joigus

As to emergence there is, I think, a dilemma between principle and practice that I think overrides almost any other consideration. Directionality of emergence is very clear in principle, but there are insurmountable difficulties in practice to ellucidate causation. Atoms make a dog. Dogs don't make an atom. It's what Weinberg called 'arrows of explanation.' That's very clear in principle. Even though it's impossible in practice to tell anything about dogs from the laws of atomic motion. It's very clear to me that dogs emerge from atoms; atoms do not emerge from dogs. I happen to know that some very philosophically-minded people think otherwise, which are the ones that @TheVat characterises as 'strong emergentists.' I think they can do that, only because the 'arrows of explanation' are invisible to all intents and purposes. It's a hopeless problem, so there is room for people to exploit this practical disconnect, interpret it as fundamental, and what's more, invert the 'arrows of explanation.' In this example, I think people who hold this view are disregarding an approach that's much more plausible: feedback mechanisms. Those are compatible with molecular determinism, IMO. Even though they're extremely complex. An algorithm to run on a machine that proved beyond any doubt that there must be such a thing as a dog based on the quantum laws of motion. That would be a sight to behold. But I wouldn't wanna be the person analising the data. This would-be machine would have to prove the logical necessity (from the atomic laws, to be kept in mind) of giraffes, and T-rex, and gut bacteria, and... covering all the organisms that ever were, that ever will be, and that would have been.

Sorry, I misinterpreted the whole sequence of events. I suppose I'm just tired. It's a very clever solution anyway. It still rests on sqrt(2) being irrational, but that's easier than pi.

Very dexterous! +1.

Allegedly, communication breakdown due to mutual misinterpretation of terms and implied meanings and responses thereof can be a good example of an emergent phenomenon. I beseech you both to come back to good terms ASAP. We all value both of you.

OK. I see. Why didn't you just say that was your answer from the beginning, instead of giving it piecemeal? You had this 'question' all ready with the answer and all. Then you ask a question pretending not to know the answer. The answer is kinda obvious TBH. You can find about infinitely many possibilities to do that. I gave you one that's pretty obvious too. About as obvious as the fact that \( \sqrt{2} \) is irrational, which you haven't proved either. Yes, that's a result in number theory too, and you're resting your answer on shoulders of giants. (pi)rational cannot give you a rational is pretty obvious to me too. Proving it rigorously is another matter. You, though, for some reason, don't like the argument. You prefer yours, (which is to come pretty soon.) You reappear then in intervals of less than one minute declaring that your answer is the answer, and it's simpler than everybody else's. Your answer in every step is, of course, flawed unless you provide the looping argument which is your final effect. Then you pull the rabbit out of the hat that you've been silent about for the whole conversation. Voilá! --Applause. 👏👏👏 To me, it's been a considerable amount of time down the drain. I have better things to do. Cute. Thank you.

Not quite: If it's not, then it's rational. Then you use your algebra and you prove that some 'rational' you've found, raised to sqrt(2), gives a rational. You need to prove that \( \sqrt{2}^{\sqrt{2}} \) is irrational in order to make that claim. See my point? So you're back to square one, which is what I tried to tell you: Number theory. ??

How do you know \( \sqrt{2}^{\sqrt{2}} \) is irrational?

When I read, I was bracing myself. And then..., And I only have this to say: I think that largely depends on the system, and the laws being dealt with. I really don't think it's possible to establish the terms for emergence to occur in a completely unambiguous way. Let alone given that physicists and biologists, as StringJunky noticed, do not completely agree on what's required. In the context of physics in particular, I'm familiar with the following difficulty that I've pointed out before on a related thread, and that's due to Leonard Susskind, AFAIK: When we reach a fundamental level of description, what is 'the system' and what are 'the parts' is not even clear. I remember the examples Susskind used were bosonization (a fundamental fermion can be considered as a couple of bosons with a kink between them = a twist in space-time) and dualities in QFT (a theory in a certain limit looks very much like another theory when we consider different limits). (Not a literal quote; rather, my re-phrasing of the observation.)

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?