Dear Jean-Claude,

You asked:

"... What is the nature of the connection between Euclidean geometry and differential geometry? "

The answer is quite simple:

• All kind of geometries are locally Euclidean

Formalism is different, but, as you know, the differentiation as mapping is a linear one, so everything is Euclidean...

@Demetris Christopoulos

OK, in your view "everything at the end is Euclidean" (I paraphrase)... So what would be your answer to the main question:

Is differential geometry more general or just complementary to Euclidean geometry?

Also I wonder from musing over that question: why do we apparently need differential geometry for a more thorough description of the natural world?

Thanks for your interest.

Dear Jean-Claude,

Differential Geometry is actually Euclidean Geometry after using Linear Algebra and more specifically: linear mappings.

What is the differentiation process?

Isn't it a linear mapping or not?

So, by using extra math tools we have generalized the concepts of Euclid in order to cover more cases...

Hello Mohamed Hassani. - You have framed the discussion for us in a very rigorous manner by bringing in the historical development of the three geometries that we have in mathematics as much as the specific subject matter that they are each dedicated to.

I must say that it is not always obvious that “L’Analyse”, as the French educational system has the Functional aspect of calculus (L’Analyse Fonctionelle), is really a geometric development in mathematics inasmuch as it represents a graphical study on the Cartesian plane of algebraic polynomials and the other functions. Which becomes quite clear under the denomination of Analytic Geometry as you rightly point out. It is also interesting to note that in the British system, calculus was originally named differential calculus at its birth, although this denomination now seems to put us closer to differential equations specifically which leads to Differential Geometry (DG).

So we seem to have a triplet development of Geometries in mathematics: Euclidean geometry, which was axiomatized at the turn of the 19th century and abducted into mathematics from physical science where it used to belong, Analytic geometry and Differential geometry. Much of the latest developments in DG have been undertaken by physicists in the pursuit of resolution of fundamental physical problems, which by the way they left unresolved for the most part, and with apparently very few contributions from modern-day mathematicians. In that trend, the relationship between Euclidean geometry and DG has not been, by my reckoning, properly studied or wrongly visualized to say the least.

If DG is essentially the study of the evolution and inter-dependencies of forms within surface sheets of n-dimensions, it does not seem to have been built on the establishment of primitives of forms, or a hierarchical order of irreducible forms if you will, which would confer the most solid foundations to the science. By contrast, our understanding of the scalar order in mathematics has given us Number Theory, which has elicited for us number systems, each and every one of which being endowed with a defining radix and ensuing calculational methods. Nevertheless, along with the scalar order, there does exist a geometric order in Nature. Euclidean geometry seems to give us better answers to this fundamental question than DG, with all the sophistication that it has over Euclidean geometry. The fact that DG is telling us that there is, i.e. more symmetry in a circle than there is, say in an ellipse or a parabola, is in my book a serious problem that reveals a blatant misunderstanding of the prime symmetries innate to the geometric order. One should remember this, the most extensive developments in DG achieved lately in mathematical-physics have not given us a single bit more knowledge about mother nature… Seems that DG needs to put its house in order, before it can truly help us resolve such problems as the true structure of coordinate space toward Unity in Physical Law, or help us visualize the true structure of the projected physical meta-dimensions. As a tool for implementation of virtualization and simulation in technology, DG is probably quite useful, but anything beyond seems to leave it in total exhaustion. Something with it must be seriously wrong or perhaps far too incomplete…

Dear Prof. Jean-Claude,

I wish to tell you and the other readers on this string that Prof. Einstein was the first according to me to have made a distinction but not Prof. Schroedinger in Physics.

Soumitra K. Mallick, Somak Raychaudhury, Sandipan Mallick

for RHMHM School of Mathematical Sciences Thinking

USA, Japan, India

Dear Joseph Jean-Claude,

The essence of your question is quite similar to my research domain.

First of all, let's understand the terms we use:

1. Euclidean geometry is related to certain mathematical structure - Euclidean space.

2. Analytic geometry refers to the methodology in studying of some structure by means of linear algebra.

3. Differential geometry refers to the methodology in studying of some structure by means of differential and integral calculus.

Non-officially Euclidean / non-Euclidean geometry differs from Euclidean / non-Euclidean space in the way these mathematical structures are constructed: from some axiom set (in case of the term "geometry") or be some algebraic model (in case of the term "space"). But either way a structure is constructed, it has the same properties. So, there is no distinction between Euclidean geometry and Euclidean space, other than logical foundation.

The same mathematical structure (Euclidean space) can be studied via different methodologies, including analytic geometry and differential geometry. On the other hand, different mathematical structures can be studied by the same methodologies. Namely geometries with infinitesimal structure like Euclidean space (elliptic and hyperbolic spaces) are usually studied by differential geometry for the reason they are not linear. Other linear spaces with different infinitesimal structure than Euclidean (Galilei and Minkowski spaces) are usually studied with analytic geometry, because the toolchain of differential geometry is essentially built from the needs of locally Euclidean spaces.

The illusion of superiority of differential geometry over analytic geometry has the historical cause. When Bolyai and Lobachevsky published their results on hyperbolic geometry, it was not considered to be connected with the reality. When Riemann published his works on differential geometry, he essentially established this connection.

Now the main question: Which methodology (analytic or differential geometry) is more appropriate for studying of the universe, in sense it can construct new mathematical structures, which resemble to physical reality? The answer is: None constructs something essentially new. If your objective is to find the way to construct new mathematical objects, you may want to refer to Klein's Erlangen Program. Unfortunately, modern geometry has too poor toolchain for studying many of these mathematical structures (e.g. De Sitter, Anti de Sitter spaces).

One approach of construction and study of all these spaces by means of analytic geometry can be found in my PhD thesis (Preprint Analytic Geometry of Homogeneous Spaces

Dear Joseph Jean-Claude,

Very good question indeed.

From analysis of the manner in which language allows the establishment of abstract concepts, Euclidian geometry turns out to be first level abstractions from generalization of observation of our environment.

Abstract localization in space was then introduced by adding the abstract concept of 3-dimensional orthogonal axes following the first observation by Descartes of the first level abstract Euclidian geometric concepts.

Observation of this 2nd level then allowed extending the 3D concept to infinity in both directions with the introduction of curved geometries by Lobachevski and Rieman.

Further geometries were then developed from observation of these third level abstract concepts.

So from this perspective, the answer to your question is yes. All non-Euclidian geometric concept can only be successive generalizations grounded on these first level nonverbal abstract concepts, themselves ultimately stemming from generalizations abstracted from perceptions of our senses.

Recent research has demonstrated that nonverbal mathematical abstract concepts are stored in the brain in areas different from the verbal areas, and also different from the areas where the nonverbal images stemming from the perceptions of our senses are stored:

Amalric, M. & Dehaene, S. (2016). Origins of the brain networks for advanced mathematics in expert mathematicians. Proc Natl Acad Sci U S A, April 2016

http://www.unicog.org/publications/Amalric%20Dehaene%20fMRI%20of%20math%20and%20language%20in%20professional%20mathematicians%20PNAS%202016%20plus%20SI.pdf

Best Regards

André

@Alexandru Popa - After reading your interesting thesis, I came to the view that what has been missing in Geometry is a study of the nature or essence of forms, while the perspective of space has remained prevalent. You emphasize structure as the inner element. We may well debate the relationship between the notions of structure and form, but I think that structure arises from form. If you emphasize structure, then space appears as a dimensional object that can readily be studied thru the nth-space hierarchy (even though nobody can produce a working visualization of any space beyond 3-space). But if you emphasize form, then the notion of state of symmetry becomes a geometric primitive as the primary descriptor of form. This notion may not have received all deserved attention from all three geometries, probably because it is a complete indiscernible, while structure or dimension is a discernible. Yet I hold that the very state of symmetry represents one of the two major components of both abstract and coordinate space in Geometry. I discuss state of symmetry, form and dimension or degree of freedom on p.2 to 5 of the referred paper as a primer. Also consideration to states of symmetry as geometric primitives may help us firmly determine which between differential geometry and Euclidean geometry is more generic or more fundamental than the other.

Dear Jean-Claude, in frames of Euclidean geometry we cannot describe gravitational interactions, because gravitation is linked immediately with temp of the time. The base of Special Relativity is pseudo-Euclidean space, where the time flows uniformly. The base of General Relativity is pseudo-Riemannian space, where the temp of time depends on gravitational potential. The time can be stopped (state of collapse) in the strong gravitational field. If this field is weak., the time flows almost uniformly.

Dear Joseph Jean-Claude,

"What is the nature of the connection between Euclidean geometry and differential geometry ?" Since your question is fairly well presented, let me follow your own lines for discussion. Before starting, we must agree on the usual definition of an n-dimensional Euclidian space, which is Rn endowed with the metric induced by the quadratic form x12 +...+xn2 .

1) "We know that Riemannian geometry generalizes Euclidean geometry to non-flat or curved spaces. Yet Riemannian spaces still resemble the Euclidean space at each infinitesimal point (in the first order of approximation)".

This is true, but needs to be made much more precise. I apologize in advance for the inevitable technicalities.The main object of study is a differential manifold of dimension n, i.e. a set M "locally diffeomorphic" to an Euclidian space Rn, in the sense that every point x of M admits an open neighbourhood Ux which is mapped via a smooth (i.e. Cinfty ) diffeomorphism fx onto an open subset of Rn. Of course the "local charts" (Ux, fx) must be compatible in an obvious sense, so that they can be "glued" together to provide an "atlas" describing the whole manifold. This may seem unduly sophisticated, but, after all, we use local charts and atlases everyday when moving on the surface of the Earth.

At every point x, the variety M admits a tangent vector space Tx(M) of dimension n. A Riemannian metric on M is a family of smoothly varying inner products (i.e. bilinear forms associated to positive definite quadratic forms) on the tangent spaces Tx(M). Riemannian metrics are thus infinitesimal objects, but they can be used to extend to M various the usual concepts based on length, area or volume . Of course, an archetypical example is given by a submanifold of a Euclidian space Rn (see also below).

But there is no reason to restrict ourselves to Riemannian metrics. The previous formalism allows us to define pseudo-Riemannian metrics in exactly the same way, just replacing the above inner products by non degenerate bilinear forms. A well known example in dimension 4 is the Lorentz form x2 + y2 + z2 - t2.

2) Coming back to Riemannian manifolds, I certainly cannot buy the over-simplifying assertion that "everything at the end is Euclidean" (even if it is a paraphrase), because the "local-global" apparatus introduces a new paradigm. This can be actually tracked back to Riemann, when a distinction was made between intrinsic and extrinsic in differential geometry. The intrinsic approach is via manifolds, as recalled above. Whereas from the extrinsic point of view, curves, surfaces, hypersurfaces, etc. are considered as lying in an ambient Euclidian space. Of course this distinction cannot be totally unequivocal, manifolds can also be used in the extrinsic approach, as (at least theoretically) a theorem of Whitney (resp. Nash) guarantees that any differential (resp. Riemannian) manifold can be embedded into an euclidian manifold RN for N large enough. But it seems rather unnatural, when studying a curved manifold, to begin by embedding it into a flat space. And all the more unnatural as, in many cases, such an embedding is not a structural part of the manifold. But the most convincing illustration of the intrinsic point of view, I think, comes from cosmology, this scientific discipline which proposes global theoretical models of the Universe. What is the Universe ? In ancient Greek, tò hólon ("all things"), or equivalently ho kósmos (the world, the cosmos). If we limit ourselves to the world of physics, an acceptable definition would be the set of all observables, i.e. "all physical quantities that can be measured. Also, physically meaningful observables must satisfy transformation laws which relate observations performed by different observers in different frames of reference" (Wiki.) In a non mathematical language, this is roughly the definition of a manifold ! And by the very meaning of tò hólon, this manifold cannot be embedded into any larger RN. The most well known (even to the layman) example is of course Einstein's 4-dimensional spacetime in the theory of Relativity, which is locally the flat space R4 with the Lorentz form, and globally the associated pseudo-Riemannian manifold.

3) "Is differential geometry more general or just complementary to Euclidean geometry ? " After all the considerations above, I think you should invert the terms of your question. Differential geometry is neither a generalization nor a complement of Eucldian geometry. Euclidian geometry, although it has been discovered first, is just a particular - and tiny - part of differential geometry ./.

Dear Thong Nguyen Quang Do,

Please have a look at Figure 2 of next:

Data A detailed critical review of reported event GW150914 that L...

You'll see that things are much simpler.

Everything is Euclidean...

Dear Demetris Christopoulos,

Not being an expert in physics, I cannot say anything valuable about your "critical review" of the Ligo/Virgo experiment. Being a mathematician, I wonder how you can say that "everything is Euclidian" since a Lorentzian manifold cannot be Euclidian. This is not an experiment here, but an axiomatic definition.

Dear Thong Nguyen Quang Do,

Please don't be lazy, I just asked you to view "Figure 2: General form of mappings between Differentiable Manifolds" of that publication.

You can see it here...

Thong Nguyen Quang Do - Say you: “But it seems rather unnatural, when studying a curved manifold, to begin by embedding it into a flat space.” This well reflects the fact that you mainly look at the question from a technical perspective, and understandably so because you are a mathematician. I look at the question both from a technical point of view and an epistemic view with a critical sense of differential geometry science itself.

If we acknowledge that dots, lines, angles, surfaces and volumes are the basic elements of Euclidean Geometry, then we ought to acknowledge as well that the only way that a DG point can have a neighborhood is for the dot to be intrinsically at least bi-dimensional (bi-linear) or somehow endowed with further Euclidean faculties. In other words, since a dot can never be even minimally bi-linear as a dot, it is only as a result of dimensional indiscernibility that a manifold irreducible can encapsulate meta Euclidean “geometries”, which are the only elements that can make up the neighborhood of the irreducible, at any level. I don’t think that the distinction between intrinsic and extrinsic space is very auspicious in DG, because it masks the question of the nature of the internal tapestry of space. I think the best proposition of DG is that of morphism and morphic evolution of surfaces, sheets or surface sheets. In abstraction, extrinsic space (dimensional order) must be made up of intrinsic space elements (states of symmetry).

If you view space as a foliate bundle and attempt to describe the irreducible of that tangent vector space, you will find that the foliate order must be composed of Euclidean conics and conic sections as well as hyperbolic curvatures, in the role of the primitive spectrum of that order. And that is the essential reason why “any differential (resp. Riemannian) manifold can be embedded into an euclidian manifold RN for N large enough. “ as you rightly report. In other words, and counter-intuitively so, curved space can be demonstrated to be a derivation of flat space (and not the other way around), because Euclidean metric is the mother of all metrics. I make a formal demonstration of this proposition in the referred paper from a function and functional approach.

I think your view of the matter sits along the line of dimensions or the dimensional aspect of space. I do not think that the dimensional order in the “global” perspective is the correct barometer to gauge the relationship between the two Geometries, simply because there is no extrinsic meta-space beyond 3-space in all geometries, such abstractions being representationally vacuous in DG and only useful for virtualizations to be implemented within the very 3-space, as I said before. That does not mean though that I reject the possible physicality of meta-spaces beyond 3-space, just that DG as is does not lead to those heights.

We can discuss the detail of the metrics, if you like. But that area is even more contentious…

Thank you for taking interest.

Cordially.

Hello Demetris Christopoulos

You say to Quang Do: “Please don't be lazy, I just asked you to view "Figure 2…”

I understand that we can quickly become impatient in public debates. But let’s keep respect to one another for the benefit of a good profitable conversation.

I have read a good chunk of your paper. I am following your argument…

Cordially.

Dear Demetris Christopoulos ,

I've overcome my lazziness to take a look at your Fig.2, but unfortunately I can't see where you aim to come to. As I understand it, your figure simply recaps compatibility conditions in the transition from a local chart to another, and this, for a general differential manifold M , with or without any additional metric (be it riemannian of pseudo-riemannian). We all agree, I think, that locally M behaves as an Euclidian Rn, but the main point is to glue all the local charts together, and the notion of manifold was created for that purpose. Loosely speaking, the local study allows to use differential calculus in the neigbourhood of a point, while doing global geometry requires , in addition, some way to relate the tangent spaces at different points - and the possibility to do so reflects partly the structure of the manifold. A typical example lies right at the start of the study of a riemannian manifold. I recall that a riemannian metric on M consists in a family of smoothly varying inner products (i.e. bilinear forms associated to positive definite quadratic forms) on the tangent spaces Tx(M). Why such a complicated statement when one could say intuitively that "all the tangent spaces behave like Euclidian spaces" ? Because such a loose formulation can be wrong , because there is in general no canonical metric on a riemannian manifold. Why ? When you say "Euclidian space Tx(M)", perhaps you think automatically of the classical form x1 2 +...+ xn 2 , and any positive definite quadratic form a1x1 2 +...+ anxn 2 on Tx(M) can be reduced (by the usual diagonalization process) to that one. True, but only in Tx(M). The transition from a local chart at x to another at y being imposed by the global structure of the manifold, there is no guarantee that it will transform the reduced form in Tx(M) into the reduced form in Ty(M). Actually the existence of a riemannian metric on a given M is not at all obvious, it depends on properties of M pertaining to topology (paracompactness) and differential analysis (partition of unity). Summarizing, and putting technicalities aside, an Euclidian space (very rigid) and a Riemann manifold (very supple), even if they are locally "the same", are not globally similar. That is why, I repeat, I cannot buy your motto that "at the end everything is Euclidean".

Another point concerning your Fig.2 : it shows how the local charts of a given manifold M are glued together, but it says nothing about how two manifolds can be compared. This concerns the so called functorial point of view, a big word to mean that in a category consisting of objects and structures, one must study not only the objects, but also the morphisms, which are maps which respect the structures. In the category of differential manifolds, morphisms must naturally respect the structure of the tangent spaces, so a Lorentzian manifold, as I said, cannot be isomorphic to a Riemannian manifold. And, at the end, the spacetime of GR is not euclidian, even locally ./.

Dear Thong,

Thank you for overcoming your laziness, I appreciate it.

Look, I don't want to bother readers with technical details, but my summary follows:

• since we are using vector spaces for mappings starting from any "X", we essentially are Euclideans, with a different way
• anything local defined in Calculus must obey the demand of differentiation which actually is again a kind of "local Euclideanization"
• if we abandon the differentiation requirement all Christoffel symbols for partial derivatives break down
• then the entire building of Riemannian applications like GR also breaks down...

Sorry, but everything is Locally Linear (ie Euclidean)

Only some general topological spaces are not, but currently no evidence for connection with natural reality exists...

So, it doesn't matter if you buy Euclid or not, reality simply doesn't care about you and me...

Dear Joseph Jean-Claude,

Although I feel uneasy when debating scientific matters using an imprecise language (this must be a professional quirk of mine), I'll try here to comment in detail your last message to me. I apologize in advance if in the course of the discussion I express here or there my frank opinion in a non censored way.

If we acknowledge that dots, lines, angles, surfaces and volumes are the basic elements of Euclidean Geometry, then we ought to acknowledge as well that the only way that a DG point can have a neighborhood is for the dot to be intrinsically at least bi-dimensional (bi-linear) or somehow endowed with further Euclidean faculties. In other words, since a dot can never be even minimally bi-linear as a dot, it is only as a result of dimensional indiscernibility that a manifold irreducible can encapsulate meta Euclidean “geometries”, which are the only elements that can make up the neighborhood of the irreducible, at any level.

Your assertions look at the same time vague and arbitrary. What do you mean by « intrinsically at least bi-dimensional » when you don’t even define the notion of dimension ? I guess you want to say that your dot must be taken in an affine space (which is just a vector space with a distinguished origin) in order to be able to talk of neighborhoods. First, note that neighborhoods are topological objects, not necessarily metrical, a fortiori not necessarily euclidian. I completely disagree with your assertion that meta Euclidean “geometries” are the only elements that can make up the neighborhoods of the irreducible [manifold]. Second, as recalled in my recap, neighborhoods can be defined on manifolds without embedding them into a vector space RN. You could theoretically perform such an embedding thanks to the theorems of Whitney/Nash, but this would not necessarily be related to a structural part of the manifold, as you can see with such un-intuitive manifolds as Grassmanians or configuration spaces. As for your dimensional indiscernibility, which I interpret as the indetermination of the dimension N above (am I right ?), it only stresses the un-natural, un-necessary character of the embedding process. This leads us naturally to the opposition intrisic vs extrinsic.

I don’t think that the distinction between intrinsic and extrinsic space is very auspicious in DG, because it masks the question of the nature of the internal tapestry of space. I think the best proposition of DG is that of morphism and morphic evolution of surfaces, sheets or surface sheets. In abstraction, extrinsic space (dimensional order) must be made up of intrinsic space elements (states of symmetry).

On the contrary, I think that it is the intrinsic study of manifolds which reveals, as you say, the internal tapestry of space. My favorite example is Gauss's Theorema Egregium on the the curvature of surfaces. the theorem states that Gaussian curvature can be determined entirely by measuring angles, distances and their rates on a surface, without reference to the particular manner in which the surface is embedded in the ambient 3-dimensional Euclidean space. In other words, the Gaussian curvature is an intrinsic invariant of a surface. According to Wiki., the theorem is "remarkable" because the starting definition of Gaussian curvature makes direct use of the position of the surface in space. So it is quite surprising that the result does not depend on its embedding in spite of all bending and twisting deformations undergone. NB : your last sentence is all Chinese to me !

If you view space as a foliate bundle and attempt to describe the irreducible of that tangent vector space, you will find that the foliate order must be composed of Euclidean conics and conic sections as well as hyperbolic curvatures, in the role of the primitive spectrum of that order. And that is the essential reason why « any differential (resp. Riemannian) manifold can be embedded into an euclidian manifold RN for N large enough » as you rightly report.

This is again Chinese, worse, the assertion that the foliate order, etc. is totally gratuitous. Prove it, or at least give an idea of the proof. And I must confess I’m shocked by the way you wave hands in pretending (?) that your proposition should "explain" two of the most elaborate theorems of DG. Excuse my bluntness here, but we are holding a scientific discussion, not a hallway conversation.

In other words, and counter-intuitively so, curved space can be demonstrated to be a derivation of flat space (and not the other way around), because Euclidean metric is the mother of all metrics.

Same comment. Can be demonstrated ? You just beg the question ( « pétition de principe » in French).

I think your view of the matter sits along the line of dimensions or the dimensional aspect of space. I do not think that the dimensional order in the “global” perspective is the correct barometer to gauge the relationship between the two Geometries, simply because there is no extrinsic meta-space beyond 3-space in all geometries, such abstractions being representationally vacuous in DG and only useful for virtualizations to be implemented within the very 3-space, as I said before.

Weird. I never suggested that the dimensional order, etc. Where did you read that ? Here : « And by the very meaning of tò hólon, this manifold cannot be embedded into any larger RN » ? Perhaps, to avoid confusion, I should have said « any larger manifold ». Or here : « The most well known (even to the layman) example is of course Einstein's 4-dimensional space-time, etc. » ? Dimension is not the reason, it’s just that a lorentzian manifold cannot be be isomorphic to a riemannian one. Finally – or again ? – I can’t understand your assertion there is no extrinsic meta-space beyond 3-space in all geometries […] as I said before. Where did you say it ? More important, where did you prove it ?

In conclusion, if it appears that we don't speak the same language (which is very probable), I suggest to close the discussion, with no feelings harmed.

Dear Demetris,

Thank you for your message. I agree that reality doesn't care about you or me buying Euclid or not. However, in the domain of science (hard science at that !) where A and non-A cannot be true at the same time, I feel disatisfied that we couldn't agree on an unequivocal answer. So, after some brainstorming, let me try one last explanation.

I take for granted that locally, we both think that a Riemannian manifold M and a Euclidian space are "the same" via local charts. But as for me, I insisted that they cannot be so globally, because their structures depend on the way these local charts are glued together to form an atlas. And as for you , you insisted that the demand of differentiation is actually a kind of "local Euclideanization", and if we abandon the differentiation requirement all Christoffel symbols for partial derivatives would break down, and then the entire building of Riemannian applications like GR would also break down, so everything must be locally linear (i.e. Euclidean). But, speaking of GR, I had previously drawn your attention to the fact that the Einstein spacetime is a Lorentzian manifold, which cannot be the same as a Riemannian manifold because locally, the signatures of the associated quadratic forms on the tangent spaces are resp. (+,+,+,-) and (+,+,+,+). So how do we lift the paradox (= apparent contradiction) ?

Your identification "linear = Euclidian" is obviously wrong (linear algebra is independent from inner products), but the paradox certainly lies deeper. After some thought, it occurred to me that both of us never took care to pinpoint the normed vector space in which we perform differential calculus. This cannot be the tangent space because of the different signatures, as shown above. Actually we do calculus on the normed space where the transition functions live (see your Fig.2) , call it FM , where the index M refers to the structure of the manifold under study. That space FM can be termed Euclidian because the norm of the functions is defined using the Euclidian structure of both the domain of definition and the target. But recall that the transition functions are not only smooth maps, they must also respect the structures on the tangent spaces provided by the quadratic forms, so FM is different according to the structure of M , and the paradox is lifted : the manifold may be non-Euclidian (e.g. the Lorentzian manifold of GR) although the function space FM is always Euclidian. What do you think ?

Yes, Thong Nguyen Quang Do, much of what I say is Chinese to you, because as it appears, you have not allowed yourself to read not even the first few pages of the paper referred in the question. You should allow the possibility of a different view about the objects of Geometry. Emotions are a wonderful thing, but it is only to the extent that we can control them that we can build something, individually or collectively. So I would hate to see you leave the conversation out of anger from disagreement or criticism.

Now let’s go. To your points.

A point that encloses a neighborhood is a skewed abstraction, whatever the nature of the neighborhood, whether topological or metric. Because in pure abstraction, a dot must be uni-dimensional. Any way you look at a dot, for it to be so it must be visualized as an absolutely monolithic element. There is in abstraction no sub-order over which a dot would acquire dimensionality or multi-dimensionality in and of itself. It must be monolithic or atomic onto itself. Hence you can see why the notion of a point in a manifold that contains a Euclidean or metric neighborhood is a troubling notion to me.

How about a point with a topological neighborhood? If a topology, like a Mobius strip for instance, must describe a full form then there is no lower geometric dimension to that entity, simply because the form must always be considered in total. So much so that it is sustained by one invariant, which if broken the form no longer exists. So therefore a Mobius strip is its own irreducible. From there we can only consider composite topologies made up of these strip units. I think that you would not know what a point on a Mobius strip means geometrically, because the point alone makes no sense if you don’t relate it to ALL the other points of the strip. That is why in topology it is structure that matters. And that is why a manifold with irreducible points each endowed with topological neighborhood is a troubling notion to me.

The reason why in DG manifolding points have a neighborhood is simply so that they get to have a way to connect to each other thru the device of morphism. This is the map. A foundation not too solid. Or do I need to “prove” for you the requirement why a dot must be quintessentially monolithic? There are indeed other fully logical or formal ways of building a cohesive foundation for a faithful geometric description of foliates.

On the Gaussian curvature, the Theorema Egregium, though correct, is not that impressive to me, because it only reflects the fact that true 3D coordinate space is isotropic and homogenous. First off, a Gaussian curvature is not a real geometric curvature category in and of itself. We speak of a Gaussian curvature only because of the way that we arrive at the description or identification of the curvature, which is from functional tangential behavior. But the elements that make up the curvature are Euclidian in nature, they are the natural basic forms treated in Euclidean geometry, to which we have to add other special forms of hyperbolic nature or unit-forms well captured in Analytic Geometry, although those have never been typified or normalized. It is only because of the universality of this spectrum of unit-forms, and because all forms are made up of these irreducible unit-forms, that Theorema Egregium is true. So it is not at all surprising that “local” curvature described by functional tangential behavior per DG is directly integrable into a “global” Euclidian space. I do not just pose this, but you would see how I support this contention, as you challenge me to, if you take the time to go over the referred paper.

Say that you wake up in the morning and look back at the blanket on your bed. You see a mosaic of interesting patterns in there in terms of folds and shapes and intertwining folds, you want to communicate a description of this interesting mosaic to someone else. If you try to describe them directly with discrete Euclidean elements in a suite, you realize that you can’t go far and would likely have to spend the rest of your life in the undertaking. Gosh, you need a more sophisticated methodology, you realize that you need to able to explain shapes and folds in terms of their curvi-linear evolution throughout this “blanket” manifold. That is what DG does for you by studying all shapes of the foliate thru their curvi-linear evolution from an inner functional perspective of the curvature sectionals. The entire toolset of DG is geared to that end, theorems or axiomatic definitions, morphic relationships, mapping methodologies, etc.

However what the artifices of DG and their successors may not have quite realized or pondered enough is the fullness of the following:

1.- A tangent vector space implies a functional view of the geometric space under study (that I think they got).

2.- The implied function or functional contains the very geometric design of the curvi-linear evolution of the foliate.

3.- The curvilinear rundown of all functions is a serial suite of Euclidean sectionals and other typified hyperbolic sectionals (in 2-space or 3-space).

4.- Euclidean Geometry with the inclusion of those Hyperbolic normals represent and present the Eigensolution to Differential-Geometry-laden geometric problems. One can view this as a result of the hierarchy that exists between all Three Geometries.

So you see, I have a radically different view than yours about the relationship between EG and DG. For me it is not that you can drop EG within DG. In fact the three Geometries are complementary to each other, that may be why it is possible to build a single unified Theory that can provide more answers than they each can independently. That’s what I do in Quanto-Geometry.

Hope my answer is less vague and satisfying to you, even though you may still disagree. Hope as much that you stay in the conversation.

Respectfully.

Dear Thong,

Nice analysis, my congrats.

Look, it is a deep concept and I cannot discuss it with many scientists, you see they don't have the background that you have.So I am happy to discuss with you.

I agree with you on the global atlas to be different from a vector space.

I am also happy that you understood my point on Euclidean locality.

My key point is next:

• Despite what the initial manifold is, our tools (Calculus etc) can only be applied upon differentiable sub-parts of those manifolds where we can find those mapping functions of Fig.2
• That is a very strong demand and alternatives have not been connected to natural reality, even if they are alive for many decades
• So, I tend to believe that our "instance of universe" (:our local iinverse, see more here:

  Article A new cosmological paradigm: universal locality

  ) is essentially an Euclidean one

I am waiting for new phenomena that cannot be explained by using linear dependent tools and then I'll revise my view.

Do you have any such candidate phenomena in mind?

Dear Demetris,

As I said before, I'm not an expert in physics, so you'll have to take the following observations as coming from a layman.

Despite what the initial manifold is, our tools (Calculus etc) can only be applied upon differentiable sub-parts of those manifolds where we can find those mapping functions of Fig.2. That is a very strong demand and alternatives have not been connected to natural reality, even if they are alive for many decades

Yes, but differentiability has been a prerequisite ever since the beginning of mathematical physics : Galileo discovered experimentally the kinematics of a material point in free fall and on an inclined plane, Kepler the laws governing the movement of planets, but only the invention of differential calculus by Newton did allow to explain these laws, and moreover to predict new ones. Why does the differentiability hypothesis work so well when applied to reality ? This is the well known « unreasonable efficiency » of mathematics, a philosophical question to which we have no answer. Newton himself said that his goal was to explain « how », not « why » (which he left to God). In other words, if he could express his views today, that we should be content to build models of reality, and not pretend to explain its essence.

So, I tend to believe that our "instance of universe" (= our local universe, see more here: A new cosmological paradigm: universal locality ) is essentially an Euclidean one

I thought we had agreed that a Lorentzian manifold cannot be Riemannian ? That Euclidianity is a characteristic of a certain technical apparatus (differential calculus), not of the geometry of space/time ? In order not to be disturbed by parasitic questions on the definition of the Universe, of a manifold, etc., let us stick to the local point of view, i.e. that of a flat 4-dimensional vector space equipped with a quadratic form of signature (+,+,+,-), the so-called Minkowski « space time » which is the framework of Special Relativity. Whatever you could say, this space is not Euclidian. I put quote marks to stress that this is just a mathematical definition, without any hidden assumption on the nature of space/time. I don’t feel qualified to discuss on the physical principles on which Newtonian and SR mechanics are founded, but as a mathematician, I’m quite satisfied that both theories can be derived from the study of two groups of symmetry, the orthogonal group (= group of Euclidian isometries) in the Euclidian case, the Lorentz group in the SR case. And don’t forget that the two theories are not antagonistic, the second being just a refinement of the first. It is remarkable that already SR not only lifted the paradox raised by the Michelson-Morley experiment, but also predicted a host of new phenomena which are now an udeniable part of our every day reality (like it or not; just think of The Bomb !)

The passage from SR to GR seems natural to me, mathematically speaking. Recall the definition of the Universe as the set of all observables, i.e. "all physical quantities that can be measured. Also, physically meaningful observables must satisfy transformation laws which relate observations performed by different observers in different frames of reference" (Wiki.) Starting from this, if we take into account the principle of SR that the speed of light is an upper limit, then the necessity to « relate observations (…) in different frames of reference » automatically imposes the notion of local charts on a manifold. Because of the impossibility to transmit informations instantaneously, there cannot exist a global chart, only an atlas obtained by pasting local charts together. This is the main purpose of the local-global approach.

On the experimental side, the problem with GR is, by the very nature of the theory, the scarcity of measurable confirmations. One can cite the perihelion of the planet Mercury, the phenomenon of gravitational lenses, and also the less well known relativistic correction included in the GPS system (required not by the speed of the satellites (SR), but the gravitational potential (GR)). As for the Ligo-Virgo Nobel prize, I agree with the critics that the Stockholm committee was too hasty, that only 2 measurements don't make an experimental confirmation.

I am waiting for new phenomena that cannot be explained by using linear dependent tools and then I'll revise my view. Do you have any such candidate phenomena in mind?

Wishful thinking. All our more or less high brow modelizations must start from concrete observations. But we perceive reality through our limited senses (first of all, sight), and I think that « linearization » is our first abstract - and already highly so - conceptualization of this reality. Contrary to the general belief, science evolves not by intuitive leaps, but by successive approximations. Remember Newton saying that he could see so far only because he could climb on the shoulders of the giants who came before him. The natural consequence is that linearization, in my opinion, will remain at the core of any modelization of our Universe. Let me give a final example. Quantum Field Theory QFT - widely popularized by the detection of the Higgs boson some years ago - is a highly elaborate, un-intuitive construction, and yet it ultimately rests on linearization in the sense discussed above (via groups of symmetry), as can be seen in the recapitulating chart below :

Quantum electrodynamics QED Group U(1) Electromagnetic force

Unified weak-electro theory SU(2)xU(1) Electro. and weak forces

Quantum Chromodynamics QCD SU(3) Strong force

Standard model of QFT SU(3)xSU(2)xU(1) 3 unified forces ./.

Dear Thong,

Well, as I understood from your post, you have accepted the nice mathematical formulation of relativity theories as an accurate description of natural reality.

Sorry but hat's not the truth.

I have to mention my relevant work again: Data 111 years of Magic are enough: Let us return to Science now

Working Paper Beyond Electromagnetic Theory

where I explain why relativity is relative by itself (to the choice of the beams used for making measurements)

The fact that we have not yet found radiations with speed C>>c does not mean that those radiations do not exist.

Just consider as "proxy" the neutrinos speed, which is always slightly more than c.

And think that neutrinos are neutral particles.

And think that neutrinos have mass.

And recall my result for the narrow broad of application for relativity (only charged particles accelerating inside EM field)

As for general theory, the so called "space-time" has not been proven to exist independently of a mathematical frame.

And the "successes" of theory:

• Perihelion of Hermes can be explained without general theory
• Black holes and other exotic entities have never been observed
• "Gravitational waves" are a financial scandal that Americans pretend taht they do not see it for other reasons..

Now at the end of your mail you presented me the "ultimate arguments", those of symmetries, mainly coming from Noether's work.

Well, I'll by pass the "Higgs issue " for the moment (it is similar to LIGO "detections"), because I have not examined it carefully (you see CERN did not give me access to its raw data).

It is a common secret that CERN research has stuck and no one of the super symmetrical predicted particle has been found, although the Energy threshold has been reached.

That is fine for me...

And one last point: do not confuse the linearity with Lorentz transformations, are they already linear ones?

Dear Demetris,

As I acknowledged many times, as a mathematician, not a physicist, I cannot (and would not) go into any discussion about what "reality" is (*). Because I think that our abstract constructions can only be approximations of "real reality" (!), I feel satisfied when these approximations, without pretention to universality nor eternity, are accurate enough in what they describe and predict to be confirmed by experimental observations. Until some new facts appear, raising a paradox (not a contradiction) which needs to be lifted. Newtonian mechanics have undergone such a crisis, but to my knowledge, SR has not up to now. I deliberately put apart GR because, as I said, of the inherent scarcity of experimental observations. However I'd like you to give some precisions on what you said about these :

The fact that we have not yet found radiations with speed C>>c does not mean that those radiations do not exist.

True, but not very relevant, I think, because this kind of remark applies to any scientific discipline. The Newtonian universe lived in peaceful harmony until it was shaken by the M-M experiment, the discovery of radio-activity, etc. I guess that when (if) new radiations with speed C>>c are discovered, then nobody serious will contest the necessity of a lifting (not its abandon) of the relativistic model.

Just consider as "proxy" the neutrinos speed, which is always slightly more than c. And think that neutrinos are neutral particles. And think that neutrinos have mass.

Yes, neutrinos are neutral and massive particles. I don't see the point. Your assertion that their speed is always slightly more than c baffles me. If this is a fact, it should be documented. All I know is the controversy some years ago about an experiment in this direction, and which came to an end when the (swiss ?) team at the origin of the announcement admitted a technical error.

As for general theory, the so called "space-time" has not been proven to exist independently of a mathematical frame.

This is an intriguing remark about "existence". Has any one ever witnessed the existence of the ether outside the Newtonian mathematical frame ? And yet Newtonian mechanics - until they partially failed - were considered as the master key to the Universe. Remember Newton's epitaph by Alexander Pope: "Nature and Nature’s Laws lay hid in Night / God said, “Let Newton be!” and all was light." More seriously, since our approximations of the Universe are abstract constructions, they can prove their efficiency only through their effects in "real reality". Has any one ever seen a mass, an acceleration ? What we witness are the physical effects of these concepts: weight, change of speed, force... If a force (conceptualized as a vector) had an existence simply because we can draw one, then space-time exists because we can write down the equations which define it. Again we must come back to Newton's motto: "how", not "why". And of course this "how" cannot be "from here to eternity".

And the "successes" of theory:

Perihelion of Hermes can be explained without general theory. I really would like to see such an explanation. But even in such a case, my position would remain the same: better a single comprehensive explanation than multiple explanations of multiple facts. By the way, you don't say a word about the multitude of concrete applications of SR which can be witnessed in daily life .

Black holes and other exotic entities have never been observed. By definition black holes cannot be observed directly, only indirectly via their effects on their neighbourhoods. Same remark as above: concerning GR, how about gravitational lenses or relativistic correction for GPS ?

Now at the end of your mail you presented me the "ultimate arguments", those of symmetries, mainly coming from Noether's work.

Neither the end nor the beginning. Emmy Noether's theorem (precisely, on the equivalence between the physical laws of conservation and the invariance of the Lagrangian of a system under certain transformations of the coordinates) was only the tip of an iceberg which was revealed in Felix Klein's Erlangen Program which would influence all the development of physics and mathematics at the beginning of the 20th century. Briefly speaking: a physical/math. stucture is mainly reflected in its group of symmetries.

"Gravitational waves" are a financial scandal. I'll by pass the "Higgs issue " for the moment (it is similar to LIGO "detections"), because I have not examined it carefully (you see CERN did not give me access to its raw data).

I've already given my feeling about the Nobel prize given to Ligo-Virgo, mainly because I find this to be too hasty. I must confess that I was skeptical about the measurements themselves, until I attended (as an amateur) a series of conferences at the Henri Poincaré Institute in Paris, where the lecturers exposed in great detail the procedures, the techniques of measurement, the analysis of data, all the mathematical apparatus to eliminate statistical errors, etc. They also distributed a more than 100 pages long booklet containing numerical data with technical and theoretical comments, especially in the talk "Ondes gravitationnelles et analyse de données", by E. Chassande-Mottin. Ref. : "Ondes Gravitationnelles", Séminaire Poincaré XXII, December 2016. You can try and look for this on the Web, where you'll also surely find other proceedings of analogous conferences all over the renowned universities of the world.

It is a common secret that CERN research has stuck and no one of the super symmetrical predicted particle has been found, although the Energy threshold has been reached.

You're too vague about this. If you allude to the Higgs boson, I certainly can't buy your assertion about the CERN being stuck. If you read French, I could send you some documents.

And one last point: do not confuse the linearity with Lorentz transformations, are they already linear ones?

No, there is no confusion. Following the Erlangen Program, the symmetry group of a vector space V is the group of bijective linear transformations of V, usually called the linear group GL(V). If you put an additional structure on V, e.g. an Euclidian inner product , then you must further require that your linear transformations respect the inner product, and the symmety group of (V, ) will be the orthogonal group O(V), a subgroup of GL(V). Idem for a Lorentzian vector space. So much for vector spaces with additional structures. Coming to general . manifolds, we use local charts to identify neighborhoods on the manifold with open sets of a certain Rn via (local) diffeomorphisms. The transition maps between the local charts are also diffeomorphisms, which for the moment need not be anything else. All this is actually contained in your Fig.2. But a manifold admits tangent spaces at all points, and naturally, the transition maps induce between the tangent spaces bijective linear maps, called linear tangent maps, which are approximations (at the first degree) of the transition maps themselves. Now, if you put on the tangent spaces an additional structure ( e.g. Euclidian or Lorentzian), you must require that the linear tangent maps respect this structure, and you get a collection of (compatible in the local-global sense) orthogonal or Lorentzian groups.

(*) See the appended file on what "math. reality" could be

Dear Thong,

I respect your declaration of not being a Physicist and I'll not continue the discussion about nature and physical theories.

However we agree on the mathematical dominance of linearity which roughly can be taken as Euclideanity...

So fro me the answer to the original question is that:

• Differential Geometry is a more powerful presentation of Euclidean Geometry that can cover much more cases

I enjoyed the conversation with you!

My regards.

Demetris

However we agree on the mathematical dominance of linearity which roughly can be taken as Euclideanity...

No, I regret to say, definitely no. Linearity is a property by itself, independent from any other structure you could add. And it is as "natural" to add a structure coming from a simply non generate quadratic form (e.g. Lorentzian) as a structure coming from a definitite positive quadratic form (roughly speaking, Euclidian).

So for me the answer to the original question is that Differential Geometry is a more powerful presentation of Euclidean Geometry that can cover much more cases

I agree with the first part of your sentence, not with the second. There are "many" non degenerate quadratic forms on Rn , whereas, up to equivalence, there is only one positive definite (just think of the signature). If you had said that the Euclidian structure is the best (and only one) natural structure for doing differential calculus, I would have signed with both hands. But Euclidian Geometry cannot even lift the paradox raised by Michelson-Morley experiment. Or if it can, show me.

@Thong Nguyen Quang Do - >>>>> More seriously, since our approximations of the Universe are abstract constructions, they can prove their efficiency only through their effects in "real reality". Has any one ever seen a mass, an acceleration ? What we witness are the physical effects of these concepts: weight, change of speed, force... If a force (conceptualized as a vector) had an existence simply because we can draw one, then space-time exists because we can write down the equations which define it.

Dear Thong,

I think that we have not defined clearly the term linearity.

• You define it by the inner product and relevant quadratic or other bilinear form.
• I define it as the vector space R^n which is the base of all other buildings

As for the Physics, I wrote it, forget it.

You can read however my works (in profile) and get some hints...

I wrote, but did not forget (smile). I cleatrly defined linearity exactly as you say, independently of any additional structure. For example, 5 days ago : Your identification "linear = Euclidian" is obviously wrong (linear algebra is independent from inner products)

Demetris Christopoulos, say you to Thong Nguyen Quang Do: “ I respect your declaration of not being a Physicist and I'll not continue the discussion about nature and physical theories”, in response to Thong reiteratively self-censoring his views about physical science: “As I said before, I'm not an expert in physics, so you'll have to take the following observations as coming from a layman.”

I want to say that there is nowhere in the natural world, in structure or composition, a label that says “only transparent to the very learned” or “only for those who went to this or that big school”. Empirical Knowledge of the intimate nature of the world knows no degree, no school, no competency by specialization. Much to the contrary, outsiders who insisted on their alternative trail, and insiders who were able to take the risk of going against established knowledge with own different perspectives as mavericks, have demonstrated to be the real artifices of the development of knowledge and progress in the history of science; examples abound.

As I discussed in my recent paper: “Human Ascent to the Stars - https://www.researchgate.net/publication/327686842_Human_Ascent_to_the_Stars_From_Scientific_Knowledge_to_Consciousn”, the parcelization of knowledge in silos of expertise and experts, has silenced in public discourse all discussions of Unity in Physical law and Humanism (or even Unity in the physical sciences themselves, I argue), as acknowledged by famous physicist Richard Feynman. Again, we don’t see in nature any border walls of any class anywhere, what we see is that everything is vastly connected or inter-connected in a continuum. And so the inability to perceive and understand this Unity is very deep lack of knowledge in and of itself.

So therefore we should all take full advantage of the wonderful exchange tool of online forums to speak uncensored on any topic of interest, without fear of being chastised or being ready to chastise. Nobody knows everything and we are all entitled to our own mistakes, so we can at least learn from them. Just that one ought to try to be as rational as possible…

Cheers.

Dear Thong,

We are playing with words.

Euclid was the first who stated the axioms of a linear space, see here:

http://ttic.uchicago.edu/~madhurt/courses/reu2013/class723.pdf

The vector space is a linear space where we have defined a metric in order to measure the length of vectors.

However, when I write about linearity I refer to the first historically defined concept and not about the rigorous modern definition.

So, everything is Euclidean... (based)

Well... But as I said, I'm a mathematician, and in mathematics, everything must be rigorously defined, especially after Bourbaki. One practical reason is to avoid misunderstanding. A second reason - which is a reason of principle - is that, when working from axioms, one should always proceed from "general" to "particular". This is not a dogmatic position, just a matter of "economy of thought" (Occam's razor if you want : simplest is best).

In our case, this means that if Euclid started with definitions which unduly mixed "affine" and "metric" notions, well, we must dare say that Euclid was wrong, and is partly responsible in muddling up things for generations of geometers. Just think of the controversial definition of "parallelism". In linear algebra, a "line" is a 1 - dimensional vector space, i.e. the set of scalar multiples of a given "guiding vector" , and two lines are parallel iff they admit a common guiding vector. End of the story. An extra Euclidian structure only adds complication and confusion : How to define parallelism of lines in a 3 - dimensional space ? Problems remain even in a 2 - dimensional space (a plane!). If one must use Euclidian metrics, a possible definition would be that two lines are parallel iff they are orthogonal to a same third line, but then the 5th postulate naturally pops up, and Euclid himself was dissatisfied with it. Recall that non Euclidian geometry could see the light only when pionneers such as Gauss, Bolyai or Lobachevsky dropped the 5th postulate.

A final remark. I've read the link to uchicago.edu which you provided, but couldn't find any hint that Euclid was the first who stated the axioms of a linear space. The lecture gives the definition of an Euclidian space as a vector space equipped with a bilinear form such that the associated quadratic form is positive definite (in my language), but the notions of vector space and linear maps are taken as already granted. I interpret this as the fact that the lecturer shares the common opinion (among mathematicians) that Euclid was not the first who stated the axioms of a linear space. I even think that the ancient Greeks could not, because of their conception of science and nature (I could elaborate on that), have a clear idea of linearity.

./.

Dear Thong,

One PS for my previous post:

• The linear space or vector space R^n does not need any norm inside it.That is needed only when you want to define distances or length of vectors.

And for the rest of your post:

Could you provide me the first mathematician who founded the linear space, if he was not Euclid?

And are you serious that my link had to do with that fact?

I mean, do I need to provide evidences about the pioneer Euclid?

The linear space or vector space R^n does not need any norm inside it.That is needed only when you want to define distances or length of vectors.

I agree, that's what I've kept repeating since the beginning.

Could you provide me the first mathematician who founded the linear space, if he was not Euclid ?

I don't know, this must have been the result of a long common maturation, without necessarily an explicit formulation at a precise moment. At any rate, the concept of a vector (viewed physically as a force, or vice versa) was familiar in Newton's time. Here I must stress that I'm not either an expert in history of mathematics. But it is well known that the Greek mathematicians were first and foremost geometers, not algebraists (they couldn't invent the necessary notations); that the Arab mathematicians invented algebra (which is an arabic word); and that Descartes allowed the two disciplines to merge by introducing the so-called cartesian coordinates. But even at Descartes' time, the distinction between "affine" and "metric" was not clear, as shown by the famous controversy between Descartes and Fermat about the notion of "tangent". Descartes actually thought of his coordinates as - here you are ! - Euclidian coordinates, more precisely his space was a Eucldian vector space in the sense we agreed on previously. Even more precisely, his main tool was the Eucldian inner product, which allowed him to express orthogonality in an algebraic way, and naturally he viewed the "tangent" with the eye of Euclid (this is truly reminiscent of what I said of the "parallels"). Whereas Fermat, who was a number theorist, hence without Euclidian bias, advocated a "topological" view, interpreting the tangent as the "limit position" of a secant, without any metric involved. And, mathematically speaking, Fermat was right.

And are you serious that my link had to do with that fact ? I mean, do I need to provide evidences about the pioneer Euclid ?

I don't understand your first sentence, but, as for the second, you should be aware that the prevailing opinion among mathematicians (at least among searchers in maths.) is that Euclid was actually less a great pioneer (as was Archimedes for instance) than a great organizer of the mathematical knowledge of his time, which he assembled and synthesized in the manner of Bourbaki at the middle of the 20th century. It is not a coincidence if Bourbaki's monumental treatise bears the name of "Elements de Mathématiques".

PS : I take this opportunity to make more precise my previous assertion that the ancient Greeks, because of their conception of science and nature, could not have a clear idea of linearity. I should have said more clearly could not have singled out the concept of linearity. I could elaborate if you are interested.

@Igael Azoulay True, my assertion was too much concise.

Euclidean geometry is more coherent. The axiom of parallelism makes exactly this difference.

Demetris Christopoulos - Thong Nguyen Quang Do

I guess the question is: is the affine space primordial? I don’t think that mathematics can answer that question. Because the problem is that abstract mathematical space pretends to be a model of coordinate space while having been inferred from coordinate space. As presented, before an affine space can exist Euclidean space must exist. In other words, for ds to exist as a differential between two most affine points in a space of some kind, there must exist both the angular element implicit within ds and the length element of the affine differential. Both of these, angle and length, are basis Euclidean geometry elements. You can argue that the angular element has an alternative origin as given from functional tangents that describe curvature in the linear algebraic language. But the backdrop of this is the Cartesian plane, which is an evolution of the Euclidian plane thru bi-axial metric correlation, whatever the Norm. As to the length element, if you consider the circle as the most basic curvature, one must realize in this context that it is essentially defined by a line, its radius. It is not the other way around.

You can also argue that the Lorentzian manifold is real and who is to say that the linear Euclidean configuration of coordinate or “real” space is primordial over the curvilinear foliate structure of space in the large-scale cosmic realm? I would argue against that that the most encompassing attribute of “real” space is its three degrees of freedom which Euclidean geometry is sufficient to describe. Space preserves this attribute across all physical scales. This invariance is even primordial over Lorentz invariance.

The least an abstract geometric order or hierarchy can do is to follow or allow the geometric hierarchy that we observe in nature. Now I must say that although my view seems to be aligned with Demetris’s, I will not agree with a fixed Euclidean background to coordinate space toward an aether theory, if that’s where he is headed with the linear argument.

My particular take.

@Thong Nguyen Quang Do - I am a total ignorant of the history of mathematics (some anecdotes aside). But it seems to me that you don’t need to introduce the concept of a vector space in order to demonstrate parallelism. Simply set two points at the same distance on each of the lines, and join them orderly one to one (without inter-cross) from one line to the other. If the two joining segments are equal in length, then the two lines are parallel. When you introduce the concept of a linear vector space, you introduce an additional angular element, just as you do if you introduce a third line as a prime independent element in order to assess orthogonalism (an angular quality), killing occam’s razor in this case.

I don’t quite see what that says about Demetri’s linearity but… there you go…

I think that we are playing with words again and again.

Euclid was the first who stated the concept of linearity, although he did not defined it as "linearity".

Extensions and improvements have been done from then.

Now we are talking about algebraic geometry, differential geometry and more generally about topological spaces.

But, all mathematics and physics are based on R^n and all tools are working on it.

If we want to "strain at a gnat" we can do it for ever...

Demetris Christopoulos - Demetris, do you agree with the linear algebra notion that a line is a one-dimensional vector space? Why am I wrong in thinking that a vector cannot be used to define a line because the length element of the vector already implies a line segment or a finite line? Does the fact that we can abstract out the length element into a scalar mean that the scalar can notionally or axiomatically substitute the actual metric dimension of the length? While the line is the primary space where a vector or all vectors live, can we turn around and make the vector the origin of the line? Is it or is it not more natural to give the vector an existence as a progeny of the linear concept instead? Furthermore, if what defines a vector is a modulus and an angle, does the vector concept not presuppose the necessity of a second intersecting line from which the angle would emerge by association with the first line?

Was just thinking...

Dear Joseph,

Frankly I lost your point!

Could you focus in one single question?

Thank you

Demetris Christopoulos - Hello Demetris.

Just having fun! This is not a quiz but a conversation. You could have answered any of those questions that you liked. I was just picking on what Thong Nguyen Quang Do has said: “In linear algebra, a "line" is a 1 - dimensional vector space, i.e. the set of scalar multiples of a given "guiding vector…” I turned to you my reaction to his comment since you were specifically making the case for “linearity”. I am interested in trying to see where you are taking us with this argument other than crediting Euclid as pioneer of “linearity”. But of course anyone else is free to pitch in.

Cheers

Joseph: The question seems to be formulated as a false choice. The answer is none of the options. When Gauss defined the curvature of a surface, for example, he made explicit use of the way the surface sits in space. Evidently, this is an extrinsic “curvature as bending” rather than the intrinsic “curvature as stretching” that he also measured.

The latter is in the basis of spacetime, the geometry of special relativity (avoiding Einstein first approach) and general relativity. Notice that this metric is not positive definite, unlike the metrics considered when thinking about the geometry of three-dimensional space. But distance is positive and agrees with our notion of distance in flat space if it is restricted to a purely spatial interval.

The possibility that distance can be zero or negative is the main reason why the geometry of spacetime differs from the geometry of four-dimensional space.

More at

https://www.researchgate.net/post/On_the_fusion_of_time_and_space_in_special_relativity

Dear Ed Gerk,

As far as mathematical definitions/ results are concerned, I feel allowed to discuss/correct possible approximations/errors :

1) I can't understand your assertion that "when Gauss defined the curvature of a surface, for example, he made explicit use of the way the surface sits in space. Evidently, this is an extrinsic “curvature as bending” rather than the intrinsic “curvature as stretching” that he also measured". There is only one definition of the Gaussian curvature, which is the product of the so called principal curvatures. Perhaps you were were misled by these. According to Wikipedia : "At any point of the surface, we can find a normal vector that is at right angles to the surface; planes containing the normal vector are called normal planes. The intersection of a normal plane and the surface will form a curve called a normal section and the curvature of this curve is the normal curvature. For most points on most surfaces, different normal sections will have different curvatures; the maximum and minimum values of these are called the principal curvatures." This notion seems to "make explicit use of the way the surface sits in space", but no, actually the gist of the Theorema Egregium is that, although "the starting definition makes direct use of the position of the surface in space, it is quite surprising that the result does not depend on its embedding in spite of all bending and twisting deformations undergone" (Wiki). In other words, the Gaussian curvature is an intrinsic invariant of a surface.

2) I think you should avoid using the words metric or distance when there is a risk of confusion (for instance when discussing about the relativistic spacetime, see below). A distance is a real valued function d(x,y) defined on any set E of points, satisfying 2 axioms (i) d(x,y)> or =0, and d(x,y)=0 if and only if x=y ; (ii) d(x,z)< or = d(x,y) + d(y,z) (triangular inequality). The pair (E,d) is called a metric space. The most widely known metric space is the Euclidian spaceE= Rn, where d(x,y) is defined as the square root of the sum of the squares of the coordinates. This is not of course the only one, there are other more general Archimedean metric spaces, and even non-Archimedean metric spaces (for example, the p-adic numbers in Number Theory, p being a prime).

In particular a distance cannot be negative (just think: analysis would not be possible without axiom (i)). On the physical side, I find it very confusing to say that "the possibility that distance can be zero or negative is the main reason why the geometry of spacetime differs from the geometry of four-dimensional space". The Newtonian 4-dimensional space, I guess, is the usual Euclidian space R3 together with a 4th independent dimension, time. Whereas the Minkowski spacetime is R4 equipped with the quadratic Lorentz quadratic form of signature (+,+,+,-). The two spaces are mathematically of two different kinds, in particular the Newtonian space cannot be a subspace of the Minkowski spacetime (it could be so only if t=0 constantly), it is just an "approximation" (when v/c is small).

3) You speak of geometry without further precision, but I guess we all are in the same situation. To come back to the distinction between vector spaces and Euclidian vector spaces which were at the core of some previous discussions, just consider the geometry of the plane. The simplest example would be the affine space R2 , i.e. the vector space R2 with a distinguished (but arbitrary) origin, in which the most "elementary geometry" would consist in picking points and drawing lines. Here is a famous problem from the Mathematical Olympiads some years ago: " Given any subset E of n points in a plane, not all on a line, there is a line which contains exactly 2 points of E ". This looks like an elementary affine riddle. Putative solvers are welcome, but if they get tired, I suggest them to google the theorem of Sylvester-Gallai (circa 1893). The point is : this is not an affine question, but an euclidian one ! The problem even goes back to the definition of what a "minimal geometry" should be, beyond the usual incidence axioms of points and lines. The famous finite Fano plane (*) contains only 7 points and 7 three-point "lines"; any 2 points determine a unique line, but there is no 2-point line. This shows that the S-G. theorem is not purely affine ./.

(*) I don't know how to draw the Fano plane here, but just look at

https://en.wikipedia.org/wiki/Fano_plane

Thong and all: I did NOT give the "definition of the Gaussian curvature", as you wrote, that is an unfounded assertion, could create a false controversy, a tempest over a cup of coffee, common in RG, to be met with a smile. This is just the first of your unforced errors, unfortunately. Sometimes, first contact is last contact.

My use of "this" was in reference to this discussion, unmistakably, not to a distant, deceased person, not to Gauss. You probably misread.

I think the term you'd rather use, or are used to hear, is in the context of "distance" as "Euclidean distance", but I just used "distance" in a natural meaning in spacetime, in a reductio ad absurdum. Your reaction means that you bought the plot, but that was just the point of reaching the absurdity, which disproves the point, standard use in maths. Here is the Wikipedia for you: https://en.m.wikipedia.org/wiki/Reductio_ad_absurdum

Let us not engage in fruitless controversies in RG. I find it remarkable how many lines of conflict you found in less words used. Relax. Take a sip of coffee.

Citing Wikipedia is not recommended even in high-school level. Anyone can edit, including Wikipedia editors, and change willy-nilly what another wanted to note correctly. Look for Wikipedia definition of vector cross product, and follow it over time -- knowing that it is not a vector, not open for discussion. A dialogue of the deaf, search for comments on RG and ex-wikipedia editors.

Everything I wrote is confirmed. You may use what you want.

Thong and all, As Joseph invites, let us have some fun. I could sustain my assertions by a detailed argumentation, but I would not write anything new, and someone could still misread, and create more RG traffic, but not the useful one. Seems good for RG points, but actually it is more work than any gain's worth.

Instead, I added context below, in parenthesis and bold, in what I wrote first, no omissions, but hopefully avoiding my readers from falling off the proverbial chair, in tight curves.

Here we started (now with added comments) ...

The question seems to be formulated as a false choice (good for traffic, not so good for significance). The answer is none of the options (which is not offered...).

When Gauss defined the curvature of a surface, for example, he (Gauss) made explicit use of the way the surface sits in space. Evidently, this (i.e., the way the surface sits in space, not Gauss, not his celebrated and well-understood theorem, we are not wet behind our ears here, not anyone or anything else) is an extrinsic “curvature as bending” rather than the intrinsic “curvature as stretching” that he (Gauss) also measured (yes, he measured many different things, the old man).

The latter is in the basis of spacetime, the geometry of special relativity (avoiding Einstein first approach) and general relativity (both use the same spacetime, one flat and the other curved). Notice that this metric is not positive definite (I mean what I say), unlike the metric considered when thinking about the geometry of three-dimensional space. But distance is positive (excludes zero, all is fine...so far...) and agrees with our notion of distance in flat space if (Big IF) it is restricted to a purely spatial interval.

The possibility that distance (as would be measured by ds2, formally) can be zero or negative (Horror! Tight curve. Hang on! That is how bad it really could get. Heresy Warning!!) is the main reason (Reach Absurdum!) why the geometry of spacetime differs from the geometry of four-dimensional space (Don't go there, a fate worse than death awaits you).

More at (with proper context, this is a matter of ontological discussions, its nature may not exist -- but does.)

https://www.researchgate.net/post/On_the_fusion_of_time_and_space_in_special_relativity

Cheers, Ed Gerck

Hello Ed Gerck. Glad that you are having fun! Have to say though that more than traffic what the question for this conversation is looking for is to afford us all, including me, the opportunity to increase our scholarship thru solid exchanges on a topic of common interest.

We have already hit Theorema Egregium in this conversation. But since you bring it back, I will say this: Good that we have this Theorem as a starting point for differential geometry, but it does not have foundational or paradigmatic value, because it obscures at least two deeper geometric orders that it does not have visibility into. To whoever does not so believe, I will ask them two questions:

1.- What is the boundary between the “intrinsic space” inside of the foliate and “extrinsic space” outside of it within the “global space”?

2.- What physical property of “intrinsic space” is the generator of the curvilinear or curviplanar behavior of the foliate, keeping in mind that mathematical tensor categories are only descriptive instruments?

On the metric, you seem to suggest the possibility of negative dimensions or a null dimension. And I am not quite sure what you mean by 4-dimensional space, it does not look like you are referring to the usual (+++, -) but something like [++++, (-)] or whatever else. Whatever your hypothesis, my stance on the issue of the metrics is that for any proposed metric to have a compelling value, it has to have first answered the question posed by Martin Reese as to why the observable universe has a dimensional set of 3 (degrees of freedom D = 3) as a fundamental physical constant whose value needs to be theoretically resolved or derived. Have you? The paper referred in my question for the conversation unequivocally answers these questions.

Cheers.

Joseph and all, Thank you. You ask two deep questions but the answer is common to both, and disarmingly simple.

First, to understand the Theorema Egregium, one has to hear Gauss account how he came to it, which thought will help in also in the other questions you list. Gauss had the habit of " being like a fox" that hides its trail. To preserve originality and/or to have time to work on it, rather than being pressed to work on it, maybe. What counts is that we need to see his setting, context, and thoughts. I refer to the historic records, that everyone can access. In case of difficulty, I will be happy if you or someone else repeat it here, as not everyone is a mathematician, even though everyone here is assumed to be a researcher.

Last, let me take some stones out of the way. I did not suggest it, maybe you read expressively, on "the possibility of negative dimensions or a null dimension." I am sure what is 4-space and I mentioned two possibilities. I did imply that it does not end there, that I expect to be more than 4 dimensions.

I deny that "the observable universe has a dimensional set of 3 (degrees of freedom D = 3) as a fundamental physical constant" -- it is wrong, it is dismissed experimentally by the fusion of space and time, but sometimes accepted as an ontological question. It could be, but the question denies itself.

Cheers, Ed Gerck

Ed Gerck, you wrote to Thong: "Citing Wikipedia is not recommended even in high-school level". There is this practice on RG to discredit people’s views by rebuffing citations from Wikipedia. Not saying that’s what you are doing here, but I think this is unfair to Wikipedia. Even though in the pre-internet era, nobody in scholarly literature would cite an encyclopedia like Encyclopedia Britannica i.e., nobody thought either that these volumes had no value. For most people, they were a source of access to universal knowledge. Most articles on Wikipedia are written by qualified people as well who consult textbooks or other scholarly books for such writing. And Wikipedia has an editorial system that helps cure these articles. So I don’t think it is wrong in conversations like this to cite Wikipedia as an expedient 3rd party reference in order to support an argument. I don’t see how a reference to a $500 publication by Elsevier or Springer on a library book shelf would help in that case. Biggest problem I find with Wikipedia articles is that many authors simply take back tidbits or passages from textbooks and do not provide enough of an explanatory framework for their articles to be completely intelligible. In others, the language is so complex that the article is of not much help to the topic. I have to concede that there are bad articles as well, on account of inconsistency in the way they are developed, but I can’t say I have seen many of them.

So I think nobody here should refrain from citing a Wikipedia passage to support a point. Anybody can do further scholarly research for accuracy if need be. Obviously learning High School kids are not intellectually mature enough to be properly discriminating about these Wikipedia articles (though they use it A LOT!). That is obviously not the situation that applies for online science forums with people in the know.

Just my personal view.

Joseph, Regarding use of Wikipedia, the reason it earns no trust is disarmingly simple: From the moment you cite an article to me, someone else may change it nonsensically, and I would be asked to read not what you sent, but nonsense.

Wikipedia earns no trust, uses a wrong long-term business model, it is outdated, the people there know it, but they don't care, it seems, as long as the product sells enough.

Hear also from other people at RG, and understand why it is banned even in high-school, it goes against basic notions of trust and copyright. (yes, there is a valid reason for copyright).

No amount of Wikipedia editors will solve it, it is not instantaneous, and they do form claques, making the problem worse, as cited in my long-standing experiment, with falsities often masquerind as truth, even supported by sock-puppets and groups in anti-science movements, political views, and fanaticism such as "special relativity is wrong", a single photon has mass, a vector cross product is a vector, etc.

But Wikipedia is good because it teaches the researcher to not trust anyone. Trust yourself -- look for references that are not controlled by conflicts of interest and outright fraud, such as knowing it has no trust. Hhaving correct information is something you have to decide. But others see it intersubjectively.

More at https://www.researchgate.net/post/Should_Wikipedia_be_used_only_for_careful_personal_education

Not my personal view only (:-

Just to remind that if we take the variable (i*t) instead of t, then the infinitesimal ds^2 is exactly the same in two spaces, so insertion of time did not alter the fundamental structure of linear space...

Demetrius: It does change, because i2 = -1.

Let us not start a controversy over an ontological choice that is, at the end, false, or cite reports that "jump the shark." Not published by reputable third parties is a warning sign, but not the first or the only.

Some people tried to change it and published, in using Cl(3,1) and STA, in real algebra of multivectors, but it does not work physically, though mathematically. Baumgartner tried to filter it out after Cl(3,1) but you cannot filter out after the fact, the error goes through.

Time is a dimension and can transform into space in spacetime, and vice-versa, as we see in experiments and theory. Time dilation happens with length ccontraction - it is not one or the other.

Ed Gerck, I will keep your last but very valid observation about Wikipedia in mind (ability to change content extemporaneously in a self-serving manner). I am aware that many people seek to have a presence on Wikipedia and some manage to create articles incognito about themselves or their own research, or skew a presentation toward their own views or publications. I think that your observation mostly applies to that type of articles. They are not too hard to spot. For settled science, I think that most articles will tend to be free from this problem. As to the High School kids, there seems to be no amount of prohibition that will keep them from consulting it, though they know they can’t plagiarize it.

Keep in mind that nothing in this life will ever be perfect. Springer and Elsevier are not either, as we all know! It will be hard to stop the Information revolution.

Cheers.

Joseph: It is not a question of seeking perfection, where the best is an enemy of the good enough, but seeking the good enough -- where Wikipedia disqualifies itself, despite all public warnings in at least 15 years, including internal, and its actions with pretending group consensus.

Wikipedia can still be used in the personal sphere. Just very carefully, as you can see in simple topics such as the vector cross product, which is not a vector, as well-known.

Ed Gerck - I hear you. You probably know more about their internal policy than I do since you’ve been tracking them for long. At the moment Google is serving content from their Knowledge Base left and right, which makes it difficult to ask others not to reference them even in casual conversations. And I can assure you there is much more harm coming from certain quarters of the scholarly world than from Wikipedia. Given how difficult it is to build such platforms with little support from those who first benefit from it, let’s not detract the overall effort. But given the imperfections that you rightly pointed out, let’s call them work in progress…

That was a parenthesis. Let’s now go back to the business of geometry.

Cordially.

Joseph: Yes. In closing this off-topic, but which is important in citations, I offer an anecdote.

Years ago, a friend who taught one of my relatives, asked me why I did not write an errata to the book he was using to teach, after my comments on the future contradictions students would face, of the notation used. My reply was that students should learn to have a healthy distrust for books, and get more references to weigh in. Therefore, I did not write the errata. It would curtail self-initiative.

Dear Thong – You write: "the point is : this is not an affine question, but an euclidian one ! The problem even goes back to the definition of what a "minimal geometry" should be, beyond the usual incidence axioms of points and lines. The famous finite Fano plane (*) contains only 7 points and 7 three-point "lines"; any 2 points determine a unique line, but there is no 2-point line. This shows that the S-G. theorem is not purely affine."

I confess that my interest in mathematics is only to the extent that it can help address and solve physical problems. So I am particularly interested in the confluence between physics and mathematics, that is, mathematical physics. I believe that David Hilbert 6th Problem (bold call for mathematical axiomatization of physics never addressed in both sciences) exactly sits at the confluence between physics and mathematics. In this context the question of defining what “a minimal geometry” is acquires particular significance. That minimal geometry is what I call the most primal geometric order. I do not believe that the order of points responds to that order. A group or coalition of points represents for me a scalar order and stands for the primitives for cardinality and ordinality (Number Theory foundation). When you join two points, you are adding or associating geometrization to the scalar order. Because, for me, geometry starts with space or spread which must therefore have a line as its irreducible. A space has to be pre-determined for 2 points to mean anything geometrically. If the space is flat, 2 points means a line or linear distance within the space. It the space is curved, 2 points means a curvilinear distance within the space or a geodesic. Affine or metric is not defined or determined by the points per se but by both the spread and nature of the space.

One has to be careful to not use superstructures such as the Freno plane to define irreducibles, just because of certain peculiarities of geometric nature they might present. This sort of error has plagued 20th century constructions thrown on top of differential geometry, and that is why they have not come even close to deliver on David Hilbert’s call for axiomatization of physics, despite their complexity and apparent sophistication. It is clear that “affine” is going to be more “resolved” than “metric” because affine is the result of curvilinear behavior or tensorial curvature experienced by the space. Euclidean metric has no rectilinear constraint whereas differential metric is restricted by the states of symmetry encapsulated within the curvature. Distance can never be 0, not even in the most monolithic black hole, because it would mean a hole or gap within the fabric of space, which in itself would be a form of space. But if one wants negative distance in abstraction, one can always claim that it matches space spreads in anti-particles i.e., but then you have to answer the question, why mother nature overwhelmingly prefers matter over anti-matter. Bottom line is that spread or spatial spread can never vanish, because it is a fundamental component of mother nature’s physics at ground level.

My take.

Ed, funny enough, whenever they are trying to use general relativity for computational reasons they:

• either divide the "spacetime" in 3+1 dimensions aka onion
• or they go back to Newton and use the PostNewtonian approximation theory

Read more here for numerical relativity: Data A detailed critical review of reported event GW150914 that L...

Those are enough for showing the useless of relativity "spacetime" hypothesis...

@ Joseph Thank you for your answer to Ed's gratuitous - in my opinion - attack against Wikipedia. I myself had had no intention to reply, because - in my opinion again - exaggerated ukases tend to verge on bad faith. But you happened to express my thought in your comment : Even though in the pre-internet era, nobody in scholarly literature would cite an encyclopedia like Encyclopedia Britannica., nobody thought either that these volumes had no value. For most people, they were a source of access to universal knowledge. Most articles on Wikipedia are written by qualified people as well who consult textbooks or other scholarly books for such writing. And Wikipedia has an editorial system that helps cure these articles. So I don’t think it is wrong in conversations like this to cite Wikipedia as an expedient 3rd party reference in order to support an argument. About the intrinsic character of gaussian curvature, I could have scholarly cited such classical (but heavy!) textbooks as Berger's or Spivak's. But since " Wikipedia is not recommended even in high-school level", I resign myself to append a few excerpts of academic courses or research articles (with "bubbles" to signal the relevant passages)

In order not to rebuke everybody, I also add an introductory paper from the AMS (after all, some people may be content with this kind of information, the AMS sigle serving as a guarantee).

@Ed Examples of what I called ukases :

1) When Gauss defined [his] curvature of a surface, for example, he made explicit use of the way the surface sits in space [true]. Evidently, this is an extrinsic “curvature as bending” rather than the intrinsic “curvature as stretching” that he also measured [what does this mean?]

2) ... as you can see in simple topics such as the vector cross product, which is not a vector, as well-known. Seriously ?

3) Everything I wrote is confirmed. Really ? Fortunately we work in a scientific domain which is founded - as I've already said - on the principle that A and non-A cannot be true at the same time. This allows me to ask you to sustain your assertions by a detailed argumentation. Then we could at last discuss.

Demetris: Thanks. To save space, I rejoinded by adding to my first reply.

Demetris Christopoulos - Ed Gerck - Sometimes it takes an analogy to make oneself clearest as possible. Like this one:

Mrs. Nature used to live in a large gated property in the countryside. Her neighbors thought that she does not seem to age, because her appearance strangely has never changed thru the many decades they knew her. Because of that and other things about her, they have come to surname her Mrs. Anomalous in their separate gatherings. But one day, while she was coming out of the gate, Newt the Scientist, her closest neighbor, happened to just be coming around by the sidewalk. Nice weather today, Newt said! Yes, gorgeous sunny day, she answered. But later in the day it will be raining with thunder. Really, Newt said! I got a few things to do today. By the way, what time is it by your watch? She looked and looked at him, quite puzzled. Of course she never used any clock and does not know what one might look like. As Newt the Scientist insisted and became quite agitated, she became uncomfortable and slowly withdrew from the gate, closing it back. Newt the Scientist thought she was being disrespectful and could not help hitting her with a few profanities. Mother Nature, knowing who she was to him, started sobbing as she was walking back. Then all of a sudden the weather changed and it started pouring with thunder and lightening (and you know why). Newt the Scientist was now running back to his home looking for shelter. So furious he was that he could not help damning her one more time: F-c-k you, old Anomalous sorcerer! You said later in the afternoon, you don’t even know your own Time!!!

Now, Ed and Demetris (and all the other physicists out there who swear by Time in one form or another), if you do not agree with the morale of the story, I will tell you this: Look for a natural unit of Time, wherever your musings and inquests as scientists take you, take your Time, no hurry. When you have found it, send it to me... please. I will personally go back to Mrs. Nature’s estate to hand it to her! Yeah, we will finally make that old ignorant quit her fr*ken Anomalous habit of timelessness!

PS: Meanwhile you can read any of my related papers to see how I am scathingly busting the b*tts of Mr Time and his Mirages.

https://www.researchgate.net/profile/Joseph_Jean-Claude, Demetris Christopoulos , and all: Thank you. The universe has no watch, has no time. Time is just an illusion, albeit a persistent one, in this material world. I will take a look at your papers. In physics, time dilation and length contraction are non-zero for non-comoving observers, a fact. They do NOT happen for comoving observers, a fact.

This means that if you observe time dilation you must observe length contraction, and vice-versa. There is a fusion, as in spacetime description, of space and time, and only that fusion -- no con-fusion here -- has physical significance, that we live in at least a 4D world. This is an experimental fact.

We can argue further, that is an ontological question. Some think that this is now unanswerable, but this very question, if posed, makes one realize that we live in, at least, a 5D world. Seems like a good enough approximation.

Hello All. I am trying to keep the conversation focused on the mathematical side of things per the question we are debating. But it keeps shifting to the physics side of things. I guess discussing the metrics and their application in physics is unavoidable in this context.

Demetris wants to demote the Ligo/Virgo experiment by going to the fundamentals and showing that the relativistic computations are unsatisfactory or must be wrong. I too would have liked them to tell me that they have directly measured something instead of “we’ve been mostly computing things for you”. But the Euclidian premises that Demetris is upholding may need consolidation.

Ed wants to demote Relativity by literally fusing time to space, but the metrics he proposes are utterly confusing and uncertain, because “we live in at least a 4D” in one moment, and the next moment “we live in, at least, a 5D world”, and all along he denies that "the observable universe has a dimensional set of 3 (degrees of freedom D = 3) as a fundamental physical constant" because of course if you fuse the one dimension of time into the 3 dimensions of space, the remaining spatial dimensions cannot be 3!! (says he.)

Thong wants to give the superstructures like the Mobius strip or the Fano plane preponderance over the basis elemental entities of Geometry and view those as Geometric Primitives (perhaps this is the view of the whole field). Thong has no problem with physicists imposing a spacetime metric into mathematics which should instead propose a generic model with highly abstract components, of which the physical time dimension in particular could become an image.

What has been forgotten all along in mathematical physics is that Einstein was keenly aware that there was a problem with all these metrics incorporating the time variable because he had identified time as a fictitious variable. He further knew that the EFE’s were not solid and expressed hope that they would be improved along the way. Sadly more 100 years after his proposal, we are still talking about spacetime, because generations of physicists and mathematicians have been unable to either seriously evolve Einstein’s vision or develop David Hilbert’s vision of mathematical physics axiomatization in the least amount.

I conclude by reminding people that Einstein did confess this: “Not for a moment did I doubt that this formulation [the so important stress-energy tensor] was merely a makeshift in order to give the general principle of relativity a preliminary closed-form expression. To this day, nobody knows what this tensor really means physically. And believe me you, the metric tensor either.

Joseph: My position was noted incorrectly, I never said I want to "demote Relativity" -- fusing time and space, is relativistic in essence. To be certain, I say the experimental evidence points to at least 4D, maybe 5D, naybe more in progression. This is not, as some want, an ontological doubt -- the nature of our reality is at least 4D. We have evidence, experimental, and mathematical models, both agree in at least 4D.

Ed, in Relativity, time is entangled to space. Not fused or conflated with space. So if you literally want to “confuse” them as you say, you certainly do not accept Relativity as it. I thought that you wanted to supersede its vision, my mistake.

String theory has the natural world with at least 10 or 11 dimensions, I believe. It is not a fact but a view. A view that is now going to be totally discredited given the LHC verdict.

The only thing that is a fact are the 3 degrees of freedom of spatial height, width and length that are all around us. Before that fact meets the scientific standard of empirical observable, it does the common sense standard of human perception. Perception can be sometimes distorted or misleading, but on this one no human has ever disagreed. If you disagree, you will probably be the first one to.

Everything metric beyond 3D is a view of the mind. We cannot even correctly understand phenomenology in 3 spatial dimensions, let alone 4 or 5 spatial dimensions which we cannot even mentally represent. Not that I object to the possibility, but at this stage it is a much better intellectual investment to limit ourselves to unravel 3D.

Joseph: You are not a physicist. I am. I used standard language in physics of Minkowski spacetime, included in his original terms. I would also really warn anyone not to use "entangled" in that context.

There is is an extrinsic “curvature as bending” and an intrinsic “curvature as stretching”. Intrinsic geometry is needed, it is not just mathematics, this is about the nature of things -- intrinsic is different nature from extrinsic. One can model that topology with a thin piece of dough, bending versus stretching, for curvature -- it is different. It is not just mathematics, the contrary is false physically.

The Gaussian curvature can be determined purely in terms of lengths measured within the surface — that is, in terms of the induced metric and its derivatives — and so is a property intrinsic to the surface itself, as opposed to an extrinsic property that depends on how the surface is embedded into an external R3, how it sits in space.

BTW, it may seem contrasensical but it is useful to study general relativity in order to understand special relativity. One can then more natutally seek to understand the correct place for curved spacetime, in terms of extrinsic versus intrinsic, before one inserts there the flat spacetime and differentiate from geodesics, which will be used to model accelerated motion and extend the idea of "straight lines" in the absence of external forces. The spacetime is the same, not the metric.

On September 21, 1908 Hermann Minkowski, an already famous mathematician, began his talk at the 80th Assembly of German Natural Scientists and Physicians with the now well-known introduction:

"The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality."

Ed Gerck, you write: “Joseph: You are not a physicist. I am. I used standard language in physics of Minkowski spacetime, included in his original terms… “. This is how bad it gets when frustration sets in out of inability to convince the audience. But, you know, I too have atoms, particles, molecules, quantum energy packets, etc. in my physical body. And I have been researching them for the last 30 years every minute that I was researching the same in the natural world. I don’t think I need any entitlement for such, neither to share my findings. A bit more knowledge about the history of science would tell you that outsiders or mavericks have always played a key role in the evolution of scientific knowledge because of the fresh and unexpected insight they bring at critical times to matters obscured by the pontiffs’ conservatism, standard formulations or thinking habits. Think of Pierre de Fermat, just a lawyer, but a colossal figure in Number Theory, or Georg Ohm, not your typical insider, who taught himself mathematics, but with momentous contributions to electrodynamic physics, recognized of course after the fact as usual, and so many others I could cite. So shove your presumption. It is not helping you, and it is not helping us here learn something more beyond what we know, which is the whole point of this humble conversation.

If you attempt to be more rigorous, in your appreciations of “facts, evidence, experimental methods” as claimed instead of “it is so because I’ve said it”, you are going to start being useful to the science that we all love. You may have to direct your advice on the use of the word “entangled” or “entanglement” back to Einstein and Schrodinger because they were the ones who introduced its use, but quite frankly everybody in physical science understands that in a cosmological context it means binding time to space and in the quantum context it means a time-free and non-causal bound between two conjugate particles. If there is a particular distinction to be made in the cosmological context, your statement of “confusion between space and time” has not borne out that distinction. As to Minkowski, ask him if he had found a natural unit of time or a natural unit of conflated space-time, to support his wrong proclamation and his wrong metric! From where I stand, I can firmly state that you only need a metric solely based on the 3 observable spatial dimensions in order to unravel the entire set of fundamental physical constants (dimensional and dimensionless), which are the invariants that sustain all the symmetries in nature’s physics. Because I’ve done it!

With all due respect to you and no desire for polemics.

Regards.

Joseph: I am in support of DYI science, multiple ways of knowing, and interdisciplinary work. When I wrote that I am a physicist, it was a declaration of MY insufficiency, not yours. I should speak differently to biologists, for example, as I have to speak to you.

I have to use standard language, though, in physics. Or, there would be a free context, no context, misleading context, or whatever. How can anyone check? It would become a moving target.

I am very happy and relaxed with any argument in the audience, I derived the very equations of what Michael Polanyi described, and I know that self-organization requires freedom to experiment. Nothing that was said by me was out of frustration, which does not exist, because unpremeditated results are the very nectar of discussions!

Joseph and all: Now that we can all breath fresh air, knowing that is always MY limitation that I have to speak in physics terms, and that is a burden to knowledge, you say we live in a 3D world, and I can calculate how that could be true, or 2D, or 1D. So, it is fine.

I can also experiment with relativistic quantum mechanics (RQM), and yet also have publications that deal with non-relativitic quantum mechanics, and the predictions were very close to experimental values -- but I will not drink the Kool-aid and say that this is how the real world is, it is an approximation.

Ed Gerck - Your clarifications, well taken. Happy that your desire to have fun has not abated.

Now, if you agree that time does not exist, then the next step is to eliminate it from all formal descriptions of dynamic systems. The “standard language” as you said is the language of the physics of time, all out. The only head in physics who had challenged this notion, and successfully so, is by the name of Erwin Schrodinger. That is why and the only reason why we have Quantum Mechanics. When the time variable unexpectedly disappeared from the Hamiltonian equation later known as independent of time, cornerstone of QM, he remained unafraid and defiant and courageously gave us QM. In order for us to have intelligibility into the quantum realm, he had to kill the impostor of time, which Einstein did not do for his description of the Cosmological realm and preferred to give us entanglement of time to space. So that had become the “standard language”. Wrong. Insufficient. Misleading. Not withstanding all the respect that I have for his genius. And more than a century later, the “standard language” is at the heart of every physics theoretical stream, defeating time and time again our inquests into the innate structures of nature’s physics.

There is no conflating time to space, simply because there is no time anywhere to grab and conflate to anything. For the description to be authentic and assertive, it ought to be radically time-free. Yes, we can reach a workable time-free description of macroscopic dynamic systems, as difficult as that may appear. But you ought to be willing to abandon the “standard language”.

A couple more things about the “standard language”. The standard language in physics is really the language of time. It is because of the “standard language” that Hendrick Lorentz could not come up with the theory of Relativity, being cooked up in the classical standard language of time. It is against him as a mentor that Einstein had to muster his intellectual resources to come up with the Relativity formulation, where he "dared" orchestrate a diminution of the notion of time thru the view of its entanglement to space. Likewise, it is the standard language of classical physics that had Max Planck underestimate his own revolutionary vision of energy quantum, because his mind was tied up to the notion of frequency continuum. He sit on the concept for quite a while, did not quite know what to make of it, thinking whether it was even correct or not, and it took others, Einstein among them, to successfully apply it to physical electrodynamics at large.

The call for superseding the "standard language" is here with us today more than ever, because the challenges we now face after more than a century of fundamental physics are so very deep!

When tring to translate from the professional use in physics, the old translator motto applies, "Traduttore, traditore" - best said in Italian, meaning "translator, traitor."

Different languages, like Portuguese and Korean, do not have a word for trust, and it is not the same to use confiança in Portuguese, for example, in physics, maths, or cybersecurity.

Conversely, Portuguese has a word in saudade, that English cannot grasp. Maybe Portuguese speakers find a comparison there, for what they miss in trust. Every language we studied, has similar cases, and justifies neologisms. Let us have saudade of conversation, if you know what I mean.

Hello All - I am going to rekindle this conversation from the notion of “minimal geometry” as expressed by Thong, a very important but unsettled notion in mathematics. It also becomes an important notion in the mathematical physics context of research in quantum gravity (ultimate structure of space), in my view.

Number Theory has clearly identified the minimal scalar hierarchy which must be upheld for the construction of number systems and their associated operators, which is the number base of a specific radix. That base rests on the notions of cardinality and ordinality as first principles. We do not have a similar hierarchy in Geometry. We do not have a match to the construct of a number base in Geometry, which I think is important and necessary.

Is geometry the science of forms or the science of space? Could it be instead the science of spaceforms? I offer the following two questions in order to initiate our reflections on the subject (food for thought).

1.- Consider a circle and a line, how would you rank them with respect to one another if you were to construct a primal geometric order of your own design? Is there perhaps other distinct irreducible elements that you could add to your primal order?

2.- Do you agree that all the points on a circle are absolutely interchangeable with one another as much as all the points on a (non-finite) line are all absolutely interchangeable with one another? If you agree, why is it so?

Hope we keep having fun!

I'd like to share a joke in the topic. A mathematician, a physicist, and an engineer, share the same cabin in a train. As the train moves along, the engineer says, "There are sheep here", the physicist explains "There is one black sheep in this space", and the mathematician corrects both, by saying "There exists at least one half of a black sheep."

Ed, the engineer is an applied physicist. The physicist is an applied mathematician. The mathematician is an applied philosopher. The philosopher asks the question: does the color of the sheep matter? To be a black sheep or not to be a sheep at all? Such questions he asks in the ignorance that his philosophy is a mere application of the omniscient knowledge of the Greek God Pan, the Sheep God!

Post Script: LOL!

Joseph and all: Thanks. Just trying to fan the same flames of dialogue. We can define and fine tune cooperation to produce an unexpected effect, a collective effect, as different people, at different times, doing different things, for the same objective. Even those who seem NOT to cooperate, are cooperating.

Ed Gerck - Some people believe that they are different or special and that what they do is different from what everyone else does or can do. Their minds are closed onto itself and reflect a self-centric state of symmetry because it is an incarnation of the circular manifold centered on a single focal point or center of symmetry (origin of the circle). Other people are keen to relate, empathize and share with other people whom they consider their kindred despite apparent differences; those individuals can be said to bear an open-mind because theirs reflect the totalitarian state of symmetry incarnate in a linear (or flat) manifold. The symmetry modulus or number of centers of symmetry of such geometries is a non-finite quantity. Overall this characterization of the mind in human groups is timeless (does not change with time or at different times).

There is however a deeper order of symmetry which makes those two types of human mentation or psychologies similar to one another. It is that same order of symmetry which makes a toddler, who is new to life and can only move around by crawling or with help, completely similar or comparable to an elderly who is at the other side of life and can now barely move around on his/her own and only with help of some kind like the toddler. The knowledge of and in-sight into this order of symmetry is the beginning of wisdom. Transcendental geometry perhaps…

Joseph: Let transcendal geometry help us understand the differences between extrinsic forces, and intrinsic. A student who does 100% of the work by extrinsic forces, may only perform in that type of environment, where he sits in. Another student, who responds to intrinsic forces majoritarily or totally, is mostly shielded from the environment. Many have found that, maybe your experience?

Very interesting comment (above). On the process of learning, I have an equally interesting analysis on page 238-239 of this writing:

https://www.researchgate.net/publication/322756041_Chapter_7_-_Quantum_Phenomena_in_the_Macrocosm

Better than any explanation I could give here.

Cheers.

Ian Malloy – You write: I would argue that Euclidean geometry is more a geometry of space than form. And I suspect that you could have added: while Differential Geometry is more a geometry of form. Either way it is an interesting view. I would like to see further elaboration, if you will.

However, my head is still turning around that one comment: Not only would it [base notation] clarify analytic solutions of coordinates, it may also introduce nontrivial pushforward and pullback. What in the world did you mean by that?

Cordially.

Ian Malloy – You write: “If thinking differently is elegant, I'll choose to think differently.” - It would be interesting to establish a parallel between the use of such attributes as elegant, different, trivial or non-trivial, etc. to characterize mathematical categories and their use in common language. One may wonder if there is a mathematical basis to the common notion of elegance i.e.? And how do we justify that a mathematical expression, either of the geometric or numeric order, is an elegant or beautiful description, a trend that has been accentuated in String Theory apparently for the justification of such a nomenclature? No one seems to see beauty in straight linearity as much as they do in curvatures or complex curvilinear patterns.

Arguably the notion of symmetry, an innate geometric notion, has much to do with it. I offer that symmetry at first order is about same or interchangeability. My sense of the [base notation] that you proposed is one that could be applied to the number of times that sameness appears in a structure. I call each of these cardinal levels or degrees of symmetry a specific state of symmetry. To my mind, this Spectrum of States of Symmetry, of pure geometric nature, constitutes the primitive Norms for our qualifications of beauty or elegance, wherever we invoke them (informally or formally). What we have qualified in physics as a symmetry is only an extrapolation of this Primitive, when we apply motion to a structure, and attempt to view how structural sameness is preserved thru the motion.

As to different, fair to say, I think, that it is best defined in relation to sameness or symmetry, by opposition thereof. So the differential order is a way to approach or make explicit sameness thru minimalist approximations (ds or ds2). Implicit (or intrinsic) is the very state of symmetry, the Beauty within. So at the end Differential Geometry seems to be the method to analyze what our innate God-given sense of Beauty is, Beauty being defined as the geometry that Euclidean Geometry directly puts before us. But I suspect that Thong Nguyen Quang Do would disagree.

Ian, pun all intended?! Took a laugh out of me!

Using set theory, Euclidean geometry ⊂ differential geometry. There is no notion of intrisic geometry in Euclidian geometry, for example.

Here, Ian Malloy may mean that when he suggests that Euclidean geometry is more a geometry of space, meaning N dimensional Euclidean space, than form, meaning non-euclidean forms as such, simple as circular coordinates.

In the comparison idea of Joseph Jean-Claude, with a view for physics, Euclidean geometry can be discarded, as nature is definitely non-Euclidean, even locally.

For example, in cosmology, comoving observers (i.e., at an identical point) can separate, considering Hubble flow. This is not Euclidean. On the other hand, considering mathematics and not physics, comoving observers may not separate, as in SR and GR spacetime.

The thread may, as desired by Joseph, consider both goals, but I want to propose that they are not compatible, and that is good.

Mathematics is made up, but physics has to stay in a standard language, where nature is the arbiter, even if the perfect mathematical edifice of SR and GR may fail. We see that also in multivector and Clifford algebra, such as Cl(3,1), which are perfect mathematically, but fail physically.

That does not mean that, somehow, maths > nature. Due to Gödel's theorem, math cannot be complete and consistent at the same time, nature is both. There is no halting problem in nature, for billions of years, for example.

Thus, physics, not being limited by human history and imagination, and mathematics, not being limited by phenomena, could play a joint role here by extending our exploration beyond that which we can observe or imagine.

Most physicists are willing to accept the existence of things that cannot be proven. Most mathematicians are willing to accept the existence of things without any relation to phenomena — e.g.things that we, at present, cannot observe or construct in the physical world.

Whether or not we can observe something directly, or even indirectly, contemplating its possible existence may allow us to understand how it might play a role in how the universe works, or not.

Everyone's contribution, is thus useful in the aggregate. Everyone is cooperating, willing or not. The discussion is just asked, by RG rules, to not include ad hominem attacks, they do not advance any argument.

Ed Gerck - Let’s start by laying down our vision of the two geometries so we can unequivocally understand each other. Euclidean geometry is a simple geometry of space, space in 1 dimension, 2 dimensions and 3 dimensions. It does not pretend to define space intrinsically. It only considers partitions of space, and how the partitioning components or elements relate to one another thru the metric of distance or spreads and angles essentially. These partitions of space create forms or express themselves in observable forms to which we give specific names such as a trapezoid or a cube. A generation of mathematicians have axiomatized the province of these elements and by doing so have removed geometry from its former realm of physics or physical constructs and formally inducted geometry into mathematics. For Euclid, geometry was a science of construction or if you will architecture. Following its complete axiomatization, Euclidian geometry became mathematical science. The idea evoked by Thong of a “minimal geometry”, which has not been found so far, refers to the basis elements of Euclidean geometry. For me. That’s why at some point we discussed whether or not a coalition of points can set the basis for this “minimal geometry” and I argued that the first element in a “minimal geometry” must be a line and not a point or set of points.

In that context, Ed Gerck, I cannot accept your idea that “nature is definitely non Euclidean”. In that very same context, I cannot quite accept the idea that motion is a qualifier in geometry either. That comoving objects may or may not become separate or behave dynamically independently from one another is not a geometric artifact, whether of Euclidean or non Euclidean nature. Motion, as much as time, are not mathematical entities in the rank of primitives. They are physical entities. And the problem we have in science is that that distinction is not being stringently upheld. And instead of having mathematics as a towering analytic model, we have an amalgam of mathematics and physics, which very clearly transpires thru the metrics, and represents in no small part the culprit in our inability to describe nature faithfully and accurately. The problems that plague theoretical physics are there for anyone who needs a reminder. This is not to detract mathematical physics as a field, but we ought to remain clear of what is what. To support my view, I remind people that the dilation of time is just a view, because time has admittedly no ontology and therefore no possible ensuing phenomenology, and the contraction of space does not exist in the absolute but only in the observer’s perception as all relativists will agree. The notion of spacetime is the worst visualization that could ever be produced in mathematical physics. For a key basis notion in mathematical physics, it’s a compounded mistake.

In good physics, we use mathematics to make projections. If these projections reasonably call for the possible existence of things that are at the moment unobservable or indiscernible, we have to give them credence, in the hope that as we go they can be firmly established empirically. The problem we have nowadays in physics is that most physicists want the rest of the world to believe in these projections as fact, like the dark matter non-sense, or the delusion of retro-causation for quantum entanglement, and that they further give for real the virtual stuffs that come out of their equations like the virtual particles they want the world to believe are real, lest they point the gun at you!

Mathematics is about modeling things, she is not and should not be concerned about the (physical) existence of things. Physics should be about assessing the existence of things and formally describe them as much as provide the tools and means to manipulate them. We should be very careful about how we put together mathematical physics constructs. And I think that none of it can be lent any definitive credence, even less taken for the ultimate truths, if they don’t meet David Hilbert’s test of axiomatization of physics thru theoretical derivation of the fundamental constants and symmetries subtending the physics of Mother Nature.

Cordially.

Joseph: You wrote, "... as all relativists will agree." That is not correct, at all. Physics is not voting, either. Nature is the only arbiter.

Note that comoving and non-comoving, as well as length contraction and time dilation, are not "just words" or optical illusions -- what they denote can cross a barrier, can produce thermodynamic work. They define different physics, different truth-conditions, not just different truth values.

Ed, you keep repeating this, but have not answered the question that I have asked about the natural unit of time or conflated space-time.

If you believe that for an observer in the frame that contains the rod or the rest frame, the rod has contracted just because there is another remote observer in motion with respect to the rest frame, then you believe that the laws of conservation have vanished in the rest frame, and that space in that frame has become spontaneously anisotropic and inhomogenous. Most relativists do not believe so, at least the ones I talk to. I further read in a textbook off my shelf:

“The length L0 of an object measured in the rest frame of the object is its proper length or rest length. Measurements of the length from any reference frame that is in relative motion parallel to that length are always less than the proper length… However a more precise statement is that the object is really measured to shrink…”

If you only reference a remote reference frame to the rest frame, you might come to believe that the object has ontologically contracted. But if you consider the possibility of an observer in the rest frame as well, and that for this non-moving observer the object is observed with proper length, then you will understand that an object cannot simultaneously have a set proper length for one observer and a different length for a remote moving observer. How many different lengths can an object intrinsically have at the same time!!??

Furthermore if you consider yourself as the object under observation, and you are walking on the sidewalk alongside a bunch of cars that are going up and down the street (parallel path), then depending on their relative speed with with respect to you, the drivers will perceive you at a contracted height that is not your proper height. Now, since you think that observed contracted length or height is real, the question for you is : have you ever felt that you are shrinking as you are walking on the sidewalk in the city where you live, as cars are speeding up past you? How many times have you contracted or shrunk and grow again so you can achieve the height that you now have? Should we now invent a biology of growing against the unrelenting contraction experienced by our bodies in the course of our lives in cities and towns? You can see that this view is untenable when you really think about it.

That is why I credited ALL relativists to have the good sense to consider length contraction as OBSERVED but not REAL, and not just the ones I talk to. But I stand corrected in erroneously stating ALL, because you have just proven that there is at least one who does not believe so! And that is fine…

Ian Malloy - Would you agree that if a point is dimension-less, as you stated, then it cannot be a geometric element? Because geometry is first and foremost about spread, aka space. If a point may be taken as a primitive of space, then would you agree that an atom of space must therefore exist? The question is why have we not discovered it thus far? What's taking us so long?!

Joseph Jean-Claude : I suggest that reading your choice of physics books wiil not get us far, nor you citing them. Your SR, as you cited, does not work for accelerated or arbitrary motion, does not use 4D. It seems to use 3D + 1D.

Some of us live in a 3D+1D universe, others in a 4D universe. We can relay our experiences to each other, and even with all goodwill, our 3D+1D or 4D experiences cannot be communicated to each other, a theorem in topology tells us.

We're a little bit like a fish and a bycicle, from a German proverb, moving in different ecosystems. I remain, friendly yours, Ed Gerck

Ed Gerk - For the sake of accuracy, I have to clarify that the book I cited was not my choice but my college teacher’s choice, many many years ago!

I don’t think that we are in different ecosystems, just that we have different views about the fundamentals of mathematical physics.

But again, exchanging views can never hurt and, if any other benefit, making us revisit once more the basic notions and reflecting on their implications is a good exercise.

Very cordially.