• ssu
    8.3k
    For summer enjoyment, I'll put here one of my threads from the old PF in a bit revised form from 2010:

    The god's at Olympos were bored again.

    To break up the divine boredom, Pallas Athena (onwards just Athena) looked for something to amuse her fellow gods and especially Zeus, her father. She noticed two philosophers, Plato and Zeno of Elea walking on a beach debating philosophy. The goddess of wisdom had an idea to amuse her companions and asked Themis, the goddess of divine order, to help her to create some fun. Themis agreed and the two goddesses appeared in front of the two startled and scared philosophers with Athena then creating an abundant number of small totally similar looking Viking dogs (a small compact Swedish dog breed fitting for this story) onto the beach. It seemed that the number of the dogs couldn't be counted and the beach with the dogs seemed to go forever. This created a huge disorder on the beach, which the Greeks didn't like. Athena then told the following task for the two philosophers that they would have to solve: they would have to tell a way for Themis to feed all the dogs on the beach without any dog being left out hungry and Themis would make this instantly to happen. Yet Athena informed that these weren't ordinary dogs and defined them the following way:

    1. The dogs are totally similar in every way except that every dog eats a different quantity of food. All the dogs eat the same food, which is divisible and there is enough of it for every dog.
    2. If there was a quantity that could be defined to be different from all other quantities, then there is a dog that would eat this quantity. There are no limitations on the quantities (physical or other), and hence on the dogs.
    3. How much the dogs or a dog eats cannot be compared to anything else than to another dog or dogs and their quantities of food.

    Then both of the goddesses disappeared back to Olympos.

    Plato, a man of action, after recollecting himself after the scary encounter thought for while looking at the dogs and then picked one that was nearest to him. He lifted the small dog towards the sky and yelled: "O mighty Themis! Goddess of law and divine order, by this Viking dog that I have picked up, I will measure all the dogs and bring order to this Apeiron and hence put all the dogs into order so that they all can be fed!" So Plato's idea was for Themis to find the dog that ate twice the amount that the dog he carried ate, then the one that ate three times more, four times and so on. And also those that eat less than Plato's dog were put into order along the same lines with finding first the dog that ate half and then the one that ate one third of the amount that Plato's dog ate. Once these were put into the line, then came the dogs which ate quantities between these dogs. Plato then thought about it and understood that for every dog there would be dogs that would eat more as there would dogs that would eat less and even less and explained this to Zeno.

    Zeno of Elea had been silent until he asked Plato: "Aren't you forgetting at least two dogs? The one that eats less than every other dog and the one that eats the more than any other dog?" At first Plato thought Zeno was either joking or that he hadn't heard what he had just explained. So he explained again to Zeno that there cannot be a dog that eats the most, because there is always a dog that eats more. Zeno said he understood Plato's point, yet wasn't convinced. Zeno started to argue that Plato, by randomly picking up a dog and then starting to count from it the various quantities other dogs, was missing at least these two dogs, one that ate the least and one that ate the most. After all, didn't their amounts that they ate differ from all the other dogs? Zeno continued that nowhere in the instructions had the goddess Athena said to pick up one dog and start counting with it. With this action Plato had forgotten the second rule and had got fixated to his picked up dog. Plato got so offended by this that he accused Zeno of Elea to be a sophist. The philosophers couldn't agree and a lot of confusion prevailed among them.

    And the gods were amused. Perhaps they are amused even now.

    20--rotuesite2.jpg?itok=ymMBpr0f
    It would be nice to hear what people here think about Zeno's dogs, which obviously are strict inequalities or inequations. And please answer the poll, there's no wrong answers in PF. And if I'm way wrong here and made a mistake, please let me know!
    1. But what do you think? (7 votes)
        Plato is right. Zeno's dogs cannot exist.
        14%
        Zeno is right. Even if incomparable and uncountable, by the loose definitions Zeno's dogs exist.
        57%
        Both are wrong.
          0%
        I have a different view about the whole story, ssu.
        29%
  • Metaphysician Undercover
    12.8k
    Plato is right. By definition #2, there are no physical limitations. A dog that eats the most, and a dog that eats the least implies two physical limits, which violates #2
  • ssu
    8.3k
    That is the obvious standard line, been for thousands of years. But it's a limitation, when you start from Plato's dog.

    Yet doesn't the dog that eats more than any other dog define it different from all other dogs? No matter if there's an Apeiron (endless amount) of dogs that eat less.
  • Ludwig V
    1.2k
    There are real mathematical and philosophical issues about infinity. I think they may be insoluble. But philosophers are suckers for an insoluble problem.
    The gods, the dogs, the beach and the food all exist but only in your imagination. The conversation between Plato and Zeno happened but only in your imagination. Athena and Themis are proxies for you. The rest of the gods are proxies for those of us who are enjoying the mischief that you have created.
    I fear the dogs will starve to death. But they are innocent victims of your imagination and the incompetence of Plato and Zeno. Can you imagine a way to rescue the dogs? Can you resist that picture? Please? (I don't care whether Plato and Zeno starve to death - they're choosing to co-operate with you, so they deserve whatever happens to them. Anyway, they're dead already.) :grin:
  • ssu
    8.3k
    I fear the dogs will starve to death.Ludwig V
    Fear not, the dogs too are imaginary. And yes, it's a story I invented.

    And for Zeno's two dogs, later people (now mathematicians) have put them on a bit different diet. One (guess which one!) has for as food an axiom and the other happily gets it's food in either the surreal or hyperreal system. But they are not with the other dogs that stem from Plato's picked up dog. The question is if they would like to be with the other dogs. Could they be?

    So there not dying from starvation. So it's kind of a happy ending?
  • Ludwig V
    1.2k
    So it's kind of a happy ending?ssu
    I suppose it is, if you think the misery of two dogs a satisfactory price for the happiness of the others. I'm sure it would get a majority vote from the dogs.

    I would suggest the transfinite system as a home for the other dog - since it's the last one and w (omega) is the limit of the series.

    It seemed that the number of the dogs couldn't be countedssu
    I'm pretty sure that was an illusion. After all, each dog can be counted and the counting can continue for as long as there are any dogs that have not been counted.
  • ssu
    8.3k
    I would suggest the transfinite system as a home for the other dog - since it's the last one and w (omega) is the limit of the seriesLudwig V
    After all, each dog can be counted and the counting can continue for as long as there are any dogs that have not been counted.Ludwig V
    Your second statement goes with the lines of Plato then. Poor of Zeno's dogs.

    And with the transfinite, Cantors set of theory of ever larger and larger infinities, it could be argued that this is somewhat similar method to adding. But as I learned from the forum (and got hold of a great book about Cantor) he did think also about Absolute Infinity, but the deeply religious mathematician held it for only God to know.
  • Ludwig V
    1.2k
    Your second statement goes with the lines of Plato then. Poor of Zeno's dogs.ssu
    Well, I don't know how this works. I have imagination deficiency. Doctors have tried for years to cure me. Don't worry, it's not fatal.

    I know that there is no smallest member of a convergent sequence and no largest member of an increasing sequence (I've forgotten the proper term for that.) You may like to consider the possibility that Zeno's dogs don't exist. (After all, he told lies about Achilles and the tortoise.)

    It's true that both sequences have a limit, but a limit is not a element of the sequence. So if they do exist they are not members of the pack. The other dogs would tear them to pieces.
  • ssu
    8.3k
    You may like to consider the possibility that Zeno's dogs don't exist. (After all, he told lies about Achilles and the tortoise.)Ludwig V
    Did he? Or did he try to make an counterargument to Plato? During the time, you tried to make questions that the one answering you would make the argument. So could it be that Zeno was arguing that by Plato's reasoning you get into the silly ideas like the Achilles cannot overtake the tortoise. Or the Arrow cannot move. Remember, the story is told by Plato, not by a third actor.

    If there's isn't that dog that eats the least, then Achilles passes the tortoise. But if you think about Plato's dog and that it can define everything else, then you get the problem. And before you think Zeno's dog that eats the least still isn't a dog, think about it another way:

    Infinitesimals do not exist in the standard real number system, but they do exist in other number systems, such as the surreal number system and the hyperreal number system, which can be thought of as the real numbers augmented with both infinitesimal and infinite quantities; the augmentations are the reciprocals of one another.
    In Abraham Robinsons nonstandard analysis that dog that eats the least exists and is fine.

    So why don't we have it in our standard real number system?

    Because of Plato. (Or at least, because of Plato in this story)
  • Ludwig V
    1.2k
    Remember, the story is told by Plato, not by a third actor.ssu
    Oh, you're imagining that you have discovered a previously unknown manuscript. Who wrote it - Plato, Zeno, Themis, Athene, Zeus? Or a rat, skulking in a corner.

    Did he? Or did he try to make an counterargument to Plato? During the time, you tried to make questions that the one answering you would make the argument. So could it be that Zeno was arguing that by Plato's reasoning you get into the silly ideas like the Achilles cannot overtake the tortoise. Or the Arrow cannot move. Remember, the story is told by Plato, not by a third actor.ssu
    I thought it was Zeno who got the silly ideas. But then, perhaps this is a non-standard analysis.

    In Abraham Robinsons nonstandard analysis that dog that eats the least exists and is fine.ssu
    So long as it is a non-standard dog, I guess it'll pass muster.

    then you get the problem.ssu
    I'm glad of that. It doesn't mean I have any answers.
    I'm afraid I have enough trouble getting my head around standard analyses. If I take on non-standard analyses, my head will go pop.
  • ssu
    8.3k
    Oh, you're imagining that you have discovered a previously unknown manuscript. Who wrote it - Plato, Zeno, Themis, Athene, Zeus? Or a rat, skulking in a corner.Ludwig V
    Well, the reasoning of the Eleatic school isn't this, but do notice that Zeno's paradoxes are handled by limits ...or infinitesimals. So it begs the question.

    You cannot take Plato's dog, add the food of the dog which eats less than every other dog, and then get more than Plato's dog eats. If you would get a different amount of food, then that could be divided even smaller portions and the dog that eats the least wouldn't be the one eating the least. So the question here is: while the dog that eats less than any other dog, does this the definite it separate from all other dogs?
  • Ludwig V
    1.2k
    You cannot take Plato's dog, add the food of the dog which eats less than every other dog, and then get more than Plato's dog eats.ssu
    I'm sorry. I just don't follow this. Is there a typo somewhere?

    If you would get a different amount of food, then that could be divided even smaller portions and the dog that eats the least wouldn't be the one eating the least.ssu
    Nor do I follow this. But I can agree that if you mess about with the food, some other dog might get less than the dog that eats less than any other dog.

    "Plato's dog" is the dog that Plato chose. Let's call the dog that eats less than any other dog "Dog One" and the dog that eats more than any other dog "Dog Two". and the dog that gets less than Dog One "Dog Three".

    In that Case, Dog One would no longer be the dog that eats less than any other dog, because Dog Three is getting less than Dog One. We will have to take the rosette off Dog One's collar and pin it on Dog Three's collar. What's the problem?

    while the dog that eats less than any other dog, does this the definite it separate from all other dogs?ssu
    Each dog is an individual, so we will always be able to find a unique description or assign a unique name to each dog. Unfortunately, we won't be able to assign a number to each dog in the order they were created, but we can assign a unique number to each dog according to how much they eat, starting with Dog One. That won't work if you start messing about with how much they eat.
  • ssu
    8.3k
    I'm sorry. I just don't follow this. Is there a typo somewhere?Ludwig V
    Nope, this is basically Plato's argument in the story: increasing the food or decreasing the food size you always get a new dog's meal. So he reasons that there cannot be the dog that eats the most or the least. Well, in finite dogs this holds true, but notice that Zeno's dogs aren't finite. Hence if you add to Plato's dog the amount the dog that eats the least, you would still have Plato's dog eats. Addition and substraction breaks down, or simply is confusing. The best example of this is the Hilbert Hotel, when it comes to the dog that eats the most.

    "Plato's dog" is the dog that Plato chose. Let's call the dog that eats less than any other dog "Dog One" and the dog that eats more than any other dog "Dog Two". and the dog that gets less than Dog One "Dog Three".Ludwig V
    But then "Dog One" would eat more than "Dog Three", so how could it be the one that eats the least? Remember, it eats less than any other Dog. I think here it's easier to say that the dog Plato picked up is "Dog One", if you think about it.

    That won't work if you start messing about with how much they eat.Ludwig V
    In Mathematics there is this well ordering theorem, so we can assume we can put them into order. Plato did it with his Dog 1, then on one side the dogs that eat more, and on the other side the dogs that eats less.
  • Sir2u
    3.4k
    2. If there was a quantity that could be defined to be different from all other quantities, then there is a dog that would eat this quantity. There are no limitations on the quantities (physical or other), and hence on the dogs.ssu

    Plato is right. By definition #2, there are no physical limitations. A dog that eats the most, and a dog that eats the least implies two physical limits, which violates #2Metaphysician Undercover

    While it is stated that there are no limits, that does not mean that there is no dog that eats the most or the least.
    It is doubtful that the food could be broken down to anything less that a molecule and still be counted as food even though the food is dividable. That would be the food for the dog that ate the least.
    And because the food is dividable to share amongst the little beasts, that would limit the amount that could be eaten by the one that ate the most. No dog could eat all of the food as there would be none for the rest of them.
  • L'éléphant
    1.5k
    2. If there was a quantity that could be defined to be different from all other quantities, then there is a dog that would eat this quantity. There are no limitations on the quantities (physical or other), and hence on the dogs.ssu
    Zeno is right. Not by reason of counting. Rather, by rule #2, the one that eats "the most" and the one that eats "the least" are conceptual quantities that differ from any other quantities already given.

    It is always valid to say "there is at least one dog that eats the most" and "there is at least one dog that eats the least".
  • javi2541997
    5.3k
    Zeno is right. Not by reason of counting. Rather, by rule #2, the one that eats "the most" and the one that eats "the least" are conceptual quantities that differ from any other quantities already given.L'éléphant

    I agree. I had identical thoughts, but I couldn't find the perfect words to express them as you did. :sweat:
    Yes, I am one of the 60% of voters that chose the second choice.
  • Ludwig V
    1.2k
    In Mathematics there is this well ordering theorem, so we can assume we can put them into order. Plato did it with his Dog 1, then on one side the dogs that eat more, and on the other side the dogs that eats less.ssu
    Yes. That will work fine if the criterion for their order can't change. But you have posited that they can change how much they eat. You need another, independent, criterion for "same dog".

    If you move the dog that no longer eats the same amount to its new position, you've no criterion to establish whether you moved the same dog or a different one. (Watching it eat won't help. However much it eats, you need to know whether it is now eating the same amount as it did before or a different one.)

    Perhaps our dog just automatically changes its position, as soon as they start eating differently. But if one dog can change how much it eats, other dogs can. But you have no way of telling whether they have changed or not and so no way of telling whether they are eating differently.

    Rather, by rule #2, the one that eats "the most" and the one that eats "the least" are conceptual quantities that differ from any other quantities already given.L'éléphant
    Yes. But there is the supposition that how much they eat can change. To establish individuation, you need an additional criterion that is not empirical.
  • ssu
    8.3k
    Zeno is right. Not by reason of counting. Rather, by rule #2, the one that eats "the most" and the one that eats "the least" are conceptual quantities that differ from any other quantities already given.

    It is always valid to say "there is at least one dog that eats the most" and "there is at least one dog that eats the least".
    L'éléphant

    I agree. I had identical thoughts, but I couldn't find the perfect words to express them as you did. :sweat:
    Yes, I am one of the 60% of voters that chose the second choice.
    javi2541997
    Absolutely fantastic! :grin:

    And yes, #2 gives us the opportunity to do this. Not by reason of counting as obviously both of Zeno's dogs are literally uncountable. Notice how different the story would come if Athena would have said: "By using this dog and what it eats, put all the other dogs into order."

    As with the story, I argue that this has been a real problem that Math at least over 2400 years.

    Now the thing is that our confusion hasn't stopped us for 2400 years as we are already using both of Zeno's dogs in a variety of way in mathematics, but since this question isn't answered how Plato's and Zeno's dog should logically coexist, we simply have different names. In the case of the dog that eats the least, usually called an infinitesimal, we have non-standard number systems (surreal and hyperreal numbers) and Non-standard analysis.

    And not only that. When you think that in the story "the least eating dog's meal" < "every other dogs meal", think about how Dedekind cuts are used to define real numbers. The name's cut comes from using greater than and less than signs ">" and "<".

    Yet the other Zeno's dog and it's problematic diet comes immediately into question when people tried to form the foundations of mathematics like with set theory. The great Cantor understood what in the story is Plato's argument. For example, when he had the idea of sets having Power set, which are bigger than the set itself, he understood how the power set for "set of all sets" is a bit problematic. And thus Cantor's set theory is hierarchial. Yet what is interesting (and what I can thank the PF forum leading me to a great book) is that Cantor didn't reject the notion of Absolute Infinity, or what in the story would be the dog that eats the most. As he couldn't describe this infinity, he thought it was something that God would know. Or God. Something that for a deeply religious person is something important (unlike for the many reading Cantor's texts). Still, with Frege's simple idea of Basic Law V invites immediately the other of Zeno's dogs to the picture.

    As you can notice, in the poll of the story I didn't actually ask if Plato is right or both are right. And this is the real question for mathematics: how can both Zeno's dogs and Plato's dogs coexist in peace?
  • LuckyR
    438
    Plato is right. By definition #2, there are no physical limitations. A dog that eats the most, and a dog that eats the least implies two physical limits, which violates #2


    Incorrect. There are an infinite number of quantities between 1 bowl of food and 2 bowls, just as there are between 1/infinity and infinity.
  • ssu
    8.3k
    Yes. That will work fine if the criterion for their order can't change. But you have posited that they can change how much they eat.Ludwig V
    Hopefully I didn't. All the dogs eat exactly a defined amount of food different from any other dog, not less, not more.

    (BTW, if someone is puzzled why I did choose Plato and not Aristotle, it's because we learn about Zeno of Elea through Plato's writings. Of course the objection is more Aristotle, but the time gap is even bigger between Zeno and Aristotle. Some reality to a story with Greek goddesses meeting philosophers. :joke: )
  • Metaphysician Undercover
    12.8k
    But it's a limitation, when you start from Plato's dog.

    Yet doesn't the dog that eats more than any other dog define it different from all other dogs? No matter if there's an Apeiron (endless amount) of dogs that eat less.
    ssu

    I don't understand what you're saying here. Can you explain?

    There are an infinite number of quantities between 1 bowl of food and 2 bowls, just as there are between 1/infinity and infinity.LuckyR

    How is that relevant?
  • LuckyR
    438


    Because having a dog at the low end (say, 1 bowl) and another at the high end (2 bowls), doesn't preclude having a limitless number, ie it's not limiting.
  • ssu
    8.3k
    I don't understand what you're saying here. Can you explain?Metaphysician Undercover
    Ok, If you start from Plato's dog as the measure for all dogs, let's call it dog 1, you get dog 2 (that eats twice the amount), dog 3 (eating triple amount), dog 4 and so on. And obviously for any dog n, then there's a dog n+1 and so on. And from this, in reality Aristotle (not Plato, in reality) would talk about only a potential infinity. And this idea stayed until Cantor, for example Gauss wrote in 1831: “I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics. Infinity is merely a way of speaking” and Kronecker, who vigorously disagreed at Cantor famously said "God created the integers, all the rest is the work of Man".

    This is what I tried to refer to, when I said " it's a limitation, when you start from Plato's dog." Perhaps better wording would be simply a rejection.

    I think others have here explained why in the story dog eating the most and least aren't limitations starting from @Sir2u, @L'éléphant and @Lucky R.

    But don't worry, this is only perhaps the most heated and difficult issue in mathematics, ever.

    It is doubtful that the food could be broken down to anything less that a molecule and still be counted as food even though the food is dividable. That would be the food for the dog that ate the least. And because the food is dividable to share amongst the little beasts, that would limit the amount that could be eaten by the one that ate the most.Sir2u
    What I was trying to say with rule 3, things like physical dimensions or other physical aspects wouldn't be taken into question (as Viking dogs do also take space and also Ancient Greece had a limited area) as the amount of food the dogs eat can be compared to only to the dogs.

    No dog could eat all of the food as there would be none for the rest of them.Sir2u
    Hope you here notice the incommensurability between what the dog that eats more than everybody else and any dog that can be measured by Plato's picked up dog. And what is "all the food" for the dogs since the food can be compared among the dogs? You cannot double the food amount of all dogs, or take half the food away from every dog. Just as with whatever dog Plato picks up, he'll by his definition pick up Dog 1. There's enough of food, the goddesses made sure about that.
  • Metaphysician Undercover
    12.8k
    This is what I tried to refer to, when I said " it's a limitation, when you start from Plato's dog." Perhaps better wording would be simply a rejection.ssu

    I can see how starting (in this case at a dog), is a limitation, because the start produces a particular perspective. However, this is not a limitation on any quantity, physical or otherwise, as dictated by the premises: "There are no limitations on the quantities (physical or other), and hence on the dogs." The starting point, "Plato's dog" is a limitation on the act of measuring, imposed by choice, it is not a limitation on any dogs.
  • ssu
    8.3k
    The starting point, "Plato's dog" is a limitation on the act of measuring, imposed by choice, it is not a limitation on any dogs.Metaphysician Undercover
    On the other hand, with Plato's dog, we can do something as important as count and measure. The first thing that mathematics evolved from, and something that smart animals can also in their way do.

    And in my view this measurement creates the confusion. Here with dogs that simply cannot be measured as their definition relies on this (if you could measure it, they wouldn't eat the least or most as Plato is totally right in this way). And I think this is the problem when we want to view mathematics as a logical system, but start from the natural numbers and assume something like addition is a meaningful operation with everything. Yet Mathematics, as a logical system, holds true mathematical objects that aren't countable or directly provable. I think we are still missing something very essential here.
  • Metaphysician Undercover
    12.8k
    On the other hand, with Plato's dog, we can do something as important as count and measure. The first thing that mathematics evolved from, and something that smart animals can also in their way do.ssu

    Yes, Plato's dog is the point of comparison, the paradigm we might say, and this is the basis of measurement.

    And in my view this measurement creates the confusion. Here with dogs that simply cannot be measured as their definition relies on this (if you could measure it, they wouldn't eat the least or most as Plato is totally right in this way). And I think this is the problem when we want to view mathematics as a logical system, but start from the natural numbers and assume something like addition is a meaningful operation with everything. Yet Mathematics, as a logical system, holds true mathematical objects that aren't countable or directly provable. I think we are still missing something very essential here.ssu

    Doesn't mathematics start with the unit, one, as the point of comparison, just like Plato\s dog, and from here we allow an unlimited number of units and also unlimited divisions of the unit. The actual problem is when we try to measure the system of measurement. The system of measurement is designed to allow for the measurement of any possibility, hence the unlimited, or infinite, numbers, and this makes it inherently unmeasurable. Then we need to go to another system of comparison, another paradigm other than measurement. This produces a problem because anything unlimited is fundamentally unintelligible because the way that the intellect apprehends things is through their limits. So the goal of measuring the system of measurement is self-defeating.
  • ssu
    8.3k
    Doesn't mathematics start with the unit, one, as the point of comparison, just like Plato\s dogMetaphysician Undercover
    I'm not so sure that mathematics starts from exactly one thing. :smile:

    The actual problem is when we try to measure the system of measurement.Metaphysician Undercover
    Well, think in the story about how much all dogs eat, then remember the rules.
  • Sir2u
    3.4k
    And what is "all the food" for the dogs since the food can be compared among the dogs?

    1. There's enough of food, the goddesses made sure about that.
    ssu

    Let me ask you a question. Is the 100% of the food is for 100% of the dogs. It makes no difference the actual quantity of the food, only the correspondence of food to dogs.
  • Metaphysician Undercover
    12.8k
    Well, think in the story about how much all dogs eat, then remember the rules.ssu

    There is no general statement about how much all dogs eat. It is explicitly stated "every dog eats a different quantity of food", and " There are no limitations on the quantities". Therefore we cannot make any inductive conclusion about "how much all dogs eat", because each eats a different amount, nor can we make a conclusion as to how much all the dogs would eat, because this is stated to be unlimited.

    What are you trying to get at?
  • ssu
    8.3k
    Is the 100% of the food is for 100% of the dogs.Sir2u
    I can't fathom it would be for anybody else.

    It makes no difference the actual quantity of the food, only the correspondence of food to dogs.Sir2u
    I think so. As I said: if you double the amount of food to every dog, it doesn't matter as they can be only measured to each other. There would be no difference. Notice that measuring is possible with the random dog that Plato picked up. Yet If you give all the dogs just the amount as Plato's picked up dog eats, that would leave a lot of dogs hungry and a lot with way more food they eat. That would create a mess.
  • ssu
    8.3k
    What are you trying to get at?Metaphysician Undercover

    To show one way how an at least 2400 year old (but likely older) difficulty in mathematics emerges, which hasn't gone away. You should read the answer that I gave to @L'éléphant and @javi2541997 here. It gives also a question for further thinking.
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