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.

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 ∞ + ∞ = ∞.

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.

It is my belief, also, that although both groups are called democracies, group 2 may behave much better in cases of hardship (like natural disaster, poverty, war or some other crisis). Culture, identity and compassion may really play a role in these small democratic nations when they will face hardships. — Eros1982

I agree. In group 2 social cohesion and solidarity is far more easier to prevail. And usually group 2 countries are far more smaller, which makes democracy easier. Small size makes even other systems quite OK for the citizens under them, in fact monarchies like Monaco and Brunei can prevail quite well because it's totally possible for any citizen simply to meet the monarch and confide his or her problems to this. And when the tiny nation is prosperous and the monach isn't a madman, why not sustain that monarchy? Just think about how nice it would be if you have problem and you could simply get a time with the US President and he would look at what he could do to help you.

With regard now group 1, I think if the countries of this group face some kind of hardship, their people will show all kinds of negative behavior just because they were taught that civilization means living well and calling the police every time you have issues with your neighbor. From the moment you don't live well in group 1 and you cannot rely on the police, you either run away or you should watch your neighbor 24 hours a day. — Eros1982

It surely is a thing of simple size matters. Yet there are real differences with cultures and how they approach the idea of the collective and what's the role of the individual towards the nation. The US is highly individualistic and basically doesn't trust it's own government as much as in some other countries. In the US people have guns to protect themselves from criminals (basically other Americans) and value this gun ownership as an example of their freedoms. In Switzerland and in Finland they have a lot of guns too, but in both countries the guns aren't for protecting your home, but for hunting and protecting the state. It's just one example, but the difference is notable because it comes to other things than just the size of the country:



And it's telling that the above documentary gets a lot of flak in the US. But this was just one example how states differ from each other.

This experience of detesting contemporary American movies makes me ask the same question all the time: why in the hell people in other countries spend so much money and energies in order to see, advertise and idolize (contemporary) American cinema?

The only logical answer I come up with is "mass control". — Eros1982

Well, another reason is that making movies is actually very expensive. If you make a movie in Finnish, basically there's only +5 million people who understand Finnish. If it's a very good movie, some foreigners will see it, but not many. Think about it like Minnesotan's making movies for only Minnesotans to watch, with Minnesotans speaking a totally different language from other Americans. This is the reason why English dominates and why even the Hollywood studios themselves have centered on making "Blockbusters" and only make few "Art Films" that require a bit more to follow than just eat your popcorn.

US culture industry has a big leverage on the rest of the world. — Eros1982

Let's start from some facts: There are so goddam many of Americans compared to any other Western people. And not only that, but your are very wealthy consumers. Thus you are the biggest domestic market there is. And this means that many talented foreign directors and actors are very welcome to work in Hollywood, just as many scientists and successful entrepreneurs (like Elon Musk etc) come to the US, because the US has the resources.

Then you speak English, which was spoken thanks to the British Empire in a lot of other places. (Now if people in the US would talk not English, but French or Spanish, then either of those two be easily the lingua universalis of the World.)

In conclusion, I tend to believe that materialism and policing may have a greater saying in our modern western world than "the global culture" which I see it as being imposed on us (and easily replaceable). — Eros1982

The US surely polices competition when it comes to it's strategic interests. And my father in his time joked about the American legal battle against NIH-products (NIH meaning "Not Invented Here"). Yet all of this is actually quite limited, when tariff barriers don't exist. Especially in Latin America there is this idea of this nearly omnipotent US guarding everything in it's interests, but it isn't so. Not all largest companies in the World are American in every sector. Just take for example forestry and paper companies. You would assume just by thinking where the large forests are and think about the sizes of the countries, it would be that American, Canadian and Russian companies would be the largest. Close, but that isn't the picture, in 2022 by revenue the list was as follows.

1. Oji Paper Company (Japanese)
2. Stora Enso (Finnish)
3. West Fraser Timber Co (Canada)
4. Weyerhauser Company (United States)
5. Universal Forest Products (United States)
6. Masco (United States)

The largest US company is only on 4th place and for many it would be surprising that the largest are a Japanese and a Finnish company, which are very small in size compared to Canada and the US. But this is how globalization works. You'll find that in many sectors there are large companies that aren't American.

I can't imagine a scenario with economies and surveillance performing very poorly and with people in USA or France being in "peace" due to their "democratic/egalitarian/cosmopolitan" values and "compassion". Till, I can imagine that scenario as plausible for some smaller nations which have been lucky enough to not look like France or USA today (though I guess there must be only a handful of such nations in the western world). — Eros1982

Not quite sure what you mean here. Well, many countries don't look like the US. But what is surprising is just how similar to the US the whole of Latin America is. You have these interesting subtle differences between American countries and European countries.

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.

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.

What do you think yourself then? (Or if you have already given a satisfying view, please refer on what page you did it.)

It should be totally evident to everybody that when discussing the foundations of mathematics, philosophy is unavoidable. You simply cannot "just stick to the math" and not take a philosophical stance in my view.

Hence this thread is totally fitting for a philosophy forum.

A transfinite number isn't a natural number, so it doesn't get attached to (aligned with) a dog. Nor could it be. — Ludwig V

Well, a dog eating ⅚ of Plato's dog's food amount isn't either a natural number, so would you deny it to be a dog? And what about transcendental dogs? They are finite, but the dog that eats π amount compared to Plato's dog?

(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).

That will take you, and even the gods, an infinite time. — 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.

If you choose to call ω completed or actual, that's your choice. I can't work out what you mean. I don't know enough to comment on Cantorian set theory. — Ludwig V

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:

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.

Empiricism (as embodied in the principle of testability) is just a temporary stopgap solution in science. What they really want, is the complete axiomatized theory of the physical universe. So, what they really want, is provability:

- - -

At this level, science and mathematics will be merged into one. They actually want to get rid of empiricism and testing and science as we know it today. However, in absence of the ToE, they simply cannot. — Tarskian

What if the positivist are indeed partly right, but they won't get the answer they would want to hear? Hasn't this been obvious starting from Hilbert? He got answer, but not those one's he wanted to hear.

What if this merging of science and mathematics can happen, yet not in the way mathematicians or especially positivists want it to happen? What if a lot of science and even something as distant as the social sciences is indeed mathematical, but in the part of math that is not provable or computable?

Just make this thought experiment: What if an area of study of reality is indeed mathematical, but firmly in the non-computable and non-provable, but perhaps in the "true and expressible" (as Yanofsky put it in the text that you referred in the OP)? How will this show itself?

In my view, one thing would be certain: those people studying that part of reality and it's phenomena aren't computing data or making functions or other mathematical models about reality. They will just smile if you ask if they could explain the phenomena they are investigating by forming a mathematical model of the phenomena.

Do you know about the democratic peace theory? — Linkey

Yes, but I don't unfortunately believe it.

United Kingdom declared war to Finland in December 5th 1941. I assume the both countries were then democracies even back then. (And do notice that the US never declared war to Finland, it only severed diplomatic ties as late as 30th July 1944, only few months before Finland declared war on it's de-facto ally Germany.)

And republics in Latin America have gone to war with each other, latest being the Cenepa war in 1995. And basically both Pakistan and India have been democracies, even if Pakistan has had it's share of military rule. Hence I would argue that being democracies lowers the risk of war between countries, but it doesn't erase the possibility.

Democratic countries unite instead of dissipating, and the people in the West must try to make the Russians know about that. — Linkey

Well, sorry, democracies seem far more weaker and undetermined than they actually are.

And I would urge that this is something that Russians themselves have to do. You already have had a proto-democracy in your history in the state of Novgorod, so you could easily built on that and finally overthrow the idea that Russia needs a Tzar or otherwise it collapses, which I view as nonsense.

as I have suggested, the US should declare that they will build military bases on Taiwan unless a referendum is performed in PRC with a suggestion to unban youtube. I think this is really a strong idea: as far as I know, many people in China (probably most) don't like the censorship in their country and the social credit system. — Linkey

Unfortunately those actions would only consolidate the position of the Chinese communist party and it's supporters. There would be many in the West who would see this as an imperialist attack on China and reckless warmongering.

Sorry, but the only ones that truly can liberate the Russians are the Russians themselves and so it is for the Chinese too.

Plato and Athena would not know this until after they stop counting (that is, if they could stop counting). — L'éléphant

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.

The largest natural number is the number that is larger than all the other natural numbers and has no natural number that is larger than it. But every natural number has a natural number larger than it. So there is no largest natural number. — Ludwig V

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.

There is a number that is larger than every natural number.
That number is ω, which is the lowest ordinal transfinite number, which is defined as the limit of the sequence of the natural numbers. — Ludwig V

(First of all, notice that ω here refers to the largest Ordinal number. In the story it would mean that you put all the dogs that food amount is exactly divisible by dog 1's food (let's call them positive dogs) in a line from smaller to bigger, and then start counting the dog line from their places on the line, from the first, second, third, fourth... and then get to infinity in the form of ω. Notice it's different from cardinal numbers.)

But back to the story: Then doesn't that ω in the story relate to distinct dog? You even referred yourself of ω being a number. 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:

Modern derivative and integral symbols are derived from Leibniz’s d for difference and ∫ for sum. He applied these operations to variables and functions in a calculus of infinitesimals. When applied to a variable x, the difference operator d produces dx, an infinitesimal increase in x that is somehow as small as desired without ever quite being zero. Corresponding to this infinitesimal increase, a function f(x) experiences an increase df = f′dx, which Leibniz regarded as the difference between values of the function f at two values of x a distance of dx apart. Thus, the derivative f′ = df/dx was a quotient of infinitesimals.

Forgive my stupidity, but I don't understand what a completed infinity is. — Ludwig V

Well, you already referred to completed infinity or actual infinity with the example of ω as that is Cantorian set theory. Here's one primer about the subject: Potential versus Completed Infinity: its history and controversy

I live in Russia (please note that I support Ukraine). — Linkey

If so, please be careful @Linkey. And welcome to the Forum.

I am sure that Russians will vote in this referendum to end the war. If the war continues, Russian soldiers will be unable to fight, because they will suffer from cognitive dissonance - what are they fighting for? For censorship and repression? — Linkey

All the Russian emigrants living in my country that I've spoken to don't like what Putin did by attacking Ukraine, many were simply horrified, but then again they don't live Russia. Only once have I seen in 2014 in Helsinki two young Russian men openly in public wearing the black orange stripes of the ribbon of Saint George. Yet 2014 isn't 2022 or today.

Yet I think there are still Russians who support the war simply fearing what will happen to Russia if the war is lost. You see, Russia isn't a normal nation-state, it still is built on an Empire. That's the real problem. Still many Russians believe Catherine the Great's words: "I have no way to defend my borders but to extend them." This pure imperialism hasn't yet died in your country.

And the worst thing is that now other countries simply won't trust Russia. You did totally surpise the West with the collapse of the Soviet Union, but it was Russia itself wanting the destruction of the Empire.

Assume if Putin's regime falls and new not so hostile towards the West administration takes over. Well, a lot of people in the West won't believe that this administration will continue to hold firmly power and assume that we can in the West might (again) wake up with coup again in Moscow and a new regime that builds statues for Putin the Great and declares to the Russian people how evil the West is and how it's real intention is to destroy Russia.

I hope I will not violate the forum rules, if I propose the easiest way for the West to defeat Putin and Xi. First, the United States should reconsider its nuclear doctrine, and declare that the use of US nuclear weapons is possible only in the form of a symmetrical response. If Putin nukes one city, the United States would nuke one Russian city, if Putin nukes ten, the United States would nike ten, and so on. — Linkey

With nuclear weapons there's always strategic ambiguity: you won't really tell what you're response is and even if you tell it, it's likely that others won't believe you. And you don't want to tie your hands. Now it is likely that a nuclear exchange might well become a tit-for-tat, isn't at all sure that nuclear war would go this way. Once you have crossed the line and have used nukes, it's a whole new World: use of nuclear weapons is normal. People will adapt to it.

how are you supposed to be a part of the same "demos" with these (distant to you) people? How is democracy supposed to work in such a scenario (that seems very plausible in many developed countries)? — Eros1982

Before going further, Let's remember first that democracy is a system of government and a state or a country is a different thing. Even if the OP doesn't take this into account, I think it is very important to understand that "people not feeling part" of a country is a very alarming issue for any state, be it democratic or not.

First and foremost the "demos", meaning the people, is inherently important for any state or country to exist independent of the system of government. The people that make the inhabitants of the state have to share an idea about their state. This is why for Empires and states that have in themselves clearly separate people with separate languages and cultures, even religions, have structural problems today. And even quite established democracies like the United Kingdom or Spain can have secessionist movements. Empires like Russia and China have obvious problems and have resorted to what some can rightly call genocidal actions (Russians with the Chechen's and China with the Uighurs).

This wasn't what the OP had in mind, but I think it's very important to understand this aspect before answering further the OP.

Now we come with the question what happens in countries where there are no dominant cultures and apart from abiding to state laws, no traditions and no values are taken to be the norm. — Eros1982

This is something that is argued to happen especially if what is promoted is "multiculturalism". And that multiculturalism destroys the norms, traditions and the values.

It would be good to observe first how actually norms and traditions change before talking about their destruction. Because I would make the claim there indeed still are norms and even traditions.

Then the question about "no dominant culture". Well, our global culture has morphed into something quite similar to a dominant culture. We read the same books, listen to the same music, look at the same films. How our own "nation state culture" survives in the Global village is a difficult question. And this isn't about just the system of governance either. I would argue that this globalization and this melting pot of cultures is the real force behind how the specific culture that a specific people falls from a dominant position it perhaps enjoyed earlier.

And then you have nations and civilizations which at a point do not know anymore what they want (apart from economic growth). Who do you think will prevail? The crazy theocratists who have some definite goals or the moderate guys whose only daily dilemma is to live a pleasant life (only) or to suicide? — Eros1982

A democracy following it's will of it's people will look quite clueless about what they want simply because the people will have different opinions and goals. And this is what always should be remembered about democracies: they appear far weaker than they are.

On the other hand, totalitarian systems look far more stronger than they actually are. The collapse of the Soviet Union is the best example of this. Never had an empire collapsed due to the bankruptcy of it's ideology as peacefully and rapidbly as the Marxist-Leninist experiment did. Yet unfortunately the "normal" way how Empires fall through war and blood is now played in the war between Russia and Ukraine, something that the last Soviet leadership was able to dodge and what the current revanchist Kremlin wanted to do.

That sounds like the "New Math" they had when I was in school. I loved it but it was a failure in general.

I don't think they teach basic arithmetic anymore. It's a problem in fact. — fishfry

There's many things they don't teach in school when looking at what my children have to study. Usually the worst thing is when the writers of school books are too "ambitious" and want to bring in far more to the study than the necessities that ought to be understood.

Here's the general theorem in the setting of category theory. It's called Lawvere's fixed point theorem. Not necessary to understand it, just handy to know that all these diagonal-type arguments have a common abstract form. — fishfry

I looked at this. Too bad that William Lawvere passed away last year. Actually, there's a more understandable paper of this for those who aren't well informed about category theory. And it's a paper of the same author mentioned in the OP, Noson S. Yanofsky, from 2003 called A Universal Approach to Self-Referential Paradoxes, Incompleteness and Fixed Points. Yanofsky has tried to make the paper to be as easy to read as possible and admits that when abstaining from category theory, there might be something missing. However it's a very interesting paper.

In it he makes very interesting remarks:

On a philosophical level, this generalized Cantor’s theorem says that as long
as the truth-values or properties of T are non-trivial, there is no way that a
set T of things can “talk about” or “describe” their own truthfulness or their
own properties. In other words, there must be a limitation in the way that T
deals with its own properties. The Liar paradox is the three thousand year-old
primary example that shows that natural languages should not talk about their
own truthfulness. Russell’s paradox shows that naive set theory is inherently
flawed because sets can talk about their own properties (membership.) Gödel’s
incompleteness results shows that arithmetic can not talk completely about
its own provability. Turing’s Halting problem shows that computers can not
completely deal with the property of whether a computer will halt or go into
an infinite loop. All these different examples are really saying the same thing:
there will be trouble when things deal with their own properties. It is with this
in mind that we try to make a single formalism that describes all these diverse
– yet similar – ideas.

The best part of this unified scheme is that it shows that there are really no
paradoxes. There are limitations. Paradoxes are ways of showing that if you
permit one to violate a limitation, then you will get an inconsistent systems.

And I would really underline the last chapter above. The issue is about limitations and if you end up in a paradox, you simply have had an inconsistent system to start with. Usually in the way that your premises or the "axioms" you have held to be obviously true, aren't actually true, not at least in every case. Hence an outcome similar to Russell's paradox is simply a logical consequence of this. Also understanding that these are limitations doesn't mean that the consistency of mathematics is brought to question. I think on the contrary: you simply have to have these kind of limitations for mathematics to be logical and consistent.

(If anybody is interested, there are some classes by Yanofsky in Youtube, for example Outer limits of reason. I haven't watched them yet, so I cannot rate them.)

Not only is it one problem, I think it's been the largest problem there has been in mathematics. Just look at the long historical debate around the mathematics of continuous change and simply the history of Analysis. Yes, we use infinity as a limit point in calculus and Zeno's paradoxes are solvable by modern calculus, yet the philosophical reasoning remains open. People wanted for set theory to be the basis of mathematics as it would have given a foundation to analysis.

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.

IMO those concepts are far too subtle to be introduced the first day of foundations class. Depending on the level of the class, I suppose. Let alone "Introduction to mathematics," which sounds like a class for liberal arts students to satisfy a science requirement without subjecting them to the traditional math or engineering curricula. — fishfry

There's a lot that in mathematics is simply mentioned, perhaps a proof is given, and then the course moves forward. And yes, perhaps the more better course would be the "philosophy of mathematics" or the "introduction to the philosophy of mathematics". So I think this forum is actually a perfect spot for discussion about this.

Of course it would be a natural start when starting to talk about mathematics, just as when I was on the First Grade in Finland the educational system then had this wonderful idea of starting to teach first grade math starting with ...set theory and sets. Ok, I then understood the pictures of sets, but imagine first graders trying to grasp injections, surjections and bijections as the first thing to learn about math. I remember showing my first math book to my grand father who was a math teacher and his response was "Oh, that's way too hard for children like you." Few years later they dropped this courageous attempt to modernize math teaching for kids and went backt to the "old school" way of starting with addition of small natural numbers with perhaps some drawings and references about a numbers being sets. (Yeah, simply learning by heart to add, subtract, multiply and divide by the natural numbers up to 10 is something that actually everybody needs to know.)

Truth versus provability is not a suitable topic near the beginning of anyone's math journey. IMO of course. — fishfry

It sure is interesting. And fitting to a forum like this. If you know good books that ponder the similarity or difference of the two, please tell.

I'm not sure how the subject came up. — fishfry

From the OP at least I made the connection.

It's interesting to know that all these diagonal type proofs can be abstracted to a common structure. They are all saying the same thing. — fishfry

That's what really intrigues me. Especially when you look at how famous and still puzzling these proofs are...or the paradoxes. Just look at what is given as corollaries to Lawvere's fixed point theorem:

Cantor's theorem
Cantor's diagonal argument
Diagonal lemma
Russell's paradox
Gödel's first incompleteness theorem
Tarski's undefinability theorem
Turing's proof
Löb's paradox
Roger's fixed-point theorem
Rice's theorem

Of course in mathematics a lot theorems have corollaries, but I would just point out to what these theorems are about: limitations in proving, limitations in computation and a paradox, that basically ruined naive set theory and spurred the creation of ZF-logic. All coming from a rather simple thing.

Going back to the OP and the article given there, perhaps in the future it will be totally natural (or perhaps it is already) to start a foundation of mathematics or a introduction to mathematics -course with a Venn diagram that Yanofsky has page 4 has. Then give that 5 to 15 minutes of philosophical attention to it and then move to obvious section of mathematics, the computable and provable part.

Thank you, @fishfry

It seems that from you I get extremely good answers. Yes, Lawvere's fixed point theorem was exactly the kind of result that I was looking for. It's just typical that when the collories are discussed themselves, no mention of this. I'll then have to read what Lawvere has written about this.

And that not necessary is important for me. This is what @TonesInDeepFreeze was pointing out to me also. I'll correct my wording on this.

Thanks for the educative response. But it seems that you went back to my earlier post.

So just to make things clear, I'll ask again:

But if you start from that there is no bijection, and then prove it by:
If there is a bijection then there is a surjection
There is no surjection.
Therefore, there is no bijection.

Isn't that a proof by contradiction? — ssu

Now why I'm ranting so much about negative self-reference or diagonalization, which I acknowledge I haven't accurately defined, is that it crops so easily in many important findings. Yet what is lacking is a general definition.

Here's a video explaining this perhaps better than me:

Could this be put to even more simple terms?

Wrong.

Aren't you forgetting the oldest monotheistic religion, the one of the oldest Empires and Rome's old nemesis, the Fire worshipping Persians? Zoroastrianism is the oldest monotheistic religion as it is roughly 500 years older than the Jewish religion. But because Islam conquered the Sassanid Empire, we don't hear much about them. But there are a few still even today alive and worshipping the old religion.



I'm always fascinated by the idea that even if there wouldn't be Islam, or the Sassanid and Byzantine empires had stopped the spread of Islam, Iranian still today would be seen as different from us, the Western people as likely they would be all Zoroastrians.

It can be interesting to consider how far philosophy/rationality can lead us towards an understanding of God. Perhaps some type of prime mover necessarily exists. — BitconnectCarlos

The study of religion is bit different from the attempt to prove God's existence. The questioning doesn't even start from the obvious question: Is there a God?

Why would continental Europe agree to that? — Tzeentch

If the US walks away from Europe, then naturally Continental Europe would love to have the support from the UK. Two aircraft carriers are always welcome.

Does infinity actually mean that there is always one more, or does it just mean the possibility of it? — Sir2u

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.)

Then why isn't Plato's way the proper way? There's no need to determine the dog which eats the most or the dog which eats the least, just keep feeding in the way Plato described. — Metaphysician Undercover

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:

Nonstandard analysis is a branch of mathematical logic which introduces hyperreal numbers to allow for the existence of "genuine infinitesimals," which are numbers that are less than 1/2, 1/3, 1/4, 1/5, ..., but greater than 0. Abraham Robinson developed nonstandard analysis in the 1960s. The theory has since been investigated for its own sake and has been applied in areas such as Banach spaces, differential equations, probability theory, mathematical economics, and mathematical physics.

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.

I agree roughly with what you wrote, but aren't you going a little light on the UK?

It was their errand boy that went to Ukraine to boycot peace, acting diametrically against Ukrainian and European interests to score brownie points with the Americans. — Tzeentch

During Trump's office, the British Parliament understood quite clearly that if Trump really walks out of NATO, they have to take more role in Continental Europa. I don't think that has changed, from the tanks that the UK had, Challenger tanks are now in Ukraine. Sure, UK wants to be the closest ally of the US, but Trump will shit on every ally it has, except Isreal. In the case of Israel, the US is it's ally, not the other way around. This isn't because of the Jewish Americans voters, but because of the many millions more of pro-Israeli Christian voters in the US.

Especially from a Finn I would expect a certain critical stance towards those pushing for war, since your nation will be on the frontline paying the heaviest price if the worst comes to pass. — Tzeentch

My friend @Tzeentch, we have discussed much in the Ukraine, and if this thread comes too popular or the heated, likely it will whisked away to the Lounge as the Ukraine conflict -thread.

But to others, the actual story both Sweden and Finland did everything to keep the relations normal with the cranky neighbor in the East. And we really did, the whole term of Finlandization was invented for the Finnish situation. But there's a point until you try to be neutral and cordial and keeping up friendly relations to your cranky threatening neighbor. That point was crossed over in February 24th 2022. That it was it. Finland and Sweden abandoned both their neutrality, as Russia is obviously a threat to them.

I'm writing this just a few kilometers from the Russian border. There is NO traffic over the border, for years I haven't seen a single Russian truck and if you want to go to Russia, you have to go through Turkey. The Finnish Armed Forces have put the training cycle to a totally different gear to prop up the deterrence. Russia is spreading bullshit propaganda to it's people over the border that Finland is planning to invade Russian Karelia. Russian Foreign minister Sergei Lavrov yearns for the days of Finlandization and talks about it's opposite "Estonization" which for the Lavrov means Russophobia. (See here)

Well, you reap what you sow.

In my view, Europeans should not focus on which clown is driving the clown car, nor on anything the clowns are saying.

The only thing that matters is Washington's actions, and what we can reasonably glean to be Washington's interests in order to predict their future actions. — Tzeentch

I agree with this, with the addition that perhaps we should listen what the US is saying and try do cooperate with country. The boisterous rhetoric of Trump can be put into one category, it's basically intended for his own base, the actual actions are another issue.

Godel wrote his proof of God for the same reason as why he wrote all his other proofs: because he could. — Tarskian

Notice that he wasn't an atheist and he did believe in God.

Because it's stupid and pointless if there is no God. — bert1

Really?
Acting righteously, being good to other people all those things...you do because only because of God?
And without God it would be pointless?

This has become more actual again now that Biden turns out to be a demented nutjob holding onto power for no apparent good reason, making sure the Democrats will lose. Now that Trump is pretty much a shoe in, what should the EU do and what can we expect with respect to, for instance, Ukraine?

@ssu @Tzeentch thoughts? — Benkei

First of all, Biden has been already for a long time a demented politician holding for power and totally incapable of seeing that he himself is not up to the job. Let's not kid ourselves with that.

Trump has been leading the polls and with this pace, he will likely be the next president. But still much can happen.

In my view, the best thing is to just SHUT UP and calmly observe the garbage fire called the 2024 elections. Respect the US political process enough to not to intervene, at least not publicly. The worst thing would be for the EU to publicly support Biden. That would anger many of those that won't vote for Trump and simply offend Americans and likely it will offend Trump even more. Then understand that you will have a weak US administration entangled with domestic problem whatever happens. Only react if Trump attempts to do changes to the relationship, which he surely will do. Then don't budge. Trump is a bully and his followers just love how "the establishment" hates him. Yet he really is no Hitler. His real chance for a self-coup went as he couldn't even control his own security staff that drove him to the White House and not to Capitol Hill as he wanted during that infamous day.

If Trump wins, you will have an more erratic administration as earlier as Trump will likely bring on far more of his totally loyal yes-men into his administration. This actually didn't happen in the first administration: Trump for example brought on totally reasonable military officers (with the exception of Mike Flynn, who lasted 24 days) and they made Trump's foreign policy to be quite standard US policy (what else?). Trump is a great populist, but he lacks leadership qualities. He has the attention span of a five year old, if the topic isn't about himself. Yet after being left alone in the Florida mansion, I think Trump has a lot of built up anger and he has been contemplating "the next time..." for years now. And even if he know does have the best A-team that Republicans can offer, he can be and will be Trump as he is unmanageable. And Trump can promote his most loyal nutjob followers to prominent positions to make a total shit show.

With Biden winning, if that would happen, then it's old-Brezhnev-time for the US where you cannot know who actually is in charge. But it's a collective effort. And if Biden would come down or is hospitalized, then the Democrats have the opportunity to get a candidate that could win Trump.

The only thing here for the EU is to check really it's defense policy and in this field make more cooperation with the UK. As the UK never did leave NATO, defense cooperation would be a natural start for the EU to warm ties with the UK. Yes, NATO isn't in any way attached to the EU, but here the lines organization lines don't matter so much and Austria wouldn't mind.

Zeno completely comprehended Plato's reasoning, although he did not convey the correct response. Instead, Zeno assumed that Plato had forgotten two elementary dogs, which is incorrect. Plato merely dismissed them as irrelevant to his argument. However, those two dogs, the one that eats the most and the other who eats the least, exist for both Plato and Zeno. Right? :smile: — javi2541997

Not under the assumption that quantities are unlimited. — 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.

Maybe I asked the wrong question.
If all of the dogs are fed, is there anything left over? Until it is time to feed them again at least. Or does the food continue to be 100% even if some of it is removed? — Sir2u

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.

Act righteously and divine favor will follow. "Reasoning God's existence" is not a biblical concern at all. — BitconnectCarlos

Exactly. So I'm puzzled by those who want to give a proof of God, because they usually are religious people. Why not simply follow the given manuals and act righteously?

It's garden variety modus tollens:

If there is a bijection then there is a surjection
There is no surjection.
Therefore, there is no bijection.

No need for a reductio assumption. — TonesInDeepFreeze

Yes,

But if you start from that there is no bijection, and then prove it by:
If there is a bijection then there is a surjection
There is no surjection.
Therefore, there is no bijection.

Isn't that a proof by contradiction. That was my point.

And you see now that a reductio argument is not needed; indeed Cantor did not use a reductio argument. — TonesInDeepFreeze

OK, so let me try get your viewpoint here: having the list g and constructing the real that is not on the list isn't itself using reductio ad absurdum. Yes, this obvious to me also.

However, I still insist that to prove that there's no 1-to-1 correspondence between exist between the natural numbers and the reals is a proof by contradiction, where you use what was done above. So when I talked about diagonalization, being an amateur here, I also referred to consequences and this (the list, anti-diagonal construct which isn't on the list) being used as part of a wider proof. For you diagonalization is just the part with the list g and the construction of the real that isn't in it. I can understand that totally.

Either this is the issue, or then I have to try to spend even more of your precious time.

:grin:

He's assumed to exist. — BitconnectCarlos

And what is his follower assumed to do? To reason God's existence? Or perhaps to do something else?

If that's the case then both Plato's dog and Zeno's dogs are irrelevant, all one needs to do is point the dogs to the food and tell them to go to it. — Metaphysician Undercover

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.

Ok, I think I now understood you. By g you referred to the list, while I thought you meant the constructed real that isn't on g.

I don't say that it is wrong. I just say that it is highly confusing. — Tarskian

Math is confusing. It's far more closer to philosophy than mathematicians and logicians want to admit.

For example, I've followed Chaitin's story and his efforts with the Omegan number and AIT, on and off for now about 20 years and seen how he has been gotten even ad hominem attacks (actually from my own university, from where I graduated). Only now it seems that Gregory Chaitin is getting respect.

OP is another crank (like PL) hiding behind fancy mathematical and logical language to push his nonsense, this time the nonsense being religious proselytising, as can be seen from his other posts. — Lionino

I wouldn't go for ad hominems, but for me this thread is informative. So hopefully nobody is banned and the tempers don't rise too much.

I myself follow the rule that if two or more PF members saying you are wrong (not just that they oppose you view in some way) with nobody agreeing with you, then you might really look sincerely where this error would lies.

That is the lowest temperature realizable from our methods of measurement. In other words it is a restriction created by our choice of dog to use for comparison, the movement of atoms. It does not mean that a lower temperature will not be discovered, if we devise a different measurement technique. — Metaphysician Undercover

And I thought in my ignorance, that there's at least this obvious limit in Physics! Of course, what is Physics else than the study of change and movement? So there's big problems to get funding for a research on the effects of temperatures of negative millions of Celsius. Fortunately there's an actual reality to seek something else.

That is exactly what I am suggesting. Plato was given the task of measurement, and he took that task and proceeded. — 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.

The "other two dogs" referred to by Zeno is a sophistic ruse, just like Plato says. Zeno could have said, "let me know when you get to the dog that eats the most, and the dog that eats the least", and Plato could have said "OK". Problem resolved. — Metaphysician Undercover

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.

Consider this example, suppose we want to set a scale to measure all possible degrees of heat in the vast variety of things we encounter, a temperature scale. We could start by determining the highest possible temperature, and the lowest possible temperature, (analogous to Zeno's dogs) and then scale every temperature of every circumstance we encounter, as somewhere in between. — Metaphysician Undercover

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).