Two Philosophers on a beach with Viking Dogs
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.
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!
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.
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!
Comments (87)
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.
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:
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?
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.
Quoting 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.
Quoting 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.
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.
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:
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)
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.
Quoting ssu
I thought it was Zeno who got the silly ideas. But then, perhaps this is a non-standard analysis.
Quoting ssu
So long as it is a non-standard dog, I guess it'll pass muster.
Quoting 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.
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?
I'm sorry. I just don't follow this. Is there a typo somewhere?
Quoting 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?
Quoting 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.
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.
Quoting 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.
Quoting 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.
Quoting Metaphysician 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.
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".
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.
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.
Quoting 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.
Quoting 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?
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.
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: )
I don't understand what you're saying here. Can you explain?
Quoting LuckyR
How is that relevant?
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.
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.
Quoting 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.
Quoting 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.
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.
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.
Yes, Plato's dog is the point of comparison, the paradigm we might say, and this is the basis of measurement.
Quoting 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.
I'm not so sure that mathematics starts from exactly one thing. :smile:
Quoting Metaphysician Undercover
Well, think in the story about how much all dogs eat, then remember the rules.
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.
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?
I can't fathom it would be for anybody else.
Quoting 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.
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.
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?
Yes, I already read that, and I didn't see much to disagree with, except your question at the end.
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.
Under my very basic sense of logic or math, I still root for Zeno because of the following: 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?
I agreed. Even if the dogs are uncountable, at least one will eat the most, followed by the least. But this is only a very basic concept of mine. I can't keep debating with logic or numbers, as you did. But, sure, I believe Zeno's two dogs must exist since there is always a "most" and a "least," correct?
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. Alternatively, we could start with one temperature, the freezing point of water for example (analogous to Plato's dog), and scale the temperature of all other things we encounter relative to this. Whether there is a hottest or coldest possible temperature is irrelevant to this alternative way of scaling.
Incidentally, I think this issue is relevant to the way that we judge goodness and badness in moral actions, and create codes of ethics. Some would argue that we need a best, the omnibenevolent God, and a worst, the evil devil, and all moral acts are judged in relation to these two. Others however, argue that we take any random act, and judge whether other acts are better or worse than it. I would argue that the latter is the common way that people make decisions. If a person is inclined toward a particular act (this represents "Plato's dog"), they will look at other possibilities, and judge these possibilities, each one, as to whether it is a better or worse course of action in relation to the one that the person is inclined toward. The person will choose accordingly. I believe that it is not often, that in making a decision, the person judges the possible act as to whether it is closer to what God would choose, than what the devil would choose.
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.
Quoting 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.
I'll try to show just how problematic this is even with Plato's dog and the multiples of this dog.
Let's start Plato's dog, dog1 and all those dogs that eat exactly some multiple times it's food (dog1, dog2, dog3, dog4, dog5, and so on). Let's pick three dog from this collection of dogs (or set of dogs) and have dog a, dog b and dog c that
dog a + dog b = dog c
Now if we know two dogs, we can compute the third one in the equation. So if dog a is actually dog2 and dog b is dog3, then you can come to the conclusion that dog c is of course, dog5. We can solve the equation. However, if we have an inequation like:
dog a + dog b < dog c
We don't know what dog c is exactly, even if we would know that the others (a and b) are dog2 and dog3. The only thing we can say then is that dog c can then be dog6, dog7, dog8 or a dog that eats a higher multiple than that of dog1's food. And that's it. We cannot calculate what dog c is. Dog c obviously exists (as it belongs to this set of dogs and if it's dog6 or higher) as the multiples of dog1 go on and on and never stop.
Just how confusing this becomes is when we notice that actually our definitions of Zeno's dogs are inequations:
Zeno's least eating dog eats < any other dog there exists eats
Zeno's most eating dog eats > any other dog there exists eats.
Yet we can intuitively think that Zeno's dogs exist and we have a place for them. We can assume a well ordering using the amount of tood the dogs eat as did Plato ( dog1 < dog2 < dog3 ). Yet consider then putting Zeno's dogs on each ends of the lines. What happens? You cannot pick any dogs between them. You have lost all ways of measurement. Or in other words, you cannot pick the next dog from Zeno's least eating dog or the previous dog before Zeno's dog that eats more than everybody.
And then, if you think that there's just two Zeno's dogs, how about then all the transcendental dogs between them.
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).
And the second rule states that there are no constraints on quantity (physical or otherwise), and hence on dogs. So Zeno is right here. There will always be one dog who eats the most and another who eats the least, which I believe is relevant to this issue, and Plato overlooked these two dogs in his counting.
Quoting ssu
Exactly. As you mentioned, the rest of the dogs are simply transcendental in the situation. What I know for certain is that there will be at least two dogs: one that eats the most (let's call him dog >) and one who eats the least (let's call him dog <), but I'm not sure who dog b is, because the latter is just transcendental to the scale. What I can't do, if I understand Zeno correctly, is start counting by dog b or another random dog "x" because my numbering will be irregular due to forgetting those two dogs. The one who "starts" and the one who "ends," or, to put it another way, the one at the bottom and the other at the top.
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. Notice there is no such limit to the hottest possible temperature, because we move to different measuring principles.
Quoting javi2541997
That is exactly what I am suggesting. Plato was given the task of measurement, and he took that task and proceeded. That the task will never be completed because the quantities are unlimited, is irrelevant. Therefore whether or not there is a dog that eats the least or the most, is also irrelevant.
Quoting javi2541997
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. 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.
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.
Quoting 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.
Quoting 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).
In fact your own argument that absolute zero being only a measurement problem is somewhat similar here, it's a limit for modern measurement as atoms cease to move. Yet if we define Physics to be only "atoms moving", then there's a categorical denial of your idea of lower than absolute zero temperatures. Luckily Physicists understand that they are only making models and theories and these that we hold now to be true can be proven wrong and new better models can be invented.
Got it. 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:
I've read some speculations showing that the hottest temperature will actually end up being the same as the coldest temperature. Strange.
Quoting ssu
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. When they all get fed the task is complete. Since it is stipulated that the quantities are unlimited, the task will never be completed, some dogs will not finish eating before Plato and Zeno pass on.
Quoting javi2541997
Not under the assumption that quantities are unlimited.
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 the task is to give a goddess a way to "sort them out", then it's not a reply to have "the gods sort them out". Remember if there is an endless amount of food, there is also an endless number of dogs.
Quoting Metaphysician Undercover
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.
Quoting Sir2u
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.
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.
Ok, so if there is no surplus and no dog that goes without food then there has to be one that will eat more than the rest and one that will eat less.
The only other option would be an infinite amount of food and an infinite number of dogs. And as you said there is a limited area in Greece, so that does not seem likely.
And even if there were an infinite number of dogs, there would still have to be the one that eats less than any other dog.
I guess thats why he [Plato] explained again to Zeno that there cannot be a dog that eats the most, because there is always a dog that eats more. I am still confused regarding Platos argument. Yes, it is clear to me that Plato roots for infinity counting of dogs, but I think he forgets two basic dogs: the one who eats the most (the dog at the peak) and the dog who eats the least (the dog at the bottom). I think the argument of Zeno is more plausible. 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. He sees infinity towards the maximum. But what about the dog who eats the least? If there is always a dog that could eat less, there will be a dog who will eat nothing at all. How can it be possible to find a dog who will eat less than previous dogs and so on? I think this has to be switched and follow Zenos point of limited counting: Zeno's least eating dog eats < any other dog there exists eats, and then start to count. Agree?
Does infinity actually mean that there is always one more, or does it just mean the possibility of it?
I think Plato means that there is always one more dog. If we take this to maximum, it could be, somehow, plausible, and I guess I have to agree with Plato.
But I cannot see it in regression. It is not plausible to think that there is always one dog who eats less than the previous one. Yes, I know that Athena stated there was enough food for every dog. But my point is that, sooner or later, we will reach the bottom and there will finally be a dog who eats less than all the infinity intermediate dogs. I think this is more plausible than to think that there will always be one dog and another in both extremes continuously in a loop.
Well, if it's so, then the counterarguments of the actual Zeno of Elea gave us are quite relevant.
And if you think that is nonsense, how about then the idea of the infinitesimal? Obviously something that created a huge debate at the time of Newton and Leibniz. The idea of an infinitesimal comes closest to the other of Zeno's dogs in the story. Remember that there's Robinsons non-standard analysis. Here's from Wolfram Mathworld:
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).
And how about then calculus or mathematical analysis in general? It's very useful, an important area of mathematics. But can you put it on a sound footing just with assuming Plato's potential infinity? Some argue, and in my view convincingly, that set theory was intended to put finally analysis on a firm footing by set theory. But then set theory itself stumbled into paradoxes.
The point of the story is that this problem hasn't been solved. And it comes down to the problem in the story.
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.
So potential infinity means that there is always more eating dogs ...and less eating dogs, that this process doesn't stop. Thus there cannot be the dog that eats the most or the least. This is in the story Plato's argument.
And actual infinity is the completed infinity. In the story it's basically the more eating Zeno's dog. Think about it this way: All the dogs eat something. If all they eat something, doesn't this the mean there exist the amount of food that all the dogs eat? If so, by rule #2, then there's a dog that eats it. That in the story is Zeno's argument.
(And again I tip my hat to the reasoning that L'èléphant gave on page 1.)
That's one problem. I'm sure there's many more. Find another, and start another thread.
And furthermore, I think that today we might be closer to a solution on these open questions because we are already comfortable of there being the non-computable and non-provable but true mathematical statements. This is actually a real sea change from the time when the paradoxes of set theory were found over hundred years ago or what people thought earlier. The existence of non-computable and even non-provable mathematics would have been quite a heresy in earlier times, but now we start to accept this. (See for example another current PF thread talk about this and about Noson Yanofsky's paper "True but unprovable" here.)
The non-computability of Zeno's dogs in the story should be (hopefully) obvious. But this non-computability goes a lot more further. Set theory shows this well and the problems that naive set theory had even more.
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.
That's no great trick. Every dog eats differently than all the other dogs.
Quoting 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.
I assume that we can take Plato's dog as dog 0, and allocate the natural numbers with the dogs that eat more than Plato's dog. (And similarly with the dogs that eat less than Plato's dog. Yes?
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. That follows from the definition of infinity.
It looks as if you are not aware of how the mathematics works in this context. Forgive me if I'm wrong.
There is a number that is larger than every natural number.
That number is ?, which is the lowest ordinal transfinite number, which is defined as the limit of the sequence of the natural numbers.
See Wikipedia - Transfinite numbers
A parallel argument (suitably adjusted for the different context of a convergent sequence) applies to the dog that eats the least amount of food.
Quoting ssu
Forgive my stupidity, but I don't understand what a completed infinity is.
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.
Quoting Ludwig V
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.
Quoting Ludwig V
(First of all, notice that ? 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 ?. Notice it's different from cardinal numbers.)
But back to the story: Then doesn't that ? in the story relate to distinct dog? You even referred yourself of ? being a number. Why then couldn't it be a dog on the beach?
After all, limit sequences are the way we also defined the other of Zeno's dogs. Yes, we refer to limits and only non-standard analysis to infinitesimals, however the modern calculus does go the lines of Leibniz, who used the infinitesimal, which is the least eating Zeno's dog in the story:
Quoting Ludwig V
Well, you already referred to completed infinity or actual infinity with the example of ? as that is Cantorian set theory. Here's one primer about the subject: Potential versus Completed Infinity: its history and controversy
Forgive my stupidity, but I don't understand what a completed infinity is.
Quoting ssu
Well, it's your story. You are the only person who can provide an answer.
Quoting ssu
Quoting ssu
A transfinite number isn't a natural number, so it doesn't get attached to (aligned with) a dog. Nor could it be.
Quoting ssu
I was careful to notice that - and. at least by implication, the cardinal numbers.
Quoting 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.
Quoting 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.
Quoting ssu
If you choose to call ? 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.
:up:
Well, a dog eating ? 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 ? amount compared to Plato's dog?
(And here I have to make a correction to above. As all dogs do eat something, we have a problem with the non-existent dog that doesn't eat anything, as that is part of the natural number (natural dogs) and I should have referred to positive integers (positive dogs, not natural dogs).
Quoting 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.
Quoting 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:
I remember I referred to that specific dog in our previous exchange. I said that following Athenas rule, it is not possible to think that there will always be a dog that will eat less than the previous one, and so on. Athena stated that there is enough food for every dog. So, lets say, there is a dog who eats 15 pieces of meat, and there is another dog who eats only 0.0001 pieces of that meat. Those are the two dogs that Zeno was referring to: the dog at the top and the dog at the "bottom, but why do you count a non-existent dog? If there is enough food for the dogs, there isn't a dog who doesnt eat anything at all.
I mean, following the premises of the OP it is not possible to imagine a dog who doesnt eat anything.
It all comes down to rule2 and how we interpret rule1. By rule2 if there is an amount, there's a dog for it. If nothing is an amount, then there is a dog for that. Now if rule1 eating means that a dog cannot refrain from eating, then obviously it's a non-existing dog with a non-existing amount of food. Now if we want to include that in the or not is in my view a philosophical choice (and in reality it took a lot of time for Western mathematics to accept zero as a number).
And notice that the debate about just what we do accept as numbers (or mathematics) has continued and hasn't faded away. For example the Ancient Greeks didn't view like us rational or irrational numbers as being numbers: for them there were numbers and then the idea of ratios. What is accepted and what is not continues with Finitism even today, as the Cantorian set theory does still give rise to opposing arguments (especially of larger and larger infinities), even if a they are views of the minority.
For example if we want have the ability to measure the food amounts, just look at the following Venn-diagram and notice at how limited "constructible lengths" is in the diagram. As I stated to @Ludwig V, just having finite, but transcendental numbers like ? or e that aren't Constructible numbers already gives the problem of Zeno's dogs, even if we would dismiss the two Zeno's dogs mentioned.
I don't recall mentioning any non-existent dogs, nor any that don't eat anything.
Quoting ssu
I didn't realize, though I should have done, that you are placing the dogs in a single continuous order. But you have defined two infinite sequences, with a common origin. So the start of your Grand Order is not defined, any more than the finish. Your ordering means you have to start from a dog that you cannot identify.
Quoting ssu
You didn't mention them. In any case, they would naturally eat transcendental food - not being able to digest natural food. As for the dog that eats ? amount of food, it will have its place in the order, so there's no problem.
I don't know the math well enough to be sure, but I think it is possible to place numbers like ? or sqrt2 in order among the natural numbers. So every dog will have a different place in the order, depending on how much they eat. So dogs numbered ? etc. will be like every other dog in having a number assigned according to how much they eat. Each dog will be different from every other dog and each dog will be the same as every other dog. It depends how you look at it.
I though it might help to quote the rules again:-
Quoting ssu
Quoting ssu
Quoting Ludwig V
Quoting Ludwig V
Notice that ? isn't constructible, but the square root of two is if irrational, is not transcendental.
By accepting transcendental dogs and their transcendental food, I argue that you have already accepted (perhaps unintentionally) the existence of Zeno's least eating dog. Because if we can put ? exactly on the number line, the I would argue that you can put Zeno's least eating dog exactly on the number line too. Real numbers are constructed by either Dedekind cuts or Cauchy sequences. Both use systems of going closer and closer, which simply begs there to be Zeno's dogs. In a way, with real numbers you have a lot more dogs that basically have a lot of similarities to Zeno's dogs, so much that they could be argued to be Zeno's dogs.
By the way, @Ludwig V has stated something interesting:
Quoting Ludwig V
It is true that my knowledge of mathematics and logic is pretty limited. Yet, if I understand the rules of this entertaining game correctly, the counting starts with two identified dogs. The one at the top (the dog who eats the most) and the one at the bottom (the dog who eats the least). Honestly, I think those two are always there but it is a mistake to try to identify them with numbers. I follow Zenos point as indicative. This is why Plato was wrong in this game. He forgot to count the two 'axiomatic' or 'affirmative premise' dogs. I don't even sure what to call these two (maybe Teo and Sarah :lol: ). As ssu pointed out, the transcendental dogs are the sole obstacle in following Zeno's point. These exist, but everything becomes complicated if we are fixated with labelling the dogs in numerical sequence.
The transcendental food was a joke, playing on the absurdity of transcendental dogs. I must be more careful about jokes.
What is the criterion for Zeno's least eating dog?
Is there an infinite number of dogs?
What is the difference between transcendental dogs and ordinary dogs?
Quoting javi2541997
That is only possible if there is a finite number of dogs.
There cannot be a dog that eats the most - there's bound to be another one that eats more. Similarly for the dog that eats the least. Infinity doesn't follow the normal rules.
Quoting javi2541997
Well, strictly speaking they are identified by the amount of food they eat, which determines their position in the line.
The numbers identify their position in the line.
So, since they are identical in every way, apart from the amount of food they eat, there is no other way to identify them.
It is easy to think that they must exist, but if the line is infinite, any specified dog has another dog after it.
Sorry, I was foolish in trying to follow usual norms when infinity is involved. :sweat:
Quoting Ludwig V
Ah, the so-called non-existing dog is the one who doesnt anything at all. I get it now. But I assumed every dog ate at least a bit.
Quoting Ludwig V
Yes, this is how I see the tricky game. If I'm not mistaken, the dog who eats less than the preceding dog would be represented by 0.00000000 , and so on. However, this dog does exist. It consumes something, even when it is infimum.
Is there a non-existing dog? If there is, it doesn't exist. If there isn't, it doesn't exist.
Quoting javi2541997
Exactly.
Quoting javi2541997
I've been bitten by that infinity more times than I can count. All common sense has to go out the window. It is possible to get used to it.
Dont get me wrong. I explained myself mistakenly. It is true that you didnt mention the non-existing dog, and I think Athena never thought about it either. But since this mysterious dog showed up in this game yesterday, I started to think about his interference in the counting. Well, if we imagine there is actually a dog who doesnt eat anything, it means that it should be represented with a zero (0) in the counting. As ssu pointed out, it took a while for Western mathematics to accept zero as a number. According to this issue, maybe Plato would never have taken the dog who doesnt eat anything into account, but yet it is clear we should take the dog into account, and thus, the dog exists. Right?
That's a complicated thought process. This is a story. It was made up. Speculations about what Athena thought or didn't think beyond what we are told in the text can be plausible or implausible but there's no criterion for truth or falsity. The same applies to ideas about what Plato would or would not have done. For what it's worth, I don't think the real Plato would have done any of what the story attributes to him. But it doesn't matter. But there's no truth or falsity beyond what is stated in the the text - and what follows logically from that.
The mathematical "problem" is based on truth and the only question is what is consistent or not consistent with that structure.
A non-existing dog doesn't exist. The clue is in the description. That's all that needs to be said - unless you want to visit Meinong's jungle.
I'm sorry to be a bit abrupt, but if you don't keep your feet on the ground, you're bound to lose contact with reality.
You are right. I am the one who apologises for derailing the topic in an inconsistent scenario. I thought the non-existence of a dog was a fascinating topic to discuss, but I admit that I overreacted.
Actually not.
The counting starts from the dog that Plato defined to be 1. The action itself defines the whole system of counting, hence the one dog that Plato picks up is always 1. Even if we assume that there really would be amounts that the dogs eat prior Plato choosing to pick up the one closest. For example, if the dog that Plato picked up would be the finite, but a large number in the octodecillion range or a bigger finite one like the one called Big Hoss, created by Jonathan Bowers, then this still wouldn't matter. You cannot increase the amount of food that the dogs eat by multiplying every dog's meal by two or by Big Hoss as the food cannot be measured anything else by the dogs.
And with Zeno's dogs you cannot count. How would you pick the next dog from the dog that eats the least? Or how would you pick a "second most" eating dog? We have to remember that Plato is correct. Just think of a finite line you draw and put at the start zero and in the end ? (or ? with ordinals). Between those two are all finite numbers (finite ordinals with the case of ?). Good luck trying to pick a certain finite number from the line.
Quoting javi2541997
Bravo.
In fact, what is really radical in the story is the "dog that eats the most", because current set theory doesn't accept that. Cantor said this to exist, but it was for God to know. Hence I had in the vote options the possibility "I have a different view about the whole story, ssu" in mind here.
Cantor's set theory can count the ordinals onward from ?. Yet do notice that when it then counts with infinities as like with finite numbers, it immediately (in my view at least) confronts the argument of Plato (that there cannot be an actual infinity) with the set of all ordinals and hence has get's the The Burali-Forti Paradox. Now when you think about this for a moment, that there cannot be the largest ordinal, because every ordinal has a larger ordinal number, it's quite similar to Plato's rejection in the first place of there being the dog that eat's the most.
However, the dog that eat's the least is quite understandable and with nonstandard analysis, we have even an equivalent number. So the question is open here in my view.
I don't think you have derailed anything. If there's any derailing going on, it's me that's doing it.
Quoting ssu
You can do that, but it's very misleading. It suggests that an infinite line is just a very long line. That's wrong. The best way I can think of is to draw your line and put your ? or ? at the end of it, but remember that those symbols mean that the line goes on forever - it has no end. That's why we always just write down the first few elements of the sequence and then ... or "and so on". That's not just an abbreviation or laziness or lack of time. It's telling you that the sequence has no end.
Quoting ssu
I don't know about all those theorems. I know I should, but I had a deprived education.
But what strikes me about your Grand Oder is that the only fixed point you have is Plato's dog. It is the only possible origin for the ordering of the dogs that eat more than Plato's dog, in which case we have to call it dog 0. But it is also the only possible origin for the ordering of the dogs that eat less than Plato's dog. We can call it Dog 0 or Dog 1, but either way, it won't look much like a single order from the dogs that eat less to the dogs that eat more.
The short version of this is that you have to start both sequences from a point in the middle of the line.
Well, we can talk about the set of all natural numbers ?, right? I don't think that it's misleading.
Notice that it's just a model showing just how strange Zeno's dogs are. Just think of the line resembling all the dogs in a well ordered line starting from Zeno's least eat dog and ending in the dog that eats the most, you could draw it like this:
0 _____________________ ?
Now in the line are all the finite Viking dogs. Can you pick any from the line? No, of course not. Plato's counterargument still holds. The simple fact is that if there would be a dog that eats half the amount of the dog that eats more than any other dog, then it couldn't be the dog that eats the most: we could immediately create a dog that eats more, by multiplying the "half eating dog's food" by more than 2. This is why I argue that with infinite you cannot start counting. This also shows why 1+ ? = ? and ? + ? = ?.
That's exactly what I have been trying to say all along! :smile:
And here's then the problem: not only Plato started from counting, but even today Set Theory starts from counting too with the Peano Arithmetic. It really starts with the construction of von Neuman ordinals and with these you get the natural numbers. And the counting goes on in Set theory with larger and larger infinities. And when this is taken to be the building block of all mathematics, then you get into paradoxes like the Burali-Forti Paradox and to avoid the paradoxes you have to make a quite elaborate definitions like that you cannot talk about set of all sets, but of proper classes.
Now we can see just how heretical Zeno's dogs are even today for set theory, because Peano axioms give a successor function to get the next natural number and (if I'm correct) this addition to larger entities is used even with infinite quantities. Yet you cannot count to Zeno's dogs as they are basically given by an inequation: least eating dog < every other dog there exists and most eating dog > every other dog there exists. Notice that here the signs are "<" and ">" which aren't the same as "=". I'll try to explain why this is important to the story.
Let's assume A, B and C are distinct numbers and belong to the set of Natural numbers, hence they are finite. If you have the equation:
A + B = C
And if you know what two are, you will know what the third one is. So if A is two and B is three, then you know that C has to be five. But notice what happens when we change this to an inequation:
A + B < C
Can you know or compute C, if you know both A and B? No, if A and B are as above, then only thing you know is that C can be a natural number 6 or 7 or 8 or larger. It might be six, but then it might be three googol also.
I see your point.
I don't quite get that "fork" argument. The notation using lower case beta for a member of the set and upper case beta for the set is confusing, and I think there's a typo in the statement of the paradox. But I know better than to challenge an accepted mathematical result.
Wikipedia defines proper classes as "entities that are not members of another entity."
That's always a good solution to a difficulty - slap a name on it and keep moving forward. Sometimes mathematicians remind me of lawyers. That's what happened with sqrt2 etc. Also when defining the limits of infinite sequences.
Quoting ssu
Yes. I always thought that was the point. Why should everything have a definite, computable result? Stating the range of a result is not pointless.
I think it's good to go this through here. So the basic problem was that "Naive Set Theory" of Frege had this Basic Law V, an axiom schema of unrestricted comprehension, which stated that:
This meant that there was no limitations on what a set could have inside it and Russel could then form "the set of all sets that do not contain themselves as elements", which is a contradiction. Yet notice the problems of Zeno's dogs had already been found when thinking of the set of all sets. There was the Burali-Forti paradox of the largest ordinal (explained earlier) and what is named Cantor's paradox of there not existing a set of all cardinalities (hence Cantor understood that if set of all cardinalities is accepted, then what would be the cardinality of this set?). This simply goes back to in the story of Plato's rejection of Zeno's most eating dog, just in a different form.
And basically what is lacking here is that with Zeno's dogs addition simply doesn't have an effect. This is why idea of infinitesimals is rejected in standard analysis. Because these infinitesimals cannot be used as normal numbers.
In fact you yourself brought up an old thread of four years ago, which is topic sometimes even banned in the net as it can permeate a nonsensical discussion. And that's the topic of
1 = 0,999999...
Ok, if modeled into the story, you could then find the least eating Zeno's dog eating it's meager rations in the end of that line depicted with "...". OK, why has this be exactly equal to one? Well, if we would assume that
1 > 0,999999...
This would simple mean that Zeno's infinitesimal dog would eat a finite amount, and hence it wouldn't be the least eating dog as Plato's arguing is true about the finite is never ending. With the infinite, ordinary arithmetic breaks down.
So basically the problem is that Zeno's dogs, what I could dare to call infinitesimal and Absolute Infinity, are obscure mathematical entities (and even quite heretical entities) as we don't have the idea just how normal arithmetic breaks down and how then they could be part of "the other dogs". Hence I would state that there's something missing in math.
Quoting Ludwig V
Unfortunately... yes.
In fact, in a great presentation of how Cantorian Set Theory counts past infinity and creates larger and larger infinities is from a popular Youtuber Vsauce below. One should view it altogether as it's a good presentation, but notice just what he says about mathematics from 12:19 onward as this just shows how much mathematicians have become lawyers (or basically have outsourced the foundations of mathematics to logicians).
Believe it or not, I can see that.
Quoting ssu
I'm a bit confused about infinitesimals. Are they infinitely small? Does that mean that each one is equal to 0 i.e. is dimensionless? Is that why they can't be used in calculations? (I thought that Newton used them in calculus and Leibniz took exception.)
Quoting ssu
Well, actually, someone else mentioned it. I misunderstood what it is about and off we go. Once I realized it was about the sum of an infinite sequence, I withdrew, with some embarrassment. But I've learnt some interesting snippets.
Quoting ssu
There is another way, mentioned in the video. Just relax and live with your paradox. It's like a swamp. You don't have to drain it. You can map it and avoid it. Perhaps I just lack the basic understanding of logic.
:grin:
Quoting Ludwig V
Both Newton and Leibniz figured out the way to make a derivation by using infinitesimals.
Let's say that we want to make a derivation of x^2 = 2x With infinitesimals it goes like this:
If dx is an infinitesimal change in x, then the corresponding change in y is dy = (x+dx)^2 - x^2, so
dy/dx = (x+dx)^2 - x^2 / dx = 2x(dx)+(dx)^2 / dx = 2x + dx
And because dx is so infinitesimally small, then we can ignore it and dy/dx = 2x.
And here's the problem: if we just ignore dx, then it would be zero, right?. But then again, we cannot divide by zero! So it has to be larger than zero, but then it also has to act as zero. That's the confusion and And this is actually similar what problem I stated earlier: Zeno's least eating dog has to eat something, but then if let's say eats from Platons dog 1, then the food hasn't decreased! (Remember, 1=0,9999...) Because if it would have decreased, then obviously this amount could be divided into smaller amounts.
And hence we use limits.
Quoting Ludwig V
Well, in my view mathematics is elegant and beautiful. And it should be logical and at least consistent. If you have paradoxes, then likely your starting premises or axioms are wrong. Now a perfect candidate just what is the mistake we do is that we start from counting numbers and assume that everything in the logical system derives from this.
And if someone says that everything has been done, that everything in ZFC works and it is perfect, I think we might have something more to know about the foundations of mathematics than we know today.
Quoting ssu
I don't get this. There's enough food for all the dogs, so why does it have to take some from Plato's dog? If it does, then of course the amount of food for Plato's dog has decreased, but the food supply is infinite, so the amount of food available overall hasn't decreased. What's the problem?
Quoting ssu
Right from the beginning, 2,500 years ago, people have been thinking that everything has been done and is perfect. But then they found the irrationality of sqrt(2) and pi. A paradox is not necessarily just a problem. Perhaps It's an opportunity. Oh dear, what a cliche!
Writing x^2 means x². A bit lazy to use this way of writing the equation.
Quoting Ludwig V
Exactly. With limits we want to avoid this trouble. Yet it isn't actually a paradox as infinitesimals are rigorous in non-standard analysis.
Quoting Ludwig V
It doesn't. This isn't part of the story, I just wanted to describe the seemingly paradoxical nature of the infinitesimals. And hence when infinitesimals had this kind of attributes, it's no wonder that bishop Berkeley made his famous criticism about Newtons o increments (his version of infinitesimals):
Quoting Ludwig V
I agree. Perhaps they admit that there's just only some minor details missing, that aren't so important.
Quoting Ludwig V
I think it's already satisfying to know just what issues we don't know, but possibly in the future could know. And I think there's still lot to understand even from the present theorems we have.
Why do I say so?
Let's take the case how Set theory gives us the actual infinity and various sizes of infinity with Cantor's theorem. What in Cantor's theorem is used is Cantor's diagonalization (or Cantor's diagonal argument). Yet using the diagonalization method we get also many other very interesting theorems and proofs and also paradoxes, which in my opinion are no accident. We get things like:
Russell's paradox
Gödel's first incompleteness theorem
Tarski's undefinability theorem
Turing's proof
Löb's paradox
These all actually tell us of limitations. And hence it shouldn't be any wonder that if we talk about Zeno's dogs, there are obvious limitations to our finite reasoning.
Thanks. One has to do something when one doesn't have the keyboard for the symbolism. Handwriting is much more flexible.
Quoting ssu
I didn't realize that argument was so powerful.
He was a great wit. I'm still trying to make up my mind whether he was a great philosopher or a complete charlatan - even possibly both. This comment is typical. It is very sharp, very pointed. But the calculus is embedded in our science and technology.
Quoting ssu
Yes, I see. You can remove an infinitesimal amount from a finite amount, and it doesn't make any difference - or does it?
Quoting ssu
What do you mean by "actual infinity"?
Quoting Ludwig V
Calculus or analysis is the perfect example of us getting the math right without any concrete foundational reasoning just why it is so. Hence the drive for set theory to be the foundations for mathematics was basically to find the logic behind analysis.
Of course engineers don't care shit about logical foundations if something simply works and is a great tool.
Quoting Ludwig V
To my reasoning it doesn't. And both Leibniz and Newton could simply discard them too with similar logic.
Quoting Ludwig V
I'll give the definition from earlier:
For example Cantor uses actual infinity as the talks about the set of natural numbers being the same size that rational numbers, yet them being smaller than the real numbers. All of these sets are of finished "actual infinity", not the potential infinity as the Greeks thought.
Oh dear! That's a real can of worms, isn't it? Some philosophers would argue that the engineers have got it right. Perhaps it is best to start with the foundation of philosophy - a question. "What do you mean by a foundation?" But I do know that some mathematicians regard philosophers in much the same light as they regard engineers. Still, it's all great fun and often elegant and beautiful; I don't want t be a grinch.
Quoting ssu
Yes. I remember. I don't think I ever replied properly. I can see why those definitions might seem reasonable. But it seems better to me to say that "potential", "actual" and "complete" have no application here. On the other hand, I can see that there are real problems here, so I'm not sure that these labels matter very much. Do they solve any problems?
Quoting ssu
The trouble is that, like plastic, if you discard them, they just come back to haunt you. Perhaps Berkeley had a point. Perhaps the concept of incommensurability could help here?
Definitely.
It should be obvious that with infinity or anything infinite, you have incommensurability that you don't have when just handling finite numbers. But once you have incommensurability, what else you don't have?
To show that the idea of Zeno's dogs least and most eating dogs, infinity and infinitesimal as a number, isn't so far off and how non-standard analysis has made the infinitesimal a come back is explain for example in the following video: