Runner's Paradox: Finishing the Race in d Metres

[Hurkyl]

This whole conversation has me thinking along those lines though, and I must confess to already have a strong belief that time and space must both be quantized.

I would too like to see a discrete model of space-time (marginally different than "quantized", but I think you mean discrete anyways)... but I don't go promoting the idea because there's no proof.

I still believe that I'm on the right track thinking in terms of what Einstein said that distance is what we measure with a rod, and time is what we measure with a clock.

Right. In fact, in my casual reading of both GR and QM, the issue of what constitutes a "rod" and "clock" is one of the more interesting and still active research topics. One interesting fact is that, according to QM, any physical clock has a finite probability of giving the same reading twice!

So yes, it's bogged down in the messiness of the real world. But unless you want to simply be a Hilbertian formalist, you're kind of stuck with that.

I am a formalist! (at least to some extent)

I put a very clear demarcation here; it is the purpose of mathematics to tell us what are and are not the consequences of a set of axioms, and it is the purpose of science to determine which axioms the subject of interest models.

(but the two are not disjoint)

I take Zeno's Paradox to simply be saying that if you attempt to actually add an infinite number of points to a line you can never succeed. Not even by using pure logic! Logic itself is telling us that this is an impossible task. Set theory only serves to verify this conclusion.

Logic does not tell us this is an impossible task; logic merely tells us that induction is not sufficient to prove it possible.


Incidentally, the question about lines is not whether we can place a set of points on a line, but a question about the set of points that are on the line. So while induction cannot (by itself) prove that we can place infinite points on a line, it can prove that there are an infinite number of points on a line.

(given the reponse written while I wrote this, I think this warrants proof, so I'm writing it now)

 

[Hurkyl]

(the context is Euclidean geometry)

Theorem: Any line has an infinite number of points on it.

Proof:

Let L be any line.
Choose P on L.


Lemma: For n > 1, there is a set of n points on L, S_n, such that there is a point Q_n in S_n such that for any other point R in S_n, R*Q_n*P. (That is, Q_n is between R and P)

Base case:
An incidence axiom guarantees that any line has at least two distinct points. One of those points has to be different than P; call it A.

A betweenness axiom guarantees a point Q such that A*Q*P. Define S₂ := {A, Q}. This satisfies the statement of the lemma for n = 2.

Inductive step:
Let S_n satisfy the statement of the lemma. The same betweenness axiom guarantees the existence of a point Q_n+1 such that Q_n*Q_n+1*P.

For every point R in S_n different from Q_n, it is true that R*Q_n*P. We also have Q_n*Q_n+1*P, thus we can conclude R*Q_n+1*P. (I don't know wehre my text is, I can't remember if this is an axiom or a theorem) This allows us to conclude that R is distinct from Q_n+1

Define S_n+1 := S_n U {Q_n+1}. This set satisfies the statement f the lemma.

Thus, the lemma has been proven for all natural numbers n > 1.


Let T be the union of all of the S_n for natural numbers n > 1

Every point in T lies on L (because it is an element of one of the S_n's)

|T| >= |S| = n for all natural numbers n > 1, so T cannot be finite.

Therefore T is an infinite set of points on L, and this concludes the theorem.

 

[drnihili]

Originally posted by NeutronStar
I don’t see how you can say that. The truth value of each sentence is dependent on the very set that it is a member of. Or should I say, the membership of each element in the proposed set is dependent on the truth value of a statement about the structure of the set itself. If that isn't self-reference I have no idea what is!

Self reference is when a sentence refers to itself as in "This sentence is false". If you want to prohibit reference to any set of which the sentence is a part, then you won't be able to say anything about sentences at all. Not even innocuous things like "the first sentence of this post begins with the letter 'S'."

But whatever, this isn't a thread about self-reference. I was merely pointing out that there are lots of cases where we can find things such that if they are true of n, they are true of n+1 and yet they are not true of the natural numbers taken as a set.

 

[drnihili]

Originally posted by Hurkyl
II am a formalist! (at least to some extent)

I put a very clear demarcation here; it is the purpose of mathematics to tell us what are and are not the consequences of a set of axioms, and it is the purpose of science to determine which axioms the subject of interest models.

(but the two are not disjoint)

That's fine. In that case you have to agree that calculus (nor any branch of mathematics) cannot by itself resolve Zeno's paradox. Any resolution would have to consist not only of a set of axioms and cosequences but also an interpretation of the axioms in terms of the real world and also an argument that the axioms correctly model the world. The first of these tasks might, perhaps, be mathematical, but the second clearly isn't.

So from your revised formalist perspective, you still owe an argument that you have an accurate model of what's going on. At this stage it's entirely unclear how such an argument might go.

 

[Hurkyl]

Zeno's paradox is a pseudoparadox; it derives no contradiction. There's nothing to resolve.


As to whether an infinite divisbility model correctly models the world, that's a completely separate question.

 

[drnihili]

Originally posted by Hurkyl
Zeno's paradox is a pseudoparadox; it derives no contradiction. There's nothing to resolve.


As to whether an infinite divisbility model correctly models the world, that's a completely separate question.

Ah, no answer. Not surprising I suppose.

 

[NeutronStar]

drnihili wrote:
Self reference is when a sentence refers to itself as in "This sentence is false". If you want to prohibit reference to any set of which the sentence is a part, then you won't be able to say anything about sentences at all. Not even innocuous things like "the first sentence of this post begins with the letter 'S'."

I was referring to logical self-reference not just any self-reference. Your example, "The first sentence of this post begins with the letter 'S'." is merely as statement. It may be true or false. It doesn't demand that its truth value be dependent on the condition stated.

Now if the sentence were to be rewritten as ""The first sentence of this post begins with the letter 'S' only if this statement is false." Then we would have a logically self-referenced situation. It's true that the self-reference would be with respect to the entire post, and not merely with respect to the sentence itself. But it is still a form of logical self-reference none the less.

drnihili wrote:
Consider an omega sequence (a set ordered as the natural numbers) of sentences where sentence S_n is given by
"There is an m>n such that S_m is false."

In this situation each element of the proposed set is making a demand that a certain truth value exist with respect to other specific elements within the same set to which this element belongs. I clearly see the logical self-reference here with respect to the set as a whole. I don't understand how you can seriously deny this. [?]

 

[drnihili]

I have no idea what you mean by "logical self-reference". However I can assure you that it's pretty uncontroversial that there's no self reference there. Try looking up the original article (by Yablo in Analysis. The whole point of the paradox is to avoid self reference. If you can find a critic who has claimed that Yablo failed in that, I'd be interested.

I grant you that there is a certain oddity to the sentences, but the oddity has absolutely nothing to do with self reference. You also said

Your example, "The first sentence of this post begins with the letter 'S'." is merely as statement. It may be true or false. It doesn't demand that its truth value be dependent on the condition stated.

which strikes me as exceedingly odd. Of course the sentence's truth depends on the condition stated. How could it be otherwise?

**Edit**
Are you prehaps worried because the sentence refer to the truth or falsity of other sentences rather than making direct claims?

 

[NeutronStar]

drnhilili wrote:

NeutronStar wrote:
Your example, "The first sentence of this post begins with the letter 'S'." is merely as statement. It may be true or false. It doesn't demand that its truth value be dependent on the condition stated.

which strikes me as exceedingly odd. Of course the sentence's truth depends on the condition stated. How could it be otherwise?

My point is that your example here would not be an example of logical self-reference. It would merely be an independent form of self-reference.

The only time that statements get involved in logical self-reference is when they start making statements about the state of their own truth value, or the truth value of other statements that are directly related to them (like other members of the same set that they themselves belong to).

If expert logicians believe that Yablo's Paradox is not a form of logical self-reference then I have lost complete faith in expert logicians. I can see the logical self-reference clear as day. Not as a self-reference of any particular element onto itself, but as a self-reference relative to the set as a whole.

All of the elements in the set make up the entire set. Therefore if any specific element within the set demands that the truth value of a statement contained in any other element in the same set is dependent on its position relative to the original specific element,… Well, it's clear to me that any set composed on such a rule is definitely referencing its own composition to decide a particular truth state about its own composition.

If that's not logical self-referencing then I don't know what is.

 

[HallsofIvy]

drnhili wrote:
quote:
--------------------------------------------------------------------------------
Originally posted by Hurkyl
Zeno's paradox is a pseudoparadox; it derives no contradiction. There's nothing to resolve.


As to whether an infinite divisbility model correctly models the world, that's a completely separate question.
--------------------------------------------------------------------------------



Ah, no answer. Not surprising I suppose.

You mean- no answer that you understood.

 

[drnihili]

Originally posted by HallsofIvy
You mean- no answer that you understood.

No, I mean no answer. I understood what was said. If you'll take the time to read my post just prior to the one you quoted from Hurkyl, you'll see that I point out that a certain kind of explanation is required based upon Hurkyl stated position. Instead of providing such an explanation Hurkyl merely retreated to his "psuedoparadox" refrain. Then he goes on to say that the question of modeling is a separate one, which is of course absurd given his earlier post about the demarcation.

So, he provided no answer to my post. He has yet to provide any justification for his position that Zeno's is a pseudoparadox. His demarcation post does finally admit that calculus is not sufficient to resolve it. But faced with that admission he simply retreats to a domatic position and avoids the question at hand.

Please understand, I am not saying that calculus is somehow flawed or wrong. It's just not the right sort of theory. Neither is arithmetic, but that doesn't mean arithmetic is wrong.

But Hurkyl somehow knows he's right. 2000 years of study and effort have all been misguided, he's got the one true answer. And it's so obvious he doesn't even need to provide justification for it.

 

[Hurkyl]

If you'll take the time to read my post just prior to the one you quoted from Hurkyl, you'll see that I point out that a certain kind of explanation is required based upon Hurkyl stated position.

I'd like to see, in your own words, what you think my position is.

 

[drnihili]

Originally posted by Hurkyl
I'd like to see, in your own words, what you think my position is.

Your position in this regard is that science and mathematics have different purposes. Mathematics is supposed to tell us what follows from what. Science is supposed to tell us what axioms correctly model the real world (or some relevant portion thereof). Further, you believe that science and mathematics have some overlap, at least in practice, but probably also in theory.

So, how close is that?

 

[Hurkyl]

Well, yes.

But I was asking about my position with regard to Zeno's paradox; the position to which you make an accusation like

"But Hurkyl somehow knows he's right. 2000 years of study and effort have all been misguided, he's got the one true answer. And it's so obvious he doesn't even need to provide justification for it."

 

[drnihili]

Originally posted by Hurkyl
Well, yes.

But I was asking about my position with regard to Zeno's paradox; the position to which you make an accusation like

"But Hurkyl somehow knows he's right. 2000 years of study and effort have all been misguided, he's got the one true answer. And it's so obvious he doesn't even need to provide justification for it."

Ah, since you quoted a place where I was talking about your position on science and math, I assumed that was the postion you were asking about.

I'm on my way out the door now, so I can't say much more than that you think there is no contradiction involved. When I get back I'll try to give a fuller account.

 

[drnihili]

Ok, as I understand it, your position is roughly the following.

1. The consistency of calculus demonstrates the consistency of an inifinite divisibility model.

2. Induction is sufficient to show that all of the ever decreasing segments lie on Achilles' path from 0 to d.

3. Since d is the limit of the series of segments, completing the series is sufficent for reaching d.

(I agree with you on 1, 2, and 3 by the way.)

4. The above points suffice to show that Achilles can reach d by covering each of the ever decreasing segments in order. (At least barring a fully rigorous derviation to the contrary.)

(Point 4 is where I see the disagreement)

This is all from memory of posts, so I may be mistaken at one or two spots, but that's where I see your position.

Oh, you also seem to have a general background belief that all paradox is the result of a lack of clarity.

 

[Hurkyl]

4. The above points suffice to show that Achilles can reach d by covering each of the ever decreasing segments in order. (At least barring a fully rigorous derviation to the contrary.)

You sure you stated this right? If Achilles does cover all of the ever decreasing segments (in order), then we just have point 3.

(actually, this is contingent on the assumption that paths are continuous, and that there aren't any silly tricks like Achilles ceasing to exist at the very instant he would be at d)

Oh, you also seem to have a general background belief that all paradox is the result of a lack of clarity.

I think this warrants further explanation.

Mathematicians have known about the various paradoxes for a while, and to the best of my knowledge they have sufficiently tweaked the formalism so that all known paradoxes are either impossible to state (such as the liar's paradox) or have been reduced to pseudoparadoxes (such as Russel's paradox).

The prototypical example of "lack of clarity" is that of the Twin paradox; the applied formulas require that the reference frame be inertial, but this condition is overlooked and the formula is applied in a noninertial frame. A number of paradoxes arise in this manner merely by making logical mistakes.

 

[drnihili]

Originally posted by Hurkyl
You sure you stated this right? If Achilles does cover all of the ever decreasing segments (in order), then we just have point 3.

(actually, this is contingent on the assumption that paths are continuous, and that there aren't any silly tricks like Achilles ceasing to exist at the very instant he would be at d)

Yes, I stated it correctly, but there is sufficient ambiguiity in the English to warrant further explanation. A lot depends on emphasis that isn't recoverable from text.

I think you hold that 1-3 demonstrate that the series is completable, that by running one segment at a time Achilles can run all of the segments.

I think we're all in agreement that if Achilles completes the series, then he will have arrived at d. The question is whether Achilles can complete the series. I think you hold that 1-3 provide sufficient argument that he can.

 

[drnihili]

Originally posted by Hurkyl
I think this warrants further explanation.

Mathematicians have known about the various paradoxes for a while, and to the best of my knowledge they have sufficiently tweaked the formalism so that all known paradoxes are either impossible to state (such as the liar's paradox) or have been reduced to pseudoparadoxes (such as Russel's paradox).

The prototypical example of "lack of clarity" is that of the Twin paradox; the applied formulas require that the reference frame be inertial, but this condition is overlooked and the formula is applied in a noninertial frame. A number of paradoxes arise in this manner merely by making logical mistakes.

All I can say is that the best of your knowledge isn't very good on this point. The Liar paradox continues to be an active area of research and there is no generally accepted solution to it. Tarski's hierarchical approach and Kripke's fixed point theories are perhaps the most commonly cited, but not far behind are defalationsist and various context dependent approaches including at least one based on set theory with an anti foundational axiom. Not all solutions make the paradox impossible to state, revision theory and paraconsistency are good examples of theories that aim to preserve the paradox while removing it's sting.

There are wide variety of ways of tweaking the formalism to avoid the repurcussions of the Liar in formal systems. Even so, it remains a very open question which, if any, of them correctly models the Liar in natural languages. It would be a gross misrepresentation to say that Liar paradox has been shown to be the result of a lack of clarity. Three are some theories that take that stance, but by no means all of them, nor even the most important of them.

If by Russell's paradox, you mean the Barber paradox, then I could perhaps agree. But that was never really meant as a full blown paradox anyway. It was merely an example of the problems with unrestricted comprehension. Set theory was modified to have only restricted comprehension and thus avoided inconsistency. However the root problem remains, and set theory's response just isn't very illuminating. It is by no means clear that restricted comprehension is ultimately the right answer. Though admittedly it does pretty well for most things. I suspect that Neutron Star would have similar misgiving about resorting to restricted comprehension as he does about resorting to the empty set.

In general, initial mathematical responses to paradox are primarily frantic attempts to avoid inconsistency. They generally succeed in doing so, but it doesn't follow from that that the paradox has been laid to rest or shown to be just the result of foggy thinking.

 

[Hurkyl]

The question is whether Achilles can complete the series. I think you hold that 1-3 provide sufficient argument that he can.

Ok, this is where you have me wrong.

When we went over Zeno's paradoxes in my Greek philosophy class, they were presented as being an actual contradiction in Greek times; geometery permitted space to be infinitely divisible, but Greeks held infinite sequences to be impossible, and Zeno could derive from that a contradiction... we know Achilles can get from here to there, but we know Achilles cannot get from here to there because of the infinite sequence of tasks in the way.

Most of the time, when the paradox arises on forums, the poster tends to be of that same persuasion, so I typically start off by addressing the typical reasons why one might think an infinite sequence of tasks is impossible... thus the discussion about calculus and infinite series.


You, however, are asking a completely different question. The root of your question is why should one accept a given model of reality. Dressing it up in the guise of Zeno's paradox only serves to obfuscate the issue. And bundling your question up with Zeno's paradox connotes that there's some aspect about the paradox that is relevant to your question, causing me to respond attempting to show that your question + Zeno's paradox is no deeper than your question by itself. (I did comment on this earlier, but it seems to have been missed)

IOW, if you're serious about pressing the "Why should we use model X to describe reality", you really should ask it in its own context, at which point I make the boring statement that mathematics doesn't care about your question, science gives only empirical evidence that model X is practical, and that there has not yet been given any good reason to accept any other particular model. Certainly not logical proof of anything, but I think you're savvy enough to realize you're asking an unanswerable question. (though I do think it was bad form to complain that the unanswerable question was, in fact, not answered)

All I can say is that the best of your knowledge isn't very good on this point. The Liar paradox continues to be an active area of research and there is no generally accepted solution to it.

I beg to differ; mathematics has accepted a solution to it. P(Q) is simply not in the language of mathematical logic. I'm not trying to imply that this is the final word on the issue, just that in typical mathematical fashion we have taken a minimalist structure that avoids all the contradictions yet permits us to do all the logical steps we like to do in mathematics. (though possibly not metamathematics... I don't know all of the gritty details I would like to know about that subject)

However the root problem remains, and set theory's response just isn't very illuminating.

But it is practical. As with logic, the idea of ZFC was to take a minimalist approach that permits us to do everything we like to do, but not have enough power to derive the contradictions. I don't see anything wrong with that, and unless you really want to work with pathologically large sets, nothing is lacking in the solution either.


In summary, one can answer all the paradoxes. It might not be the best answer, but it is an answer, and it is probably contained in any better answer to boot.

 

[NeutronStar]

On the topic of Paradoxes

On Restricted Comprehension

drnihili wrote:
I suspect that Neutron Star would have similar misgiving about resorting to restricted comprehension as he does about resorting to the empty set.

Unfortunately I'm not familiar with the idea behind restricted comprehension so I can't say.

On Russel's "set of all possible sets" paradox

As far as Russel's Paradox is concerned I was actually thinking that Hurkyl was referring to Russel's set of all possible sets. I do see this as a legitimate paradox, but I also see it as being caused by the flaws associated with the empty set. Repair set theory and Russel's Paradox is not longer possible.

On Russel's "Barber" Paradox

"I shave all those men, and only those men, who do not shave themselves."

As I see it, the paradox of which set the barber himself belongs to is based on the assumption that the above quote is true. My simple solution is to say that the barber's statement is simply incorrect, it can't be true. Where's the paradox? The barber is simply mistaken. That's all.

On The Twin Brothers Paradox

The twin brothers paradox is only a paradox if we insist on keeping absolute time. Since we have accepted the concept of time dilation there is no paradox here. It is well understood, and has just kept its original title as a paradox. No modern physicist sees it as a real paradox.

We may still have questions concerning the actual physics of how time dilation is accomplished, but in general we have accepted that this is the case.

On Zeno's Paradox

I'm afraid that I must take a different stance than Hurkyl on this one. I do see Zeno's paradox as a valid logical paradox. I don't accept the methods of calculus as a solution because, as I understand them they do not profess to solve the problem of completing an infinite number of tasks. They simply give the results of what would happen should they somehow be completed.

In some ways, I might consider mathematical induction in this case. However, as I pointed out many posts ago, assuming the Achilles can make the first step is really cheating, because if we turn the problem around Achilles can never even start the race. In order to start he would first need to step half-way to the first point that he is trying to step to! Ouch! Actually I see this as a mirror-type self reference. (trying to touch a mirror at a point other then were your finger is reflecting on it!) That's not a paradox, its just an impossibility. Unless you silver the mirror on the back side and use very thick glass!

Like the mirror reflections I see Zeno's Paradox as a valid reflection of reality. I take Zeno's Paradox quite seriously. And just like the Twin Brothers Paradox, which is only a paradox is you are unwilling to let go of absolute time, I see Zeno's Paradox as only being a paradox if you are unwilling to give up the idea of space and time being continuous. Give that up and Zeno's Paradox becomes nothing more than another strange property of reality. It's no longer a paradox if you accept that space and time must be finitely divisible.

Finally, to Hurkyl I would like to say the following:

Hurkyl wrote:
I would too like to see a discrete model of space-time (marginally different than "quantized", but I think you mean discrete anyways)... but I don't go promoting the idea because there's no proof.

However, you appear to be going around promoting the idea that space must be infinitely divisible. Yet, you don't seem to have any real proof for that either!

It's true that I can't prove that space can only be finitely divided. But I can't prove that it can be infinitely divided either! So I promote what I believe to be the most likely case based on other things that I know.

As I have mentioned, I have reason to believe that a finite line can be logically said to only contain a finite number of points. This supports my case logically (I don't claim that it is a proof). However, I haven't seen anything that I would consider to be a stronger proof to the contrary.

I have reason to believe that our conception (and experience) of time and space is based solely on bound quantum states.

I have reason to believe that bound quantum states are quantized.

I have reason to believe that any idea of absolute space must be abandoned.

In short, I have more reasons to believe that space and time must be finitely divisible than I have to believe otherwise. So while I can't prove which case is true, I promote the idea that space and time are quantized and finitely divisible. It seems the more reasonable thing to do based on my current understanding of things.

 

[Hurkyl]

Unfortunately I'm not familiar with the idea behind restricted comprehension so I can't say.

It's the axiom of subsets thing I mentioned earlier. Unrestricted comprehension means that it is rigorously justifiable to construct the set S such that:

P(x) := x in x
S := {x | ~P(x) }

And then, it is in the language of logic to allow one to ask if S is in S. Both true and false lead to contradictions, thus the problem.


The restricted comprehension axiom (axiom of subsets) only gives us the ability to say

S := {x in A | P(x) }

so with this axiom one cannot construct the problematic set. Now, the axioms of ZF don't say you cannot construct that set; instead the paradox becomes proof you cannot construct the set in ZF (Assume you can, derive contradiction, thus you can't), and thus we say that ~P(x) is a proper class; we can only refer to this class with logical propositions, we cannot make a set contining all of its elements and analyze the whole thing with set theory.

I see Zeno's Paradox as only being a paradox if you are unwilling to give up the idea of space and time being continuous.

I only see it as a paradox if one insists on holding the notion of continuous time and space along with the notion that infinite sequences of tasks are impossible. Really, discrete space is not free from pseudoparadoxes; we can again go with a modified Zeno for this one: if distinct units of time are separated from each other, rather than being connected in a continuous fashion, then at each instant of time no motion can occur; there is no difference between a stationary arrow and an arrow in flight, so motion is impossible!

(incidentally, one reason science thus far rejects discrete spaces is that every discrete space model proposed predicts blurring of distant objects that simply isn't observed)


Incidentally, QM and GR are why I argue that infinite divisibility does seem to be the right answer; both derive their results by presuming space and time are continuous, and thus far, no discrete model has duplicated their success... and certainly by no means has the continuous model been disproved.

 

[drnihili]

Originally posted by Hurkyl
I beg to differ; mathematics has accepted a solution to it. P(Q) is simply not in the language of mathematical logic. I'm not trying to imply that this is the final word on the issue, just that in typical mathematical fashion we have taken a minimalist structure that avoids all the contradictions yet permits us to do all the logical steps we like to do in mathematics. (though possibly not metamathematics... I don't know all of the gritty details I would like to know about that subject)

I'm guessing that you're restricting mathematical logic to a sort of standard first order logic as one might see it in a beginning logic course? If so I grant you that the Liar isn't stateable in that language, but so what? Remove the words "true" and "false" from English and it's not stateable in English either. If your only point is that for an paradox we can restrict the expressive power of our language so that the paradox is not stateable, then so what? That's guaranteed by the construction of a null language.

The fact is that mathematics does not typically restrict itself to that sort of logic. But even if it did, there's nothing here about a lack of clarity or precision. There are perfectly clear mathematical treatments of the Liar which don't make it unstatateable. That you choose to dismiss the majority of the field of logic is irrelevant here. Fortunately mathematics is not generally so cavalier.

But it is practical. As with logic, the idea of ZFC was to take a minimalist approach that permits us to do everything we like to do, but not have enough power to derive the contradictions. I don't see anything wrong with that, and unless you really want to work with pathologically large sets, nothing is lacking in the solution either.

Certainly one can choose to work in such a system. Of course one thus gives up a lot of interesting mathematics. However, choosing to work in a restricted system does not amount to solving the paradox.

In summary, one can answer all the paradoxes. It might not be the best answer, but it is an answer, and it is probably contained in any better answer to boot.

Again, merely restricting the expressive power of a language does not amount to answering a paradox. If anything it amounts to a refusal to answer it. As for being contained in any better answer, that's simply false as even a cursory glance and the literature would show.

 

[drnihili]

Post deleted by the moderator.[/color]

 

[drnihili]

Originally posted by Hurkyl
Ok, this is where you have me wrong.

When we went over Zeno's paradoxes in my Greek philosophy class, they were presented as being an actual contradiction in Greek times; geometery permitted space to be infinitely divisible, but Greeks held infinite sequences to be impossible, and Zeno could derive from that a contradiction... we know Achilles can get from here to there, but we know Achilles cannot get from here to there because of the infinite sequence of tasks in the way.

[...]

You, however, are asking a completely different question. The root of your question is why should one accept a given model of reality. Dressing it up in the guise of Zeno's paradox only serves to obfuscate the issue. And bundling your question up with Zeno's paradox connotes that there's some aspect about the paradox that is relevant to your question, causing me to respond attempting to show that your question + Zeno's paradox is no deeper than your question by itself. (I did comment on this earlier, but it seems to have been missed)

IOW, if you're serious about pressing the "Why should we use model X to describe reality", you really should ask it in its own context, at which point I make the boring statement that mathematics doesn't care about your question, science gives only empirical evidence that model X is practical, and that there has not yet been given any good reason to accept any other particular model. Certainly not logical proof of anything, but I think you're savvy enough to realize you're asking an unanswerable question. (though I do think it was bad form to complain that the unanswerable question was, in fact, not answered)

I beg to differ; mathematics has accepted a solution to it. P(Q) is simply not in the language of mathematical logic. I'm not trying to imply that this is the final word on the issue, just that in typical mathematical fashion we have taken a minimalist structure that avoids all the contradictions yet permits us to do all the logical steps we like to do in mathematics. (though possibly not metamathematics... I don't know all of the gritty details I would like to know about that subject)

Let me just put it this way. As a person with a PhD focusing on Logic who has been empolyed teaching the same for several years, and as a person who has been doing active research on paradoxes for over a decade and who has been focusing on aspects of Zeno's paradox for about 5 years now, I can assure you that your apparent grasp of the fundamentals here is somewhat less that complete. I would suggest that before you go around trying to correct others, that you first acquaint yourself with the related literature.

It's all fine to have a discussion that is based solely on the arguments at hand. I actually prefer that. But when you start making general claims about a field, and when you take it upon yourself to correct others views about the state of that field, it's best to have some acquaintance with the field that goes beyond a brief exposure as part of one class.

Most of the time, when the paradox arises on forums, the poster tends to be of that same persuasion, so I typically start off by addressing the typical reasons why one might think an infinite sequence of tasks is impossible... thus the discussion about calculus and infinite series.

May I suggest that in the future you simply read what others post and attempt to understand them, rather than just adopting a pro forma approach based on the assumption that you think you already know what they are going to say. I think you'll find that it makes discussions more productive for all concerned. You may even learn a bit in the process.

As for the current discussion, I had a try at stating your position, how about you see if you can state mine?

 

[Hurkyl]

I'm so astonished by your arrogance that at this time I cannot respond to the bulk of your last couple posts without personal attack.

However, choosing to work in a restricted system does not amount to solving the paradox.

It's no longer a contradiction, paradox solved. The restricted system is sufficient to do any mathematics I have ever used and read.

If this does not count as a solution, then maybe you should explain what you mean by "solution".

May I suggest that in the future you simply read what others post and attempt to understand them,

To be frank, you do not have the gift to say a couple sentences and have everyone who hears magically understand precisely what you mean. You might recall that I have spent quite an effort to get you to explain better what you mean (which you have, for the most part, steadfastly refused to do).

Maybe you should heed your own advice. For example, did you notice that you went off on your lecture to explain to me the composition fallacy after I had already brought it up in response to the gods with vows paradox? (though I didn't refer to it by that name)

As for the current discussion, I had a try at stating your position, how about you see if you can state mine?

Aside from the assertion I don't have a clue about what I speak, I honestly don't know what your position is or what point you're trying to make. My most recent best guess was that you were trying to bring up the issue of how we know which model describes reality, but you haven't responded to any of my comments attempting to draw a discussion out of you on the topic.

 

[drnihili]

Originally posted by Hurkyl
I'm so astonished by your arrogance that at this time I cannot respond to the bulk of your last couple posts without personal attack.




It's no longer a contradiction, paradox solved. The restricted system is sufficient to do any mathematics I have ever used and read.

If this does not count as a solution, then maybe you should explain what you mean by "solution".

Perhaps you should instead set out what mathematics you have used and read and what your basis is for saying that the logic is sufficent. I've given you references to the literature and provided the basis for my own claims to know what the field contains. Thus far all you've done is stated that you once took a philosophy course that mentioned Zeno. If that is your only basis for the sweeping claims you make, then the arrogance that should astonish you is your own.

To be frank, you do not have the gift to say a couple sentences and have everyone who hears magically understand precisely what you mean. You might recall that I have spent quite an effort to get you to explain better what you mean (which you have, for the most part, steadfastly refused to do).

Maybe you should heed your own advice. For example, did you notice that you went off on your lecture to explain to me the composition fallacy after I had already brought it up in response to the gods with vows paradox? (though I didn't refer to it by that name)

Whether what you brought up was the fallacy of composition or just a worry about induction in a particular case is not decipherable from your posts. In any case, whether you previously brought it up is irrlelevant to whether you committed it. When I see someone blatantly committing a fallacy, I think it's appropriate to point it out. If you were aware that you were committing it, then perhaps you owe an expanation of why you did.

And no, you've spent little to no effort looking for explanations. By your own admission (repeated now) you reacted to a thread based upon what you had seen in similar past threads. As for your critique of my writing style, I'm entirely uninterested. When you become an editor, let me know.

Aside from the assertion I don't have a clue about what I speak, I honestly don't know what your position is or what point you're trying to make. My most recent best guess was that you were trying to bring up the issue of how we know which model describes reality, but you haven't responded to any of my comments attempting to draw a discussion out of you on the topic.

Then again I recommend the art of reading. Careful reading that pays attention to what is said. And when you really feel like you don't have a clue about what another person is saying, that's not generally a good time to start telling them that they've got the position wrong.

Now, lest we devolve into simply expressing disgust with each other, let me say again what the issue is, from my point of view of course.

The question is how does Achilles complete an infinite sequence one element at a time, given that the sequence is unbounded.

Caluculus can tell us how a finite segment can be composed of an infite sequence of ever smaller finite segments, but it says nothing on the issue of how one can proceed through the sequence from start to finish. You can't simply appeal to induction in this case, since induction equally shows that the end can never be reached, that there will always remain an infinite set of segments to traverse no matter how many have already been traversed.

So, in short how does one reach the end of a sequence that has no end, but only an upper bound? Not, what is the upper bound, but how can one reach it one element at a time?

 

[Hurkyl]

Now, lest we devolve into simply expressing disgust with each other

Before moving on, I'd like to point out that it's extremely bad form to get in your last insults in the same breath you suggest we stop insulting each other.

The question is how does Achilles complete an infinite sequence one element at a time, given that the sequence is unbounded.

Caluculus can tell us how a finite segment can be composed of an infite sequence of ever smaller finite segments, but it says nothing on the issue of how one can proceed through the sequence from start to finish. You can't simply appeal to induction in this case, since induction equally shows that the end can never be reached, that there will always remain an infinite set of segments to traverse no matter how many have already been traversed.

So, in short how does one reach the end of a sequence that has no end, but only an upper bound? Not, what is the upper bound, but how can one reach it one element at a time?

I will restate what was my primary question on this. What precisely do you mean by "How can one ...?" An attempt at precise definition or some explicit properties would be nice, but barring that, picking some thing and answering "How can one do this thing?" just to give an example may be helpful.

 

[ahrkron]

Originally posted by Hurkyl
Before moving on, I'd like to point out that it's extremely bad form to get in your last insults in the same breath you suggest we stop insulting each other.

I completely agree.


Stick to arguments. Credentials have no weight in any discussion, and showing them tends to deteriorate discussions.

Instead of saying "I've studied so and so for so many years", it would be much better for all if you just share some of your allegedly solid understanding. Some may like your arguments, some may not, but such evaluation should come from the arguments themselves, regardless of the time anyone has spent on them.

 

[drnihili]

Originally posted by ahrkron
I completely agree.


Stick to arguments. Credentials have no weight in any discussion, and showing them tends to deteriorate discussions.

Instead of saying "I've studied so and so for so many years", it would be much better for all if you just share some of your allegedly solid understanding. Some may like your arguments, some may not, but such evaluation should come from the arguments themselves, regardless of the time anyone has spent on them.

Credentials have weight when the issue turns to the question of what the accepted view in a field is. Ideally, this thread should have stayed on the topic of Zeno's paradox and whether it is resolvable. However, it instead strayed to the topic of what the accepted view in a field is. For example, my claims that the Liar paradox is still widely considered to be unresolved requires some substantiation. That substantiation can only be supplied by reference to literature or some other statement of authority as the question is not answerable by reason alone. Likewise, Hurkyl's claims regarding the status of paradoxes, the proper interpretation of Zeno, and the status of mathematical logic cannot be resolved by pure reason. They require some documentation. When documentation is not supplied, credentials are the only thing saving the discussion from brute assertion.

I supplied my credentials only when you specifically brought up the idea of doing so via PM. I do find it passing odd that you should now dismiss them.

So let's put credentials behind us if you wish. But let us also put behind us the sorts of claims that can only be resolved by appeal to them. Or at the very least let's supply some sort of reference to the literature when we make sweeping claims about the status of a field.

 

[drnihili]

Originally posted by Hurkyl
Before moving on, I'd like to point out that it's extremely bad form to get in your last insults in the same breath you suggest we stop insulting each other.

qed

I will restate what was my primary question on this. What precisely do you mean by "How can one ...?" An attempt at precise definition or some explicit properties would be nice, but barring that, picking some thing and answering "How can one do this thing?" just to give an example may be helpful.

And I'll repeat my answer from before.

If Achilles accomplishes an infinite series of tasks, there must be some action of his which counts as completing all the tasks. But none of the tasks can be that action as each of the tasks leaves an infinite number remaining. So, if Achilles accomplishes all the tasks, then there must be something he does beyond the tasks themselves in virtue of which he can be said to have completed them all. By the description of the problem, there is no such action.

If there were such an action, then it would be theoretically possible for Achilles to accomplish each of the tasks and yet still fail to complete all of them. This is absurd. Hence there can be no such action.

I obviously can't give you an example of completing an infinite sequence as the possibility of doing so is what is at issue. In a finite case examples are easy to come by. Suppose that Achilles' task is to count all the integers form 1 to 100,000 in sequence. There is an action he does which counts as finishing the sequence - the act of counting the number 100,000. By doing that act he finishes the sequence.

So, either tell me what action of Achilles' counts as finishing the sequence or explain why no such action is needed.

 

[Hurkyl]

A few questions (to try and flesh out the meaning of things):


If, by some means, it could be proven that Achilles could run a straight line from point A to point B, would that be sufficient to prove that Achilles can complete the infinite sequence of tasks:

{ Go from A to the 1/2 point between A and B,
Go from the 1/2 point to the 3/4 point,
Go from the 3/4 point to the 7/8 point,
... }


If so, would the above be an answer to the question "How can Achilles complete this infinite sequence of tasks?"


If Achilles counted:
1 - 2 - 3 - 4 - 6 - 7 - 8 - ... - 99,998 - 99,999 - 100,000
I presume that this would not count as finishing the task of counting from 1 to 100,000. Can you flesh out the steps in your proof with some rigor? Or if not, would it be correct to say that the completion of the overall task was accomplished, not because it can be said Achilles completed a final task, but because it can be said every task in the list was accomplished?


And in light of the previous observation, would it be correct to say that the sequence of tasks does not need to have a final task in order for Achilles to complete the sequence?


And finally, is the inquiry of the form:

Can Achilles complete the infinite sequence of tasks required to get from point A to point B?

or of the form

Does there exist points A and B such that Achilles can complete the infinite sequence of tasks required to get from point A to point B?

 

[drnihili]

Originally posted by Hurkyl
A few questions (to try and flesh out the meaning of things):


If, by some means, it could be proven that Achilles could run a straight line from point A to point B, would that be sufficient to prove that Achilles can complete the infinite sequence of tasks:

{ Go from A to the 1/2 point between A and B,
Go from the 1/2 point to the 3/4 point,
Go from the 3/4 point to the 7/8 point,
... }


If so, would the above be an answer to the question "How can Achilles complete this infinite sequence of tasks?"

No, there's no question about whether Achilles can run from A to B. That is physically demonstrable. What is not demostrable is whether his run can be understood as the completion of an infinite sequence of shorter runs. Infinite divisiblity gives some reason to think that it can, but does not give an answer to the question of what counts as completing the series.

If Achilles counted:
1 - 2 - 3 - 4 - 6 - 7 - 8 - ... - 99,998 - 99,999 - 100,000
I presume that this would not count as finishing the task of counting from 1 to 100,000. Can you flesh out the steps in your proof with some rigor? Or if not, would it be correct to say that the completion of the overall task was accomplished, not because it can be said Achilles completed a final task, but because it can be said every task in the list was accomplished?

I'm not entirely sure what you're asking here. Are you asking whether the entire sequence may be taken as a single task? If so, then the answer is that given the 100,000 tasks already on the list there can be no further task which counts as completing them all. The reasons for this are given in my previous post. Given this your proposal in the last sentence is a non-starter.

And in light of the previous observation, would it be correct to say that the sequence of tasks does not need to have a final task in order for Achilles to complete the sequence?

I don't know which sequence you are referring to here. But I am not presupposing that a sequence must have a final member in order to be completed. It is of course easier to understand how such a list sequence might be completed. Understanding what completion of an unending sequence amounts to is part of the current problem though. Appeals to supertasks are fruitless for reasons already stated.

And finally, is the inquiry of the form:

Can Achilles complete the infinite sequence of tasks required to get from point A to point B?

or of the form

Does there exist points A and B such that Achilles can complete the infinite sequence of tasks required to get from point A to point B?

Of those two the former is better, though I prefer the inquiry in the form "How can Achilles complete the infinite sequence of tasks required from point A to B" as it emphasizes that more is required than a simple yes or no response.

 

[Hurkyl]

Infinite divisiblity gives some reason to think that it can, but does not give an answer to the question of what counts as completing the series.

Are you merely asking for a definition of what counts as completing an infinite sequence of tasks?

I'm not entirely sure what you're asking here.

Mainly I'm asking for you to present the gritty details of a proof.

Your post appears to equate "the completion of a finite sequence of tasks" with "the performance of the final task".

My example was to demonstrate that you're leaving something fairly important implicit.


Is "performing the final task" synonymous with "completing a sequence of tasks"?

Is "all of the tasks in the sequence have been completed" sufficient to conclude "the sequence of tasks has been completed"?

If so, then the answer is that given the 100,000 tasks already on the list there can be no further task which counts as completing them all.

If I understand you correctly, you are justifying this with "So, if Achilles accomplishes all the tasks, then there must be something he does beyond the tasks themselves in virtue of which he can be said to have completed them all. By the description of the problem, there is no such action."

I don't see how that follows from the description of the problem. Unless you've been leaving it implicit this whole time, the problem does not state "Achilles does not perform any action you can imagine other than those that complete a single task in this sequence... not even actions that can be described as being composed of the aforementinoed allowed actions".

And it's unclear precisely what you mean by "action" anyways. Do actions occur over extended periods of time? Can actions occur instantaneously? Must the completion of a task (or sequence of tasks) be describable by an action? What about the converse?

Of those two the former is better, though I prefer the inquiry in the form "How can Achilles complete the infinite sequence of tasks required from point A to B" as it emphasizes that more is required than a simple yes or no response.

Is the quoted statement synonymous with:

"Can Achilles complete the infinite sequence of tasks required from point A to B? Support your answer with proof."

 

[drnihili]

Originally posted by Hurkyl
Are you merely asking for a definition of what counts as completing an infinite sequence of tasks?

No, I am not merely asking for a definition. A definition might be part of an answer, but it certainly wouldn't be an answer all by itself. At the very least it would require also demonstrating that Achilles' run could satisfy the definition. It would probably also require at least a prima facie argument that the definition was apt.

Mainly I'm asking for you to present the gritty details of a proof.

Your post appears to equate "the completion of a finite sequence of tasks" with "the performance of the final task".

No, my post explicitly denies that equation.

My example was to demonstrate that you're leaving something fairly important implicit.

If you believe I'm leaving something implicit, then come right out and spell it out. Provide, as you put it, the "gritty details" of a proof on the matter. At the very least provide a clear statement of what you take to be implicit and some reason for thinking that it is implicit based upone what I've said.

Is "performing the final task" synonymous with "completing a sequence of tasks"?

As stated in my previous post, I am not assuming that a sequence needs a final element to be completable. Hence I am not even holding that the two phrases are coextensive, let alone synonymous.

Is "all of the tasks in the sequence have been completed" sufficient to conclude "the sequence of tasks has been completed"?

Quantifiers in English are notorious for introducing ambiguities, but I'll provisionally say yes given something like a substitutional interpretation of them.

If I understand you correctly, you are justifying this with "So, if Achilles accomplishes all the tasks, then there must be something he does beyond the tasks themselves in virtue of which he can be said to have completed them all. By the description of the problem, there is no such action."

No, if you'll look at the context in which that quote occurs you'll see it's part of a proof by cases. The first case deals with one of the tasks counting as completing the list. Because that case does not hold when the sequence is infinite, I then take the case in the list is completed by some action which is not one of the tasks. However in the finite sequence scenario, the first case does hold.

The problem with a super task, as we have seen before, is that it is empty. As a result it has no explanatory value.

I don't see how that follows from the description of the problem. Unless you've been leaving it implicit this whole time, the problem does not state "Achilles does not perform any action you can imagine other than those that complete a single task in this sequence... not even actions that can be described as being composed of the aforementinoed allowed actions".

No, that's not implicit. Achilles can do lots of things. If you believe he does something beyond the tasks in the list, then please state what it is. Then we can see whether the task could be usefully used to resolve the issue.

And it's unclear precisely what you mean by "action" anyways. Do actions occur over extended periods of time? Can actions occur instantaneously? Must the completion of a task (or sequence of tasks) be describable by an action? What about the converse?

I am willing to countenance both instantaneous and extended actions. As for what an action is, I think the intuitive notion is sufficient for the nonce. If there is some specific ambiguity or vagueness you're worried about, then say what it is and I'll attemtp to clarify it. But at this stage your response is sounding more like a blanket call to define all terms. This of course cannot be done, so pace specific worries about particular words, I'm not going to attempt it.

Is the quoted statement synonymous with:

"Can Achilles complete the infinite sequence of tasks required from point A to B? Support your answer with proof."

No, though I would probably accept "Can Achilles complete the infinite sequence of tasks required from point A to B? Support your answer with an explanation."

 

[NeutronStar]

Finishing The Race

I'd like to make a couple of comments on the topic. However, before I do I feel a need to touch on the topic of credentials, simply because it had been brought up previously. I personally have no formal degree at all. However, it would be a gross mistake to take this to mean that I have no formal education. I have actually spent more than enough time in college classrooms to earn a PhD had that been my goal. And much of that time was spent as instructor rather than student so no credits were accumulated during those years.

Furthermore, just recently I have returned to college to basically start over from scratch at an undergraduate level. Why? Because I am more interesting in gaining a solid understanding of the fundamental concepts than I am about gaining credentials that supposedly show that I know what I am talking about when in fact I don't.

Albert Einstein has clearly shown us that a sincere interest to understand a problem, and a genuine dedication to focus your mind on the fundaments is more productive than merely going through the motions to earn credentials. When Einstein wrote his papers on the photoelectric effect, and on Relativity many PhD's did not want to believe him. They were taught to believe in absolute time and space and they did not want to go against what they thought they knew. They were wrong. Einstein was right. So much for credentials.

Now for the problem at hand I can't help but offer the following observation.

There is necessarily a distinction between infinite divisibility and unbounded divisibility. Unfortunately I can’t point to a formal distinction between the two because I'm not even sure whether mathematical formalism has made such a distinction. But if they haven't, they most certainly should.

A while back I posted a comment about "The Points Salesman". He has an infinite deck of cards (an infinite set) and each card (each element of the set) represents a finite number of points. By allowing the runner to chose a card (any card) prior to the race we have a distance that is finitely divisible but it also has unbounded divisibility. There is no bound on how fine we can divide it up. Yet, there is a bound on how we can divide it up - we must divide it finitely.

However, Zeno comes along and rejects the Points Salesman approach. Instead he demand that you continually redefine your line while you run the race. How does he doe this? By demanding that you step to the half-way point to the finish line with each step that you take. This forces the infinite property of the deck (set) onto the cards (elements). But in reality none of the cards (elements) actually has the property of being infinite.

So from a mathematical point of view I can clearly see what Zeno is doing. He is forcing the property of the set onto its elements. Yet in mathematical formalism we have no rule or axiom that allows us to do this!

Zeno is forcing infinite divisibility onto a case of finite but unbounded divisibility.

This is my purely logical resolution to the paradox. It resolves the paradox with respect to our understanding of mathematical formalism. Even though our mathematical formalism is not very rigorous in such matters!

However, I am much more interested in the reality of Zeno paradox. Achilles was a real person in the real universe running a real race to a real finish line. And the cheering from the crowd only serves to emphasize that fact that he can indeed complete the race.

So my question is this: As physicists how can we explain the finite unbounded divisibility of the material universe. And is it really unbound? As far as I'm concerned it is obvious that it is not infinitely divisible. For if it were, we would need to conclude that an infinite number of task can be finitely completed. That flies in the face of our very meaning of infinity. So for this very reason we must reject it. We simply cannot accept the idea that infinity is finite. That is a contradiction of concepts. So we must come up with a concept that has no contradiction. And if we simply accept that space cannot be infinitely divided we have solved the paradox. Thus, this is my solution.

I actually think that Zeno was quite the genius to figure out how to create an illusion of forcing the property of a set onto its elements. But there is really nothing in mathematical formalism that permits us to do this. Even using Zeno's very method to build another set all we end up with is yet another infinite set containing an infinity of finite elements. Zeno's method does not actually transfer the property, it merely give the illusion of property transfer.

 

[HallsofIvy]

Albert Einstein has clearly shown us that a sincere interest to understand a problem, and a genuine dedication to focus your mind on the fundaments is more productive than merely going through the motions to earn credentials. When Einstein wrote his papers on the photoelectric effect, and on Relativity many PhD's did not want to believe him. They were taught to believe in absolute time and space and they did not want to go against what they thought they knew. They were wrong. Einstein was right. So much for credentials.

Yes, Einstein's papers on relativity (not so much the photo-electric effect) were controversial and resulted in a lot of argument- as they should.

One of the reasons that Einstein's theories were given that consideration was that he HAD the "credentials"- a Ph.D from a prestigious German University and several prior publications.

 

[sage]

ok this is too long for me to read. i have skimmed through some of the arguments though. here is my question. suppose there is before me an infinite number of apples in an infinite basket. my task is to pick up the apples one by one and throw each into a second infinite basket. my task will be regarded as complete when the first basket is
completely empty. now when will i finish my task if (1)the time i need to pick up an apple and throw it in the second basket remains constant throughout the duration of my task? and (2)the time taken decreases by half for each successive apple i choose?
i have given here two conditions. the second case seems to me to be somewhat equivalent to the xeno's problem. the question is whether it is possible to complete the said task in a finite time and if so how is that physically possible?

 

[drnihili]

Originally posted by sage
ok this is too long for me to read. i have skimmed through some of the arguments though. here is my question. suppose there is before me an infinite number of apples in an infinite basket. my task is to pick up the apples one by one and throw each into a second infinite basket. my task will be regarded as complete when the first basket is
completely empty. now when will i finish my task if (1)the time i need to pick up an apple and throw it in the second basket remains constant throughout the duration of my task? and (2)the time taken decreases by half for each successive apple i choose?
i have given here two conditions. the second case seems to me to be somewhat equivalent to the xeno's problem. the question is whether it is possible to complete the said task in a finite time and if so how is that physically possible?

As stated neither of your problems has an answer. This is because it is possible to remove an infinite number of apples from the basket and still have an infinite number remaining. In order to get a solution we need to assume that the apples are ordered, say by each haveing a label with a number on it, and that you take them out in order.

Given that revision, and given the assumption that it is possible for you to finish the tasks at all, the answers are:

1) When eternity has passed.
2) In twice the time that it took for you to move the first apple.

Now, let me add a further twist. Suppose you place a coin on a table, and each time you move an apple you turn the coin over. The coin starts out showing heads, and so shows tails after the first apple is moved, heads after the second, and so on. When you finish your task, is the coin showing heads or tails?

 

[dschou]

zeno will finish the race quite quickly since the calculator he uses to determine the length of each step has limited precision and will quickly underflow (after about 330 steps for a TI-83). thus an age-old paradox is unraveled with the advent of the semiconductor.

:)

 

[HallsofIvy]

I'm glad we've got THAT settled!