joigus

Yes, in fact 'assumptions' or 'concepts' go far beyond the realm of pure mathematics, so these definitions that try to be so broad really don't finish the job of specifying the matter IMO. What I tried to do is to picture the two distinct attitudes that govern all of maths. The idea that everything in maths is either algebra or analysis is not mine AAMOF, but I can't remember where I picked it up. But the feeling, when you're doing maths, of 'I'm doing algebra' or 'I'm doing analysis' is very clear in your mind when you're doing it.

Let me try that: I would say that mathematics is the science of skillful operations with concepts and rules invented just for the purpose of skillful operations with concepts and rules. Nah, it doesn't work for me either. But hey, what do I know.

I deeply and wholeheartedly appreciate Wigner, and he was sure much a better mathematician than I will ever be. But somehow the definition: "mathematics is the science of skillful operations with concepts and rules invented just for this purpose (mathematics.)" (My addition in parenthesis and my emphasis.) leaves something wanting for me.

Broadly speaking, mathematics is the science of how little you can assume in order to be able to say anything at all, and how much you can say after having assumed this and that. The first one is algebra. The second one is analysis. <joke> Those stand for the two A's that are in all the words mathematics, algebra, and analysis. <end of joke> The whole ramifications are more or less elaborate variations on these two basic themes. Some branches incorporate both.

To correct myself: as soon as the system is above 2 degrees of freedom and the evolution equations are non-linear (@studiot mentioned it before.) ==> We've got chaos. As a thought byproduct that might be relevant: It's interesting to notice that, in a way, that's what the Classical atomists, like Democritus, Leucippus, and Lucretius did: Given that there are these things and there are these behaviours (Democritus' clepsydra,) it stands to reason that the world is made of atoms. Now that's not an algorithm, but use of common --if learned-- intuition. How does intuition lead you to a well-founded hypothesis about what things are made of? Maybe some day we can teach intuition to computers, amplifying their reasoning abilities. Do I digress?

Well put.

Amazing. I was listening to this song no more than 3 or 4 days ago. I've just remembered and was about to post it. And I see your post. !!

I think it can, and I think it has --promising steps-wise. 1) Chaos with strange attractors. 2) Theory (and experiments) on open systems with structure formation or so-called self organisation. It is key in these systems that they are open --they exchange energy, momentum, and angular momentum with their surroundings. That's also another reason why I think the path is not following the traditional conservation laws, but other organising mathematical entities, perhaps based on hidden correlations. 'Hidden' here means we still don't know what they are.

Thank you. I suppose you mean logical reversibility... I can see no reason why you couldn't in principle prove (work for another supercomputer) that given that there are dogs, there must be something like atoms. Seems to me that it would be far more difficult to do, though. I don't know if that can be formulated as a theorem either. Suppose this: You feed the data to the supercomputer that there must be dogs. But dogs is not enough. You feed the data that there must be gut bacteria and black holes (interpolate from three different scales). The SC starts crunching numbers and.. bingo ==> There must be atoms that couple with a certain range of coupling constants, and so on. That's not impossible. Sounds reasonable. But atoms give dogs, and gut bacteria, and black holes seems inscapable. That's within the range of what I called 'in principle.' Yes, I'm aware of your observation before. The problem with this, I think, is that there are so few conservation laws that, as soon as the system goes above 3 degrees of freedom, you've got chaos. And chaos displays emergent structures (strange attractors, Studiot's catastrophic points --think phase transitions, etc.--) that are not related with conservation laws AFAIK, and I think it's safe to assume, as far as anybody knows. A discussion of what is a symmetry and what's not would lead us far too far, but I think has to do with why your point is missing something. Is there a way to interpret non-analitically-integrable variables as obeying some kind of hidden symmetry that we haven't been able to recognise so far just because it's not one of the garden-variety symmetries that we know and love?

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?