Possible lingual solution to paradoxes.

[tar]

kristalris.

 

There are a number of paradoxes that I prefer to look at, from a lingual, sensible solution point of view. If something does not add up, or does not work or creates a logical impossibility or contradiction then its a clue to me, that there is work to be done, in rephrasing the question. Putting the two gears on the rod, so to speak, and seeing how that analogy or model works out.

 

In my OP I talked about a situation where the faster moving thing would never overtake the slower moving thing, as that they were "locked" into their relative positions, by the setup of the problem. Any movement of the one, would be exactly linked to the movement of the other and any progress one made was matched proportionately to the progress of the other.

 

In the Zeno paradox such terms as "by the time...the other would have". These words link the two together in such a way, as they can not independantly proceed as a tortoise and a warrior in the real world, would, in an independant fashion, normally proceed.

 

The words of the setup, precude one from considering actual things, like where would the warrior be after running at warrior speed after 1 minute and where would the tortoise be, after 1 min of traveling from his/her "100 meter" head start position at tortoise speed.

 

If the bind up in the situation happens in one's head, whether it is on an analogy level, or a logical level or a mathematical modeling level, then the problem is in ones head, and not on the race track. The useful models that we make in our heads, whether they are lingual or done with symbolic logic or done with mathematical expressions, are, on a base level, only useful, if they have an analogy in reality. If they do not have an analogy in reality, then they are imaginary, and don't require actually fitting or working in reality, as is the case with actual stuff, that operates in a cause and effect, positional manner.

 

As Kant would have said, in regards to his table of Judgements, which cover everything that one can think about an object in general, which equates to everything that can be said about an object in general and hence correlates to his "categories", our judgements are based on our two a priori intuitions, that of space and that of time.

 

Thusly our perception of the world, our judgments of the world and what we can say in general about any object is inherently a "positional" thing. The very neuron firings upon which our perception and judgements and thoughts and language are based, have to happen at a particular place, over time. The chemical environment and connections and pathways of perception and memory and thought being real, actual situations and events that are occuring in the real world in possible, fitting ways, must "make sense" and not be impossible. So we have a solid basis, upon which to build a lingual understanding of a situation. And any symbols that we use, or analogies that we engage, are already sensible, to some degree or another, in that they both mirror reality, and are composed of real positional events and configurations happening over time.

 

We, on a certain level, must already have a feeling for the function and the relationship, the quantity and the quality and the mode, before we write the equation that expresses it. So the meaning comes prior or together with the symbol that stands for the thing. And a lingual solution is just as good and solid as the mathematical one that symbolizes said meaning.

 

And if the meaning is impossible, you can say that it is impossible, in words or symbols, but if the thing is possible, there are probably words we can find to say something understandable about it.

 

And if the warrior will overtake the tortoise, then no symbol system that expresses an inability of the warrior to overtake the tortoise, is correctly put together, or signifying the correct aspects of the situation. And one should rightly rethink the situation and say something meaningful and understandable about the situation, and not get hung up on words or symbols that don't fit together right.

 

Regards, TAR

[md65536]

There are a number of paradoxes that I prefer to look at, from a lingual, sensible solution point of view. If something does not add up, or does not work or creates a logical impossibility or contradiction then its a clue to me, that there is work to be done, in rephrasing the question. Putting the two gears on the rod, so to speak, and seeing how that analogy or model works out.

But in this case it literally does add up. It is reasonable, and resolved, mathematically. You have not even demonstrated that your "lingual solution" can convey the meaning (or an understanding) of the problem, let alone resolve it.

In the Zeno paradox such terms as "by the time...the other would have". These words link the two together in such a way, as they can not independantly proceed as a tortoise and a warrior in the real world, would, in an independant fashion, normally proceed.

Those aren't the words that are the problem.

 

You probably can describe the paradox without math, and resolve it in words. Every time the warrior catches up to the tortoise, the latter has moved on. Or to make it more purely linguistic: Every time the warrior's position is made to mean the same as the tortoise's position, the tortoise's position has been made to mean something different. Or something like that. The paradox is that the warrior's position will never mean the same thing as the tortoise's, even though we know they must mean the same thing in the end.

 

The solution is that "never" in this context means not after any (finite) number of iterations. It does not mean "not after any (finite) amount of time". A linguistic confusion of the implied meaning of the word "never" causes the paradox, and sorting out the meaning can resolve it.

 

But is that just a description of the math, in words? Is there a linguistic understanding of numbers of iterations and the meaning of "finite" etc without the math? Does the solution make sense without considering the math? The meaning of "An infinite number of iterations as described can be completed in a finite time" is proven and understood in the math; are the words alone satisfactory?

 

 

I feel that the more words you've used, the more you've wandered away from the meaning and understanding of the problem. That's not a math vs words thing... I think it's analogous to writing pages of numbers and connecting them with relations, saying "The mathematical answer is in here somewhere, I've just got to write the numbers until the solution presents itself." You can question the concept of meaning, etc, but I don't see you getting any closer to describing anything related to Zeno's paradox. So I think that your attempt at reasoning through the paradox, with a goal of shunning the math, has not been successful.

Edited June 5, 2014 by md65536

[kristalris]

Tar I agree with md65536. As soon as you start to use to much words to clarify the position words lose from the mathematics. Words only win when you don't get yourself lured into the deep thought of a seeming contradiction that a paradox should be understood to be. An apparent seeming contradiction can immediately and logically be taken as to be just that: a seeming contradiction.

 

This proves the strength of lingual common sense over unnecessary and thus dangerously over-complicating in effect simple problems via mathematics or long lingual reasoning. Dangerous as in lessening the probability of reaching a stated goal to which the reasoning was used.

 

In effect several stated paradoxes in physics suffer the same problem. Such as entanglement. The problem nearly always lies in taking a wrong turn at the start of the game: in the assumptions or garbage or non garbage taken into the logic. There is a religious lure that draws the majority in to a problem that isn't there if treated correctly from the start.

 

Thus the rule of science should be on Zeno as a metaphor for all such paradoxes: if you spot a non intuitive solution having thus a possible contradiction always check all your assumptions, for you probably have made a mistake there. You can check this via changing the prior assumptions and see where that leads in comparison. In Zeno thus you have a intuitive solution namely taking it as a fact that Achilles will overtake the tortoise => no problem => then don't create one.

Edited June 5, 2014 by kristalris

[tar]

md65536,

 

Shunning the math is only required in the case where the math comes up with the wrong answer.

 

Where the math says the warrior overtakes the tortoise, then the math has a good chance of not being shunned, but used to describe the situation correctly.

 

There are some logical binds or situations one can get into, that one can think their way out of. When one suceeds in thinking their way out of the bind, so that everything adds up and works and the right analogies and transforms have been applied to the right peices at the right time so that the model reflects the reality, then the problem is solved, and one can say "oh, that makes sense, that is how it works." Nice. I get it.

 

In your post you say that the word never is used in two senses, one in a time duration sense and one in a number of iterations sense. Exactly my point about sorting out where the concepts are linking the warrior and the tortoise and where the math locks a situation in, and where the math is inappropriately applied.

 

I still remember my general thinking in a math class in high school, where the paradox was presented. I was imagining the progression of the warrior getting closer and closer to the position of the tortoise, and every time the warrior took a warrior step the tortoise was not yet completed taking his/her slower and shorter(in distance) tortoise step. It was not many warrior steps, until the warrior had tranversed the intitial head start, and was now negating several tortoise steps with every warrior step, up until the warrior was indeed within a single step of the tortoise, at which point his next step, would place his foot ahead of the spot on the ground that the tortoise's next step was heading. Once this particular warrior step was completed, the warrior would have overtaken the tortoise, regardless of the fact that the tortoise was continually advancing during the overtaking step. So I saw no paradox. Never did. Never will. Because there is no paradox there. The paradox is only in one's mind, if they make a mistake in what is standing for what.

 

Regards, TAR

[studiot]

Like many I have been following this thread with interest and would like to make the following comments.

 

1) The OP did not contain a question, but since this is the speculations thread I assume that the statements made were a proposed analogy for explaining Zeno, and this is borne out in the Title statement, which is also not phrased as a question.

 

2) The thread deteriorated into a dispute about mathematics v linguistics.

 

3) Now since there is nothing that can be said in mathematics that cannot be said linguistically this is or was a pointless diversion.

 

4) Since there are those who will wish to argue against (3) they should note that this may involve going back to the basic axioms of maths, which are all stated linguistically and developing the necessary mathematics for resolving the problem. Using Maths is of course more concise. So a linguistic answer may be very very long winded but it can be done.

 

5) The most interesting point is that neither side has actually resolved Zeno. Not even the 12 minute youtube video did that. So far only part of the solution has been presented.

 

6) No, calculus is not required, but a study of convergence either mathematically or linguistically is. A good textbook for this is the classic by Ferrar at Oxford, suprisingly entitled " A Textbook of Convergence ".

 

I look forward to someone presenting a complete solution.

[kristalris]

Studiot you are dodging the issue. You have to point out what is wrong with intuitively providing a complete solution to Zeno via taking it as a fact that Achilles - given certain assumptions - will overtake the tortoise?

[studiot]

 

Studiot you are dodging the issue. You have to point out what is wrong with intuitively providing a complete solution to Zeno via taking it as a fact that Achilles - given certain assumptions - will overtake the tortoise?

 

 

Why am I dodging the issue?

 

No one, not even Zeno, pretends that Achilles does not catch and overtake the tortoise. That is not the issue.

He just saddles up his trusty kangaroo and hops right past.

 

 

Zeno's issue is/was "Where is the mistake in my homework, Miss? I can't see it"

 

In order to explain the inadequacy of Zeno's reasoning we need to do something equivalent to that presented in physica's video, and then some, because the good doctor of maths in that video didn't complete his 'proof'.

This can be done either mathematically or liguistically, following the same route.

 

What I have not done, but left in the air for the moment, is state what I think is missing from existing explanations in this thread.

What do you think that might be?

Edited June 5, 2014 by studiot

[Prometheus]

No one, not even Zeno, pretends that Achilles does not catch and overtake the tortoise. That is not the issue.

 

 

Just an aside; i think Zeno offered his paradoxes to support the view taught by Parmenides that change is impossible, and existence is timeless, uniform, necessary, and unchanging. His view was that what we perceived was an illusion.

[studiot]

 

Just an aside; i think Zeno offered his paradoxes to support the view taught by Parmenides that change is impossible, and existence is timeless, uniform, necessary, and unchanging. His view was that what we perceived was an illusion.

 

 

The ancient Greeks did a lot of thinking, some of perhaps due to the peculiar quality of the wine made from the grapes that grew on Mt Olympus.

 

They were particularly fond of paradoxes, Zeno remaining the most famous in modern times, but there were many chroniclers.

 

Most of their 'paradoxes' can be resolved by adding our further knowledge to their less developed state knowledge.

 

But I am not really a classics scholar so I don't know the full story of Zeno.

 

Today we have our own 'paradoxes' to wrestle with such as quantum tunneling and quantum entanglement. Perhaps later generations will have the proper understanding to fully resolve these.

 

For example Proclus has that " Not every triangle is also a trilateral figure", simply because they did not fully understand angles.

[kristalris]

 

Just an aside; i think Zeno offered his paradoxes to support the view taught by Parmenides that change is impossible, and existence is timeless, uniform, necessary, and unchanging. His view was that what we perceived was an illusion.

I agree with Zeno then. And I thus can agree with what you think he meant.

[md65536]

Exactly my point about sorting out where the concepts are linking the warrior and the tortoise and where the math locks a situation in, and where the math is inappropriately applied.

Where is the math inappropriately applied? What situation is locked in? Are we talking about an accurate mathematical representation of the paradox, or something else?

 

Math describes the situation and resolves the paradox, I can't see anything inappropriate about that. The math certainly doesn't "lock in" something incorrect.

[kristalris]

 

Why am I dodging the issue?

 

No one, not even Zeno, pretends that Achilles does not catch and overtake the tortoise. That is not the issue.

He just saddles up his trusty kangaroo and hops right past.

 

 

Zeno's issue is/was "Where is the mistake in my homework, Miss? I can't see it"

 

In order to explain the inadequacy of Zeno's reasoning we need to do something equivalent to that presented in physica's video, and then some, because the good doctor of maths in that video didn't complete his 'proof'.

This can be done either mathematically or liguistically, following the same route.

 

What I have not done, but left in the air for the moment, is state what I think is missing from existing explanations in this thread.

What do you think that might be?

Contrary to what you stated earlier, the issue on Zeno is IMO about the divide between pure mathematics / pure logic and reality. Achilles and a tortoise have no place in that and are thus a problem in reality. Put in other words: on the game / chess board of reason on a pure logic problem the ultimate arbiter is mathematics even though lingual logic can also get quite far. On a question about reality it is intuitive lingual set up as a garbage or non garbage position that decides the issue where mathematics does not even come into play for otherwise you end up in a Escher like conundrum, that Zeno I believe was trying to teach us.

 

Only when you have an observed non intuitive solution to a problem do we have a true paradox, to be taken as a seeming contradiction by scientists in search for an intuitive i.e. simple solution. This by as matter of course (or for Yanks as a standard operating procedure) then check the prior set up: or prior assumptions, by changing these in order to see what that as a testable thought experiment leads to.

[md65536]

2) The thread deteriorated into a dispute about mathematics v linguistics.

 

3) Now since there is nothing that can be said in mathematics that cannot be said linguistically this is or was a pointless diversion.

A linguistic description of a mathematical result is still the result of math. The results of calculation can't always be obtained through linguistics alone. But this isn't about math vs linguistics. I think everyone involved has admitted that the paradox resolution can be described in words. The problem I think is more about ignoring analysis of the details of the paradox (avoiding math is just one way to do that). Instead of pinpointing the problem (in words or math), TAR is doing the opposite, considering Kant and the nature of thought and looking at the big picture, settling on a "solid basis, upon which to build a lingual understanding of a situation"... and it doesn't work... so maybe he can go broader... smash the impossible concept into atoms and spread them around and look for the problem there. I think that's what TAR's lingual solution is: to restate a paradox with less meaning and more convolution (it all depends on how the brain works, etc), until it is vague enough that there is no longer anything left that is both meaningful and puzzling.

[studiot]

 

Math describes the situation and resolves the paradox, I can't see anything inappropriate about that. The math certainly doesn't "lock in" something incorrect.

 

 

I've yet to see it in this thread.

 

 

A linguistic description of a mathematical result is still the result of math.

 

Nonsense, it's the other way round.

 

Every symbol in maths has a name that can be spelled in English.

Every line in maths can therefore be written as a proper english sentence.

 

English is composed of proper english sentences.

Edited June 5, 2014 by studiot

[kristalris]

Studiot and md65536 you still don't get it. Mathematics inherently doesn't solve the garbage in problem. Zeno tried clearly in vain to teach you this. You may only in pure mathematics go into the stated problem, for only then can it have meaning. Thus you must then exclude Achilles and the tortoise for they have no meaning in pure mathematics. That you maybe can solve the purely mathematical / pure logical problem also in lingual logic is beside the point.

[tar]

MD65536,

 

I like your breakdown of what it is that I am attempting to do, and not suceeding in doing.

 

There is an "angle" there, that you describe, that of being both puzzling and meaningful, that may be in play here, on a couple of levels. One, why the difference of opinion can cause people distress, as is evidenced by the heat of the debate on a deep level, that seems to be engendered when one fellow/gal takes away, or attempts to take away the other's meaningful puzzlement. And two, in regards, to the nature of a proof or a solution, the question or problem first has to be commonly understood, as to which portion or piece of the situation is the puzzling, and meaningful part.

 

Breaks down on one level to whether or not one "gets the joke". And I think, in the case of Zeno there are different levels involved in the joke. Different levels of puzzlement, and different levels of meaning.

 

Was thinking today about meaning and double meaning and triple meaning, in regards, to this discussion, and was considering how a simple statement like "I went in and back out" can have many different meanings and might have two or three meanings appriopriately understood between two parties at the same time. That is, could be taken to mean you are referring to the room, or the fray, or the job or have some sexual connotation where communication of which and what aspects of meaning are being expressed and understood depend upon context and the next sentence, nods and winks and looks and whether or not the phrasing and word choice and order and such "work" on both or all levels being engaged on.

 

Perhaps the Greek wine, as well has something to do with this discussion. I gave up drinking back when I was in the Army in Germany, during the Iranian crisis...so many years ago. I do not get high, or drunk, and gave up a life-long nicotine addiction two months ago. The only mind altering drug I use now is caffine. In terms of activation theory, I pretty much have to solve the puzzles that are puzzling and meaningful to me. I already "did" the convergence one. I "got it" with the one where the ideal ball bounces halfway up from the drop hieght, and then halfway up from the halfway point, and bounces an infinite number of times, before it comes to rest in a finite period of time. I get that joke. I see how that mathematically works out. The puzzlement it causes me is that I know the actual count of bounces, in a real (not ideal) ball cannot actually be infinity, so I consider what constraints reality would put on the actual test or experiment, that would create an actual finite count, should the test be made with a hypothetical, "actual" ball. That is, there are some other real contraints one can work with, like planck's length and one's definition of a bounce, that would eliminate the requirement or possibility of a next bounce, and thus end the count, before infinity is reached.

 

So, perhaps, in terms of me not yet presenting any solution to the puzzling and meaningful aspects of the paradox in question, perhaps it would be good, at this point, for someone who is puzzled by some aspect of the paradox to state the question or problem or puzzlement, in either words or symbols or both, and I will attempt to answer the puzzlement in words or symbols or both.

 

I have this feeling that there is some aspect of this discussion that is similar to a "strawman" argument, in that a question is posed that is easy to knock down, just for the fun of knocking the thing down.

 

In this regard, we already know the warrior is going to overtake the tortoise, so "duh", you think we might be able to construct a proof to show how that can happen?

 

Regards, TAR

But then again, I already presented my solution in the OP. You have to put two gears on a rod and figure that, to properly model the tortoise and the warrior. You can't be putting two dots on a ball and turn the ball expecting to get the one dot to overtake the other.

As to reality being an illusion, it can not be an illusion, because then it would have to be somebody's illusion, which would require there to be somebody real to have the illusion. So one can just cut to the chase and be convinced that reality is actual by virtue of there being at least one, but rather evidently 8 billion instances of real entities that can experience the place. This places both the place and the experiencer in the category of "existing".

Edited June 6, 2014 by tar

[studiot]

 

I already "did" the convergence one. I "got it" with the one where the ideal ball bounces halfway up from the drop hieght, and then halfway up from the halfway point, and bounces an infinite number of times, before it comes to rest in a finite period of time. I get that joke. I see how that mathematically works out. The puzzlement it causes me is that I know the actual count of bounces, in a real (not ideal) ball cannot actually be infinity, so I consider what constraints reality would put on the actual test or experiment, that would create an actual finite count, should the test be made with a hypothetical, "actual" ball. That is, there are some other real contraints one can work with, like planck's length and one's definition of a bounce, that would eliminate the requirement or possibility of a next bounce, and thus end the count, before infinity is reached.

 

This is the nub of my comment, since this is only half the story.

 

This is also what I mean whe I say that I don't see any mathematical solutions posted in this thread, since you are all only discussing half the story.

 

Think carefully about Zeno's concise description, then compare with the comments in this thread.

Why are most of the posts are much longer than Zeno?

It is because most are distracted by 'reality', 'hypothetical', infinity and whether or not it is reached and so on.

 

A hint there is not one 'infinity' inherent in the question but two different ones.

[md65536]

One, why the difference of opinion can cause people distress, as is evidenced by the heat of the debate on a deep level, that seems to be engendered when one fellow/gal takes away, or attempts to take away the other's meaningful puzzlement. And two, in regards, to the nature of a proof or a solution, the question or problem first has to be commonly understood, as to which portion or piece of the situation is the puzzling, and meaningful part.

I guess that's one of the benefits/drawbacks of linguistics. There's so much room for expanding meaning and interpretation. Four people have given answers and no one agrees with anyone else. The meaning of the problem isn't even settled on.

 

Physics gives a definite prediction and the corresponding math is explanatory and unambiguous (not that Zeno's model perfectly represents reality, or that the problem couldn't be modified to be modelled differently, eg with quantum time or something, but that's beside the point).

 

I disagree that the gear analogy is related to the paradox. I don't see it. However since there's no physical paradox, what you said---(paraphrased) that the paradox is only in the understanding---must be true, so I guess if you resolve the paradox in a different way with a different meaning, even if it makes no sense to me, even if I think it's simply brushing aside the paradox, I suppose that still resolves it for you. If the presence of the paradox is subjective, depending on the understanding of it, then the presence of the "lingual solution" is too.

 

Still I don't see it that way. The mathematical description of the problem, whether expressed in numbers or symbols or words, is unambiguous and so is the resolution.

[tar]

MD65536,

 

I was not, and am not attempting to make the question a non question. It is a good question but deals primarily with a human's understanding of space and time, and as I understand Kant, these things are basic, a priori understandings that we have a grasp of, with no explaination required or possible. As in "what is time?"...well its that thing we all know as past present and future...you know...time. or "What is space"...well its that place where everything is located. Neither space or time are thusly made up of "sub" understandings. They are instead the basic understandings that have no component understandings that one needs to cobble together to reach a composite understanding of some sort. They are instead the starting points upon which all other understandings and judgements and categories of the predicates that we can associate with objects, are built. Everything we can say about an object in general, first relies on the fact that you know what space and time are, and I know what space and time are and we can say something about the way space and time are put together, because of this common a priori understanding.

 

Calculus deals exactly with space and time. So when it comes to describing movement, how it is possible, what it is and what we mean by speed and distance and acceleration, and the change in position of something over time, calculus provides the answers to Zeno's questions. And the convergence understandings take care of the different "kinds" of infinities that Studiot is hinting at.

 

But this can also be handled linquistically because the infinitely many, infinitely small divisions of time, that one must proceed through, to get to the point where the warrior overtakes the tortoise, or the motion starts, or the arrow moves, are actually over in just a few moments, not an infinite amount of moments. The whole infinite set of steps is completed in 10 or 15 seconds, and the warrior is past the tortoise, and the arrow has left the bow, traveled through the air and is sticking in the target.

 

Some part of the debate that caused Zeno to come up with these logical problems had to do with trying to prove the world was one thing, or the world was several. Philosophical and religious differences of opinion on these types of questions are still evident, in the world, and on this board. Immortal, a banned former member here was very much an "illusion" proponent, someone who would say, that there is no movement, that everything is one, and everything is an emination from the one.

 

So the question has been put to me here to show where Zeno's logic is wrong, not that it is wrong. We already know the tortoise will be overtaken. The puzzlement, is how is that possible, when all these infinite divisions of time must be transversed, prior the overtaking moment.

 

Was reading today the Wiki article on Zeno's Paradoxes. It said all the things we are talking about here. It, on the whole is already solved, already understood. Each of the differing explainations of what are the puzzling and meaningfu parts that have been recently expressed on this thread are already in the article.

 

But, the clincher for me, the thing that "proves" what everybody is saying, is that we already know what time and space is and know exactly what can and does happen when the two get together. And therefore anything that would suggest that what is perfectly possible and understandable is impossible, must be sight of hand, and someone has switched the cup the pea is under without noticing the switch.

 

Studiot and I are rather sure all the infinite steps will be completed in a short period of time, because each successive step is a tinier and tinier slice of time, so tiny that you can pack an infinite number of the slices in a very brief amount of time...so the warrior can overtake the tortoise as soon as that brief moment has passed.

 

Regards, TAR

[studiot]

Yeah, Tar you are getting towards your lingual solution. +1 for good thinking.

 

 

Two key phrases

 

"time and space"

 

Yes there are not one but two infinite sequences involved with finite limits.

 

"The whole infinite set of steps is completed in 10 or 15 seconds,"

 

Yes Zeno's analysis only acounts for part of the real number line axis that we call time and to resolve the 'paradox' we need to determine what happens on the rest of it.

Edited June 7, 2014 by studiot

[tar]

Studiot,

 

What happens on the rest of it, is the idea I was trying to loosen up, or let loose, in regards to the OP analogies.

 

I agree with MD65536 that the analogies are not completely right and considering the gears might work, but the spots on the Earth do no completely mirror the "incorrect" setup. The problematic setup, the one that has the warrior never overtaking the tortoise, is not exactly like the two spots on the Earth that will always be the same distance from each other, regardless of the fact that one is moving faster than the other, but attempting to match up such an analogy with the original setup gives one the opportunity to evaluate which and what one is holding constant, and which and what one is allowing to proceed.

 

This is important in considering what functions you are entertaining. Which quanties are increasing or decreasing at what rates, and to what limits.

 

In a video that was posted at some point on one recent thread or another recently related to the Zeno Paradox, a young man was describing a mathematical "proof" of something. He was doing rather well, and came to the rub and then made a switch that I did not follow. Made a switch that was not forced or required or necessary by the lead up. Made a switch that required one already understood the nature of time and space and the overtaking principles under discussion. Like a bait and switch operation. Like he didn't prove a darn thing we didn't already accept as obvious. As if he said "see the piece of candy is getting closer and closer to your mouth as I move it closer and closer to your mouth, but it will not actually ever get into your mouth, unless I decide to give it to you". "There, you can have it now."

 

Regards, TAR

[tar]

An interesting "other" situation just occurred to me, that might be helpful, in two ways, in unraveling this "mystery". One in that it deals directly with timing, and two it deals with how a human gets a word out of their mouth.

 

Studdering.

 

I do not know much about studdering, but I remember reading an article in "Discover" that was discussing the "timing" aspect of forming a word, and it is difficult for a studderer to "practice" not studdering...(and this is paraphrasing just what I took from an article read many many months ago, but) because by the time you have done all that it has caused you to studder.

 

So if you can not say a word until after you have planned the whole thing out, in terms of the shape of your mouth and position of your tounge, and the amount of air you are going to force through the chamber and so on, there is no chance that you are going to be able to listen to what you are saying "at the same time". It would be easy to logically consider this situation interms of what has to happen inorder for speech to occur, and through Zeno type reduction ad absurbium decide that it is therefore impossible to utter a clear word. Impossible to think, impossible to formulate a thought, transform the thought to sound, and create the sound, "all at the same time". Therefore speech is impossible. Except we know it is not impossible, and we type out sentences at the exact same time as we are thinking the thoughts. Perhaps we think the thought first and then think about how to put it into words, and then consider how to type it and then type it, but if we were not thinking the thought, while we were typing it, how would the keys be getting pressed in the right order? There is still a chicken and egg type conundrum one faces when logically trying to sort the thing out, where one just can not throw up their hands and say it is impossible, when it so obviously must have a way in which it gets done, and the "timing" aspect that is built into a human moment is a VERY important consideration when attempting to sort the thing out.

 

We have a predictive motor simulator, that "practices" combinations of motor neuron firingings before we actually send the signals. We have "learned" what combinations and timing of said motor nuerons result in what motions, and hence we can walk and catch a moving ball and such, because we unconsiously or subconsciously predict and plan these things out "as" we are doing them. Thus when living in a moment we probably give the moment its whole duration of 2 1/2 seconds to "be" all together considered the same time. So in Zeno, when we hear a phase like "by the time the warrior gets to the tortoise's postion, the tortoise would have moved on" we have given the tortoise a moment to have moved on, that is in actuality a moment already spent by the warrior, and a moment that has already been spent by the tortoise, and we should not give the tortoise "another" moment to spend moving forward.

 

So moment size is crucial in determining what position change can occur within the moment, and we can't be giving the warrior different size moments than the tortoise. We can only be allowing the tortoise to move 1/10 the distance the warrior does, in any given same size moment. So "by the time" the warrior moves 200 yrds, the tortoise would have moved 20 yds, the tortoise would be 120 yds from the starting line, and the warrior would be 200 yds from the starting point, and the warrior would have overtaken the tortoise. You would not even need calculus as an algebra equation would have the warrior overtaking the tortoise.

 

Regards, TAR

-funny- firingings-guess there is such thing as finger studder. Fuddering?

[studiot]

Gosh Tar I thought you were getting somewhere, but you have gone of rambling down the byeways of I know not where again, and overcomplicating things.

 

Quite simply Zeno is correct. Achilles does not catch the tortoise.

At least not in the time period that Zeno considers.

 

Zeno's mistake is to think that the interval in time he considers is all of time, whereas all he considers is the period from the satrt of the race to the time when Achilles is abreast of the tortoise (and technically still has not overtaken it).

 

His analysis says nothing about the time after that any more than the time before the start of the race.

 

However, buried in your last two posts you referred to a conjurors switch. This is true.

Most descriptions of the analysis start off (correctly) considering time as the independent variable, but suddenly switch to considering distance. You cannot change horses midstream.

[Prometheus]

 

The ancient Greeks did a lot of thinking, some of perhaps due to the peculiar quality of the wine made from the grapes that grew on Mt Olympus.

 

 

And thank Zeus they did. Whatever their motivations they have taken mankind forward with their thinking.

 

I found this website, which though doesn't tackle the 'Achilles and the tortoise' paradox, it tackles similar ones. Together with the other chapters, it provides a good account of our understanding of infinity.

[tar]

Prometheus,

 

Well thanks for the link. I got a little bogged down with some of the latter more complicated thoughts of Cantor, because I lost him on the second diagonal. I did not agree that the numbers he created by changing the digits found on the diagonal of the first countable list were "new" numbers. They were merely numbers now put out of order from the orignal count. They still could have been in the list already, and therefore not a "new" number. Calling it a new number is implying it had not already been counted, and by my thinking, the "new' number was in the list already in some other position in the count, and thusly by making the transfer of digits, down the whole list, you wind up with exactly the same number of numbers, but you have succeeded only in pairing each up with a new counting number. Breaking its match with the first counting number you associated it with by saying "OK, now this "old" number, which was before in position 6,456,399,049 is now in position 4 in the count. You have not created any "new" number, as that particular sequence of digits was already accounted for.

 

So, permit me to introduce another aspect to this controversy. We cannot say that we have a new and better understanding of numbers, or of infinity, based on the understanding that only a subset of "we" might have.

 

It does not make sense to me that one infinity can be larger than another, because infinity already is at the max of quantity. You cannot beat that, you cannot exceed that, because if you could, your first claim of being at the maximum possible count, would have been incorrect.

 

Regards, TAR

Reminds me of a joke/story we used to tell about our plumber/handyman, a Pennsylvania Dutch fellow who proudly exlpained to us how he had helped his son with his math, by explaining infinity to him. "You take a straight line, and you draw it on the page and imagine it still going off the page, out the door, over the field, down the hill, past Sieselsville and keep going and going...all the way to Topton." (a town about 12 miles away)