joigus

Ditto. We can only attempt at an explanation when we have a theory (a law), plus its domain of applicability. We can take Newton's law of gravity, or GR, and try to explain planetary motion, or why the Moon is slowly sidling away from us by tidal interactions. To summarise, we can explain relatively complex phenomena in terms of simple laws by means of a mechanism spelled out in terms of that simple law. But fundamental physical laws have no mechanism. Richard Feynman The Feynman Lectures on Physics Vol. II

In geometry, there is no zero. That's either algebra or analysis.

Thank you. https://en.wikipedia.org/wiki/Chatoyancy From 'cat's eye.'

I don't seem to be able to find it now. It must have been a university professor. What I have been able to check though is that many people seem to factually use that dichotomy of mathematics into analysis and algebra, because there are many questions concerning it. It's true that geometry is a completely different animal. I would say that the starting point for the three of them is axioms --that's just what mathematics does. But both analysis and algebra posit them as identities based on definitions, while geometry posits them as giving rise to formulas that claim to state relations in a world of 'visual entities' (Pythagorean theorem, Thales' theorem etc.) Those belong in the realm of perception, or intuitions about space, I would say. From these reasonable intuitions, you are compelled to transform them into algebraic statements that you later use to derive further theorems, and solve problems.

Is there any way that you could identify the species?

I've been pondering this for quite a while. I'm not enamoured of the 'inverse' approach, but I don't think it's impossible. But the way I see it, you would need more 'points' to interpolate. Levels of self-organization appear at different scales --already I know @studiot doesn't find this plausible--. Let's say: cells, multi-cellular organisms, planetary biota, and so on. At stellar level you would reach the point where no longer is there self-organisation. Instead, what you get is qualitatively different systems that, not only don't give rise to self-organisation, but actually erase information from their environment, and give it back to the universe completely thermalised (collapsing stars). I'm not saying it's plausible, I'm just saying the next Boltzmann of this world may be able to outline something like that. Very interesting. Let me keep thinking about this. One difficult aspect about the principle of least action is that, while its application is quite useful and simplifying in many cases, its meaning is obscure at best. It's a very abstract principle of physics.

Gorgeous. Looks like Minoan art.

Yeah, you're right. I forgot about geometry. I kind of think of geometry as a whole different class of its own. In fact geometry went through a process of trying to unify it all of its own, the Erlangen program, by Felix Klein. To me geometry is kind of a bridge between physics and pure (abstract) mathematics. But I don't know really. I think Poincaré tried to base all of maths on the concept of group. I'm not an expert, but I don't think he was successful. Poincaré and I think otherwise. And Euler agrees with us. Now serious. I wish I could remember where I picked up that dichotomy into algebra and analysis. I'll look it up. And this doesn't give me much hope: https://math.stackexchange.com/questions/1392273/algebra-and-analysis

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.)