For over two thousand years, Zeno's paradoxes disturbed the rest of the best thinkers, who had difficulty reconciling their strange conclusions with our understanding and knowledge of the concept of motion
Marius Cohen. Published in the February 2008 issue of Galileo magazine (issue 113)
historical background
Zeno was a pre-Socratic Greek philosopher, who lived in Elea in southern Italy in the fifth century BC, and is considered the father of philosophical dialectics (which was also adopted by Socrates, one of the greatest philosophers of ancient times, who met Zeno in his youth). Under the influence of his teacher Parmenides, who claimed that the world is uniform and unchanging and that multiplicity and change are nothing but an illusion, Zeno devised a series of paradoxes, the purpose of which is to show that if multiplicity or change is assumed, a contradiction is reached.
In total, dozens of paradoxes are attributed to Zeno, but only some of them have been preserved to this day (and there is no certainty that Zeno himself actually wrote all the paradoxes attributed to him). Below we will present his three most famous paradoxes, through which he tried to show that the assumption that motion is possible leads to a contradiction. Through them, Zeno hoped to show the correctness of Parmenides' position that the world is static, and that movement is nothing but an illusion (the presentation of these paradoxes is also a complement to the paradoxes of infinity, which were presented in the two previous lists in this section).
Achilles and the tortoise
Let us imagine a running competition between the mythological Greek hero Achilles, who is considered an outstanding athlete, and an average land turtle, who is far from being a worthy opponent. Achilles, who is convinced of his victory, allows the tortoise to start the race from a point closer to the finish line, which we will call point A.
When the whistle blows, the two competitors begin their movement to the finish line, and Achilles, with light and sure steps, reaches point A with ease. However, the tortoise, despite its great slowness, managed to move a certain distance in the meantime (albeit significantly less than what Achilles traveled at this time), and reached another point, closer to the finish line, which we will call point B.
Achilles notices when they reach point A, that the tortoise is still ahead of him, and continues his fast run, which brings him to point B in a short period of time. However, the tortoise, who did not rest for a moment, had time to advance an additional distance in the meantime (of course shorter than the one between points A and B), and reached point C, which is closer to the finish line than where Achilles is at that moment.
The athlete of course notices that the turtle is still ahead of him, and continues his fast run, which brings him to point C in a flash. But while Achilles was running from point B to point C, the tortoise also had time to advance a little, and reached point D, which is closer to the finish line than point C, where Achilles is now, so the tortoise is still ahead of the Greek hero. And so the two continue their movement, when every time Achilles reaches the previous point where the tortoise was, the tortoise manages to make another small distance. Since Achilles must always pass through the previous point where the tortoise was, and in this time (however short it may be) the tortoise is sufficient to advance a little further, it is found that Achilles will never catch up with the tortoise, and this is contrary to our certain knowledge that if such a contest had taken place in reality, the swift Achilles would have won the The slow turtle with ease!
The movement of a body is not determined in single moments. A body is in motion if at two adjacent moments it is in different places
Image: stock.xchng
according to the movement of the hands
An analogous version of the Achilles and the tortoise paradox, and which we witness in everyday life, is the movement of the hands of a wall clock. At 12:00 the two hands point in the same direction (12), and from that moment a race begins between them, as the minute hand (Achilles) tries to catch up with the hour hand (the tortoise), which is given an advance of a full round at the beginning of the race.
Since the minute hand moves at a speed 12 times higher than that of the hour hand, then by the time the minute hand completed one revolution (60 years), the hour hand had enough time to advance 5 years, and it points in the direction of the number 1 on the clock face. When the minute hand also reaches the same place, and points in the direction of the number 1, the hour hand has managed to move forward for another half of the year, and it is still leading. And so on and so forth, similar to Achilles' attempt to reach the tortoise, the minute hand tries to reach the hour hand, but every time it reaches the previous position of this hand, the hour hand manages to advance a little further, and it follows that the minute hand will never be able to reach it. This is in complete contradiction to our knowledge that he does this repeatedly during the day.
The mathematical solution
This paradox, like Zeno's other paradoxes dealing with movement, has troubled the best thinkers for over two thousand years, as no flaw has been found in the logical course of the argument (Achilles must, on his way to the finish line, reach the previous point where the tortoise was, while the tortoise just needs to move a little further) , although it leads to an absurd conclusion.
Only in the last few centuries, with the development of infinitesimal mathematics, and the introduction of the concept of "infinite convergent column", it seemed that the solution to the paradox had been found: even though in Achilles' attempt to get the tortoise he had to travel an infinite number of sections (from the starting point to point A, from point A to point A B, from point B to point C and so on), and infinite periods of time (for Achilles to go through these sections of road), because due to the fact that the sections of the road and the periods of time are getting smaller and smaller at a constant rate, their sum is nevertheless finite!
Suppose, for example, that at the beginning of the race the tortoise is 100 meters ahead of Achilles, and that Achilles' running speed is 10 times greater than that of the tortoise. Under these conditions, while Achilles passes the first 100 meters, the tortoise is enough to advance 10 meters. When Achilles passes even these 10 meters, the tortoise is only one meter ahead of him. While Achilles moves another meter, the tortoise moves another 10 centimeters, and so on.
It is easy to see that the total distance that Achilles travels until he reaches the tortoise is (in meters): , and it is possible to show that even this expression includes an infinite connected, its value is a meter, and this is the distance that Achilles travels (under the described conditions) until he reaches the tortoise.
The same is true of the time periods required for Achilles to pass these road sections: if we assume that the athlete's running speed is 10 meters per second (and the turtle's - 10 meter per second), then he covers the first hundred meters in 10 seconds, the next road section (XNUMX meters) in one second, and so on. The infinite column is obtained (in seconds): , and it is possible to show that the value of this infinite column is seconds. That is, Achilles catches the tortoise in this period of time, and immediately after it he is already ahead of him on the way to the finish line.
In the same way, it can be shown that the number of years that the minute clock passes until it catches up with the hour hand (12 times slower than it) is: . Since this hand moves one year per minute, this is also the number of minutes it takes to reach the hour hand. The advantage of the hands version of the paradox is that the path the minute hand takes until it reaches the hour hand also represents the time it takes to do so, and this makes it easier to understand the phenomenon of convergence (which, as mentioned, we can see with our own eyes by carefully examining clock with hands).
The dichotomy paradox
This paradox, also called the "Race Track Paradox", has two versions (corresponding to different interpretations of the text appearing in Aristotle's writings), but they are similar to each other in their logical course and conclusion, and therefore we will present here only one of them (sometimes called the "progressive version"): A person wants to get from point A to point B. For this he will first have to go halfway. Then he will have another long way to go (the second half), which in order to go through it he will have to first go through half of this one (that is, another quarter of the original way), and then the remaining half of the way, and so on and so forth.
All in all, he will have to travel an infinite number of road sections, since after each road section he passes (half of the remaining road), he still has a certain distance left to cover, and therefore he will never reach his destination. Since this logical course does not depend on the distance between points A and B, it cannot reach its destination even if the path it intends to take is extremely short. Conclusion: it is impossible to travel any distance - movement is an illusion.
As mentioned, only understanding the nature of converging columns made it possible to solve these paradoxes satisfactorily. The paradox of the dichotomy is also resolved when it is shown that the infinite number of road sections that a person must pass on his way from one point to another converges to the length of the entire road (which is a finite size), and the sum of the time periods it takes him to travel these road sections also converges to a finite number.
If we assume, for example, that I have to cover a distance of 3.6 kilometers, and I walk at a speed of one meter per second, then I will cover the first 1.8 kilometers (half way) in half an hour; I will cover half of the remaining distance (900 meters) in fifteen minutes; I will cover the next 450 meters in an eighth of an hour (7.5 minutes), and so on and so forth. The total amount of time it will take me to travel an infinite number of sections of this road is (in hours): , and it is possible to show that this infinite sum converges to one hour. That is, the time it would take me to go through infinite sections of these roads is nevertheless finite.
In reality, the speedy Achilles would easily beat the slow tortoise!
Image: stock.xchng
The Arrow Paradox
Let's think about the movement of an arrow in its trajectory: since any movement, no matter how short, takes a certain amount of time, then at some given moment (which lasted 0 units of time), the arrow is at rest. In other words: since a given moment has no duration, the arrow is not enough to move it. This is true for each of the moments in which the arrow is in its trajectory, however, if the arrow does not move in any of these moments, then it is not moving at all, and therefore its movement is nothing but an illusion.
The arrow paradox also waited over two thousand years until the concept of movement was clarified, and it was possible to offer a solution: the movement of a body is not determined in individual moments. A body is in motion if at two adjacent moments it is in different places. We are indeed talking about the speed of a body at a given moment, but this speed is determined by the ratio between the distance that this body travels between two adjacent moments and the period of time that passes between these moments (even if it is very small). According to the theory of limits, which kinematics (the theory of motion) makes use of, it is possible to choose for this purpose moments as close to each other as we wish, but still, in these two moments the arrow will be in different places (even if very close), and therefore it has a speed in each of the moments in which He is on his way.
Dessert
In the dessert corner, this time we will present another paradox of Zenon's (apparently), and it is the "paradox of infinite division": since geometrically it is possible to divide any unit of length into two, it is possible to continue such a division of any given (and finite) distance to infinity (such a division does not need to be carried out in practice , and it does not take time; it is a principled geometric division).
There are two possibilities regarding the result of such a division: either the length of each of the resulting infinite parts is 0, or it has some length different from 0. However, in the first case, Zenon argued, the common length of all the parts is also 0, because the sum of any number of parts whose length is 0 must also be 0, while in the second case their length is infinite, because the sum of an infinite number of parts whose length is different from 0 must be infinite . Each of these two possibilities is contrary to the fact that these parts were obtained from the infinite division of a finite given distance.
68 תגובות
fresh:
I hope it is clear to you that this does not give any validity to your mistakes that remain mistakes.
Also in response 48 I talked about the fact that there could be a quantification of the space and I really meant what serious people think about the matter.
http://www.physorg.com/news/2011-03-space-chessboard.html
The whole universe can be seen as a kind of cellular automaton
See the Wikipedia entry
http://he.wikipedia.org/wiki/%D7%90%D7%95%D7%98%D7%95%D7%9E%D7%98_%D7%AA%D7%90%D7%99
In my opinion, only solution #1 is correct, there is no physical existence for the conceptual concept called infinity. (At least not minus infinity in terms of the magnitude of scales, plus infinity maybe possible)
Regarding the "dessert", 3 possible solutions:
1. It is impossible to divide infinitely. There is no real definition of infinity. This solves all the paradoxes you will try to put here.
2. There are numbers called "aspiring to 0", like 0.0000000000000…5 (where there are infinitely many zeros). If you multiply them by infinity they are not completely zero, so you don't get 0, but they are not big enough to get an infinite number.
3. The wrong assumption is the definition of multiplying at infinity. think i'm stupid Try the calculator 0 times infinity, then wake up.
I think that's enough reasons.
I'm not wasting your time, no one is forcing you to read my comments.
Tal:
It turns out you didn't understand
Michael, I understand that you switched to psychology and have a professional opinion on the writers of the messages on the site.
fresh:
You don't work for me and it looks like you never will work for me either.
You're just wasting my time.
I don't want to be superior to those who are able to solve much more complex questions, so I don't need to solve any question that an anonymous surfer puts to me, I don't work for you to the best of my knowledge.
fresh:
As mentioned - you want to tower above those who are able to solve much more complex questions, but the need to prove your skills to justify their attention is an obstacle because you are not able to do this.
That's why you say you don't want to do it.
I'm sorry but it's a waste of my time.
You have a right to think so. And I don't want to convince you and prove anything to you, even more so I'm not trying to prove to you my ability to solve elementary school movement questions.
fresh,
We circle around the same point.
Do you realize that a finite section can be divided into as many small sections as we like?
Do you realize that the sum of infinitely many terms of a descending column can equal a finite number?
Once you understand that, everything else will follow.
fresh:
You are rambling and it is not at all surprising that instead of solving an elementary school question that requires a numerical answer, you prefer to talk meaningless nonsense about a question whose answer is not numerical and that I did not refer to you.
You just don't understand what you are doing.
pleasantness
A meter (or any other finite length) can be passed because it is a finite length. An infinitely small length cannot be crossed because you have nowhere to start, there is no starting point from which you can start the road, and therefore you will not be able to cross the road either.
fresh,
You wrote: "And if the first section is as small as we want, it can never be passed." How did you get to that?
It is certainly possible to pass it - and it will take zero time (or as little time as we want).
Your arguments indicate a fundamental misunderstanding: you are convinced that it is not possible to go through a finite time, an infinite number of smaller and smaller segments.
It's a mistake. You can mark a section a meter long, go through it easily, in a finite time, whether it is divided into billions of billions of small sections, but a finite number, or whether it is divided into an infinity of infinitely small sections.
It seems to me that inside you, you are still convinced that there cannot be a situation in which the sum of infinite terms is equal to a finite number - indeed, it is not easy to internalize this, but that is the situation.
To Michael
You are entering into another discussion here, of whether mathematics represents reality 1 to 1 or whether it represents a very, very, very accurate approximation of it (or both depending on the subject) This is a philosophical question that I think will never have an answer.
That's why I made a distinction in my comments between infinity as a mental or mathematical idea, and as such this infinity only exists in our thoughts and not in reality. and actual infinity in reality, which I don't think exists. The Pythagorean language is true and root 2 exist in the mathematical sense without any doubt. And in the real sense they exist, but reality does not have infinite precision after the decimal point, therefore the root of 2 in reality differs slightly (but very, very slightly) from the mathematical root of 2.
fresh:
You overlap all the way.
Remember when you cited Wikipedia as proof that the term "convergence to infinity" is wrong?
Remember that I showed you that in the same place that you brought as proof the exact opposite is actually proven?
So what? Couldn't you check? Of course you could but you are so sure of your mistakes that you don't see fit to check.
Remember when I confronted you with the claim that Zenon would not have accepted your "explanation"?
Remember how you ignored it until now?
So what? You couldn't explain how Zenon would accept your words?
Of course you couldn't. Your words are wrong.
And what about all the other things I said?
I have to say I was quite scared to post the trivia question I posted above.
This is really a question that a child in elementary school should know how to answer and I was worried that you would be able to answer it and create a presentation as if you actually understand something.
It was a tightrope walk:
The more trivial the question - the more I risk that you will solve it - on the one hand - but the fact that you don't solve it better demonstrates your misunderstanding.
The more complex the question is, the less likely you are to solve it, but that means less because it is a question that many do not know how to solve.
I thank you for your cooperation.
Shifting the discussion to a semantic matter - even now - as in the previous incarnation of this stupid discussion - shows that at least you were not able to solve the question immediately.
An elementary school question! Rabak!
This is already becoming a boring argument.
It is not acceptable to say convergence to infinity, it is acceptable to say divergence, but this is only a semantic matter and not an essential one. It's clear what you mean when you say "converses to infinity" simply in my opinion it's an inaccurate wording, which is why they invented the phrase "entertaining" in the Hebrew language. In discussions, I try to see beyond the semantics, and understand the essence.
And besides, you have the right to disagree with me, I'm not trying to convince you, I'm just saying what I think in the clearest way.
Friends:
I have heard about the idea of the quantification of space a long time ago but I have never come across such a simplistic and wrong interpretation of it as Raanan is trying to sell here.
The interesting thing is that according to the interpretation that Ra'anan tries to give to the term, at least one of the following two things must exist:
A. Pythagorean theorem is wrong
B. The root of two is a rational number
Those who are frustrated by the fact that the previous question was addressed only to refresh are welcome to try to understand for themselves why I made the above claim
fresh:
certainly exists.
Also the fact that there is no convergence to infinity exists. Even on Wikipedia! It's written there in hidden ink above the text that says convergence to infinity exists.
You have no idea about math. point. (period - I didn't mean you)
I am even willing to bet that you are not able to solve a question that is in elementary school material.
Knows what?
Let's check.
Here is a question about elementary school material.
She deals with movement - your favorite field.
We'll see you solve the question.
If you solve it by chance - please explain the solution in the terms in which you "solve" Zenon's paradox:
I ask all the others not to interfere, this is a really easy question and there is no need for you to try to solve it because the fact that you succeed will not prove anything. On the other hand, I'm pretty sure that Rahan isn't even capable of that:
One man finishes his work every day at 16:00 p.m.
At the end of work, he gets on the train (exactly at 16:00) and gets off at the station closest to his home.
His wife leaves the house with her car and arrives at the train station with him.
He gets off the train straight to the car and they go home.
One day (on the holiday) there was less work and he left work an hour earlier (ie - if you don't know subtraction, at 15:00 p.m.).
His wife did not know about this and left the house at the usual time, so when he reached the train station, he started walking towards the house.
At some point along the way he met his wife, got in the car, and they returned home.
When they got home he looked at the clock and saw that they had arrived 10 minutes earlier than usual.
Assuming that the speed of the train is constant, the speed of the car is constant, the walking speed is constant and no time is wasted on stops - how long did he walk?
Really not inventing anything, everything already exists.
fresh:
Ok - you are not ignoring the mathematical subject but inventing new (wrong) mathematics
To Noam
Of course Achilles will pass the tortoise at the end of time, we all agree on that because it exists in reality, but not for the reasons you mentioned, those are the reasons I mentioned.
When you say that "the periods of time needed to move from section to section are also rapidly infinitesimally small" you assume that it is even possible to move from section to section (when each such section is infinitely small as we wish) and this is not true because it is impossible to move from section to section as long as you have not passed The first section, and if the first section is as small as we like, it can never be passed. In addition, you assume that there is such a thing as a small period of time as we wish, there is no such period of time.
Not only do I not ignore the mathematical subject, but the mathematics supports and explains what I say.
pleasantness:
There is nothing to argue with.
Everything has already been told to him many times and he is not letting go. This is not the first discussion that this is so.
He just heard something about the quantification of space and decided to link it to Zenon's question without any justification.
He is wrong but since he is locked - nothing will help.
Note that he ignores the entire mathematical side and the fact that the mathematics he ignores is flying rockets to the moon and solving any problem that can be formulated in its terms.
He also ignores the other claims that are raised - such as the one that his "solution" would not have satisfied Zenon in any way.
I think he's just a teenage boy who thinks he knows everything better than everyone else.
fresh,
When you say: "He will never overtake him" you mistakenly assume that it will take an infinite amount of time for him to overtake him.
This is a mistake, the periods of time needed to move from section to section are also rapidly infinitesimally small, and the sum of these time sections is *finite**, meaning that Achilles will reach the tortoise after a finite amount of time!
No problem, we'll leave the physical part. If there is an infinite number of *not physical* but mental segments (ones that are really as small as we want all the time for eternity) that Achilles has to go through until he gets around the tortoise, he will never get around him, not only will he not be able to get around the tortoise, he won't even *be able to start * the road because he cannot have a point from which he can start running! Because he will never reach a section from which he can start running, because in order to reach the section from which he will start running, he needs a section of finite length from which it is possible to move on and in our mental infinity there is no such finite section and therefore he will not be able to start the journey. But since we know that in reality it is certain that Achilles will start the road and also overtake the tortoise, there is no escaping the conclusion that reality does not necessarily have the characteristics of such a mental infinity.
fresh,
Leave for a moment the physical side, it prevents you from seeing the fundamental matter.
If we return to the carpet, note that the argument is true even if it can be divided into an infinite number of small and increasing particles
(quickly enough).
It's not easy to internalize this, but ** a sum of infinite segments that are getting smaller and smaller, is absolutely finite**.
And contrary to what you wrote before, in the calculation of the limit the column is not made finite at any stage - the infinite sum of the members of the column is simply calculated!
The concept of the limit, and the calculations of an infinite descending column, are what the Greeks lacked, so it seemed to them that there was a paradox here.
To Noam
In my opinion, it is impossible to divide a carpet of finite size into an infinite number of physical parts (not because we do not have a sufficiently sophisticated machine that can cut such small sizes, but because space-time itself cannot contain smaller particles) and therefore all the more so that it is impossible to sum up those infinite parts that do not exist in reality. (And of course in our thoughts we can divide to infinity, but this mental division does not exist in reality)
There is no problem to pass a section with a finite length, and if it is really short, it is all the more easy to pass it. On the other hand, going through a section with infinite width is a bit problematic, I would say...
fresh:
Of course, your "explanation" doesn't explain anything because Zeno's paradox, to the extent that you decide to see it as an expression of reality, holds for each of those tiny sections you talk about.
How do you think they can be passed?
I'm sure Zenon wouldn't have accepted your explanation either (and the fact is that he didn't offer it).
extension:
Therefore Achilles will cross the tortoise after a completely finite section of the road, even though it consists of infinite sections, and in a completely finite time.
fresh,
You miss the most important point: the sum of infinite parts **not necessarily** equals infinite ways, that's exactly the point!
Read the carpet example again: an infinite sum of carpet particles does not equal an infinite area, but a completely finite area: 2 square meters
It is equivalent to a road segment finite in length, even though it consists of an infinite sum of rapidly decreasing segments.
fresh:
This does not mean anything.
I promise you that even Hitler's words will be remembered in another 2000 years
You have the right to think that Zeno was talking nonsense, I just hope that you too will be remembered for your words even in 2000 years as they remember Zeno's "nonsense"...
fresh:
Some people are unable to let go of their mistakes.
Don't get me wrong.
In the end it may turn out that space is quantum but Zenon did not prove it. He was just talking nonsense.
Lenaam
The solution to Zeno's paradox is that within a finite section there is a *finite* number of *physical* sections (as opposed to an infinite number of sections that can exist in our thoughts but not in reality), and because a finite number of sections *can* be passed (especially if they are very, very short) so Achilles Will indeed pass the turtle as we know it should be. But if there is an infinite number of physical sections (even if they are terribly small) that Achilles has to pass until he overtakes the tortoise, he will never overtake him, because he has infinite sections to pass (infinite path)! And since we know that Achilles will outrun the tortoise in reality, this necessarily means that there is no such thing as an infinite number of physical segments within a finite segment.
fresh:
Because of laughing at the column thing, I didn't see that you wrote about Zenon again.
I repeat and suggest you read what I wrote about paradoxes because you probably don't understand something very basic.
A paradox is an expression of a fault in thought.
You can call it beautiful but it still expresses a mistake.
As soon as someone reaches a conclusion that contradicts reality - all he proves is that he is wrong somewhere.
In the link you refused to read I wrote where he is wrong.
Ok, fresh, now I also checked and even according to the link you provided, you are wrong.
In the link to "Convergence of an infinite column"
Write:
"Also the harmonic column [formula that cannot be copied here] diverges (or converges to infinity)."
fresh:
I'm not that interested in what the society that writers on the Hebrew Wikipedia know or don't know.
The expression "converses to infinity" is accepted (even if the ignoramuses who write the Hebrew Wikipedia do not know it - and by the way - I did not check) and it is indeed equivalent to the expression "entertains to infinity" and all this does not lead to your conclusion that there is no infinity.
It's a shame to argue, go to Wikipedia and you'll see that convergence is only for a finite number. There is entertainment to infinity, and there is a "changing" column for the example you gave 1-1+1-1 when the amount changes all the time.
http://he.wikipedia.org/wiki/%D7%98%D7%95%D7%A8_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)
Zenon didn't claim anything, so I can't justify him even if I wanted to, he just posed a beautiful paradox, which I think proves something about reality, and the insights that came later with the differential calculus to solve the paradox only *prove* what I claim about the nature of reality. And so it is not clear to me why you think I am contradicting insights that came to them later?
fresh,
The solution to Zeno's paradox is that it is possible to go through an infinite number of points, in a completely finite time - this of course follows from the fact mentioned earlier, that there is definitely a situation where an infinite sum of smaller and smaller segments is a completely finite number.
And regarding the nomenclature:
The expression "converges to infinity" is accepted and meaningful.
A column or series can converge to infinity.
This is a private case of entertainment and not every column or series is entertaining - converging to infinity.
The column (1, 2, 3, 4,…..) converges to infinity (the series of its members also converges to infinity in this case).
The column (1, 2 -, 4, 8 -, 16, 32-……) is entertained but does not converge to infinity or anything else.
The series (1, 1 – , 1, 1-, 1, 1- ….) also diverges without converging to infinity.
fresh:
I see you decided not to read the correct explanation.
This is of course your right, but you must understand that the mathematicians of our time (and I allow myself to count myself among them) have progressed far beyond Zenon.
There is something recursive in a situation where - in order to justify Zenon's claims - you are not ready to accept the insights that came to them after him. It seems like you really stopped time.
There is no such thing as convergence to infinity, this convergence is only to a finite number, when an infinite column grows/shrinks all the time it is said to be entertaining. What Zeno proved (in my opinion) is that it is not possible for space-time to be divided into scales that can shrink at will, because otherwise Achilles would never have gotten the tortoise and the universe would be static without movement. Time-space must inevitably converge to very small but finite scales (pieces/quanta).
fresh,
It is possible to visually illustrate an infinite column whose sum is completely finite:
Let's imagine a rectangular carpet, 2 by 1 meter, that is, its area is 2 square meters.
We will cut it in half and place a 1 square meter square on the floor.
We will cut what is left in half, and add half a square meter to the floor
We will cut what is left in half and add a quarter of a square meter to the floor
In this way we will continue ad infinitum to cut in half what is left, and add half of what is left to the floor.
Obviously, no matter how long the process lasts, the total area of the carpet on the floor will never exceed the original 2 square meters.
And in fact, the sum of an infinite number of carpet parts, which are getting smaller and smaller, is equal to the final sum of 2 square meters.
(And if you say that it is impossible to divide a carpet into an infinite number of small parts, then the exact same argument applies to a geometric area consisting of an infinite number of dimensionless points.)
point:
It's a third - not two thirds 🙂
Every infinite decimal number (and in particular the periodic fractions) represents an infinite convergent column.
For example - the number pi is 3 + 0.1 + 0.04 +0.001 + 0.0005……
After all, the simplest example of an infinite series that converges is any number.
For example the number two thirds. Written as 0.3+0.03+0.003...
fresh:
If you want to better understand the paradox of Achilles and the tortoise and other paradoxes I suggest you read this response of mine:
https://www.hayadan.org.il/toward-infinity-0703081/comment-page-1/#comment-40965
fresh:
What do you mean?
After all, there are series that do not converge to a finite limit and the sum of their terms is infinite.
In this case the limit that is discovered is infinity.
We discover a finite limit in some amounts and an infinite limit in other amounts.
Does the fact that one sum converges to a finite limit show in your opinion that there is no "infinity" even though another sum actually converges to infinity?
A. You have the right to object
B. I never claimed that there is no connection between the mathematical and physical world.
Raanan, I disagree with your last words. It also doesn't quite agree with what you wrote in the first paragraph. that there is actually no connection between the mathematical and physical worlds.
It doesn't matter if you set a limit or reveal a limit. Because if there *is* a limit, there is no infinity, but only finitude. And this finitude is precisely a discrete/quantum space-time (made of small pieces with a finite size and physically indivisible beyond the same size, although in our minds we can continue to imagine division at will)
fresh:
This is not true.
When you say that an infinite series converges, you don't say it because you put a limit on it, but because you can prove that despite the infinite number of its members, the sum of all the members is finite.
The border is not set but discovered. It is not an artificial limitation but one that is there by the very definition of the problem.
This is the reason why the sum of all (infinity) negative integer powers of 2 converges and in contrast the sum of the inverses of all naturals does not converge and it will not help how much we want to put a limit on it.
Hi Raanan, if I were to respond to the article, I would write roughly what you wrote..
Zeno's paradoxes are the most beautiful paradoxes that exist in my opinion. And everyone is actually talking about the same subject, which is the existence of a discrete space-time, that is, made of small particles of finite size (as opposed to the illusion we have as if time-space is continuous and that it is possible to divide the spatial dimensions or time ad infinitum).
His paradoxes are not paradoxes at all in my eyes, they are proof that it is really impossible to divide a section in an infinite way *physically*, (as opposed to mentally, in thought everything is possible) because if it were possible then really Achilles would never have passed the tortoise, and we would live in a static universe Absolutely no movement.
The whole mathematical subject of an infinite column that can miraculously converge to a finite and defined sum, is possible only with the help of the mathematical concept - limit. And the essence of the concept of the limit is to set a limit to some length, a small but finite length. That is, first the infinite column is made finite and then it is claimed that it converges to a finite size. And of course it will converge, because they made the letter of the column finite by the concept of the limit. A column that is truly infinite (without using the concept of the limit that limits this infinity) can never converge to a finite size.
To my father Blizovsky, Violelli, what-does this remind you of my brother Solinka Shalom Kupferberg-Nachshon..you really..must meet him on the occasion..the invitation..
You should meet him...it will bring you many genuine laughs..
Well, this is my brother Solly's nice phone..don't forget..ho.ho.ho.right in his soul
Shytopnik-graceful-kibbutznikk-funny.
To Emanuel the most whistling in the world - apropos - what is the wind.. for a loud whistle.. how
The Kantian spiral began..how the mica was born..and emitted the sunlight..the apricot.
Hanzo Hi, the genealogist, the celebrated... there are some registration duties at Yad Vashem Chaya, Avraham
who perished..in Auschwitz..God knows..they are the ones who raised my father in his childhood..
Address: 25 Trobrova
Warsaw, Poland———and what is standing there now? If not the Central Bank of Poland????
Well..well..I'm not A.S.K-Central ??.
And ask..why Amalia..has a senior, independent international account..in Jerusalem.
Well..well..so what's the connection??.
Valpier Paulin-director of the Acropolis.. what happened? ..philosophy ..really..really
It's really.really.important..by the way, and of course you remember..with hermetic love
Platonic..and your mouth of the desert is amazing..I gave you all in the academic fellowship
Eb Kars, a valuable book on astrology, in Spanish..ahh..company number 6
Simple, one does not forget friends on the way... a great manager, the greatest... replies.
It didn't work in all browsers, but I'll forward your request to Remy anyway.
Avi:
I hope that the option of a direct link to the response was removed only for correction purposes and will return soon.
I found it useful despite the bug in it (I simply chained the number of the comment that she was correct in to the address of the page where she was wrong) and therefore I would be happy if it remained valid even in this state.
Age:
I must admit that I do not understand exactly what you are asking.
If you are asking how it is possible for a series with infinite members to give a finite sum then the example you gave is exactly that and as Yehuda pointed out, the sum is X. The fact that the terms of the series tend to zero is a general property of series whose sum of terms converges (converging series).
In this case it is a geometric column and it is easy to prove the convergence and calculate its limit (although it is difficult to write the proof here because of the poor support of the text editor for formulas) but you must understand that the possibility to calculate the limit is not a condition for convergence.
I don't know if I even came close to answering your question because, as mentioned, I didn't understand it.
As for the connection to differential and integral calculus, indeed, the recognition of the existence of infinite convergent series is still far from the discovery of these methods, but other historical facts really indicate that at least some of the ancient Greeks already used methods that today we would associate with these fields.
Archimedes, for example, found an ingenious way to calculate the volume of the intersection between two equal, perpendicular cylinders whose main axes meet. The way he used it shows that he knew how to calculate what we would call an integral today and that he did it according to the same ideas that were later used to define the integral.
Gil, quote:
"Because the sum of an infinite number of parts whose length is different from 0 must be infinite." End quote.
Incorrect statement. For example, the sum of the series you gave as an example converges to X.
I imagine you know that so you don't understand the connection.
Sabdarmish Yehuda
Regarding the last paradox, how does differential calculus deal with it? I know qualitatively because it is close to this the definition of the concept of the limit and in fact the formulation of the phenomenon described in the paradox as a series of members that converge in a scheme as a column to a certain discrete value.
Can any of the readers explain this quantitatively?
x/2
x/4
x/8
...
And after n such divisions when n tends to infinity we will get sections
Those with a length of XNUMX:
x/(2^n)
Aiming for zero, obviously
How does this actually resolve the paradox? "And if in the second case their length is infinite, because the combination of infinite parts whose length is different from 0 must be infinite."
Thanks in advance to the respondent(s)... (:
It is necessary to preserve all kinds of "authentic" manuscripts that are suddenly discovered like an unknown work by Mozart that happened to be in his grandmother's apartment or something like that. And of course also this document of Archimedes.
but…. I will look for the book and read it.
I am sure that if the differential calculus was developed in intelligence, it would break out and be known to all
Apparently the mathematical problems encountered by the Ion sages were simple and did not require differential calculus.
Friend, someone break this shard already!
Sabdarmish Yehuda
It is true for Yaron and Yehuda that it seems as if they understood Hadova, so it is not true, in fact, it is the ancient understanding of gravity and how bodies act at speed or the rotation of the planets, how despite understanding the subject quite well in relation to the means they had, they could not calculate it but only in theory that cannot be calculated, if you want another really good example, read Fermat's Last Judgment book, it is about a riddle that was not solved for a long time and took 600 years to solve, and despite the means and the high level of sweetener used by the solver of the riddle, this does not mean that the person who wrote the riddle understood the sweetener that is needed to solve the riddle
to Judah,
I recently read an interesting book - "The Codex of Archimedes" about the discovery of an ancient manuscript of Archimedes, and among other things the book includes geometrical proofs written by Archimedes from which it can be concluded that he understood and used differential calculus... worth reading
In my opinion, this whole story of Achilles and the tortoise shows how weak philosophy is compared to science (including mathematics): Zeno encountered a mathematical problem which in essence is quite simple. But, instead of going and looking for a logical solution that aligns with reality, as a scientist would do, he went looking for far-fetched and baseless explanations, which led him to all kinds of conclusions that are clearly wrong. The only value of Zeno's "solution" is that it can bring the listener to reflection. It is quite similar to the fact that I will go to the grocery store to buy milk that costs 5 shekels, I will pay 10 shekels and the seller will tell me that I do not need more. So I tell him - but wait, I paid you 10 shekels, I need to get an excess! Then he will tell me that I must understand that excess is actually an illusion, and that in the general essence of the universe we are all meaningless, and that each of us perceives the concept of existence in a different way, and therefore I should not receive excess. So what, is he right? He is just philosophizing.
When Newton and Leibniz (who was also a philosopher, wasn't he?) encountered the exact same problem, their solution gave birth to a branch of mathematics without which the world would look very different today. If that's not a philosophical influence on reality, I don't know what is.
This just proves how close the Ions were to the development of the differential calculus. It is difficult to know how the mathematical-scientific world would have developed, if the differential calculus had already been known two thousand years before Leibniz-Newton.
may we have a nice week
Sabdarmish Yehuda
From my reading - "shiiit" is a bit of a dig, but ahhh, and the part about dividing halfway has been in my psychometrics endless times....just for general information