Just thought i could create this thread, given i am presenting those 9 remaining paradoxa anyway. It is likely there were more, but only these were reproduced in texts we have, mainly in Aristotle's physics
The thread is not just meant for information presenting, but also any discussion you may feel like having on this subject..
The Paradoxa are named thus not because Zeno deems them themselves as something paradoxic (i mean he was not of the view they aren't valid), but due to his work with them being aimed at 'presenting that the enemies of the view of Parmenides (that all is One) inevitably have more paradoxes in their own view that there are Many things and not One'.
Although in general one can categorise those paradoxes in terms of the focus being on volume/space, time or motion, in reality they are all about the core ideas of Infinity and The Single point, as is the rest of the eleatic philosophy.
Zeno is speaking a bit about that lost work of his in the dialogue between Socrates and Parmenides.
An important note before i start the descriptions of the paradoxes: Zeno (and Parmenides) do not argue that we sense things in this way, but that our senses are illusionary anyway, and therefore it is more important to focus on what the ideas tied to those senses of movement, space and time are. This follows both from the 120 surviving lines of Parmenides' own work, titled "On Nature", and second-sources such as Plato's "Parmenides".
Ok, here are the 9 surviving paradoxa now:
1) The paradox of Dichotomy
If one has to get from point A to point B (eg in a line), he will first have to get to half the distance between A and B. But prior to that he must reach half of that half distance. And so on. So there is an infinity of distinct spots to be reached.
2) The paradox of Achilles
If Achilles was in a race with a tortoise, and the tortoise was allowed to have a small headway prior to the start of the race, Achilles would still be behind the tortoise for an infinity of distinct moments in time, or parts of that course. This is so given that at the start the tortoise is at point A, and Achilles at point A-X. In any next moment when Achilles has reached the original distance (ie in moment #2 he will be in X) the tortoise will have moved for a smaller bit past that original distance, and thus an infinity of such moments is there if focused upon.
3) The paradox of the moving groups (so called 'the Stadium', cause that would be where it took place)
If we have three groups of equal length, A,B,C, and one of them is immobile (B), and A moves to the opposite direction that C does, then A will have moved past C at half the time it took for it to move past B. This highlights that time was relative to the other parameters set, so if it was counted as a measure of movement for a distance then that distance itself could always have qualities altering the count (in this case the distance A=B=C being surpassed in both X time (for the immobile B) and X/2 time (for A or C which are moving to opposite directions to each other). The paradox is about relativity of setting a paragon (time in this case) as 'independent' of other parameters accounted or not.
4) The paradox of the Position (or of the Void)
If something can move then it follows that 'it moves to somewhere it was not already, and from somewhere it no longer is', and thus the position it takes cannot be tied to any place it moves, cause it does not grant position to that place nor does the place grant it a position it moves with. The argument concludes with the note that 'for a position to exist it has to have itself an external to it, other position, and that position of a position to have another position, and so on'. So this paradox is close to the previous one, about the setting of paragons in a system which seems to not include its own set things in the first place, and external categories or factors will be needed.
5) The paradox of the still arrow
Anything which occupies at a point of time an equal space/volume to its own at rest, is again at rest in the distinct point in time. So an arrow in movement is still immobile for any theorised part/instance of that movement. (This remnant of a paradox seems to allude again to a lack of an overarching/external system next to which the phenomenon of movement we note with our senses would logically have to be viewed as movement).
6-7-8) The Paradox of Size, and Finite or not, if there are Many things and not One
a)Zeno argues that if there are indeed many things, and not just all being one as Parmenides claims, then those "many" things must be always a finite number. But if we assume there are a set or finite number of things then there would be no variations of them or others in between them, unless the original set is not part of the subsets, in which case the many things are infinite anyway. (a simple example from arithmetic being that while in the group of positive integers there are just two in 1-2, there are endless numbers of decimals in between already).
b)If there are many things and not one then each thing must have volume, and thus have also edges to that volume. But if something has an edge as a volume then that edge will also have its own edge, and so on, and therefore if indeed there are Many things they must also include things which have No volume at all, and also things which has Infinite volume, cause only this way can seperation into many hold true. But if something has nothing of a quality then it is (absolute) zero, and zero not existing is a core Eleatic argument. (this would need elaboration, juxtaposing it with the atomic theory of Democritos and more aptly Anaxagoras, given in the later something can have an infinite amount of different constituent particles, which often border absolute zero).
9) The paradox of the falling grains making no sound
Given that while a large amount of something when falling will make a sound, while a single grain or less will not, it is paralleled to adding absolute zero to arrive at a non-zero amount. While the paradox is not about the trivial note that the human ear will not pick up sounds after some point, it seems tied to the idea of some inherently indistinct barrier/limit point, which itself is not factored in the phenomenon itself (if viewed as a parable it can be parallel to one noting that one was trying to find the ratio of sound to single grain, but the answer was in a wholly different, external order, such as sound-sensory qualities).
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Ending note
While sadly most web pages fall for the wrong (and boring/simplistic/against the original texts anyway) account of the paradoxes as being brought by through absurd lack of noting how the senses work, as i wrote in the start of this thread this argument literally is not worth one's time (finite or not
). Zeno and Parmenides are always about the idea of infinity, not the senses.
But i want to past the following interesting bit i read on the web, while searching for other lists of the paradoxa:
From
http://www.franzkiekeben.com/zeno.html (no idea who he is, but i liked his comments there).
And indeed, if something literally goes on for an infinity of parts, the 'end' of it is no simple matter. Indeed in such an end things are not what they are in any other point of the progression, which is the most interesting in this math examination in my view
(echoes an analogous point in the fibonacci series, related to phi, and
some specific further point in it having to be a rational number despite being filled with phi in multiple ways..).
(for some reason "mate" works here, I'm going with it)
) knows of all concepts, let alone how they would tie into another. I have to suppose (intuitive) that only a fraction of all concepts in math by now are needed for advancement/breakthrough on any single one of the main (known) standing problems (eg prime theories, to use one with a few famous problems attached to it).
