• ssu
    8.3k
    If there is enough food for the dogs, there isn't a dog who doesn’t eat anything at all. 
    I mean, following the premises of the OP it is not possible to imagine a dog who doesn’t eat anything.
    javi2541997
    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 π 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.

    transcendental_numbers-v4_fig-2b-1.jpg
  • Ludwig V
    1.2k
    As all dogs do eat something, we have a problem with the non-existent dog that doesn't eat anything,ssu
    I don't recall mentioning any non-existent dogs, nor any that don't eat anything.

    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?
    I would not deny it to be a dog and I would be happy to assign a natural number to it depending on where it comes in the ordering.
    ssu
    So, let’s 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.javi2541997
    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.

    And what about transcendental dogs? They are finite, but the dog that eats π amount compared to Plato's dog?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 π amount of food, it will have its place in the order, so there's no problem.
  • Ludwig V
    1.2k
    As I stated to Ludwig V, just having finite, but transcendental numbers like π 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.ssu
    I don't know the math well enough to be sure, but I think it is possible to place numbers like π 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 π 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:-
    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.ssu
    Once these were put into the line, then came the dogs which ate quantities between these dogs.ssu
  • ssu
    8.3k
    The whole story is about the problem of definition that math has. And for the Grand Order you refer to, there is the Well Ordering Theorem. In the story it would be simply that since every dog eats more or less than other dogs, they can be put into an order of dog1<dog2<dog3<dog4. Of course, from this we get to interesting challenge that the Axiom of Choice gives to mathematics.

    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 π amount of food, it will have its place in the order, so there's no problem.Ludwig V
    I don't know the math well enough to be sure, but I think it is possible to place numbers like π 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 π 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.Ludwig V
    Notice that π 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 π 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.
  • javi2541997
    5.3k
    Thanks, ssu. A great and very well-written post. I appreciate your teachings. :up:
    By the way, @Ludwig V has stated something interesting:

    Your ordering means you have to start from a dog that you cannot identify.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 Zeno’s 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.
  • Ludwig V
    1.2k
    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.ssu
    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?

    The one at the top (the dog who eats the most) and the one at the bottom (the dog who eats the least).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.

    Honestly, I think those two are always ‘there’ but it is a mistake to try to identify them with numbers.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.
  • javi2541997
    5.3k
    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.Ludwig V

    Sorry, I was foolish in trying to follow usual norms when infinity is involved. :sweat:

    Well, strictly speaking they are identified by the amount of food they eat, which determines their position in the line.Ludwig V

    Ah, the so-called non-existing dog is the one who doesn’t anything at all. I get it now. But I assumed every dog ate at least a bit.

    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.
    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.
  • Ludwig V
    1.2k
    Ah, the so-called non-existing dog is the one who doesn’t anything at all. I get it now. But I assumed every dog ate at least a bit.javi2541997
    Is there a non-existing dog? If there is, it doesn't exist. If there isn't, it doesn't exist.

    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.javi2541997
    Exactly.

    Sorry, I was foolish in trying to follow usual norms when infinity is involved. :sweat: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.
  • javi2541997
    5.3k
    Is there a non-existing dog? If there is, it doesn't exist. If there isn't, it doesn't exist.Ludwig V

    Don’t get me wrong. I explained myself mistakenly. It is true that you didn’t 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 doesn’t 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 doesn’t eat anything into account, but yet it is clear we should take the dog into account, and thus, the dog exists. Right? 
  • Ludwig V
    1.2k
    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 doesn’t 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 doesn’t eat anything into account, but yet it is clear we should take the dog into account, and thus, the dog exists. Right?javi2541997
    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.
  • javi2541997
    5.3k
    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.Ludwig V

    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.
  • ssu
    8.3k
    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).javi2541997
    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 ω with ordinals). Between those two are all finite numbers (finite ordinals with the case of ω). Good luck trying to pick a certain finite number from the line.

    Honestly, I think those two are always ‘there’ but it is a mistake to try to identify them with numbers.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 ω. 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.
  • Ludwig V
    1.2k
    I am the one who apologises for derailing the topic in an inconsistent scenariojavi2541997
    I don't think you have derailed anything. If there's any derailing going on, it's me that's doing it.

    Just think of a finite line you draw and put at the start zero and in the endssu
    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 ω 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.

    The whole story is about the problem of definition that math has. And for the Grand Order you refer to, there is the Well Ordering Theorem.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.
  • ssu
    8.3k
    You can do that, but it's very misleading. It suggests that an infinite line is just a very long line. That's wrong.Ludwig V
    Well, we can talk about the set of all natural numbers ℕ, 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 ∞ + ∞ = ∞.
  • Ludwig V
    1.2k
    This is why I argue that with infinite you cannot start counting. This also shows why 1+ ∞ = ∞ and ∞ + ∞ = ∞.ssu
    That's exactly what I have been trying to say all along! :smile:
  • ssu
    8.3k
    That's exactly what I have been trying to say all along! :smile:Ludwig V

    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.
  • Ludwig V
    1.2k
    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.https://en.wikipedia.org/wiki/Class_(set_theory)ssu
    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.

    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.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.
  • ssu
    8.3k
    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.Ludwig V
    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:

    For any two concepts it is true that their respective value ranges are identical if and only if
    their applications to any objects are equivalent.

    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.

    That's always a good solution to a difficulty - slap a name on it and keep moving forward. Sometimes mathematicians remind me of lawyers.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).

  • Ludwig V
    1.2k
    This simply goes back to in the story of Plato's rejection of Zeno's most eating dog, just in a different form.ssu
    Believe it or not, I can see that.

    This is why idea of infinitesimals is rejected in standard analysis.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.)

    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.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.

    That's always a good solution to a difficulty - slap a name on it and keep moving forward. Sometimes mathematicians remind me of lawyers.
    — Ludwig V
    Unfortunately... yes.
    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.
  • ssu
    8.3k
    Believe it or not, I can see that.Ludwig V
    :grin:

    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?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.

    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.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.
  • Ludwig V
    1.2k
    I'm afraid I don't know what "^" means. But the paradox in the concept of the infinitesimal - that it both is and is not equal to zero - Is not difficult to grasp - and I realize that that's what the concept of limits is about. (But the idea of a limited infinity is, let's say, a bit counter-intuitive.)

    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!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?

    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.
    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!
  • ssu
    8.3k
    I'm afraid I don't know what "^" means.Ludwig V
    Writing x^2 means x². A bit lazy to use this way of writing the equation.

    But the paradox in the concept of the infinitesimal - that it both is and is not equal to zero - Is not difficult to grasp - and I realize that that's what the concept of limits is about.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.

    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?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):

    “They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the Ghosts of departed Quantities?

    Right from the beginning, 2,500 years ago, people have been thinking that everything has been done and is perfect.Ludwig V
    I agree. Perhaps they admit that there's just only some minor details missing, that aren't so important.

    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!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.
  • Ludwig V
    1.2k
    Writing x^2 means x². A bit lazy to use this way of writing the equation.ssu
    Thanks. One has to do something when one doesn't have the keyboard for the symbolism. Handwriting is much more flexible.

    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.ssu
    I didn't realize that argument was so powerful.

    “They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the Ghosts of departed Quantities?
    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.

    I just wanted to describe the seemingly paradoxical nature of the infinitesimals.ssu
    Yes, I see. You can remove an infinitesimal amount from a finite amount, and it doesn't make any difference - or does it?

    Set theory gives us the actual infinityssu
    What do you mean by "actual infinity"?
  • ssu
    8.3k
    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.

    This comment is typical. It is very sharp, very pointed. But the calculus is embedded in our science and technology.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.

    Yes, I see. You can remove an infinitesimal amount from a finite amount, and it doesn't make any difference - or does it?Ludwig V
    To my reasoning it doesn't. And both Leibniz and Newton could simply discard them too with similar logic.

    What do you mean by "actual infinity"?Ludwig V
    I'll give the definition from earlier:

    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.

    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.
  • Ludwig V
    1.2k
    Calculus or analysis is the perfect example of us getting the math right without any concrete foundational reasoning just why it is so.ssu
    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.

    All of these sets are of finished "actual infinity", not the potential infinity as the Greeks thought.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?

    To my reasoning it doesn't. And both Leibniz and Newton could simply discard them too with similar logic.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?
  • ssu
    8.3k
    Perhaps Berkeley had a point. Perhaps the concept of incommensurability could help here?Ludwig V
    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?
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