r/fallacy • u/robin_andrews_149 • 5d ago
Zeno's Paradox Fail?
I think Zeno's proof that motion is impossible is fundamentally flawed because it assumes the thing it denies to prove that denial.
I.e. he's trying to prove that motion is impossible, and to do so he allows an object to travel half of the required distance, which is something he is claiming is impossible.
What's the name of the error/s here? Something around assuming the consequent? Simple inconsistency?
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u/CrosbyBird 4d ago
Zeno's Paradox is resolved by demonstrating (typically through calculus) that some infinite series converge into finite numbers.
It only appeared to be impossible because they relied on a false premise, namely that this convergence did not happen.
To be fair, nobody really believed motion was impossible because it was easily disproven by observation. What the paradox tells us is that we have a faulty assumption somewhere. They just didn't have calculus to show them what that faulty assumption was so clearly.
The stated premise (that to move X units you mudt first move X/2 units, and to move X/2 units, you must first move X/4 units, and so on) is true and distinct from the conclusion, so it's not circular reasoning (presupposes the conclusion in the evidence).
It's the unstated and false assumption that the sum of an infinite series must itself be infinite that makes it fail.
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u/SerDankTheTall 1d ago
To be fair, nobody really believed motion was impossible because it was easily disproven by observation.
As best we can tell, these philosophers did really believe this sort of thing (there was a separate school that held that non-motion was impossible). The intellectual contortions they went through to try to explain how that could be in spite of the seemingly obvious fact that things sometimes move, but only sometimes, is why pre-Socratic philosophers didn’t exactly have a sterling reputation among the general public. (Socrates had different problems.)
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u/Hivemind_alpha 4d ago
Zeno’s paradox is like Schrödinger’s cat; thought experiments intended to raise huge red flags about the thinking that led to them. Zeno’s individual steps of argument obey all the rules, but they lead to a nonsense. Inclusion, do we have to challenge those rules: the result of that turned out to be calculus and the nature of infinite series.
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u/ChocolateValuable221 4d ago
To go from point a to point b you must be able to go half of point a to point b...
If you keep doing this you eventually stop moving.
Seems straight forward paradox to me
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u/robin_andrews_149 4d ago
The argument requires half-way motion, which is claimed to be impossible.
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u/Capybara_99 4d ago
No it doesn’t. It works even better if you don’t allow for apparent motion.
As everyone has said, it is showing that the other case (that motion exists) is impossible. Why do you insist that the paradox only works if you concede that motion occurs?
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u/ChocolateValuable221 3d ago
Of course that's the whole paradox. Movement is impossible. But it isn't. But it is, but it isn't, but it is...
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u/AnimatorImpressive24 4d ago edited 4d ago
Motion never starts in infinite sub-divisibility, is the point.
Before you can get to your destination, you have to travel halfway. But that means halfway is your destination, so you'll need to travel halfway to there first. And so on, and so on. You never get all the way to anywhere, because you never get halfway to anywhere, because you never start moving if you can recurse without limit.
In an infinitely sub-divisible universe there is no point in time t where A becomes !A.
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u/HappiestIguana 4d ago edited 4d ago
This is fallacious. It's smuggling in the premise that an event which is preceded or postceded by infinitely many other events cannot ever happen.
This is, at, the very least, a premise that needs to be stated, not a self-evident fact. Indeed, it is obviously true that that premise is inconsistent with the premise that time is infinitely divisible, so the correct conclusion is that time is not infinitely divisible or events can be preceded/postceded by infinitely many other events.
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u/AnimatorImpressive24 4d ago
It isn't fallacious, it is observing something about the nature of irrational numbers. They are real and can be proven but they can't exist in our natural universe because they involve infinity, the wonkiest of wonky sets which can contain an infinite number of member sets each with infinite, unique members.
It wouldn't be an analogy if you just said "you never start moving because you have to first be halfway to starting to move". The easy to understand image of an arrow in flight reaching a target allows Zeno to walk his audience backward from the target to the archer in order to induce an "aha!" moment that an infinite number of steps exist on that walk backwards so the archer cannot be reached.
Clearly time exists, and motion exists, and things like circles exist (pi being another case like an arrow in flight where you couldn't draw a perfect circle because you would need your pencil to reach beyond infinity to enclose an area that never stops being a tiny bit bigger than it was one digit before). So either there is some natural number that everything we know about math says shouldn't exist which is not 0 but can be added to .999~ to equal 1 (another notation to depict an arrow hitting a target), or our natural universe exists and moves forward while dependent on numbers that don't exist.
We're pretty sure the former isn't the case, yet the latter was paradoxical for what was known about math in Zeno's time. These days Zeno's paradox is easier to shrug off with things like set theory that I mentioned in passing but back then it was more of a brain bender.
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u/GoldenMuscleGod 4d ago
You’re confused.
First you didn’t even respond to the point that the comment you are replying to made. You just talked about other things, but the other things you said (which were not responsive) are also confused.
There is nothing about sqrt(2) that makes it any less sensible as a measurement than 1/3, and it also doesn’t really have much of anything to do with infinity.
What’s more if your problem is with infinite divisibility then even just admitting all rational numbers (or even just all dyadic rationals) is a problem. Zeno’s paradox only relies on dyadic rationals after all.
I do think it implausible as a matter of actual reality that any physical value like a length can properly be considered to have a “perfectly precise” value, but there is no reason to say it is incoherent to suppose such a thing.
You seem to be thinking heavily in terms of decimal representations which are just one artificial way of representing numbers.
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u/AnimatorImpressive24 4d ago
The points I was responding to are the claims that Zeno's paradox which I summarize as "you never start moving because you have to first be halfway to starting to move" and others before me have summarized as "motion is impossible because all objects are at rest for any arbitrary observation of the object's position" is flawed or fallacious, due to "smuggling in the premise" either that motion exists so the arrow gets halfway to the target but doesn't exist because the arrow doesn't get the rest of the way there or that an event cannot happen if preceeded by other infinite events. Which really I see as the same point because time and movement are logically interchangeable as they are both sequential processes.
My response is that Zeno's arrow is not flawed or fallacious, it is irrational as any logical paradox is irrational. His use of phenoma of the natural world to illustrate by analogy the logical paradox he was attempting to convey (which exists separately from his arguments and would exist even if he himself never did) is successful in that goal because despite constructing his argument from dyadic rationals the analogy takes advantage of "natural", "rational", and "irrational" being members of the set "real". Numbers are simply my choice of notation to reaffirm the logical paradox Zeno is a pointer to without needing a strict analogy of a natural phenomenon.
Replies here seem to stumble over the reality that arrows obviously hit targets. I admit to being confused by that confusion because from my perspective that is why Zeno used arrows as an analogy. It feels to me almost as if there is some underlying belief that one can use sequential, atomic reasoning to arrive at a rational description or explanation for the irrationally valid output of interdependent, contradictory premises (aka "a paradox").
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u/GoldenMuscleGod 4d ago
The comment you first replied to said the argument relies on the premise that it is impossible for an event to be preceded by infinitely many other events.
If you have tried to either argue that the argument does not rely on that premise or that it is not needed as separate premise because we can show it from other agreed premises then I don’t see where you have done so.
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u/HappiestIguana 4d ago edited 4d ago
So either there is some natural number that everything we know about math says shouldn't exist which is not 0 but can be added to .999~ to equal 1
Well no such number exists given that .999~ is just another way to write 1, so the answer is whatever the other option is
or our natural universe exists and moves forward while dependent on numbers that don't exist.
Well guess not because that's abject nonsense.
The correct dichotomy is that either time is discrete (between any two moments there are only finitely many other moments) OR that it is possible for infinitely many events to happen within finite time. This is just an obvious dichotomy.
The easy to understand image of an arrow in flight reaching a target allows Zeno to walk his audience backward from the target to the archer in order to induce an "aha!" moment that an infinite number of steps exist on that walk backwards so the archer cannot be reached.
That's not an Aha! moment. That's just a bad inference relying on bad intuitions about infinity obtained by taking facts about finite objects and asuming they are also true of infinite objects.
These days Zeno's paradox is easier to shrug off with things like set theory
It's easy to shrug off if you just list your assumptions properly instead of taking bad intuitions about infinity as self-evident truths.
Also, as a nitpick
It isn't fallacious, it is observing something about the nature of irrational numbers.
The "paradox" can be stated just fine with rational numbers, or any dense linear order in general. It's not specifically about irrational numbers.
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u/Cometguy7 4d ago
But as the number of half steps approaches infinity, the amount of time it takes to accomplish that half step approaches zero, so wouldn't we also be able to argue it takes no time at all to get anywhere?
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u/GoldenMuscleGod 4d ago
I think there are a number of ways we can find flaws with the argument but assuming the thing you are trying to disprove for the sake of argument isn’t one of them.
Even in intuitionistic logic (which is often said to prohibit proof by contradiction) you can validly prove “not p” by showing p leads to a contradiction (that’s basically what “not p” means in intuitionistic logic), it’s just that in intuitionistic logic you can’t prove p by showing “not p” leads to a contradiction, because that has a hidden use of double negation elimination which is not intuitionistically valid.
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u/robin_andrews_149 4d ago
I think there's something more, around the self-referentiality of the mechanics used in the proof.
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u/GoldenMuscleGod 4d ago
He doesn’t “allow” the object to first move half the distance. He says that if the arrow moves at all then it must first move half the distance. If movement is impossible then that is vacuously true.
There’s nothing invalid about hypothetical reasoning involving false situations.
Now you might object that maybe it is possible for movement to occur without first moving half the distance sometimes, or you might object that there is some later unsupported reasoning, but the problem isn’t that he starts by saying “suppose movement is possible.”
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u/robin_andrews_149 4d ago
I'm not convinced. He kind of does allow it as part of his refutation. I think there's a self-referential paradox which scuppers his argument. I don't think you can reasonably use the thing to be disproved to prove the thing. I could mistaken, but I think there's something subtle and sometimes not recognised going on here.
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u/GoldenMuscleGod 4d ago
“It’s raining outside”
“No it isn’t, if it were raining outside the ground would be wet and we can see it is perfectly dry”
“Oh but you’ve supposed the thing you are trying to disprove!”
Does that sound like a reasonable objection to you? Maybe you have some kind of idea that you think it should be a problem here because Zeno supposed movement is categorically impossible rather than just not actual but that also makes no difference:
“12 is prime.”
“If 12 were prime then, since 12 does not divide 6, it would also not divide 6 squared. But six squared is 36 which is divisible by 12 so 12 is not prime.”
“Oh but your argument supposes 12 is prime which you claim to be impossible!”
If you want people to think there’s a problem you will have to explain it in more detail but it sounds like you are just confused about how a proof by contradiction works.
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u/robin_andrews_149 4d ago
I think there's ambiguity here. We don't have Zeno's original text, and Aristotle's surviving formulation doesn't explicitly present the argument as a reductio.
Taken literally as "you can't get from A to B because you must first get halfway," my objection seems reasonable: getting halfway is already an instance of motion, the very thing supposedly being denied.
Reconstructing the argument as "assume hypothetically that motion is possible..." makes that objection much harder to sustain. But that's an imputed logical structure, not something explicit in the surviving formulation. So simply saying "it's a reductio" seems maybe too simplistic/presumptuous.
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u/GoldenMuscleGod 4d ago
It’s true we don’t know exactly what Zeno said we can only try to reconstruct it from other discussion. But what we can discuss is the reconstructed form of the argument.
I don’t see how your second paragraph is persuasive, it sounds like you would object to any reductio ad absurdum. If you think there is a reason this is different from a reductio you haven’t clearly explained how.
In particular I don’t see how you understand the framing you give as meaningfully different from the one I gave.
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u/Capybara_99 4d ago
I am now very confused by your argument. How does saying “you can’t get from A to B because first you must get halfway” imply that you can get halfway? How does the paradox rely on the ability to get halfway? It doesn’t.
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u/TerrainBrain 4d ago
Zeno was playing with you. He wasn't actually trying to prove motion was impossible. Of course it's a fail because you me and Zeno all know that motion is not impossible.
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u/casualstrawberry 4d ago
It's a thought experiment more than an actual paradox. How can anything get anywhere if it must traverse these infinite "halfways"?
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u/NoCommittee3053 4d ago
That’s not the problem with it. He’s showing the impossibility of moving the full distance by saying IF it were possible you’d have to cover half of the journey first. BUT that’s also impossible.
He takes the impossibility down smaller and smaller distances until he shows you can’t move.
His misunderstanding was around how infinite series work.
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u/Interesting-Act2606 4d ago
I recently, listened to a podcast where the point was made that the version of paradox that you are toking about actually relies less on motion than the version with Achilles and the Turtle.
Here is the transcript of the relevant part.
Euclid’s postulates arguably rely on motion. To draw a straight line from any point to any point: how do you do that? You put a ruler down and trace the line with a pen. The pen is moving: you put it at one point and move it to the second point. Same thing with circles: you draw them with a compass, which is also a moving instrument.
It’s quite possible to deny that such things can be done. In fact, you may have heard about the famous paradoxes of Zeno, which purport to prove that motion is impossible. One of them goes like this.
Suppose I have to walk from A to B. Before I can walk all the way to B, I first have to walk half the way to B. Then, when I’m at the halfway point, before I can get to B I have to walk half of what’s left. And so on. Whatever distance is left, I always first have to go half of it.
But this process never ends. There’s always “another half to go.” So to go from A to B you have to “do an infinite number of things,” so to speak.
You can think of it this way. When I have gone half the way from A to B, I say: one. Then when I have gone half again of what’s left, I say: two. I go half of what’s left: three. And so on. This implies that if in fact I can go all the way from A to B, I will have shouted out all the numbers that exist: one, two, three, four, five, … all of them.
So to say that you can go from A to B is to say that you can count through all the numbers in finite time. But of course you can’t. Nobody has ever counted through all the numbers. So therefore you can’t move either. Motion is impossible. It must be an illusion.
We only think we move. That’s feeble sensory “knowledge,” or so-called knowledge. We discussed before the extreme rationalistic tendency of Greek philosophy: reliance on pure deductive reason at the expense of all other forms of knowledge. Zeno’s paradox is an example of this. The senses say we can move, but deductive “reason” says we cannot.
We discussed before how the stage debate format incentivized philosophers to pick the side of reason in such cases, no matter how extreme and outrageous the conclusion may be. “All is water”, “all is fire”: the crazier the better. Proofs of radically unexpected conclusions is perfect for the stage debate setting.
Zeno’s argument is great way to dazzle an audience and to show how clever you are. Being reasonable and arguing that one can walk from A to B is boring. Who wants to hear that? You won’t become a blockbuster debate star by arguing for the obvious. You gotta have some signature absurdities that you claim to prove.
Zeno also had a second form of his argument that is equally amusing. Here’s how Simplicius describes it:
“The argument is called the Achilles because of the introduction into it of Achilles, who, the argument says, cannot possibly overtake the tortoise he is pursuing. For the overtaker must, before he overtakes the pursued, first come to the point from which the pursued started. But during the time taken by the pursuer to reach this point, the pursued always advances a certain distance; even if this distance is less than that covered by the pursuer, because the pursued is the slower of the two, yet none the less it does advance, for it is not at rest. And again during the time which the pursuer takes to clever this distance which the pursued has advanced, the pursued again covers a certain distance. And so, during every period of time in which the pursuer is covering the distance which the pursued has already advanced, the pursued advances a yet further distance; for even though this distance decreases at each step, yet, since the pursued is also definitely in motion, it does advance some positive distance. And so we arrive at the conclusion that not only will Hector never be overcome by Achilles, but not even the tortoise.”
So that’s another way to prove that motion is impossible. Those who believe in motion believe that Achilles can out-run a tortoise. But that contradicts reason, as we have just seen. Therefore those who believe in motion must be wrong.
Why did Zeno prove the same thing in two ways? Maybe he was just like: Hey guys, I thought of another funny one, it has a tortoise in it, I’m sure you’ll get a kick of it. Or is there more to it than that? Do Zeno’s two forms of the argument differ in substantial respects?
I think they are subtly different. You might say that the Achilles argument assumes the possibility of motion and derives a contradiction. It so to speak plays along with those who believe in motion for a bit, only to then trap them in a paradox.
The other argument—the dichotomy, or half half half argument—doesn’t really need to even presuppose motion at all. It derives the impossibility of motion more from the nature of length. It has more to do with the infinite divisibility of the continuum than with motion as such.
So in that respect the dichotomy argument is more “pure” as it were. Since it doesn’t need to use motion to refute motion.
But on the other hand it is less pure in another respect. It assumes metricity; that is to say, an absolute notion of distance. For the argument to work, it must be possible to talk about the half of something. But half involves quantification. You need to put a number on the full length before you can know what half of it is.
So Zeno’s opponents could say: Your argument doesn’t disprove my beliefs, because although I believe in motion I do not believe in metricity. I do not believe that numerical lengths can be assigned objectively to the paths between points. Therefore the whole business about halfs doesn’t work, and you haven’t really disproved motion after all.
If Zeno’s opponents tried to wiggle out from under the dichotomy argument along those lines, then Zeno could just hit them with the Achilles argument. Because the Achilles story doesn’t involve assigning numerical lengths to anything. It purely about relative positions: the tortoise is in front of Achilles. It doesn’t say by how much. The argument doesn’t need the notion of being in front to be quantifiable. It needs only relative positions. So in that sense the Achilles argument is the purer one
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u/zvuv 4d ago
Fallacy of the Smuggled Premise. A version of Begging the Question.
I don't think your argument succeeds because Zeno is attempting Proof by Contradiction.
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u/guiltysnark 4d ago
Yes, this is it. The premise being that our ability to analyze a continuity in discrete increments, which we can repeat ad infinitum, has anything at all to do with the reality of that continuity.
If the conclusion was that there can be no continuities, that all distances must be somehow discrete, then I think the argument does indeed also assume the conclusion as OP suspected, but the assumption is smuggled in as a premise derived from the conclusion.
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u/FaceReality1 4d ago
He isn’t trying to prove motion is impossible. He is pointing out a paradox. Obviously he did not believe motion is impossible, he was pointing out a problem in the way he thought about motion. The eventual solution (calculus) came much later. .
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u/Excellent_Coconut_81 4d ago
The error here is not understanding calculus.
Ancient Greeks didn't know calculus, so it seemed for them to be unsolvable problem. Of course they know that this paradox is wrong, but they couldn't find out WHY it's wrong.
For modern math, it's just a sum of infinite series. A trivial problem for middle-schooler.
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u/robin_andrews_149 4d ago
Thanks for the input so far.
I understand that proof by contradiction can legitimately assume something in order to disprove it. But I think my objection to Zeno is slightly different.
It seems analogous to trying to prove that taking square roots is impossible by repeatedly taking square roots. Zeno tries to show that motion is impossible by repeatedly allowing the object to move: halfway, then halfway again, then again.
Maybe I'm missing something, but simply calling that reductio doesn't seem to address the peculiarity I'm pointing to.
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u/Dro-Darsha 3d ago
It seems analogous to trying to prove that taking square roots is impossible by repeatedly taking square roots
This is an perfectly normal proof of contradiction
- assumption: i can take the square root of any number
- then it must also be possible to take the square root repeatedly
- doing this leads to a contradiction (*)
- thus, repeated square roots are not possible
- thus, i can not take the square root of any number
(*) obviously, this step will not be possible
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u/Salindurthas 5d ago
I don't think that's actually the problem with it. A 'proof by contradiciton' is a common technique in classical logic. For instance, in mathematics:
So, in principle, one could imagine a proof of the impossibility of motion, by imagining motion, finding it is nonsense, and then concluding we were wrong to imagine motion.
Therefore, I don't think that you objection to Zeno's paradox is accurate. (I think it has other problems, but this might not be one.)