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. — ssu
Please, I value everybody's contribution as I cannot overstate here just how difficult and open ended question this is. Yet it's very simple and you can think about it even without a long background in math. That's the real beauty of math, at it's most beautiful, it's elegant and simple.You both had a very interesting exchange. I am sorry, ssu. His reply to me and Elephant was awesome, but I didn't know what to answer back because I do not have a big background in math and logic. The replies by MU are pretty good too. — javi2541997
I agree too, wholeheartedly. But notice how radical (or outrageous to some) our view is, actually. Plato's rejection is totally logical. And think just where we come with our own thinking. If the other of Zeno's dog more than any other dog, there cannot be a dog or a collection of dogs that eat more, right? It absolutely eats more than any dog, I would boldly argue.But, sure, I believe Zeno's two dogs must exist since there is always a "most" and a "least," correct? — javi2541997
Err, isn't there actually an absolute lowest temperature, - 273,15 Celsius? We cannot talk then about a temperature of - 2 000 000 Celsius or lower temperatures to my knowledge. So this isn't similar to the problematics of the Zeno's dogs in the story (or at least the other one).Consider this example, suppose we want to set a scale to measure all possible degrees of heat in the vast variety of things we encounter, a temperature scale. We could start by determining the highest possible temperature, and the lowest possible temperature, (analogous to Zeno's dogs) and then scale every temperature of every circumstance we encounter, as somewhere in between. — Metaphysician Undercover
And then, if you think that there's just two Zeno's dogs, how about then all the transcendental dogs between them. — ssu
Err, isn't there actually an absolute lowest temperature, - 273,15 Celsius? We cannot talk then about a temperature of - 2 000 000 Celsius or lower temperatures to my knowledge. So this isn't similar to the problematics of the Zeno's dogs in the story (or at least the other one). — ssu
Are you suggesting that it is irrelevant to Plato whether there is a dog who eats the most and another who eats the least? Well, maybe. — javi2541997
But the rules stated by Athena say: ‘All the dogs eat the same food, which is divisible, and there is enough of it for every dog’ but Zeno argues (and I agree with that) that 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. — javi2541997
And I thought in my ignorance, that there's at least this obvious limit in Physics! Of course, what is Physics else than the study of change and movement? So there's big problems to get funding for a research on the effects of temperatures of negative millions of Celsius. Fortunately there's an actual reality to seek something else.That is the lowest temperature realizable from our methods of measurement. In other words it is a restriction created by our choice of dog to use for comparison, the movement of atoms. It does not mean that a lower temperature will not be discovered, if we devise a different measurement technique. — Metaphysician Undercover
Even if this was for javi, here's my point: That wasn't the task. The task was to feed all the dogs. Plato tries desperately to please his goddesses by taking a dog as the measurement stick (dog?) and tries to get some order to the dogs. Will he accept even irrational dogs, I don't know. But transcendental dogs surely are something he didn't know and the reals are the problem. But they are should I say in the realm of being Zeno's dogs.That is exactly what I am suggesting. Plato was given the task of measurement, and he took that task and proceeded. — Metaphysician Undercover
I have to point out this: Zeno understood Plato's argument. Indeed you cannot reach Zeno's dogs from Plato's dog because of Plato's argument. It is quite valid. Or to put this in another way, the whole definition of Zeno's dogs relies on that they cannot be reached by measurement (or counting).The "other two dogs" referred to by Zeno is a sophistic ruse, just like Plato says. Zeno could have said, "let me know when you get to the dog that eats the most, and the dog that eats the least", and Plato could have said "OK". Problem resolved. — Metaphysician Undercover
Zeno could have said, "let me know when you get to the dog that eats the most, and the dog that eats the least", and Plato could have said "OK". Problem resolved. Instead, Zeno said you are "forgetting" these two dogs. But Plato is not "forgetting" them, he has not yet found them, so there is no need for them to have ever entered his mind. — Metaphysician Undercover
And I thought in my ignorance, that there's at least this obvious limit in Physics! Of course, what is Physics else than the study of change and movement? So there's big problems to get funding for a research on the effects of temperatures of negative millions of Celsius. Fortunately there's an actual reality to seek something else. — ssu
The task was to feed all the dogs. — ssu
However, those two dogs, the one that eats the most and the other who eats the least, exist for both Plato and Zeno. Right? :smile: — javi2541997
That's why the task was for the philosophers "to tell a way to feed all the dogs on the beach without any dog being left out hungry and Themis would make this instantly to happen".If that's the case then both Plato's dog and Zeno's dogs are irrelevant, all one needs to do is point the dogs to the food and tell them to go to it. — Metaphysician Undercover
Zeno completely comprehended Plato's reasoning, although he did not convey the correct response. Instead, Zeno assumed that Plato had forgotten two elementary dogs, which is incorrect. Plato merely dismissed them as irrelevant to his argument. However, those two dogs, the one that eats the most and the other who eats the least, exist for both Plato and Zeno. Right? :smile: — javi2541997
Plato doesn't accept the existence of Zeno's dogs. Or in reality, Aristotle and many in the following Centuries believe that there is only a potential infinity, not an actual infinity. Many finitists still this day don't believe in actual infinity, perhaps any infinity altogether. And Absolute Infinity is even more controversial.Not under the assumption that quantities are unlimited. — Metaphysician Undercover
There doesn't have to be any surplus, as this is done once. The task is that the philosopher is to define in some way all the amounts of food and hence all the dogs, that they don't leave some dogs out. As no dog eats the same amount, then it's easy for the goddes to put the dogs in an growing or decreasing line based on their amount of food.Maybe I asked the wrong question.
If all of the dogs are fed, is there anything left over? Until it is time to feed them again at least. Or does the food continue to be 100% even if some of it is removed? — Sir2u
That's why the task was for the philosophers "to tell a way to feed all the dogs on the beach without any dog being left out hungry and Themis would make this instantly to happen". — ssu
There doesn't have to be any surplus, as this is done once. The task is that the philosopher is to define in some way all the amounts of food and hence all the dogs, that they don't leave some dogs out. As no dog eats the same amount, then it's easy for the goddes to put the dogs in an growing or decreasing line based on their amount of food. — ssu
and @Metaphysician Undercover @Sir2uPlato doesn't accept the existence of Zeno's dogs. — ssu
On the other hand, Plato argues that there cannot be a dog that eats the most, because there is always a dog that eats more. — javi2541997
Does infinity actually mean that there is always one more, or does it just mean the possibility of it? — Sir2u
Well, if it's so, then the counterarguments of the actual Zeno of Elea gave us are quite relevant.Then why isn't Plato's way the proper way? There's no need to determine the dog which eats the most or the dog which eats the least, just keep feeding in the way Plato described. — Metaphysician Undercover
It sure sounds a lot like the other Zeno's dog, doesn't it? And why is then non-standard? Well, basically because of Aristoteles and his following (or Plato in the story).Nonstandard analysis is a branch of mathematical logic which introduces hyperreal numbers to allow for the existence of "genuine infinitesimals," which are numbers that are less than 1/2, 1/3, 1/4, 1/5, ..., but greater than 0. Abraham Robinson developed nonstandard analysis in the 1960s. The theory has since been investigated for its own sake and has been applied in areas such as Banach spaces, differential equations, probability theory, mathematical economics, and mathematical physics.
If it would only be possible that there could be a dog, but there wouldn't be that next dog, then obviously the number of dogs on the beach would be finite.Does infinity actually mean that there is always one more, or does it just mean the possibility of it? — Sir2u
The point of the story is that this problem hasn't been solved. And it comes down to the problem in the story. — ssu
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