Genady

No, I disagree with your addition to my statement, namely "In other words the applied maths."

"Now how about properly addressing my points and making a discussion of it ?" Let's do. "Well I think you implied that in your words..." -- This is in reference to "fundamental". No, I did not. "Pure Maths, by definition, does not include Applied Maths, which you also referred to in relation to 'concepts'." -- Agree with the first half. Regarding the second part, No, I did not. "... "skillfully operating with it to get deep and rich theorems" In other words the applied maths." -- This, as you said in your last post, "suggested they are applied maths." They certainly are not. The whole section about the Fourier transform. -- You don't like to call it a concept. Just cross it off my list of examples. It is not too dear to me. "Lecture 1 refers to 'analysis' and 'synthesis'..." -- It refers specifically to the direct and inverse Fourier transforms. "... my main objection to your 'rule based' approach ..." -- I don't find "my approach" to be an interesting subject for discussion.

"Your 'definition'" in my post above was in reply to the previous post by @joigus, not @studiot. I don't think that pure mathematical theorems such as Pythagoras is applied math.

Newton asked about an agent mediating an act of one body upon another, conveying action and force from one to another, acting according to certain laws as opposed to an "innate" action at a distance. Today we know such an agent. The agent is gravitational field and it acts according to the laws of GR.

Please, do, and let me know. I want to read more about it.

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.