The tortoise and the hare

What is the basis for reason? And mathematics?

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Philosophy Explorer
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The tortoise and the hare

Post by Philosophy Explorer » Mon Jul 16, 2018 12:07 am

This is Zeno's most famous paradox which has a twist.

The hare (whose initials are wf) bet the tortoise he can beat him in another race and let the tortoise start with a ten foot lead. The tortoise (whose initials are ak) said let's talk about it first (because he wanted to show off his knowledge of philosophy).

So wf said go ahead. Now ak said you're ten times faster than me so when you move up ten feet, I'll be a foot ahead of you. wf responded "so?" ak said so when you move up that foot, I'll be ahead 1/10 foot so that no matter what you move, I'll be ahead. wf responded "not quite because the distance keeps diminishing between us and I will catch up." To which ak responded "Except you'll find in each case, time also diminishes so you'll have no time to pass me by." To which wf responded "Except that's the time that corresponds to the distance we moved up and we'll reach a point where I'll catch up to you." to which ak responds "That point is dimensionless and it doesn't exist." to which wf asks
"How do you account for me winning all the races?"

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Re: The tortoise and the hare

Post by vegetariantaxidermy » Mon Jul 16, 2018 3:55 am

I thought it was Aesop.

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Re: The tortoise and the hare

Post by -1- » Mon Jul 16, 2018 9:58 am

Philosophy Explorer wrote: ↑
Mon Jul 16, 2018 12:07 am
This is Zeno's most famous paradox which has a twist.

The hare (whose initials are wf) bet the tortoise he can beat him in another race and let the tortoise start with a ten foot lead. The tortoise (whose initials are ak) said let's talk about it first (because he wanted to show off his knowledge of philosophy).

So wf said go ahead. Now ak said you're ten times faster than me so when you move up ten feet, I'll be a foot ahead of you. wf responded "so?" ak said so when you move up that foot, I'll be ahead 1/10 foot so that no matter what you move, I'll be ahead. wf responded "not quite because the distance keeps diminishing between us and I will catch up." To which ak responded "Except you'll find in each case, time also diminishes so you'll have no time to pass me by." To which wf responded "Except that's the time that corresponds to the distance we moved up and we'll reach a point where I'll catch up to you." to which ak responds "That point is dimensionless and it doesn't exist." to which wf asks
"How do you account for me winning all the races?"

πŸ‡ΊπŸ‡²PhilXπŸ‡ΊπŸ‡²
"That point is dimensionless, so it does not exist." That's the first I heard that someone would deny the existence of points.

A point is not physical; it is geometrical. As such, it exists, in geometrical considerations. The existence of wf and ak are happening in space; and space (as a geometrical object) certainly has points of which it consists.

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Re: The tortoise and the hare

Post by -1- » Mon Jul 16, 2018 10:00 am

vegetariantaxidermy wrote: ↑
Mon Jul 16, 2018 3:55 am
I thought it was Aesop.
Aesop's was a parable. Xeno's was a paradox. They both used the hare and the tortoise in their respective stories.

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Re: The tortoise and the hare

Post by Noax » Mon Jul 16, 2018 11:20 am

Philosophy Explorer wrote: ↑
Mon Jul 16, 2018 12:07 am
... to which wf asks
"How do you account for me winning all the races?"
Because the finish line is (unstated, but presumably) beyond the calculable point at which the two pass each other.
I counted at least three mathematical errors in the text of the OP. There's probably more.

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Re: The tortoise and the hare

Post by Philosophy Explorer » Mon Jul 16, 2018 2:59 pm

Noax wrote: ↑
Mon Jul 16, 2018 11:20 am
Philosophy Explorer wrote: ↑
Mon Jul 16, 2018 12:07 am
... to which wf asks
"How do you account for me winning all the races?"
Because the finish line is (unstated, but presumably) beyond the calculable point at which the two pass each other.
I counted at least three mathematical errors in the text of the OP. There's probably more.
Wrong Noax. ak says the passing point doesn't exist which implies the race track doesn't exist since it's made up of dimensionless points so wf asks how does he win all those races if the race track doesn't exist? While based on that paradox of Zeno, I took a few liberties in changing it so there are no math errors.

So what are the so-called math errors?

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Re: The tortoise and the hare

Post by Noax » Mon Jul 16, 2018 9:24 pm

Philosophy Explorer wrote: ↑
Mon Jul 16, 2018 2:59 pm
So what are the so-called math errors?
Surely Zeno didn't word the problem with such sloppiness. I isolated four:
... so that no matter what you move, I'll be ahead.
This is incorrect and doesn't follow from the preceding argument. Example to the contrary: If the hare moves 20 feet, it will lead the tortoise by 8 feet.
"Except you'll find in each case, time also diminishes so you'll have no time to pass me by."
Time required to pass by diminishes to an arbitrarily low figure. Time available to pass by diminishes to some nonzero value. This means there is plenty of time to pass by, the opposite of what is being said here.
"Except that's the time that corresponds to the distance we moved up
No, it corresponds to the distance remaining to the passing point or to the finish line, or maybe to the next place that the tortoise was at the beginning of the iteration. The comment was unclear, but none of these is the distance either of them has moved up, which is something that increases, not diminishes. Maybe this line just needs some grammatical clarity.
"That point is dimensionless and it doesn't exist."
Where does geometry state that points are nonexistent, or that existing things must have extension in at least one dimension? Were it true, your entire argument could have rested on this, without all the mathematical paradox crap. "There is a point at which we'd pass, and points don't exist, therefore the passing point doesn't exist and we can't pass."

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Re: The tortoise and the hare

Post by surreptitious57 » Mon Jul 16, 2018 9:54 pm

Noax wrote:
Where does geometry state that points are nonexistent or that existing things must have extension in at least one dimension
In quantum mechanics elementary particles are so infinitesimally small that they cannot be seen and so they are treated as points
If something exists then it must have at least one dimension because a zero dimensional thing by definition cannot exist in reality

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Re: The tortoise and the hare

Post by Philosophy Explorer » Mon Jul 16, 2018 11:19 pm

Noax wrote: ↑
Mon Jul 16, 2018 9:24 pm
Philosophy Explorer wrote: ↑
Mon Jul 16, 2018 2:59 pm
So what are the so-called math errors?
Surely Zeno didn't word the problem with such sloppiness. I isolated four:
... so that no matter what you move, I'll be ahead.
This is incorrect and doesn't follow from the preceding argument. Example to the contrary: If the hare moves 20 feet, it will lead the tortoise by 8 feet.
"Except you'll find in each case, time also diminishes so you'll have no time to pass me by."
Time required to pass by diminishes to an arbitrarily low figure. Time available to pass by diminishes to some nonzero value. This means there is plenty of time to pass by, the opposite of what is being said here.
"Except that's the time that corresponds to the distance we moved up
No, it corresponds to the distance remaining to the passing point or to the finish line, or maybe to the next place that the tortoise was at the beginning of the iteration. The comment was unclear, but none of these is the distance either of them has moved up, which is something that increases, not diminishes. Maybe this line just needs some grammatical clarity.
"That point is dimensionless and it doesn't exist."
Where does geometry state that points are nonexistent, or that existing things must have extension in at least one dimension? Were it true, your entire argument could have rested on this, without all the mathematical paradox crap. "There is a point at which we'd pass, and points don't exist, therefore the passing point doesn't exist and we can't pass."
You haven't read carefully enough because on your first point, I meant within the first ten feet.

I'll comment on your three other points later.

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Re: The tortoise and the hare

Post by vegetariantaxidermy » Tue Jul 17, 2018 7:57 am

-1- wrote: ↑
Mon Jul 16, 2018 10:00 am
vegetariantaxidermy wrote: ↑
Mon Jul 16, 2018 3:55 am
I thought it was Aesop.
Aesop's was a parable. Xeno's was a paradox. They both used the hare and the tortoise in their respective stories.
Zeno's was Achilles and the tortoise.

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Re: The tortoise and the hare

Post by Noax » Tue Jul 17, 2018 11:37 am

Philosophy Explorer wrote: ↑
Mon Jul 16, 2018 11:19 pm
Noax wrote: ↑
Mon Jul 16, 2018 9:24 pm
... so when you move up that foot, I'll be ahead 1/10 foot so that no matter what you move, I'll be ahead.
This is incorrect and doesn't follow from the preceding argument. Example to the contrary: If the hare moves 20 feet, it will lead the tortoise by 8 feet.
You haven't read carefully enough because on your first point, I meant within the first ten feet.
Read your own comment more closely then. It is already talking about subsequent iterations, not the first ten feet, and it was supposed to point out that in any subsequent iteration, 'I'll be ahead', which would be true. But you stated something else instead, that 'I'll be ahead' "no matter what you move". That wording is demonstrably wrong.
vegetariantaxidermy wrote: ↑
Tue Jul 17, 2018 7:57 am
-1- wrote: ↑
Mon Jul 16, 2018 10:00 am
Aesop's was a parable. Xeno's was a paradox. They both used the hare and the tortoise in their respective stories.
Zeno's was Achilles and the tortoise.
Ah, thank you. I thought the hare didn't sound right. The hare symbolizes arrogance (I'm so much better I can win even if I nap), but Achilles symbolizes undefeatability, which I guess more emphasizes the paradoxical point Zeno was trying to make. But Zeno's paradox wouldn't be so well known if it was so full of invalid mathematics like this. Taken literally, it is probably pretty pathetic that either Achilles or a hare would be only 10x faster than a tortoise.

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Re: The tortoise and the hare

Post by Philosophy Explorer » Tue Jul 17, 2018 12:43 pm

Noax wrote: ↑
Tue Jul 17, 2018 11:37 am
Philosophy Explorer wrote: ↑
Mon Jul 16, 2018 11:19 pm
Noax wrote: ↑
Mon Jul 16, 2018 9:24 pm

This is incorrect and doesn't follow from the preceding argument. Example to the contrary: If the hare moves 20 feet, it will lead the tortoise by 8 feet.
You haven't read carefully enough because on your first point, I meant within the first ten feet.
Read your own comment more closely then. It is already talking about subsequent iterations, not the first ten feet, and it was supposed to point out that in any subsequent iteration, 'I'll be ahead', which would be true. But you stated something else instead, that 'I'll be ahead' "no matter what you move". That wording is demonstrably wrong.
vegetariantaxidermy wrote: ↑
Tue Jul 17, 2018 7:57 am
-1- wrote: ↑
Mon Jul 16, 2018 10:00 am
Aesop's was a parable. Xeno's was a paradox. They both used the hare and the tortoise in their respective stories.
Zeno's was Achilles and the tortoise.
Ah, thank you. I thought the hare didn't sound right. The hare symbolizes arrogance (I'm so much better I can win even if I nap), but Achilles symbolizes undefeatability, which I guess more emphasizes the paradoxical point Zeno was trying to make. But Zeno's paradox wouldn't be so well known if it was so full of invalid mathematics like this. Taken literally, it is probably pretty pathetic that either Achilles or a hare would be only 10x faster than a tortoise.
No you're just trying to put your own spin on what I've posted and trying to extend it. Show me where I was talking about the hare passing the tortoise which isn't my concern (but yours apparently) as a race can refer to any part of a racecourse, but the additional information clearly covers where the hare is trying to catch up to the tortoise so you do need to learn how to read,

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Re: The tortoise and the hare

Post by Philosophy Explorer » Tue Jul 17, 2018 1:13 pm

vegetariantaxidermy wrote: ↑
Mon Jul 16, 2018 3:55 am
I thought it was Aesop.
Who's to say that Zeno didn't substitute Achilles for the hare. This isn't important to the OP (but makes the story more interesting).

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Re: The tortoise and the hare

Post by Arising_uk » Tue Jul 17, 2018 4:49 pm

Philosophy Explorer wrote:... "That point is dimensionless and it doesn't exist." to which wf asks
"How do you account for me winning all the races?"

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Well ignoring that the 'time' doesn't diminish it'll be because there are no 'dimensionless points' in reality.

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Re: The tortoise and the hare

Post by Noax » Tue Jul 17, 2018 10:46 pm

Philosophy Explorer wrote: ↑
Tue Jul 17, 2018 12:43 pm
No you're just trying to put your own spin on what I've posted and trying to extend it. Show me where I was talking about the hare passing the tortoise which isn't my concern (but yours apparently) as a race can refer to any part of a racecourse, but the additional information clearly covers where the hare is trying to catch up to the tortoise so you do need to learn how to read,
I did not used the word 'pass' or 'passing' anywhere in that post. You talked about being ahead, and that it wouldn't happen "no matter what you move", which is incorrect. I'm commenting on what you actually said. If you meant something else, perhaps you should have said the thing you meant.

My third point admits to unclarity. It really wasn't clear what you were trying to say, so I commented on various interpretations of the words. Lacking a clear statement, it wasn't a clear mathematical error.
Arising_uk wrote: ↑
Tue Jul 17, 2018 4:49 pm
Well ignoring that the 'time' doesn't diminish it'll be because there are no 'dimensionless points' in reality.
I took the example to be mathematical. There is sort of a different analysis and set of errors if the example is taken to be reality with a talking hare and tortoise. Creatures (especially hares and walking pigeons) don't act like ballistic mathematical points, so the velocity of the front of the subject (usually the thing measured in a race) is not constant as is presumed in the analysis. The infinite series breaks down eventually to periods where the hare moves but the tortoise doesn't, and only if either actually has a location, which they probably don't in reality, but there is a measurement device at the finish line, so that suffices if determination of the winner is desired.
The argument is presented as a mathematical one, so I was commenting on that approach.

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