# Singing bottles, negative kelvins and resolving Zeno’s paradox

Picking up hiking keeps leading to beautiful conversations and thought experiments on the way. Yesterday, on our longest hike so far (over 8 km, partially in the sand), as we passed wild cows and Icelandic horses roaming free, it was a treat to hear Simon think out loud about all of these curious things:

Why his water bottle sings at a rapidly lower frequency after he takes a sip but why the frequency doesn’t change as dramatically when the bottle is half-empty? Is it because the frequency wavelength increases exponentially with tone?

Do ordered particles inside an infinite line of a laser beam mean laser has negative kelvin temperature? Or is its temperature undefined?

And as we got seriously off track as opposed to our original route, Simon began contemplating about objects catching up and whether they would ever catch up/ collide. He worked out a formula to calculate that time of collision as the difference in positions divided by the difference in speeds (what he called the algebraic approach) and the same formula changing the reference frame so that one of the objects appears stationary (relativistic approach).

As were watching a tiny duckling try to catch up with its siblings in the pond, Simon realized the catching up problem is actually the same as Zeno’s paradox (you know, the famous one about Achilles and the Tortoise). We continued talking about this the whole time while walking back and I even filmed a small part of our conversation as Simon explained how he would resolve the paradox:

The Zeno argument works, but now a more philosophical question arises: how do you define summing an infinite number of things?

Me: I though it was the difference between mathematics and physics, because in physics you can’t have an infinite number in between two other numbers.

Yes, in physics you can’t actually have this paradox, because at a certain point — the Planck length or Plank time — you can’t divide up space or time anymore. There’s just the smallest length possible or the smallest time possible. Even in math, if you have a sum like 1 + 1/2 + 1/4 + 1/8 +… you might think it’s slightly less than 2 because it never quite reaches there. If you make a list and if you chop up the sum at different positions then it gets close to 2 yet never quite reaches it. But the thing is, if you pick any number less than 2 as the answer, then there would always be more terms in that list that are closer to 2. So it can’t be less than 2, it must be 2.

The other objection is that the sum of infinitely many numbers must be infinite. And that’s also not true. Because if it were any more than 2, the sum would stop before it ever reached there. If you say the sum is 2 or higher, that means there must be terms of that list I just mentioned in that region. And there aren’t any.

So that was our intuitive explanation how you mathematically rigorously define adding infinitely many numbers together. And that actually resolves the paradox if you think about it, because if you do that you get the same answer as in the algebraic method and the relativistic method, that I mentioned earlier.

We’ll keep on hiking, even after the coronavirus crisis subsides and we can resume our usual summer activities again. It’s just so much fun to pick Simon’s brain in the wind.