## Zeno's paradoxes in the modern era

• 199
Zeno made several arguments against the possibility of change and this sparked various responses from his contemporaries (Aristotle, for example) and from philosophers in the twentieth century (Russell, for example).

My question is related to what their standing is today?

Does anyone philosopher still think that they prove that change is impossible?

Or have the philosophers and mathematicians solved the paradoxes.

• 2.1k
All of the Zeno paradoxes that I know have the following form:

1. Prove that in order for one's position to change, one must first do an infinite number of 'things'

2. Assert that one cannot do an infinite number of 'things' in a finite time.

From 1 and 2, conclude that one's position cannot change.

The resolution is to observe that, if 'things' is defined in the way that it needs to be in order for the proof of 1 to succeed, there is no reason to accept assertion 2.

I don't think modern mathematics or philosophy, or even calculus, is needed in order to perform that analysis and identify the reliance on the nebulous notion of 'things'. Aristotle's propositional logic suffices.

The current status of the (veridical) paradoxes is what it always has been: they eloquently demonstrate that, when one reasons to a conclusion that contradicts what one confidently observes to be the case, there must be a flaw in the reasoning, and one has to carefully examine it in order to locate it.
• 199
Interesting. In the Stanford encylopedia, they mentioned that Aristotle tried to resolve the paradox by introducing potential infinites. They state:
"However, Aristotle did not make such a move. Instead he drew a sharp distinction between what he termed a ‘continuous’ line and a line divided into parts. Consider a simple division of a line into two: on the one hand there is the undivided line, and on the other the line with a mid-point selected as the boundary of the two halves. Aristotle claims that these are two distinct things: and that the latter is only ‘potentially’ derivable from the former. Next, Aristotle takes the common-sense view that time is like a geometric line, and considers the time it takes to complete the run. We can again distinguish the two cases: there is the continuous interval from start to finish, and there is the interval divided into Zeno’s infinity of half-runs. The former is ‘potentially infinite’ in the sense that it could be divided into the latter ‘actual infinity’. Here’s the crucial step: Aristotle thinks that since these intervals are geometrically distinct they must be physically distinct. But how could that be? He claims that the runner must do something at the end of each half-run to make it distinct from the next: she must stop, making the run itself discontinuous. (It’s not clear why some other action wouldn’t suffice to divide the interval.) Then Aristotle’s full answer to the paradox is that the question of whether the infinite series of runs is possible or not is ambiguous: the potentially infinite series of halves in a continuous run is possible, while an actual infinity of discontinuous half runs is not—Zeno does identify an impossibility, but it does not describe the usual way of running down tracks!"

Why do you think Aristotle invented potential infinite to get out of the paradox?
• 2.1k
Why do you think Aristotle invented potential infinite to get out of the paradox?
Because, while the logical discipline that he had developed was sufficient to identify the flaw in Zeno's reasoning, Aristotle did not spot how that could be done. So he instead opted for a much more elaborate and philosophically controversial approach.

Nevertheless, his counter-argument looks reasonable to me. However, it doesn't seem to me that Aristotle's notions of 'potential infinite' and 'actual infinite' are essential to his chosen rebuttal. It suffices for him to observe that running down the track and marking the ends of every sub-interval is different from running down the track without doing that, and the runner that gets from A to B does the latter.
• 2.9k
Or have the philosophers and mathematicians solved the paradoxes.

No.
• 199
Can you explain further?
• 2.9k
It would appear that in order to move from A to B, one would need to arrive at 1/2 the distance between them, and so on and so on. That paradox is unsolved.
• 199
How do you look at the distinction between potentially infinite lines and actually infinite lines? Are they truly distinct?
• 2.1k
That question can only be fairly answered by an Aristotelean. I am not one, but I think there are plenty on this board. IIRC @Metaphysician Undercover is one (apologies in advance if I have misread your position MU).
• 5.6k
Differential Calculus. Problem solved.
• 271
Differential calculus provides an approximation by substituting a curved line with straight ones like so:

It doesn't solve the problem. It makes the problem so small so that it is no longer visible.
• 5.6k
Before an object can travel a given distance d, it must travel a distance d/2. In order to travel d/2, it must travel d/4, etc. Since this sequence goes on forever, it therefore appears that the distance d cannot be traveled. The resolution of the paradox awaited calculus and the proof that infinite geometric series such as sum_(i=1)^(infty)(1/2)^i=1 can converge, so that the infinite number of "half-steps" needed is balanced by the increasingly short amount of time needed to traverse the distances.

Wolfram.
• 5.6k
An approximation? Your differential calculus is different to mine, then.
• 2.9k
How does calculus help?
• 5.6k
• 7.9k
2. Assert that one cannot do an infinite number of 'things' in a finite time.

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.

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
• 7.9k
That solution begs the question as it takes as a premise that some finite distance can be travelled.
• 7.9k
The current status of the (veridical) paradoxes is what it always has been: they eloquently demonstrate that, when one reasons to a conclusion that contradicts what one confidently observes to be the case, there must be a flaw in the reasoning, and one has to carefully examine it in order to locate it.

Perhaps the flaw is the premise that motion is continuous? Perhaps motion is possible precisely because distance isn’t infinitely divisible.
• 5.6k
But that assumption is made in the description of the problem.
• 7.9k
I’m looking more at the dichotomy paradox. We can’t say that it’s possible to count in ascending order the rational numbers between 0 and 1 by using that calculus as it would beg the question to assert that we can count from 0 to 0.5 in n seconds, from 0.5 to 0.75 in n/2 seconds, and so on.

For the same reason we can’t assert that we can move through the divisions between 0m and 0.5m in n seconds, between 0.5m and 0.75m in n/2 seconds, and so on.
• 2.9k
So we move from considering distance to considering time? We still have infinite time intervals taking place during a finite span.

To escape the paradox one has to change something fundamental about the way we think of motion.
• 5.6k
We still have infinite time intervals taking place during a finite span.

Yep. That's not a problem.

½+¼+⅛+...=1.
• 5.6k
I don't follow what you are claiming here. May I ask, have you studied differential calculus and limits?
• 2.9k
It's not a problem if you let go of the notion that motion is a sequence of events. Otherwise, yes, it's a problem.
• 5.6k
I might ask that more generally. Is it the case that those here who maintain there is a paradox have studied limits and differential calculus? Let's stick tot he dichotomy paradox for this answer...
• 2.9k
Yes.
• 5.6k
SO you understand the idea of an infinite number of steps with a finite sum?

How does that not solve the paradox?
• 2.9k
I don't understand how anyone could think differential calculus solves the paradox while maintaining that motion is a sequence of events.

Odd.
• 5.6k
motion is a sequence of events.

Where's that fit? Are you wanting to treat distance over time as discontinuous? That'd be novel.
• 2.9k
Motion involves a sequence of non-overlapping intervals. That's the basis of the paradox. ??
• 5.6k
Well then, let's drop that assumption. End pf paradox?
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