joigus

Let me add something that involves probabilities. I don't think it's very likely that picking r and s irrational, at random, you can get rs to be rational. The simple 'probabilistic' argument being that the cardinality of irrationals is aleph 1, while that of rationals is aleph naught, which means that there are incommensurably more irrationals than rationals. So, what are the chances. But I do think there are infinitely many occurrences of (irrational)irrational that are. LOL. Good one! I do think it's equally simple, tho. No serious. Good one. Now try to do something that simple with \( \left( -1 \right)^\pi \)!

Ah, OK. Yes, that can happen. Studiot gave an important clue, I think. Think Euler. I think you will agree that \( \log_{\pi}2 \) is irrational. Take \( r=\pi \) and \( s=\log_{\pi}2 \). Then, \[ r^{s}=\pi^{\log_{\pi}2}=2 \] The fact that \( \pi^{x} =2\) cannot be solved with x rational should be easy to prove by contradiction. Edit: Actually, I don't think it's 'easy', it's a somewhat elaborate result of number theory.

You said an irrational raised to an irrational. (-1)-i is a negative integer number raised to an imaginary (complex) number. Both can be done. One is more sophisticated. Which one is your question? For example, \( \left(-1\right)^{\pi} \) presents its own challenge. Which one are you interested in? It's better perhaps to tackle directly \( z^{w} \) with \( z,w\in\mathbb{C} \).

The fact that sometimes you need infinitely-many degrees of freedom, or perhaps, a large enough number is not essential, IMO for qualitatively different features to appear. Examples: 1) The 2-body problem in celestial mechanics is always solvable, the 3-body problem is always chaotic (because the equations are non-linear from the get-go. 2) An entangled system of 2 identical particles displays correlations that a 1-particle system cannot reproduce, because it violates Bell's inequality. And I agree with Studiot that the Navier-Stokes equation is quite transparent as to its meaning. It's consistent with the conservation of mass (continuity equation), conservation of angular momentum, energy, etc. So the behaviour of the constituents is apparent in the form of the equation, yet there are consequences of the equation (turbulent regimes, and so on) that have no correlate to the behaviour of one particle.

I'm not sure that these categories apply with all generality. For example: Temperature or phase transitions have been known for millenia, though they've been understood very recently. And I wouldn't hesitate to call them emergent, but never novel. Take @studiot's example of the arch, which resists a simple mathematical description, and yet the Mycenaean Greeks already used similar principles (corbelled roof) more than 3000 years ago. So it's not novel, but there is no doubt that the bricks are doing something as a 'congruence of individual behaviours' --if I may be allowed to use such mouthful-- to produce something that's not implied in their behaviour as individualities. As to 'unpredictable'... well, it depends. Entangled states (if we allow them to be considered an example of emergence) are completely unpredictable. But the archs of the aqueduct of Segovia will be there tomorrow, I'm confident to assure. 'Irreducible' is perhaps the one that's closer to the mark, IMO. It should be understood, though, that the sense in which we say it is: Whatever these qualities (emergent) are, they're not present in the parts. If the process can be analysed, and some kind of reasoning can be applied that proves that emergent phenomenon must be the case, it should be far from obvious. Example: thermodynamics. It's very far from obvious that the thermodynamic variable that quantifies both heat and irreversible work must be related with internal degrees of freedom we cannot see directly. I'm not sure I'm being helpful at all. It's kind of the way I understand the concept. I know it's not too far from the standard way, but there's plenty of room for nuances.

Good point. I was thinking about it myself a moment ago. Is it really useful? Maybe thinking that a particularly difficult concept can emerge from a sub-level can be inspiring. It certainly inspired the likes of Boltzmann and Gibbs to found the statistical-mechanical version of thermodynamics, which has farther-reaching consequences than Carnot and others' version. For the most part, when people talk about this or that concept as emergent, it sounds to me either as motivational or as an afterthought.

I think @StringJunky is implying a necessary condition for a phenomenon to be emergent, not a sufficient one.

I see. OK. We all've got words that kind of set off our 'philosophical alarms.' In my case it was that you seemed to imply 'irrelevant.' (My emphasis.) But I understand what you mean now: Irrelevant for the business of handling the emerged laws within their domain of applicability. Am I getting closer?

Yes. Mine is more specific: The emergent system must be simpler to describe than the parts, and, most importantly, the criterion I proposed does not ignore that the system is made up of 'parts.' Eise, I think, prefers to place a black box around the 'simpler parts' in order to describe the laws of the composite system without referring to the simpler parts, because they are 'irrelevant.' If I understood him correctly.

A minimal criterion would be that you need: 1) Composite system 2) Parts making up that system and relations between them The law of behaviour for the composite system is simpler (requires fewer parameters) than the laws of behaviour of the parts. Such laws are qualitatively different. Meaning: the patterns of behaviour change too with respect to the parts.

I always had problems with Wigner's motto. I would totally agree with the great man, had he picked a word other* than "unreasonable". To me, it's not unreasonable. A big part of understanding Nature is about quantifying it, and then measuring it. So, were the great Wigner alive, and would he bother to listen to me, I'd probably ask him, 'What do you mean "unreasonable?". I think this is very much in the vein of what Swansont said. * Alternative list: amazing fortunate etc.

The fact that some emergent phenomena can be formulated without referring to the more elementary level doesn't mean that we shouldn't aspire to it. I think we should aspire to it. We can't simply do away with the reductionist approach just because, oh, it can be formulated otherwise, so why bother? If the range of phenomena stubbornly resists that approach, so much the worse for our understanding --example, the weather and Navier-Stokes eq.--. But I'm sure understanding something about methane, and CO2, and conservation of energy, etc. doesn't stand in the way of understanding broadly what's going on with the weather (climate patterns). As to time and emergence --I'm familiar with Smolin's view, not so much with Rovelli's, although they are in the same front, I think--, I think it's a distinct possibility, but I don't expect it to be anything like the picture of 'a thing made up of tiny little things' in the way thermodynamic variables are. Although this is just a hunch on my part, granted. I'm not going down the rabbit hole of free will now. The wave function picture of quantum mechanics is definitely not, in its present state, an example of emergence. That doesn't mean it's not gonna be some day. The density-matrix picture you can consider as emergent. It's a statistical mixture of wave functions (so-called mixed state, made up of so-called pure states.) I hadn't seen Studiot's comment, which already goes in this direction. I was editing my post. I can elaborate more, if anyone's interested.

Indeed. The OP clearly means inertia, rather than gravitational mass. There's also the confusion between acceleration due to a force (that can never exceed the speed of light) and acceleration due to space expansion (that can). Further, there's confusion between what's accelerated and what's got the mass in F=ma. There could hardly be more elements of confusion.

“There are more things on heaven and earth, Horatio, than are dreamt of in your philosophy” comes to mind...

Why? Because no one knows what it is? I would agree that it's not very useful to think of dark energy in terms of mass, but what principle of physics are you invoking here?: No one knows what something is, therefore it can't have mass?

I think you come across as a deep observer of Nature. When we've disagreed, it's always been constructive and enriching. So thank you too!

Or perhaps a pothos leaf?

They are a family of paleoantropologists; Louis and Mary Leakey, as well as Richard's wife, Meave. Richard Leakey was an out and out field scientist. He was raised in the African savanna, rather than an academic environment. Neither he or his parents, probably, changed the big picture of the evolution of Homo sapiens by themselves. Although his father, Louis, is credited with having found a fossil that nobody knows where to put, Homo habilis. They mostly dug the ground, described what was there, and kept looking. But they were people who knew the environment inside out. R. Leakey was very involved in wildlife management and conservation strategies too. Very interesting people all of them. Here's a sample of what Richard Leakey was. Attenborough, Dawkins, Goodall, and Leakey discussing strategies to "save the planet." (2005.) The Selfish Green: IMHO, the contributions of Goodall and Leaky are priceless, and invaluable in their effort to tie these other two great minds to the ground, if I've understood them correctly. "It's very much part of being a primate, and the whole business of being a primate is to get other primates to do what you think is right" --Richard Leakey

https://www.theguardian.com/science/2022/jan/02/fossil-hunter-richard-leakey-who-showed-humans-evolved-in-africa-dies-at-77?utm_source=dlvr.it&utm_medium=facebook&fbclid=IwAR1PhZqET2nyh3CUhPbAvWs63MVs4l5o5mEP2j6bbH6iEqYIPvmCararT5w Sorry about the bad news. An excerpt of his book with Roger Lewin, Origins Reconsidered, is forever etched in my memory, where he paints a vivid picture of a young Homo erectus dying, and slowly, through more than a million years, becoming the fossil he and Alan Walker discovered at Lake Turkana, and forever after called the Turkana boy: This image of a wait-a-bit thorn springing from what once was a boy's head I find bordering poetry.

Very nice account, very informative, and very clear. Thank you. A simple calculation involving Boltzmann's constant gives you a temperature of a couple thousand Kelvin. Dividing 178000 J by Avogadro's number, and further dividing that by Boltzman's constant produces about 2141 K to break the bonds regularly. So no wonder there's a lot of it about. Breaking sigma bonds is not at all like breaking dipole-dipole hydrogen bonds, as in making water vapour... Does that make sense?

Very interesting. Thanks. My intuition is that the feeblest temperature background would split this tiny little weirdo of a molecule...

Whining won't make any of your 'proposals' more compelling. Denying evidence is not a good idea either. And ignoring the maths --the fact that non-linear equations are generally impossible to solve-- is not the solution. Don't know what to make of the attempt to trash one of the most brilliant physicists in living memory.

A mechanism can only be proposed once you have a theory. At least in physics and chemistry. I've told you in another thread, but you didn't answer. Newton's laws of motion don't come from a mechanism Maxwell's equations of electromagnetism don't come from a mechanism Einstein's equations of gravity don't come from a mechanism And so on. In the case of cosmology, the big bang is an inevitable consequence of Einstein's equations plus reasonable --and observationally sound-- cosmological hypotheses (cosmological principle, Hubble's law...). When extrapolated backwards in time, an expanding universe leads to a time in the remote past when everything was much, much closer together. There's your bang. Seems to make sense, doesn't it?

My sarcasmometer is going through the roof!

You're right if what you mean is that there are physical reasons to reject those solutions: A static universe would be unstable. But solutions they are. Einstein looked for that, and found it, because he had the prejudice of a static universe. In fact, the whole reason why he introduced the cosmological constant is precisely because it allowed him to tailor-make the universe as static. He did find that solution, but it's a freak universe. Then he regretted that he could have found the expansion of the universe as the biggest blunder of his life. Or so the story goes.