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Mathematical Paradox

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BaalChatzaf
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16 Feb 2016, 10:30 pm

Deltaville wrote:
Ever heard of this problem?

Let us a assume a frog jumps half a distance to it's destination. Therefore, its jumps would account as: 1/2 the distance + 1/4 the distance + 1/8 the distance and so forth.

Essentially, 1/2n

Would it ever reach its destination? Well I have computed that formula numerous amount of time and mathematically, it always results as impossible to reach.

Or perhaps not...

Some theoretical mathematicians have argued theoretically, after a googleplex of times, the destination would be reach as the resulting step is a positive integer. Infinitely small, yes, but nevertheless a positive integer.

Do the same laws of mathematics still apply when one reaches an enormous, unfathomable value?

Please discuss.


Zeno did not know about infinite series and convergence. The sum of the powers of 1/2 starting with the first power is 1. The series converges.


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17 Feb 2016, 6:21 pm

BaalChatzaf wrote:
Deltaville wrote:
Ever heard of this problem?

Let us a assume a frog jumps half a distance to it's destination. Therefore, its jumps would account as: 1/2 the distance + 1/4 the distance + 1/8 the distance and so forth.

Essentially, 1/2n

Would it ever reach its destination? Well I have computed that formula numerous amount of time and mathematically, it always results as impossible to reach.

Or perhaps not...

Some theoretical mathematicians have argued theoretically, after a googleplex of times, the destination would be reach as the resulting step is a positive integer. Infinitely small, yes, but nevertheless a positive integer.

Do the same laws of mathematics still apply when one reaches an enormous, unfathomable value?

Please discuss.


Zeno did not know about infinite series and convergence. The sum of the powers of 1/2 starting with the first power is 1. The series converges.


So it converges.

So what?

That doesnt answer the question.


The question is "is there an actual number (other than infinity) of hops that will enable the frog to reach the end of the line? And if so how do you calculate that number?".



marshall
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18 Feb 2016, 12:50 am

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This is just a variant on the 2500 year old "Zeno's Paradox" from ancient Greece.

Forget about frogs hopping. Just use yourself walking, or driving a car. The same thing applies.

Lets say you wanna get from your sofa to the fridge: you first have to reach the point halfway between your sofa and the fridge. But before that you have to reach the point halfway between your sofa and that halfway point, but before that you hafta reach the point halfway between your sofa and that second halfway point (the quarterway point), but before that...and so on. You would have to hit every point between the sofa and the fridge. And since there are an infinite number of geometric points along the line between your sofa and the fridge (just like there are an infinite number of points on any line) ****it would take an infinite time to accomplish that****. Which means you cant move from your sofa to the fridge. Indeed ALL motion is an illusion. There is no motion. Nor time itself. That according to the ancient Greek philosopher Zeno in 490 BCE.

The bold part is the fallacy. :p A finite interval of time can also contain an infinite sequence of "time points". It makes no sense to believe that a finite distance can be filled with an infinite amount of points, yet a finite time interval can't. There is no contradiction.

edit: Apparently bold doesn't work within quotes :x What is wrong with bbcode! I used '*'s instead.



marshall
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18 Feb 2016, 1:05 am

Anyways, Godel's incompleteness theorem is more disturbing to me. It is a theorem that implies, among other things, that there exists mathematical statements about natural numbers (i.e. the numbers 0,1,2,3,....) that can neither be proved nor disproved. If A is such a statement, then is A true or false? According to the theorem its impossible to know. It isn't that statement A is nonsensical either. The theorem shows a perfectly logical fact that must seemingly either be true or false, yet it's impossible to prove either. I can't really wrap my head around it.



BaalChatzaf
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18 Feb 2016, 7:38 am

Deltaville wrote:
Ever heard of this problem?

Let us a assume a frog jumps half a distance to it's destination. Therefore, its jumps would account as: 1/2 the distance + 1/4 the distance + 1/8 the distance and so forth.

Essentially, 1/2n

Would it ever reach its destination? Well I have computed that formula numerous amount of time and mathematically, it always results as impossible to reach.

Or perhaps not...

Some theoretical mathematicians have argued theoretically, after a googleplex of times, the destination would be reach as the resulting step is a positive integer. Infinitely small, yes, but nevertheless a positive integer.

Do the same laws of mathematics still apply when one reaches an enormous, unfathomable value?

Please discuss.


Those goes back to Zeno (4th century b.c.e.) It is a convergent series which adds up to 1. The Greek mathematicians know little or nothing about limits until the time of Archimedes.


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naturalplastic
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18 Feb 2016, 5:22 pm

marshall wrote:
Quote:
This is just a variant on the 2500 year old "Zeno's Paradox" from ancient Greece.

Forget about frogs hopping. Just use yourself walking, or driving a car. The same thing applies.

Lets say you wanna get from your sofa to the fridge: you first have to reach the point halfway between your sofa and the fridge. But before that you have to reach the point halfway between your sofa and that halfway point, but before that you hafta reach the point halfway between your sofa and that second halfway point (the quarterway point), but before that...and so on. You would have to hit every point between the sofa and the fridge. And since there are an infinite number of geometric points along the line between your sofa and the fridge (just like there are an infinite number of points on any line) ****it would take an infinite time to accomplish that****. Which means you cant move from your sofa to the fridge. Indeed ALL motion is an illusion. There is no motion. Nor time itself. That according to the ancient Greek philosopher Zeno in 490 BCE.

The bold part is the fallacy. :p A finite interval of time can also contain an infinite sequence of "time points". It makes no sense to believe that a finite distance can be filled with an infinite amount of points, yet a finite time interval can't. There is no contradiction.

edit: Apparently bold doesn't work within quotes :x What is wrong with bbcode! I used '*'s instead.


Doesnt answer the OP question.

But that IS one way of looking at it. Infinite points in space divided by infinite moments in time equals finite time.

Well darn!

That was one of my favorite paradoxes!

But now it's a......paradox lost! :D



marshall
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18 Feb 2016, 11:05 pm

naturalplastic wrote:
Doesnt answer the OP question.

But that IS one way of looking at it. Infinite points in space divided by infinite moments in time equals finite time.

Well darn!

That was one of my favorite paradoxes!

But now it's a......paradox lost! :D

I think all paradoxes result from a fault in the way we think about something. Most aren't outright contradictions/inconsistencies, but rather counter-intuitive truths. There are a few paradoxes like Russel's paradox that are outright contradictions, but these are the results of faulty/inconsistent hypothesis. In any case, trying to find an explanation to a paradox doesn't make it any less interesting in my mind. Counter-intuitive ideas are part of what makes math interesting.



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20 Feb 2016, 7:33 pm

I think that this a matter that cannot be settled, with all due honesty, I just do not believe that mathematics is a topic that treads with smoothness on a forum devoted to Asperger's and Autism related issues. To be frank, I regret bringing these topics into existence.


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marshall
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21 Feb 2016, 1:08 am

Deltaville wrote:
I think that this a matter that cannot be settled, with all due honesty, I just do not believe that mathematics is a topic that treads with smoothness on a forum devoted to Asperger's and Autism related issues. To be frank, I regret bringing these topics into existence.

I don't know what I said. Sorry if I offended you.



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21 Feb 2016, 9:21 am

marshall wrote:
Deltaville wrote:
I think that this a matter that cannot be settled, with all due honesty, I just do not believe that mathematics is a topic that treads with smoothness on a forum devoted to Asperger's and Autism related issues. To be frank, I regret bringing these topics into existence.

I don't know what I said. Sorry if I offended you.


You did absolutely nothing wrong. You were, in fact, the most constructive member on this thread. Another member just abused me.


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21 Feb 2016, 3:00 pm

naturalplastic wrote:
marshall wrote:
The bold part is the fallacy. :p A finite interval of time can also contain an infinite sequence of "time points". It makes no sense to believe that a finite distance can be filled with an infinite amount of points, yet a finite time interval can't. There is no contradiction.

edit: Apparently bold doesn't work within quotes :x What is wrong with bbcode! I used '*'s instead.


Doesnt answer the OP question.

But that IS one way of looking at it. Infinite points in space divided by infinite moments in time equals finite time.

Well darn!

That was one of my favorite paradoxes!

But now it's a......paradox lost! :D


Agreed. That's quite brilliant in its simplicity.


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22 Feb 2016, 12:41 pm

100000fireflies wrote:
The way i see it, fancy physics calculations aren't needed here. There is a premise to be taken as a given - that every step the frog makes is halfway between him and the current destination. At that point equations aren't needed as it's just logical common sense. You stated the frog can never hop more than half way. So no matter how many hops, how long the length, with that premise, the frog will never get there. Any equation 'proving' the contrary doesn't match the premise.

Essentially, to me, the original post read the same as "Jane is taller than Mike. Mike is taller than Steve. Who is tallest?" Clearly, Jane.

Hypothetical situations in which, if the premise were a fact, what is the conclusion? In this case, it's just a hypothetical poor frog destined to spend his life half hopping. That said, again i've never heard the question before, so maybe, like the Earth tunnel one, there's a bit more behind it than mentioned - but as it was stated, the frog can't get there.

/my two cents

I always took it as a philosophical question. Nothing more. No math. Just the idea that you can keep halving an existing distance and never reach the end point because it will always be a 'half distance.'


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29 Feb 2016, 9:54 am

Deltaville wrote:
cberg wrote:
Phenomena portraying absolute series strikes me as the computational singularity, anything can be in-vivo vivisected to Planck depth provided the scientists with the funding manage to stay there rather than using all that same equipment for cryptographic brute-force exploits. Are we examining the sum wave state of Zeno & Euclid?


Planck's length is only relevant if you only wish to discuss quantum foam. This was my specialty when I did physics, it is irrelevant to this problem, as a real number can get much smaller than Planck's length.

Actually, strings are even more elementary than quantum mechanics.


But we are talking about a real frog, not a real number, no?

A real frog cannot make a jump smaller than the Plank length, can it?

I thought Rudin's answer was correct in anything but a question about an unreal frog and you need only apply the same rules that are used for unicorns in such cases.


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29 Feb 2016, 10:57 am

Adamantium wrote:
A real frog cannot make a jump smaller than the Plank length, can it?

I thought Rudin's answer was correct in anything but a question about an unreal frog and you need only apply the same rules that are used for unicorns in such cases.

If you treat it philosophically, it can be virgin unicorns. Everything has a half distance philosophically. In reality, nothing can be smaller than itself. So, Plank lengths do not apply to anything in reality as far as this question goes.


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02 Mar 2016, 7:05 pm

zkydz wrote:
In reality, nothing can be smaller than itself.

Hmmmmmmmmmmmmmmmmmmmmm.

How is this peculiar and highly questionable statement relevant?

I'm guessing the limiting factor in frog jumps has to do with frog motor neurons and muscles. I suspect there is a minimum muscle twitch size that can move the body of a frog in a way that fits the general concept "jump" and that motion is on a much, much bigger scale than Planck lengths....



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02 Mar 2016, 7:24 pm

A frog jumping is just a metaphor. And a good one to quickly visualize the set up the problem. But once your imaginary frog starts getting down to smaller than millimeter distances you dispense with the frog in your mind's eye and think of it as a black dot jumping forward on graph paper eternally halving its distance to the edge of the page, and for the squares on the graph paper representing ever smaller units of measurement.