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March 19th, 2018, 02:27 PM   #31
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Originally Posted by AplanisTophet View Post
On the flip side, you (we) question what is between $r$ and $r’$ when the rationals are aligned in the game. That distance would be the same distance that separates the philosophically unaligned rationals when $r$ and $r’$ are aligned, right?
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Originally Posted by v8archie View Post
Not necessarily.
Not necessarily? I say absolutely: the distance between $r$ and $r’$ when the rationals are aligned per the game’s instruction is exactly the same distance between each pair of unaligned rationals when $r$ and $r’$ are aligned. The question is does that distance exist?

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I see it as just a distance
Oh, I guess there is a distance that exists then?

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not any sort of minimum possible distance.
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.

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To me it has no philosophical meaning because no such "minimum" exists.
I see nothing philosophical about noting that the rationals cannot align when $r$ and $r’$ are aligned. I see nothing philosophical about asserting that “the distance between $r$ and $r’$ when the rationals are aligned per the game’s instruction is exactly the same distance between each pair of unaligned rationals when $r$ and $r’$ are aligned” either.

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All these questions boil down to asking about the bottom turtle.
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?
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March 19th, 2018, 05:12 PM   #32
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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?
But of course Achilles can catch the tortoise.

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.
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March 19th, 2018, 05:35 PM   #33
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Originally Posted by AplanisTophet View Post
Not necessarily? I say absolutely: the distance between $r$ and $r’$ when the rationals are aligned per the game’s instruction is exactly the same distance between each pair of unaligned rationals when $r$ and $r’$ are aligned.
The distance $(r-r')$ can be close to zero or close to $\frac1{10}$ or close to $\frac23$. Any fraction between zero an 1 does the same job. You still have an infinite quantity of rationals unmatched. There isn't a "best" solution.

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You can’t have your cake and eat it too on this one.
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.

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I see nothing philosophical about noting that the rationals cannot align when $r$ and $r’$ are aligned.
Granted. That's just a result.

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I see nothing philosophical about asserting that “the distance between $r$ and $r’$ when the rationals are aligned per the game’s instruction is exactly the same distance between each pair of unaligned rationals when $r$ and $r’$ are aligned” either.
Well neither "distance" is well defined. In mathematics that makes them not equal. Philosophically and intuitively they seem equal, but intuition is spectacularly bad at this game.

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But what if Achilles really does catch that tortoise, wouldn’t that be something?
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 real-world intuition to guide you. It's a recipe for disaster in terms of actual meaning, I fear.
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March 19th, 2018, 07:04 PM   #34
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Originally Posted by AplanisTophet View Post
This approach is no different than noticing that when $r$ and $r'$ are vertically aligned, none of the rationals will be. It's like using the death star to blow up a peanut, imho.

Ask yourself, what fills the space between the unaligned rationals when $r$ and $r'$ are aligned? That is more or less the starting point for the problem. My longer exposition to a8archie then follows, as he went right to that point (which is what anyone should do...).
This approach offers a mathematical proof of your claim that the rationals are not aligned. You were the one who asked for a proof. If it seems like "blowing up a peanut" with a death star its because the linked article is doing mathematics and in math we make things precise and prove things rigorously.

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.
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March 19th, 2018, 07:47 PM   #35
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If what you meant is "can you offer an easy proof of this fact" the answer is no
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.
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March 19th, 2018, 07:51 PM   #36
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(b) you cannot get the correct answer by this infinite process because such a process is invalid.
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.
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March 19th, 2018, 09:22 PM   #37
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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.
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March 19th, 2018, 09:26 PM   #38
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The distance $(r-r')$ can be close to zero or close to $\frac1{10}$ or close to $\frac23$. Any fraction between zero an 1 does the same job. You still have an infinite quantity of rationals unmatched. There isn't a "best" solution.
Your comments about there not being a best solution are right on track as you and I both know. We know that when $r$ and $r’$ are aligned, each pair of unaligned rationals must have a space between them (otherwise they would align), and that space is equal to the $|r-r’|$ you appear to be referring to should the rationals be aligned (ie, I say “appear” because $(r-r’)$ would be negative at the start of the game…). A space therefore exists, be it a space measurable by a real number or otherwise (such as an infinitesimal). Our inability to calculate a distance that is a real number with which to measure that space is a limitation of modern mathematics though, not an indication that the space doesn’t exist as you propose. Quite plainly, it is not philosophical to assert a space exists that must exist. Rather, it is necessary if we are to be mathematically sound. There is a big difference.

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I can. I maintain that there is no distance to talk about here.
No, you cannot maintain that there is no distance between the unaligned pairs of rationals when $r$ and $r’$ are aligned while simultaneously asserting that the unaligned pairs of rationals are in fact unaligned. So sorry, there will be no cake and eating it too in that case. You can maintain that we cannot calculate that distance and then ask why.

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Well neither "distance" is well defined. In mathematics that makes them not equal. Philosophically and intuitively they seem equal, but intuition is spectacularly bad at this game.
They don’t have to be well defined in the sense you are referring to in order to be equal on a non-philosophical level because their equality is easily proven. Let figure A represent an alignment of $r$ and $r’$ with each pair of rationals unaligned and let figure B represent an alignment of each pair of rationals with $r$ and $r’$ unaligned. Where $q,p,v \in \, (r,r+1\, ) \cap \mathbb{Q}$ and $q',p',v' \in \, (r',r'+1\, ) \cap \mathbb{Q}$, we can represent the space between each unaligned pair of rationals when $r$ and $r’$ are aligned, which is equal to the space between $r$ and $r’$ when each pair of rationals is aligned, simply as $x$:

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.

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Originally Posted by v8archie View Post
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 real-world intuition to guide you. It's a recipe for disaster in terms of actual meaning, I fear.
Yes, [Achilles] does [catch the tortoise] in the real world. Similarly, motion can be applied to a piece of paper in the real world just as we do in my game. In the theoretical world Achilles cannot catch the tortoise using Zeno’s method because he just gets infinitesimally closer to it. Likewise, we cannot calculate the distance between each unaligned pair of rationals when $r$ and $r’$ are aligned because our distance gets infinitesimally small. But, in the real world Achilles does catch the tortoise just as sliding a piece of paper would seemingly cause it to pass through the best solution that we cannot calculate.

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?
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March 19th, 2018, 09:27 PM   #39
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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.
Exactly my question for you now (see above), yes!!
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March 20th, 2018, 03:40 AM   #40
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Originally Posted by AplanisTophet View Post
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.
The reason we can't solve for $x$ in your diagram is that you can't give a value for $q$ and $q'$. If you could, the calculation would be simple.

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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