Apparent Paradoxes

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Apparent Paradoxes

Unread postby Keith Ness » Fri Dec 09, 2016 12:01 pm

There are some tricky paradoxes in public discussion, for example, Zeno’s impossibility of motion, whose compellingness may influence people to give more weight to unscientific thinking than they should. However, some years ago I came up with the following resolution to that paradox, which also suggests a general method for resolving apparent paradoxes.

Zeno’s paradox of infinite subdivisions resulting in the impossibility of motion is the same as saying that, because a segment from point A to point B can be infinitely subdivided, the segment’s true width is infinite. It is confusing two different characteristics, that of distance and scale. It is also interesting what happens to the argument when you call out the amount of time it takes to cross each subsequent subdivision: it takes one hour to go from point A to point B, half an hour to go halfway from point A to point B, a quarter of an hour to go a quarter of the way from point A to point B. The logic is restored or maintained. However, if there are an infinite number of prerequisites to fulfill, and each prerequisite takes a certain amount of time greater than zero to fulfill, then it will take an infinite amount of time to fulfill those prerequisites. However, there is no infinitely small scale. There is only an infinite number of scales, each one of which is finite, or, in other words, has a finite number of points in the segment. Furthermore, if motion is impossible, then how would he demonstrate his premise that we first have to go halfway?

When Zeno says that, in order to go the whole way, you first have to go half way, and so on, the fact is that, actually, you don’t go half way first, and then the whole way, you go the whole way and half way simultaneously. Zeno might then counter-argue that you have gone half-way and the whole way in the same amount of time. To which I reply that I didn’t say that you went the whole way and half way in the same amount of time, I said you went those two different distances simultaneously; that is, the graph of one is superimposed over the graph of the other.

However, even with all the above, I still somehow feel like I am just working around the edges of Zeno’s problem, like I still haven’t dealt directly with the sheer compellingness of Zeno’s simple statement quite to my satisfaction. It seems like he’s making the mathematical error of adding 1 to ½ when he makes that first changeover, when he says, “In order to go the whole way, you first have to go half the way.” Is he doing that? And if so, what limitation of human neural logic structures makes such an error so difficult for us to detect (it’s not limited to a particular language; he did it in ancient Greek, and his works were compelling enough to survive thousands of years and across multiple languages, including English)?

Well, let’s see… Hmm, for some reason, I am consistently drawn back to the factors of time and speed as I think about this. Allright, let’s see if further work with them might lead us to an even greater clarity... Ahh, yes, when you add in the time factor to Zeno’s argument as follows, the warning sign, the distinction, the potential for two different meanings in his statement, becomes clearer still: “If point A is the starting point, and point C is the destination, and point B is halfway between points A and C, then, in order to spend 1 hour going from point A to point C, you have to first spend ½ hour going from point A to point B.” The natural response to that phrasing is, “Well, wait a minute, are you adding those two times together, or including the smaller time in the larger time?” The use of time measurements, three distinctly and conveniently labeled points, and two explicitly defined segments really addresses his seemingly very innocent use of the cute little term, “first,” making clear that the term is actually quite dangerously vague, and that he is actually quite thoroughly exploiting that dangerous vagueness to generate his quite magically erroneous conclusion. More specifically still, he is using one meaning of the word “first” to derive his premise, and another meaning of the word “first” to derive his conclusion, even though the two meanings contradict each other, and relying on his very smooth and simple transition from premise to conclusion to make it very difficult for people to pin-point this fact.

So yes, it turns out to be a problem of semantics and logic, just like, “What will I have for lunch today? Well, let’s see. Nothing is better than a ham sandwich, and a ham sandwich is better than anything else, so I’ll have nothing.” Of course, some such problems are more difficult to see than others. In some cases, two or more meanings are bound so tightly and illusorily into one that it is difficult to realize that they are more than one meaning, and likewise difficult to extricate them into their separate forms, yielding apparent paradoxes where the best answer to their premises is, “That depends on what you mean.”

From this resolution and other experiences, I have found that apparent self-contradictions/paradoxes generally arise from insufficient specificity, and are resolved by sufficiently increasing specificity.

“Who’s on first?”
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Re: Apparent Paradoxes

Unread postby jacmac » Sat Dec 10, 2016 12:23 am

the math language of the paradox seems to describe a curve approaching a limit, but the reality is two straight lines (constant speeds) that intersect where achilles catches the tortoise.

https://en.wikipedia.org/wiki/Zeno's_pa ... rtoise.gif

It is basically the same reason the "big bang" is nonsense.
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Re: Apparent Paradoxes

Unread postby dd6 » Tue Dec 13, 2016 8:07 pm

Interesting points. One possibility is that there is a minimum divisible unit of distance. So you get to 1 millionth of a cm left to go, and with his paradox ok, then you go half that distance to 1/2 a millionth of a cm.

But if 1 millionth of a cm is the minimum divisible distance, then the next minimum movement distance takes you to the end point.
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Re: Apparent Paradoxes

Unread postby D_Archer » Wed Dec 14, 2016 3:27 am

Zeno paradox is a teaching tool, to separate the weed from the chaff.

If you do not understand it there is a mystery if you do understand there is no mystery and no paradox.

Solved by Miles Mathis > http://milesmathis.com/zeno.html

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Re: Apparent Paradoxes

Unread postby Chan Rasjid » Thu Dec 15, 2016 12:22 am

Zeno's paradox is indeed a true paradox that has no solution with certain conditions.

No motion is possible for a point to move from point A to point B. In order to reach B, it has to pass through the midpoint M; there are infinite number of midpoints to cover and so the point cannot reach point B. This is one argument that strictly has no logical flaw. I think it may be possible to find another argument that the point can also reach point B, also rigorously logical thus giving rise to a true paradox.

The reason for the true paradox is the point itself being a paradox. A point is a thing with zero space width. For a hare chasing a tortoise, the positions of the hare and tortoise are not well defined points.

EDIT: the other argument is this. If it can reach the first midpoint M, then it must have passed the midpoint of A and M; this imply that motion is possible, it can reach any point after A,etc...

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Re: Apparent Paradoxes

Unread postby D_Archer » Thu Dec 15, 2016 2:23 am

Chan Rasjid wrote:Zeno's paradox is indeed a true paradox that has no solution with certain conditions.

No motion is possible for a point to move from point A to point B. In order to reach B, it has to pass through the midpoint M; there are infinite number of midpoints to cover and so the point cannot reach point B. This is one argument that strictly has no logical flaw. I think it may be possible to find another argument that the point can also reach point B, also rigorously logical thus giving rise to a true paradox.

The reason for the true paradox is the point itself being a paradox. A point is a thing with zero space width. For a hare chasing a tortoise, the positions of the hare and tortoise are not well defined points.

EDIT: the other argument is this. If it can reach the first midpoint M, then it must have passed the midpoint of A and M; this imply that motion is possible, it can reach any point after A,etc...

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did you read the article by Miles?

Zeno is solved, it is stated that you can reach the midway point (M), that is the start of paradox, becsuse if you can reacha midpoint you can also reach an endpoint, otherwise you would never reach the midpoint as well, because you would only reach half of the half, or half of the half of the half, ie you would not get anywhere, which is not true in reality.

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Re: Apparent Paradoxes

Unread postby Chan Rasjid » Thu Dec 15, 2016 11:38 am

D_Archer wrote:
Chan Rasjid wrote:Zeno's paradox is indeed a true paradox that has no solution with certain conditions.

No motion is possible for a point to move from point A to point B. In order to reach B, it has to pass through the midpoint M; there are infinite number of midpoints to cover and so the point cannot reach point B. This is one argument that strictly has no logical flaw. I think it may be possible to find another argument that the point can also reach point B, also rigorously logical thus giving rise to a true paradox.

The reason for the true paradox is the point itself being a paradox. A point is a thing with zero space width. For a hare chasing a tortoise, the positions of the hare and tortoise are not well defined points.

EDIT: the other argument is this. If it can reach the first midpoint M, then it must have passed the midpoint of A and M; this imply that motion is possible, it can reach any point after A,etc...

Chan Rasjid.
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did you read the article by Miles?

Zeno is solved, it is stated that you can reach the midway point (M), that is the start of paradox, becsuse if you can reacha midpoint you can also reach an endpoint, otherwise you would never reach the midpoint as well, because you would only reach half of the half, or half of the half of the half, ie you would not get anywhere, which is not true in reality.

Regards,
Daniel

I wrote earlier that it is a true paradox as both arguments, for an against the possibility of going from A to B, are both logically rigorous. The reason for the existence of the paradox is the concept of a "point". A point has no physical reality, whereas moving from real position (not point!) A to real position (not point!) B is a real world phenomena.

In your argument, you assert : "it is possible to reach point B". It can be shown that this leads to a contradiction. If it is to reach B, it must have past through the first midpoint. Then, there is the next,... and the next midpoint, add infinitum. This implies the motion will never reach point B. This contradicts the first assertion "it is possible to reach point B".

The paradox is all because we take motion, which is a real world event, for motion of a fictitious thing.

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Re: Apparent Paradoxes

Unread postby saul » Fri Dec 16, 2016 12:54 am

Wait a minute, are we still discussing Zeno's paradox?

Perhaps no motion is possible after all :)
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Re: Apparent Paradoxes

Unread postby Roshi » Fri Dec 16, 2016 2:42 am

There is no time:
viewtopic.php?f=10&t=16118

http://phys.org/news/2011-04-scientists ... nsion.html
http://phys.org/news/2012-04-physicists ... space.html

there is only our measurement of it. This measurement is a number. We count any cyclical process we want to choose, and call it "time".That's not "time", it just our count, a number. This count does not "move the world forward". The world moves, and everything happens because of thousands of other causes and laws, not because we count - the rotation of the Earth for example.

Zeno's problem is that there is an infinite number of "time intervals" needed for one to go through the infinite points from A to B. But what is a "time interval"? It's a count. A number. It's nothing physical, even if we count a cyclical process, that process is not driven by our count. A caesium atom does not vibrate because "a second has passed", but science today considers it does. The caesium atom does it's thing because of other causes, we just count what it does and say "that's 1 second!".
An object moves from A to B because of other causes, not because "time has passed".

Zeno's paradox: We cannot work with infinities in real world applications - like when traveling from A to B. Or when drawing a circle. A circle should not exist in the real world, because it's circumference is 2*pi*R, and that number has an infinity of decimals. Also it's radius is circumference/2*pi. It never ends to a finite determined value as we can see a circle drawn on paper.

Infinity should remain in the domain of mathematics, if he wants to divide his distance from A to B he should choose a number of divisions. Else - we can solve it only if we consider a limit to our division, the smallest "count of time" but why would there be a limit? Time is just a number and it can be divided forever. We just need to choose a divisor, no matter how big, and separate pure mathematics that works with infinity from reality.

In other words: let Zeno choose his divisor, in turns. First he divides by 2, then by 3, then by 4, 5, 6, and so on forever. Each time the result is a finite number of intervals, no matter how much he divides. But he cannot say "I divide by Infinity, and you need to go through an infinite number of points", that is going out of touch with reality.

No matter how big the universe is, even if it is infinite, there is always a finite distance between 2 points, else we arrive in wonderland, and we can move forever in one direction in one frame of reference, without moving, because "this distance is infinite...". It cannot be, if it is a distance it is finite, and non 0, and not negative.
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Re: Apparent Paradoxes

Unread postby Chan Rasjid » Fri Dec 16, 2016 6:45 am

Roshi wrote:...
Zeno's paradox: We cannot work with infinities in real world applications - like when traveling from A to B. Or when drawing a circle. A circle should not exist in the real world, because it's circumference is 2*pi*R, and that number has an infinity of decimals. Also it's radius is circumference/2*pi. It never ends to a finite determined value as we can see a circle drawn on paper.
...

Zeno's paradox is big time debate after thousands of years! Just check the wiki; the comments of big names are given. It shows the greatness of the ancient Greeks - not the present loser Greeks who just can't balance their books!

Mathematics is a study of abstract objects and structures; it is very good as a tool in physics which is a study of real world phenomena. It gives us numerical values for quantitative comparisons. The circle is an abstract geometrical construct - nothing physical. But surveyors, builders do need to draw real geometrical objects like lines and circles on the ground - these drawings are physical representations which are real. Geometry helps in the real world so we don't repeat building a leaning tower like Pisa.

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Re: Apparent Paradoxes

Unread postby jacmac » Fri Dec 16, 2016 6:55 am

Roshi says:
There is no time:
there is only our measurement of it.


I agree. Also, there is no inch, mile or meter. These too are names of our measurements so we can communicate.
Very useful, but not real in and of themselves.
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Re: Apparent Paradoxes

Unread postby webolife » Sun Dec 25, 2016 8:10 pm

Zeno's paradox is non-sense.
Once an object moves at all, it has traversed an "infinity" of points. Its motion is not impossible, OBVIOUSLY, so it is the infinity of points that is an imaginary construct. Infinity is in all its applications mathematical, not physical.
This is one reason I believe that a finite universe is the most sensible option.
Truth extends beyond the border of self-limiting science. Free discourse among opposing viewpoints draws the open-minded away from the darkness of inevitable bias and nearer to the light of universal reality.
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Re: Apparent Paradoxes

Unread postby neilwilkes » Sun Jan 08, 2017 11:39 am

webolife wrote:Zeno's paradox is non-sense.
Once an object moves at all, it has traversed an "infinity" of points. Its motion is not impossible, OBVIOUSLY, so it is the infinity of points that is an imaginary construct. Infinity is in all its applications mathematical, not physical.
This is one reason I believe that a finite universe is the most sensible option.


And this is why it is not quite nonsense - indeed it is very valuable.
When I was back in junior school I had a long talk with my teacher about this one (although I did not know it as Zeno's paradox then) and if memory serves it was along the lines of "are numbers infinite or not?"
The pointer to infinity is that no matter what number you can think of, I can always, always make it larger either by addition or multiplication (which is in actuality just a complex form of addition) - so to be able to touch another person - and therefore breed - however far they are away from me the distance cannot be covered because the halfway points must be infinite if numbers are infinite. A distance is 1 foot between 2 people but before they can touch they must close the distance to 6 inches, 3 inches, 1.5 inches ad nauseam. Time (which is an artificial construct created by humans to try & break things up into manageable pieces) is irrelevant here. The distance between 2 people is in theory mathematically infinite according to pure logic.
However observational evidence plainly falsifies this idea so it has to be rejected - even though pure logic and mathematics states the opposite.

Plato stated we can only understand creation (by which he presumably meant the universe) by means of pure reason and Einstein said the same - yet pure reason & mathematics can lead you badly astray.

Observation and experimental evidence is king - or should be.
Thought experiments are often misleading and no indicator of reality.
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Re: Apparent Paradoxes

Unread postby saul » Tue Jan 10, 2017 9:52 pm

neilwilkes wrote:
webolife wrote:Zeno's paradox is non-sense.
Once an object moves at all, it has traversed an "infinity" of points. Its motion is not impossible, OBVIOUSLY, so it is the infinity of points that is an imaginary construct. Infinity is in all its applications mathematical, not physical.
This is one reason I believe that a finite universe is the most sensible option.


And this is why it is not quite nonsense - indeed it is very valuable.
When I was back in junior school I had a long talk with my teacher about this one (although I did not know it as Zeno's paradox then) and if memory serves it was along the lines of "are numbers infinite or not?"
The pointer to infinity is that no matter what number you can think of, I can always, always make it larger either by addition or multiplication (which is in actuality just a complex form of addition) - so to be able to touch another person - and therefore breed - however far they are away from me the distance cannot be covered because the halfway points must be infinite if numbers are infinite. A distance is 1 foot between 2 people but before they can touch they must close the distance to 6 inches, 3 inches, 1.5 inches ad nauseam. Time (which is an artificial construct created by humans to try & break things up into manageable pieces) is irrelevant here. The distance between 2 people is in theory mathematically infinite according to pure logic.
However observational evidence plainly falsifies this idea so it has to be rejected - even though pure logic and mathematics states the opposite.

Plato stated we can only understand creation (by which he presumably meant the universe) by means of pure reason and Einstein said the same - yet pure reason & mathematics can lead you badly astray.

Observation and experimental evidence is king - or should be.
Thought experiments are often misleading and no indicator of reality.


Bravo, well put.

This reminds me of my favorite quote:

"If simple perfect laws uniquely rule the Universe, should not pure thought be capable of uncovering this perfect set of laws without having to lean on the crutches of tediously assembled observations? True, the laws to be discovered may be perfect, but the human brain is not. Left on its own it is prone to stray, as many past examples sadly prove. In fact, we have missed few chances to err until new data freshly gleaned from nature set us right again for the next steps. Thus pillars rather than crutches are the observations on which we base our theories."
-- Martin Schwarzschild
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Re: Apparent Paradoxes

Unread postby Keith Ness » Mon Mar 27, 2017 2:12 pm

Thanks for all the feedback, I didn't expect so much. It's funny, through the link to Miles Mathis above, I found that, although I have no formal training in calculus and I'm just using logic, I agree almost point-for-point with Miles Mathis on the issue. The only potential significant disagreement I can see is that I have no problem adding to and subtracting from infinity. For example, in your set, you have a ray (infinite in one direction) pointing west, then you add to your set a segment (finite in both directions) pointing east; your set is still infinite, but it has changed in some meaningful way. But that may just be a matter of semantics.

Anyways, it seems pretty clear to me that even without reducing to zero, Zeno can demonstrate that there are an infinite number of halfway points to cross, so the problem is not about zero, it is that he tempts us to believe that an infinite number of halfway points can not yield a finite, fixed distance greater than zero, when it actually can. It can because the infinite number of finite distances to halfway points he proposes all superimpose over each other in space (and this is a fact which Mathis does not mention in the link). It's similar to thinking that an infinite universe must be infinitely bright when actually, a hyperbolic curve (of amplitude, for example) can increase forever without ever crossing a fixed, finite asymptote.

Furthermore, as I've said elsewhere in this forum, I consider logic to be a prerequisite of science, and, "...the statement, 'there was no time before time X,' is using a time coordinate system to describe a time before time X, and is therefore self-contradictory; the same logic applies to space." Further still, since both an ultimately finite and an ultimiately infinite universe appear unfalsifiable by direct observation currently, we default to mere simplicity, and it should be clear from the above self-contradction that an ultimately finite universe has a lot more explaining to do than an ultimately infinite one.

Side note: I'm trying to go on a long-term video diet, so I may be slow to reply, especially if I have to do online research to reply...
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