Mathematical Paradox

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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

Last edited by 100000fireflies on 03 Feb 2016, 4:24 pm, edited 2 times in total.

Now this version has more complexity and is no longer just a common sense answer.

Mathematically, a real number can be as limitless small, ad infinitum; even smaller then Planck's length, although this serves no pragmatic purposes.

Secondly, there is plenty of solid mathematics behind string and quantum foam, I studied them myself.

This^.

There is no paradox. The problem is bogus. The frog obviously never reaches the destination because there will always be an infinitesimal distance remaining after each halving of the distance.

However Rudin's notion of using the Planck length is an interesting possible way to do an end run around Zeno's Paradox. If the the theoretical Planck length is real then there would be a finite lower limit to distances. The fact that "there are real numbers smaller than the Planck length" would be irrelevent. Niether frog nor physicist would be able to subdivide the distance less than the Planck Length. So then the problem would be how many halvings would it take to subdivide the distance down to the Planck Length: which would be a finite number.

The problem is in maths, there is no smallest unit, a point is purely relative to another. in these problems each subsequent step is merely relative to the step before it. that's why you can never reach the "end" because as long as there is a distance you just [numerically] traveled there is a smaller distance. Can't look at things as concrete but as purely relative and informational
Right?

As this problem has a known solution, it cannot be plausibly described as bogus.

Moreover, it is indeed possible to subdivide Planck's length and still result in a real number, XeR, but only for numerical or theoretical functions. However, as mentioned its sole function is restricted only to number theory. Numerical uncertainty is the driving force in which at a very small value, the quantity is fundamentally uncertain because of its extreme value.

Secondly, this so-called 'bogus' problem is nothing more than a derivative of a problem used to teach convergent series to Calculus students, and it merely attempts to take a second look at it.

Thirdly, I finished a university degree in both math and physics, so if you are implicitly insulting me, please stop.

Whether it's 'bogus' or not, as written, is it a true problem any more than Jane being taller? I see the paradox in naturalplastic's version, but none in the original. As stated though, perhaps there was information missing, but as written, it seems pretty straight-forward.

Re: Planck's - as i see it, if that were indeed the smallest finite length, whether dividing can result in a real number or not wouldn't matter. If it were to be used as described for Zeno's paradox, it could not be divided or you'd be right back to the paradox with infinite points. No?

As for the degree, you've said that; several times. I get that you're proud of it and i was too; completing a degree is an accomplishment. After a while though, it starts to come across as if that makes you better or smarter than everyone else.

Last edited by 100000fireflies on 04 Feb 2016, 4:54 am, edited 1 time in total.

Actually a degree does indicate higher intelligence then average, so it is not a far cry to claim that it gives you authority in certain disciplines. When I finished my undergrad I was given a chance to start work for my PhD in Theoretical Physics right away (skipping grad), although I declined to do so in favor of law.

Seriously? That's your reply? I tried to be tactful. And you retort that you are smarter than everyone else followed by more of why you think you're better?

Guess what. I got a degree too. So have a lot of people on the board. Only we don't feel the need to announce it at every moment, including when it's completely irrelevant to the subject. Not to mention that quite a few people here who didn't get degrees don't lack the required intelligence. They merely lack the keep your sh*t together executive functioning skills that deeply interfere with obtaining a degree.

And, no, getting a degree doesn't make one an expert nor an "authority" - it makes one a graduate. Working in that field for many years after the degree is where the experts come from.

I was being tactful and respectful. Succinctly put, getting a degree is the quintessential sign of intelligence; getting a graduate degree like law or the upper sciences means you are definitely in the triple digit tier of the IQ range.

Edit: When I was 15 my IQ was tested to be 136, and I am not openly saying this fact over and over again. Not saying that fact over and over again certainly attests that I am not a narcissistic.

-Seb

This is the classic Xeno Paradox. Yes, the rules of mathematics still apply if you introduce the idea of limits.

If the OP does have an actual solution to this that is not "infinity" then I take back saying "its bogus".

But without a lower limit in length of hops I dont see how it could have any other solution but 'infinity'.

What I wanted to bring into focus on this question was whether numerical fluxes occur near asymptotes that reasonably could bring about numerical uncertainty. Yet all I get are these negative responses from everyone.

1. The higher degree correlation is for the general population, not people with severe executive functioning issues and other significant co-morbidities. It has no relevance here.

2. No one said you weren't smart or hold many qualities. But you're not the smartest cookie in town. You don't even have the highest officially tested iq on this thread nor would you be the only one asked to join mensa.

The point is, a degree is not synonymous with authority others should blindly follow, and being smart is not synonymous with being better. If you don't want ongoing implied or direct insults (whether there was one here or not is irrelevant, there have been elsewhere), nor people no longer replying to your threads (and not because you've outwitted everyone), then maybe let people discover it naturally, not via redundant brute force announcement. The latter strongly comes across as pure arrogance, not intelligence, and will only progressively piss people off.

For the original question stated, it would be irrelevant. Jane is taller and the frog will half hop forever.

For the paradox version naturalplastic posed, i still don't see how it would matter? When i go from the couch to the fridge, whether i know the number of points i cross or not, there's a halfway to the destination. When i get there, there's another halfway ahead. And so on. Meanwhile, there are infinite points on any line. So whether i'm at point 999,556,743 or 12,456,465 how would it matter? I'd still have infinity to traverse.
How do you see it as influencing?

If there were finite points on a line, then the precise quantity would matter. But as there isn't, i don't see it. But again, as you mentioned it, maybe you're seeing a facet i'm overlooking??

Not sure what you mean by Jane is taller. Difference in magnitude is an incorrect premise in relation to this problem, although I am not exactly sure what you mean to be honest.

As I have mentioned numerous times on this thread, if you were to use the ratio limit test for convergence, you are right the limit as x->infinity is 1. That means you are correct, it would not be possible for the frog to every reach 1 in finite amount of moves as the distance does halve itself every subsequent step. This 'force field', as some call it, is known as an asymptotic barrier.

But these are two types of asymptotic for practical purposes, the other is a value where the denominator is 0 and it is impossible even theoretically to cross that threshold, because any value divided by zero is inheritable unknowable and undefined. If you were to find the limit of 1/x for instance, the function on the right would extend all the way up to infinity, and the function on the left would go all the way down to negative infinity.

However mathematician have long known that asymptotes can in fact be crossed, and thus are de facto regions in which functions cannot cross rather then de jure regions. For instance, some cubic functions do in fact cross the horizantal asymptote initially.

Therefore, some super computers, given the fact that 1 may not be in fact an absolute limit, have computed the so called Xeno's paradox and miraculously, after a gazillion (I cannot even write such a huge amount ) amount of subsequent steps miraculously, the value of 1 is crossed. Who knows why?

Perhaps maybe you are right, they are nothing more then errors in decimal places, or perhaps, mathematics is a field that is more uncertain then we ever imagined.

This is the kind of discussion I wanted to probe in this thread, but it didn't go the way I hoped it would.