# Why does Zeno's paradox seem valid but remain obviously wrong?

Look at this argument:

The 1st introduction: We may present time on an axis; therefore any time interval is a segment, thus consisted of infinite number of points, on this axis. The 2nd introduction: At any time, a point on the time axis, the displacement of any particle is zero. Conclusion: The displacement of any particle within a time interval is zero.

But clearly, the conclusion is wrong. What's the problem then? Myself think the argument is valid. There must be something wrong with the introductions.

• You must approach the concept with derivatives and you will understand
– user14065
Mar 16, 2016 at 16:41
• Nothing wrong with these arguments and this is why they will not go away. It's our ideas of time that are problematic. This is what the logic tells us.but we tend not to listen.
– user20253
Feb 29, 2020 at 12:58

The paradox shows that one cannot argue with infinite sets analogous to finite sets.

The correct method to deal with the paradox is calculus. Accordingly one has to integrate the velocity at each each point of time to obtain the distance covered by the arrow

s(t) - s(t_o) = Integral from to_0 to t [ d_tau v(tau) ], v = velocity, s = distance, t = time.

• Isn't calculus based on the idea that space is modeled by an uncountable continuum? Do you have evidence that this model applies to the real world exactly, as opposed to being an approximation? Mar 16, 2016 at 15:28
• @user4894 Calculus builds on the limit concept. In case of integration: The integral of a continous function is the common limit of finite upper sums and finite lower sums. Calculus has proven useful in mathematical physics; take any differential equation. Mar 16, 2016 at 15:35
• @user4894 Calculus normally does make the assumption that space is continuous, but it can easily be reformulated for different types of spaces, e.g. a rational space. See analysis and topology. It's not known whether space is in fact continuous, or if not, what kind of discontinuity it exhibits. General relativity and quantum mechanics make different assumptions about the nature of space, part of why it is hard to reconcile them into an overarching theory.
– Era
Mar 16, 2016 at 16:10
• @user4894 time is also not a continuum under the new theories (you have the jiffy) so the paradox no longer applies, since there's not such POINT in time, but a minimal interval. However, in the universe stated in the paradox, calculus is still an explanation.
– user14065
Mar 16, 2016 at 16:43
• @JoWehler Useful is not the same as true. Integration theory also allows the Banach-Tarski paradox, are you claiming that's a fact of physics too? Mar 16, 2016 at 16:58

I'm not sure why you think

Zenos paradox seems valid but is obviously wrong

Aristotles account of the paradox, is the first sustained thinking on this paradox that has come down to us and still bears thinking on, even today. He reports that the usual answer, which everyone knows, but no-one appears to think that Aristotle knew and understood it, is as he said 'adequate'. But its not the full story.

Aristotle suggested that the problem lay in how we thought of reality and space; the former, in his ontology, was through through actuality and potentiality; and the latter, in his notion of place. (He asked, for example, does place itself have a place?!)

When we consider, than in modern physics, that even the notion of a particle loses its primacy (for example in relativistic quantum mechanics we discover a single particle theory is simply not viable. Instead, one has a many particle theory) and that, again, in the quantum notion of space, the notion of a point place also loses it primacy; we see just how astute the early thinkers on space and time were (much more so, than certain modern, cheap and shoddy thinkers)...