to_hobbes

I just can't agree with "math is flawed". There are paradoxes in mathematics, but according to Godel's incompleteness theorem, all complete theories must have some inconsistencies (paradoxical statements).

Mathematicians understand this and treat paradoxes as potholes in the road to developing a complete proof -- paradoxes may be inevitable, but you can avoid them if you are careful.

Just because there are paradoxes does not mean math is fundamentally flawed. It just means that there are many ways to prove certain things, but if you don't choose the proper list of proof steps carefully, you could end up proving something to be both true and false at the same time, which would make your proof invalid.

Anyway, except for this, I like what you said about our limited dimensions.

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(By the way, it's all in your head.)

Shadee

It's not that you can't calculate at what point he will catch up with different methods of math. The problem behind Zenon's paradox is that if you have to travel half the distance to get from point A to point B then you must also first have to travel half of half of that distance, and half of that etc, so if you actually put the numbers in it becomes endless. So if you calculate it that way you'll never even reach the second half which means that that specific calculation even though correct in math isn't the way things work in real life because you will move past the endless division point and get to your destination.

As for calculus, I'm sure you're right when you say improper use.
I'm not a math whiz myself at all but someone who claimed to be showed me examples once which seemed to make sense. After doing some searching on-line just now though I'm sure the examples I've seen cross some rule or another with 0/0 being indeterminate, 1/0 undefined and dividing by zero illegal and all.

Oh no I never meant to say math is fundamentally flawed, just that there are some holes and paradoxes which make certain calculations unreliable. And just like the whole dimensional thing I believe that once we learn more and start seeing things in different ways it can be improved upon. In fact some of it might change completely.

Basically the only point I was trying to make was that we don't know everything, and just because we've developed a system to calculate the things we are able to observe now doesn't mean the world can't be a lot more complex than we think. So proving things within our own mathematical systems and comprehension is possible but that doesn't mean it's the absolute truth.

romansh

Hi Shadee
in Zenon's paradox we choose an arbitary pattern of decreasing distances (or times) and then try to read something significant into that pattern? In your second post you chose half the distance, in the first post you chose to where the turtle/tortoise is at any given point of time. The point is you are framing the question in ever decreasing increments of time and distance. If you choose constant increments of time we don't have this discussion.

In my opinion there is nothing paradoxical about the question. It is us choosing a time frame that does not allow Achilles to catch up. Like I said it easy to calculate how long it will take Achilles to catch up. Now if we chose an infinite set of numbers that add to less than that time we will never see Achilles catch up.

Now if you are refering to an infinite set of numbers that increasingly become smaller eg .... one, half, quarter, eighth, sixteenth...etc that approach a value of two? Adding a series of infinite numbers that become infinitesimally small.

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Shadee

Actually in the my first post about Zenon's paradox I didn't give any description I just mentioned the words "Zenon's Paradox".
And it's not a time frame that doesn't allow him to catch up the numbers are the distances he needs to traverse. To travel 1 meter he first needs to travel half a meter, to travel half a meter he first needs to travel a quarter of a meter etc. And that can go on forever because the number of halves of halves of halves he needs to traverse to get anywhere is infinite.

romansh

Sorry you are right .... it must have been the description of it (I read) on the net of Zenon's paradox where Achilles runs to where turtle started, then runs to where turtle is at that time and so on.

Eitherway we are choosing a time frame where Achilles won't catch up and dividing that time into infinitesimal chunks of our choosing. We set up Achilles not to catchup, whilst we are looking! No paradox here.

Sorry again .... in this case time and distance are related .... time = distance/velocity

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There is a theory which states that if ever anybody discovers exactly what the Universe is for and why it is here, it will instantly disappear and be replaced by something even more bizarre and inexplicable. There is another theory which states that this has already happened. ........... Douglas Adams

romansh

Hi Shadee
I was thinking of a better way to explain what I am trying to say. I thought an example would be best
Imagine the the turtle is in top form and can move at 0.1 m/s
Achilles is going for a gentle stroll ~ 4 km/h..............1.1 m/s
Achilles is not confident and gives the turtle a head start of 10 m

Achilles will have caught the turtle in 10 m / (1.1-0.1) m/s = 10 s
and have will walked 10 s x 1.1 m/s = 11 m

So try using Zenon's parlor tricks with 12 m or any distance greater than 11 m; Achilles catches up with the turtle. A similar argument can be with any time greater than 10 s. Zenon's paradox apears to be so only because insufficient time is allowed.

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There is a theory which states that if ever anybody discovers exactly what the Universe is for and why it is here, it will instantly disappear and be replaced by something even more bizarre and inexplicable. There is another theory which states that this has already happened. ........... Douglas Adams

Shadee

Well basically the idea of Zenon's paradox is that Achilles never really gets anywhere because he gets sucked into all the .0000 type decimals. The thing is in that kind of reasoning the turtle would be in the same predicament so the whole turtle part doesn't even really matter.

In this paradox time and speed don't matter either.
The reasoning is that there are an infinite number of existing points on the line between point A and Point B. If you want to move one inch then to get there you need to first traverse 0.5 inch and before you can reach that you need to first travel half of that which is 0.25 inch but before you get to even that you need to travel half of that which is 0.125 and that goes on forever.

In calculation that is of course correct, there are an infinite possible points on a straight line between point A and B but in real life we do somehow seem to move past those.

romansh

I disagree .... Achilles does not catch up because he is not given enough time. give him 10.000000000000000000000000000001 s he overtakes the turtle (add all the zeros you want). Remove the final 1 and Achilles has drawn level. Why does Zenon stop an infinitesimally short of 10 s in this case? The paradox is if there is one there is no intention of getting to 10 s.

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There is a theory which states that if ever anybody discovers exactly what the Universe is for and why it is here, it will instantly disappear and be replaced by something even more bizarre and inexplicable. There is another theory which states that this has already happened. ........... Douglas Adams

Shadee

Like I said before Zenon's paradox ultimately isn't about time or speed so forget those completely. It's about being able to split a certain =distance= in half, then splitting the half in half, splitting that half in half and that goes on endlessly because you will never get to 0 by dividing like that. so there's an endless about of halfway points he needs to reach for him to get anywhere and since it's and endless amount he's not going anywhere theoretically.
But in real life, he would of course pass the turtle eventually if he was faster.

I'm not sure how to explain it better.
As for you disagreeing, this wasn't an opinion of mine, I was just trying to explain the paradox to you.

Basically Zenon's point was that numbers are infinite but that this doesn't always make sense when compared to real life.

romansh

OK .... so using the example I gave and 11.1 m, Achilles catches up even if you continually split the difference in half.

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There is a theory which states that if ever anybody discovers exactly what the Universe is for and why it is here, it will instantly disappear and be replaced by something even more bizarre and inexplicable. There is another theory which states that this has already happened. ........... Douglas Adams