Briefly put, you may recall, the simulation argument goes like this: There are a number of possible accounts of reality. According to one particularly popular one, the naïve realist account, we are living in a universe that is external to ourselves, and whose nature accords with our own perceptions of it. In other words, the Thing-in-Itself exists objectively in much the same form that it appears to us, after it has passed through our sensory apparatus.
Another possible account of reality, however, might be that we are actually encased in a tub of life-sustaining juices somewhere, and that a hyper-realistic computer simulation of the universe as we perceive it is being wired directly into our brains, such that we think we are seeing it for real.
So far, so what. You are probably thinking that the second account of reality could be the correct one, but we would have way of knowing, and you have already long since put to rest your own adolescent sense of wonder at these kinds of would-be skeptical mind-blowers.
Ah, but the simulation argument has a further pseudo-logical step to take. Not only are we to admit that the simulation scenario is possible, which we are already prepared to do, but we have also to admit that it is likely. Why?
Because the first account of reality, which we have termed that of the naïve realist, is only one account. But there are an infinite number of possible variations of the simulation account. We could, after all, be living in a simulation created by our own future civilization. We could be living in the simulation of a simulation created by that civilization. We could be living in a simulation of that simulation. And so on in infinite regress.
Given this infinite number of possible simulation-universes, therefore, what warrant have we to think we are living in the one real universe. That would be rather arrogant and anthropocentric of us, don't you think? - something like the medieval cosmologists who imagined that the universe must revolve around Earth.
But let's leave the personal attacks out of it, shall we? We can simply do the math and it will tell us we are in a computer simulation. After all, of all the possible universes, we have no a priori grounds for supposing we are in the real version of the universe instead of one of the simulations. And since we've established that the number of simulation-universes is infinite, then our odds of having been born into one of them are infinitely greater than our having been born into the real universe. Thus, we are all but certainly living in a simulation. Q.E.D.
There you have it, folks. The simulation argument. As introduced to the world originally by one Nick Bostrum, and gradually popularized by undergraduate philosophy clubs and pseudo-profound podcast hosts the world over.
Now, most people possessed of common sense will immediately perceive that this argument is absurd - indeed, that there is something flagrantly logically unlawful about it. This can be shown by the fact, if by no other, that it is so simple to come up with parody versions of it. Indeed, something like this argument could be used to "prove" any assertion on any subject.
Suppose I wish to tell you that your table, despite appearances, is not made of wood. You may reply that you purchased it, and were told that it was made of wood. I may say that the person who sold it to you may have been lying. Or they may have been mistaken. Or the table may have been made of wood at the time you bought it, but it has since been replaced - through the handiwork of thieves - by a convincing plastic replica.
Already, then, we have three possible non-wood accounts of the table to your naïve wood position, and I'm sure I could think of others. So, the non-wood hypothesis is already three times as likely as the wood alternative. I'm not saying that any of them is definitely true, but there are more of them, so they are starting to become more likely in sum than the wood-hypothesis.
I just need to spin out a few more of these non-wood hypotheticals to make it vanishingly unlikely that your "wood" table is in fact made of wood.
The fact that such a maneuver is obviously ludicrous and an abuse of the laws of probability may lead us to wonder why we should still be talking about this argument one way or the other. It is often worthwhile, however, to examine the precise ways in which faulty arguments break down, as this helps us to more accurately describe the true laws of reason in future.
Plus, the simulation argument, as laughable as it may seem to you or me, is still being presented as if it had logic on its side (including on the podcast that first led to this rant), and it is still regarded as a serious argument, if not quite by serious people, at least by a large number of unserious people who take themselves very seriously.
So what exactly is wrong with the argument? The answer, we perceive immediately, has something to do with probability. If we take any chance event, say, the casting of an ordinary unweighted six-sided die, there is a sense in which any of one of the possible outcomes is "unlikely." The odds against the die coming up "three" are one in six, thus the odds are against it.
This is true. Score 1 for the simulation argument. Or for the non-wood hypothesis. But there is also an important sense in which the die coming up three is as likely as any other outcome. This is the point the simulation argument misses.
While it may seem that this is such an elementary matter in the study of probability that it must have been ironed out early in the development of the field, this is not necessarily the case. It would seem from reading J.M. Keynes' Treatise on Probability that it took rather a long time to clear up this particular confusion - in part because the early mathematicians of probability wished to use the theory to make precisely the sort of unlawful, Bostrum-like metaphysical claims of which the simulation argument is just the most modern example.
Our author even refers to cases in which these mathematicians tried to smuggle in theological assertions ("establish[ing] the existence of God from the premiss of total ignorance," as Keynes puts it) through the backdoor of mathematics, in flagrant defiance of Hume's essential distinction between "matters of fact and real experience" and matters of the relations between ideas. We will see momentarily how they attempted to do so, and why it was illegitimate in much the same way as Bostrum's maneuver.
What follows may be familiar and unsurprising to students of probability, but it was news to me.
From reading Keynes, it becomes clear that the problem with the simulation argument - and all its kindred, past and present - stems from an abuse of one fundamental principle that emerged early on in the study of probability. Sometimes called the "principle of insufficient reason" (Keynes says "principle of non-sufficient reason," when referring to earlier thinkers), the Bloomsbury philosopher and future vanquisher of classical economics rechristened it, in the Treatise now before us, the "principle of indifference" - which I believe is still the term in use today.
Simply put, the principle holds that, when we are confronted with a range of possible chance outcomes, and we have no positive reason to infer that any one of them is more likely than the others, then each outcome may be treated as equally probable to all the others.
Stated in this form, the principle may appear undoubtedly true - if not downright tautological. If I roll a die, and I know that it has six faces, and I have no reason to suppose that it is unfairly weighted, I am rationally justified in assuming based on the evidence so far that each face of the die has a one-in-six chance of coming up.
This may seem obviously true by definition, and therefore uncontroversial and dull. But oh the abuses to which it can be put, when improperly understood!
After all the principle has the mysterious property (or at least, the appearance) of allowing the mind to draw mathematical conclusions from the fact of ignorance, rather than of knowledge. This has endeared it to the mystic-minded throughout modern history. Hence Keynes' metaphysicians seeking to summon God out of sheer ignorance - a post-scientific revolution version of apophatic theology, if you will, which Keynes describes under the tongue-in-cheek heading of the "alchemy of logic." And hence our Nick Bostrum coming up with an idea of a simulation, and working his way around from this to the assertion that it exists in fact.
Keynes examines in some detail several hypothetical ways in which a botched version of the principle can be used to reach self-contradictory results -- as well as at least one real-life example, in which apparently "serious" philosophers and scientists tried to use the principle of indifference to make a positive assertion about the chemical composition of the star Sirius - based on no experimental data, observation, or evidence.
In order to show us how an imprecise and insufficiently qualified version of the principle of indifference can lead us into error, Keynes asks us to consider the following line of argument.
Suppose we are asked to guess where a random person chosen from the global population lives, based on no other information about them or their place of residence, apart from the fact that it is somewhere on planet Earth.
Applying the principle of indifference, we may say that they are just as likely to live in France as to live in the British Isles (since we have no reason to think otherwise). We aren't able to assign a numerical value to the probability of either case yet, since we don't know many options there may be, but we do know that the two are equiprobable, using the principle of indifference. Thus, we can say that the likelihood of the person living in France or the British Isles is in each case 1/x.
Now, we ask ourselves what are the relative odds that the person lives in England versus Ireland. Again, we have no reason to think that one is more likely than the other. So again, the probability in each case is 1/x. And if we were to ask the probability, in this case, that the person lives in either England or Ireland, the answer would only plainly be 2/x.
Here we begin to see the problem. We've already said that it is just as likely that the person lives in France as in the British Isles, and we set both at 1/x. Yet here we have an argument that seems to prove that the likelihood of the person living in England or Ireland - both parts of the British Isles - to be twice that amount.
Plainly, we realize, we need to do a better job of defining our units, before we can justly apply the principle of indifference. Instead of simply asking "where" a person lives, we need to ask "in what country?". But then, we would also need to establish that each country has exactly the same population size, before we can assert that a given person - of whom we know nothing other than that they are an inhabitant of Earth - has an equally likely chance of living in any one of the countries.
The principle of indifference works, then, as applied to a set of outcomes that we know to be in some sense equivalent. It does not work when we are dealing with units of varying size. Or, as Keynes puts it, "the Principle of Indifference is not applicable to a pair of alternatives, if we know that either of them is capable of being further split up into a pair of possible but incompatible alternatives of the same form as the original pair."
A paragraph that appears just before this is also worth excerpting in full:
"The examples," writes Keynes, "in which the Principle of Indifference broke down [such as the British Isles example], had a great deal in common. We broke up the field of possibility, as we may term it, into a number of areas by a series of disjunctive judgments. But the alternative areas were not ultimate. They were capable of further subdivision into other areas similar in kind to the former. The paradoxes and contradictions arose, in each case, when the alternatives, which the Principle of Indifference treated as equivalent, actually contained or might contain a different or an indefinite number of more elementary units." [Emphasis in original.]We may paraphrase these remarks by concluding that the principle of indifference only applies when we are dealing with a set of outcomes that cannot be broken up into smaller units. How do we know when we have gotten to such a point? This occurs when each potential outcome that we are considering as a unit is equivalent.
Equivalent in what regard, though? We can't say in size (even two countries with the same geographic area will not have the same population, and six sides of a die are not equally likely to come up if we know that one of them has been weighted). We are forced to say - equivalent in likelihood. There is no other ultimate quality we are considering.
And here, I'm afraid, we are thrust back upon a fear that was only warningly teased above - that we may have strayed into the realm of outright tautology. If the principle of indifference is asserting nothing more than that a series of equiprobable outcomes are equiprobable, then it is asserting nothing at all.
Let us leave this to Keynes to sort out, however. I'm sure he does, somewhere, in the inscrutable purely formal sections of the Treatise. I prefer to get back to the simulation argument, because I believe we are closer to seeing now the exact manner in which it is mistaken.
Before we turn that corner, however, let us examine one other of Keynes' examples of faulty applications of probabilistic reasoning. This is the instance mentioned above in which mathematicians proposed by sheer power of ignorance to summon to mind the chemical composition of the distant star Sirius.
How did they work this magic? Well, they reasoned that the answer to the question: does Sirius contain the element iron? has two possible answers: yes, or no. We have no reason to prefer one of these two answers to the other, so we apply the principle of indifference, out of our state of ignorance, and say that the probability that the answer is yes is 1/2. Thus, Sirius is 50% likely to contain iron (or maybe it's another element - I forget - but the same logic applies).
This is ridiculous enough. But, never content to rest upon their laurels, the Bostrums of the world and their forbears must push their conclusions to the limit. The Sirius-minded people we are considering here decided that the same principle could equally well be applied to all 68 of the other known earthly elements at the time. Each of them, in turn, also has a 50% chance of showing up in the star Sirius.
Can we conclude from this that it is vanishingly unlikely that Sirius contains none of these elements, since each of them has a 50% chance of being there, therefore Sirius is almost certain to contain at least one terrestrial element? Just try and stop them! And so our scientists have discovered that Sirius almost certainly contains one of our known Earth-elements, and all without having to so much as peer into a telescope! Just as Nick Bostrum has summoned into certain existence whole universes of AI-produced virtual reality simulations, just through the power of his brain!
Why do these arguments not actually work, despite their pseudo-mathematical structure? Because, as we have already seen, the principle of indifference (and I am skeptical - as indicated above - as to whether it should really be treated as anything more than a circular statement) only applies, if it applies at all, to a set of outcomes that we know to be equivalent to one another.
It cannot actually summon knowledge from ignorance, therefore, as it purports to do. The appearance that is can do so descends simply from the fact that it was originally formulated in reference to games of chance (involving cards and dice), in which we do in fact already know something about the likely outcomes. And the thing we know is precisely that which the principle of indifference sets out to demonstrate: namely, that the possible outcomes of a dice-throw are equivalent to one another in probability. Even to assert this, however, we already have to know that the specific set of die has not been tampered with or weighted. So there is really nothing we know, having applied the principle, that we did not know at the outset.
Thus, we cannot assert that Sirius is just as likely to contain iron as not, on the bare fact that we do not currently know what elements it might contain. (Of course, astrophysicists are now in a better position to guess at the star's composition than they were in the nineteenth century, but this has been due to advances in observation, experimental techniques, and space exploration, not to any pseudo-probabilistic jugglery.) We have not established that "Sirius contains iron" and "Sirius does not contain iron" are two equiprobable outcomes. Simply asserting that they are achieves nothing.
So too, with the simulation argument, we are dealing with something like the British Isles/France problem. The argument claims that the number of potential simulation-universes is infinite, or at least some indefinitely large number. Let us call that x. Now, we have admitted already that we have no a priori reason to suppose we are more likely to live in the real universe than in one of the simulations. So, we can say that our likelihood of living in the real universe is 1/x.
The argument then adds up all the simulation universes, and it finds that the likelihood of our living in one of them must be (x-1)/x. We then compare this to the probability of the real universe, which we said was 1/x. And since x, we have established, approaches infinity, then the odds of our living in a simulation become overwhelming.
I hope we perceive already the way in which this argument, however momentarily compelling it may seem, is inevitably entangled in the same problems Keynes pointed to above. Because precisely the same argument that Bostrum has just used to "prove" that we do not live in the real universe could equally well be used to "prove" that we do not live in any particular one of the simulation-universes.
Let us see take one of the hypothetical simulation-universes for our example - say, the one in which our own future civilization has created a convincing simulation of our experience of the "present."
We have no reason to say that this is a false account of reality. So, applying the principle of indifference, we can say that it is just as likely as all the other accounts. And so, we can say that the odds of this version of the simulation being the true one is 1/x. Now let us add up the probability of all the other potential accounts of reality. The sum again turns out to amount to (x-1)/x -- hence it is overwhelmingly likely that one of them is in fact the case.
We have just "proved" that this particular version of the simulation-universe is so unlikely as to be almost certainly false. And we could go through the list for all the other infinite number of potential universes too, had we but world enough and time, and "prove" by the same crock-logic that they are all nearly "impossible" as well.
We are back to the problem that the British Isles and France are not comparable units. We are dealing, in Keynes' words, with a situation in which "the alternative areas were not ultimate. They were capable of further subdivision into other areas similar in kind to the former."
Bostrum has transgressed Keynes' dictum: "the Principle of Indifference is not applicable to a pair of alternatives, if we know that either of them is capable of being further split up into a pair of possible but incompatible alternatives of the same form as the original pair." For the various simulation-universes are certainly capable of being split up into an infinite number of further pairs, just as the British Isles can be.
Bostrum seeks to divide up the potential universes into "real" (of which there is only one example) and "simulation" (of which there are an infinite number). He neglects the fact that the category of "simulation" is divisible into an indefinite number of units that are equivalent in "size" (by which we mean here probability) to the one "real" universe alternative.
He is not justified, therefore, in saying that the real universe is unlikely simply because his invented alternatives outnumber it - any more than we would have license to say that rolling a three on a die is less likely than its alternatives, simply because we have gathered up those five alternatives and relabeled them as "not-three."
All Bostrum can justly assert, therefore, is that the naïve realist account of the universe is only one among many, and we have no way of knowing for certain if it is the true one. But it is at least as likely as any other account.
The simulation argument has added nothing, therefore, to what has not already been established by the skeptical questioning that has provided the fuel of all philosophy since the field was invented. It simply raises once again the tired old imponderables of how can we know the world is real, or anything about it, with any degree of certainty. And as with all such ancient riddles of philosophy, these questions are most likely unanswerable, or else can only be answered by discovering that they cannot meaningfully be asked.
And speaking of ideas that end up lending a lot less to the field of knowledge than at first appears, the principle of indifference too -- at the end of our journey today -- comes to seem like little more than a restatement of its own premises. Despite all the excitement it has generated through the centuries, it cannot actually create knowledge from ignorance; it can only assert equiprobabilities among outcomes that were already known to be equiprobable.
When we don't know anything about the likelihood of a given outcome, the principle of indifference cannot help us derive it. If we do not know anything about whether or not we are in a simulation, and if there is no way of testing the hypothesis, and we have no evidence or means of obtaining evidence even potentially at our disposal, then we are not in a position to make any claims at all as to the probability or otherwise of finding that we live in one -- no more than we can justly assert that the star Sirius has a 50% chance of containing iron, based on nothing more than the fact that we don't currently know whether it contains this element or not.
Keynes quotes George Boole on the subject, and his words come to us in this hour as a torch of reason glowing amidst the dark night of obscurity and faulty logic: "[T]he state of expectation which accompanies entire ignorance of an event is properly represented," he writes, "not by the fraction 1/2, but by the indefinite form 0/0."
Were sweeter fractions ever written?
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