Genady

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.

From the article I think he means, mathematical concepts. The "miracle" is its appropriateness etc.

62 years ago, on February 1960, Eugene Wigner concluded his article of the above title: "The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve." I think it might be interesting to discuss, if there is a better understanding of this miracle now. (Leave the issue of deserving it alone, please )

I did. There is no contradiction. In "differential geometry of curves in three dimensions" their curvature is an extrinsic curvature. What I refer to is an intrinsic curvature, which is a curvature of a one dimensional curve itself, not as being embedded in a higher dimensional space. Generally, extrinsic curvature is intrinsic curvature plus something (could look up the formula.) Example: intrinsically, a plane and a cylinder are both flat, with zero curvature.

I don't think one-dimensional line may have a torsion. Seems to me you need some points off axis to twist them relative to each other. (I don't see a connection between the graphs and the curvature question, sorry.)

We need to distinguish between intrinsic and extrinsic geometry. GR and differential geometry refer to the intrinsic one. Intrinsically, all 1D manifolds are flat = have zero curvature = locally indistinguishable, regardless if they look straight or curved extrinsically, or even close on themselves like a loop.

Just a little anecdote here. I was once "attacked" by somebody for saying that CMB data indicate that our universe is perhaps flat. That person decided that I'm a flat earther.

Here it is: Isn’t it conceivable that spacetime is actually flat, but the clocks and rulers with which we measure it, and which we regard as perfect in the sense of Box 11.1, are actually rubbery? Might not even the most perfect of clocks slow down or speed up, and the most perfect of rulers shrink or expand, as we move them from point to point and change their orientations? Wouldn’t such distortions of our clocks and rulers make a truly flat spacetime appear to be curved? Yes. Figure 11.1 gives a concrete example: the measurement of circumferences and radii around a nonspinning black hole. On the left is shown an embedding diagram for the hole’s curved space. The space is curved in this diagram because we have chosen to define distances as though our rulers were not rubbery, as though they always hold their lengths fixed no matter where we place them and how we orient them. The rulers show the hole’s horizon to have a circumference of 100 kilometers A circle of twice this circumference, 200 kilometers, is drawn around the hole, and the radial distance from the horizon to that circle is measured with a perfect ruler; the result is 37 kilometers. If space were flat, that radial distance would have to be the radius of the outside circle, 200/π kilometers, minus the radius of the horizon, 100/π kilometers; that is, it would have to be 200/π – 100/π = 16 kilometers (approximately). To accommodate the radial distance’s far larger, 37-kilometer size, the surface must have the curved, trumpet-horn shape shown in the diagram. If space is actually flat around the black hole, but our perfect rulers are rubbery and thereby fool us into thinking space is curved, then the true geometry of space must be as shown on the right in Figure 11.1, and the true distance between the horizon and the circle must be 16 kilometers, as demanded by the flat-geometry laws of Euclid. However, general relativity insists that our perfect rulers not measure this true distance. Take a ruler and lay it down circumferentially around the hole just outside the horizon (curved thick black strip with ruler markings in right part of Figure 11.1). When oriented circumferentially like this, it does measure correctly the true distance. Cut the ruler off at 37 kilometers length, as shown. It now encompasses 37 percent of the distance around the hole. Then turn the ruler so it is oriented radially (straight thick black strip with ruler markings in Figure 11.1). As it is turned, general relativity requires that it shrink. When pointed radially, its true length must have shrunk to 16 kilometers, so it will reach precisely from the horizon to the outer circle. However, the scale on its shrunken surface must claim that its length is still 37 kilometers, and therefore that the distance between horizon and circle is 37 kilometers. People like Einstein who are unaware of the ruler’s rubbery nature, and thus believe its inaccurate measurement, conclude that space is curved. However, people like you and me, who understand the rubberiness, know that the ruler has shrunk and that space is really flat. What could possibly make the ruler shrink, when its orientation changes? Gravity, of course. In the flat space of the right half of Figure 11.1 there resides a gravitational field that controls the sizes of fundamental particles, atomic nuclei, atoms, molecules, everything, and forces them all to shrink when laid out radially. The amount of shrinkage is great near a black hole, and smaller farther away, because the shrinkage-controlling gravitational field is generated by the hole, and its influence declines with distance. The shrinkage-controlling gravitational field has other effects. When a photon or any other particle flies past the hole, this field pulls on it and deflects its trajectory. The trajectory is bent around the hole; it is curved, as measured in the hole’s true, flat spacetime geometry. However, people like Einstein, who take seriously the measurements of their rubbery rulers and clocks, regard the photon as moving along a straight line through curved spacetime. What is the real, genuine truth? Is spacetime really flat, as the above paragraphs suggest, or is it really curved? To a physicist like me this is an uninteresting question because it has no physical consequences. Both viewpoints, curved spacetime and flat, give precisely the same predictions for any measurements performed with perfect rulers and clocks, and also (it turns out) the same predictions for any measurements performed with any kind of physical apparatus whatsoever. For example, both viewpoints agree that the radial distance between the horizon and the circle in Figure 11.1, as measured by a perfect ruler, is 37 kilometers. They disagree as to whether that measured distance is the “real” distance, but such a disagreement is a matter of philosophy, not physics. Since the two viewpoints agree on the results of all experiments, they are physically equivalent. Which viewpoint tells the “real truth” is irrelevant for experiments; it is a matter for philosophers to debate, not physicists. Moreover, physicists can and do use the two viewpoints interchangeably when trying to deduce the predictions of general relativity. Thorne, Kip. Black Holes & Time Warps: Einstein's Outrageous Legacy (pp. 400-401).

@Markus Hanke Got it. Thanks a lot! Could you also shed light on my "another thought" (in a post way above)? Here I repeat it:

Did you mean these "low prime factors" -- see the previous post? If so, I apologize. My mistake, I didn't understand you. You were right. From this point, you need only one "trick" more, and not a very uncommon one.

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

Yes, this is certainly so in the classical EM. However in QED these EM gauge transformations are coupled with local phase transformations of charged particles and this creates the mechanism of their interactions. This gauge symmetry I am asking about.

No, no, this thought has nothing to do with gravity (and I don't have any personal attraction to gravity, pan intended.) It is here only because it is about space, but this space is not affected by gravity at all. The "internal space" is a technical term in particle physics. In math it called "bundle space".

I don't think this is correct. The Friedmann equations (GR in a homogenous isotropic universe) have only expanding or contracting solutions, no static ones. Even adding a cosmological constant in GR didn't help -- it allowed only an unstable equilibrium.

And another thought. We've said, together with Thorne, that gravity can be thought as spacetime curvature or as effects on all clocks and rulers, and these two descriptions are equivalent. Yes, but... If we want to use QFT in gravity, there is a prescription of how to do so in a curved spacetime: replace partial derivatives with covariant derivatives, stick square root of metric determinant in Lagrangian (there is one more step, don't remember now) and you got it. How to do it if gravity does not curve spacetime, but rather affects rulers and clocks?