Genady

I agree with everything @MigL says here, with two small additions: what I refer to as "propagation of the field" is an idealized situation of a change from zero field to non-zero field = from no-field to field = from flat to curved; this is perhaps semantics I don't know if the answer to GR is necessarily Quantum gravity

Newton didn't ask for an explanation of gravity or for its fundamental underlying mechanism. My point is that we got the answer to what Sir Isaak asked for. The answer is GR. @Markus Hanke says, that GR is not final in its own right. Susskind, Zee, and other authors say that many, or most theoretical physicists think so. Just 150 years more, since 100 years have already passed ... @geordief says, that it is wrong to think of gravitational field as a physical object. I don't know what physical object means in this context. Are gravitational waves detected by LIGO "just a set of measurements in space and time" or a physical object? Didn't these waves mediate between those colliding stars and us? Gravitational field, "like any field" ( @geordief ), propagates with finite speed and carries energy. Doesn't this tell that it is (they all are) more than just a set of measurements in space and time?

I've found this 13 cm snail crawling at the depth of 3 m not far from shore. I didn't tattoo it.

Ahh, that problem... (my emphasis.)

The entire Analysis vs. Synthesis topic in this lecture takes 2 minutes, between 21:00 and 23:00. You can jump straight there: Stanford Engineering Everywhere | EE261 - The Fourier Transform and its Applications | Lecture 1 - Previous Knowledge Recommended (Matlab)

What problem?

Now go ahead and listen what he says about this topic.

I have already answered the point regarding the Fourier transform. Copying it here: 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. I will not reply any more to your questions about me, my knowledge, my education, etc. It is not your business, neither it is the purpose of the forums.

And I apologize for that. Yes, I took the course. I know that it was the only thing the prof referred to in Lecture 1.

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.