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I thought id ask how you would explain what is commonly referred to as "Zeno' Paradox of Achillies racing the turtle" in Buddhist terms.

For those who aren't familiar the paradox is like this Youtube Animation;

  • Achillies is supposed to race a turtle.
  • Achillies gives the turtle a head start of say 10 meters.
  • Achillies runs at 10 meters per second.
  • The Turtle runs at 2 meters per second.
  • Achillies starts the race from the point A as the turtle starts the race from point B which is B=A+10meters.
  • By the time [+1 second] Achillies reaches the point B, the turtle will have moved from the point B at a speed of 2 meters/second and after 1 second will have reached the point C=B+2meters, moving B to C and remain ahead of Achillies by 2 meters.
  • By the time [+1.2 second] Achilies moves A to B and B to C, reaching the point C; the turtle will have moved from the point C at a speed of 2 meters per second and after 1.2 that it took Achilies to cover the 12meters going first from A to B then from B to C, the turtle will have moved from B to C and further from C to D wherein D=C+0.4meters, still ahead of Achilies by 0.4 meters.
  • By the time [+1.24 seconds] Achillies reaches the point D, the turtle will have moved from the point C at a speed of 2 meters per second and after 1.24 seconds that it took Achilies to cover the 12.4 meters going first A to B, then B to C and furthermore from C to D; the turtle will have moved B to C, C to D and furthermore from D to E wherein E=D+0.08meters, still ahead of Achilies by 0.08 meters.
  • This is then supposed to go on indefinitely with Achillies paradoxically never reaching the turtle due to infinite division of space & time into smaller intervals.

The reason i am asking is because i have not seen Buddhists do this before, am curious as to whether someone here can provide a comprehensible explaination and want to develop an answer myself even tho i have an idea of how id answer this.

Let's just assume for simplicity sake that it's just a thought experiment and a ground for confusion rather than a "true paradox" of some sort.

I want substantiated answers based on texts know to be true such as the sutta-pitaka but it's not a requirement.

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I think that SN 51.15 is an example:

"If that's so, Master Ananda, then it's an endless path, and not one with an end, for it's impossible that one could abandon desire by means of desire."

"In that case, brahman, let me question you on this matter. Answer as you see fit. What do you think: Didn't you first have desire, thinking, 'I'll go to the park,' and then when you reached the park, wasn't that particular desire allayed?

I think that Brahman Unnabha argues from the pespective of paradox -- that it's infinite i.e. "endless".

And Ananda replies something like, "When you arrived, that was the end of that" -- which is IMO more or less a description of how a real arrow behaves, without getting caught in the mentally-constructed paradox.


I notice I've changed the formulation of the paradox -- not "Achilles and the Tortoise" but just an arrow -- so I'm actually referring to Zeno's Dichotomy paradox.

IMO they're about the same paradox: because you can simply subtract the tortoise's speed from Achilles's so that it's the description of the relative motion, between a moving object and a (relatively) stationary target.

Doing that, i.e. subtracting to transform one to the other, is a safe/valid transformation -- except at speeds approaching the speed of light.

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Nagarjuna's Mūlamadhyamakakārikā (MMK) "Chapter 2 - Critique of Walking" is the same as Zenos paradox that you outlined. If you Google Zeno and Nagarjuna on Motion you will see many articles written by Translators and Philosophers comparing the two.

Both thinkers use the idea to prove that conventional reality is not the ultimate truth, but the ultimate truth they posit is a polar opposite. For Nagarjuna it is Emptiness and for Zeno one unchanging being.

The idea doesn’t end in disproving motion only, check this video to see how Zeno expound the idea to cover wider topic. Nagarjuna also use it to disprove a lot of ideas that we think as truth, he also use the same method to disprove the momentary consciousness posited by the Abhidhammist (i.e Theravada Buddhist).

So, you may ask, is the paradox proven? No it remain a paradox.

In the west there has been many attempt to prove it, but at every age it resurrect itself as unproven. In recent history, the Giants of mathematician gather in Vienna to come up with an all-encompassing mathematical proof, but a man by the name Kurt Gödel dug out Zeno and called the whole project a bluff. Again with the advance of computer some tough computers will be the answer, but Alan Turing showed up and pulled the Zeno card on computers too.

Quoted below is Candrakiri Lucid commentary on MMK.

At this point, someone says, «By negating production, you have established that interdependent origination has the quality of not being ceased and so on. Nevertheless, in order to prove that interdependent origination has neither coming nor going, you must state an additional argument to refute the action (kriyā) that is going to and fro (gamanāgamana), which is commonly accepted in the world.»

In response, we say that if going (gamana), exemplified by the action of walking, 1 were to exist, then one would necessarily conceive of walking in relation to the spot on the path where one has just walked; or one would conceive of it in relation to the spot where one has not yet walked; or one would conceive of it in relation to the spot where one is currently walking. But none of this makes sense. So he says:

2.1. First of all, it is not the case that one is now walking where one just walked; one is also not walking where one has not yet walked; and without the spots where one already walked and where one has not yet walked, one has no knowledge of a currently walked upon spot; the currently walked upon spot is thus not being walked upon. Here, “where one just walked” expresses that part of the path on which the action of walking has just ceased. That which is now affected by the action of walking is expressed by, “one is now walking.” It is incoherent to speak of “where one just walked”—meaning the place where the action of motion has just ceased—with the phrase “one is now walking,” which expresses a connection with a presently existing action of walking. Therefore, it is incorrect to say, “One is walking where one just walked.” The phrase “first of all” indicates the order in which the various possibilities are negated.

So too, “one is not now walking where one has not yet walked.” The phrase “where one has not yet walked” expresses a part of the road where a future action of motion has not yet occurred. The phrase “one is now walking” expresses a present action of motion. Therefore, since the future and the present are completely distinct, it does not make sense to say, “One is now walking where one has not yet walked.” If one has not yet walked upon it, how can one be now walking on it? And if one is walking on it, how can it be a spot where one has not yet walked?

Furthermore, there is no action of walking on the currently walked upon spot because “without the spots where one has already walked and where one has not yet walked, the currently walked upon spot is not being walked upon or known.”

In this context, the spot that the walker has traversed is for him a walked upon spot; and the spot that he has not yet traversed is for him a not yet walked upon spot. But in isolation from the walked upon spot and the not yet walked upon spot, we do not see any third spot that would be “the currently walked upon spot.” And since that is the case, “one has no knowledge of a currently walked upon spot.” Here, na gamyate (i.e., “is not being walked upon”) is taken to mean “is not known.” Therefore, since one has no knowledge of it, there is no currently walked upon spot. Hence, that spot is not affected by the activity of motion; that is, it is not currently walked upon. Hence, there is also is no action of walking even in the currently walked upon spot. But someone might say, «There is a spot that is tread upon by the feet of the walking walker. That is the currently walked upon spot!»

This is not the case because the mover’s feet are a conglomeration of atomic particles. The area behind an atom located on the tip of the toe is included in the spot already walked upon, relative to that atom; but the spot in front of an atom located at the back of the heel is included in the spot not yet walked upon, relative to that atom. Feet cannot exist without the atomic particles of which feet are composed. Therefore, there is no currently walked upon spot without being either a spot already walked upon or one not yet walked upon. And just as one analyzed the feet, so should one analyze the atomic particles in terms of the relation between their front and back parts. Moreover, if someone says that the currently walked upon spot is the spot that one has halfway walked upon, then one uses the answer that has been given in the analysis of production. 3 Therefore it is established, “The currently walked upon spot is not being walked upon or known.”

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  • Hi. I just wonder what is "momentary consciousness posited by the Abhidhammist (i.e Theravada Buddhist)" a reference to? I've never seen this momentariness of which you speak in the Theravadin Abhidhamma, which book/passage are you referring to? Thanks – deadmanposting Jul 19 at 17:54
  • Ok i see it is a reference to post-canonical Theravada, i am thinking this is probably a doctrine that might have some basis in 5th century commentary Vsm and finally expounded in the Abhidhammatha Sangaha of 10th century. Afaik It is foreign to the early Theravada and is not in their canonical Abhidhamma – deadmanposting Jul 19 at 19:19
  • Please read starting from Paragraph 5, the section from Paragraph 5 in Part-2 is applicable to Theravada exposition as it is today..For Theravadins they called the theory of momentariness Khāṇavāda..... Part 5 on 'Causation' also show the same method employed to counter 'momentary dharmas' idea in the Abhidharma. – Epic Jul 19 at 19:47
  • Thanks seems like a nice summary – deadmanposting Jul 19 at 20:04
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I think one needs to put this paradox in terms of Sabbe Sutta;

The Blessed One said, "What is the All? Simply the eye & forms, ear & sounds, nose & aromas, tongue & flavors, body & tactile sensations, intellect & ideas. This, monks, is called the All. [1] Anyone who would say, 'Repudiating this All, I will describe another,' if questioned on what exactly might be the grounds for his statement, would be unable to explain, and furthermore, would be put to grief. Why? Because it lies beyond range."

Having put it in terms of the all it proves that the problem can only be rightly described in terms of what is seen & thought about and thus in terms of the workings of perception.

The thought experiment itself is of a race analogue to what can be seen. He is essentially here thinking about what is seen, ie seeing people racing. His paradox relies on slipping by the Sabba Sutta, he postulates existence of objects repudiating the all and not subject to the "mechanics" of perception.

What one sees that one thinks about, here one sees the two racing and thinks about it in terms of velocity, distance and etc.

What Zeno does is assuming that perception too arises and ceases in progressively smaller intervals but this is an unsubstantiated assumption because;

  • it contradicts the norm; if there is an analogue race he can't actually percieve 'Now im percepient of the turtle being this far ahead and now i am percepient of it being that far ahead' ad infinitum.

What he'll have to admit is that at some point he won't be able to be percepient of who's actually ahead due to the workings of perception.

Thus the problem becomes that of an inadequate thinking about the workings of perception.

One can here go on to explain how perception is dependently originated and how mind arises as one thing and ceases as another but it's unnecessary i think.

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  • I didn't understand the explanation, i.e. "if there is an analogue race", why he can't (as Zeno claims) perceive ad infinitum, why/how it contradicts the norm, and what that has to do with the Sabbe Sutta. – ChrisW Jul 19 at 9:12
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    The thought experiment itself is of a race analogue to what can be seen. He is essentially here thinking about what is seen, ie seeing people racing. His paradox relies on slipping by the Sabba Sutta, he postulates existence of objects repudiating the all and not subject to the "mechanics" of perception. – deadmanposting Jul 19 at 9:21
  • He is talking about a world outside of a perception frame of reference. – deadmanposting Jul 19 at 9:23
  • I cannot directly see viruses with my own eyes, but I can use a scanning electron microscope to observe viruses. So, I think seeing viruses is still acceptable according to Sabbe Sutta. If the reducing distances are still larger than the Planck constants for space and time, then being able to see or observe the race is more of a question of technology than perception. – ruben2020 Jul 19 at 11:14
  • Fifty years ago, we could not observe gravitational waves, but today we can do so using LIGO. Fifty years ago we could not observe the Higgs boson, but today we can do so using the LHC. Fifty years ago, we could not observe black holes, but today we could do so using the global network of deep space observatories. So, it's about technology. – ruben2020 Jul 19 at 11:20
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The Hero & The Tortoise Paradox etc are more about measurement & distance description: eg, infinitesimal division of distance, walk 10 meters in ten seconds: the time is finite & the distance is finite, yet both the time interval & distance interval may be divided into arbitrarily many pieces(which dont even have to be identical in magnitude); cf an existential goal: may arrive there after a while, but the time interval and progress interval can be divided into infinitesimal bits(which dont have to be identical in magnitude). Maybe SN51.15 would be an example of an illustration of this, where the distance is a different sort of path than literally a sprinting racetrack. Maybe a key point in both would be that progress doesn't have to be measured in ever smaller increments, and that a result is a result, however the interval is measured. And the Hero passes the tortoise after a certain time interval: the measurement of either the time or the distance can be divided into arbitrarily many small bits, but the whole amount is the whole amount, sort of like a meter ruler: it's still a meter whether the divisions are in cm or mm or microns, etc.

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The canonical Buddhist answer will be from Acintita Sutta:

"Conjecture about [the origin, etc., of] the world is an unconjecturable that is not to be conjectured about, that would bring madness & vexation to anyone who conjectured about it.

And also the Parable of the Poisoned Arrow from MN 63.

In physics, this is answered using the Planck constant for time and space.

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