Simple Solutions to Zeno’s Paradoxes?

This post is a celebration of nerdery; it’s a sign that our hard labors are nearing their end for the season. I need mental exercise to keep my soul content much like Dani needs physical exercise to keep her body content.

I got plenty of exercise in my tasks of our divide-and-conquer homesteading efforts this season, so I’m beginning to revel in sitting my butt on the couch, reading, and thinking. I charged up my ancient Kindle (for other geek reasons, which I will not get into here) and found on it somehow Richard Dawkins’ The God Delusion. I’ve listened to him lecturing and speaking countless times. But I had never read any of his books so I thought I’d give it a try.

I quite enjoyed it. With clarity and wit he tackles the ponderous subject, making it approachable and digestible. Definitely worth a read if that sort of thing interests you. Dawkins mentioned Zeno’s Paradoxes at one point and I had to pause my reading to go off on a Zeno’s Paradoxes rabbit trail. To date the only person I’ve heard discuss them was Alan Watts (who focused on the race between Achilles and the tortoise). These puzzling word-pictures have stimulated minds for millennia. At first blush they seem like they must be wrong. But it can be a bit of a challenge to say in what manner specifically they are wrong.

I mulled the three most famous ones over (discussed in detail in the Wikipedia article linked above ). I resolved them in my mind and decided to see what other solutions had been offered. To my surprise most explanations are strictly mathematical and usually fairly (i.e. unnecessarily) complicated. A little more digging revealed no published solutions like mine.

Since I have not seen a written record of the following solutions to the Achilles/Tortoise and Dichotomy paradoxes, I would like to offer them as examples of how understanding and insight into even challenging academic problems can be achieved; and often much more simply than one might suspect. Insight and understanding can strike anyone and any time.

The paradox goes something like this:

Achilles can run 100 times faster than the tortoise. They decide to have a race. Because Achilles is so much faster, he gives the tortoise a 100 meter(or whatever) head start. They start the race. When Achilles runs 100m, the tortoise has covered an additional 10m, and is still ahead. When Achilles covers that new 10m distance, the tortoise has again covered new ground. The trend continues: each time Achilles covers the distance the tortoise has already traveled, the tortoise has covered new ground, (however diminishingly small) in that same time. So the tortoise is always in the lead and Achilles is always catching up. Therefore, Achilles can never overtake the tortoise and win the race.

That’s the famous paradox. It feels wrong, doesn’t it? It seems to fly in the face of obvious experience. But it also seems very soundly argued. If it is false, where is the mistake? If you don’t want me to spill the beans, stop reading here and go think about it. Can you come of with a solution? I promise you there are more than one! Most people feel they know the right answer: of course Achilles can overtake any tortoise. So where is the mistake in the argument? Pause and ponder. Then read on!

I was surprised to find only complicated mathematical solutions out there. The way I’ve worded the paradox maybe you’ve already thought of the following one, which I think is the simplest: stride length.

At each moment of time both characters are moving forward. But each is moving with their own stride and rhythm. When the distance between them becomes about the length of Achilles’ stride, things cease to behave like the paradox’s narrative would lead us to believe. As Achilles catches up to the rear of the tortoise, they both continue to make strides. Even if we allow for the tortoise to make more rapid strides than Achilles (which is very unlikely) the difference in stride length is the most important factor. In a single stride or two (with a stride length of 1m-2m) Achilles will pass the tortoise no matter how fast its legs are moving (within reason) as each tortoise stride – what do you suppose a tortoise sprinting stride looks like? – is going to move it only a few centimeters ahead.

In Zeno’s “Dichotomy” paradox, something similar is afoot. Using the narrative from the Wikipedia article:

“Suppose Homer wishes to walk to the end of a path. Before he can get there, he must get halfway there. Before he can get halfway there, he must get a quarter of the way there. Before traveling a quarter, he must travel one-eighth; before an eighth, one-sixteenth; and so on. This description requires one to complete an infinite number of tasks, which Zeno maintains is an impossibility.”

Take a break if you like and think about it. Can you find a problem with this scenario? I found many refutations that, again, were more mathematically complex that necessary. Mine is not complex at all.

If Homer wishes to walk to the end of a path, he will begin by taking a single step. No matter how long the path or how short his stride, in that first step he will complete “an infinite number of tasks” to use the wording above. Divide Homer’s stride length by the length of the path. That fractional length of the path covered will be gargantuan compared to the infinitesimally small fraction Zeno wants us to start with (effectively any number divided by infinity).

The addendum narrative offered in this write up of the paradox is:

“This sequence also presents a second problem in that it contains no first distance to run, for any possible (finite) first distance could be divided in half, and hence would not be first after all. Hence, the trip cannot even begin. The paradoxical conclusion then would be that travel over any finite distance can neither be completed nor begun, and so all motion must be an illusion.”

Preposterous. Only a fool (and more will be said on this later) would consider language more real that the real world. Of course it contains a first distance: take a step. If Zeno can say “No, no! That first step doesn’t count because I can subdivide the distance it covers.” Then I can say “No, no! First means first. A step taken is a binary event; it either happens or it doesn’t. If it happens, it covers a distance.” If it happens and you say “It didn’t actually happen because it’s an illusion.” then I can say “You’re delusional and not worth listening to.” Zeno’s assertion is comprised of only words, and there is no evidence to support his position; only more words. If that is sufficient “evidence” to support his thesis, then I can successfully rebut his thesis with mere words: “It’s wrong.” Further I can back up my rebuttal with physical evidence. Again if he protests saying reality is an illusion (note he cannot offer any evidence of this), he’s either delusional and can be ignored, or he’s right and there’s not point in having the conversation in the first place. More will be said on that later, as well.

Further, that first step can not only be divided, but has an infinite number of potential subdivisions within it that can be “accomplished” in the process of taking just the one step. Therefore the first step does accomplish an “infinite number of things” quite easily.  To conclude, whatever that first step accomplishes, however small, it is not near zero. Divide the path length by Homer’s stride length and you can even know how many steps it will take him to walk this path of “infinite tasks.” Whatever the number is, it is not infinity.

I will admit Zeno’s third paradox (the arrow paradox) does actually require some more complex math (or at least some very deliberate defining of many words) to resolve. But the manner of Zeno’s confusion (or delusion) is the same. But it most assuredly can be resolved. Can you find a way to resolve it?

If you know the definition of a derivative in calculus and understand a little about measuring position (i.e. distance) with respect to time, a solution is obvious.  Likewise linguistic resolutions can be found by paying special attention to terms like position, time, instant and motion.

I hope you have enjoyed the distraction these ancient puzzles offer. I enjoy their consideration well enough, but they are also aggravating for several reasons. First they’re abysmally wrong. Zeno’s shared Parmenides opinion that there was no distinction of separate things (i.e. monism), there was no such thing as “change” and, consequently, motion was an illusion.

The first point can be quibbled about all day long, but the rest is not only wrong, but completely, backasswards, dead wrong (Parmenides and Zeno would agree their ideas fly in the face of everyday experience, but they were sure they were right). The error is the conclusion is built on unexamined premises. Another point of irritation is no critique seems to clamor for examination of these premises. Greek philosophy is replete with erroneous thinking firmly grounded in unexamined, false premises (the Platonic idea that “thought” is more “real” than anything physical, for instance).

The Greek philosophers were misguidedly obsessed with dividing things up. From Democritus’ atoms, to Zeno dividing distance (or time in the case of the arrow paradox), to the dividing of body and spirit, form and matter; division seems to be on the collective ancient Greek brain. I’m not sure why, perhaps overactive cognition in the same vein that gives rise to language?

Greek philosophy also happens to be the foundation of western philosophy; that foundation, unfortunately, is often factitious and/or erroneous (e.g. Aristotle’s false dichotomy of form and matter, most of what the Pythagoreans espoused). This laid traps that continue to plague us today and therein lies the next point of aggravation. For over 2,000 years the flawed roots of Greek philosophy guided the development of western thought. Additionally bolstered by (tacit or explicit) acceptance of Abrahamic theology, this has left us moderns enmeshed in a tangled net of miscalibrated cognition.

I am a pragmatist by nature (and not pertaining to moral philosophy). Unfortunately the trajectory of Greek philosophy has yielded so many lines of western, academic philosophical thinking that amount to arguing about how many angels can dance on the head of a pin. To all these questions I have to say: Who cares? Not only is it a poor use of time and energy to talk about it, why does the conversation so rarely turn to address the question “Are there any compelling reasons to believe in angels in the first place?” Surely that should be the initial point of discussion. But it isn’t, and quickly a heated discussion ensues regarding the potential nature of angels in order to shed light on how many can dance on the pin…

For instance some philosophers have argued that reality is nothing more that the dream of someone or something other than ourselves. There are threads of this from Plato to Descartes. Bostrom’s “Simulation Argument” is a contemporary example. I’ve heard heated discussions along these lines from amateur students of philosophy on more than one occasion. Such an assertion is pointless to discuss for multiple reasons. Why argue? If you want to put forward the premise that we are all living in the “perfect” dream (or simulation) of something else, it’s easy to frame things in such a way that it’s impossible to falsify the thesis. So be it. Even if you were forced to conclude that the dream/simulation is possible (or very likely as one interpretation of Bostrom’s simulation argument can yield), what does that mean for the way you live your life? It doesn’t mean anything! And this is just one question garnering a lot of thought and attention.

Is there a way out of this discursive morass? I think so. When I was first exposed to eastern philosophy I very quickly realized it’s wisdom and practicality; I lamented I hadn’t been exposed to it sooner. I think the entire western world would benefit greatly from a healthy steeping in it.

Zeno argued there’s no such (real) thing as change and motion is an illusion. Eastern philosophy argues there is nothing but change; there is nothing immutable in the cosmos. Science has only (relatively) recently corroborated this ancient point of eastern thought. Motion is change. If Zeno and Parmenides might argue that all perceived reality is an illusion and (along with Plato) that our thoughts and ideas are the ultimate reality, Eastern philosophy says the one thing you can’t rely on are your thoughts and that the most important illusion to overcome is the illusion self.

The most notable work in western philosophy (as opposed to a scientific practice like psychology) in regards to the self was Descartes’. Doubting to the extreme he concluded “Cogito ergo sum.” But for all his heroic doubt, Descartes didn’t go far enough. In a way he hit bedrock, but – as if successfully employing Douglas Adam’s instructions on how to fly – ultimately he missed. He took the fact of “thoughts arising” as the positive evidence that he (his ego self) existed. So close, yet so far away.

I don’t speak Latin so I can’t render the truth he could have reached in Latin. But in English he should not have concluded “I think therefore I am.” He should have concluded “There are thoughts.” or more simply “Thoughts are.” Eastern philosophy is completely at home in this untethered (from the standpoint of the ego) reality. It’s not reality that’s the illusion; the illusion is idea that the thoughts have a thinker, the experiences have an “experiencer”, or the senses have a “senser.”

Psychology has yielded some illuminating insights into the nature of the self (e.g. Jung’s Individuation). But much like the Greek forerunners, the insights often come shackled to large amounts of dross.

Lest anyone think I am erroneously arguing from (eastern philosophical) authority, let me offer the following: A wise man once said that we should never accept anything just because it was told to us by an authority figure. He said we should take ideas, test them for ourselves and then reach our own conclusions. I can’t think of a better rule of thumb. The man who offered this sensible advice is today called the Buddha.

Buddha taught dharma. Alan Watts argued that the best translation of dharma in English is not “Law,” nor “doctrine,” but “method.” The more I have dug into this the more I have come to agree with it. The Dalai Lama wrote in The Universe in a Single Atom how application of dharma and how employing the scientific method (especially as espoused by Karl Popper) are very similar. The main difference is while scientific inquiry is directed from the observer outwardly, Buddhist contemplation is directed inwardly.

It’s not a perfect equivalency, of course, but the point is well taken. Depending on your philosophy-of-science background there are certain rules that must be followed when formulating hypotheses. In Buddhism, there are much more lax rules, or the lay person can practice introspection and observation without hypotheses altogether. Hypotheses aside, the central method is: careful observation, noting the evidence, and then following where the evidence leads regardless of your preconceptions.

Of course good things have come from western philosophy. Science and democracy are awfully neat, for instance. And eastern philosophy is the source of some pretty silly ideas, too (arts for the preservation of “vital energies” or alchemy, for instance). But on the whole I find eastern philosophy to be far more salient, reasonable and, frankly, helpful in everyday life than western philosophy.

Both have their places and both exude their influences. But in a world defined by the clashing of cultures as globalization sets in, the more grounded, reasonable, salient, and pragmatic philosophy can be, the better. Ecology is a western science. But aboriginal peoples across the globe and across time were well versed in it. Eastern philosophies of Taoism and, in turn, Buddhism (Chan and Zen especially) tapped into these deep truths as well.

But those truths are all too often lost in the hustle and bustle of civilization and progress. Will global humanity will attune to the reality of an endlessly-interdependent existence? Or will we stamp forward in ignorance and, quite possibly, remove ourselves from the face of the planet? Time will tell.

If we care for others (both today and the future generations of them) it would behoove us to cultivate a sustainable here-and-now and then pass the art of that cultivation on to the next generation.


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