Zeno’s paradox revisited

Sitting in Tokyo over an innocent bowl of sake, my philosopher friend from London brought up Zeno’s paradox. Damn philosophers. Always something. He knows more about ancient Greek philosophy than I do, but let me have a go at it from the poet’s side of the field.

Space

I guess Zeno’s paradox (5th century BC) comes in various forms, but I think of it in terms of space. Paradoxically, mathematically, motion is impossible. To move from point A to point B, we have to cross midpoint C. But to move from point A to midpoint C, we have to cross midpoint D. Etc. But since simple geometry tells us that there are an infinite number of points between any two points, we can never get to the nearest midpoint.

Or forget about midpoints. In order to move we must cross an adjacent point. But there are an infinite number of points between us and any adjacent point. In today’s computer programming lingo, we’d have to execute an infinite number of tasks before reaching the adjacent point, which is impossible.

There are only two conclusions I can draw from the paradox. Either it shows us that motion is truly impossible or it shows us the limits of logic – that logic can solve a lot of local problems but there are points at which it fails as a conduit of knowledge and results in an absurdity.

Time

My tipsy interlocutor pointed out that the paradox works along a time axis as well. The idea of a linear flow of time is equally impossible, as we’d have to move past the adjacent moment, which is impossible. However, conceptualizing it along a time axis opened a different tangent of thought for me.

My more devoted readers will note that I’ve looked at the following William Faulkner quote HERE as a way of theorizing time:

“The past is never dead. It’s not even past” (Requiem for a Nun, 1951).

Pondering the Faulkner quote led me to consider that our conventional way of looking at time – with the past as a thread disappearing into some distant place that no longer exists – is actually counterintuitive. Doesn’t it make more sense to see the past as something very much still with us, but at a depth, providing the real-time substructure of the present, just as the rings of a tree do not disappear as years go by but rather continue to provide the real-time substructure of the tree? In the same way, the “past” is not gone, but is right here, at a depth, providing in real time all the folds and substructure without which the present would collapse.

So if Zeno’s paradox suggests that we cannot move along a linear path of time, does the tree ring model of time show us a way out of the paradox? On the one hand, it seems to do so, as it shows we can conceptualize the manifold of time without requiring a linear flow. On the other hand, we still need some kind of wiggle room, as time, though not extending backward into some now-absent past, does recede to the center (of the tree) or the depth (on which the present stands). Would Zeno be able to grant us so much without giving up his precious paradox? To untie this further knot in the fabric, we need a to add a third category to space and time. And here it comes …

Imagination

Kant, in the Critique of Judgment (1790), speaks of the dynamical and mathematical sublime, and makes a rigorous case for the power of human reason as the sublime human faculty. In the mathematical sublime, for example, we might look up on a starry night and imagine how many stars are up there. The imagination, however, can only stretch so far and is overwhelmed by the sheer numbers in the scenario at hand. Reason, however, can step in and calculate numbers beyond what the imagination can fathom (estimating that there are something like 1024 stars in the universe). It is reason that inspires the highest awe in Kant.

Now let’s use Zeno to turn Kant on his head. Reason leads you down the rabbit hole of Zeno’s paradox, and there you get stuck. No motion. But where reason folds into absurdity, imagination steps in and liberates us. We imagine ourselves in motion. We imagine ourselves moving through time. And if reason can’t back that up, that’s reason’s problem. And if the flow of our experience into the future is an imagined flow, so much the better. Without imagination, perhaps Zeno’s paradox would hold. Reason is trapped in what is; and what is, is fixed. The world as a static object of knowledge. But imagination is the one faculty that allows us to project and manifest all manner of possible futures. Imagination creates destiny, and imagination is what moves us toward that destiny.

So philosophers and scientists, keep up the good work but go to the back of the bus. Poets, artists, and mythmakers, move forward.

“In art and dream may you proceed with abandon” (Patti Smith)

“Imagination is more important than knowledge. For knowledge is limited, whereas imagination encircles the world” (Albert Einstein)

“Artists are here to disturb the peace” (James Baldwin)

Now for my real poetry, click the book cover below.

mountain lantern light
breaking through bamboo and ice
a thousand angels

* * * Click covers for more links * * *

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8 thoughts on “Zeno’s paradox revisited

  1. Because of man’s continuous drive to seek answers that can withstand the test of time, man’s dream of contradicting each other will rise to the challenge to dispute what had previously been established as truth. If one is willing to share common ground without voicing doubt most likely indicates, one is too complacent to consider more challenging options.
    The strength of the curious mind lies in its willingness to revise its theories, and only through the analysis of our own experiences can we answer a question truthfully.

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  2. “…there are points at which it fails as a conduit of knowledge and results in an absurdity.” I couldn’t agree more with that statement, but then I loved your description of time and the past being present in “depth.” What a wonderful characterization. It feels exactly like that to me, and I’ve never thought about it that way – deep instead of linear. I’m a little mind-blown by that perspective. And imagination… speaks for itself. Einstein was an avid supporter of the power of the imagination, which I’ve always found so interesting. I love his quote and think of often while debating with empiricists. 😀 Great post. Thanks.

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  3. My husband was just telling (reminding) me a few days ago that nothing we see around us is solid matter. It’s all just vibrating at such a level that it appears to be solid (sic). Now I have to muse on the nature of time and space? lol. I guess it keeps the synapses healthy at least, right? 🙂

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    • My understanding is this. Based on standard atomic theory, space contains particles (which have mass — i.e. matter) and waves (which are pure energy with no mass). The number of particles in any solid body suggests that that body is about 99.9999999999996% empty space. Quantum physics revises that to suggest that the model of the universe as empty space with lots of tiny particles inside, or as separable units of mass (particles) and non-mass waves, is outmoded. The “stuff” of the universe has both characteristics of what we used to call “particles” and characteristics of what we used to call “waves.” Really, it’s more like a smear across space, with no clear division between space that is empty and space that is occupied by particles. Thus, to ask how much of a solid object is empty space is to ask a meaningless question, since neither “solid object” nor “empty space” are actual features of reality. But I am struggling here. Your husband is the better source, since he said in 15 words what took me 500 🙂

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  4. Finally got around to reading this. As I kinda mentioned, am not sure it’s a limit to logic per se as much as thinking/reasoning, insofar as they are differentiable, which to some extent I would contend that they are. Conversely, is imagining something distinct from thinking/reasoning about something? Imagining some things, yes, others, no?

    Finally recalled my “solution” to the paradox. Movement is clearly possible since X is now at point B instead of point A where it was previously. Doesn’t matter if it’s static at any point in between A and B. Actually the first time I ever thought about it, tho, so probably easily shot down.

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    • My reaction to your response reminds me of Wm. Blake’s reaction to “Paradise Lost.” He said that Milton’s vision of divine reality was rich and accurate (corroborated by Blake’s own visionary experience), but for one mistake: Milton had accidentally named Satan “the Messiah” and accidentally named the Messiah “Satan.” Your comment makes good sense to me if I reverse your terms, “logic” and “thinking/reasoning.” To wit, the paradox shows the limits of logic (as it is logic that tells us that we cannot move without crossing an adjacent point). However, reasoning is a larger dialectical process that has logic as one of the mechanisms under its scope. From this point of view, we might reason out that motion is possible, even though a narrow application of logic precludes it. Reasoning can give us the bigger picture.

      However, it is still not clear to me that reason can accomplish this feat without an act of imagination, as I can’t elucidate any clear rational principle that might explain our escape from the paradox. So now the relationship between reasoning and imagination becomes the point at issue.

      Meanwhile, your “solution” seems no more than the common sense rebuttal that Dr. Johnson gave to Berkeley’s theory that the material world was an illusion. You might recall that Johnston kicked a stone with force and said, “thus I disprove it.” Thus, likewise, Jabunta. Movement is possible because we clearly see it happen … or did we imagine that it happened? Might it be that imagination is the force that rolls through all things and makes all things happen, with the machinery of reasoning as a lagging indicator, and poor pitiful logic still further back? Something for you and Dr. Johnson to ponder while Boswell and I have a pint and a game of backgammon.

      Now throwing everything I just said in the garbage, if I try to break the paradox without (overt) recourse to imagination, I’d have to start with Seliza’s comment above. Classical physics modeled the universe as empty space filled with tiny points or particles. This world view is a premise of Zeno’s paradox. But quantum physics suggests that the model of the universe as empty space with lots of tiny particles inside, or as separable units of mass (particles) and non-mass waves, is outmoded. The “stuff” of the universe has both characteristics of what we used to call “particles” and characteristics of what we used to call “waves.” Really, it’s more like a smear across space, with no clear division between space that is empty and space that is occupied by particles. So Zeno’s paradox only exists in that clumsy, fictional world of some vast space filled with billions of discrete points. That, my dear Thrasymachus, is an imaginary world, and Zeno’s paradox is as trapped in it as your countryman’s Alice is in her looking glass. You and I, our world is not so imaginary, is it?

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