That doesn't refute my argument for everything, including fiction, has to be coded if reality is a simulation and if noncomputable irrational numbers exist in fiction, that too requires to be coded and we run into the same problem of a program that's got to be infinite in size and that means it'll never be finished/completed and so can't be compiled/translated into an executable file. Reality can't be a simulation. — TheMadFool

The fact that the mathematical existence of noncomputable numbers follows from the rules of standard math, doesn't imply that any noncomputable process is instantiated in the real world.

To sum up, the existence of irrational numbers that aren't formula-based proves that the reality we're living in isn't a simulation because the program required to encode for them would have to be infinite. — TheMadFool

Good post. I especially appreciate your pointing out that some irrationals like sqrt(2) or pi are computable and actually encode only a finite amount of information.

Now your point would stand IF (big if) you can demonstrate that any noncomputable real number is instantiated in nature. Till you can do that (and you can't, nobody can), you have no argument. I don't happen to believe the universe is a computer, but I still can't endorse your argument that would seemingly support my belief. Because for all we know, noncomputable real numbers are nothing more than an artifact of our system of mathematics. Constructive mathematicians don't even believe in them.

As an analogy, a story about Pegasus, the flying horse, does not show that our theories of biology are wrong. Rather, Pegasus exists only in fiction; as do, according to some, noncomputable numbers.

The universe of M is {0}, and e is interpreted as {<0 0>}.

What do the angle brackets mean? Perhaps you can explain your example. I don't understand it.
— GrandMinnow

The reason I didn't pedantically spell out that argument is that it takes but a nanosecond of reflection to see that yes, of course, "ExAy yex" has models. — GrandMinnow

But without further qualification, those models are in no way sets. And you refuse to provide such context. And if you mean something like NF, that's a pretty sophisticated concept that does involve stratification in order to avoid Russell's paradox. I have given your posts quite a bit more than a nanosecond of reflection and I don't believe you are making your case in the least.

Granted, that doesn't capture an ordinary concept of sets, — GrandMinnow

We're in agreement. You just refuted your own point. I'll leave it at that. We're not going to reach mutual understanding because you're stretching a point for its own sake, failing to provide context, and you have already refuted the point you claimed to be making. You said "There exists a set such that every set is a member of it," but in the end you agree that your claim is not about sets as commonly understood; and you have failed to provide any context in which your claim could be taken about sets.

Ironically I gave you such a context, NF, but you don't want to go down that road. There's no other road you can take.

All that said, please slow down and read exactly what I post at exact face value. — GrandMinnow

I most definitely have. I gave the NF article a pretty good read. You should do the same to get a clue about what you think you're talking about. You've already agreed you're not talking about sets, which refutes your own point.

as I mentioned in particular it contradicts the axiom schema of separation. — GrandMinnow

You surely said no such thing, in particular or in general. Here is a quote of your entire post.

One doesn't have to provide much argument that the following claim onto itself is not self-contradictory:

(1) There exists a set such that every set is a member of it.

However, it does contradict the claim that:

(2) For any property and for any set, there is the subset of that set with the members of the subset being those with said property.

You can have (1) or you can have (2), but you can't have both. That is the basic upshot of Russell's paradox applied to sets. — GrandMinnow

I wonder if you can explain what you have in mind. You can't just say there's a set of all sets and that "One doesn't have to provide much argument" to justify it, and provide no argument, and then claim you said something you didn't say. Really wondering what your post was about. I did give the example of Quine's NF but evidently you're not talking about that. So what is the context of your remark? Not giving you a hard time for the sake of it but trying to get you to explain your cryptic remark, which is false without additional qualification. If you deny specification you haven't got a theory of sets, unless (as in NF) you stratify your sets. But why do you think you made a point about specification when you clearly didn't? Am I being unfair in challenging you here?

Separation and specification are subtly different according to Wiki. It's surely not the case that "one doesn't need to provide much argument" here. Quite the contrary IMO.

But I said ONTO ITSELF. — GrandMinnow

Where? Those words are clearly not in your post. Am I missing other posts of yours perhaps?

As the Wiki article on NF points out, your claim falls to Russell's paradox unless you stratify your sets, and that DOES require some explanation.

One doesn't have to provide much argument that the following claim onto itself is not self-contradictory:

(1) There exists a set such that every set is a member of it. — GrandMinnow

Perhaps you could humor me and provide "much of an argument." Such a set would violate regularity/foundation. Are you perhaps making reference to Quine's New Foundations or some other alternate axiomitization of set theory? Reading the Wiki article in its entirety, or at least to the point where they said the category of NF sets is not Cartesian closed (*), convinced me that "much of an argument" must indeed be given to put your remark into its proper context. I could be wrong, curious what's in your mind with this post.

(*) This means that the sets of NF lack products and exponentials. There's not always a Cartesian product of two given sets; and/or there is not always the set of functions from one set to another. In computer science terms, you can't always curry functions. These are not sets as generally understood except perhaps by specialists in NF, as I understand it.

Just look at the proof that I am presenting without bias and without preconceived notions. — Philosopher19

I have been doing that. I can't add anything to what I've said other than that you should carefully examine the proof of Russell's paradox. And you should carefully examine your own argument, to see that you repeatedly claim that x is a set but you never present an argument to that effect.

Of course it doesn't. Your posts are uniformly excellent. — jgill

A number of posters around here would beg to differ, but thanks for the kind words.

┌Ф┐ — Martin Raza

Attack of the killer robots?

┌Ф┐┌Ф┐┌Ф┐┌Ф┐┌Ф┐┌Ф┐┌Ф┐┌Ф┐┌Ф┐

As does yours — jgill

I'm not the one who has a problem with it. Russell's paradox is a deep argument. Frege was a smart guy (he invented the universal and existential quantifiers) and he missed it. It's worth discussion. I daresay @Philosopher19 is not the only person who's ever experienced confusion about the subject. SEP has 7448 words on it, they must think it has some importance. It's the paradigmatic example of all the self-referential arguments such as Cantor's diagonal argument (which influenced Russell to think of it), Gödel's incompleteness theorems, and Turing's Halting problem. It's worthy of discussion IMO regardless of the circumstances.

This website is not the proceedings of the Royal society. I often wish for a more high toned conversation around here, especially on mathematical topics; but we take what we can get. You should see the politics forums. I hope my participation in this thread doesn't inconvenience or distress you too much. It's the forum software that bumps active threads, I have no control over that. I couldn't help calling out the irony that someone who doesn't want to see this thread on the front page, bumped it to the front page themselves. How self-referential.

Yes. Off the first page of TPF. — jgill

Your bump pushes it to the front page.

The following is proof:
I find that if I say x isn't a member of itself, I am being paradoxical because x is a set. — Philosopher19

You claim x is a set but it isn't. You have no proof that x is a set.

I find that if I say x is a member of itself, I am not being paradoxical because x is a set. Do you see? — Philosopher19

Is x the collection of all sets that are not members of themselves? If so then x is not a set. You claim it is but haven't proved it is. If x is the collection of all sets that are not members of themselves, then x also isn't a set even though you claim it is. You've used x to mean both of those things at various times. In both cases they are collections, or classes, or extensions of a predicate. But they're not sets.

You find that x both is and isn't a member of itself.
— fishfry

How do you get to this???? — Philosopher19

By Russell's paradox. Say x is the set of all sets. Then let y be the set of all sets that are not members of themselves. Is y a member of itself? If it is, then it's NOT a member of y. But if it isn't, then it IS a member of y. So y is a member of itself if and only if it isn't a member of itself. That's a contradiction.

If x is the set of all sets that are not members of themselves, just run the same argument on x.

x = the set of all sets. Is x a set? Yes. — Philosopher19

No, it can't be. I just showed that if x is the set of all sets, then we can form y and derive a contradiction. If x is the set of all sets that don't contain themselves, then x itself leads to a contradiction.

Thus x is a member of itself. Is x not a member of itself? Yes it is a member of itself because it is a set! Let's try the alternative. x is not a member of itself. Why not? No reason can be given. The set of all penguins is not a member of itself. Why not? Because a penguin is not a set. See? — Philosopher19

You need to work through the proof of Russell's paradox carefully. You're just repeating incorrect ideas.

I find that x is a member of itself. That is all I find. — Philosopher19

You need to work through the argument.

Not at all. If I am, then I'm an idiot. I just want efficiency and truth. — Philosopher19

I have made no characterizations. It's a tricky argument. You need to go through it for yourself carefully.

Suppose x is the set of all sets that don't contain themselves. Then we ask if x contains itself? If it does, it's NOT a set that doesn't contain itself, so it DOESN'T contain itself.

See what just happened? If x contains itself then x doesn't contain itself.

On the other hand suppose x doesn't contain itself. Then it must be a member of x. So if it doesn't contain itself it does contain itself.

Having just shown that x contains itself if and only if it doesn't contain itself, we have a contradiction. Therefore there is no such set as x.

Now if you prefer to let x be the set of all sets, we let y be the set of all sets that don't contain themselves and we get a contradiction from y. So again, x can't be a set.

Sorry for all the confusion. I edited the post you quoted to explain a contradiction as "(statements which are always false)" — fdrake

Darn. I am afraid I have to be picky again. A contradiction is not false. A contradiction is a pair of syntactic derivations, one of some statement P and the other of not-P. There is no truth or falsity in syntax.

2 + 2 = 5 is false in every model of the Peano axioms, but it's not a contradiction.

Gordon Ramsey's Perfect Scrambled Eggs recipe to allow you to also cook Gordon Ramsey's Perfect Beef Tenderloin. — fdrake

His cooking videos are very good. But when he's doing those restaurant rescues and verbally berates some underage female waitperson, he's a bit of a bully. He has daughters himself and should know better.

(4) A system is consistent if it proves no contradictions (statements false in every model). — fdrake

I got in trouble here. A system is inconsistent if it proves both P and not-P for some statement P.

But if a system is inconsistent, it has no model at all; so the claim that a contradiction is false in every model doesn't make sense to me.

"The semantic definition states that a theory is consistent if it has a model ..."

https://en.wikipedia.org/wiki/Consistency

I looked up Q, which turns out to be Robinson arithmetic. I've heard of it but I'm not familiar with it so it's best if I don't try to add anything to the discussion beyond the minor correction I noted earlier.

To my knowledge, Russell's paradox concludes that you cannot have a set of all sets because he fails to non-paradoxically define a set of all sets that are not members of themselves. — Philosopher19

You're turning the argument on its head to confuse yourself. If we can't define such a set without creating a contradiction, then there is no such set.

No one is disputing that there can be no set of all sets that are not members of themselves that is itself, not a member of itself (call this absurd set y). But this paradox in no way logically amounts to saying that there is no set of all sets. x is the set of all sets. This set contains all sets including itself. No paradoxes. — Philosopher19

If you call x the set of all sets, you quickly get a contradiction. You find that x both is and isn't a member of itself. Therefore there is no such set. You keep claiming there is but you have not provided proof.

For the sake of argument, let's say x and y are not the same. I have no problem in saying that y is absurd. But there is still a set that contains all sets that are not members of themselves. x contains them all does it not? If it does contain them, then it contains them. Why does x have to be not a member of itself??? Why are we trying to force a paradox where there is none? A set is a set. It doesn't matter if it's a member of itself or not. If it truly is a set, then it is clearly a member of the set of all sets. — Philosopher19

There is no set of all sets. You keep claiming there is, but you have not provided a proof. On the contrary, the assumption that there is a set of all sets leads to a contradiction. Therefore there is no set of all sets.

Yes, x also contains other sets (actually it only contains one other set...which is itself). But it still contains all sets that are not members of themselves. — Philosopher19

The CLASS, or COLLECTION that contains all sets may indeed be formed. It just turns out to not be a set. And you haven't shown that it can be. You keep claiming it without proof.

Regarding sets, Russell misunderstood semantics and logic. I understand he was an important philosopher, but he made a mistake. — Philosopher19

Not in this instance.

It is absurd to say that there is no set of all sets. Now are we in agreement? — Philosopher19

You trollin' me?

Incompleteness says that there are tautologies which can't be derived (as theorems). — fdrake

I'm no expert on this stuff but every tautology has a proof. Incompleteness involves propositions that aren't tautologies, whose truth value varies with the model. For example the axioms for a group say nothing about whether the group is Abelian. The statement "xy = yx for all x, y" is true in some models and not in others. So the group axioms are not complete.

A proposition is a tautology (true in every model), if and only if it has a proof. That's Gödel's completeness theorem.

https://en.wikipedia.org/wiki/G%C3%B6del%27s_completeness_theorem

I wrote a reply and then, in the process of editing out my typos, I came to understand your argument. Rather than rewrite it I just let it all stand and interspersed notes that fix it all up. Therefore this is about twice as long as it needs to be. But here it is. The bottom line is that you assume x is a set but never prove it; and Russell shows that it can't be.

Thank you for replying and I understand where you're coming from. I will try to convey to you my understanding more specifically hoping that specification saves naive true set theory. I will ask questions to see exactly where it is that we are in disagreement. — Philosopher19

Ok. But the core of the issue is the same as Russell noted in 1901. The "set of all sets that are not members of themselves" both is and isn't a member of itself, a contradiction. Therefore there is no such thing. This is not going to change. It would be helpful if you would carefully review the argument yourself and I urge you to do so.

For the sake of argument, assume we have the set of all sets. Call this x. — Philosopher19

I'm perfectly willing to agree, as this is the assumption that will soon lead to a contradiction.

x is a member of itself because it is a set. No paradoxes so far, agreed? — Philosopher19

Ah, perhaps I see the problem. Why do you think a set is a member of itself? Are you possibly confusing set membership with the subset relation? It's true that every set is a SUBSET of itself. But no set, in the presence of the axiom of regularity, is an element of itself. And even without regularity, a set can be a member of itself. But it doesn't have to be.

Is this the confusion? The set of natural numbers $\mathbb N$ is a SUBSET of itself; but not a MEMBER of itself. That is, $\mathbb N \subset \mathbb N$, but $\mathbb N \notin \mathbb N$. If this resolves your question, we're done. Because saying that x is a set does NOT in any way imply that it is a member of itself. That's an error. So if this is the problem, we're done. I'll continue, but let me know if this was the issue.

Since x contains all sets, do we agree that x contains all sets that are not members of themselves? — Philosopher19

Yes, certainly. x contains all the sets that are members of themselves and all the sets that are not members of themselves. This will lead to the conclusion that x both is and isn't a member of itself.

The set of all penguins, is a set. — Philosopher19

I have a problem. This isn't even naive set theory; it's high school set theory. Penguins are not elements of sets. In math, sets are generally "pure" sets, meaning that their only elements are other sets. There are alternate versions of set theory in which sets can contain urelements; that is, things that are elements of sets that are not themselves sets. However even in set theories containing urelements, I do not believe that penguins or any other natural objects can be urelements. I have to plead ignorance though, I don't know much about sets with urelements.

If you ask a biologist, they'll tell you that a penguin belongs to the family Spheniscidae, of the order Sphenisciformes, class Aves, phylum Chordata, kingdom Animalia. "Today I learned," as they say. Biology isn't my thing. But the classifications in biology are not mathematical sets. We could call them high school sets, but they are not even naive sets, let alone axiomatic sets. So this analogy is going in the wrong direction already. The collection -- not set, collection -- of all penguins is, as we all just learned, the family Spheniscidae. It's not a set. Whatever point you're trying to make, I'd prefer if you make it with actual mathematical sets. The set of all real numbers, or the collection of all topological spaces, which ISN'T a set (for the same reason as the set of all sets isn't a set).

This set is not a member of itself precisely because it is a member of the set of all sets. — Philosopher19

Now this I do not understand, even if I grant that the collection of all penguins is a set. Because the set of all sets contains all the sets that ARE members of themselves, AND all the sets that ARE NOT members of themselves. So even if I grant, for sake of argument, that the collection of all penguins is a set, that does not make it not a member of itself. Even granting your example, this statement makes no sense. Some sets are members of themselves (in non well-founded set theories) and some aren't. So just because something's a member of the set of all sets doesn't mean it's not a member of itself. How did you conclude that??

NOTE (after I've been through this a few times and think I understand your argument). If we assume no set is a member of itself, that still doesn't show that the collection of all non-self-containing sets is a set. You haven't shown that. And if you assume it is, you get Russell's paradox.

That is: You assume x is the set of all sets. You assume (as we all do all the time) that no set contains itself. Therefore x is also the set of all sets that don't contain themselves. HOWEVER! You still have your ASSUMPTION hanging around. And it falls to Russell's paradox. You assumed x is a set. What you have actually shown is that the collection of all sets is the same as the collection of all the sets that don't contain themselves. But you haven't shown that these collections are sets; and in fact they are not.

By this I mean It is specifically a set, not a penguin. Are we sill in agreement? — Philosopher19

I agree that the collection of all penguins is not a penguin. But I agree NOT because the collection of all penguins is a set; but rather, because the collection of all penguins doesn't happen to be a penguin. Your logic is off the rails at this point.

The set of all animals, is one set that contains the set of all penguins. — Philosopher19

As we've seen, a biologist would not agree. The concept of penguin is a subconcept or subcategory of the concept or category of animals. But they are not sets. The predicate "is a penguin" does imply the predicate "is an animal," I agree with that. But these are not sets. Still, for sake of discussion I'll grant your premise. I still fail to see your point and your claim that these classes do or don't contain themselves simply by virtue of being sets, is wrong.

This set is also not a member of itself, precisely because it is a member of the set of all sets. — Philosopher19

This is just wrong. The set of all sets (which we assume for sake of argument exists) contains some sets that DO contain themselves; and other sets which DON'T contain themselves. Just because some object (the collection of all penguins) happens to be a member of the set of all sets, doesn't let us conclude which is the case. It might contain itself or it might not. In this case the collection of all penguins does not happen to be a penguin; but that's a fact of nature and NOT just because it's a set.

Thus, by definition, any set that truly is a set (as opposed to a penguin or animal), and is not a member of itself, is not a member of itself precisely because it truly is a set and is thus a member of the set of all sets. Agreed? — Philosopher19

No, and I don't follow your reasoning. If the set of all sets contains all sets, then it contains all the sets that ARE members of themselves along with all the sets that AREN'T members of themselves. Given a set, we have to examine it carefully to determine whether or not it's a member of itself. Of course in standard set theory we have the axiom of regularity so no set is a member of itself. But if we drop regularity, then some sets ARE and some sets AREN'T member of themselves. We can't determine which is the case merely by knowing something is a set. Even granting all your premises I don't follow your reasoning. The set (if we call it that) of all penguins is not a penguin; but not because it's a set; but rather, because it doesn't happen to be a penguin!

NOTE: See Summary at the end. If you are assuming that by definition no set is an element of itself, that's perfectly fine and is the standard assumption in math. But that does not mean that the collection of all these non-self-containing sets is a set. You haven't shown that. What your argument shows is that the COLLECTION of all the sets that don't contain themselves, is identical to the COLLECTION of all the sets that there are. This is true. But you haven't shown that either of these collections are sets.

If agreed, — Philosopher19

I'm afraid not. Even granting your premises there is something terribly wrong with your reasoning. Some sets are and some sets aren't members of themselves. So even if you convince me that the collection of all penguins is a set, that doesn't tell me whether it's a penguin or not. I have to consider the specific case.

then can you see how the set of all sets that are not members of themselves, can only be (by definition) x? — Philosopher19

No. And the problem is that I have no idea how you got to this point. x includes all the sets that aren't members of themselves, AND all the sets that ARE members of themselves. x contains all possible sets, right?

NOTE (now that I think I understand your argument): You are right that if we assume no set is a member of itself (as we normally do), then the COLLECTION of all non-self-containing sets is identical to the COLLECTION of all sets. But you haven't shown that these collections are sets. You're only assuming that; and your assumption is wrong, as shown by Russell.

I will specify this some more: All sets that are not members of themselves, truly are sets. — Philosopher19

Well yes, because you said they're sets. All the fish that are left-handed are fish. Why? Because we said they're fish! A set that's painted green is a set, because we stipulated that it's a set. You've made a vacuously true statement. A set is a set, so of course a set that is painted green, or flies through the air, or is not a member of itself, is a set. We haven't said anything!

What is the set of all these sets? Can the answer be anything other than x? — Philosopher19

Yes. The set of all sets that are not members of themselves is different from the set of all sets; because x contains all the sets that are not members of themselves AND all the sets that ARE members of themselves. I don't follow why you don't see this.

A set either contains itself or it doesn't. The collection of ALL sets contains the ones that do and the ones that don't. But that collection isn't necessarily a set, and can't be. And you haven't shown that it is.

NOTE (all these notes were written after I came to understand your argument, apologies for all these interspersed notes). YOU ARE CORRECT. If no set contains itself and x is the collection of all sets, then x is also the collection of all sets that don't contain themselves. BUT the claim that x is a set was an ASSUMPTION, which you haven't justified. And Russell's paradox shows your assumption is wrong.

The set of all sets which contains all these sets, is a member of itself (because it truly is a set). — Philosopher19

No, it's not. Russell shows it's not a set. It's a collection that's not a set.

NOTE. You have CORRECTLY shown that the collection of all sets is the same as the collection of all non-self-containing sets. But you ASSUMED that this collection is a set and Russell shows that it's not. You made an assumption but never justified it.

Where do we have a paradox in what I have proposed? — Philosopher19

You didn't get there yet. You have shown that the collection of all sets is the same as the collection of all sets that don't contain themselves, under the axiom of regularity. But then you assumed that x is a set, and you never justified that. And if you then apply Russell, you'll see that x can't be a set. Just a collection that isn't a set.

Summary:

* One issue as I mentioned is that you may be confusing subsets with elements. This is a fairly common point of confusion and if that's the case, let's focus in on that.

* The business with the penguins was not helpful to me, it added confusion.

* You are assuming the axiom of regularity (perfectly normal, we all do that all the time in standard set theory) and therefore saying that everything that's a set, doesn't contain itself. This, I agree with. But that doesn't mean that the collection of all the sets that don't contain themselves is a set! That's a logic error. This is the part that I didn't realize on my first couple of readings.

That is: Let us adopt the axiom of regularity, so that no set contains itself. We can therefore form the COLLECTION (not yet proven to be a set) of all the sets that don't contain themselves; and this will indeed turn out to be the COLLECTION (not yet proven to be a set) of all possible sets. I think this is the argument you're trying to make.

But you haven't shown that either of these collections is a set! And you can't, because any such attempt runs into Russell's argument.

* So the bottom line is that we assumed the set of all sets exists, and, assuming that no set contains itself, you are correct that it must be equal to the set of all sets that don't contain themselves. But then we just apply Russell's argument to show that the set of all sets that don't contain themselves both does and doesn't contain itself, a contradiction.

* So what you need to do is, AFTER you have made your argument: that x is the set of all sets and also the set of all sets that don't contain themselves; you have to apply Russell's paradox to see that x both is and isn't a member of itself. Therefore x isn't a set. It's merely a collection, the extension of a predicate. It's not a set.

x, the set of all sets that are not members of themselves, is thus a member of itself. — Philosopher19

That right there is the contradiction. x is a member of itself if and only if it's not a member of itself.

Since the assumption that there is a set of all sets leads to a contradiction, we must therefore reject that assumption, and admit that no such set exists.

It's always worth pointing out in these discussions that self-containing sets are not contradictory. We generally assume an axiom, the axiom of regularity, that outlaws self-containment as well as circular chains of containment such as $a \in b \in c \in d \in a$

If we instead choose to allow self-containing sets, the resulting system is logically consistent. The study of such systems is called non well-founded set theory.

In modern set theory we avoid Russell's paradox by saying that we can not form a set merely out of the extension of a predicate, like $x \notin x$. Rather, we must start with a set we already know to exist, and then apply a predicate to it. This is the axiom schema of specification. It's a schema because it stands for an infinite collection of axioms, one for each predicate.

As an example of how this works, what is the set of all natural numbers that are not members of themselves? Well, $0 \notin 0$, so $0$ is in the set. $1 \notin 1$, so $1$ is in the set. $2 \notin 2$, so $2$ is in the set. Continuing in this manner we see that the set of all natural numbers that are not members of themselves is ... drum roll ... the set of natural numbers! No paradox. Specification saves the day.

So, really I don't have any long lasting bad feelings towards Fishfry — Jack Cummins

If you did you wouldn't be alone here! At least you didn't call me a racist like some other guy did once.

I was honestly just trying to provide a classic reference to the subject you introduced. It's all good. Peace.

Gadamer presents some interesting ideas about art in Search for a Method. — Pantagruel

How dare you provide a reference that the OP might find interesting.

Here's the Wikipedia link to your reference. That should really send him over the edge.

https://en.wikipedia.org/wiki/Truth_and_Method

Gadamer draws heavily on the ideas of Romantic hermeneuticists such as Friedrich Schleiermacher and the work of later hermeneuticists such as Wilhelm Dilthey. He rejects as unachievable the goal of objectivity, and instead suggests that meaning is created through intersubjective communication.

Reminds me a little of (my shallow knowledge of) Kierkegaard.

@Jack Cummins, there's nothing wrong with someone supplying you with an on-point reference to your topic, especially when they are trying to help you broaden your knowledge of a subject in which you expressed interest. If someone does that just say, "Thank you for the suggestion," and move on if you don't like Wikipedia. . You can't control what others say on an online forum and when you do, you provoke them to poke you back. In case you were wondering why. As far as wishing someone would give you a summary of Snow's thoughts, that's exactly what's on the Wikipedia page I showed you.

I do appreciate that you gave a link to many references but what you did in giving me a link is exactly the way I described in the bad arguments thread today. — Jack Cummins

You're the one who started the bad arguments thread? I was just about to nominate you for it. Go read the C.P. Snow article and learn something. I gave you an on-point, highly relevant and famous reference to your topic. It is in fact the single most famous essay on the topic. It was clear to me that you probably hadn't heard of it, so I pointed it out to you. I don't see why that bothered you, but whatever.

Thanks for your link to wikipedia but I do think that wikipedia is only a basic discussion and I was hoping that this site is able to go a bit deeper in philosophical exploration. — Jack Cummins

Did you utterly fail to note the reference I gave you to C.P. Snow's famous essay?

I am asking this question because in the philosophy discussions of the present time there appears to be scientific enquiry on one hand, and the views expressed in the arts falling into another category. — Jack Cummins

Been done.

https://en.wikipedia.org/wiki/The_Two_Cultures

"The Two Cultures" is the first part of an influential 1959 Rede Lecture by British scientist and novelist C. P. Snow which were published in book form as The Two Cultures and the Scientific Revolution the same year.[1][2] Its thesis was that science and the humanities which represented "the intellectual life of the whole of western society" had become split into "two cultures" and that this division was a major handicap to both in solving the world's problems.

masturbation in itself is nothing to be particularly ashamed of, it’s really the pairing with porn that makes it a concerning matter. — Ignance

That's an interesting point of view, can you explain please? You are saying that if someone with a vivid imagination creates their own stimulation in their mind, that's ok; but if someone without such a good imagination uses the interwebs to dial up some visual stimulation, that's "concerning." Can you say why, and exactly what the concern is? Let's assume that the performers and production crew are all consenting adults and are compensated fairly.

Also, one should not be "particularly" ashamed of wankification; but that perhaps they should be just a little ashamed? How much shame is required, exactly? I need to know right away. "Asking for a friend."

Are there situations where its allowed to erase a memory from someonelse's mind? Imagine if it would be possible where are the borders to who and who not? — LiveAnotherDay

Standard anesthesiology technique. You're sedated but aware, and afterward you don't remember the experience.

https://en.wikipedia.org/wiki/Twilight_anesthesia

If you had a quantum theory of special relativity you'd get the Nobel in physics without having to wait for a vote of the committee.
— fishfry

It already exists, it's called the Dirac equation, and he did get the Nobel prize (so you're right about that). It's general relativity that's proving a hassle — Kenosha Kid

Today I learned! Thanks.

I have a quantum theory of relativity — Enrique

In that case I can just wait to read about it in the papers. You'll be more famous than Einman.

Three of the fundamental equations of quantum physics are:
E=mc2, — Enrique

$E = mc^2$ is an equation of special relativity, a classical theory. If you had a quantum theory of special relativity you'd get the Nobel in physics without having to wait for a vote of the committee.

This mistake, which can fairly be characterized as a howler, enabled me to stop reading right here. The only reason I didn't mention this yesterday was that you described your initial post as humorous, so I thought it was a deliberate joke. But if you're serious, you're seriously misinformed.

I see @Kenosha Kid beat me to the punch on this observation. Nevermind.

Thanks for the detailed response. The videos I've seen on Godel is the In Our Time one from philosophyoverdose and the ones from actualized.org, all on Youtube. I was under the impression that Godel used the ambiguity of the Liar Paradox to formulate codes in mathematics. This reminded me of Russell's paradox. In common language "The set of all sets that do not contain themselves" would contain itself if it didn't contain itself. A set containing itself would be very strange though and I don't think this paradox is a contradiction in numbers but in language. IF Godel's theorems were based on language, then they could be revised like Russells theorem. A fault in language should have no effect on numbers. I did say God (whether we are God or not) should be able to prove everything in mathematics. Self reference might be an illegal move in mathematics and could possibly be godel's problem. I don't know. Actualized.org had a video on relativism and mentions a Quine paper from the 20's where he discusses talking with aliens. Godel, being a Platonist, is setting up a whole theory he thinks is true for all species and divinities for all time. But an alien might have a numbering system wherein there is no self reference. As a genuine question, is it possible Godel put too much of human language into mathematics? — Gregory

I want to say something about all this (and your other two posts) but it seems too difficult to me to respond point by point and the more I mull it all over, the farther back all this becomes. So let me just write down a couple of thoughts that are on my mind and maybe something will resonate. This isn't meant to be a comprehensive response to everything (or anything) you wrote; it's just a few thoughts that I might as well write down now to avoid the risk of never writing anything at all.

1. Why should man-made proof be able to approach what God knows about math? What is a proof? A proof is a finite-length string of symbols. Just as "The Tao that can be told is not the eternal Tao," the truths that can be proved are not all of the truths. God knows all the mathematical truths; and humans are restricted to finite-length strings of symbols. Why on earth would you or any human arrogantly expect that we can reach God's knowledge with finite strings of symbols that we ourselves made up?

I am arguing here that your expectation that we should be able to prove everything that God knows, is unrealistic and unreasonable. After all, we should not expect to know everything that God knows using our puny human minds; any more than a caterpillar or a bat can know everything God knows.

Icarus couldn't fly with wings made of feathers and wax; and we can not know what God knows using only finite strings of symbols.

2. Of more interest to our discussion than Gödel's first incompleteness theorem, is Tarski's undefinability theorem. Tarski proved (and in fact Gödel proved on his way to proving his own theorems) that no system can define or characterize all its truths.

Let me give a paraphrase of the idea. Consider the natural numbers. God surely knows every true fact about them. In fact God keeps a list of all true facts about the natural numbers: 2 + 2 = 4, 3 x 5 = 15, "there is no natural number solution to $x^n + y^n = z^n$ for n > 2," and so forth.

Each of those truths can be written as a formal statement in the language of math; and each formal statement of math can be assigned a positive integer, its Gödel number. I'll omit the details but just ask if you're curious. It not difficult to show that every formal mathematical expression can be uniquely assigned to some positive integer.

So God knows a set of positive integers that encode all mathematical truths and nothing else.

What Tarski proved is that there is no mathematical statement or formula or algorithm that can crank out or describe that set of numbers! That is, mathematical truth is essentially random.

This, I think, gets at the heart of what you're saying better than banging on Gödel's incompleteness theorem, about which there is already way too much misunderstanding in the world. We cannot write down a formula that describes all the truths of a system. God knows those truths, but they lie beyond any formulaic enumeration. That's a fact, not a mistake or a quirk of language.

3. This bit about the liar paradox. The liar paradox is about semantics. incompleteness is about syntax. Incompleteness is about the limits of what we can do by pushing formal symbols according to formal rules. Analogizing it to the liar paradox is on the one hand helpful, because it's easier to visualize. But it's also very much NOT helpful, because syntax is not semantics and incompleteness is NOT the liar paradox.

The Wiki article on Gödel's incompleteness theorems explicitly mentions this point:

"Gödel specifically cites Richard's paradox and the liar paradox as semantical analogues to his syntactical incompleteness result in the introductory section ..." My bolding. It's a bit like the rubber sheet and bowling ball visualization of Einsteinian gravitation. On the one hand it's a nice visualization because it illustrates how mass distorts spacetime in a way that can be understood by a child. On the other hand, it's profoundly wrong. What makes the bowling ball push down on the rubber sheet in the first place? Meta-gravity? Of course not. The bowling ball and rubber sheet is a popularized analogy that falls apart if you think about it much.

Likewise, the liar paradox is an analogy. It's not the incompleteness theorem. The former is semantic; the later, syntactic. The Wiki article expands on this point:

"It is not possible to replace "not provable" with "false" in a Gödel sentence because the predicate "Q is the Gödel number of a false formula" cannot be represented as a formula of arithmetic. This result, known as Tarski's undefinability theorem, was discovered independently both by Gödel, when he was working on the proof of the incompleteness theorem, and by the theorem's namesake, Alfred Tarski."

In short, your belief that incompleteness is just the liar paradox is erroneous. The liar paradox is a conceptual aid to understanding, but it's not what incompleteness is about. You're confusing the bowling ball and rubber sheet model with Einstein's actual theory of general relativity.

4. There are other versions of the incompleteness theorem that don't even involve the liar paradox as an analogy; they're completely different approaches. Among these are Rosser's trick and Chaitin's proof of the incompleteness theorem. From the Wiki article on incompleteness:

"Gregory Chaitin produced undecidable statements in algorithmic information theory and proved another incompleteness theorem in that setting. Chaitin's incompleteness theorem states that for any system that can represent enough arithmetic, there is an upper bound c such that no specific number can be proved in that system to have Kolmogorov complexity greater than c. While Gödel's theorem is related to the liar paradox, Chaitin's result is related to Berry's paradox."

These are technical matters that I'm not qualified to discuss and that I don't mean to throw at you. I only mean to say that incompleteness is a deep truth about formal systems that has been approached and proved in several different ways, and not all by analogy with the liar paradox.

To sum up:

* Your claim that incompleteness is a word game is simply false. You're confusing syntax with semantics, and helpful analogies with the more complicated ideas they analogize. I daresay the authors of the videos you viewed may well have themselves been confused on this point.

* Why the heck should man have any hope of knowing what God knows, from just using our pitiful finite strings of symbols? God is not nearly so limited.

ps -- I didn't touch on your misunderstandings of sets. Briefly, what we tell people about sets in high school are not sets. Set is an undefined term. I better say this again so that I'm perfectly clear. There is no definition of what a set is. It's helpful to think of a set as a collection of objects, but you have to realize that this is only an intuitive approximation. Another bowling ball on a rubber sheet.

Sets are characterized by the axioms that say what sets do. Sort of like the Supreme Court's famous definition of pornography, that they know it when they see it. Mathematicians know a set when they see one. But there's actually no definition at all. I'm sure that must come as a suprise but it's true. Sets are not "well-defined collections," or groups of discrete elements lined up like soldiers, 1, 2, 3, etc. Your ideas about sets aren't directly related to incompleteness so I'll leave that for another time. Let me just add that Wikipedia says that "a set is a well-defined collection of distinct objects ..." and that is just factually false. Even their disclaimer that they're talking about naive versus axiomatic sets is a little off the mark. They're really talking about high school sets and those are of no use to us at all.

Being interested in different forms of relativism, I am interested in logical paradoxes and tonight watched a video on Godel. — Gregory

From the rest of your post I gather that you either misunderstood a sensible video or else believed a nonsensical one. Can you please link the video so I can determine which is the case? Nothing in your post remotely corresponds to Gödel's incompleteness theorems.

From what I know, his incompleteness proofs are very dependent on the use of human language. — Gregory

Not so. Gödel's incompleteness theorems are works of pure mathematical logic. They could be, and have been, presented purely symbolically. They could be valdated by a computer. Can you explain why you erroneously believe what you said? Perhaps you saw a video that compared incompleteness to the liar paradox. This is a handwavy and inaccurate description. The incompleteness theorem is simply about the limitations of axiomatic systems in determining mathematical truth.

I am not sure there can't be an intelligent life form, following what Quine wrote on that, which couldn't, if it thought purely in numbers, prove to itself all of mathematics. — Gregory

Can you explain what Quine wrote about intelligent life forms violating known results in mathematical logic? What would it mean to think purely in numbers? Are we doing sci-fi speculation? Kirk and Spock encounter an alien race that thinks purely in numbers, and Kirk must find a way to make it confuse itself before it destroys the Enterprise? While putting the moves on Yeoman Rand? Quine wrote about this? Reference please. What you said truly makes no sense in the context of incompleteness.

Such an ideal might be impossible for a human, but if Spinoza were right God is closer us than we are to ourselves, — Gregory

Anything has distance zero from itself, and no distance culd be less than that. It's difficult to know what Spinoza could have meant here unless he's speaking metaphorically. What exactly are you talking about and what does this have to do with a technical result in mathematical logic?

But you did say something interesting that gets to the heart of your misunderstanding of the first incompleteness theorem.

ergo if we are God we could prove all of mathematics. — Gregory

No. Let us assume, for sake of discussion, that:

* There is a Platonic realm of mathematics in which every well-formed mathematical statement has a definite truth value. For example the Continuum hypothesis (CH) is either true or false. There either is or isn't a set whose cardinality lies strictly between that of the natural numbers and the real numbers.

* Let us further assume that God has access to this Platonic realm.

* If you will grant me that God, for all his or her powers, is nevertheless bound by the laws of reason and logic; then God knows the answer to CH, but God can NOT prove CH from the axioms of ZFC; nor from any axiom system whatever that doesn't essentially posit either the affirmation or denial of CH. And even then, that new axiomatic system must itself still be incomplete, unless it is inconsistent.

What the first incompleteness theorem says is that no axiomatic system complex enough to model the arithmetic of the natural numbers, can be both complete and consistent. Even God must be bound by this result as long as God is contrained by logic and reason.

It may well be true that God knows the truth value of CH; but God can not PROVE CH from any axiomatic system that does not already include a new axiom (beyond ZFC) that implies CH. And that new system must itself be either inconsistent or incomplete.

Let me state this again. Incompleteness is about the limitation of axiomatic reasoning to determine mathematical truth. It says that there are always truths outside the reach of any axiomatic system (for sufficiently complex axiomatic systems). That's all it says. God would be bound by it even if God knew all mathematical truths. I hope I've made this clear.

Whatever video you watched either grossly mis-stated incompleteness, or you grossly misunderstood it.

At least that is how I see it, because relativism for me is such that the relative is in Becoming, never in the Absolute. — Gregory

You're simply misunderstanding incompleteness. It says nothing about relativism or absoluteness. Gödel himself was a Platonist and believed that CH had a definite truth value. All he did was show the limitations of axiomatic reasoning to determine mathematical truth.

Now as for sets, each one can be considered nominally as a succession of units — Gregory

You are confusing sets with well-ordered sets. And even well-ordered sets are not entirely built from succession, but also by taking upward limits. Your concept of sets is greatly at odds with how mathematical sets are understood. What you said here can't be made to correspond with mathematical set theory at all.

But if a set is a succession of units, how can you describe the set of real numbers in this manner?

or taken as a whole so that it includes itself. — Gregory

A set that contains itself violates the axiom of regularity. No set contains itself in standard set theory.

It is true that mathematicians study non well-founded set theory, but this is off to the side of mainstream set theory.

I think of the latter as including the nature of the set. — Gregory

If you prefer non well-founded set theory that's fine, but I suspect you are just making up your own concept of sets.

The set of all sets that do not contain themselves would be a series, maybe infinite, of individual units. — Gregory

Sets have no inherent order. Once again you are making up your own concept of sets that have nothing to do with sets as commonly understood.

The"set of all sets that do not contain themselves" is not a valid set because it is a set specification that does not conform to the standard rules of set formation; in this case, the axiom schema of specification. Specification says that to form a set from a predicate, we have to first start with a known set and then cut it down via the predicate. We can't just state the predicate without any enclosing set, without creating a contradiction. That's what Russell's paradox shows.

The set of all sets that do contain themselves likewise does not require that a set contain and not contain itself. — Gregory

True enough, if only by accident. The set of all sets that contain themselves is the empty set, in the presence of the axiom of regularity.

It would have merely all the individuals in addition to their groupings. — Gregory

I'm afraid that doesn't mean anything. What are groupings in this context?

But I am open to relativism in mathematics and if someone has a proof of it i'd be interested. — Gregory

All these proofs are on Wikipedia. I can't imagine what video you saw that gave you these wildly inaccurate ideas, which can best be characterized as "not even wrong."

I get that you're sincere, but your ideas are not in accord with mathematics nor do you seem to understand what incompleteness is about.

Godel seemed to mesh language with numbers so tightly that Becoming seemed to enter mathematics. — Gregory

You have stated this twice without context, argument, justification, or evidence. And it's not true. The burden is on you to explain yourself.

However, the Absolute in my opinion must be consistent with proving everything logically and mathematically. — Gregory

Why? Does the Absolute get its marching orders from misunderstood Youtube videos? The fact is that axiomatics are not sufficient to determine all mathematical truth.

Maybe I am entering Leibnizian territory, — Gregory

Can you be more specific? Leibniz did a lot of different things.

but I find this topic to taste like steel and I like it. — Gregory

What? What does this even mean? What does steel taste like? What does it mean to enjoy the taste of steel? What does that have to do with anything? I enjoy the taste of burritos, but that doesn't imply any particular result in set theory.

Unless you mean set theorist John Steel. I like puns too, were you testing your readers?

If we understand ambiguity in language, I don't see how Godel or Russell could prove that contradiction lies in the heart of numbers — Gregory

But nobody claims they did any such thing. Russell showed that naive set theory, in particular unrestricted comprehension, leads to a contradiction. And Gödel showed that no axiomatic system complex enough to include the natural numbers can be both consistent and complete.

Neither result bears on "contradictions in the heart of numubers." That's something you're making up; either as a result of misunderstanding a video, or believing a bad video, or both.

Incompleteness simply says that axiomatic systems (of sufficient complexity) are not capable of determining all mathematical truth, unless they're inconsistent.

It's weird. Since posting my opinion of Harris a week ago, my Youtube feed is suddenly full of Sam Harris clips. Maybe it's a coincidence. A Harris clip showed up by chance and since this thread was on my mind, I clicked on it. And whatever you click on in Youtube, you get a lot more like it right afterward. That's how their algorithm is coded. Or maybe deep in the bowels of Google's servers, my comments here are linked to other aspects of my online identity, and they factor that into my Youtube suggestions. In theory that shouldn't be possible but Google does a lot of business with the government and I'm sure they have access to data they shouldn't have. Or they could use AI to cross-reference my writing style, that would be doable with only publicly-obtained data. Writing style analysis is pretty advanced these days.

https://www.storyfit.com/blog/new-ai-emma-identity-detects-distinct-writing-styles

In any event, I have for the past couple of days been watching a lot of Sam Harris clips; and I think I can give a better answer to what it is that I don't like about him. Some things I like. He has a calming delivery and he gets off some good lines and has a way of verbally. clarifying the obvious. I just don't consider him particularly smart. Or at the very least, not particularly deep or interesting.

I originally said that I don't consider him very bright. And what I mean is, he's witty, but it's the wit of a precocious 16 year old who just discovered that the world doesn't work the way they were told. So he makes a joke about praying over your breakfast pancakes to turn them into God, in order to mock the Catholic belief that the wafers are literally the body of Christ.

Well ok, he's right about the analogy. The pancake image is funny. He has a great deadpan delivery.

But it's essentially a puerile observation. Many volumes have been written across the ages about the meaning of the Eucharist. Harris offers no scholarly insight into the practice. And say what you will, there are 1.2 billion Catholics in the world. You can't dismiss their earnest and heartfelt beliefs with pancake jokes. A philosopher has to account for the undeniable power of religious faith in the hearts of so many of the world's people.

My recent Sam Harris binge has confirmed my original opinion. Puerile is the word. Childish, silly, trivial. He is entertaining and satisfies our pseudo-intellectual urges. But a deeper question would be why 1.2 billion people derive personal value in their lives from the wafers. I'm not a Catholic and I'm not religious. But I recognize the awesome power, for good and for evil, of religion in the world. Dismissing religion as superstitious claptrap makes some people feel good about themselves. But if we are to claim to be philosophers or "public intellectuals," we must give a thoughtful, intellectually satisfying account of those 1.2 billion. This, Harris does not do.

Harris is superficially clever but lacking in depth; and ultimately intellectually unsatisfying.

Like the OP, you are applying a lay concept of a hierarchy. I am defending something else. — apokrisis

"We're not worthy!" :-)

Yeah, I give up. If you don’t get network theory, then I’ll leave it there. — apokrisis

You should get some self-awareness. When you know you're wrong, you get personal. Saw that before.

So it had nothing to do with going non-hierarchical and everything to do with creating a new virtual stage where the information was divorced from the physics. — apokrisis

You are making my point.

People let rip in this new world. And as is natural, hierarchical order resulted. We ended up with the influencer economy, Trump, cancel culture, and all those other good things. — apokrisis

The Internet didn't abolish human nature. And cancel culture doesn't need the Internet. Chairman Mao's cultural revolution did fine without it. You are listing all the hierarchical ills of the world and claiming them as evidence that the Internet is hierarchical. That's a terrible debating point. What does Trump have to do with it? You know, American politics is somewhat anti-hierarchical. That's yet another example I could give. Federalism. The president is not boss of the states. Of course in recent decades the Feds have learned to pressure the states by withholding funds and so forth, but our system has much more local autonomy than most other democratic systems, by design.

How exactly is Trump a debating point in favor of the thesis that the Internet is a hierarchy? Can you see that your enthusiasm for your thesis is causing your logic to be a bit weak?

I was thinking of this as well - along with the many images shared on social media by supporters of Trump atop a tank in a cartoonish fashion, things exploding everywhere, Trump holding a bazooka or rifle. We also have neocon backers of Trump - the American Enterprise Institute is one, I believe. And then we have the Erik Prince association with the Administration through DeVos. Trump's book, Time to Get Tough, has a chapter simply entitled, "Take the Oil." I also recall a sort of to and fro "saber rattling" between Trump and North Korea. I also recall when Trump first announced his "America First" campaign in such a way that basically said something like we're steering the world order unilaterally - that's how it sounded to me at the time anyway - with the fairly ridiculous sounding sugar coating "as all nations should put their nations first" - I wondered how "peaceful" he struck the rest of the world watching. Trump kicking around the "China Virus" in concert with Pompeo reportedly encouraging the G7 to refer to it as the "Wuhan Virus" also raise an eyebrow. — Kevin

Trump: No new wars. Obama: Syria, Libya, Yemen, Somalia Niger, probably a few others we don't even know about. Trump hasn't taken anyone's oil. Watch what he does, not what he says. His words are negotiating tactics designed to keep his opponents off-balance. He has not started any new wars. That is a LOT more than you can say for the Obama/Hillary/Kerry foreign policy. And if those people return to power? It's bombs away.

It might be helpful to talk about this in the language that network theory has created for itself. — apokrisis

I've pretty much said my piece on this. The OP said it was hard to think of a big man-made system that's not hierarchical. I offered the Internet not only as an example of a non-hierarchical system, but also one that was easy to think of.

The link you posted claiming the Internet is hierarchical failed to offer evidence or make a case. I agree that the hardware of the Internet, running over the existing 20th century telecommunications network, is hierarchical. The Internet as a whole, though, is peer-to-peer; as our conversation illustrates. I don't have to go up my management chain and down yours in order to speak to you. Everyone in the world is directly connected to each other.

Of course hierarchies have arisen in the software layer (Facebook, etc.), but they don't invalidate the basic point.

The OP hasn't seen fit to reply to me, and I don't have sufficient passion for the subject to reply in detail to your interesting points. I made my points in my initial post and have nothing more to add. The Internet is a peer-to-peer system, despite the hierarchical hardware and the hierarchical Domain name system. And therefore not everything we make is hierarchical. There's at least one exception. We live in an age of disintermediation, or at least so the early Internet theorists believed. You can buy a Gutenberg Bible, you no longer need a priest to tell you the word of God. The printing press was a great blow to hierarchy of its day. Of course governments and corporations are getting a pretty good stranglehold on the Internet these days, which does support your point.

Yes most definitely. He has not started any new wars.
— fishfry
Got very close with Iran. Really close. — ssu

Most definitely. Bolton was lusting for war and Trump wouldn't let him have it and fired him. Trump blusters about military strength and then avoids war. That's his style. Ignore everything Trump says, watch what he does. He's the most peace-oriented president we've had since Eisenhower, another guy who understood that you achieve peace by making your war threat credible.

In other words I disagree that Trump was ever close to war with Iran. He was never close to war with Iran. His words were not intentions, they were negotiating maneuvers in the service of peace. The actual results bear that out.

The Dems are chomping at the bit for more wars. The selection of Biden is a huge win for the war party. Don't you remember the 2016 GOP debates when Trump knocked Jeb! out of the contest by attacking him for his brother W's war?
— fishfry
He as the neocons were Republicans, as you likely know. — ssu

Yes indeed. Recall that in the 2016 GOP primaries, Trump blew Jeb! off the stage by going after 43's disastrous invasion of Iraq; an unthinkable heresy in the GOP up till then but a hugely popular position with the American people. There is an unholy alliance of GOP neocons and Dem neolibs wanting more wars. That's why you have all these generals throwing rocks at Trump in the media. A Biden win puts the war wing of the Democratic party in charge of the country. Not a pleasant thought if you value peace. Obama's foreign policy represented Bush's third and fourth terms, and those are the people hoping to get back into power.

If you look at the actual record, Trump is the peace candidate. The Dems and the left do the country a disservice by failing to see that. As far as the GOP in general, I have nothing good to say about them, if that was your point. They love the wars too. Not much of a constituency for peace in DC. That's one of the reasons Trump won. Peace is very popular with the people.

Or at least humans will be happy to pay the data centre electricity bills. — apokrisis

Your post overlapped my edits, I'll try to respond and hope you're rereading the last part of my post. I better leave this for tomorrow. FWIW I came up in the microcomputer business at around that time. So my origin is very "anti-IBM" which perhaps also explains a lot. IBMers wore suits, we had beer busts. And packet-switched networking.

But if IBM had managed to stay in control, then you would have been stuck with corporate information systems and not evolved to those new levels of information flow. — apokrisis

Hence the evolutionary advantage of non-hierarchical systems! You are talking me back into my original point!

Ok time for bed.

I actually did edit an IBM mainframe journal for a couple of years so interviewed guys like Gene Amdahl and Bill Gates. It was right at the time that IBM was losing the battle to impose its proprietary hierarchical SNA cooperative processing architecture on the data processing world. — apokrisis

Cool experiences. And now, I have to reframe my understanding of that history. Are you saying that in the end, hierarchies won again? That TCP/IP in effect turned into SNA? That they built an open system but it inevitably turned into a hierarchical one? So they might as well not have bothered? Ok that's clearly not true. So it brings up the question: What did the open peer-to-peer idea bring to the table, in somehow augmenting or improving the hierarchy? Does your theory account for the fact that a hierarchical hardware layer with a peer-to-peer software layer seems to be the winning ticket?

Of course IBM would have loved to have controlled the world's networking standards ... TCP/IP foiled that dream. Will you agree that there is something, even if I'm not nailing it yet, non-hierarchical about the Internet, even today? And that it's an important component?

TCP/IP says that any hardware whatsoever that adhered to publicly published protocols, could participate. In fact you could write your own commentary on the protocols, improvements, new protocols. Any kind of hardware at all. And you don't have to know anything about the network. you just need the address of your destination and the network delivers it. And no one part of the network has to know all the routing. Each node only knows where to send each packet next, and the network itself dynamically adjusts its routing tables as new nodes appear or leave.

All that was a profound advance in networking and it did take over the world. I contend that Its openness was a crucial aspect of its market victory.

There is something to that. I am sort of coming around to my original position. The non-hierarchical nature of the software at its core, has to be acknowledged. You can't just say it's a hierarchy because that's how the packets flow. It's a lot different than a pure hierarchical network.

Does you theory accommodate this situation?

ps -- It's the dynamic routing. A hierarchical network has to do all the routing at the top. It can't compete with dynamic peer-to-peer routing. So at the software level. the Internet works because it is essentially anti-hierarchical. No one node knows the network. No one node controls the network.

I have convinced myself my original point was right. That's a different position from where I was when I started this post so nothing's set in stone. But you have to acknowledge the importance of anti-hierarchy. The Internet's hardware layer is hierarchical; and the software is anti-hierarchical or peer-to-peer. Yeah there are big routers in the middle but they don't control things, they just keep track of their own local sphere of knowledge.

It is the same as the wealth inequality story. — apokrisis

Right, same reason utopian communities always fail. In the end, lords and serfs is a law of nature. How depressing. I think you made your point. It's not all an evil plot. It's an engineering principle of how things flow. Still ... discovering that your worldview comes from IBM mainframes explains a lot :-)

Am I being too subtle? — apokrisis

I think your IBM/SNA background gave you a certain worldview. It's a different reality than the packet-switching approach of TCP/IP. Funny that it's influenced your understanding of philosophy.

The hardware was hierarchical because that's just the naturally efficient way to organise the world so it can handle a traffic of data. — apokrisis

I'm going to concede this point. Even if the AT&T's of the world didn't exist we'd have to invent them. Local aggregation will layer its way up to form a hierarchy, as you say.

Then the software was the attempt to create a new flat virtual realm on top - a unstructured network. — apokrisis

Ok then we are in agreement. The original aim was anti-hierarchical, even if it didn't quite work out. Must that necessarily be the case? Perhaps, and if so I'd need to concede the software point too. Just as utopian human communities always fail. Equality only works in theory; and in the end you always have lords and serfs.

And yet once this software started to handle real world activity, it then developed a hierarchical pattern of activity. As again, that just is what is natural. The flat network became a hierarchical network of networks, with some networks much bigger than most of the others. — apokrisis

So it was all inevitable, and not an awful contingent perversion of the original idea. You're probably right. The utopian vision of the Internet failed; and it wasn't an accident or a plot of the telcos and the government; but rather some sort of structural law of nature in favor of hierarchies, if I can put it that way.

But then you'd say that SNA is a limiting case of TCP/IP; and that, I can't agree with. The TCP/IP packets don't know and don't care how they get where they're going; and surely there's something essential in that.

Check out constructal theory for the generality it this. — apokrisis

Found a couple of interesting references. I see the point you're making. In the end, the packets have to organize themselves into hierarchical flows no matter how utopian the original intent.

Thanks. I'm quite familiar with all that. I was around when IBM was pushing LU6.2. — apokrisis

Ok. Then you must agree with my point that the software protocols are peer-to-peer and the opposite of hierarchical. And I concede your point that the present hardware infrastructure is hierarchical.

One still must draw a distinction between a rigid communication hierarchy, in which A talks to B who talks to C in order to communicate with D; and a peer-to-peer network, in which any node can freely communicate with any other. It's a matter of which level you view it from I suppose. You agree at least with this much.

Ah .... explains a lot. SNA is a hierarchical network. Your formative conception of networking is hierarchical. If you'd come to TCP/IP first (as I did) your philosophical understanding of networking might be different. TCP/IP is the exact opposite of SNA. Hierarchical networking failed (at the software level) and peer-to-peer took over the world. Of course from a practical level I suppose hardware is inherently hierarchical, since network traffic must be aggregated at each level. Is this the emergent hierarchy you're talking about? Still, conceptually, TCP/IP is peer-to-peer. I hope you'll work extra hard to overcome your SNA bias here.

So the peer-to-peer is implemented at a software level ... not the hardware level? — apokrisis

Yes. TCP/IP is a pure software networking protocol. It's independent of any particular hardware implementation. As you may know, every single Internet packet includes the source address and the final destination address. The routing is entirely up to the network. All Internet communication is essentially peer-to-peer. No packet has any idea how it will be routed. It only knows its ultimate destination. This is very basic technical information about how the 'net works.

And the article was about the hardware level. — apokrisis

We have huge telecommunication companies so it's no surprise that the current hardware implementation is hierarchical, as I mentioned in my initial post -- precisely to avoid this objection. How the Internet happens to be implemented is separate from its essential peer-to-peer nature. See for example Comer, Internetworking with TCP/IP, the standard text on the subject.

https://www.pearson.com/us/higher-education/program/Comer-Internetworking-with-TCP-IP-Volume-One-6th-Edition/PGM138190.html

And then at the software level - given a carefully-levelled playing field - we find, as I said, a scalefree network structure emerging? — apokrisis

I don't see that at all, except as a byproduct of the contingent hardware implementation. One could in theory imagine a fully-connected graph of nodes, in which AT&T and the other monster telcos would not have a death grip on human communication. One could argue that this is exactly the vision the original Internet developers had in mind. One wouldn't have to argue too strenuously, since they explicitly intended a pure peer-to-peer network.

One with a fat tail distribution of connectivity? — apokrisis

Again, this is a byproduct of the contingent hardware implementation resulting from the existing hierarchical structure of huge telecommunication companies. It's neither required by the fundamental protocols nor is it necessary to the functioning of the Internet. One could argue compellingly that it's become counter to the original intent of the Internet.

One where no designing hand was involved and yet a hierarchical distribution of "significance" was formed? We see agglomeration and disintermediation as the signature of the dynamics? — apokrisis

Only because the telecommunications behemoths already existed and got their (grubby) hands on it; to the detriment of the original open and free aims intended by the original academic designers.

Perhaps you are arguing that the original Internet designers were naive academics and should have seen it all coming. Or perhaps since the original Arpanet was funded by the government, the naive academics were tools and their vision of unfettered open communications among all humanity was a delusion. You'd have a point, based on how things turned out.