joigus

Not a lot, to me at least. For example, as a definition of 'energy': Energy doesn't always result in translation. OTOH, not an operational definition. As to 'action': Yes, good physicists are in that habit, because they have good reasons to think space is finite, and action in a confined space leads to quantised energy, for the simple reason that space-time confinement leads to a periodicity. Continuous energy is probably just a theoretical extrapolation. Same reason why angular momentum cannot be even conceived of but as quantised, because it's the conjugate momentum to an angle, which always restricted to a confined space \( \left[0,2\pi\right] \). Action again: Time, mass, and length are not derived from action. It's the other way around. As to the conclusion: This doesn't even make a smidgen of sense to me. I'm sorry. But I do have a sense of what the problem is with this kind of definitions/'derivations': They lack the operational point of view, on which all of physics rests, they engage in a loose runaway of concepts and statements, and consequently they lead to whatever preconfigured picture was already in the author's mind.

I know du Sautoy from the documentaries. 'The Self-Made Tapestry' does ring a bell, perhaps you mentioned it before. I have no idea about cement mixer patterns, but sounds interesting.

That seems to be the trend. The discontinuity of the fossil record is very important to keep in mind. I think we're in for more surprises. Dmanisi was another big big surprise in a different direction. Thanks for the link to Jebel Irhoud.

https://phys.org/news/2022-01-earliest-human-eastern-africa-dated.html?fbclid=IwAR3qOIHmKKO6EsILQUbh6DCngk5MjeQl3pI-ugX_n6yrHc5k-WPp5GhYkEM

But I don't disagree with this. In fact: Let me add another definition of maths that I've heard to Marcus du Sautoy, if I remember correctly: Maths is the study of patterns

Loved the poem. Thank you. Somehow I don't see zero so much as a geometric concept. I think it's more of an algebraic concept. You can do quite a bit of geometry without it (similarity of triangles, Thales theorem, Pythagorean theorem, and so on.) In a curvilinear space you don't really have zero curvature; and in a pseudo Euclidean space there are infinitely many points that have zero "distance" with respect to any one point. I think it's more of an auxiliary concept than really central to geometry. You can do some geometry without mentioning zero. You can't really start doing algebra or analysis without it. I don't disagree with this at all. Maths is a tool. And we'd rather use maths to make a hammer than use hammers to do maths. I even think maths is at the basis of language. Even people who say they hate maths, I think, have a simpler, more basic way of mathematically understanding the world. Perhaps less sophisticated, refined, or whatever.

Today I've learned about the phenomenon of chatoyance or chatoyancy: https://en.wikipedia.org/wiki/Chatoyancy and a beautiful sea snail called Voluta musica that displays this effect. It is an optical effect consisting in certain 2D patterns being perceived as 3D --if I understood it correctly. Thanks to @Genady and @StringJunky. 👍

Ok. I trust you and Euclid. But keep in mind that when Euclid wrote his Elements, there was no distinction analysis/algebra/geometry, so when he wrote that, he didn't make a clear distinction was probably trying to introduce the minimal elements of analysis necessary for doing geometry. Obviously you cannot do anything at all in maths if you don't start out with some elements of algebra and analysis. But zero is the distance between two points only when they're the same point, and as for a coordinate, it doesn't mean anything that its value happens to be zero. So I don't think zero is a relevant part of geometry. There is no 'zero point,' as opposed to the real number zero in analysis, or the element zero in a ring (algebra), etc. That was kind of my point --no pun intended. Edit (addition): On the other hand..., 'that which hath no part.' I don't know what to do with that. I don't think Euclid was in his finest hour when he wrote that.

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?