What the...... Those 2 triangles are definately distinguishable, one is equilateral and one is a right angle. They have different symmetry groups for starters, and you can prove that one has more symmetry elements than the other. Fix the angles and they are not isomorphic. If this is saying what I think you are saying then you are wrong. Circles, squares, equilateral triangles all have properties that are invariant on scale transformations. The ratio of circumference to diamter is pi no matter the scale.
Xgen, Here are some questions for you. Let A be a position in quantized space. How many other distinct positions are adjacent to A? (I.e. how many discreet places exists at a distance of 1 quantum from A?) Let A and B be positions in quantized space separated by a distance of 1mm. Can you explain how that distance is composed of an arrangement of 1 quanta distances? Please do so without appealing either explicitly or implicitly to angles as you have stated they make no sense at the quantum level. If AB and CD are macro distances of precisely the same length, are they each composed of the same number of quanta? Conversely, if AB and CD are macro distances composed of the same number of quanta, do they have precisely the same length? The answers to the questions will help me understand what your view of quantized space is. Then maybe I can answer your question more clearly. ***Edit*** In rereading your post, it occurs to me that you may not be advocating quantized space. Instead you might be asking what I meant by redoing geometry to get rid of irrationals and how it might solve the problem. If that's what you were doing, let me know and I'll try to say something about it.
Purported Solution http://www.deltalink.com/dodson/html/zeno.solution.html Stumbled across it in a search. Thought I'd share.
From http://www.andrews.edu/~calkins/math/biograph/199899/topparad.htm Zeno's Paradoxes: Dichotomy Paradox - Before an object can travel t, it must travel t/2, before it can travel that it must travel t/4, so on and so forth. This continues infinitely. Achilles and the tortoise Paradox - Achilles runs ten times as fast as the tortoise. The tortoise has a ten unit headstart. Achilles can never catch the tortoise. Arrow Paradox - An arrow flying, at a given time, has a certain position. So does a motionless arrow. How do you tell which one is moving? Stade Paradox - To assume (remember to assume....) that space and time can only be divided by a definite amount.
Re: Purported Solution Here's a quote from that link, followed by an explanation of why this is not a solution. So far as I know, Zeno does not explicitly make that assumption. However, I'd have to reread the original (or as close as I can get) to be sure. In any case, the assumption is not essential to the paradox. The only assumption is that in oder to step completely through a sequnce one element at a time, one must at some point be closer to the end of the sequence than one was at the beginning. Nothing in the proposed solution addresses this assumption or shows how it is wrong. This is a case where a mathematical "solution" goes awry because of a failure to understand the philosophy behind the original problem. The math is perfectly fine. It does answer some important questions. However it doesn't address Zeno's paradox at all.
Re: Re: Purported Solution You did notice that is was a solution to the second of Zeno's paradox's as listed in my last post.. rather than the first which is the topic if this thread right? I only put it up because it's directly related.. not that it's the answer to this thread. I'm not much of a mathemetician, what's the limit of RSUM(1/(x+1)), etc.) as x goes from 0->infinity?
Re: Re: Re: Purported Solution Yes, I noticed, though it's really only a matter of perspective. THe same issue underlies both paradoxes. However, the dichotomy is a bit more compelling to the modern ear. The limit of 1/x as x tends to infinity is 0. Oh, and the paradox that started this thread is a combination of 1 and 2 on your list. It's the dichotomy done backwards.
Re: Re: Re: Re: Purported Solution I was trying to ask what the limit of (1/(n+1) plus itself as a reimann (or however you spell it) sum from 0 to infinity. You know what I'm asking? Ack, I need to know how to use that math forumala HTML junk to ask my question correctly... hehe.. maybe I just need to be a better mathematician. Please Register or Log in to view the hidden image! EDIT: Okay, I'm a little slow. It's the geometric series I'm talking about and they say in converges to 2.
It may be a bit easier for people to see why calculus is irrelevant if we concentrate on the dichotomy. THe dichotomy shows that movement is impossible because it cannot start. At best calculus shows us where Achilles is when he just before he starts to cross the road, but it doesn't explain how he can start without taking a first step. Likewise with Achilles and the Tortoise, or the reverse dichotomy. Calculus tells us where Achilles is just after catching the TOrtoise or crossing the road, but it doesn't tell us how he manages to do so without taking a last step.
But doesn't calculus in essence state that it doesn't matter, we know he has crossed the room? In essence aren't you asking the ridiculus question "what is infinity minus one?".
Re: Re: Re: Re: Re: Purported Solution Ah, ok. And here I was busily trying to figure out how to do Riemann sums. It was fun though. For simplicity's sake we normally deal with the geometric series as x goes from 1 to infinity. That sums to 1 and we just choose our units as the total distance to be covered. That way all specific instances of the paradox can be described with the same equations.
I dunno, I'm all mathematically ignorant. I figured out it's gotta be the riemann sum of (1/2n) as n goes from 1 to infinity. Is that a geometric series? That's what would describe Paradox 2 right? Wouldn't taking that limit resolve the paradox if it's not divergent? That's what led to my infinity minus one question above.
We always knew he crossed the room. Even Zeno knew that. That's why it's a paradox. There are some similar problems that calculus solved. For example, the problem of infinite divisibility. Calculus shows us how a finite distance can be conceived of as the sum of an infinite sequence of ever decreasing intervals. An infinite number of intervals can sum to a finite distance. In this way Calculus actually shows that Zeno's description isn't the problem. If Achilles traverses each of the infinite number of ever decreasing intervals, he will have done so in a precisely definable finite distance. But notice that all we get here is a conditional. We still need to show that Achilles *n* traverse each of an infinite number of ever decreasing intevals. That's where the problem is. None of the intervals are a final interval. So there is no interval such that Achilles crosses the road (or catches the tortoise) upon completing that interval. So long as Achilles has just finished an interval, he has not yet crossed the road. But notice that all Achiles does is complete intervals one at a time. So there's nothing he does that can count as completing his crossing of the road. Imagine you have a bag with an infinite nuber of beans in it, and you have to empty the bag one bean at a time. Fortunately you are able to remove each bean in half the time it took you to remove the previous one. If it takes you 30 seconds to remove the first bean, we can calculate that you'll be all done in 1 minute. (I'm sweeping some issues about cardinality and transfinite subtraction under the rug here.) But how do you finish? You never take the last bean out of the bag. There is no last bean! No matter how many beans you remove, there are always the same number left. So you take out beans faster and faster, never getting any closer to the bottom of the bag, until suddenly, without taking out the last bean, the bag is empty! How is that possible? If Achilles crosses the road, then there must be something that he does in virtue of which he crosses the road. But nothing he does counts as crossing the road. So he can't doesn't cross the road. But of course we know he does cross the road. Ergo, paradox.
All that sums do is tell us how long it will take to cross the road *IF* it can be done. They don't tell us how to count to infinity one number at a time.
Understood, but that was the point of my posts above the posts above regarding perspective. No one said much so I wasn't sure if I was on the right track but you just basically agreed with me as far as I can tell.
I think this is the post you're referring to. I agree with what you say here except for perhaps one thing. As a matter of description, there certainly are paradoxes in our theories. There is a deeper question about whether all such paradoxes can or ought to be resolved eventually. The majority opinon seems to be that we should seek theories that are free from paradox. Paradoxes show that our theories are inconsistent, that they entail a contradiciton. There is a small minority who want to embrace contradicitons, claiming that some contradictions are true. Graham Priest is the founder of Paraconsistent Logic, which tries to make this claim plausible. There are even some who think that it is impossible to rid our theories of paradox for one reason or another, even though paradoxes show that our theories are mistaken. However we're now afield into discussion of paradox quite generally and away from Zeno. I'm in the middle of moving to a different state. After I get there I'll start a paradox thread, if no one beats me to it.
Please post a link here if you don't mind. I'll be most interested in what you've got to say on the issue. As a note: I think of these types of paradoxes as the collapsing of an instance of a wave function into a determined result. Sadly, I only mostly know what a wave function is. I'm just using it in the context I've heard it used regarding quantum physics and the likes. I've gathered that in essence a wave function is function with a set of possible outcomes, the outcome based on the perspective of an observer and the act of observation itself. The act of observation causing it to "collapse" and perspective (circumstance) in essence choosing a particular result.
It seems to me that paradox is the antithesis of reason. When we try to eliminate paradox we are attempting to make our picture of the universe make sense. So I guess it follows to ask: What is so reasonable about reason? Why is a reasonable view of the world preferrable to a paradoxical one? For me, the answer has to be its operational usefulness. If Zeno can create a mathematically consistent paradox, I choose to consider his math flawed(maybe even all math), because the paradox is not a useful concept for me. Maybe every time I approach my bathroom door, I actually fall into some kind of positional singularity then come out the other side into a seemingly identical universe so I can brush my teeth. Please Register or Log in to view the hidden image! This may, in fact, be what happens, but it doesn't help me with my dental hygiene.
Interesting perspective. Please Register or Log in to view the hidden image! I would disagree... I'd say that it defines the extents of objective reasoning. Note that paradox only arises in hypothetical situations and is always (except for extreme cases and none are confirmed) confined to abstract space. Further note that if you choose a perspective the paradox is resolved. It may not be a usefull concept for you, but that doesn't mean it's impossible for it to have use.
