Does QM offer a solution to Zeno's paradoxes?

[SimplePrimate]

TL;DR

Achilles and the Tortoise
In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.

— as recounted by Aristotle, Physics VI:9, 239b15

I understand that QM identifies the nature of the universe as being granular (discrete?) rather than continuous. In Xeno's race, it seems Achilles problem is he must divide distance an infinite number of times before chasing down that speeding tortoise.

But I understand that 'Plank Length' is held to be a minimal distance in QM. So I'm wondering does this give a 'granular' quality to distance - enabling Achilles to cut his work down by 'rounding up' each time he reaches a halfway point?

 

[Demystifier]

I understand that QM identifies the nature of the universe as being granular (discrete?) rather than continuous.

Actually it does not. Possibly quantum gravity does that, but we don't really understand quantum gravity yet.

 

[PeroK]

Summary:: Achilles and the Tortoise
In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.

— as recounted by Aristotle, Physics VI:9, 239b15

I understand that QM identifies the nature of the universe as being granular (discrete?) rather than continuous. In Xeno's race, it seems Achilles problem is he must divide distance an infinite number of times before chasing down that speeding tortoise.

But I understand that 'Plank Length' is held to be a minimal distance in QM. So I'm wondering does this give a 'granular' quality to distance - enabling Achilles to cut his work down by 'rounding up' each time he reaches a halfway point?

For anyone who has learned modern mathematics, Zeno's paradox is no paradox at all. This is not 400 BCE and we are not limited by the knowledge and mathematics of Zeno and Aristotle. This is 2021 and we have modern mathematics, including analysis, calculus and differential equations.

 

[SimplePrimate]

Hey PeroK, as always I'm happy to discover I'm wrong. Except knowing this is not so useful until I know WHY I'm wrong. If you could at least give me a lead on the where to pin-point the answer I'd be very grateful.

 

[PeroK]

Hey PeroK, as always I'm happy to discover I'm wrong. Except knowing this is not so useful until I know WHY I'm wrong. If you could at least give me a lead on the where to pin-point the answer I'd be very grateful.

Well, there must be plenty of solutions to Zeno's paradox online.

The simplest answer, IMO, is that Zeno didn't notice that adding smaller and smaller time intervals may never add up to a given unit of time. E.g. the next $1s$ cannot be reached by adding $\frac 1 2s + \frac 1 4 s + \frac 1 8 s \dots$.

In other words, Zeno's model of time is not a model that leads to time passing beyond a certain small limit.

If, however, you add $\frac 1 2 s + \frac 1 3 s + \frac 1 4 s \dots$, then you do end up reaching any future time.

The mathematics of this is that the second series diverges, but the first series converges.

https://en.wikipedia.org/wiki/Harmonic_series_(mathematics)

 

[PeroK]

Indeed, a sharper version of Zeno's paradox is that simply the next $1s$ cannot pass.

 

[martinbn]

The paradox can be modified a little. Knowing that the resulution lies in the fact that one can add infinitely many numbers and still get a finite number, say $\frac12+\frac14+\frac18+\cdots = 1$. Then do the following. Switch the light on for half a second, then switch it off for a quarter of a second and so on. After one second has passed, is the light on or off?

 

[SimplePrimate]

OK thanks PeroK, I did of course look it up on line (this is on line too right?)

You are right in that there are many 'solutions'. 'Proposed' solutions at least. Wikipedia alone lists nine that are still in competition. https://en.wikipedia.org/wiki/Zeno's_paradoxes. But I assume these aren't all equally correct solutions.

The problem has been returned to as recently as 2003 in the work of Peter Lynds of 2003 relating to the Heisenberg uncertainty principle.

So for me the problem seems not quite ready to consign to the past.

 

[PeroK]

So for me the problem seems not quite ready to consign to the past.

Then it never will be!

 

[SimplePrimate]

Then it never will be!

That's rhetorical right?

 

[PeroK]

That's rhetorical right?

Here's my thought for the day.

Whenever science and mathematics touch on practical matters there is little if any debate about progress. We can take medicine, engineering, transportation, communication etc. There is no contest between what the ancient Greeks were able to achieve against modern science.

But, when it comes to the fundamentals like mathematics itself, you can believe that there is nothing to choose between the ancient Greeks and modern mathematicians. Ancient Greek dentistry might be out of fashion, but that doesn't stop you believing that Zeno could still stump the modern mathematician.

I believe that it is absurb to draw this distinction and imagine that only in practical matters does modern science hold sway. It makes no sense to believe that we have advanced only in practical matters and in matters of pure mathematics we have not.

Indeed, I might claim a certain intellectual laziness in believing otherwise. I suspect you wouldn't do without the benefits and advantages of modern technology, but you don't feel the need to accept the modern mathematics that underpins such technological progress. It's easier to keep Zeno on the table, rather than put in all the hard work of learning the mathematics that built on what the ancient Greeks started.

In my opinion, Zeno is out of date in the same way that the horse-drawn chariot is out of date. And, as far as this site is concerned, we are here to teach and promote modern science and mathematics as it is best understood. And that best understanding does not have an open question over Zeno's paradox.

That said, there are modern questions about the nature of time, especially in terms of a fundamental theories from which spacetime might emerge. But, this is not something that Zeno knew anything about and recasting Zeno as a player in this arena is an artificial nod to ancient wisdom.

 

[SimplePrimate]

That's OK PeroK, I've not disputed your mathematical solution. I don't hold any of the anti-math, anti-technological views you have ascribed to me. I don't champion the Ancient Greeks

These maybe valid concerns, but much better directed to those who hold these opinions.

 

[vanhees71]

But, when it comes to the fundamentals like mathematics itself, you can believe that there is nothing to choose between the ancient Greeks and modern mathematicians. Ancient Greek dentistry might be out of fashion, but that doesn't stop you believing that Zeno could still stump the modern mathematician.

That's also historically not right. Even if you forget about the tremendous extension of mathematical topics discovered (or invented?) since the ancient Greeks, there has been a lot to be discovered about an as classical subject as Euclidean geometry. Famously Hilbert figured out where the gaps of the ancient understanding of this subject were, particularly the need for a more definite and rigorous treatment of the "continuum properties" for 20th century mathematics as compared to ancient mathematics.

 

[PeroK]

That's OK PeroK, I've not disputed your mathematical solution. I don't hold any of the anti-math, anti-technological views you have ascribed to me. I don't champion the Ancient Greeks

These maybe valid concerns, but much better directed to those who hold these opinions.

That's fair enough. I apologise.

 

[SimplePrimate]

That's fair enough. I apologise.

Thanks

 

[Nugatory]

And with an accepted apology we will close the thread.