Dear Akira Kanda,

Geometry in Euclidean representation had not any numerical characteristics. It is a reason, why the geometry in the Euclidean representation cannot be generalized.

"Geometry in Euclidean representation had not any numerical characteristics."

Well, it involved length as numbers and it brought the insight that fractions are not sufficient for this.

As to the question of this thread, Tarski proposed a version of Euclidean geometry based on two fundamental concepts: betweenness and equidistance. But in dimensions >=2, it is possible to define betweenness in terms of equidistance. So technically speaking, it becomes monistic.

Dear Marcel,

I have two questions: (1) Are betweenness and equidistace numerical characteristics? (2) How can one express dimension via betweenness and equidistance?

Yuri,

Betweenness is a ternary predicate and equidistance is a quaternary predicate (it produces an equivalence relation between line segments, which are pairs of points in Tarski's approach). A total order relation can be defined on the set of pairs (line segments, or rather, their "intrinsic length") by a clever intuition about equidistance in the plane. This order on "length" is used for a definition of betweenness. So there is no a priori concept of number, but if I remember well, it leads to a real closed field (a first-order equivalent of the real number system). Of couse one needs an arbitrary choice of unit length (i.e., scalability, in Akira's view).

As to the second question, I should consult the details of Tarski's aioms to see exactly how dimension is determined from them.

Dear Akira Kanda,

I myself try not to use frame of reference in the formulation of basic statements of a theory. Nevertheless, I try to answer your questions.

!. First question “. For mathematics, there is only one 3D space up to homeomorphism. So, it is hard to understand why in Physics, there are infinitely many reference frames.”

I answer by means of a question: “Why mathematicians think, that any conceptions, that has been invented by them, may have a relation to real phenomena?

2. In geometry, no point moves. This is because mathematics does not deal with time which transcends geometry. Motion is a concept defined upon time. Why in Physics reference frames move?

There are no moving points in 3D-geometry. But at consideration of 4D-geometry, where one of coordinates is time, there are moving points. Physics have a right to use any means of description. They may use English, Russian, Chinese, or any other language.

3. . In geometry, one can not "move" 5 on real line to the position of 3 and vice versa as it violates the topology of real line. So why reference frames move?

The mathematicians have invented topology, and they must to consider this problem themselves. A scientist, which wanted to understand a physical phenomenon, must  reduce the number of fundamental concepts, but not  increase them. The ideal situation takes place, when there is only one fundamental concept (monistic conception). However, in reality such an interesting situation takes place. One constructs a geometrical conception which is inadequate to real space-time geometry. Thereafter one invents additional concepts and values, which could be improve the existing inadequate conception.

4. I did not understand your fourth question

Dear Akira Kanda,

Frankly speaking, I did not understand, what do you like to say by means of your examples.

I should like to note only, that the boundary between dynamics and space-time is mobile. Beginning from nineteenth century this boundary moves from space-time to dynamics. The main stages of this motion are: (1) conservation laws came to space-time from dynamics , (2) special relativity (3) general relativity, (4) theory of Kaluza-Klein. The further motion of the boundary stopped, because our knowledge of geometry were insufficient. This tend of boundary motion is determinant in the elementary particle theory.

Dear Akira,

Thanks for your mathematical insights. In regard to your critical comments on Einstein relativity, it would help if can free oneself from the concept of a mathematical point. Relativity can easily be formulated in terms of waves, matter waves, phonons photons etc.

Some time ago this point (no pun intended) was discussed in another thread and the conclusions, if I remember correctly were that: "The concept of a mathematical point is unphysical and starting any fundamental description of physical reality with points and trajectories (although classically very successful) would nonetheless be severely limiting and not fundamental".

Akira,

I guess one can get a long way via a rather humble approach, based on the Lebesque measure (quantum theory) and the simple concept of bound states and an absolutely continuous spectrum. Such situations may in addition permit realistic continuations in to the complex energy plane, which provides a first trivial step towards a formulation of open systems and non-hermitiean dynamics. In this scenario physics takes on a different interpretative perspctive that appears to work excellently in my field theoretical chemical physics and quantum chemistry.

Akira,

From a mathematical perspective QM is not inconsistent, since it is based on the axioms of linear algebra concatenated by topological spaces like the Hlbert space. To be inconsistent you must mean something physical, which is entirely dependent on the physical model that you want to describe using the formalism of QM!

I do not think that any quantum physicist or quantum chemist would say that QM will predict no trajectories in a Wilson chamber. Note that we always provide a physical model, which if needed will be matter waves. However, if further needed we can always find the appropriate interpretation when we reach the borders between QM an classical.

The double slit experiment is a wonderful illustration of relevant quantum effects.

One can of course rightly question modern QED and general relativity conundrums. This is part of the scientific development, which evolves deductively, like the universe, snd the mental part depending on social phenomena, art, music, language including math.

There is nor problem to include gamma factors where appropriate in QM.

Personally I do not believe in adding any axioms to Peano's arithmetic! Our deductive axiomatic scientific development whether in natural science or in syntax like math, has to abide by Gödel's theorem(s). Your proposition would add fire to a smouldering chaos.

Akira,

I do not agree. The SE as obtained from the Klein Gordon equation is a straight forward connection. If you look for general invariance principles you start at the more sophisticated theory.

If you look hard for inconsistencies you can just start from propositional logic!

Contradiction is a contradiction! I am surprised how you carelessly throw around reflexive statements that have no meaning.

If you are looking for a way to criticise physicists in their use of algebra or group theory you should first ransack the mathematicians that developed and made the vehicle ready for consumption.

It is a curious question: If you understand the uncertainty principle. As far as I know using observables (operators) imparts by definition a mathematical relation between the conjugate partners. To use the language (math) in our physical models is of course to trivially account for this relationship and evaluate the consequences for the physical model – what is the problem?

I think your sweeping comments regarding the Wilson Chamber do not pay proper attention to the intricate interactions that take place between charged particles and the background, creating ions that act as condensation nuclei forming the mist that reveals the path of the alpha or beta particles in question.

I do not agree with your condescending remark on Gödel. I think it is the wrong track to try, as many mathematicians do, to find modifications of arithmetic in order to become immune to Gödel.

I think the proper way is to embrace Gödel and incorporate his theorem appropriately in physical models.

You may of course refuse to believe in the bending light experiment, but the present consensus is that Einstein's laws, light bending, perihelion planet motion, gravitational red shifts, time delays etc. are all according to experimental facts. I hope you enjoy the GPS!

I do not understand your critique of Einstein's STR. The Lorentz transformation is logically and uniquely derived from general postulates without the employment of the Maxwell's equations. I do not see any problems here.

Endnote. I am not defending present day physics as closed or without some fundamental problems. But to attack them by sweeping statements and inaccurate critique only complicate the situation instead of looking for better approaches in a deductive manner.

Dear Akira and Erkki,

Your discussion is rather interesting, but unfortunately, I do not see a connection of this discussion with the question: "Can we present the proper Euclidean geometry in a coordinateless monistic form? "

Akira,

You can of course drown your opponent by irrelevant problems and thought experiments. Your main problem appears to be an unhealthy mixture of mathematics (language) and physics. You can spend the day by criticising Newton, while both you and I know that science has moved far beyond this phase. I think Einstein was one of the first to realize that the force law, the momentum law and the energy law was in conflict with each other.

Another simplification in your attacks are the general consideration in too general situations of Schrödinger's equation as the latter only applies to isolated systems with no interaction with its environment. We know this is an idealization that do not hold for realistic systems.

The Wilson chamber is a particular case where there is a mixture of QM, classical physics, shock waves, nuclear decay etc. etc. + the measurement hypothesis for QM, that we know is an idealization.

On the positive side I did not know of Gellman, please give reference or better enclose a pdf.

Regarding the Lorentz transformation, LT. Einstein's first derivation concerned "On the Electrodynamics of Moving Bodies".Later derivations, Pauli, Weyl, Löwdin, "trimmed" the derivation as an example of deductive scientific thinking in providing 5 simple and evident hypotheses then deriving an impeccable result in terms of high school math. The first postulate that the velocity of light is constant in all priviliged systems was, despite its "noncommon sense" an experimental fact.

I have said earlier in threads on RG that common sense is a treacherous companion in physics. Using a succinct mathematical represntation and syntax together with a relevant physical model is by far the best strategy.

So, please do not mix math and physics in one chaotic soup!

Dear Akira,

The post, where you discuss [1] Violation of the Energy Conservation under the Conservation of Momentum is incorrect. You consider collision of two bodies. Your method of the collision description is inadequate. Your concept of collision is too rough. Consider collision of two charged pointlike particles, having an electric charge of the same sign. Take into account their acceleration at the collision, energy of the electromagnetic field, energy of electromagnetic radiation at the particles acceleration. Such a model of collision will be adequate. All conservation laws will be fulfilled.

Akira,

Your so-called case 1 is wrong. With the initial conditions given the outcome of the "scattering"experiment will not give the velocities you "guess". What is the point?

Akira,

Could you, please, inform me regarding the comment from Gellman, either as a reference or better a pdf.

Best

erkki

Akira,

Regarding case 1., see copy below. Please tell me how you you reached the "outgoing data" from the initial ones. How do you explain that the 5 kg mass moves with -10 km after the collision? If you are using some home made theory to disprove Newton, I just want to know.

Case 1: Assume that a mass A of 5(kg) was moving towards the positive direction on the x-axis with speed 40(km/h). Assume it hit a mass B of 10(kg) which is stationary on the x-axis. After the collision, if 5kg mass is moving with speed 10(km/h) in the negative direction on the x-axis and the 10(kg) mass is moving with the speed v(km/h) towards the positive direction on the x-axis. Then by the conservation of momentum law we have40×5=(-10)×5+v×10.

So, v=25. The initial kinetic energy is (5×40²)/2=4000(J). The final kinetic energy is (5×(-10)²+10×(25)²)/2=3375(J). Clearly the conservation of energy law is violated.

Yuri,

Sorry for occupying your thread with some irrelevant debates on fundamental physics.

My point regarding your question is quite simple. If we start with a background independent theory, like Einstein's theory of general relativity, one obtains a line element, cf. the one due to Schwarzschild. Then we have a space-time point "s". To this point there is corresponding conjugate quantity the energy E(s). This appears to be all that one needs to know in order to formulate the dynamics of a general gravitational system.

In a general perspective it seems as the space coordinates have disappeared although they are hidden in "s".

If I understand "monistic" properly, a fundamental text on Geometry determined by the metric is the habilitation thesis of Bernhard Riemann "Uber die Hypothesen welche der Geometrie zu Grunde liegen" (About the hypothesis which are fundamental for Geometry). There is also a logic describing the Euclidean Geometry. Basic infomation and references can be seen at http://en.wikipedia.org/wiki/Tarski%27s_axioms - the Tarski axioms. They are quite surprising. This axiomatization of Geometry has features very different from the logic of Arithmetics in the spirit of (the famous) Goedel theorems, e.g., all "sentences" can be proved to be true, or not true (decidebility) and also in an algoritmic way (in finite steps).

Dear Svatopluk,

The problem is in a usage of Euclidean geometry, but not in its construction. Let us image that we know completely the Euclidean geometry, and the Euclidean geometry is presented as a monistic conception. We can construct any geometrical object in terms of only Euclidean metric. Euclidean metric is the only information about Euclidean geometry.  All concepts of Euclidean geometry can be expressed via metric and only via metric. In particular, dimension of the Euclidean geometry can be expressed via metric. My question is: “How to express dimension via metric?”  The fact is that, the dimension is given usually at the beginning of the Euclidean geometry construction. The dimension is considered as a fundamental quantity. In the monistic conception of a geometry the dimension is considered as a derivative quantity.

Why does one need such an approach? At the monistic conception of a geometry one can easily construct any generalized geometry. It sufficient to replace the Euclidean metric by the metric of the considered generalized geometry in all definitions of geometric quantities (which are defined via Euclidean metric). My question is connected with the fact, that the dimension is considered usually as a fundamental quantity.

May be, one can take point A in the Euclidean Geometry space and construct all straight lines passing through it (by definition - shortest lines with respect to the metric) and then to search for the maxim of linearly independent ones (the direction vectors of which are linearly independent in the vector space V_A (the vector space underlying the affine space sitting in A - this is standard in affine geometry). So the metric gives really the dimension. I think, one can derive also a formula.

Tarski's axiom system rather lends itself to the use of induction: given two distinct points A, B, the set of all points equidistant to them should form a hyperplane (bisecting and orthogonal to AB) which is one dimension lower than the original space.

Dear Svatopluk,

In your post you do not fulfill the condition, that the presentation of the Euclidean geometry is monistic. (1) You should point out, how you construct a one-dimensional line, using only metric. (2) Even you construct straight lines (the shortest lines), one needs to determine their linear independence of tangent vectors in terms of metric, to construct the affine space.

Dear Marcel,

You should to determine dimension of the hyperplane, whose points are equidistant from points A and B. However, the problem of determination of the hyperplane dimension in terms of metric is the same problem as determination of the Euclidean space dimension.

Yuri, that is why I refer to the inductive nature of the dimension in Tarski's geometry. Strictly speaking, you have to pass to set-theory for a convenient notion of dimension. The simplest way to achieve a definition of "n-dimensional" within Tarski's first-order theory is something like

(forall a_n,b_n)(forall a_{n-1},b_{n-1})...(forall a_1,b_1)

[ ( a_n != b_n and a_n a_{n-1} = a_n b_{n-1} and b_n a_{n-1} = b_n b_{n-1} )

and ... and 

( a_2 != b_2 and a_2 a_1 = a_2 b_1 and b_2 a_1 = b_2 b_1 )

and

a_1 != b_1 ]

implies (exists one x) [ x a_1 = x b_1 and x a_2 = x b_2 and ... and x a_n = x b_n ] .

Here the number n is used as a meta-mathematical number, to be replaced by any of 1,2,3,4,5,6,... The length of the entire definition depends explicitly on this choice.

In my paper on Theories with the Independence Property (available in RG) I used a similar trick to describe the dimension of a Pasch-Peano geometry in first order terms, avoiding set theory.

Dear Marcel,

I did not understand your argumentation. But it is of no importance, because, as I have understood, conception of Tarski is not monistic. He uses three basic concepts: point, betweenness, congruence. My question was about monistic conception. It is connected with a construction of generalized geometries, which are constructed as a modification of the Euclidean geometry. If the Euclidean geometry is presented as a monistic conception with the basic concept: the Euclidean metric, then replacing the Euclidean metric by some another metric, one obtains a generalized geometry. Intuitively it seems, that a geometry is defined completely, if its metric is given. If the geometry conception is not monistic (three basic concepts), then one is to modify the basic concepts by concerted way. It is impossible practically.

Yuri,

"[Tarski] uses three basic concepts: point, betweenness, congruence."

"Point" is present in each geometry as an undefined (fundamental) object and does not count as a "basic concept". That leaves us with betweenness and equidistance, where (as I mentioned before) the former can be defined with the aid of the latter. My previous post shows that dimension can also be defined in terms of equidistance only, but this definition has little practical value except, perhaps for dimensions 1,2,3 or in a set-theoretic setting.

Dear Marcel,

I have understood you in the sense, that congruence is a derivative quantity constructed of betweenness. Is it true? If so, then betweenness must be some numerical quantity. If it is not so, I cannot understand, how betweenness can describe Euclidean geometry. For instance, metric is a basic (fundamental) quantity. Metric is a function of two points. It means that metric is a numerical quantity. Is betweenness some numerical function?

If speaking of generalizing Euclidean Geometry, then the Klein programme seems to me to be very relavnt to mention. Namely, that every generalized geometry can be given by giving a kind of metric or 'metric', is a bit say controversal or provocative. Consider the projective geometry. This geometry can be equivalently described by a set (or collection) of connections. I do not know whether they are all metric connection - coming from a metric and a parallels problem. If they come than sure, one may then be interested only in the metric(s) and search their properties which one can do (CP^n and Fubini-Study, chordal metric). In this case one knows more information than it is allowed in projective geometry or otherwise said, the properties (like the length of a segment) are not invariant under projective automorphisms (allowed symmetries or equvalences)  and one is lead to check metric independece. Therefore the Klein program - studying of the geometry mod out by the automorphisms and it was further generalized to what is called Cartan or generalized geometries. 

Yuri,

betweenness is defined with the aid of equidistance, not the other way round. The process is complicated and rather fundamental, because at the same time it shows that the real number system is derived as an associate of this geometry. More precisely, it produces a "real closed field", which is a "first order version" of the well-known system (which cannot be formulated in first-order terms because of "completeness").

On the other hand, in "synthetic" projective geometry and in so-called Bryant-Webster spaces (based on betweenness), it is known that the "Desargues property" ("centrally perspective triangles must be axially perpective") induces the real field, too. (I am not sure that this avoids set-theory; it's a very long time ago for me.)

So, in principle, a non-numerical geometry can produce the reals either by a (suitable) betweenness or by a (suitable) equidistance. Isn't this fascinating?

I said more than the truth in my previous post. A projective space with Desargues property leads to a division ring (not necessarily commutative) and vice versa. Hence projective spaces are a bit more "ambiguous" than some other geometries, but they share the property of implicitly inducing some kind of arithmetic, acting as a fingerprint for the given geometry.

Dear Marcel,

We discuss the problem, how to construct the Euclidean geometry, deducing from a system of axioms (Hilbert, Tarski, or anybody else). But logical method of the geometry construction is possible, only if the geometry is continuous and it is unlimitedly divisible. Real space-time geometry is not unlimitedly divisible. It is rather discrete. It cannot be constructed by the logical method. The space-time geometry is constructed as a result of a deformation of the Euclidean geometry (or other known geometry). The first geometry is constructed by a logical metho  d, which means that any geometrical object can be constructed from blocks. Method of the geometry construction from blocks (logical method) is complicated and it is not effective. We forced to use it only for construction of the first geometry. Any second (generalized) geometry should be constructed, deforming the first geometry.

For instance, we need to construct a discrete space-time geometry, where there is minimal distance \lambda. Mathematically it means

|\rho(A,B)| does not belong to (0,\lambda) for all A,B belonging to \Omega,   (*)

where \rho(A,B) is the distance between the points A and B, \Omega is the set of points, where the geometry is given. The condition (*) is a constraint. But there are to variants: (1) (*) is a constraint on \Omega at fixed distance \rho, (2) condition (*) is a constraint on \rho at fixed \Omega. In the first case,  using Euclidean distance (or Minkowskian distance) \rho, one obtains a geometry on a lattice, which is not uniform and isotropic. This is not physical. Nevertheless researchers consider only this case.

One may use the second case, using Minkowskian manifold as \Omega and world function \sigmad=1/2\rhod2 in the form

\sigmad=\sigmaM+\lambda2/2 sign(\sigmaM).

where \sigmaM  is the world function of the geometry of Minkowski.

It easy to verify, that the condition (*) is fulfilled. The discrete geometry Gd, described by the world function \sigmad is uniform and isotropic. The “curve” in Gd is defined as a broken line consisting of straight line segments T[PsPs+1]. Any straight light segment T[AB] is defined by the relation

T[AB] = {R|\rhod(A,R) + |\rhod(B,R) - |\rhod(A,B) =0}           (**)

In the Euclidean geometry the straight line segment is defined by the same relation (**). The smooth line is obtained, when the length \rho(Ps,Ps+1) of the broken line link tends to zero. In Gd such a limit does not exist, and one is forced to use broken lines as a world lines of particles. Besides, equality of two vectors AB and CD,

(AB.CD) = |AB| |CD| and |AB| = |CD|                              (***)

has many solutions for D at fixed A,B,C. (Here the scalar product (AB.CD) is expressed via the world function \sigmad). It means intransitivity of the equivalence relation. As a result the corresponding conception of the geometry cannot be axiomatizable, because in any axiomatizable conception the equivalence relation is to be transitive. It means that the discrete geometry Gd has a restricted divisibility. Multivariance of the relation (***) leads to stochasticity of the particle world lines in Gd. Statistical description of stochastic world lines leads to the quantum description in the form of the Schroedinger equation, provided \lambda2=\hbar/cb, where b is universal constant, connecting particle mass m with the length \mu of the link of the broken world line by means of the relation m=b\mu . This relation geomerizes the mass. Of course, it is more reasonable to explain quantum effects by the discrete space-time geometry with stochastic world lines, than at first to postulate continuous space-time geometry and thereafter to postulate quantum principles. The reason of such behavior is the fact, that mathematicians do not accept idea of nonaxiomatizable geometry. In turn the supposed axiomatizability of the space-time geometry is founded on the fact, that mathematicians do not know the deformation method of the geometry construction. As far as I understand, the main obstacle on the way of the deformation method is the fact, that the segment of a straight line (**) may be a surface (hollow tube) in the real (discrete) space-time. At any rate I cannot overcome this obstacle in the course of thirty years. Besides, this fact looks as “experimental data”, used in mathematics, which is unusual for mathematicians. See details in “Metrical conception of the space-time geometry” Int. J. Theor, Phys. 54, iss.1, 334-339, (2014), Electronic version http://gasdyn-ipm.ipmnet.ru/~rylov/mcstg2e.pdf .

Dear Akira,

You accuse me, that I cannot think logically and I do not know something. I shall not discuss such things. Look at my previous answer to Marcel

Akira,

Please give me the reference to the work of gellman.

Dear Akira,

You have mentioned the comment or critique of a person named Gellman, that seem to have given a critical appraisal of the physics regarding the trajectories in the Wilson chamber. You brought it up in the beginning of your postings. In order to answer your flurry of criticism I ask you again if you could please give me the reference etc.

Thanks

erkki

Akira,

You also did not answer my question regarding your disproof of Newton's laws.

Akira,

You stated: Physics is not science. it is an over glorifed Alchemy. I call it Alchemetric.

In order to have some serious dicussion you must tag down and relax! Some scientists are arrogant and some are not, some experiments are crucial in advancing physics some are not, some fields demand a lot of money others have to fight to survive – these are well known facts, but totally irrelevant for our discussion!

Dear Akira,

I think that the ResearchGate is not a place for discussion of political problems. As to your scientific discussion, I should like to note, that these discussions are very far from the question, which was put. I cannot understand, how simple problems of geometry can be explained on the basis such complicated conception as quantum mechanics.

Yuri,

"[The] logical method of the geometry construction is possible, only if the geometry is continuous and it is unlimitedly divisible. Real space-time geometry is not unlimitedly divisible. It is rather discrete. It cannot be constructed by the logical method."

I disagree with the first sentence of the quote. There are many examples of finite geometries (e.g. among projective spaces or among Pasch-Peano spaces, to name a few types). I assume that you are referring implicitly to the fact that Euclidean geometry is fundamentally involved with infinity and the real number system.

That brings me to the remainder of the quote. In mathematical terms, you want to replace Euclidean space with a differentiable manifold, where the role of straight lines is taken by geodesics. I considered a similar problem in the early nineties, when I wanted to extend my results in "convexity theory" to manifolds. I was rather disappointed when it appeared that certain fundamental properties of convexity fail under this transition.

One failing issue is that, given a triangle ABC (with geodesics AB, BC, CA) and given two points D on AB and E on AC, it is not certified that the geodesics BE and CD will meet if the manifold has dimension >= 3.

Or, if you connect B geodesically with a point E on AC and if you take a point F on BE, then the geodesic CF is not guaranteed to extend geodesically at F in order to cross AB.

A simple drawing on a piece of paper will make it clear that these are things one may expect of a working geometry. They are part of the axioms of Pasch-Peano geometry, which is a weak form of Euclidean geometry.

FInally, differentiable manifolds are only an idealization of a physical reality which you assume to be discrete. Euclidean geometry is fundamentally infinite as it implies the real number system. One could try a vector space over a huge finite field to retain something of Euclidean space. However, such a field cannot be ordered, which is essential for "betweenness".

Your aim for a "non-axiomatic geometry" is probably your way of expressing these and other difficulties.

To end with a positive suggestion, have you ever considered the notion of tolerance space? It goes back to Zeeman (and even to Poincare). A tolerance is simply a reflexive and symmetric binary relation, interpreted as "undistinghuishable". A typical example is to take a metric space and a small real number epsilon>0. Then call two points undistinghuished if their distance is less than epsilon. You can combine the ideas of a manifold and a (locally) finite universe by using a tolerance. Similarly, you can try to combine exact Euclidean geometry with a tolerance.

Dear Marcel,

If you consider any logical construction as a geometry, you may obtain very exotic situations.

I am a physicist, and for me a geometry is a method of description of the event space (space-time). Of course, a mathematician may qualify any logical construction as a geometry. If anybody objects, that it is not a geometry, because it is not available for description of a real space-time, the mathematician may say: “If it is not available, then do not use it.”  Thereafter the mathematician will continue his development of the logical construction, because it is his profession.

In reality there are different properties of the Euclidean geometry: (1) the general geometric properties, which are take place in any geometry, (2) special properties of the Euclidean geometry, which do not take place, generally speaking, in other space-time geometries. One-dimensionality of the straight line (and of geodesic in Riemannian geometry) is a special property of Euclidean geometry. In general a segment T[AB] of straight line between point A and B is one-dimensional only in the Euclidean geometry. In the coordinateless form T[AB] has the form

T[AB] ={R|\rho(A,R) +|\rho(B,R) - |\rho(A,B) =0}             (*)

where \rho is the distance. One equation is described (n-1)-dimensional surface in the n-dimensional space, generally speaking. The fact, that (*) has a solution in the form of one-dimensional line is a special property of Euclidean distance (triangle axiom). One cannot use this property in consideration of geometries, which differ from the Euclidean geometry. You have used one-dimensionality of straight line in considerations of your post.. This circumstance devaluates your considerations.

Dear Akira,

Deformation of the Euclidean geometry (as well as other known geometry) is the most general way of the space-time geometry construction. Bending is a partial case of deformation. Deformation is a change of metric in the monistic representation of the Euclidean geometry, where geometry is described completely by its metric. Bending conserves the dimensionality, but deformation does not. My initial question concerned this connection between the metric and the geometry dimension,

Dear Akira,

Motion of particles takes place in the event space (space-time). The boundary between dynamics and space-time geometry is mobile. For instance, the motion of a charged particle you may describe (1) in the electromagnetic field in the 4-dimensional space-time, (2) in 5-dimensional space-time of Kaluza-Klein, where the electromagnetic field is included in the space-time metric and the particle motion is free. Istorically the boundary moved from dynamics to geometry. There were several steps of this motion: (1) conservation laws became describe properties of the space-time instead of properties of dynamics, (2) special relativity, (3) general relativity, (4) space-time of Kaluza-Klein. In the twentieth century this motion stopped, because our knowledge of space-time geometry are insufficient.

Description of space-time geometry is simpler, then description of dynamics. Geometry can be described completely by a metric (one function of two points). Dynamics of a particle is described be several functions. Such a description is more complicated, especially if  there are many sorts of particles.

In general, mathematicians introduce new concepts and quantities, supposing that it can to explain a nature of things. I myself tend to reduce dynamics to geometry, keeping in mind, that the geometry is described by one function (metric, or world function). Properties of one function can be easier investigated, than properties of numerous concepts of contemporary geometry, or numerous concepts of dynamics.

Dear Akira,

I prefer to use a coordinateless description. In this case one may not to use coordinate transformations at all. I hope that I succeeded, working in this direction.

Yuri,

" T[AB] ={R|\rho(A,R) +|\rho(B,R) - |\rho(A,B) =0} (*) "

I'm glad you mentioned this forrnula for metric (geodesic) intervals in metric spaces. In my monograph on Convexity Theory I spent several sections on "interval spaces" and reasonable requirements for them. Most examples are derived from metric spaces (as above), median spaces, ordered spaces, lattices, semilattices, and a few more. Intervals often have dimension >1 and may behave quite differently compared with geodesics.

Pasch-Peano spaces are characterized by interval properties (general betweenness axioms) and constitute a reasonable proposal for a generalized convexity. Convex surfaces, which are the boundary of an open (standard) convex set, derive such a (local) convexity from the surrounding linear space without borrowing from the geodesic viewpoint.

Dear Marcel,

I am surprising that everybody discuss problems of logical structure of a geometry, but nobody discuss problems, connected with monistic representation of the Euclidean geometry, where all geometric quantities are derivative, and all they are expressed via metric \rho (or via world function \sigma=1/2\rho2). Any possible space-time geometry G can be constructed by a deformation of the Euclidean geometry GE. One replaces the Euclidean world function \sigmaE by world function \sigma of space-time geometry G in all geometrical concepts (segment of straight line is an example of such deformation). One does not need to prove numerous theorems.

The obtained space-time geometry G will be multivariant and nonaxiomatizable, generally speaking. But this circumstance is of no importance, because axiomatizability is needed for construction of a geometry, because the axiomatizability is an attribute of the geometry construction, but not an attribute of a geometry in itself. If one constructs the space-time geometry by means of a deformation of the Euclidean geometry (but not as a deduction from axioms), the nonaxiomatizability does not create any problems.

Yuri,

I understand some of your explanations as a proposal for metric geometry with metric intervals. Tarski's version of Euclidean geometry can be transformed to axioms that start from a metric, from which to define equidistance and betweenness. It works perfectly with the Euclidean metric. I do not see how merely copying these (or other) definitions to a Riemann metric on a manifold  will accomplish a deformation of the Euclidean geometry (as you call it). Copying original definitions to "alien" circumstances does not automatically copy or transform original results into adapted results.

In addition, I have a few remarks on a possible misunderstanding of axiomatics, appearing from your statement (quote):

"axiomatizability is needed for construction of a geometry, because the axiomatizability is an attribute of the geometry construction, but not an attribute of a geometry in itself."

Tarski's Euclidean geometry (in a given dimension, say: 2) is completely fixed by its axioms. But most types  of axiomatizations are not that categorical. For instance, "projective geometry" (say: in two dimensions) determines a large class of both finite and infinite specific geometries. Similarly, Pasch-Peano geometry involves an axiom system leaving room for a class of wildly varying examples.

So, axioms are not needed to construct specific geometries (axioms just have to be verified on the example)  nor are they an attribute of a geometry: every mathematical theory has as an axiom system (a useless example being the whole theory itself, but every reduction of the theory to a set of basic truths can be useful).

Dear Marcel,

There is a misunderstanding between my approach and your approach. You discuss different methods of the proper Euclidean geometry GEconstruction. You consider different versions of axiomatics.

However, I consider the proper Euclidean geometry GE having been constructed already. I am interested only the method of GE presentation in a monistic form (i.e. in terms and only in terms of the world function \sigmaE.) It is of no importance for me from which axiomatics GE is deduced. Maybe, a simple example shows a difference between the two problems.

Let us formulate the condition of linear dependence between n vectors P0P1,P0P2, P0Pn. Usually this condition is formulated in terms of linear operations in a linear vector space. As a result some researchers think, that linear dependence is an attribute of the linear vector space. In reality the linear dependence is a property of geometrical vectors (g-vectors) AB, defined as a ordered set of two points A and B.

The only connection between mutual orientation of two vectors AB and CD is defined by the scalar product (AB.CD), which is defined via the world function \sigma

(AB.CD) = \sigma(A,D) +\sigma(B,C) - \sigma(A,C) -\sigma(B,D)        (*)

In the GE this definition coincide with conventional definition of the scalar product. This definition of the scalar product is coordinateless. It does not refer to linear vector space.

n g-vectors P0P1,P0P2,.... P0Pn are linear dependent, if and only if the Gram’s determinant

Fn = det||(P0Pi.P0Pk)|| vanishes Fn =0

According to (*) the Gram’s determinant is expressed via world functions of point P0,P1,….Pn

In other words, linear dependence of g-vectors does refer to linear vector space. It is a property of only geometry. To express all relations of GE via world function and to obtain a monistic representation of GE, one needs to know only geometry GE itself. But there is no necessity to know the axiomatics from which GE has been deduced.

Why does one need a monistic representation of GE. The monistic representation of GE is needed to obtain space-time geometries by means of a simple replacement of the world function in all relations of GE. The fact is that, the real space-time geometry is not known. To find a true space-time geometry. one needs to manipulate easily with a change of the space-time geometry. At the monistic representation of a geometry one needs only to investigate possible world functions and to choose an appropriate world function. After this one obtains a necessary space-time geometry. Let us imagine, that for obtaining any new space-time geometry, one needs to invent a new axiomatics, and to deduce from it a space-time geometry. It is a very difficult problem. Additional problem arises, because practically all space-time geometries are multivariant and nonaxiomatizable. There are no such axiomatics, from which one can deduce a necessary geometry.

Yuri,

Your previous post did clarify some of the remaining misunderstandings, but grosso modo I already got the picture you suggested. In several of my posts I explained that (owing to Tarski) Euclidean geometry can be expressed in terms of Euclidean metric only. Tarski's axioms could be used to verify this. As Euclidean metric is readily available, you can now go ahead with it and be sure that it ultimately captures anything you would otherwise have expressed in traditional non-monistic terms -- as far as Euclidean space is concerned.

After all, the position you take is not that strange. Often, a structure is being presented and one wants to study it. Then one needs the details of its configuration. In your case, these details include: a Riemannian manifold with an appropriate tensor and derived distance (or world) function. (At such an advanced level, people do not easily talk about axioms but, essentially, this technical data functions as your working axioms.) On such a structure and with such information, you operate with geometric ideas and Euclidean mimicry and try to get useful results from it. There is no pre-given package of geometric axioms.

Seems a perfect idea to me.

Dear Marcel,

Constructing a new space-time geometry, we modify the proper Euclidean geometry GE. This modification is very simple, if we use a monistic representation of GE. This representation admits one to start the modification of GE from the already constructed GE (but not from axiomatics of GE). In practice, using the Riemannian space-time geometries, we consider different versions of the metric tensor gik, but we do not interesting in axiomatics of Riemannian geometry. Essentially we use infinitesimal world function 1/2ds2=1/2gikdxidxk. Unfortunately, infinitesimal version of world function does not realize monistic representation of GE. I am happy, that you have understood my idea. I think, that you are the first person, who did this.

Yuri,

The involvement of infinitesimals (with ds2) is not much of a disturbance from the viewpoint of logic. Indeed, modern logic justifies infinitesimals as part of an alternative first-order model of the reals. Its first-order theory cannot be distinguished from the one of "real" reals.This is one advantage of Tarski's approach: it turns Euclidean geometry into first-order logic, and hence it is just as well connected with non-standard reals.

Your project looks challenging, and risk of failure is a vital part of every challenge. Be prepared to deal with results heavily deviating from Euclidean experience.  

Dear Marcel,

As I have understood, you consider the method of a space-time geometry construction by means of a deformation of some known geometry as complicated and unreliable. However, let me note, that contemporary method of a new Riemannian space-time geometry construction is essentially the method of the known Riemannian geometry deformation. Indeed, the metric tensor gik is a parameter of the Riemannian geometry. Nobody deduced the new Riemannian geometry from axiomatics. One replaces dS2=gikdxidxk by a new value of dS2=gikdxidxk and obtains a new Riemannian geometry. It is essentially the same deformation method, which works in the region of Riemannian geometries. Replacing infinitesimal dS2 by a finite world function \sigma one obtains the same deformation method, which is not restricted now by a region of Riemannian geometries.

"[...] you consider the method of a space-time geometry construction by means of a deformation of some known geometry as complicated and unreliable".

Not exactly. I am rather suggesting that the idea of "deforming" a standard geometry may be a rather naive view. Your basic idea is, as I already said before, not unusual in mathematics: to copy a concept to circumstances other than the intended originals. In the past, I applied this "method" more often (I think) than the average mathematics researcher. The results varied from disappointing to amazingly successful.

Strictly speaking, it is not the original Euclidean geometry that you deform, but the original metric. With it, you deform the initial information. My experience with logic is, that the result of changing premises can vary wildly. There is no such thing in logic as a "continuity principle" stating that "small changes of cause make small changes of effect".

My (very mild) pessimism is largely based on a negative experience with a certain (Pasch-Peano) property that I expected to be satisfied by geodesics. The effect of such a failure may be that 2-dimensional "geometric" subspaces may not exist locally. 

At the positive side, you will probably be able extract valuable information about space/time models. Einstein's idea about nonexistence of simultaneity could point at the absence of certain 3-dimensional "geometric" subspaces of the 4D-universe.

I would certainly not qualify your approach as "unreliable".

Sik’Slk’=\deltai           Sik’=d2S(x,x')/dxidx’k

Dear Marcel,I agree with you, that deformation of a geometry changes the representation of the geometry, because after deformation the geometry acquires some new characteristics, which were absent before the deformation. Many expressions make that circumstance, that a segment of a straight stops to be a one-dimensional set. I could not overcame this obstacle during many years.

In 1059 I was a student and investigated Riemannian geometry applying it to space-time. The world function has been invented by Synge. He used it for description of the space-time geometry. I obtained equation for S=\sigma of the form

SiSik’Sk = 2S,      (*)

where Sk=dS(x,x’)/dxk,         Si’=dS(x,x’)/dx’i

World function S of any Riemannian space satisfies equation (*). The question arose immediately. If the world function S does not satisfy the equation (*), then is S a world function of non-Riemannian geometry, or is S a world function of no geometry? I was a physicist and this question means a question “ Does such a space geometry, whose work function does not satisfy (*) exist in reality?”. I could not answer this question. It was solved only thirty years later, when I showed, that in the discrete space-time geometry the segment of a straight has a shape of a hollow tube and that free particle motion in the discrete space-time geometry is described by the Schroedinger equation. "Non-Riemannian model of the space-time responsible for quantum effects". Journ. Math. Phys. 32(8), 2092-2098, (1991). Electronic version: http://gasdyn-ipm.ipmnet.ru/~rylov/nmstrq1e.pdf

It meant that the world function, which does not satisfy (*), may describe a real space-time geometry. This situation resolved my hesitations.I admit, that approach of a mathematician may be another. He may be based on the geometry representations, connected with the axiomatics of Euclidean geometry, where segment of a straight line is one dimensional set.