In the paradox of Achilles and the tortoise, Achilles is in a footrace with the tortoise...

>In the paradox of Achilles and the tortoise, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 meters, for example. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run 100 meters, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, say, 10 meters. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles arrives somewhere the tortoise has been, he still has some distance to go before he can even reach the tortoise.
Veeky Forums, this has been fucking with me. How is anything able to catch up to anything?

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because this is the way our universe works, literally, this is the only answer

Use space instead of time and you'll see how simple it is.

>Veeky Forums, this has been fucking with me.
The essential point here is that the fact that I can subdivide a finite amount of something (in this case, time, or distance) into infinitely many vanishing parts, doesn't actually mean anything.

Yep, I can subdivide something finite into infinite parts, *in my mind*. The mathematics makes sense. So what? That doesn't make the finite thing any less finite.

The scenario is expressed as a geometric sequence a(r^n) where a = 1, r = 1/2. The limit of the total sum is a/(1 - r) = 2, therefore the tortoise never gets past 200 metres and logically achilles must have passed the tortoise if he reaches this point.

Zeno's paradoxes. Unsolved before calculus. Newton BTFO Zeno.

>Plank length

The times between reaching these distances will also grow infinity small, so Achilles will catch up to the tortoise in a finite amount of time.

How does calculus solve this?

It doesn't and only brainlets think it does. Zeno basically shows that, given a continuous conception of space time, it's possible to divide a finite distance into an infinite series that has to be summed up in order to result in the original distance. This results in a natural sequence with no end point, i.e. if you stepped forward with each part of the sequence there'd be no way of saying that either the left or right foot reached the end point first, even if the speed of each step is sped up exponentially.

This is his argument used to demonstrate that conceiving of a continuum inherantly results in paradoxical ideas about physical reality, and so the universe is probably not actually continuous in a way that allows this.

This guy actually has the right idea. Zeno would have loved to find out about this.

>befote you can move from one plank length to another you must first cross half the distance between it...
:^)

Thats even worse....time would come to a standstill

>This is his argument used to demonstrate that conceiving of a continuum inherantly results in paradoxical ideas about physical reality, and so the universe is probably not actually continuous in a way that allows this.
goddam you're dumb

yes you do reach the point in a finite time despite "doing" an infinite summation, which is exactly what calculus shows

limits are not some far-fetched product of the imagination, they are well defined

if it "shows" something about physical reality, it is that you can't think of physics as a discrete algorithmic process. Despite the fact that we use such processes to simulate physics and solve physics problem, we shouldn't confuse the map and the territory.

Likewise, there are plenty of processes that work just fine in nature but produce divergent algorithms if you naively try to simulate them, simply due to the discrete nature of our calculations.

Fuck off you arrogant calc 1 student. Limits are based on convergence, they only tell you what the summation gets arbitrarily CLOSE to. It does not ever actually get there.

except, nigger, you're not actually doing a summation
see

>if it "shows" something about physical reality, it is that you can't think of physics as a discrete algorithmic process.

His point is that the ability to think about the universe in these terms is inherently allowed by the conception of the continuum, and even if you could do all of the "algorithmic process" with the each step being computed arbitrarily fast so that it finished in finite time (as would physically be the case), it's still a physical sequence with no final step. As such, to him it would make more sense for the universe to be based around a discreet makeup.

Are you a dumb person?
I think so.

It's called Quantum mechanics literally because reality is quite obviously /quantized/ that is, discrete. Pixelated. Voxelated. Something. It's discrete bro.

QM in no way imply that space in quantized
try against popsci lord

yeah and instead of that the sane conclusion is that nature is simply NOT a giant discrete computer
this is not limited to this paradox, for example edge effects that don't exist in nature will appear in some simulations and just get worse the more computation steps you add

>it's still a physical sequence with no final step
yeah so he means that if you willingly introduce a timestep instead of treating time continuously then thinking continously doesn't work
woopdeedoo
it's not an insight, it's begging the question

>yeah and instead of that the sane conclusion is that nature is simply NOT a giant discrete computer

He's not claiming this though.

he is, by saying it has to make some sort of sum, that the sum IS the physical process
in essence his whole observation is that if you think of nature as a discrete computer, then the continuum doesn't exist
which is a trivial observation

your penis is a trivial observation

and today another popsci brainlet got btfo

Actually, Newton's use of infinitesimals was decidedly ropey, and was argued against by some as "compensating errors".

It wasn't until much later when Cauchy and then Weierstrass (with the delta-epsilon approach) made the concepts of limits mathematically sound.

Yes, of course. Every time a physicist discovers a new branch of knowledge, a hundred pompous ass mathematicians have to rush in and "explain" it and "improve" it and strut around saying things like "it's trivial," and "I knew this years ago." :^)

math.stackexchange.com/questions/142932/achilles-and-the-tortoise-paradox

>that idiot was just quoting from the third comment
goddam
How do you write that and don't stop to think "wait, the idea that he does one steps on each increment is maybe retarded"?

That's not how plank length works