Genady

I didn't say and didn't mean to say anything about "fundamental" concepts. Yes, some concepts are much more fundamental than others. However, concepts can also be nested. I think that the Fourier transform concept is deeper than just a skillful operation. There is a very good class in Stanford, EE261 - The Fourier Transform and its Applications. The first half is about the concept. Stanford Engineering Everywhere | EE261 - The Fourier Transform and its Applications

Ha-ha-ha! @Yevgeny Karasik and @Euan Taras are one and same person! Yes, I do. Don't.

We, i.e. Wigner and I , consider examples of "concepts" such as: complex number, Dirac function, Lie algebra, metric, Fourier transform, vector space, etc. The focus in this "definition" of pure math is: a) inventing a concept, and b) skillfully operating with it to get deep and rich theorems. I am not sure if there is a correspondence between these two parts and the two attitudes in your "definition." It doesn't have to be.

I think this is about right: You cannot do much with poor concepts. But invent a good concept and you can go very far by skillfully operating with it. Actually, it is not very different from what you've said above, "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." Wigner calls it "concepts" rather than "assumptions" and I agree with this.

I understand that "this purpose" refers to "skillful operations" rather than "mathematics."

E. Wigner said it so well, I won't try to do better: "I would say that mathematics is the science of skillful operations with concepts and rules invented just for this purpose. The principal emphasis is on the invention of concepts. ... The depth of thought which goes into the formulation of the mathematical concepts is later justified by the skill with which these concepts are used."

There is another possibility in addition to the two directions of explanation being either bottom-up or top-down. The explanations of the laws on all levels could come from the same direction or from the same principles. For example, symmetry principles. Such principles are level independent and apply to all levels of phenomena, i.e. dogs, atoms, galaxies, etc.

Just want to share what the great man wrote about it: "That gravity should be innate, inherent, and essential to matter, so that one body may act upon another at a distance through a vacuum, without the mediation of anything else, by and through which their action and force may be conveyed from one to another, is to me so great an absurdity that I believe no man who has in philosophical matters a competent faculty of thinking can ever fall into it. Gravity must be caused by an agent acting constantly according to certain laws; but whether this agent be material or immaterial, I have left open to the consideration of my readers." (It stayed open for 250 years.)

If a research paper is good, this is about the right amount of time. Comprehending a good research paper should take time. However, my experience is from different fields than yours, so maybe it doesn't apply.

Sorry, no, I cannot. This is the only characteristic of emergence I assume: not to be a result of direct application of other laws, but to be a consequence of some elaborate and not obvious mathematical construction built upon other laws. In the examples above these constructions are: statistics, Noether theorem, antisymmetry of fermionic wavefunctions, and Lorentz invariance of wave equations.

They are emergent, according to the proposed approach, because they are consequences of some mathematical "hocus-pocus" rather than straightforward results of other laws. No, I cannot clearly define "hocus-pocus" vs. "straightforward". Maybe these are not objective attributes, but rather how the things appear to us.

I think so as well. I propose here a different approach: A phenomenon appears emergent to us if it emerges as a consequence of some mathematical "hocus-pocus" rather than a straightforward result of other laws. Such are all statistical phenomena: entropy, statistical distributions, blackbody radiation, Bose-Einstein condensate, superfluidity, superconductivity, ... Such are conservation laws emerging, via Noether theorem, from symmetries of the system's Lagrangian. Such is even a familiar phenomenon that solid materials do not collapse in on themselves or on each other. There is nothing emergent about it when it is erroneously explained by electrostatic repulsion of outer electrons in atoms. But it is emergent when viewed correctly as a consequence of Pauli exclusion principle, which in turn is a manifestation of antisymmetry of fermionic wavefunctions. Existence of mater and anti-matter is an emergent phenomenon too, because it is a required in order for quantum wave equations to be Lorentz invariant.

Mind affecting brain would mean that atoms and molecules disobey laws of physics, unless mind is a new fundamental field. (Exploring implications as well.)

The same with math and physics. If one reproduces a derivation with their own hand, it gets imprinted in memory forever. In most cases, you don't need to memorize a formula - you can just quickly rederive it.

Is there a way to predict a next branch of the tree in biological evolution?

Absolutely right. However, one shouldn't get an A for compiling works of others, unless such compiling is a goal of the assignment, should they?

The thread about wire black corals' chirality is done, but here is another picture I took back then:

G: ... if we had a gigantic computer which could simulate evolution of a system of billions molecules of water, a macroscopic behavior of water would appear in the output. M: Well, this is what we assume (of course with good reason) would happen - but how can we show this? G: By building such a simulation, or an approximation, and let it run. There is a very successful computer simulation of the Universe evolution, with creation of filaments, voids, super clusters and such, with 2.1 trillion “particles” in a space of 9.6 billion light-years across for more than 13 billion years.

Wow! It's a small world after all. This person used to be a classmate of mine around 50 years ago.

Sad. On the other note, my daughter was in Grade 12 in 1995... Now I'm starting to see why these forums are more interesting than some others.

I have tried several projects in Zooniverse. Was not impressed by a depth of their science. Simple surveys of images. I wonder if there are published research papers where the authors acknowledge citizen scientists' contribution or even specifically Zooniverse.

To make clear: not the university profs, but the high school teachers who were students in the program. Nice trick. And good for you. But this is a relatively innocent cheating. What I saw was steeling and getting credit for other people's work. Some plagiarism was just ridiculous, like copying stuff from Wikipedia. But some other was "ideological". E.g. a woman was caught copying paragraphs from published papers on biological evolution. It was so bad, that she was expelled. In her last message on the board she said that she doesn't care because she doesn't believe in this bs anyway. 50 something years ago? Looks like we are about the same age.

There is no question here, but I'd like to see comments. It is about my unpleasant experience in an M.Sc program in one of the US universities. Not an ivy league school, so relatively inexpensive. I was an out-of-state student, I guess it was even less expensive for the in-state ones. About half of the class were regular kids while another half were high school teachers who needed the degree to be able to teach in a higher education. I didn't have any prior experience in US schools, so maybe I'm not going to say anything new, but it was completely unexpected for me. The program was good, the professors were excellent, but the students... many of them routinely plagiarized in their work. In the beginning of the program everyone got a paper with explanation of plagiarism and expected degrees of punishment. I guess, the school knew about the problem. Everyone signed a statement of understanding, but they plagiarized anyway, and some actually were caught and punished. Not all, though. So, if teachers did it, it is to be expected from their students. And it is to be expected to spread out and to become a part of the culture...

It was not my answer, unfortunately. Somebody has shown it to me, and I was curious to know, how easy it is to find it.

No, I don't. Here is why. The question is: are there such irrational r and s that rs is rational? Consider two possibilities. 1. r=sqrt(2) s=sqrt(2) If rs = sqrt(2)sqrt(2) is rational then this answers the question. 2. If sqrt(2)sqrt(2) is irrational, then r=sqrt(2)sqrt(2) s=sqrt(2), and rs is rational, answering the question.