Zeno of Elea was a Greek guy who for some obscure reason really hated motion (1). To have every future Wikipedia reader know how much he hated it, he devised a bunch of paradoxes that proved motion to be an illusion (He was a philosopher on his spare time.). One of them was the Motion Paradox. Said Zeno whilst he sat in a Kumbaya circle with a bunch of his buddies:
“Suppose Homer wishes to walk to the end of a path. Before Homer could reach the end of the path, he must reach half of the distance to it. Before reaching the last half, he must complete the next quarter of the distance. Reaching the next quarter, he must then cover the next eighth of the distance, then the next sixteenth, and so on. There are thus an infinite number of steps that must first be accomplished before he could reach the end of the path. It is impossible for a human to complete an infinite number of steps and Homer is a human, therefore it is impossible for Homer to walk to the end of a path.” (1)
The best philosopher of all time, Diogenes the Cynic, who happened to be sitting at the circle at the time, said nothing. He stood up, walked around the circle, and sat back down in the same place. (Little known fact: Homer too sat at the circle and he was pissed at always being the butt of Zeno’s paradoxes)
I propose that there is still something really right about what Diogenes did- even if Zeno gives an argument, and he does not.
Maybe some of you have heard of this paradox before, or just came up with the solution to it whilst reading. To not make your suffer the suspense I want to let you know that the apparent paradox does indeed lie on a faulty assumption: The sum of an infinite number of steps can be finite, as demonstrated by Euclid, and modern calculus, and this smoking illustration.
And so, we, with our modern tools and superior intellect, have an easy way out. We see Zeno’s argument and we immediately turn it into a numbered argument:
- For Homer to walk to the end of a path, he must first reach half of the distance.
- For Homer to reach half of the distance of the path, he must first reach a quarter of the distance.
- For Homer to reach a quarter of the distance of the path, he must first reach an eight of the distance, for homer to reach an eight of the distance, he must first reach 1/16 of the distance, and so on.
- For Homer to walk to the end of a path, he must first complete an infinite number of steps [1-3]
- Homer cannot complete an infinite number of steps.
- Therefore, Homer cannot walk to the end of the path.
We assert that  is but a confusion, that [1-3], whilst leading to an infinite series, leads to a finite sum, a finite number of steps and, voilá, we cleanly get ourselves out of the paradox. Check-mate, Zeno. Please try to use your outside voice next time, Diogenes.
I want us to go a wee bit further though – I mean, backwards – and to imagine that we didn’t have our superior intellect and our superior tools. I want us to imagine ourselves as a contemporary of Zeno, hearing this story, back in the Kumbaya circle, 2000 years ago.
So here we are. Sitting, Kumabya circle, Ancient Greece, 2000 years ago. We are roughly the same people, with the same values – we put great value in having the correct beliefs, and in using arguments to get there – and we just saw Diogenes sitting after going around the circle in response to Zeno.
We are used to taking arguments seriously and thus we feel torn, bewildered even. Diogenes seems to be pointing at a clear flaw in the argument, but we can’t articulate it; we can see no false premise, no fallacious step. We start wondering if the rational, the justifiable, almost the honorable, thing to isn’t to bite the bullet and give up our belief in motion – if nothing else, at a System 2 level – and to revise our beliefs to account for that.
This would, of course, have been a mistake. But how could we have prevented it? Back then, with inferior tools, and inferior intellect?
We recall the night before this one, where, in the same circle, just below the olive tree, the town lunatic-magician Malucus made himself levitate, as evidence that he was a god, He certainly did levitate – we saw it – and certainly no human can do that, and any god could.
And yet we don’t take him to be a god, in fact, we chased him out of the circle. We assume that we have been tricked by his ingenious levers and pulleys, even if we can’t exactly pinpoint how.
We wonder if it isn’t the same with Zeno and his argument against motion.
Before figuring out the flaw in the argument – either as Greeks 2000 years ago, or just earlier at the beginning of the essay – a failure to find a flaw in the argument has 2 possible explanations. First, the failure to find a flaw might be due to the actual flawlessness of the argument: that is, the argument might really be sound. Alternatively, it might be that the argument is in fact flawed, and our failure to find that flaw is due to our own limitations. (A more sophisticated version of us would detect a subtle fallacy or equivocation, a better informed version of us would recognize a premise as false.)
In deciding how to respond to the argument, we are thus performing an inference to the best explanation, where the thing to be explained is our own inability to identify some flaw in the argument. If the better explanation of this fact invokes the actual flawlessness of the argument, then we should adopt the belief that motion is impossible and revise the rest of his beliefs accordingly. If, on the other hand, the better explanation of the failure invokes our own cognitive limitations, then we should retain his belief that motion is possible in the face of the argument. (2)
This idea can also be made into an argument. It usually goes by the name “One Man’s Modus Ponens is another Man’s Modus Tollens” (3)
We were faced with this option:
- If Zeno’s argument obtains then motion is impossible.
- Zeno’s argument obtains.
- Therefore motion is impossible. [1,2]
And decided to respond thus:
- If Zeno’s argument obtains then motion is impossible.
- Motion is not impossible.
- Therefore, Zeno’s argument doesn’t obtain. [1,2]
That is, we affirm that Zeno’s argument harbors some hidden flaw even though we can’t (yet?) say what it is. In the same way we affirm that Malucus is tricking us, even though we can’t say exactly how. We counter-argue by affirming as a premise the belief that the conclusion of the unwanted argument denies.
I agree, I agree! This type of reasoning seems dangerous. The mind immediately jumps to it being used all sorts of wrong ways: the perfect escape route against all unwanted ideas.
- If [inconvenient, apparently flawless argument] obtains then I’ll have to do this thing I sorta don’t want to.
- I’d rather not this thing I sorta don’t want to.
- Therefore, [inconvenient, apparently flawless argument] doesn’t obtain. [1,2]
In this indefensible version you would use your current belief as a premise of whichever argument as to be able to counter all arguments that would cause you to revise a belief you’d rather hold. I agree that this option both exists and is problematic as it leads to dogmatism.
(I think there is a defensible version of the above argument, differently motivated, that people actually already engage in: When the thing we sorta don’t want to is to be under extreme psychological suffering, there seems to be a reasonable pragmatic consideration about stacking engaging with arguments in an order such as to minimise extreme psychological suffering and thus maximizing our capacity to keep functioning. Us having the expectation of being smarter in the future, we can postpone seriously engaging with some arguments until a later time where we can either demonstrate their flaws or accept them without or with greatly diminished suffering. I can’t pick flaws in the simulation argument, or pascal’s mugging, or justify either Ockham’s razor or induction, and I happily carry on assuming I live in the real world, doing expected utility estimates, preferring more parsimonious explanations and expecting the sun to rise tomorrow.)
Now why does this all matter and why have I driven you through to here? Our community values updating in response to arguments. It is almost heroic to bite the bullet on difficult arguments and come to hold seemingly consistent very far off from the mainstream views (AI destroying the world, the non-existence of selves, …) We are at least thought more highly of if we have arguments for our positions than if we do not. And yet, if my reasoning throughout the essay holds, we can construct any number of arguments to which the appropriate response is not to update, to which the correct response is somewhere between outright dismissal and temporary dismissal with outsourcing to a future smarter self.
This seems like a problem. I don’t have a solution. I don’t know when engagement or dismissal or outsourcing to a future self will each be adequate.
I’m trying to do what I think Diogenes was doing: point at a problem, even if I can’t articulate a solution. Get a conversation started, have people think about this, show me I’m wrong somewhere somehow. I’m much less parsimonious and much less snarky than Diogenes but here we are.
Also, if I outright dismiss the next stupid argument you show me you now know why.
1 This is not true. Zeno probably didn’t harbor an irrational hatred for motion. I took creative liberty with this illustration. The paradox Zeno laid out isn’t precisely this one. This circle story probably didn’t happen, etc. These are merely meant to illustrate a point.
2 A cool consequence of this is that the more cognitively sophisticated one is, the more reasonable it is for one to expect having identified a flaw in an argument should there be a flaw. In this case, a failure to find a flaw is strong evidence that the argument is sound and valid, and thus strong evidence to update. On the other hand, the more unsophisticated one is, the less one should be impressed by the fact that one has tried and failed to find a flaw in a given argument: such a failure is comparatively weak evidence for the argument’s flawlessness. This means that different people might have completely different responses to the same argument, and them both be reasonable. This means that if two people have the same response, one of them might be being quite unreasonable. This also yields that the more unfamiliar one is with arguments and using arguments formally, the more one should discount arguments that lead to hard to believe conclusions, or, put differently, the ability to come to justifiably come to minority conclusions via arguments is an achievement.
3 Hi less-wrongers 😉
This essay heavily borrow from https://www.princeton.edu/~tkelly/ftawil.pdf. I developed the ideas by myself trying to articulate an itch I’ve had for a long time, then found the essay and took it as aesthetic inspiration. Thanks to M. C. for comments on the initial draft.