What’s a solution to Zeno’s Paradox that proves math/physics is a viable tool for determining the laws of reality.

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I got into an argument with a friend who says logic and reason alone cannot determine the truth, and that we need emotions too. He says that Zeno’s Paradox is proof of the shortcomings of math/physics in determining the nature of reality. Is he right about this? I thought math/physics are the holy grails for understanding the nature of the universe.

In: Physics

From a purely mathematical perspective, Zeno’s paradox is solvable. We know that the sum of the distances (so 1/2 + 1/4 + 1/8 + 1/16…) converges to 1. This means that from a mathematical perspective, the distance between me and the wall in Zeno’s paradox is a finite value.

From a physics perspective, we can say that the person is moving at a constant velocity. This means that, provided the distance to the wall is finite, we will arrive there.

Your friend is either naive or intellectually dishonest, probably both. He thinks his ability to reason is more powerful than observed reality, and that’s patently false. There are multiple infinities (see Georg Cantor and Set Theory), and the important facet here is that they can be categorized as either convergent or divergent. Cauchy provided us with a formula for determining whether a series converges or diverges. In conclusion: an infinite series can converge on a finite number. To trivialize this as “merely a theory” is overly reductive, that no mathematician or physicist would make of their entire respective fields.

Limits, infinitesimals and infinities generally solve Zeno’s Paradoxes.

The maths behind these wasn’t really sorted out until the late 19th century, which is why Zeno’s paradoxes seemed problematic for so long.

Part of the problem with the paradoxes is that they’re not necessarily stated in clearest terms, and rely on older ideas about space and time.

If anything Zeno’s Paradoxes are awkward because they’re *not* using sufficiently sophisticated mathematics and physics.

Zeno’s paradox doesn’t take the time required to cover the distances into account. Zeno assumes that Achilles takes just as long to cover each half-measure as he did to cover the previous full-measure. From that perspective, *of course* he never catches the tortoise. He’s slowing down the closer he gets to it.

If Achilles runs at a constant speed, the problem resolves itself. He covers each half-measure in half the time. Both the distance *and* the time it takes to cover that distance are going to zero.

Zeno’s premise is that it is impossible to perform an infinite number of tasks in finite time. That’s obviously true when each task requires the same amount of time, but what about when the time required to perform the task is proportional to the size of the task? As the size of the tasks becomes infinitesimal, so too does the time required to perform them.

Zeno takes a finite distance and divides it into infinitely many, non-overlapping segments. The sum of these segments, by construction, must be the original distance. In particular, the sum is a *finite* distance. Does it not stand to reason that we could do the same with time, that the sum of infinitely many, non-overlapping time segments can, at least in some cases, be finite?

Nothing in Zeno’s paradox (which btw was solved) implies that you need “emotions” to explain the world. Not having a logical explanation does not in any way imply there is a supernatural/emotional etc one. It just means that with the current toolset you can’t solve a problem.

> I got into an argument with a friend who says logic and reason alone cannot determine the truth, and that we need emotions too.

That in itself isn’t an unreasonable view. When you start looking deeply at the underpinnings of maths and science, it becomes clear that there is a lot of uncertainty about what exactly they are based on. Many of the fundamental assumptions they make are somewhat vague. One of the most basic and central ideas in science is induction – the idea that you can learn general principles by making repeated observations. For example if I keep dropping objects and they keep falling, I can conclude there is some consistent process that makes every object fall to the ground. But how many observations do I need to make, how certain do I need to be about them, what exactly is the appropriate generalisation (e.g. should I assume that things on the other side of the world fall up or down?) and what should I do if I notice rare exceptions like birds? Arguably our emotions and gut feelings do play a big role in how we decide the answers to those kinds of questions.

> He says that Zeno’s Paradox is proof of the shortcomings of math/physics in determining the nature of reality.

I don’t really see how Zeno’s paradoxes have much to do with any of these issues.

If we’re talking purely about maths, Zeno’s paradoxes were put completely to rest with the development of real analysis. You can construct mathematical systems in which (for example) something travels a finite distance in a finite amount of time but its journey is broken down into infinitely many steps. There is no contradiction there.

If we’re talking about the real world, it’s not really clear how applicable the paradoxes are in the first place. We have no idea if time or space can be broken down into infinitely many pieces. Most likely we’ll never know for certain.

> I thought math/physics are the holy grails for understanding the nature of the universe.

I don’t know if I’d go that far? They’re certainly useful for understanding many things about the universe, but there are plenty of things they can’t tell us yet (what will ultimately happen to the universe, does extraterrestrial life exist, etc.) as well as things they will probably never be able to tell us (is it moral to eat meat, how should governments be run, what is consciousness, etc.).

This is based upon a misunderstanding of science and physics.

Physics does not enable us to understand the nature of the universe. Physics enables us to predict how the universe acts. We create models of the universe as ideas and as mathematical formulas and equations. Those models let us predict how the universe acts. They do not describe how the universe actually works.

For instance Newton’s second law is: The acceleration of an object depends on the mass of the object and the amount of force applied.

We usually write that as F=ma.

It does not say why this is true or give any explanation of how force causes an object to accelerate. It just says that this relationship exists.

We hope that we can understand the nature of the universe, but the test of science is not that it explains the nature of the universe. The test of science is that it is predictive of how the universe behaves.

> He says that Zeno’s Paradox is proof of the shortcomings of math/physics in determining the nature of reality. Is he right about this?

No, he’s not. He misunderstands Zeno’s Paradox. It doesn’t mean that all of physics and math is wrong. Paradoxes are nothing but our lack of understanding how some things work. We know the arrow goes from the bow to the target. We know the Rabbit can outrun the tortuous. The notion that things have to move in the increments Zeno suggests, is not a real thing. Just because one needs to cross the halfway point, does not mean a person is limited to only going that far. So your friend is misunderstanding what is really going on. Sounds like he has some agenda to push, like religion. When people start claiming math and physics are wrong, you know they are probably pushing religion.

The solution to Zeno’s paradox is that infinities are tricky! There an infinite number of steps to go, but each step beceoms infinitely shorter! And so proper mathematics of infinities and limits are needed to be rigorous about it.

In casual discussions, its often left out that there is a rich field of logically consistent mathematics dealing with the limit of things as they tend towards infinite quantities, and those are very well understood.