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Zeno's Paradox

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psk177
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I think the essential error in Zeno's Paradox is mixing up the abstract and the concrete and therby treating an infinite as an actuality.

For instance, in an abstract race, Achilles can run any length divided into any number of finite smaller parts, but in an actual race, he must run some finite length divided into some finite smaller parts.

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It's an attack on the concept of infinitely small things. And it's a good one.

I think that if time is fluid and capable of passing in finite quantity, and space is fluid and capable of being passed through over finite time, then it follows that a runner can cross the finish line without a need for dividing by two ad infinitum. Personally, I don't think Zeno's paradox is a big problem for Rand, because it doesn't say as much about how the real world works as it says about how the real world doesn't work.

Edited by Arkanin
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I agree with the assessment that Zeno's paradox is a brilliant proof of discrete time and space. This is not what Zeno intended with the paradox, but its construction is still brilliant when interpreted correctly. The paradox can be presented as a set of premises and a conclusion:

Premise: time and space is continuous

Therefore (due to the Achilles/tortoise example): motion is impossible

HOWEVER, we know from observation that motion indeed is possible. HENCE, the premise is false. Time and space must be discrete. There must exist an indivisible smallest spacetime unit.

Where Zeno got it wrong is by taking a rationalistic turn when the conclusion conflicted with reality. Instead of concluding that the premise was wrong (which he should have) he concluded that sensed reality was wrong instead.

But when interpreted rationally Zeno's paradox is not only consistent with objectivism, but *exactly* what you would expect. Objectivism holds at its core that the basic stuff of existence is *objects*. But what is an object if not a *discrete* existent?

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Objectivism holds at its core that the basic stuff of existence is *objects*. But what is an object if not a *discrete* existent?
I assume you mean "entity" rather than "object". I don't know what it would mean for an entity to not be discrete. For example, a cup of water doesn't look like it's made up of bits, but it is, and a cup of water is different from the rest of the universe. Even if an electron isn't divisible, it's still distinct from the rest of the universe. As far as I can tell, the idea of a "non-discrete existent" is a contradiction, so I don't see how to grant the "what it" assumption. Are you suggesting that time and space are entities?
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I agree with the assessment that Zeno's paradox is a brilliant proof of discrete time and space. This is not what Zeno intended with the paradox, but its construction is still brilliant when interpreted correctly. The paradox can be presented as a set of premises and a conclusion:

Premise: time and space is continuous

Therefore (due to the Achilles/tortoise example): motion is impossible

HOWEVER, we know from observation that motion indeed is possible. HENCE, the premise is false. Time and space must be discrete. There must exist an indivisible smallest spacetime unit.

I completely disagree. Zeno's paradox shows an error inherent in Platonism: the mixing up of the abstract with the concrete. No matter how many times you divide the race-track up, in a concrete sense, you willl have some finite number of finite individual lenghts, each of which Achilles can traverse more quickly than the tortoise can. On the other hand, in an abstract race, Zeno must run an infinite number of lenghths, but he CAN, because using the concept of infinite correctly, there can be ANY number of finite lengths, but there must be SOME number of finite lenghts.

By the way, a good lecture on Zeno's paradox is this one:

http://www.aynrandbookstore.com/prodinfo.asp?number=CC22D

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I assume you mean "entity" rather than "object". I don't know what it would mean for an entity to not be discrete. For example, a cup of water doesn't look like it's made up of bits, but it is, and a cup of water is different from the rest of the universe. Even if an electron isn't divisible, it's still distinct from the rest of the universe. As far as I can tell, the idea of a "non-discrete existent" is a contradiction, so I don't see how to grant the "what it" assumption. Are you suggesting that time and space are entities?

"Discrete object" is a statement of the same kind as "existence exists." It is not a redundency but an elaboration of the underlying axiomatic nature of the concept. Just like you cannot deny existence without undermining your denial you cannot speak of non-discrete existents. The very fact that existents have identity implies that they are objects and hence discrete. Although not made explicit in her axioms of metaphysics the very name of Ayn Rand's philosophy -- objectivism -- is derived from the concept of the "object." In fact, if her metaphysics should ever be elaborated/expanded it would in my opinion be to include the axiom that existents are objects and objects are discrete.

And to answer your question: yes, time and space are built from objects. Time and space *exists*, hence it must be *something* that exists. This something must have an identity and hence be an object and hence be discrete. This follows from the axiomatic nature of existence.

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I completely disagree. Zeno's paradox shows an error inherent in Platonism: the mixing up of the abstract with the concrete. No matter how many times you divide the race-track up, in a concrete sense, you willl have some finite number of finite individual lenghts, each of which Achilles can traverse more quickly than the tortoise can. On the other hand, in an abstract race, Zeno must run an infinite number of lenghths, but he CAN, because using the concept of infinite correctly, there can be ANY number of finite lengths, but there must be SOME number of finite lenghts.

By the way, a good lecture on Zeno's paradox is this one:

http://www.aynrandbookstore.com/prodinfo.asp?number=CC22D

Well, I agree that Zeno's original *conclusion* is both rationalistic and platonic. It is platonic in the sense of confusing the abstraction of infinity with the concrete existents which are not infinite, and it is rationalistic in the sense of choosing a logical conclusion over what is observed by the senses ("motion is impossible"), which is a primacy of mind-error. But that is my point exactly: if you DON'T make the platonic and rationalistic errors you end up with a radically different conclusion from Zeno which completely eliminates the paradox. And the conclusion is that even though you in your _mind_ can hypothesize the infinite subdivision of time and space this leads to the obvious contradiction of the senses. Due to the primacy of existence your mental play with infinites must be thrown out. What remains then is the notion of a finite, discrete spacetime.

Let me also add that Zeno's experiment can be applied to consciousness and here we can empirically observe the effect of discreteness. Yes, the stream of consciousness too has a discrete existence, and it is easy to prove. It is in fact an everyday phenomenon. Consider the following experiment: a subject is shown a short flash of an image, say for a tenth of a second. (then it shown the negative image for a similar amount of time in order to eliminate retinal after-images) The image is, say, two parallel lines, one longer than the other. The subject is then asked to identify which of the two lines are shortest. Each time the subject succeeds the flash is shortened slightly. At some point the subject is unable to tell the difference between the two lines. In fact, at some point the flash is too short for awareness to register it altogether. This phenomenon corresponds to the Nyquist sampling theorem, which states that a discrete sampling of an analog signal at a sampling rate of frequency f will successfully sample all information in the signal with frequencies lower than f/2. In short, there is a threshold where the sampler no longer detects any signal. This is what happens in the mind too. If a sound is too short, we can hear it, if a detail is too small we can't see it etc. In short: CONSCIOUSNESS IS DISCRETE. This is an observable fact, but by Zeno's reasoning we could know it even before we have evidence for it because that is the way that it *must* be.

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The very fact that existents have identity implies that they are objects and hence discrete.
You didn't say anything about the question of what you mean by "object", so I'll be more direct: do you mean "entity" or "existent"? Attributes and relations are not objects, but they are existents.
And to answer your question: yes, time and space are built from objects. Time and space *exists*, hence it must be *something* that exists. This something must have an identity and hence be an object and hence be discrete. This follows from the axiomatic nature of existence.
The one thing that time and space aren't is entities: they are either attributes or relations (or entities), depending on how you are expressing them. They exist, they presuppose some entities (that is, they are aspects of the identity of the entity), but they aren't entities. Hence, I would conclude, they are not objects. What I'm objecting to is treating "existent" and "entity" as interchangeable.
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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.]

If he was right, it seems coincidental. I don't see any way to infer discrete units of time and space from Zeno's paradox.

By all means, if you have some kind of substantive logical or mathematical argument for the leap from one to the other, I would be happy to argue about it or even accept that you are right if it is very good. I do see that there is some kind of argument that you are trying to make about the nature of infinitely small units, but until there is more elaboration about that, all I can do is look stupid. :)

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"Zeno's Dichotomy paradox: Before a moving object can travel a certain distance, it must travel half that distance. Before it can travel half the distance it must travel 1/4 the distance, etc. This sequence goes on forever. Therefore, it seems that the original distance cannot be traveled, and motion is impossible."

This is the paradox of Achilles and a tortoise. It's already been answered by mathematics.

1/2 + 1/4 + 1/8 + ... = Sum[i = 0 to infinity] (1/2^i) - 1 = 1/(1 - 1/2) - 1 = 1/(1/2) - 1 = 2 - 1 = 1

Ayn Rand said she was never a great mathematician.

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"Aristotle pointed out that as the distance decreases, the time needed to cover those distances also decreases, so that the time needed also becomes increasingly small. Such an approach to solving the paradoxes would amount to a denial that it must take an infinite amount of time to traverse an infinite sequence of distances."

speed = distance / time

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(my 2cents)

For Zeno's Paradox to be true, you must "assume" that there was a starting point of stillness with "zero" movement in any direction.

Nothing is still. Everyting is in motion (even at a micro-level)... So there is no POINT A.

Everything is infinitly moving and there is no time measurement that can clock it.

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Nothing is still. Everyting is in motion (even at a micro-level)... So there is no POINT A.

Everything is infinitly moving and there is no time measurement that can clock it.

This is essentially a Heraclitean view of reality and it leads to all sorts of problems when applied consistently. Then again, how can something ever be consistent if everything is always changing?

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I've been fascinated with Zeno's paradoxes (esp. the Runner Paradox) ever since I was presented with them in an sort of Intro. to Phil. class I took in H.S.

I agree with Felix that it seems to imply a discreteness to space (and also time -- the fundamental unit of time being the time it takes to "jump" from one unit of space to an adjacent unit at the speed of light). I just wanted to add the following thought to the conversation.

It's convenient (and progressively popular in some places :confused: ) to draw comparisons between the universe and a computer system. After all, the universe contains what we call information or data (positions, velocities, etc.). And we perceive the universe to behave according to certain laws, some of which we have suitably gleaned. Though the comparison is no doubt clumsy, these laws can conveniently be thought of as the universe's software. Furthermore, it's evident that we have the capacity to mirror at least some of the universes processes in our own minds (our computer systems are powerful tools to this end as well), and by so doing we in effect reproduce the universe's own information processing, the seeming essence by which it produces change within itself. We depend on the laws we've discovered in this way to consistently make reliable and accurate predictions.

So, if we assert that when the runner moves from the starting line to the finish line, information is processed by the universe to produce his change in position, the argument for discrete space is furthered when you consider that his run would require an infinite amount of time to process -- this if in fact his run is through a continuum of infinite positions.

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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.

She gave a solution while writing about infinity in Introduction to Objectivist Epistemology:

An arithmetical sequence extends into infinity, without implying that infinity actually exists; such extension means only that whatever number of units does exist, it is to be included in the same sequence.

-ITOE p 22.

http://aynrandlexicon.com/lexicon/infinity.html

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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.

Nice reply ! :)

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This is essentially a Heraclitean view of reality and it leads to all sorts of problems when applied consistently. Then again, how can something ever be consistent if everything is always changing?

In mathematics and physics one asks what remains unchanged under some group of transformations. For example Euclidean Geometry is the geometry of transformations on the underlying space that leave distance between points unchanged (or as mathematicians like to say, invariant).

Physicists are in search of those measurable properties of the physical cosmos that remain invariant under physical changes. From this we get our well know conservation laws, among which are the conservation of energy (in the relativistic sense), momentum, angular momentum. The maximum entropy of the cosmos is a constant which the transient entropy approaches (second law of thermodynamics).

So it is possible (and even necessary) to have constancy amidst change.

Bob Kolker

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