joigus

Thank you for the detailed answers. +1 You mean lambda = 1+ √5 And lambda = 4 is the chaotic one.

That's why I said: I don't know. I wasn't aware that we were talking about the North Atlantic Gyre. Isn't that from another forum, more to do with eddies? Your point does remain indeed. And my point that chaos, the way it's normally taught at universities, is about mixing of trajectories and ergodicity, also remains. That's what my books say, that's what the KAM theorem says, and that's what I thought I knew and I learnt in the classrooms. I must confess I remember only vaguely, but the idea of it was to introduce a measure in the space of parameters of (quasi-periodic) Hamiltonians and prove that the non-chaotic systems had zero measure, while the chaotic ones had the measure of the continuum. I don't recognize that explanation on the Wikipedia article, but I'm quite sure of it. Because the characterization that you seem to be working towards speaks of more general, non-conservative systems, I'm willing to learn more from it. The only thing in which I've disagreed is about chaos being characterized only by instability (Liapunov exponents.) The example I gave, was clear enough. Also, I said, And although from a practical POV it doesn't do to treat some of such systems with Hamiltonians, the fact that everything we know about chaos, as well as the initial motivation by Poincaré's work on the stability of the Solar System, derives from a Hamiltonian, has some bearing on the question,

I think you got it wrong: Gannets and barracuda fish better together Edit: x-posted with Mordred +1. Couldn't have given you better advise.

It's like trying to find your way out of a maze, or a forest. There is no roadmap, and if you've got one it's probably wrong. But understanding topology, reading clues, minor details, can help you a lot. Do you always need a map to find your bearings? In physics the map always comes later. Not very well known fact: Einstein spent one whole year without accepting Minkowski's concept of 4-dimensional space-time (I've heard this in a classroom.) He already had all that was needed, logical fact to logical fact. In the words of Steven Weinberg: "physicists are more like hounds than hawks" Dreams of a Final Theory

And I really recommend that you read the OP: And then: (My emphasis) Now what? Do you realize you haven't understood the OP already? Don't worry, I'm waiting for you to catch up. IOW: Just because we cannot prove everything (following Gödel's theorem), does it follow that we cannot prove that 2+2=4? Should we accept that 2+2=5? Or anything else? That's called "reading between the lines." If you can't read between the lines, the OP looks like a contradiction. Which it isn't. But you must take some time to really read carefully the OP and really want to help. Just trying to appear cleverer than everybody else just because you can quote Bourbaki, or link to it, doesn't really help. Oh, really? I hadn't noticed. I thought we were talking about the history of pudding (sigh).

I already told you about this. I'm certainly not going to repeat just because you don't care enough to read.

Bourbaki, really? The OP has a simple enough question about Gödel's theorem and simple relations between real numbers and your suggestion is a treatise that proposes to redo mathematics from scratch by a group of (brilliant) mathematicians that proposed to change the whole structure of maths under a pseudonym? This is the line that you misunderstood, split into independent lines: 0+0=0 0+1=1 1+1=2 You understood: Incorrect interpretation of OP's question. Honest mistake, so far. But then, You bring up division by zero, which is irrelevant, as it was not implied by the OP. Then you bring up my level of knowledge. Not that I care. I don't. Then you bring up Bourbaki. I happen to know Bourbaki and I coincide with @studiot that it's nothing to do with the OP, nor does it have any bearing on the question, nor is it advisable in order to answer it, among other things, on account of what the OP said, very clearly, All of this when the question had already been answered to the satisfaction of the OP, as I understand. I wonder, what's next? Algebraic topology? Apparently there's no limit to how far off-topic you're willing to go in order to bring more confusion to the discussion or not to recognize that you misunderstood the initial question. Now, I suggest you ask this yourself: Am I really helping here?

Have you seen my profile?: Who says I say I know everything? It's you who seems to think that people are saying things they're not really saying. Read whatever people say and then say whatever you have to say. Enough said.

I'm well educated enough to withhold my opinion of what you (or any other member of this forum) are or are not. I will always concentrate on the arguments and document them properly wherever necessary. I suggest you do the same.

Totally agree. +1. Nobody understood division by zero here, except you, @ahmet. Plus the question, has been satisfactorily answered, I think. Unless the OP has any further question. To me, end of story. Edit: Unless you have any further comments on how Gödel's theorem could imply that we can prove 2+2=5, in which case, I unrest my case. Edit 2: It seems the OP made a mistake in the title. They meant, I think, ¿Can't we prove (from Gödel's theorem) that 2+2=4?

I think it's a good idea. +1. Modern and Theoretical Physics & Astronomy and Cosmology are especially affected in what physics is concerned. Speculations could be another one, because much of what starts on the above ones ends up there. This forums are very interdisciplinary, so maybe there are nuances between our respective current definitions, but I see no reason why I couldn't be worked out.

Ok. Before you and I get into a long-winded discussion, why don't we let the OP tell us what concept of chaos they're interested in? I have the suspicion that the subject has evolved and the word has been taken to mean different things by different communities, according to their needs. That's more or less the reason why I said, As to Hamiltonian dynamics. Again, never mind me. I understand your complaint about my not "taking" your argument. I suggested that there is no reason why Hamiltonian dynamics should not be taken as of total generality, even if (and here's the subtle point I may have forgotten to suggest more strongly or suggest at all), from a practical POV, it may not be very useful for open systems. For open systems you could always consider your system, of coordinates (q,p) (many of them, with lower-case letters), plus your "environment", of coordinates (Q,P) (also many, with capital letters). And then you could write your Hamilton equations (formally) as, \[\frac{\partial H}{\partial p_{i}}=\dot{q}_{i}\] \[\frac{\partial H}{\partial q_{i}}=-\dot{p}_{i}\] \[\frac{\partial H}{\partial P_{i}}=\dot{Q}_{i}\] \[\frac{\partial H}{\partial Q_{i}}=-\dot{P}_{i}\] While the total Hamiltonian would really mess things up for your sub-system of interest. \[H\left(q,Q,p,P\right)\neq h\left(q,p\right)+H^{\textrm{env}}\left(Q,P\right)\] with "env" meaning "environment". Any open system can be considered as nested in a larger closed system. The point is very precisely explained in Landau Lifshitz (Course of Theoretical physics) Vol I: Mechanics. Not even fluids, elastic materials, or anything really, escapes this consideration. P(t), Q(t) would make the sub-system non-conservative. Because chaotic behaviour appears in such simple systems as conservative, few-DOF systems, it's only natural to assume that it "infects" every other dynamics that we may consider. Even if it's open (system+environment can always be considered as closed). That was about all my point. But this discussion is quite academic and I wouldn't want the OP to be scared off by it. I'd rather have some feedback from the OP.

Yes, but some are detrimental for the individual, while leaving the reproductive success of the species alone (those are the parasites that thrive); and others aren't. It is entirely possible. I just hope you're wrong, although it seems to be a well-informed guess. 😬

I neither agree nor disagree at this point. But I see no reason why the theory of elasticity or fluid mechanics cannot be put under the umbrella of Hamiltonian mechanics. It's the non-conservative aspect that would make it different from the academic examples of pendula or the like, though. Your definition of chaos seems to be more general. Why would I rush to disagree with you at this point when I'm likely to learn something new?

There are several aspects of your question I don't understand. 1) How can something become lighter by means of electricity? Weight doesn't change by thrust or electromagnetism. Acceleration does. 2) You say "stimulus" as in "stimulus/response". Cybernetics? What is that stimulus? Do you mean push, transfer of momentum? 3) Thrust within. Thrust for a rocket is nothing to do with "within". The exhaust goes away. Maybe someone can understand better...

Sorry, I see it all as Hamiltonian mechanics. 😭

There are things the Bible doesn't say and almost everybody believes it does. There was no apple. It could have been a quince, or maybe a fig, as there were no apples back then in the Middle East. The Bible doesn't say it was an apple, actually. The Bible doesn't say Jonah was eaten by a whale either. The Bible doesn't say there was an angel at the Garden of Eden, but a cherub, which was a mythical animal represented very frequently in the gardens of palaces throughout the Middle East. The Hebrew Bible doesn't say that Mary was a virgin, but a "young woman." ------------------------------------------------ There are things the Bible says and few people know it does. The Bible talks about a pantheon of gods that are subservient to Yahweh. And names God both as Yahweh and El. Is it the same god? I'm not sure. Asherah, the wife of Yahweh, is also mentioned, but the interpretation was presumably changed, as it's mentioned as a synonym for "a stick" in very obscure passages, when she is known to have been a goddess, as archaeology has shown. The stick was one of the symbols of the goddess. Back to Adam and Eve: There's at least one thing the Bible says twice in different (incompatible) ways: Ezechiel 28. Two prophecies, one of them against the king of Tyre. There you can see that the king of Tyre is expelled from the Garden of Eden, on account of his sins. The cherub also appears. Very similar legend; two different narrative uses. Who was expelled from the Garden of Eden, Adam and Eve or the king of Tyre? I'm not so sure. The authors of the Bible seem not to be either. Some scholars believe the Oracles against the king in Ezechiel 28 predate the Adam and Eve story in Genesis. ------------------------------------------------ There are things the Bible says that are taken from somewhere else: The Bible takes the story of Noah from The Epic of Gilgamesh, Utnapishtim , and adapts it to its own narrative needs. ------------------------------------------------ There are blatantly obvious things the Bible is silent about: Omri, big king of Samaria, was a very relevant character of the Assyrian domination period, but the Bible only mentions him in passing, as a baddie. The Bible also plays down the role of many other kings, like Manasseh, although he made Israel into an important olive oil factory and brought a period of peace, contrary to what Hezekiah, his father, did. ------------------------------------------------ And lastly, there are many things the Bible says that cannot be true. Josuah didn't conquer Jericho, as Kathleen Kenyon has proved. Jericho was uninhabited at the time. Plus the Egyptians were in control of Canaan and had the country strongly policed from Beit She'an. I don't believe God gave the law of gravity a suspension for some hours for the benefit of his people to the detriment of the Canaanites either. Plus the Canaanites and the Israelites were the same people: No difference in material culture or belief system, as Israel Finkelstein has shown. Abraham could not have possibly used camels. Camels were domesticated about 1000 years later.

Just change the final height to minus whatever or the initial height to plus whatever and redo the calculation.

The most obvious to me is the misunderstanding of what a scientific theory is. A couple of paragraphs with words about dimensions, parallel universes and such does not make a theory. Where are your predictions? You're not addressing any questions of observational cosmology, so you have no theory, really.

I agree with @Strange that your proposal is littered with misconceptions. Also, with @MigL that a WH is a construct that works very differently to a BB. And as I was reading all your answers I have remembered the particular sense in which the term "parallel universes" appeared in the scientific literature. It had to do with structures similar to the Einstein-Rosen bridge. I think the original motivation was with D-branes, if I remember correctly. So yes, there is a specific sense in which you can talk about parallel universes, but it's not related to extra dimensions, but with curvature. The "funnels" that you see in the picture can extend as to become parallel to each other. Extra dimensions are perpendicular, rather than parallel. No. Dark energy has to do with a scalar field, which is a completely different thing than curvature as it appears in a black/white hole. Speculation in physics with just words is hopeless. You must make calculations, and understand the mathematics. And fit experimental data, of course.

Ok. It seems we disagree about this, even if only mildly. The arguments I've heard or read that have convinced me that some rituals and religious practices may have played a positive part in the remote past are those that contend that some kind of centralized authority, plus a set of rules to decide what to do could have been an efficient way for a group of people in which disagreement can easily emerge, to take a decision and stick to it. But things that stay with us don't have to be good. Parasitic entities have their own evolutionary "agenda." They grow and prosper among us. The only mistake they must avoid making is being so damaging to their host that they manage to extinguish it. Examples of it from biology are the common cold or the measles. Examples from the world of memes are faith-based religions and the Flat Earth Society.

Every universe? How many are there? What's an anti-universe? In what sense is it anti-? What anti-symmetry isn't generally accepted? Anti-symmetry in the sense of matrices or operators? If you mean it in that sense, it is not only accepted, but it is a part of the formalism of, e.g., electromagnetism or General Relativity with torsion. If you mean something like in the sense of Hermann Weyl (symmetry as an operation after which something is unchanged), I'm not aware of any extension to anti-symmetry concept. All of those sign-changing operations are symmetries. Any anti-symmetry is a particular kind of symmetry. I'm not aware of any sense in which an extra dimension can be parallel to the previous dimensions. You can define it as perpendicular, but parallel is the only thing it can't possibly be. If it's parallel, I can see no way in which it could be extra. It would be the same dimension. Also, it's not either "parallel" or "compactified." Rather: It's either infinite or compactified. People talk about parallel universes, I know, but that's a very serious misnomer. Nothing is parallel to our universe in those (hypothetical) dimensions. Misnomers are a problem with popular science (and sometimes with serious science too.) Like, for example, Hubble's constant. It's not a constant. This makes science more confusing than need be. There is, and a very strong one. The metric of space-time and the invariance of the speed of light. There are strong elements of imprecision in your questions. I would like to help, but I don't know how, because I don't understand very well what you mean. Sorry I wasn't able to prove you wrong. I hope this helps, though, to clarify your questions.

Good question. +1 1+1=2 is a definition. The axioms only require the existence of 0 and 1. Were it not for the definitions (symbols, substituters) 2, 3, 4, etc., we would have to write 7 as, 1+1+1+1+1+1+1 If the axioms (associative law for sum) didn't allow for the proof that (1+1)+1=1+(1+1), etc. we would have to distinguish between "these 2 kinds of three": One for the left sum and another for the right. This actually happens in more general algebraic systems, like groups, octonions, etc. I hope that helps.

Sorry. I meant trajectories in the phase space. (q,p) So strict rest would be a point surrounded by asymptotic trajectories (flowing away from it: stable equilibrium; or converging towards it: unstable equilibrium). Rest, OTOH, would be a trajectory in (q,t). All of them would take infinite time, as phase-space trajectories cannot cross (Liouville's theorem.) Nah, this is nonsense. Let me think about it longer. From what I've been able to look up in Euler's theory as applied to engineering, it seems to be not really about strictly statics, but small deviations from an equilibrium position. Am I right? Maybe the subject is suffering some kind of generalisation I'm not aware of. I'll stay tuned.

No. Who said that? 2+2=4 can be obtained immediately from definitions. 2+2 = 4 is the same as (1+1)+(1+1) = (1+1+1)+1 which is obviously true following the axioms. The scope of Gödel's theorem is (presumably) about divisibility, number theory, primes. Things like that. Things that are (or may be) out of reach of finite (algorithmic) proofs from the axioms. x-posted with Studiot. +1 Sorry for overlapping with your answer, @studiot.