@István Lénárt: No, you can't say that - a movement along a curved trajectory is always! an accelerated movment (so far for the physics part).

@Anatoly A Khripov: your question has many point to be considered: first off all Euclidean geometry is a mathematical model, not a physical one (it just happens to describe a lot of physical models very well). Secondly an axiom, per definition cannot be proved or disproved, you can just build a description of nature in a way that it contains as few axioms as possible. Thirdly, Euclids geometry does not describe nature correctly in the sense that it allows you to have abitrary accuracy (btw., no physical model does that - that is the big difference between the mathematical models and the physical models. Mathematical models allow you, in principle, to be as accurate as you like due to their high degree of abstraction while physical models are always idealisations of natural systems), Euclidean geometry just happens to be accurate enough for a lot of physical descriptions, by far not for all, though.

An example: when constructing the LHC at CERN the engineers had to take into account the curvature of the earth already because the machine is so large. The same holds for most of the models in cosmology. If you look deep enough into physics you will see that actually most of the universe is not accurately enough described by Euclidean geometry.

Because we believe in Euclid, and build our world accordingly.

Newton says that no force acts on a point that moves on the plane along an Euclidean straight line with constant speed. He has no other reason for this assumption than his belief in Euclidean geometry. Consider a point moving on the surface of a sphere, and say that no force acts on it if it moves along a spherical straight line with constant speed. You can build an alternative world to Newton's; and it is up to you alone which option you prefer when solving a problem.

Dear Istvan Lenart,

You tell what I tell - Euclid geometry is JUST A PHYSICAL MODEL (or some mental theory) of space as Newton`s laws is a physical model of mechanical motion. Physics deals only with its physical models (which are absent in our real world!): material point, absolute solid body, fluid continuum,  etc. 

Why does Euclid geometry not exist in real nature (as material point, e.g.)? Because any object in real life do not satisfy two axioms: Archimedes axiom (which allows for infinite division of line segment, see Heisenberg inequality), parallelism axiom (for infinite lines it is simply impossible to check!). The second objective is because it is unknown whether we can apply rules of mathematical proof to real nature objects.

My question was as follows. Let us talk only about Euclid geometry not about physics. What causes has Euclid geometry that allows the correct description of our real space in many cases? Could you indicate ANY reason?

@István Lénárt: No, you can't say that - a movement along a curved trajectory is always! an accelerated movment (so far for the physics part).

@Anatoly A Khripov: your question has many point to be considered: first off all Euclidean geometry is a mathematical model, not a physical one (it just happens to describe a lot of physical models very well). Secondly an axiom, per definition cannot be proved or disproved, you can just build a description of nature in a way that it contains as few axioms as possible. Thirdly, Euclids geometry does not describe nature correctly in the sense that it allows you to have abitrary accuracy (btw., no physical model does that - that is the big difference between the mathematical models and the physical models. Mathematical models allow you, in principle, to be as accurate as you like due to their high degree of abstraction while physical models are always idealisations of natural systems), Euclidean geometry just happens to be accurate enough for a lot of physical descriptions, by far not for all, though.

An example: when constructing the LHC at CERN the engineers had to take into account the curvature of the earth already because the machine is so large. The same holds for most of the models in cosmology. If you look deep enough into physics you will see that actually most of the universe is not accurately enough described by Euclidean geometry.

Dear Sirs,

Thank you for your attention.  Johannes Gruenwald said "Euclidean geometry just happens to be accurate enough for a lot of physical descriptions". But WHY IS IT SO!! We are not small children, we need any explanations:)

All the study of geometry related problems of the real world are local study, that is study of the matter in a small part around the point of the problem. And localy Euclidean geometry is the best fitting geometry. But globaly that geometry nay not hold. The modern differentual geonetry of manifolds is also localy study with Euclidean geometry.

OK, then the grown-up answer: if you have a completely abstract system such as mathematics with a "subset" such as Euclidean geometry you can describe an infinite amount of things with it (because as it is totally abstract there is nothing from the physical world needed to work with it). Hence the probability that there is something from nature is among the infinite amount of things that can principally be described by the means of mathematics is one.  As all the subsets share the properties of the larger set this also holds for (Euclidean) geometry.

Dear Johannes,

I did not understand anything :):)...

In a nutshell - when you have a tool to describe everything imaginable then you will always succeed in describing the part of things, which is actually there in nature.

Sounds funny at least..:) But n-fold variety also started from some basic assumptions, like a metric. So as an example let us talk about Euclid geometry.

Where is the key, the reason which allows Euclid geometry to be so powerful in many real cases?

Or we do not know at all :) ??. We just make a hypothesis that some solid, like a table can be imagined in our mind as a thing constructed by infinite number of cords each of which is stretched between the two nails. So we decide to apply Euclid geometry as a geometrical model of the table. :(:(

But what are real criteria of application of Euclid geometry as a geometrical model of a real object? I know a few ones. And they are measurements of triangle angles sum (must be 180) by Gauss or later astronomers. Is that all needed test to apply Euclid geometry? Do you know other tests?

Maybe Euclid geometry should have macroscopic physical interpretation in order to understand its power? Hilbert said that his Euclidean axiomatic system can represent what you want, like shoes, etc. Dear colleagues, what is your interpretation :)??

I think both experience and mind produced axioms ( first the Human faces a problem in real life, then he tries to solve it by measure in general then the mind tries to find something to base on, which in general independent from the situation= invariant ), There are many famous axioms which form the base of Mathematics like Peano's axioms for the Natural Numbers, Euclid axioms for Plane Geometry,...

When we were children,all us proved easily that the vertically opposite angles are equal.  I just reproved it now, I found myself using the straight angle! ! but who told us that the straight angle is 180°?, who told us that the plane is 360°?

So the plane and the half of the plane are axioms.

@Anatoly A Khripov: Frankly, I am not sure what you want to hear as I gave you the answer already - it seems somehow that some colleagues here think that my answer was some kind of scientific joke but that's not the case - that is the explanation to your question.

Dear dr. Anatoly A Khripov,

I agree with dr. Johannes Gruenwald on the fact that Euclid’s geometry is a mathematics. However, it is better to distinguish between two different approaches to Euclidean geometry.

(1) Euclid’s own hypothetical-deductive approach.

(2) Plotinus’s understanding of (Euclid’s) geometry.

(1) To see that Euclid intentionally leaves out any concern about the appearance or essence of the physical reality, his geometric constructions, traced with the help of ruler and collapsible compass, should be compared with the cosmological models of his times. Although algebra is quite old – it is traced back to the Rhind Papyrus – Euclid apparently prefers a mathematical system combining demonstrable statements (theorems) and geometric constructions. In a certain sense, his geometric constructions can be considered as specific examples (ꓱ) of the general theorems (ꓯ). Thus, Euclid’s approach to the logical understanding of our world differs from algorithmic descriptions, and in particular from algebraic and differential geometry. Perhaps, its efficacy is worth of comparison with the algebraic and local descriptions.

(2) Plotinus is a Platonic philosopher. Since he tries to give a rational form to the constituents of the universe, he deals with numbers and geometric figures. He puts geometry at an intermediate level between the sensible reality and the realm of the forms.

In my opinion, Plotinus' interpretation of geometry should be kept distinct from Euclid's, even if sometimes the problems expressed by the former author seem nearer to the modern understanding.

Dear Sirs,

I just would like to receive any relating answer to my question "Why does Euclid geometry describe well physical phenomena"? What is this reason, key, or whatever? In this discussion at researchgate I did NOT receive any idea!!!

@Anatoly A Khripov: yes you did - as I said, I already gave you the answer.

Just an advice in the art of asking meaningful questions: If you ask a question and you get an answer, which you do not understand, it makes absolutely no sense to repeat the initial question over and over again, expecting different answers. It is much better to think about the answer that you don't understand and then ask further questions regarding the specific part of the answer, which makes you problems. Everything else is just turning in circles, getting you nowhere.

"What, then, are we to think of the question: Is Euclidean geometry true? It has no meaning. We might as well ask if the metric system is true, and if the old weights and measures are false; if Cartesian co-ordinates are true and polar coordinates false. One geometry cannot be more true than another; it can only be more convenient. Now, Euclidean geometry is, and will remain, the most convenient: 1st, because it is the simplest, and it is not so only because of our mental habits or because of the kind of direct intuition that we have of Euclidean space; it is the simplest in itself, just as a polynomial of the first degree is simpler than a polynomial of the second degree; 2nd, because it sufficiently agrees with the properties of natural solids, those bodies which we can compare and measure by means of our senses." H. J. Poincaré ˙(http://www-history.mcs.st-and.ac.uk/Extras/Poincare_non-Euclidean.html)

Let me, the author of the present note, add something to his arguments, not because I think that I am greater than Poincaré, but because I do think that it is my right, it is my duty, to add new thoughts to the thoughts of any other human being who does not consider him/herself an infallible demigod. Probably I disagree with Poincaré and Anatoly A Khripov, because, after my Euclidean education at home and in my schools from primary to tertiary, I fell in love with spherical geometry, fabricated spheres to experiment with, and later on, found manufacturers to produce much better construction materials for spherical geometry. Step by step, my mental habits and intuition suffered a sea- change into another world of geometry. In many cases, spherical geometry appears to me as simpler that Euclid's flat world 'just as a polynomial of the first degree is simpler than a polynomial of the second degree.'

One more remark addressed to Johannes Gruenwald without a touch of being offended by his note above: Dear Johannes, believe me, my remark about the possibility of a spherical mechanics did not happen because I got stuck in physics in the secondary school, but because the concept of Newtonian force is not palpable reality for me, but only a possible option of the human mind to explain certain phenomena of Nature. Imagine that you live on the surface of a perfectly spherical Earth-globe,  have never seen a Euclidean straight line, the term 'straight' means an arc of a spherical great circle for you. On this surface, a perfect spherically triangular dry land, say, India, is moving on the Indian Ocean towards Asia. Where is the mechanical center of this triangle? (Remember that you cannot fly off from the surface nor can you sink into it.)

Dear dr. Lénárt,

I agree with you that, as Euclid’s geometry is a hypothetical-deductive system, not a physics, it can be applied to our world in many cases, and in many other cases spherical geometry may be more convenient. And I also agree that “intuitiveness” is a question of experience. Yet, in my opinion, one trouble with the so called non-Euclidean geometries, as well as with spherical geometry, is that they were first discovered on purely axiomatic reasons.

At a difference with Euclid’s geometry, which prescribes constructions with ruler and collapsible compass, non-Euclidean surfaces are mostly represented analytically by isometrically embedding them in Cartesian space. Thus, it is impossible to represent non-Euclidean spaces isometrically in this way. That makes non-Euclidean spaces appear less intuitive and more arbitrary from a physical point of view than “Euclidean space” itself. I suppose that spherical geometry would become much more acceptable if there were an easy way to represent it.

Felix Klein gave an alternative model of non-Euclidean geometries, by embedding surfaces in projective geometry. Giving up projective metric, and simply stating that the non-Euclidean metric can be established by means of the invariant quadric, it is possible to embed a non-Euclidean space in projective space. This would render spatial non-Euclidean geometries in a visual fashion, and make they become more acceptable, if it were known how to independently visualize projective space itself. In fact, its embedding in Cartesian space requires more than three coordinates. For example, Plücker tried to introduce coordinates in projective geometry by taking the line element instead of the point as fundamental (then four coordinates are needed).

Still, since the projective geometry based on point-plane duality principle is a synthetic geometry, it should be visualizable independently on coordinates. Possibly, this question has not been faced yet?

@István Lénárt: I'm not offended at all but for me straight means straight - if you want to see a straight line just look down to your feet ;)

Regarding the movement along a curved trajectory - it is a measured physical fact that such a movement is an accelerated one.

Dear dr Sara Liyuba Vesely, thank you for your emphatic and correct comment. Since I do not want to digress from the point of the present conversation, I only remark that your question 'Still, since the projective geometry based on point-plane duality principle is a synthetic geometry, it should be visualizable independently on coordinates. Possibly, this question has not been faced yet?' is one of the main obsessions with which I have spent five decades. The pith of my answer is that the point AND the line together can be taken for the basic element of our geometry (a pair of Riemannian opposite points AND their Plückerian line equator, that is, a polarity on the sphere). We can build a new algebraic system on this foundation which has various models, including those built on coordinates over a field. This is just the reverse of the Plücker-Riemann-Lie-Hilbert way. Again, it has no meaning whether this approach is better than another one. It is just an option, and the user must decide which is more handy to solve a given problem.

1. Maybe Johannes Gruenwald`s (and people agreed with him) answer essence sounds like the numerous successful experiments confirm the usefulness of Euclid geometry. Is that all you can say on my question? Please correct me if I am wrong.

2. Stefan Gruner, István Lénárt said the straight line (for example) is an abstraction of the properties of real bodies. It is the key for the successful use of Euclid geometry in real life. Could you explain your ideas a little bit more! Please correct me if I misunderstand you.

3. By the way I read somewhere at Internet that Poincare said that the straight line is the rotation axis of the body. So let you rotate some object and you understand what straight line means.

Dear Anatoly, let me relate an anecdote in this regard. A friend of Gauss complained how loosely the concept of straight line was treated in scientific literature. Gauss answered as follows: 'Yes, the straight line is treated in a shamefully loose way. A straight line is a line that revolves in itself.' Bolyai junior (who was angry with Gauss) read this remark and added the gloss: 'This wording is incorrect, because the revolution of a straight line has no meaning at all.' Whatever our opinion might be in this question, the fact is that two geniuses did not agree on the definition of a straight line less than two hundred years ago.

@ Stefan Gruner: Far from being well versed in philosophy, I can only add subjective remarks to your challenging arguments. In accord with what you called 'Aristotelianism', I think that abstraction can only be derived from concrete experience. You call Riemann geometry 'highly NON-intuitive' - why? Why do you think that Euclidean geometry is more intuitive than Riemann geometry? The spherical surface is finite, in contrast with the infinite plane, so Euclidean plane geometry needs far greater abstraction at the very beginning than spherical geometry. The same goes for the infinite straight line on the plane and the finite great circle on the sphere. I think that  Riemann's great achievement was to step out from a two-thousand-year-old tradition, and create new (but not exclusive) options to the already existing systems. If only Menelaosian spherical geometry had been accepted as 'intuitive' for many centuries, Riemann would have revolutionized our thinking by discovering Euclidean geometry.  From this point of view, it is very interesting to consider other cultures which developed under different traditions and convictions. For example, in traditional Japanese temple geometry (Sangaku) the planar circle played a much more important role (as compared with the planar straight line) than in European geometry. This shows that intuition is very closely connected to our traditions and experiences. For me, the hardest question in all sciences is to decide what is simple and what is complex or difficult.

@Stefan: I understand your reasoning, but disagree with your conclusion. I find the basics of spherical geometry and Riemann geometry comprehensible (not easy, because Euclidean geometry is not easy for me, either), because in my young years I was eager to find new ways, new ideas, and began to experiment on wooden balls, plastic spheres, round fruits to discover a geometry different from the one taught in my schools. These experiments changed the meaning of the words 'easy' and 'difficult' for me. Menelaos wrote Sphaerica around AD 100 which indicated the same historical era from our perspective. It is really challenging to ask why plane geometry became dominant, considering the fact that the spherical form is far more frequent in our natural environment than the flat surface. My assumption is that it was the flat sheet of paper (on which most of the scientific and educational information became accessible) that gave prominence to Euclidean geometry for two millennia.

Dear dr. Lénárt, and dear dr. Gruner,

as I am interested in your exchange of opinions, let me add some points.

In my opinion, Euclid’s geometry is a hypothetical-deductive system embracing the whole of preceding geometrical knowledge. It prescribes the “logical” use of ruler and collapsible compass for constructions. In fact, other mechanical instruments were already used to deal with the so called Delian problems (http://www.cut-the-knot.org/arithmetic/antiquity.shtml). I suggest that the “experimental” side, that dr. Gruner sees in Euclid’s geometry, consists in the fact that real drawings (on paper, as dr. Lénárt stresses) require a real ruler, and a real compass.

In Euclid’s times, there was already a Cosmology (https://en.wikipedia.org/wiki/Anaximander). But, apparently, Euclid didn’t relate his geometry to it.

There were also writings about spherical and projective geometry (https://en.wikipedia.org/wiki/Euclid%27s_Optics). But perhaps, they were ahead of their time. Anyway, it seems that they were not pursued at the time.

On the opposite, there were no differential geometries. In my opinion, Descartes rethinking of geometry was a necessary premise for their development.

@Sara Liyuba, @Stefan: In a previous message I proposed to consider geometry in other cultures. Maybe there exists another option if we want to clear up the development of a geometric system from experience to abstraction. The origins of Euclidean plane geometry and Menelaosian spherical geometry are buried in the distant past, but the formation of still another geometry, hyperbolic geometry, is much better documented.  With the excitement of a detective story we can follow the way as geniuses of the past (Proclus, al-Haytham, Omar Khayyam, Saccheri, Lambert, Legendre and others) tried to get rid of their Euclidean fetters. Only a new wave of researchers (Gauss, Bolyai, Lobachevsky) succeeded to break through the wall. I suppose that the same or similar route had been followed through earlier millennia until Euclidean and Menelaosian systems of geometry were born.

One more remark about the Aristotelian and Platonic conception - as far as I am capable of grasping them. Maybe each researcher in math (or any other creator of new knowledge) has to decide for him/herself whether new results come from experience, or express eternal-immutable ideas in adequate form. As for me, I think that new results, if any, came to me via Aristotelian experience (with effective help of intuition). The only Platonic eternal-immutable idea in the process has been  the desire to change, to discover, to improve, 'To follow knowledge, like a sinking star, Beyond the utmost bound of human thought.'

Just one more remark, after two years of this long and - as I believe - worthy discussion. A few days ago, one of the greatest historians of the 20th century, Hayden White died at 89. A quote from Wikipedia about his teachings: " He claimed that the manifest historical text is marked by strategies of explanation, which include explanation by argument, explanation by employment, and explanation by ideological implication. He argued that historical writing was influenced by literary writing in many ways, sharing the strong reliance on narrative for meaning, therefore eliminating the possibility of objective or truly scientific history. White also argued, however, that history is most successful when it uses this "narrativity", since it is what allows history to be meaningful." Well, I firmly believe that his arguments apply to geometry as well. Given the infinite flat sheet of a plane, or the surface of a sphere or a saddle, with infinitely many possible shapes in it, we are free to choose any set of elements to be called as "points" or "circles" or "triangles", and build up a system on the concepts, just as White speaks about the historical "facts" and the "narrative" that the historian invents to connect these "facts." In this context, the "basic elements" of the mathematician corresponds to the "historical facts" of the historian, and the mathematician calls "a system of axioms" what the historian calls "a narrative." Every system of thoughts, be it geometric or historic, is an attempt of the human spirit to make some human order in the world; and no system of the human spirit can boast with being the only possible explanation of the world - including Euclid's magnificent system.