Achille’s paradox… why can’t Achilles just reach the tortoise?

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Achille’s paradox… why can’t Achilles just reach the tortoise?

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Anonymous 0 Comments

The idea is that to reach the tortoise, Achilles has to cross an infinite number of subdivisions. And since we can’t go across an infinite number of anything, it’s impossible to reach the tortoise.

The fault in the paradox lies in that as the number of subdivisions grow, the length of each decreases. As the number approaches infinity, the length of a given subdivision approaches zero. So ultimately, you get an infinity times zero condition which is mathematically undefined. The paradox is actually a good example of why it’s undefined, as the distance between Achilles and the tortoise can be any finite number.

Anonymous 0 Comments

It’s a hypothetical and a thought exercise, not a literal impossibility.

So…the idea is that if Achilles sets out to catch the tortoise he’ll say “the tortoise is 1 mile away” and then no matter how fast he goes to run that mile, by the time he traverses that mile the tortoise will be a smidge further ahead. Achilles then would have to recalibrate and run the new gap, and be hit with the same problem again. The knowledge of where you are and where the tortoise is is insufficient to reach the tortoise – you can guess and overshoot (i’ll run 2 miles and be safe!) or guess and get lucky and nail it, but you can’t _plan_ to do a perfect catch because your knowledge at the time is discontinuous with the state of you and the tortoise in the future.

Anonymous 0 Comments

Because we are arbitrarily changing the definition of time flow, rigging it to suit our needs. In the Achilles or Zeno’s paradox, we’re running time through something like a 1/2^x function (don’t let the math throw you, it just simply means that each step of the “catch up game”, we add less time… first we add half a second to our timeline, then a quarter of a second, then an eighth of a second, then a 1/16th, etc)…essentially we rig it so that we never tack on enough time at the end to allow Achilles to pass. It’s almost like taking a photo of a ball in the air and saying “why is it that the ball never reaches the other side of the frame, no matter how long the photo sits on the table?” The answer, again, is that we are rigging the passage of time…in the case of a photo, we are freeze framing time; in the case of the paradox, we are simply taking exponentially smaller increments of time up up to hard limit (called “asymptote” in math) that we, by definition and design, will never reach. We are saying, “let’s figure out a way to keep adding on smaller and smaller increments of time to the end of our time total, in such a way that we’ll get closer and closer to a particular number (the point where Achilles would catch up with tortoise) but never reach it”. This is easily done mathematically, but it’s just gaming the system. It’s just a trick. It’s literally the same trick as a freeze frame, just playing with the numbers to make it seem like time is actually moving a very teensy bit, but never enough to pass a certain point. In reality though, time does not flow in exponentially decreasing units, it flows evenly at one second per second, right past the arbitrary limit that would make achilles never pass the tortoise.

Anonymous 0 Comments

Because in order to make it to the tortoise, first he has to make it halfway there.

But before he can make it halfway, he has to make it halfway to the halfway point.

But first he has to make it halfway to *that* point. And over and over and over and over it goes.

It explains something called infinite regression, and not meant to be taken literally.

Anonymous 0 Comments

It’s basically a very early version of a “Troll Machine”.

The argument for why Achille will never catch the tortoise is fundamentally flawed but the flaw isn’t declared – as u/quietmedic notes the issue is it’s taking ever smaller slices of time but presenting them as having an infinite number of smaller pieces that can never actually be run through.

Given it’s entirely possible to catch tortoises this clearly isn’t the case.

The goal is to challenge the reader to think about how the question is set up and explain why the tortoise can be caught.