How does one understand and resolve Zeno's paradox?

Question

Zeno, a follower of Parmenides, reasoned that any unit of space or time is infinitely divisible or not. If they be infinitely divisible, then how does an infinite plurality of parts combine into a finite whole? And if these units are not infinitely divisible, then calculus wouldn't work because $n$ couldn't tend to infinity.

Another way to think about it is a flying arrow must first travel half way to the target from where it begins ( the first task), then travel half way to the target from where it is now (the second task), then travel half way to the target (third task), etc... What you get is this...

$$\begin{array}{l} {d_{Traveled}} = \frac{1}{2}d + \frac{1}{4}d + \frac{1}{8}d + \frac{1}{{16}}d + ...\\ \\ {d_{Traveled}} = d\left( {\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{{16}} + ...} \right)\\ \\ {d_{Traveled}} = d\left( {\frac{1}{\infty }} \right) = 0 \end{array} % MathType!MTEF!2!1!+- % faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj % 2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0x % bbL8FesqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaq % pepae9pg0FirpepeKkFr0xfr-xfr-xb9Gqpi0dc9adbaqaaeGaciGa % aiaabeqaamaabaabaaGceaqabeaacaWGKbWaaSbaaSqaaiaadsfaca % WGYbGaamyyaiaadAhacaWGLbGaamiBaiaadwgacaWGKbaabeaakiab % g2da9maalaaabaGaaGymaaqaaiaaikdaaaGaamizaiabgUcaRmaala % aabaGaaGymaaqaaiaaisdaaaGaamizaiabgUcaRmaalaaabaGaaGym % aaqaaiaaiIdaaaGaamizaiabgUcaRmaalaaabaGaaGymaaqaaiaaig % dacaaI2aaaaiaadsgacqGHRaWkcaGGUaGaaiOlaiaac6caaeaaaeaa % caWGKbWaaSbaaSqaaiaadsfacaWGYbGaamyyaiaadAhacaWGLbGaam % iBaiaadwgacaWGKbaabeaakiabg2da9iaadsgadaqadaqaamaalaaa % baGaaGymaaqaaiaaikdaaaGaey4kaSYaaSaaaeaacaaIXaaabaGaaG % inaaaacqGHRaWkdaWcaaqaaiaaigdaaeaacaaI4aaaaiabgUcaRmaa % laaabaGaaGymaaqaaiaaigdacaaI2aaaaiabgUcaRiaac6cacaGGUa % GaaiOlaaGaayjkaiaawMcaaaqaaaqaaiaadsgadaWgaaWcbaGaamiv % aiaadkhacaWGHbGaamODaiaadwgacaWGSbGaamyzaiaadsgaaeqaaO % Gaeyypa0JaamizamaabmaabaWaaSaaaeaacaaIXaaabaGaeyOhIuka % aaGaayjkaiaawMcaaiabg2da9iaaicdaaaaa!7035! $$

But suppose we wish to calculate the area below a function between $a$ and $b$ say, the bars that compose this area consist of taking a reference point on the first bar $f(a)$, multiply it by $dx$, then using the slope $f'(a)$ as a guide, "jack up" the reference point onto the top of the next bar, multiply by $dx$, jack it up, multiply by $dx$, and repeat this until we reach the final bar (L.H.S.). The summation of all this yields the exact area.

So, it's like taking the line segment $ab$ and dividing each piece over and over again. Per division, the sizes of the pieces are half of what they were before, but there are twice as many of them as before; but as the number of divisions tends to infinity (n tends to infinity), they diminish to almost nothing such that when added back together, they still equal the length of the original line segment $ab$.

How does one understand and resolve Zeno's paradox?

Answer 1

Zeno's paradox is called a paradox exactly because there is a mismatch between a seemingly logical argument that concludes that motion is impossible, and our experience in dealing with reality, which says that there is motion.

To resolve the paradox, then, you need to figure out where the argument goes wrong. Saying that in your experience motion exists does nothing to get rid of the argument. Indeed, rather than resolving the paradox, when you flap your arms or drop any balls, you emphasize the paradox!

Finally, let me also add that with his argument(s) Zeno most likely wasn't trying to conclude that motion doesn't exist, but instead offered his argument as a reductio ad absurdum against the idea that space is infinitely divisible: If space is infinitely divisible, then [insert typical Zeno story here] motion becomes impossible. But since [flap your arms now] there is motion, space cannot be infinitely divisible.

So, you being able to drop a ball to the ground is now part and parcel of the argument! And if you want to reject the conclusion that space is not infinitely divisible, you need to show how motion is possible in such a space. Dropping balls doesn't demonstrate such a thing, because Zeno will simply say: you were able to drop the ball exactly because space (as in: the real, physical space of the world we live in) is not infinitely divisible.

Answer 2

Your response is mathematically wrong, but intuitively not that far off. The key is to note that the error in the 'paradox' is:

an infinite many tasks must be performed [CORRECT] ... an infinite many tasks to perform can never be completed [WRONG]

To get a better understanding of what exactly is the error, one must ask:

WHAT exactly is a task?

If each task needs you to expend a certain minimum time/energy to do it, then you cannot do infinitely many separate tasks. However, if the requirements of the tasks overlap, then it can certainly be possible to do infinitely many of them in some situations like Zeno's:

Task 1: Go from the start to the end point.

Task 2: Go from the start to the $\frac12$ point.

Task 3: Go from the start to the $\frac13$ point.

$\vdots$

Clearly, we can start doing all the above tasks at the same time, and eventually will complete all of them. In fact, after any non-zero amount of time (after starting), we will have completed all except finitely many of them.

Another possible definition of "task" is simply as something that you have to make true. Under this definition it is obvious that infinitely many tasks may be possible to achieve:

Task 1: You have reached the end point.

Task 2: You have crossed the $\frac12$ point.

Task 3: You have crossed the $\frac13$ point.

$\vdots$

If it is still not clear why these infinitely many statements can be made true simultaneously, simply rewrite them:

$x \ge 1$.

$x \ge \frac12$.

$x \ge \frac13$.

$\vdots$

If you set $x = 0$ at first, they are all false. If you then set $x = 1$ they become all true. You have successfully achieved infinitely many things!

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Nevertheless it is important to realize that mathematically you cannot talk about dividing a line segment into infinitely many pieces and adding them all up or whatever. There is no such thing. In mathematics what you can do is to consider limiting processes. This is why the Riemann integral has to be defined by a limit, not by adding infinitely many infinitesimal bits.

Answer 3

In this specific case, there is a nice argument using geometric series. However, as you will see this answer may be quite unsatisfying.

A geometric series is an infinite series in the form

$$ f(r) = \sum_{n=1}^{\infty} r^n $$ It is easy to prove that geometric series converge for $ r \in [0,1) $ to $$ \frac r{1-r}. $$ In your case, you have the expression, $$ d\sum_{n=1}^\infty {(\frac 12) ^n} = d \frac{\frac 12}{1-\frac 12} = d\frac{\frac 12}{\frac 12} = d(1) = d. $$ However, Zeno still rears his head if we consider $r \not= \frac 12$. In fact, the function $$ f:[0,\frac12] \longrightarrow [0,1] $$ defined by $f(r) = \frac r{1-r}$ is easily shown to be a bijection; in particular, it is onto. Therefore, we can argue, that by picking different values of $r$ that our arrow will fly any fraction of the expected distance.

Answer 4

I don't think "So, if I drop a ball from my hand, it will just stick there and only appear to hit the floor." is a valid extension of Zeno's paradox.

A more valid one might be, "If I dropped a ball it would never reach the floor, since to do so it must pass through infinitely many steps, which it can never do."

In reality, if we are to say that some distance may be divided into infinitely many intervals, then to do so, knowing that we can pass through a finite distance in a finite time, is to declare up front that it will not be considered paradoxical to pass through infinitely many parts.

Answer 5

Consider the following thought experiment, which I will frame in terms that Zeno would most likely have understood.

Take an urn, and fill it with water. It will contain a finite amount of water.

Now, transfer half the water in the urn to a second urn.

Then transfer half the water in the second urn to a third.

Repeat this process with a purely hypothetical infinite collection of urns.

How much water is present in the infinite collection of urns? Other than the initial filling of the first urn, at no time was any water added to or removed from this thought experiment.

Is it finite because we know how much was present at the start and that quantity hasn't changed, or is it infinite / otherwise impossible to determine due to being an "infinite sum"?