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However, just because one cannot measure something "exactly" (i.e., acontextually, i.e., intrinsically) doesn't mean that measurement is not possible (i.e., subjective).

EXACT measurments doen't exist since atoms are always in motion. Talk to me about the smallest possible grains of matter at absolute zero and I will concede.

Of course, unit length my foot, will always =my foot, not that the measument wouldn't change in quantum size. (did I use quantum correctly?)

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Are you sure about that? Because something tells me that if you do the following:

.5 + .25 + .125 + ........

or

E [1/(n^2)] , where 1 <= n

you'll never really reach 1. You can get arbitrarily close, but never equal.

There is a mathmatical answer to Zeno that doesn't simply parrot out E [1/(n^2)], where 1 <= n. I'm currently looking it up to make sure I don't mispeak about it.

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His whole set up was flawed because the ancient Greeks did not have sufficient understanding of the mathematics.

You all over look the fact that Zeno made the paradox because he wanted the Greeks to re-evaluate there definitions. Zeno knew that his statements were not true, but that was the point. Zeno knew that there was an answer to every paradox he presented, but on the surface each one is right. Thats what makes them a paradox. Zeno wasn't preaching that motion is impossible, he just wanted to show that by the words we use we can manipulate logic and reality. Thus, no answer to the paradox is necessary, they're rhetorical.

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This whole thread is unecesary! I explained it to my friend and quote "WHAT IS THE POINT?"

I deem discussions like this to mental workouts, but no, there is no piont other than our amusement.

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This whole thread is unecesary! I explained it to my friend and quote "WHAT IS THE POINT?"

I deem discussions like this to mental workouts, but no, there is no piont other than our amusement.

Luckily philosophers and mathematicians disagreed with your friend when they were (eg) discussing how to formulate the calculus and theory of limits.

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Luckily philosophers and mathematicians disagreed with your friend when they were (eg) discussing how to formulate the calculus and theory of limits.

Ditto.

Something funny that always comes to mind whenever people bring up Zeno's paradox is the way Peikoff describes him. I've been listening to some of his lecture courses recently and whenever he mentions Zeno, he always say, "You know, that guy who couldn't walk across the room."

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Luckily philosophers and mathematicians disagreed with your friend when they were (eg) discussing how to formulate the calculus and theory of limits.

I apologize for being murky, but what I was calling pointless was such a long discussion, we could all answer this in one post: state "before moving an inch you move .5inch etc. etc.," then state about grainy universe; but we CAN move!, so there is something here (teleportation on quantum scale?). This either goes to a different subject such as what could this teleportation be, or a refuting of grainy universe.

Aight?

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• 1 month later...
She started, but realised that before she could fully answer the question, she had to half-answer it, but before she could half-answer it she would have to quarter-answer it. At that point she realized "This is a total waste of my time", and so she wrote Atlas Shrugged instead.

HAHAHAHAHAHAHAHAHAHAHAH!!!!!!!!!!!!!!!

The standard interpretation of Greek pre-Socratic philosophy is that Zeno was trying to show that motion was an illusion, and hence that change didnt occur. One of the main 'problems' for the Greeks was how change was possible. Parmenides held that the reality 'underneath' our sensations was eternal and hence couldnt actually change - Zeno's arguments were intended to illustrate this point. Compare it (for instance) to Democritus' argument that one thing could change into another because the world was actually made up of indivisible eternal atoms, and change was only a change in the configuration of these atoms rather than anything new coming into being.

Pretty much. He was, in the tradition of Parmenides, trying to show that the very idea of change was nonsense and that everything is one.

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• 3 weeks later...

Pat Corvini of the Ayn Rand Institute recently released a lecture series that resolves Zeno's paradoxes (especially the Achilles/tortoise one) and uses Ayn Rand's theory of concepts as a general principle to deal with all paradoxes.

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As we all know, to 'be' is to be something. The law of identity. No matter now minutely you break up your measurements of reality, for something to exist it has to be as large as at least one of the increments you're using.

Say that the distance between Point A and Point B is 10 "yadayada"s (the smallest unit of measurment in use), the object traveling between A and B has to be at least 1 yadayada in size. Once the object gets fully past yadayada number 9, no matter how Zeno wishes to record that motion, it necessarily is touching Point B - the 10 yadayada mark. It has arrived.

Edited by ggdwill
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You're not thinking infinitely. If something runs off into infinity, just because we haven't defined a number for X yet doesn't mean it doesn't exist. Existence is independent of consciousness; if I don't think of a measurement of something, that doesn't mean it's immeasurable. The "foot" didn't exist in 200BC. Does that mean nothing was a foot long? NO. So why, then, must something ONLY be a "yadayada" short? Because we haven't defined anything further? If we accept that the shortest measurement possible is "yadayada" then we cannot define anything shorter. If we can (we can), then your statement is moot anyway. My new measurement is equal to 1/4*yadayada. I'll call it the metricyada.

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Quite the contrary, every time I enter in to a discussion on this board, I expect to have to think infinitely.

By "yadayada" I was thinking about things like atoms. I didn't use that term because I'm not sure exactly how people measure things at that level, or if atoms really are the smallest units known. I realize that subatomic particles do exist and can be measured, but they cannot exist independently of the arrangement know as the atom. I hoped to bypass such objections by using a variable.

Also, you cannot quantify infinity because it does not have a quantity. It is a purely conceptual phenomenon representing NO THING in reality. If you decided to call some point that infinity passed though X, and determined it's value, infinity would still have to keep going in order to be infinity. Infinity is by defnition indeterminite and non-existent. Does that mean that there is an infinite number of things in existence. I don't know. It's an interesting topic for another thread.

The only other thing I can say to your comment is yes, just because some smaller unit of measurment hasn't been thought of doesn't mean it doesn't exist and can't be used. However, conversely just because it can be thought of doesn't mean that it actually does exist. The units of measurement at the atomic level - what I ignorantly call yadayadas - were identified and used for good reason. They weren't just arbitrarily ordained as "the smallest units of measurment possible". Scientists, and the Japanese, have direct experience with what happens when they attempt to create something that is smaller.

So, as I said, in order to be something, it has to be as big as, and be composed of, at least one yadayadya. If you attempt to cut it in half, or in to quarters, all you get is a violent rearrangment of yadayadas - not metriyadas.

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So then your answer is that zeno is wrong because the smallest unit of measurement is the length of an atom. Which isn't true, because I can concieve of 1/2 of the length of an atom. Are you trying to tell me that 1 atom-length exists, but 1/2 an atom length doesn't exist? That's a fallacy. 1/4 an atom length also exists, as does 1/8, 1/16, 1/32, 1/64, 1/128, 1/256, 1/512, 1/1024, etc. Whether or not we are able to measure it is completely irrelevent to its existing. 1=.5+.5=(.25+.25)+(.25+.25)=((.125+.125)+(.125+.125))+((.125+.125)+(.125+.125)), etc... If we can see its whole, how can you claim that any part of it doesn't exist? Because it's really small?

You can't quantify infinity going away from 0. You CAN quanify it approaching 0 or well... 1, which is what Zeno is doing. It's called a limit! If we were to put it in a geometric sequence, it would be E n^2(1/(n^2)). Zeno's paradox isn't a contradiction, it's just not mentioning all the relevant information. If I'm an arrow, the distance between me and my target can be divided up infinitely. Except, I'm traveling at 200m/s. No matter how you divide up 200m/s, you're still traveling that far. 200m/s=100m/s+100m/s=50m/s+50m/s+50m/s+50m/s, etc. It's got the sum, and you can add up to that sum an infinite number of ways.

Edit: Probably a more satisfactory answer: http://mathforum.org/library/drmath/view/52507.html

Edited by BNeptune
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Two things you said are important:

So then your answer is that zeno is wrong because the smallest unit of measurement is the length of an atom. Which isn't true, because I can concieve of 1/2 of the length of an atom.

Sure, you can conceive of it all you want. That doesn't mean that it exists. There is no such thing as 1/2 of an atom. An atom is the smallest unit of existence. Nothing can be broken down any smaller. Like I said, when it is tried, all that results is a violent reorganization of atoms. To talk about something being "1/2 an atom's length" away from Point B is absurd. What's in between? Something is in between, and to be something, means to possess at least one atom. When that something and the original object switch spots, the object has literally reached Point B.

And as for this:

Zeno's paradox isn't a contradiction, it's just not mentioning all the relevant information.

See what I just said above.

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There is such thing as half an atom. Are you telling me that you can have a whole atom without its left side?

That's irrelevent. I answered the paradox correctly, so any objections you have to it are irrelevent.

EDIT: If the atom is the smallest existing particle (which it isn't, but that doesn't matter), then nothing smaller than it exists. There is still space 1/2 its length and 1/2 of the atom still exists.

Edited by BNeptune
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There is such thing as half an atom. Are you telling me that you can have a whole atom without its left side?

That's irrelevent. I answered the paradox correctly, so any objections you have to it are irrelevent.

EDIT: If the atom is the smallest existing particle (which it isn't, but that doesn't matter), then nothing smaller than it exists. There is still space 1/2 its length and 1/2 of the atom still exists.

Right, but not in any meaningful sense. That's like saying an 8 man baseball team is a baseball team, or a 1 man baseball team is a baseball team. A baseball team is, by definition, 9 (or more) men - one for each of the 9 defensive positions. An atom cannot exist with out BOTH it's left side and right side intact. Yes, you can measure 1/2 of the distance of an atom, but you will never find 1/2 of an atom floating around in reality. There is not "space [equal to] 1/2 of the lenght of an atom" in which nothing exists. That space is occupied by something, which is composed of it's own atoms which play by the same rules.

Space, in it's common usage, does not exist. It is merely a means of describing the lack of some specific element or object, while implicitly relying on the premise that that "space" is composed of something else. Even if space literally did exist, it would be something. If I moved an object from one place to another empty space, it's original location would immediately be filled with space. Furthermore, even if I were wrong about this and space actually was some sort of gap in reality, what makes it forever beyond my ability to traverse? Certainly you're not going to propose that there is something that exists that is 1/2 the length of nothing.

Edited by ggdwill
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I'm sorry, but I disagree with that. I find billions of "half an atom" in existence.

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The problem with Zeno's Achilles and the tortoise paradox is the conflation of the abstract with the concrete. In the abstract sense, Achilles must run an infinite number of half distances, but he can; "infinite" just means "some number but any number." In the concrete sense, you would have to choose an actual (as opposed to infinite) number of finite lenghts to run; and Achilles can run any number of finite lengths more quickly than the tortoise can.

Edited to correct a typo.

Edited by LaszloWalrus
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According to the quantum physics I have heard most recently, they actually say that there is a smallest unit of distance and, while it is sensible to speak of half of it, subatomic particles never travel through it. They simply are at one point and the next moment in time are one quanti removed. [Edit: So this actually means that Zeno was more or less right.]

Edited by aleph_0
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Has Ayn Rand given a solution to Zeno's Paradox? and if she has not, how might she gone about solving it? Thank to all in advance.

Occum's Razor takes care of this. "You shouldn't multiply entities needlessly".

Yes, but given the time he made this remark, it's been quite brilliant. He practically proved that if space and time could be broken down infinitely, we wouldn't live in the universe we apparently do. I think it's brilliant.

When Henry Ford developed the Model T, for it's time, it was an exceptional achievement. However, I don't see the point in pursuing the process of rebuilding an obsolete model of car in order to pay hommage to a past innovator. I don't see much difference between the analogy and it's subject, though I am sure you will differ.

Edited by dark_unicorn
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I'm sorry, but I disagree with that. I find billions of "half an atom" in existence.
What does that mean? I understand how you can juxtapose the concept "half" and "atom", but are you saying that there is a fact out there of a "half an atom"? What are its properties? How come I haven't heard about one before? Or are you redefining "half" in a special way?
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What does that mean? I understand how you can juxtapose the concept "half" and "atom", but are you saying that there is a fact out there of a "half an atom"? What are its properties? How come I haven't heard about one before? Or are you redefining "half" in a special way?

I assume we are using "atom" to mean the smallest unit of existence, or some such thing. As such, picture and atom, now picture and atom in front of the original, partially obscuring it from your view. It could be said that you see one half of the original atom, as long as an atom isn't a point...which wouldn't really make sense being 0-dimensional and all.

Assuming that we can detect an atom, we should be able to, one day at least, measure it's size by comparison or displacement or something really really sexy and scientific. Even if nothing smaller than an atom exists, you could then refer to a length that is equal to one half of that of an atom.

In fact, such a designation would also be metaphysically relevent or at least, or at least make a hell of a lot of sense. Since the atom must take up space (correct me if we are talking 0-dimensional point atoms) there must be a specific distance from it's left side to it's right. We can designate this distance to be 1 SUAT (Smallest Unit of ALL-TIME), but SUATs in integer amounts won't be capable of mathmatically describing everything. Take two spherical atoms and place them side by side, now place an atom on top of each of the original two in a gridlike pattern(

00

00). How far is it from the top right to bottom left corners of the atom stack?

Edited by LaVache
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I assume we are using "atom" to mean the smallest unit of existence, or some such thing. As such, picture and atom, now picture and atom in front of the original, partially obscuring it from your view. It could be said that you see one half of the original atom, as long as an atom isn't a point...which wouldn't really make sense being 0-dimensional and all.
I think this way of speaking only really makes sense if you have a picture in your head of fundamental particles being something like extremely small little balls, rather than (eg) localised wave packets or fluctuations in a field (which is how they are actually treated in current theory). I dont think theres any good reason to assume in advance that the fundamental constituents of the universe 'must' have a size, and if you think that anything else would necessarily be incoherent then its possible that youre still thinking of them as being like tiny solid balls.

I believe that within our most current models of particle physics, the most fundamental constituents of the universe do not have size. There are other models though (for example, the strings of string-theory are not pointlike). Having said that, I'm not 100% clear about the precise meaning of 'point-like' in the context of theoretical physics and I wouldnt be surprised if it meant something slightly more subtle than 'being a geometric point'. As far as I know (and I freely admit to not knowing much about physics, so take this post with a large grain of salt), the actual story is that although fundamental particles are pointlike, their associated wavefunctions are smeared across space.

Also even if fundamental particles did have a size, its not obvious that we would ever be able to measure this accurately, due to the uncertainty principle.

According to the quantum physics I have heard most recently, they actually say that there is a smallest unit of distance and, while it is sensible to speak of half of it, subatomic particles never travel through it. They simply are at one point and the next moment in time are one quanti removed. [Edit: So this actually means that Zeno was more or less right.]
I believe the Planck length is something more subtle than that, and doesnt imply that space is actually discrete. One theory of quantum gravity, Loop Quantum Gravity, does predict that space is discrete on the planck scale, but this is not widely accepted among physicists.

But even if this trned out to be true, I dont think it would mean Zeno was right. "Being right" should imply more than just making a lucky guess. There was no good reason for anyone 2500 years to believe that space was either discrete or continuous, given that cutting edge physics today is still undecided on the question (and anyway, he wasnt arguing that space was discrete, he was arguing that movement was illusory). If you need to refer to highly technical details of science in order to settle a philosophical argument, I think this is a good sign that something has went wrong somewhere along the line.

Edited by Hal
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I assume we are using "atom" to mean the smallest unit of existence, or some such thing.
Okay, but that only makes the distinction, whatever it is, even more obscure. Since, after all, we know that actual atoms do have stuff in them.
As such, picture and atom, now picture and atom in front of the original, partially obscuring it from your view.
In reality, I'm skeptical about doing this but let's procede. So this is the "fruit of (imaginary) method" sense of existence, namely juxtaposing these two concepts. The distractor was BN's claim 'I find billions of "half an atom" in existence'. You don't really find them, you have to create them (they do not physically exist at all), and it's pretty much guaranteed that nobody has actually so created billions of "half-atoms" just as nobody have created billions of integers by actually mentally forming specific integers. Also, with numbers we can say with justified confidence that "124,383,765" is the same number as "124,383,765" so if you think of "124,383,765" today and "124,383,765" tomorrow, you would say that you thought of the same number twice, and not that you thought of two numbers which happened to be the same. That is, enumerating the integers between one and a billion is thinking of a billion numbers, but thinking "one, one, one..." a billion times isn't thinking of a billion numbers. With imaginary atom juxtapositions, I just wonder how you would invent a billion such pictures, rather than invent a mental picture once and then replay it in your mind a billion times.
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I think this all borders if not mirrors the discussion of either an infinite regress of substances or an essentially formless primary substance, addressed by Aristotle. Perhaps should it be moved to another topic, as it seems related to but distinct from the question of Zeno's Paradox?

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