Genady

Einstein is not an idiot. Einstein is not, period. Just like that famous parrot: 'E's not pinin'! 'E's passed on! This parrot is no more! He has ceased to be! 'E's expired and gone to meet 'is maker! 'E's a stiff! Bereft of life, 'e rests in peace! If you hadn't nailed 'im to the perch 'e'd be pushing up the daisies! 'Is metabolic processes are now 'istory! 'E's off the twig! 'E's kicked the bucket, 'e's shuffled off 'is mortal coil, run down the curtain and joined the bleedin' choir invisible!! THIS IS AN EX-PARROT!!

But the second part in Dark Energy is Energy. That is what goes into that formula, Masha.

If I'm allowed to speculate here, I could imagine that in that hot dense particle soup, before Higgs and before symmetry breaking, all particles were massless. Thus, time was "frozen", like the time of a photon is.

Yes, space has mass. Substitute a dark energy for E in E=mc2 and you get the space mass. PS. As my math teacher used to say, "For my every question he has his every answer."

Why is it often said that "time itself started with the Big Bang"? If you take the scale factor in the Friedmann–Lemaître–Robertson–Walker metric to zero, then the spatial component goes to zero, but not the temporal one. In other words, space contracts, but nothing happens to time. Or, in the words of A. Zee, "In our current description, space is created at the Big Bang, but not time." (Zee, A. Einstein Gravity in a Nutshell: p. 787). So, any idea from where the notion of beginning of time in BB comes from?

Step 1: 48 + 68 + 98 = 22*8 + 28*38 + 32*8 Step 1.5: = (28)2 + 28*38 + (38)2 Step 2: = (28)2 + 2*28*38 + (38)2 - 28*38 Step 2.5: = (28 + 38)2 - (24*34)2

Yes, the third one. There is one more after that. Thanks a lot. Right. Or by endlessly repeated (esp. in a pop-science) reference to "mathematical beauty".

This is all correct. The overlap between Mathematics and Physics is far from being complete. Especially, from the Mathematics perspective (I might be just a bit biased ). Mathematics is much-much larger than the part used in Physics. The OP, however, was why there is such an overlap at all and quite a big and essential one, from the Physics perspective.

On the other hand, sometimes these numbers turn out to be not requiring any theoretical basis. For example, Kepler worked hard to find a geometrical principle explaining why Solar System has five planets. Of course, it turned out that there are more than five, but moreover, it turned out that this number as well as the planets' given orbits, are just an incidental consequences of the Solar system evolution.

The highly successful Standard Model has about two dozens of such numbers that need to be just put in by hand. One of the criteria for a good theory Beyond Standard Model is to have fewer of those.

Yes, this is one direction of thought. For example, see the reference here: I don't like this direction of thinking. I rather think in the opposite direction, similar to Penrose and to this: In other words, it is not math that corresponds to the physical world, but rather physical theories that correspond to math. Physical theories, to be good theories, need precision, rigor, clear concepts, they need to be free of human prejudices, psychology, vague language, etc. Math has exactly these attributes. Physical laws are described mathematically by necessity. To this argument (which is Penrose's one, not mine) I add a bit of biology. For example, humans, like all other mammals, have an area of the brain responsible for computing a "mental map" of the surroundings. It evolved because it gives obvious advantages to the organism. And of course it computes specifically 3D maps. Why would it do any other kind? We never needed to deal with any other kind. Thus, we are good in visualizing in 3D. Now, studying phenomena beyond our immediate experience, we stumbled upon other kinds. We cannot use our visual imagination and intuition, and our everyday language, to describe them. Mathematics is the only available tool.

May I reply point by point? Starting with the last point? Thank you. I didn't mean it as an offence. I didn't mean to shut you up. Please, continue commenting and explaining. Am I obligated to reply? That is what I think is not always necessary, is it? May I reply only on comments that are of interest to me? Somebody else might be interested and reply on other comments. In my opinion, discussions, unlike arguments, don't have to end in agreement or disagreement. They can just give a food for further thought.

Gives a pretty good idea on human social life and attitudes in the Bronze Age.

I am a new member, but already have benefited significantly by the forums. Posting my questions and answering or replying to others, have helped me to clarify my thoughts, to get new insights, and in one case even to solve an old problem. Understandably and unavoidably, there is a lot of noise there. It is not too difficult to filter it out and to get a positive effect. Thank you to the members and to the moderators.

Thank you, I will check it. Here is one of many things Penrose has to say about Platonic world of mathematics: This was an extraordinary idea for its time, and it has turned out to be a very powerful one. But does the Platonic mathematical world actually exist, in any meaningful sense? Many people, including philosophers, might regard such a ‘world’ as a complete fiction— a product merely of our unrestrained imaginations. Yet the Platonic viewpoint is indeed an immensely valuable one. It tells us to be careful to distinguish the precise mathematical entities from the approximations that we see around us in the world of physical things. Moreover, it provides us with the blueprint according to which modern science has proceeded ever since. Scientists will put forward models of the world— or, rather, of certain aspects of the world— and these models may be tested against previous observation and against the results of carefully designed experiment. The models are deemed to be appropriate if they survive such rigorous examination and if, in addition, they are internally consistent structures. The important point about these models, for our present discussion, is that they are basically purely abstract mathematical models. The very question of the internal consistency of a scientific model, in particular, is one that requires that the model be precisely specified. The required precision demands that the model be a mathematical one, for otherwise one cannot be sure that these questions have well-defined answers. If the model itself is to be assigned any kind of ‘existence’, then this existence is located within the Platonic world of mathematical forms. Of course, one might take a contrary viewpoint: namely that the model is itself to have existence only within our various minds, rather than to take Plato’s world to be in any sense absolute and ‘real’. Yet, there is something important to be gained in regarding mathematical structures as having a reality of their own. For our individual minds are notoriously imprecise, unreliable, and inconsistent in their judgements. The precision, reliability, and consistency that are required by our scientific theories demand something beyond any one of our individual (untrustworthy) minds. In mathematics, we find a far greater robustness than can be located in any particular mind. Does this not point to something outside ourselves, with a reality that lies beyond what each individual can achieve? Penrose, Roger. The Road to Reality (p. 12). We do have quite different views on mathematics, I think. My view is much closer to the one of Penrose, which is partially described in the quote above. Here we agree. And Eugene Wigner thought so as well. His article was just for posing a question to be answered. I disagree with your answer. I do agree with Penrose's attitude about mathematics and extend it to answer the Wigner's question. You have said what you wanted to say, and I have said what I wanted to. I don't think we need to keep going circles. Maybe somebody else will contribute something different.

I read that you talk here mostly about applied mathematics. It is not a part of the question. The last sentence is about mathematical processes such as proof. This also is not a part of the question. The mathematical results are. As I've answered in the beginning of the thread, the topic is, mathematical concepts. P.S. Could you please limit your posts to one question at a time? This would help to stay focused, and might eliminate a need for some other questions.

Yes, you are right about my English, and thank you for calling it "very good." I think I understood your other questions. I didn't reply, except for the "not numbers" one, because I did not understand how they are related to the topic. For example, "...they are not fundamental concepts...". I don't think so, but it just doesn't matter here. They are mathematical concepts, that is the point here. Or, "what about the (physical or engineering) subjects Mathematics cannot tackle ?" The question is about the fundamental subjects where math is extremely effective and necessary, not about other subjects. "What about the difference between synthesis and analysis ?" I don't know. What about it?

Examples of not numbers? Sure: Riemannian geometry and GR Linear algebra and quantum mechanics Group theory and elementary particles

Each example separately is not unreasonable. What is "unreasonable" (I'd rather say, asks for a root cause explanation) is the deep connection between these two originally not connected worlds, mathematical concepts and physical phenomena. No, it is not related to Euler's identity.

The Gaussian integral is not necessarily related to circles, spheres, and periodic phenomena. However, the number π is there:

Thank you again. I also tend to think so, but I don't have a proof, so good to have a supporting opinion :).

Let's take another example, number π. It appeared in math while investigating circles and triangles. But it kept and keeps popping up almost everywhere in math and physics, in places that have no direct (immediate, obvious) connections to circles and triangles. I think, with complex numbers, the π, and many other mathematical concepts we just stumbled upon something big and important -- just like looking for a shorter way to India, finding America. BTW, there are uncountably infinite number of transcendental numbers, but only two appear everywhere (almost) in math and physics, π and e.

It is strange that the article doesn't mention Plato. This is the idea of Platonic world of pure concepts. (I assume the Roger Penrose's take on it.) It exists (somehow?) by itself. In the sense that, for example, there is infinite number of prime numbers, regardless if we know that or not, and moreover, regardless of existence or non-existence of the Universe. Mathematics is an investigation of that world. Yes, he marveled at that as well. But he argues, and I agree with this, that mathematics is not developed to model them, physics does. Let's take a simpler example than Hilbert space, complex numbers. They were not developed in math to model any pattern in nature. But they are absolutely essential to describe quantum laws.

Step 1: 48 + 68 + 98 = 22*8 + 28*38 + 32*8 Step 1.5: = (28)2 + 28*38 + (38)2 Step 2: = (28)2 + 2*28*38 + (38)2 - 28*38

Do you mean you want to know my take on it? Yes. Wigner asks, why. It is not generally developed as such by mathematicians. First, I think he had exaggerated in the title, "Natural Sciences". Most natural sciences rather use applied mathematics, for obvious reasons, we need to calculate things just like in engineering. So, the question is limited not even to physics, but to fundamental physics. This is where the fundamental laws are intrinsically mathematical, are based on purely mathematical concepts, such as Hilbert space in QM.