Fear not, the dogs too are imaginary. And yes, it's a story I invented.I fear the dogs will starve to death. — Ludwig V
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.So it's kind of a happy ending? — ssu
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.It seemed that the number of the dogs couldn't be counted — ssu
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 — Ludwig V
Your second statement goes with the lines of Plato then. Poor of Zeno's dogs.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
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.Your second statement goes with the lines of Plato then. Poor of Zeno's dogs. — ssu
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.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
In Abraham Robinsons nonstandard analysis that dog that eats the least exists and is fine.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.
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.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.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
So long as it is a non-standard dog, I guess it'll pass muster.In Abraham Robinsons nonstandard analysis that dog that eats the least exists and is fine. — ssu
I'm glad of that. It doesn't mean I have any answers.then you get the problem. — ssu
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.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
I'm sorry. I just don't follow this. Is there a typo somewhere?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
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.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
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.while the dog that eats less than any other dog, does this the definite it separate from all other dogs? — ssu
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.I'm sorry. I just don't follow this. Is there a typo somewhere? — 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."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
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.That won't work if you start messing about with how much they eat. — Ludwig V
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 #2 — Metaphysician Undercover
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.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. — L'éléphant
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".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. But there is the supposition that how much they eat can change. To establish individuation, you need an additional criterion that is not empirical.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
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
Absolutely fantastic! :grin: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
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
Hopefully I didn't. All the dogs eat exactly a defined amount of food different from any other dog, not less, not more.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
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
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
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".I don't understand what you're saying here. Can you explain? — Metaphysician Undercover
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.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
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.No dog could eat all of the food as there would be none for the rest of them. — Sir2u
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
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.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. — ssu
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
I'm not so sure that mathematics starts from exactly one thing. :smile:Doesn't mathematics start with the unit, one, as the point of comparison, just like Plato\s dog — Metaphysician Undercover
Well, think in the story about how much all dogs eat, then remember the rules.The actual problem is when we try to measure the system of measurement. — Metaphysician Undercover
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
Well, think in the story about how much all dogs eat, then remember the rules. — ssu
I can't fathom it would be for anybody else.Is the 100% of the food is for 100% of the 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.It makes no difference the actual quantity of the food, only the correspondence of food to dogs. — Sir2u
What are you trying to get at? — Metaphysician Undercover
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