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
But you're missing the point, I think. We don't know when they stop counting of how much each dog eats -- whether going up or downwards quantity. They could continue counting, for all I care. But the fact remains that there is the dog the eats the most and the dog that eats the least. Plato and Athena would not know this until after they stop counting (that is, if they could stop counting). But already Zeno identified two dogs that eat differently than their dogs.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. — Ludwig V
That's no great trick. Every dog eats differently than all the other dogs.But already Zeno identified two dogs that eat differently than their dogs. — L'éléphant
There's an ambiguity in the ordinary use of these superlatives which means they cannot be meaningfully applied in the context of a infinite sequence.But the fact remains that there is the dog the eats the most and the dog that eats the least. — L'éléphant
Forgive my stupidity, but I don't understand what a completed infinity is.And actual infinity is the completed infinity. — ssu
Notice in the story Athena, the goddess of wisdom, might very well know the answer as she did use the two philosophers for amusement for the other gods.Plato and Athena would not know this until after they stop counting (that is, if they could stop counting). — L'éléphant
I think everybody understands that there is no largest finite number. Because, every natural number is finite, right? Even in the story Zeno is well aware of this.The largest natural number is the number that is larger than all the other natural numbers and has no natural number that is larger than it. But every natural number has a natural number larger than it. So there is no largest natural number. — Ludwig V
(First of all, notice that ω here refers to the largest Ordinal number. In the story it would mean that you put all the dogs that food amount is exactly divisible by dog 1's food (let's call them positive dogs) in a line from smaller to bigger, and then start counting the dog line from their places on the line, from the first, second, third, fourth... and then get to infinity in the form of ω. Notice it's different from cardinal numbers.)There is a number that is larger than every natural number.
That number is ω, which is the lowest ordinal transfinite number, which is defined as the limit of the sequence of the natural numbers. — Ludwig V
Modern derivative and integral symbols are derived from Leibniz’s d for difference and ∫ for sum. He applied these operations to variables and functions in a calculus of infinitesimals. When applied to a variable x, the difference operator d produces dx, an infinitesimal increase in x that is somehow as small as desired without ever quite being zero. Corresponding to this infinitesimal increase, a function f(x) experiences an increase df = f′dx, which Leibniz regarded as the difference between values of the function f at two values of x a distance of dx apart. Thus, the derivative f′ = df/dx was a quotient of infinitesimals.
Well, you already referred to completed infinity or actual infinity with the example of ω as that is Cantorian set theory. Here's one primer about the subject: Potential versus Completed Infinity: its history and controversyForgive my stupidity, but I don't understand what a completed infinity is. — Ludwig V
Forgive my stupidity, but I don't understand what a completed infinity is.And actual infinity is the completed infinity. — ssu
Well, it's your story. You are the only person who can provide an answer.Notice in the story Athena, the goddess of wisdom, might very well know the answer as she did use the two philosophers for amusement for the other gods. — ssu
Even in the story Zeno is well aware of this. — ssu
A transfinite number isn't a natural number, so it doesn't get attached to (aligned with) a dog. Nor could it be.But back to the story: Then doesn't that ω in the story relate to distinct dog? You even referred yourself of ω being a number. — ssu
I was careful to notice that - and. at least by implication, the cardinal numbers.First of all, notice that ω here refers to the largest Ordinal number. — ssu
That will take you, and even the gods, an infinite time. But I guess Plato, Zeno and certainly the gods, have that amount of time available, and are bored.you put all the dogs that food amount is exactly divisible by dog 1's food (let's call them natural dogs) in a line from smaller to bigger — ssu
You can start, but you can't finish in less than infinite time. And even Plato, Zeno and the gods will be bored by the time they get to the end of a second infinite count.start counting the dog line from their places on the line, from the first, second, third, fourth... and then get to infinity in the form of ω. — ssu
If you choose to call ω completed or actual, that's your choice. I can't work out what you mean. I don't know enough to comment on Cantorian set theory.Well, you already referred to completed infinity or actual infinity with the example of ω as that is Cantorian set theory. — ssu
Notice in the story Athena, the goddess of wisdom, might very well know the answer as she did use the two philosophers for amusement for the other gods. — ssu
Well, a dog eating ⅚ of Plato's dog's food amount isn't either a natural number, so would you deny it to be a dog? And what about transcendental dogs? They are finite, but the dog that eats π amount compared to Plato's dog?A transfinite number isn't a natural number, so it doesn't get attached to (aligned with) a dog. Nor could it be. — Ludwig V
Now your are putting physical limitations to the story, which didn't have them (Athena created the dogs instantly and Themis could feed them instantly also, if given the proper rule / algorithm). In fact when you think of it, already large finite number of dogs cause huge problems in the physical world: if counting or feeding a dog takes even a nanosecond, with just finite amounts of dogs the whole time universe exists won't give enough time to count or feed them. If your counterargument is ultrafinitism, that's totally OK. This is a Philosophy Forum and this issue is totally fitting for a philosophical debate. I would just argue that the system of counting that basically is like 1,2,3,4,...., n, meaningless over this number isn't rigorous. It's very logical to have infinities as mathematics is abstract.That will take you, and even the gods, an infinite time. — Ludwig V
Well, I gave you already on article going over this earlier. Just a quote from it, if you don't have the time to read it:If you choose to call ω completed or actual, that's your choice. I can't work out what you mean. I don't know enough to comment on Cantorian set theory. — Ludwig V
Potential infinity refers to a procedure that gets closer and closer to, but never quite reaches, an infinite end. For instance, the sequence of numbers
1, 2, 3, 4, ...
gets higher and higher, but it has no end; it never gets to infinity.
Completed infinity, or actual infinity, is an infinity that one actually reaches; the process is already done. For instance, let's put braces around that sequence mentioned earlier:
{ 1, 2, 3, 4, ... }
With this notation, we are indicating the set of all positive integers. This is just one object, a set. But that set has infinitely many members. By that I don't mean that it has a large finite number of members and it keeps getting more members. Rather, I mean that it already has infinitely many members. We can also indicate the completed infinity geometrically.
and @Ludwig VAs all dogs do eat something, we have a problem with the non-existent dog that doesn't eat anything, — ssu
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