March 19th, 2018, 02:27 PM  #31  
Senior Member Joined: Jun 2014 From: USA Posts: 528 Thanks: 43  Quote:
Oh, I guess there is a distance that exists then? I agree that “minimum possible distance” is undefined, though I’m not entirely sure in what context you are using it. On the one hand you say there is a distance between $r$ and $r’$ when the rationals are aligned but on the other you say there is no distance between the unaligned pairs of rationals when $r$ and $r’$ are aligned. Which is it? You can’t have your cake and eat it too on this one. Quote:
There is no bottom turtle, just as there is no point where Achilles catches the tortoise, at least according to Zeno’s calculations. But what if Achilles really does catch that tortoise, wouldn’t that be something?  
March 19th, 2018, 05:12 PM  #32  
Senior Member Joined: May 2016 From: USA Posts: 1,310 Thanks: 551  Quote:
No one can now be sure, but, based on the arrow paradox, it seems that Zeno truly was trying to argue that motion is an illusion. That is, he was arguing metaphysics. Zeno says: I can't get your answer by using this infinite process a finite number of times; therefore your answer is wrong. The response is that (a) you can get the correct answer if you use the infinite process correctly, or (b) you cannot get the correct answer by this infinite process because such a process is invalid. The "paradox" is no paradox. Rather Zeno had a mistaken intuition about infinity.  
March 19th, 2018, 05:35 PM  #33  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,671 Thanks: 2651 Math Focus: Mainly analysis and algebra  Quote:
I can. I maintain that there is no distance to talk about here. It's not well defined. But in order to talk about this concept of which you speak (but which doesn't exist) I must use words. Quote:
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He does in the real world, but that ain't the ground we're on. I vaguely recall some research on differences/similarities between the discrete and continuous mathematical universes. Perhaps it would apply if I could track it down. But essentially, I think you are mismatching concepts using you realworld intuition to guide you. It's a recipe for disaster in terms of actual meaning, I fear.  
March 19th, 2018, 07:04 PM  #34  
Senior Member Joined: Sep 2016 From: USA Posts: 635 Thanks: 401 Math Focus: Dynamical systems, analytic function theory, numerics  Quote:
If what you meant is "can you offer an easy proof of this fact" the answer is no because this fact is highly nontrivial and there are a lot of weaker statements which are "obviously" true which are false. Its important to clearly state what you are talking about. Anything less and you aren't talking about math, you are talking about philosophy (poorly) disguised as math.  
March 19th, 2018, 07:47 PM  #35 
Senior Member Joined: Jun 2014 From: USA Posts: 528 Thanks: 43  The distance between $r$ and $r'$ is irrational as stated in the game's instruction. If we move the paper a distance $r'r$ to the right, then we would be moving an irrational distance. Adding an irrational distance to the rationals in $\, (r,r+1\, )$ will move those rationals to irrational points on the number line. They therefore cannot align with the rationals in $\, (r',r'+1\, )$. Rather than go through everything you were saying, why not just use the easy, yet rigorous, proof that I already provided. Indeed, you were trying to blow up a peanut with a death star because your proof ends up the same way my above one does, only with much added, but unnecessary, complexity. That's all. No biggie.

March 19th, 2018, 07:51 PM  #36 
Senior Member Joined: Jun 2014 From: USA Posts: 528 Thanks: 43  Exactly my point, yes. We cannot solve for the distance between any two unaligned pairs of rationals when $r$ and $r'$ are aligned because the infinite process we attempt to use may be invalid.

March 19th, 2018, 09:22 PM  #37 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,671 Thanks: 2651 Math Focus: Mainly analysis and algebra 
The main problem is not that any particular process is invalid (although there is a problem there). The problem is that the "distance" is not well defined.

March 19th, 2018, 09:26 PM  #38  
Senior Member Joined: Jun 2014 From: USA Posts: 528 Thanks: 43  Quote:
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Figure A: $$\begin{matrix} r & \dots & q & <x> &\dots & p & <x> & \dots & v & <x> & \dots \\ \downarrow & \dots & \dots & \searrow & \dots & \dots & \searrow & \dots & \dots & \searrow \\ r’ & \dots & \dots & <x> & q’ & \dots & <x> & p’ & \dots & <x> & v’ & \dots \end{matrix}$$ Figure B: $$\begin{matrix} \dots & <x> & r & \dots & q & \dots & p & \dots & v & \dots \\ \dots & \swarrow & \dots & \dots & \downarrow & \dots & \downarrow & \dots & \downarrow & \dots \\ r’ & <x> & \dots & \dots & q’ & \dots & p’ & \dots & v’ & \dots \end{matrix}$$ There is nothing intuitive or philosophical about noticing that $x$ must be the same distance between $r$ and $r’$ when the rationals are aligned as it would be between the unaligned rationals when $r$ and $r’$ are aligned. This is a fact regardless of whether or not we can solve for $x$ or even whether or not $x$ is a real number as opposed to an infinitesimal. Quote:
Where we differ is that you say there isn’t a best solution while I am asserting that there isn’t a best solution that we can solve for which perhaps implies there is no best solution. My statement is much more precise, both in a purely mathematical and philosophical sense, because it lacks assumption. So far you have resorted to trying to calculate the distance we must move the paper as a real number. You have yet to comment on how the distance does not have to be a real number when motion implies that all distances, be they real numbers, infinitesimals, hyperreals (whatever math has to throw at it...), will be crossed. You also haven't commented on why being unable to solve for the distance using a convergent series implies there is no solution as opposed to merely being a limitation of modern mathematics. Those are the comments I seek, not the trivial discussion about why there must be a rational between $r$ and $r'$ if they are not aligned seeing as how it's the methodology of how we arrive at that conclusion that is being called into question as opposed to asking what that methodology is. Why do you insist on having to calculate the distance to acknowledge that it exists?  
March 19th, 2018, 09:27 PM  #39 
Senior Member Joined: Jun 2014 From: USA Posts: 528 Thanks: 43  
March 20th, 2018, 03:40 AM  #40  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,671 Thanks: 2651 Math Focus: Mainly analysis and algebra  Quote:
Asking for a value for both is the same as asking for the smallest rational greater than a given irrational. It simply doesn't exist. The question itself implies an unfounded assumption. So your answer doesn't make fewer assumptions, it just hides one that is false. Even appealing to the hyperreal doesn't help because there is no single infinitesimal that does the trick. The infinitesimals do not have well defined quantitative values. They are objects with purely qualitative properties in relation to the reals.  

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