberg uncertainty principle is the real answer…
Uh, I forgot: you can’t tell to the subject of an experiment that implies choice that you are monitoring the choices he will make. Doing this you make the subject make his choices considering that he’s being monitored.
…f***, this is Heisenberg again!
Ok, forget everything, I made the mistake to comment before reading all your comments. I’ve been just redundant.
*crawls out in shame*
Aww, don’t crawl away, Lupo! We still love you!
I mean, not in a creepy way or anything. Just… just enough so you don’t feel bad anymore?
Ok Greg, thanx, I understand what you mean (and it’s mutual, i swear).
By the way, the way you made the “timetravelling teddy bears” storyline evolve makes me think about tour comment of the 26th…
Good eye. But is it really how things work in the strip?
Well gee, if I told you that, it’d ruin all the suspense!
(I suspect we’ll find out eventually, though.)