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Comment by MarcellusDrum | Hacker News Reader

How do you prove both of these claims wrong, using calculus?

The first is Zeno's Paradox, or a variation thereof. The solution is that passing through infinitely many midpoints is fine, since the sum of an infinite number of positive numbers can still be finite. The example that solves the original Zeno's Paradox (you run half the distance, then one quarter, then one eight, etc, and never get there) is the fact that 1/2 + 1/4 + 1/8 + ... sums up to 1, since it's a Geometric Series.

The second comment touches on how just because the limit of something approaches 0, you can't treat it as 0. If you divide a line into infinitely many segments, the length of those segments approaches 0, but if you treat them as if they were really 0 you get wrong results when summing them back up. Usually that gets relevant with integrals, but in this example it happens with a simple 1-dimensional sum.

If you divide a line into infinitely many segments, the length of those segments approaches 0

Hence the real world being quantized. You can't even hypothetically divide a string into infinite points because there is a smallest distance, below which it really does round to zero, i.e., a singularity.

We have no solid evidence of space-time itself being quantized.

> The second comment touches on how just because the limit of something approaches 0, you can't treat it as 0.

I think this one is a little more subtle. You can definitely treat the limit of something as an exact value.

If you split a segment of unit length into n segments, then each segment has length 1/n. As n goes to infinity, 1/n goes to zero. But at the limit, 1/n = 0, exactly. Not approximately, really exactly, and there is nothing wrong with defining lim[n->inf] 1/n = 0 and using that in a calculation. It's no different than saying lim[n->inf] sum[k=1..n] 2^-k = 1 as you did in the first paragraph.

The problem is slightly different. The person conceives of a sequence of `n` copies of `1/n`. For any finite value of n, these add up to 1 of course; it's just the result of calculating `n × 1/n`. But at the limit, they pull a trick: they say lim[n->inf] 1/n = 0 (which is correct) therefore any sum of these values is 0 too. But we never had lim[n->inf] 1/n, we had lim[n->inf] (n × 1/n) instead!

If you put it in mathematical terms, the argument is:

   lim[n->inf] (n × 1/n) = 
   (lim[n->inf] n) × (lim[n->inf] 1/n) = 0

Which has two problems: first, you can't split up a limit like that, so the first step is already invalid. Second, even if it were valid, lim[n->inf] n = infinity, which is not a number, so you can't multiply with it, and the conclusion that the result is 0 is wrong. You might think `x × 0 = 0` for any x, but that's only true if x is actually a number; if x is not a number the answer is undefined.

This is a similar kind of fallacy that underlies the proof that pi=4, here: http://www.kevinhouston.net/blog/2011/03/pi-is-4-video/, where the author (intentionally) confuses the perimeter of the limit (pi) with the limit of the perimeter (4).

alternatively, you realize the runner/observer is not infinitely small and that effectively puts a finite, lower bounds on the steps, because most of that approach to zero is too small to matter, practically.

learning that infinite things could be finitely bounded was mind-blowing back in the day however.

You don't. Calculus gives exact measurements of the situation in terms of math. But math is fiction, one of many descriptions of reality that we have invented. Other descriptions, many of them non-scientific, work equally well.

I just threw a rock at your head. It hit your head. Therefore, motion is possible without math.

Appeal to the stone fallacy!

https://en.m.wikipedia.org/wiki/Appeal_to_the_stone

Appeal to the stone may be the most effective refutation to "very smart" A+ students with B- intellects (a.k.a. my K–12 peers)

Then again, I generally have a dim view of using deductive reasoning as the first method of thinking. Lots of petulant "my absurd conclusion is correct because you agreed to my premises and that my argument contained no obvious logical fallacies" in that camp.

I wouldn't say that math is fiction. I'd say that it's a tool that can be tested, but there isn't one tool that's correct. There can be many such as Euclidean vs Bolyai-Lobachevskian geometry. Both are non-fiction, one maybe more useful than the other.

All the tools are fiction. They are creations of man. The rock doesnt care. Its existence is not altered one iota by the tools that man use to describe its motion. Things that have no impact outside the minds of men are fiction.

If you are correct, then man must also be fiction.

I keep using my consciousness to figure out if my consciousness is real, but I think it might be biased.

well then 'fiction' is fiction so i guess everything is real

We can predict what will happen to the rock. That predict will or won't be fiction. If the rock is nahcolite we know what'll happen if we mix it with acid. The prediction will be fact or fiction.

> But math is fiction […]

Anyone interested in this can search for "Is math invented or discovered?".

Calculus explains how the sum of infinitely many infinitely small numbers can be any finite or infinite number, depending on how they are constructed.

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