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P.V. Poluyan,
Krasnoyarsk, Russia.

NON-STANDARD ANALYSIS
OF NON-CLASSICAL MOTION


DO THE HYPERREAL NUMBERS EXIST
IN THE QUANTUM-RELATIVE UNIVERSE?

This article is the text of the poster, which has been presented by the author on International Conference "Multidimensional Complex Analysis" Krasnoyarsk State University, Russia, August 5-10, 2002. The Russian version of the text is more informative. All the quotations in the article are rendered from Russian.

Author's notes:  Hyperreal numbers appear in Abraham Robinson's non-standard model of analysis as a result of extension of the field of real numbers, if the offence of the Eudocks-Archimed axiom is permitted. In other words, the hyperreal numbers - are a artificially-created abstract mathematical object, and if so, the question, stated in the title of this article, sounds at least strange.

In connection with this, it is necessary to explain what "existence" I mean, and why quantum-relative characteristics of the objective world are so important. That is why some philosophic introduction, which can be not read by readers who are not inclined to philosophy and methodology, will be made. After the introduction the material of the article goes as follows.

I. The basing of logically necessary connection between real and hyperreal numbers. Appearance of the hyperreal numbers in some concrete cases. The Maximum limit of the expending series: Fibonacci numbers and "the gold proportion", harmonic series and number "e". The compression of Dirihle function.

II. Non-standard transference. The Fractal trajectory the motion with the indefinite velocity-general definitions.

III. The destruction of the linear continuum temporally, CHRONOMETRICS. The notion AREAL MULTITUDE and its application to the time analysis. A multitude of norms.

IV. Same general theoretical concinsions. Variety of geometries and the solitude of the empirical space.

I must also note that as far as the material is represented, the readers will have thoughts about the shift to the adjacent themes, such as the p-adical analysis, the multitude theory, to the technical questions, connected with the mathematical apparatis of modern physics and others. However, we shan't consider those problems for the sake of saving time and space. On the whole, this article is just a thesis exposition of the extensive theme, which I call "NON-STANDARD ANALYSIS OF NON-CLASSICAL MOTION". I hope, that this approach will allow the researchers to develop this approach creatively as applied to the varied number of questions.


PHILOSOPHIC INTRODUCTION

We know, that physics as a strict science began with the basic work of Isaac Newton "The Mathematical bases of Natural Philosophy", basic not only for the theoretical physics, but for the classical mathematical analysis. Up to now in the manuals the notion "derivative" is explained by learners with the example of physical notions about mechanical transference of the material point and the instantaneous velocity. However, in modern, non-classical physics Newton's notions of velocity and transference are essentially modified. In relative physics not every ratio dx/dt is permissible - the maximum velocity point C is set, and in quantum mechanics the trajectory of the motion of a particle, where the moment of time and the co-ordinate are strictly bound, is replaced by quantum-wave notions with the definite ratio of Heisenberg uncertainty.

Thus, the harmony between physical and mathematical which existed in the classical science notions appeared to be disturbed. If someone asked in the century before last: "Do the differential functions exist in the Universe?" it wouldn't be difficult to define the word "existence" in such a formulation of the question. Many people believed, that "God speaks Mathematics" - Mathematics reveals the essence of the Universe, even if we don't understand it. In this case Hamilton's idea of that like geometry is the theory of space, algebra is the theory of time. The attempts of Gauss and Lobachevsky to define experimentally it Non-Euclidean geometry is adequate reality are as mach as remarkable.

Now another theory prevails: Mathematics is regarded as the supplier of the abstract construction for the theoretical modelling of the physical observation results. As Bertrand Russel said: "The Mathematical conception gives the abstract logical scheme, to which by means of proper manipulation the empiricist material can be fitted..." (B. Russel "Introduction to Mathematical Philosophy"). Now mathematics is not the language of Logos, Objective Spirit, but a symbolic science language to describe reality. In conformity with it, more and more abstract schemes, are being created, the mathematical conceptions, used by physicist - theorists goes further and further from the obvious simplicity, typical for "the mathematical bases of natural philosophy". It seems that the abstract objects take the part of the antediluvian elephants and tortoises, with the help of which ancient people "modelled" the Universe...

Since ancient Greek time there has been two lines: classical philosophy, preoccupied with searching for truth, and sophistry, preoccupied with composing inner logical schemes to prove everything whatever. Nowadays the latter dominates. It is considered that any inner non-contradictory, abstract mathematical construction can be used in physics. And it is because of this, divergence in non-classical notions of mechanical motion and initial bases of the classical analysis is not considered a serious problem. What is the problem? - to model there are mathematical apparatis of other kind, and for every case there can be found a more-or-less proper interpretation!

Can such a state of things be considered the only possible way of cognition, and the ideology proving it - the only right? My answer is no. Moreover, there are reasons to believe that this verdict is not just the opinion of a philosopher-idealist, but the reflexion of the particular intentions, typical of many people. Here I give two quotations.

Richard Feynman in his book "The Character of Physical Laws" says: "The theory, according to which the space appears to be continuous seems not right to me, because it leads to infinitely bigger quantities and other difficulties. Moreover, it doesn't answer the question what determines the size of all particles. I suspect that simple geometrical notions, spread over very small areas of space are not true. Saying this, I breach in the general notion of physics, of course, saying nothing about how to fill it in" (Richard Feynman, "The Character of Physical Law", London, 1965.)

And the following remarkable judgement was said in the famous book D. Gilbert and P. Barnice: "As a matter of fact, we don't have to consider that mathematical space-time nation of motion is physically interpreted in cases of arbitrarily small spatial and terminal intervals. Moreover, we have all grounds to believe, that striving to deal with quite simple nations, this mathematical model extrapolate facts, taken from one field of experience, particularly from the fields of motion within the limits of quantities, which are not available to our observation. Like water ceased to be water in case of unlimited special breaking up , in case of unlimited special breaking up there arises also something that can hardly be characterised as motion." [Gilbert D., Barnice P. "Bases of Mathematics. Logical Calculus and Formalisation of Arithmetic", M., "Nauka", 1979, h. 41, the first addition of the book was in 1934]

I am sorry for these big quotations, they are necessary to ground the main premises of the important problem:

1. These exists principal divergence between modern physical notions of motion and classical notion of analysis.

2. It is possible to build "mathematical model", which will fit to describe micro-motion within the limits of "quantities which are not available to our observation".

But actually the main thing concerns not a model, and not its building, but the fact that inside logic of classical mathematics itself it is necessary to find the bases for further development of the theory.

Nowadays there is an ideology which can be called model constructivism, but the real development of science goes other way, I would call this way logo genesis. That is new essences are "not thought up", but the bases capable to develop themselves in a sound mathematical science, which would be true, are found in the natural logical theory. It will take this philosophic approach we should agree with Feynman - classical analysis does not correspond to reality, but not because it is mistaken, but because in its logics logical possibilities, which allow to bring the mathematical theory into like physical notions, haven't been revealed yet. Introducing action quantity, Max Plank worried tragically, that he had to modify formulae with the reference to the experiment. Perhaps, his worries were not groundless, and the quantum number can be concluded theoretically - from logical bases, still being hideous and unrevealed I believe, that the matter is so.

But it would be too precipitate to declare that the mentioned at the beginning of the article non-standard analysis give us the necessary theoretical model and hyper real numbers are just right abstract object, which would allow to model quantum-mechanical discreteness. Thinking so, we stay within the limits of the model constructivism. Non-Archimedean analysis in its modern way is an artificial model, based on the direct negation of Eudocks-Archimed axiom, and there are no serious reasons to widen the field of the real numbers.

Indeed: what kind of numbers are they, if any sum of which cannot be more than one, and the inverses of which appear to be beyond the sigh of the infinity? Introduction of them is an arbitrary assumption, and the analysis model, neither with the empirical reality, nor with the theoretical physics.

But in the latter case we can find some interesting special features. In Einstein's Theory the rule of speed addition is used, when adding units does not lead to the endless increase of the sum, it is limited by the maximum velocity-of-light limit. But in this case the matter is not in the breaking up the Eudocks-Archimed axiom, but in the special features of Lawrence transformations, actual for pseudo-euclidean continuum of space and time. Obviously, it can be admitted, that the analogical rule of addition will work when dealing with simple quantities, such as the length or time spaces. But still, it is not clear why we must limit the endless space with some set radius, to which the sum of the added quantities would aspire. The prospect law exists, but we do understand that lessening of length within the distance is the optic illusion, but not the characteristic of the spatial metrics. 

Now let us take the quantum mechanics. It is known, that the so-called "ultra violet catastrophe" was the direct consequence from the formulae of the classical mathematical analysis -for the balance of radiation in the field of high frequencies the result was the endless quantity of energy. But the way out was found not in the modification of mathematical principles, but in realising experimental data: Max Plank's hypothesis put the limits to the endless energy subdivision - E=hn  appeared to be non-divided. And at the moment the clinical formulae of analysis being used, and what concerns all "disturbing" modern physic-theoretic learnt as Richard Feynman said, to "sweep them under the rug".

Thus, the theoretical situation can be characterised as follows. On the one hand, the classical analysis is not enough for physics, though its original notions seem so obvious and natural. On the other hand, "non-standard" seem suitable for physics: actually, endlessly small ones somehow "quant" the continuum on the micro scale, and hyper reality (the notion from Martin Davis's book "Applied non-standard Analysis) is divided into "micro world", the world of "actual scales" and the world of "cosmic" infinity. But, non-Archimedean analysis is still an artificial structure, this "non-Archimedean" logically to the endless division and the classical notion "limit". Thus, the only way out can be logical adding of a non-standard model of analysis to the classical analysis, finding out their necessary connection, and if it is necessary -their supplement. The appearance of the irrational number did not abolish the national numbers, just as that the introduction of hyper real numbers will not become a declarative model structure, but natural leading out them of the logic of the classical analysis. I am going to show that this task formulation is right. 


 I. ACTUAL INFINITY IN THE TRUE SENSE OF THE WORLD.

"Analysis" - is the calculus of infinitesimal, and the basic notion of the analysis is the notion of a real number". This statement in Gilbert and Bernice's book is followed by the more precise definition: "The notion of endlessly little and endlessly big are excluded by the theory of the real numbers in the true sense of the word from consideration". (The same edition, ch. 2, p. 4 "Non-finite Methods of Analysis")

Non-standard analysis, just the other way round, includes into consideration infinitesimal and endlessly large in the true sense of the word, hence the notion of hyperreal numbers appears. Thus, if in microscale by unlimited breaking up we really hope to get the breach of continuity of the continuum (as R. Feynman expected) and to discover "something of the kind that can hardly be characterised as motion" (as D. Gilbert and Bernice foresaw), it must be connected with the endlessly little scale - with something quite actual. However George Cantor in his "Studies of Multitudes" agreed on the irrelevance of such notions. He asked rithorically if it was not possible to continue numbers not only in the area of endlessly large, but also in the field of infinitesimal. And he said emotionally: such attempts are "violent", they have "unstable foundation, they are "groundless" at all and etc. [G. Cantor Studies of Multitudes. Russian edition in the book "New ideas in mathematics", №6, S-Pb, 1914, p.15]

Cantor formulated the direct prohibition: "There exist no linear number quantities, different from null (the numbers, presented as limited non-interrupted rectilinear segments), which would be less anyhow than the least number, that is quantities contradict the notion of a linear number quantity (G. Cantor Works on the Theory of Multitudes. M., "Nauka", 1985, p. 294). The prove of this thesis is based on Archimed axiom and his conviction that it is not natural to try to introduce actually endlessly little quantities (in his opinion-such structures "remain only on paper".)

On the same page Cantor compares his transfinite theory with the hypothesis of endlessly big and little numbers, proposed by Fontenelin his time, and notes: "One cannot say that growing last numbers reach W as close as they can it is most likely that any as big as it can be number V remain as far from W as the least number. My least transfinite number W, and thus all big ordinal numbers are exclusively beyond the endless number series 1, 2, 3, ... Fontenel's mistake was that he looked for the transfinite number within the series 1, 2, 3, ... V ..., though somehow at its end (by the way, while there is no such an end). After he introduced non-solved contra Fontenel in advance in his endless numbers, the future of his fruitless theory was evident. But if the critics, tempted by the crush of Fontenel's endless numbers, think to bring in the verdict to actually endless numbers at all, they will be stopped by my radically different from Fontenel's and quite non-contradictable theory"[the same edition, p. 294]. 

Criticising Fontenel for looking for endlessly big numbers within dispersed number series, Cantor himself places actually endless numbers "at the end" of the gathering series and successfully proves that they are not there. But actually endless little numbers can be placed beyond the number series of the real numbers, and then Cantor's verdict will be groundless. It is characteristic that in the notes to the quoted volume of very interesting to real Fontenel's book from the point of view of non-standard analysis" (the same edition, p. 408)

However, it is necessary to precise. In the above-mentioned book by Martin Davis "Applied Non-Standard Analysis" (Martin Davis. Applied Non-Standard Analysis. N. Y., 1997), hyper real numbers are interpreted as ideal elements -like endlessly remote points in projective geometry of their appearance in mathematics. But the creator of non-standard analysis, Abraham Robinson, was of different opinion himself, and if we remember the famous aphorism of L. Cronecker, natural numbers are created by God, and any the rest people made, the question about artificiality and natural origin is the matter of taste. In principle, the only criterion is the ability of this approach to enter the science. Cantor's transfinite open a wide field of applying creativity for us, do hyper real numbers give us such a possibility".

Let us see what ideological basis the traditional notional notion of endless division of non-interrupted continuum has.

Here is the division of a one-unite segment on two halves, the usual gathering series +   + +..., the sum of which is one. We can see how the points of division gather at the beginning of the segment-nil, which remain unattainable while any point of point of division is always separated from nil by a segment of a significant length.

Picture 1.

 

The process of half division is at the same time the progress of doubling, that is the initial single segment is the same segment, like any part of it, and the whole picture will be seen if we continue the series of segments in the direction of their increase: ... ⅛, , , 1, 2, 4, 8, ...

 Picture 2.

In this case   appears also unattainable, like nil. But, nil we can see with our own eyes,   we cannot, it is situated somewhere beyond the edge of the screen.

The fact that one gives us point, according to which the increase or decrease happens, it is evident. The shift of this point does not change the ration of the adjacent quantities, but the possibility of the shift itself determines endlessness of the division: any small segments, closed to 0 can be considered as one (single). We can point "0", because it is the end of a chosen single segment, in comparison with the point of endlessness" which is not the end in its definition. But to what degree is it right to understand 0 as the END? It is clear that the definition is connected with leading out segment, but its division is one thing, and building the confession of segments is the other. And if we take the growing segments the right, we small not reach "the point of endlessness", and if we take the growing smaller segments to the left, we also shall not be able to finish building structure at any point. This should not confuse us at all, as the point 0 is set on the straight line and to drew this structure the limit is set. But structuring on the straight line is the position of points in the definite concession, the matter is prettier different, when building structuring becomes real.

Let us think that segment 1/2 is built as a perpendicular to the single segment, and segment 1/4 is built as a perpendicular to segment 1/2 etc. 

Picture 3.

In this "snail" there is nothing remarkable; we can build series, which give co-ordinates of the point, to which the end of the broken line must come. But here is the question: from what side does the broken line come to this point-"from above", "from under", "from the right" or "from the left"? This building set 4 directions in comparison with the traditional one. The question is not senseless. If uniform motion of a point happens on the usual straight line, and this point successively covers half a distance, then a quarter of the distance etc., how shall we interpret uniform motion of a point in case of the rectangle spiral? If we suppose that a point moves along the broken line, and it covers the same distance for the same time, its coming to the centre of "the snail" is so, that it would be impossible to indicate the vector direction of its velocity in this point. It may seem that there is nothing strange, as at any point of trajectory breach the moving point has two inter perpendicular velocity vectors, but the limit point cannot be the breach point? To make everything said above sound more clearly, let us imagine that "the snail" does not consist only to a segment of trajectory from 1 to 0, but it has a continuation from 0 to -1.

The point o appears to be such a "place" of a broken trajectory where its rector direction becomes indefinable, and its motion in this point can hardly be characterised as motion at all. For all that, we cannot reach area of "micro scales" at any part, the difficulties appear only in the point 0. Can we say that this special feature is typical of 0 in case when the motion is set on the straight line, but not on the broken one? No, till the motion itself is connected with the moving along the segment. But if we start to speak of INSTANTAEOUS VELOCITY, of velocity in the point- there arise questions again. In fact, a problem arises: from which side shall we pull the segment, striving to 0? How should the ration of differentials look in this case?

Picture 4.

And again it may seem that there is no problem: if a point moves from 1 to -1 through 0, from common notion to particular we can conclude that at any point of the trajectory (including 0) its direction remains the same. If we suppose that in the point 0 its motion direction is indefinable, from particular notion to common one, we have to speak of indefiniteness of its motion vector on the whole along the segment and this contradicts the set velocity vector. However has such logic's the right to exist in the non-finite reasoning.

The problem becomes more acute, when in the position of non-differentiated 0 appear all the points of the trajectory. Let us see Wan-der-Warden figure, made of an equilateral triangle, when each its side is divided into 3 parts, to which one more side is added, making on each side a new triangle. 

Picture 5.

As it is known, in the limit we get a figure in each point of which "there is a breach", and the whole length of this endlessly broken line strives for the endless quantity 3(4/3)n , with n striving for . When we set the point motion along the single segment we are not confused that it is a part of an endless straight line, but can we speak of the point motion along such a trajectory in case of Wan-der-Warden figure? We can see all the points of it, but the paradox is that between any two points there is a distance, which is endlessly big. If we "straighten" the broken line between such points it will become obvious. And if we consider Wan-der-Warden figure actually set with all its curves, its straightening will give us "in the limit" an endlessly big triangle. In the opposites direction: it is possible to make of the set triangle with its single side an endlessly broken figure if to divide its sides not into 3, but into 4 adding 2 central parts as the sides of a triangle. Finally we get a figure: 

Picture 6.

*

Where the point is not a point at all, but an endlessly broken line of the finite length, situated within the limits of the point. If there is a point, which covers a usual segment (a single side of a triangle) for the finite time, it must somehow move along Wan-der-Warden figure trajectory, appeared from the triangle. I let the readers judge what is the direction of the point motion along the finite distance for the finite time. Maybe, the point, which moves in such a way, is stable, because it does not go further from its place at any distance, measured by the real number?
All in all, maybe a standard single triangle is not so simple in reality, as it is considered? There is a childish paradox, when we prove, that 2=1, as the sum of the two sides of a triangle is equal to the third one. These two sides are broken (for all this, the sum of the sides, composing the broken line, is unchangeable) and it is declared that within the limit a broken line appears which coincides with basis of the triangle. 

Picture 7.

This paradox is, as it is known, imaginary, but it can be considered more serious. Let us suppose, that we have "straighten" this endless broken line of a finite length, we can say that the two sides of the triangle are also broken lines, which can be "straighten", making two more triangles on the sides of the previous one, etc.

Picture 8.

In other words, the segment of a straight line, which forms the single triangle foundation would be equal not to the length = 2, but to the length = 2n , where n strives infinity. In fact, we make the following operation: we say that a multitude of micro triangle can be adjacent to the single triangle foundation, these micro triangles can be conclussively "straightened", forming a line of non-finite length.

If when we divide segments, we try to find the find the field where endlessly little quantities are found, in the latter cases we have endlessly big numbers -  3(4/3)n and 2n, expressing endlessly big number of single lengths with n striving for . Usually it is considered, that the power base does not play a special role as the power strives for higher and higher order [see J. E. Littlewood "Big Numbers" in the book J. E. Littlewood. A Mathematician's Miscellany. London, 1957]

But English mathematician, making "big numbers" equal, takes the word "equality" in brackets. [The Russian edition Дж. Литллвуд. Математическая смесь. М: "Наука", 1978, с.108] really our appreciation is determined by the fact that the comparison of non-finite lengths seems unbelievable. Let us draw a scheme, where the non-finite length would be seen "with our own eyes".

If a non-finite fraction (for the sake of the simplicity let us take a periodical one) 1,111111... is a gathering series 1 + 1/10 + 1/100 + 1/1000 + ... , the dispersing series 1+10+100+1000+... can be written as a number 111111..., where the first number is the number of unities, the second - the number of tens, etc. We can "grow endless tree", a count, the length of which is equal to this number. From the unity segment come 20 branches with the length of 1/2 each, the sum of which us the length of 10, from each come also 20 shoots with the length of 1/4 each, etc. If a bug crawled all the time up, it would reach the top of the tree for the finite time, as its road is 1 + (1/2 + 1/4 + 1/8 + ...) = 2. What is the number of the sums, sitting on all "last twigs" of this "witch's whisk"?

Picture 9.

The impression is being made that there are "ecological niches" for the hyper real numbers. Can we somehow "expose this enchanted ground"? What happens in the 0 area where endlessly dividing real quantities cannot reach? And what can happen with numbers i8n the transfinite area, where no big number can reach.

If we mark the points, corresponding to Fibonacci Series on the numerical line, where the next point is the same of the two previous (1, 1, 2, 3, 5, 8, 13, 21, ...), in the limit with striving for the area of growing numbers, the ratio of the two last Fibonacci numbers, as it is known, gives us j -famous irrational number 1,61803... It sets "the golden ration" - the section of the segment, the smallest part of which is related to the biggest, as the biggest one to their summoned length. It can be declared that moving along the numerical line through Fibonacci numbers; we shall discover infinitely big "segments" in the transfinite area, the ratio of which is expressed by the irrational number j .

And vice versa. It is possible to build a number of segments, corresponding to "the golden ratio" in the real area:

Picture 10.

As the ratio of the biggest segment to the adjacent smallest is 1,61803..., their summoned length in the left direction with have quite a definite utmost end point. The growing less segments will "curve" in its surroundings these segments, according to the infinite division of the non- interrupted continuum, will never stop diving. In this building the utmost maximum point will never be reached, but we can state that in this endlessly small surrounding near the utmost point a wonderful thing happens: instead of the uninterrupted continuum the reappear numbers, which would come to the utmost point like the growing less Fibonacci numbers. And as Fibonacci series begins as 1, 1, 2, 3, these numbers (and actually endlessly little hyper real lengths corresponding to them) will come to the utmost point (limit point).

I could put a "dot" here, but I want to draft some prospects of development of this approach. E. G., it is interesting to imagine how Dirihle function would look like, if its unity strove for rule and turned to the hyper real area of actually infinite unities?

In this light it is interesting to see the harmonic series of the whole numbers 1, 2, 3, 4, 5, ... Evidently, in the endlessly large limit a ratio are related to the actually infinite segment of equal length.

The process seems unchangeable here, and in reality the series of unity- length segments does not give us the utmost point, near which in the hyper real surrounding a harmonic number series is built. Fortunately, here we have properties of other kind. Though we cannot see the area where the actually little lengths, forming a harmonic number series, are situated but we can see the infinite straight line, on which even one-unity segments are marked and we can take the infinite half-straight line, beginning from any of the segments. On it the adjacent segments are related to each other as N + 1/N, where N is infinitely big number, expressing the sum of actually little lengths. That is, a geometrical progression is formed, where the multiplier is 1 + 1/N, and if the length of the first segment is one, the growth of the length happens in such a way, that the length of "the last one-unity segment" on this endless half straight line will be (1+1/N)N. It is not difficult to note that this length is e.

Let us interpreter this result.

Let us suppose, an endless number of points comes out of the co-ordinate base, the velocity of the first one is 1, and the distance, covered by them for a unity of time, are consessively different from each other, and the difference is an infinite little unity quantity. On what segment are the points in a unity time period?

When I asked this question, I omitted one thing: I did not say that it was necessary to make all vectors be directed in one direction-along the straight line. But is it possible to set single direction? 


II. INDEFINABLE VELOCITY MOTION

In the above-mentioned buildings motion, which is a moving of a point along the trajectory at same velocity, played an auxiliary, illustrative part. Now we are going to bring more sense to this notion.

Mechanics begins with the notion of the uniform constant velocity, but in case of the constant velocity striving of the related time and distance intervals for the infinite little loses its sense-all intervals are alike. And though we make intervals X and T strive for 0, we always mean that there are two points and two time moments, the range between which is uninterrupted. From the mathematical point of view all the finite segments of a straight line are equal, but what motivates the notion of "infinitely littleness"?

Nevertheless, it fits the every-day practice so well that there are no doubts. But we can change the logic connection, and say that the every-day practice itself predetermined the mathematical conception, with the help of which it is modelled. 

Surely, we can distinguish the notion of the real motion and its mathematical model but doubting if the latter is adequate, at least, we must propose the other way of theoretical modelling. And for all this, we have to begin with the same elementary premises: any kinds of the mechanical motion are moving of a material point in space (which is, to say roughly, in different time moments in different places), the position points are always divided by some distance, and position moments set the time intervals. The most interesting thing is that all these initial premises help us to form quite a different notion of the motion, opposite to the traditional one. 

So, the two points of space are given Xa and Xb, in which a material point is in two different moments of time Ta and Tb. These two, let us say "positions", let us check the ratio of the distance segment and time interval, which is called "velocity" by us. If we say in the limit of the first Newton-Galilee Law, the motion is uniform and rectilinear. It means that all such segments between the positions are strictly alike for the set constant velocity. At the same time, we think it is necessary to introduce the notion of instantaneous velocity, striving the intervals for nil, where in the limit by some strange way "infinitesimal" appears. There are two thoughts:

1. If the velocity is constant along the whole interval, it is typical of the point at any spot moment of time, at any point of the trajectory.

2. If at any moment at any point of the trajectory the velocity is the same, it is typical of the material point during all the time its motion along the whole trajectory.

Obviously, these are two different logical approaches.

Here we can remember Zenon's "Arrow". The ancient-Greek philosopher wanted to draw the theorists' attention to paradoxical fact of motion-moving: if to define velocity it is necessary to have two positions, and TWO time moments, how can we conclude that there is velocity at a definite moment and at a definite place? Nit is clear, that introduced the instantaneous velocity we "hid" this paradoxical fact. However, if dx and dt are "too little" they nevertheless, remain "segments" and "intervals". "To strive for the point-does not mean "to be at the point".

It is considered that Aristotelian physics coped with Zenonic paradox. It is clear that the philosopher was mistaken when he said that in a moment at the point there was "no motion". We admit that, if motion exists in general (in the multitude of moments and places) it exists in particular-at any single moment. If motion exists, it means that velocity also does, if motion exists in general, it does in particular, and thus we MUST admit, that a point possesses velocity at each moment and place.

Now we shall see the model of motion, where there no such logical duties. That is at each moment at each point there is only motion, but no velocity.

Picture 11.

Let velocity be a ratio at XaXb segment to the time interval TaTb.
Fixed this ratio, let us take the time moment Tc, situated between Ta and Tb at this moment the point is at some Tc and, correspondently, we have got two new segments, two new intervals. Speaking constancy, we suppose that the ratio of new segments and intervals would give us the same result of velocity. We are making the logical choice: there are two variants, either Vab=Vac=Vcb, or they are not equal. This choice seems be true. And really, if we set velocity Vab it says that there is such a velocity at the points A and B, and different from the initial value. Having chosen the point Xc and the moment Tc, we did not use all the points of space and all the moments of time. If we continue the choice of time moments, all of them give us different velocities. In other words, for the initial position Xa and Ta (and in final position Xb and Tb) we shall get new velocity values. That is the value of velocity "at a definite point at definite moment of time" - generally should be considered INDEFINABLE.

So, let us introduce the next absolute rule: no matter what the initial ratio was, "new" velocity (Xc-Xa)/(Tc-Ta) and (Xb-Xc)/(Tb-Tc) in the general case would be ANY. In other words, we declare, that any time there appear new values of the ratio DXi/DT, which in the general case should not be obliged to correspond to the previous ones, and are not obliged to be connected with them by some law. This rule must be true in case of "any little" division of the initial interval of time. And naturally, in the general case, the corresponding points of the position in space cannot lie on the one straight line, though any time they set the finite distance segments. In its turn, the particular case of the co-determined motion would be a standard uniform motion along a straight line with the constant velocity (if "any", so, possibly, "equal" in case go equality of the corresponding time intervals).

Thus, for any two-time moments there are two positions of the point in space, what sets the value of velocity exactly for these two moments. But for it any position of the point, corresponding to the time moment between the two initially chosen ones, let us find different ratios of distance intervals and time. On the whole, for any single time moment there are definite co-ordinates of the position and quite an indefinite velocity (definiteness appear, if and only if we chose one more moment-position). All the variant of non-uniform motion is also particular cases of motion with the indefinable velocity. 

In the above - stated building there is nothing unnatural, alien to the initial premises of understanding the mechanical motion-moving and to the principles of its theory, and if such an approach is logically permissible, we have no right to disregard it. And the most important thing is that this logical variant is more GENERAL as "equality" of value-is a particular case of all their possible interrelations. That is why our model of a priori is more general, because it covers the standard velocity notion.

I do not argue such a non-standard model of material point motion in space is extremely exotic. More than this, the suggested approach is completely different from the classical one: in case of the standard approach the constant velocity is the base, on witch any particular cases of non-uniform motion-with acceleration, with curved trajectory are constructed. In our case, it is visa versa. The base is the model, which can be characterised with uniform, uniformly accelerated motion etc.

The main figure of the given model is that there is no definite velocity at any point. This indefiniteness is originated in the model: between two close time moments there is always an instant, to which a new position of the point in space with the new value of velocity corresponds.

Such a succession of operations of determining the values of velocity is in principle infinite we cannot speak of any standard differentiation, any instantaneous velocity. The motion trajectory here is like the mathematical fractional broken line heretically curving at any little part. (And so-called "straight line" is a particular case of a fractional structure). At each moment a material point is at the definite place, all positions lie on the definite (fractal) trajectory. Chaotically placed positions are points in themselves, of which such a non-interrupted trajectory consists. Let us be, in this particular case, it can be a straight line with the constant value of all possible ("any") velocity, but then it is a straight line of principally another kind: the operation of differentiation for it, which leads to the instantaneous velocity, loses its sense just because the trajectory of motion and time of motion are initially set by points - completely apart, discretely. Such a motion is absolutely fractional, it is split up into the endless multitude of segments X and intervals T and not because the non-interrupted segmented is divided till infinity, but because all the points of definition form it themselves. 

So the reason of alternative of our structuring becomes evident: the traditional notion is based on understanding the segment, which set the points, limiting itself, and our non-traditional understanding is bases on the points and moments of positions, any pair of which set the segments-intervals, found between them. The common thing between these two alternative variants remains that the succession of positions in space is comfortable to succession of the time moment, which correspond to them.

I shall note that in the suggested model of a material point the elementary notion of velocity does not disappear-velocity is nominally determined for any intervals X and T, but it is impossible for the single moment points, which form these intervals, to have this value of velocity. Thus, the notion of velocity is necessary for our model, but it is just an element of the process description, and it is not its direct reflection. 

I realise how unusual the suggested motion model may seem but I want to underline once again: it is formed of the same basic notions as the traditional one (points of position in space, time moment etc). It is a logical alternative to the latter and if so possesses theoretical equal rights. Yet we do not consider its physical essence, its empirical adequacy, we do not speak of motion formula, of quantum-discredity or of the ratio of indefiniteness. Like the classical dynamics interprets different variants of motion, and the standard mathematical analysis lets us to describe them, the just introduced motion with the indefinite velocity will also demand introducing some dynamic characteristics. Yes, the position-points are scattered chaotically, fractionally in space like the broken beads - but there must be a thread to join them.

As the reader might feel, the main difficulty in this approach is the ideology of the classical mathematical analysis. It happens that its powerful apparatus is not suitable for our purposes. Let me quote Abraham Robinson's words: "We are going to show that in the present limits we can develop a number of endlessly little and big quantities. It gives us an opportunity to formulate many well-known results of the function theory in the language of endlessly small unities in the way it was foreseen in the indefinite formulation by Leybniz." [Introduction to the theory of models and meta-mathematics of algebra. M: "Nauka", 1967,p.325] and more: "Non-standard differentiated calculus can complete in simplicity with the most orthodox approach [the same book, p.340] and about integration "Our limits of dividing into intervals of an equal length is too artificial. We will build an approach, which will let us consider the more common divisions" [the same book, p.341]

The presence of the non-standard model of analysis in the modern mathematics indicates that there are no principles logical prohibitions on our way. Let the new notions of motion seem if not absurd senseless, but useless and artificial. They are just not usual and not typical.

In 1963 Leo Mozer showed that if a ray of light falls at an angle onto two glass plates, put together, a different number of possible ways appear, depending on the number of the reflections of the ray. When the value of the number of the reflections are bigger, the numbers of possible ways form Fibonacci series (The example of Martin Gardner from Scientific American. Russian translation: M.Гарднер, Математические новеллы. М: "Мир", 1974, с. 398) The suggested non-standard approach may, evidently, seem productive for the interpretation of the quantum-mechanical events, but this model of motion contradicts to the theory of relativity conclusion, where the variant of the radio dx/dt are limited by the maximum limit C - the light velocity. At the same time, the law of addition velocity, as it has been already noted, breaks Eudocks-Archimedean axiom. And though the law itself is the sequence of Lawrence's modifications for pseudo-euclidean time space, the non-standard approach let us see the main point differently.

Nothing stops us to turn over the ratio and say that non-archimedean adding of quantities is the first cause, and pseudo-euclidean space is the model, which reflects this more fundamental ratio. In other words, for any quantity from nil till infinity according to the linear law we can introduce an imaginary additional co-ordinate axis and a co-efficient of the transition of this quantity to its imaginary measure. By this we set the transition law, according to which adding single unity quantities will be realised but by the non-archimedean adding a question arises: if velocity is a ratio of a distance to the time period, how must we determine the velocity of quantity range in ration to itself. And the main thing: the co-efficient C - is the empirical constant, and it would be too independent to look for the mathematical bases for its introduction.

Nevertheless, we shall try to do it.
Let us begin with the basic mechanic notion-with the principle of relativity.

The essence of the principle of relativity is simple: there is no absolute motion, two points can be move only with regard to each other. If we take one of them for the standard point, we believe it is stable, and the second one moves with regard to the first one. And visa versa: we can take the second moving point for the stable starting point and consider the first one to be moving. The notion of motion quite naturally and necessarily requires the principle of relativity as the distance change between these two points happens BETWEEN THEM with some time. Sketchily the principle of relativity is explained with the example of two points:

Picture 12.

We take one of them for the starting point, the other moves with regard to the starting point, and visa versa. Let us imagine, in space there are two points (mathematically size less), separated by some distance. Now let us try to imagine that this distance changes... But how can we check this "change"? Anri Poincare, illustrating these cases, made the imaginary experiment-he asked: what would happen if the distance between the two points becomes twice bigger? And he answered: the world would not notice it. I think it is clear. To be able to speak of the change of the distance between the two points, there must be one wore point which would be stable with regard to one of the two given points.

Picture 13.

"Stable" means "to be situated at the same distance from it all the time". There is no difficulty, we just declare, we need not the point, but a starting system with the set length standard. We began with only two points, then added the third one and now we can speak of motion, but someone can ask: "How can we determine, that the distance between A and B is constant, and that between A and C the distance changes?" You see, we can take the distance BC for standard, and the former one can be considered changing. In such judgement there is nothing illogical. On the contrary, we have introduced the third point and the standard distance because we could not check the distance change, but we cannot check it in two ways: in one way we take AB for the constant standard and say that the point C moves away uniformly from A and from B, in the other way we take the AC distance for constant, then the former standard distance AB should be treated as changing.

Picture 14.

But we change places of the length standard, a strange thing happens. Let us imagine that "uniformly moving" C is stable and sets a distance standard = const, then "really stable" with regard to this standard would move not uniformly: B comes closer to A, slowing down all the time. In the most absurd variant it accelerates from nil till infinity, then comes from the infinity from the other side and begin slowing down till nil again - for the rest of its infinity.

The above-described conclusion seems so ridiculous, that the first wish is to give it away. The problem is, if we open inter equality of the two points in the process of their imaginary interchange in the Galilee-Newton principle of relativity, why in the logically necessary system consisting of the points should we neglect the same interchange? Logical possibilities arias not to be given away, it is necessary to try to understand what happens in this strange situation. Is the matter, perhaps, in the wrong interpretation of the result?

At first, in the "ridiculous" variant we got the notion at all the possible velocities. That is, this "crazy" point begins with the minimum distance (equal to the set one). Then it runs all the possible values of velocity till the infinity, then comes "from other side", slowing down again till nil (on condition that, we began with some moment, and the whole point while moving closer to the starting point, and coming up to it comes further from it, moving away).

At second if look at it more carefully, the standard variant is not very simple. If we have only one set uniform constant velocity, its quantity expression can be dual. Velocity as the ratio distance segment to the given time unity [m/s], and quite an equivalent ratio of the time period, spent on covering one unity segment [s/m].

Let us answer a simple question: why in the usual sense of motion is the alternative dimension excluded, why do we not express velocity as an amount of seconds, spent on covering of a unity of distance? You see this ratio is logically admitted, and mathematically it is quite individually for each concrete velocity.

Does it not surprise us, that in the stadium the judges express sports result not in the numerical value of a runner's velocity, but in the quantity of time, spent on covering a distance? You see it is the unique fact: the motion is measured not in meters for one second, but in time, which is required for covering a given distance Nevertheless, in physics the given measurement of motion with the dimension [s/m] is rejected. Why?

It is possible to give quite a serious answer to this "childish" question. People order lots of possible velocities by a principle "slower - faster", and, in compliance with this, they build them on the vector "less - more": the faster velocity is, the numerically more it is, - a lot of meters is covered for a time unity. Taking the other measurement, we shall meet a reverse ratio: a smaller number would correspond to greater velocity,- the faster a material point moves, the smaller amount of seconds is requires to cover a distance unity.

The traditional spectrum of velocities begins with nil and quantitatively grows in the process of increase - fastening of velocity (in the classical mechanics the maximum velocity limit is unlimited). The "fastest", infinitely large velocity is an infinite quantity of meters for a time unit. But with the alternative dimension [s/m] everything is precisely on the contrary: the stability is an infinite quantity of seconds, spent on covering a distance unity, so to say, the infinitely large slowness. You should admit, that to count from infinity to nil is, at least, not convenient.

It may seem that our reasoning is groundless. However, it is not so. It would be enough to say, that when Gotfrid Leibniz was creating the mathematical analysis, he thought this question over many times. He wrote: "The stability can be considered an infinitesimal velocity or the infinitely large slowness" (G. Leibniz, The compositions in four volumes. Т. 1. M.: "Мысль" p. 205. See also T. 3, p. 199.).

Leibniz has one more remarkable reasoning: he identifies zero velocity of motion along a circle with infinite velocity, when "each point of a circle should always be in the same place" (Т. 3, p. 290). That is, not only 0 m/s and   s/m (accordingly    m/s and 0 s/m), are logically identified, but also 0 m/s and    m/s in case of their cyclic motion. This last identification gives us a way out from the confusing situation.

Why it is not convenient to count the increase of velocity of motion in the measurement [s/m]? Because attributing an infinite slowness to the starting system and introducing a certain single slowness 1 [s/m] for a moving point, we shall not get a uniform scale of quantities, where it is possible to add arithmetically A [s/m] + B [s/m] = (A+В) [s/m]. That is, such an addition will contradict the natural notion of how the velocities are estimated when changing one starting system to another. But the matter would radically change, if we use Leibniz transformation.

Really, when in a classical principle of relativity we revealed the necessity of introduction of the third point which specifies a constant measurement of distance, this third point served a prototype of stability - for any period of time it "could cover" only a zero distance. If we, after Leibniz, equal stability and infinite velocity of cyclic motion, we shall find out an interesting thing: having attributed infinite velocity to such a stable point, we together with the measurement of length introduce also a measurement of a circular trajectory, the length of which is determined by a measurement of length as by radius. Then it appears, that in a measurement of slowness [s/m] this velocity will have not infinite, but zero slowness: to cover this radius it requires zero seconds. Now we can already conduct normal addition of slownesses, but a single slowness will be considered 1 second, required for covering a single circular trajectory. Accordingly, covering this trajectory for 2 seconds gives other quantity of motion velocity - a slower one etc. For all that, relativity in such circular motion is completely saved, and "slownesses" can be added arithmetically. In other words, now the normal axis is being built for slowness quantities, where the starting point goes from zero till infinity. The fact is that not velocities of linear motion strive for an infinite slowness - for complete stability - along a straight line, but velocities of motion on a single circular trajectory.

And now is the most interesting thing. If for such a quantity as slowness non-archimedean law of addition also works we shall not be able to reach an infinite slowness. There should be topside - the limit of a slownesses which is so unattainable, as velocity of light. A measure unit of this limit will be, naturally, [s/m] - that is, the quantity opposite to a measure of velocity. And if the empirical velocity limit C really exists and is measured in [m/s], there should be a certain empirical constant, measured in [s/m]. It would be very poetic to call it, let us say, "velocity of darkness", but we shall not run into such mysticism, as the required constant in physics is known, it is formed of a ratio h /e2, where e is the charge of  an electron, and h is the Plank constant. And the ratio of velocity of light to the given combination of empirical constants gives us a dimensionless quantity, called a constant thin structure. Its quantity in round figures equals 1/137, and till now attempts are being made to express this number through a combination of mathematical constants p  and  е. Now we can approve, that these attempts are not deprived of the bases.

There is a question: does all the above-stated mean, that for the abstract continuum the natural metrics and real law, which orders increase of quantity in the field of real numbers, settling down between unattainable points 0 and ? I believe, yes. But to show it precisely, it is necessary to understand: what is the linear continuum? If speaking of space, the essence of the matter is more or less understood, concerning time, the matter looks not so clear.


III. CHRONOMETRICS. AREAL MULTITUDES.

Unfortunately, the metric properties of time, in comparison with its orientation and fluidity, attract attention of the theorists in the last turn. There is an important reason: here time as such is easily identified with space - with one-dimensional linear continuum, therefore there is not anything specifically temporal here.

The attempts are known, which give a logic substantiation to that the time base is a linear continuum similar to the continuum of material numbers. Most thoroughly it was made by Bertrand Russell. The remark stated on this occasion by English cosmologist G. Whitrow in his magnificent book "Natural Philosophy of Time" seems important to me. (G. J. Whitrow, "The Natural Philosophy of Time". London and Edinburgh, 1961, Russian edition - M.: "Progress", 1964). He absolutely correctly indicates that in mathematics there are ordered multitudes of a more complex type.

Whitrow notices: "Russell DEFINES an instant as such a number of events, any two events from which are simultaneous, and there is no other event (that is, an event which is not contained in the number), simultaneous with all these events. It is supposed, that the instants determined this way, "EXIST" (G. J. Whitrow, Natural Philosophy of Time, p. 207.)

We appear in the closed circle: we are going to undertake logic research of time, and inevitably we begin to base on "empirical consciousness data", and as a result it turns out science-like translation of our subjective notions of the language of the logic terms.

Nevertheless, we shall note the importance of the question: is the continuum of time identical to the continuum of material numbers or has it some other, more complex structure? The answer to this question can make a basis of a science called "CHRONOMETRICS".

Thus, we will be interested first of all with the metric ratios, characteristic for the temporal continuum. One more basic Aspect lies in here - congruency. If to define congruency of spatial segments we can refer to the comparison of segments at their parallel transportation, to compare the temporal periods even this opportunity disappears.

The book by Adolf Grunbaum "Philosophical Problem of Space and Time" is devoted to a problem of congruency of spatial and temporal segments (Adolf Grunbaum, Philosophical Problem of Space and Time. N.Y., 1963, Russian edition - M.: "Progress", 1969.) The essence of a dilemma is: whether there is a basis for attributing internal metrics to space (and time), according to which (internal metrics) the concurrence only establishes equality of separate intervals caused by its internal quantity? In his book Grunbaum protects Reaman-Poincare position, according to which the definition of congruency is conventional. That is, the space and the time do not possess the metrics, internally typical of them. As well as the linear continuum of material numbers, where any number can be accepted for a unity of measurement beginning with 1, we add to it one more and we get 2, simultaneously receiving 1/2, provided that received 2 will be considered as one unity.

However, as it was expected, in the analysis of a problem of congruency the Grunbaum spatial ratios are more often considered, which are then transferred in the temporal sphere. And the specificity of the temporal sphere still occurs only in the analysis of anisotropy (orientation) of time and exotic variants of the closed, cyclic temporal sphere.

So, the basic problem of CHRONOMETRICS is the search for the answer to the question: is the continuum of the temporal sphere and the continuum of material numbers identical? There are 3 possible answers: both the continuums are identical, and if not identical, there are two possibilities - either the ordered temporal continuum is simpler, or it is more complex. In its turn, the simplicity of the temporal continuum can be expressed in that it is a numerical multitude: it is identical to a natural series of numbers, it has atomic structure, or it is identical to the series of rational numbers - all intervals are commensurable. In case of its "greater complexity" there are also two variants: either it is any "complexity," known to us, or some special specificity - a multitude of some special type.

When Russell wrote his research in 1914, he traditionally transferred in the temporal sphere methods, already known from mathematics, and he presented the temporal sphere itself proceeding from our sensual experience. Generally there is no other way for us: all our notions about time are the data of our experience. But all the same it is necessary to base on notions of TIME, instead of its MEASUREMENT. It is a very important clause.

The matter is that MEASURING of time is an operation completely identical to construction of a scale for any measurable quantity. However, when we build a scale of temperatures, we do not confirm, that temperature is a linear ordered continuum. Here we realise, that we order the given measurements SO, that it would be convenient to compare different temperatures of the same body in different situations or different bodies in the same situation. And in due course it is different. We implicitly assume that our procedure of measurement - putting consecutive certain lengths, determined with "din-don" of any periodic process, is TIME. The fact that time is measured by us, certainly, reflects the features of this essence, however, this essence - TIME - is not exhausted by them at all. If I put it differently, in our notions of time it is necessary to look for such its property, which is not connected with "measuring", that is, it reflects any other specific quality of time.

We shall take such a property of time for a basis, as well as its division onto the PAST, PRESENT and FUTURE. It is clear, that this division does not concern the measurement of time, but it directly concerns anisotropy, orientation of time. The novelty of my approach is that I offer to abstract from this "evidence". That is, for our analysis it is not important, that the time " flows from the past - through present - in the future". The important thing is that the uniform multitude of instants of time is somehow divided into parts (subsets).

So, we shall begin with the obvious to us all division "of a uniform flow of time" into PAST - PRESENT - FUTURE. It is clear, that, if we want to advance a bit in scientific understanding of essence of time, it is necessary once and for all to reject psychological interpretations and to admit that the division LAST - PRESENT - FUTURE is an objective property of TIME inherent in it, no matter, who perceives or participates in this process: a person-thinker, a watch-dog or a spontaneously breaking up elementary particle.

If we abstract from subjectivity, TIME will be presented to you as quite a suitable subject for the analysis, and we shall notice one of its fundamental features.

Here I want to show my respect to the past, I want to reproduce a postulate from the work "The Studies of Space and Time" by the Russian philosopher Alexander Suhovo-Kobylin, written in the end of XIX century. This studies is a part of the unpublished book "Vsemir", where the philosopher tried to formulate "Universe" with the help of binomial decomposition of the multimember of an infinite degree. Alexander Suhovo-Kobylin is known more as a Writer. I happened to study his scientific works in 1990 in the archive ЦГАЛИ USSR, where the unpublished manuscripts of this remarkable thinker are kept. Converging numbers are shown as a symbol of processing of the Absolute Idea by the author "Vsemir", here "the Philosophy of a spiral" is developed, the final numbers are taken away from infinity etc. So, in Suhovo-Kobylin's work as some refrain it is repeated: "The Time is divided into three times - present, past and future... Past passed, it is gone. The future still will be, it does not exist yet. THERE IS only present".

In logic sense the division "of this flow of instants" into three parts (three subsets) is of great interest. And, only one subset EXISTS, the two other subsets DO NOT. Future and past are NOT PRESENT because the link dividing them - the present - is supplied with "predicate" IS. So there appear abstract objects, to which it is possible to try to apply traditional for mathematics methods.

So, let's consider TIME to be a multitude of instants. Or otherwise:

1. There is some multitude, which we call "time".

2. This multitude consists of an infinite number of the individual elements, which we call "instants".

3. The elements of THIS multitude possess the original quality: if one element of the multitude IS, the other elements of this multitude ARE NOT.

Not to be confused in sensual associations connected with the words "IS" and "IS NOT", we shall define this original property more precisely. Let us say so. All the elements of the given multitude have such a feature: if one (or some) elements are REAL, all other elements of the multitude are UNREAL. And we shall call multitudes of such type - AREAL MULTITUDES. 

The term "areality" embodies two senses: this is the connection of a negative prefix "а" to a word "reality", and a reference to the biological term "natural habitat" ("area") - place of living of the certain kind of living beings). The sign of Areality:

What do we get as a result of such a definition?

Firstly, we ascertain, that the TIME, as such, suits this definition - if to consider an instant of the present the only real, all other instants in the exactas sense are unreal: the past instants were already real, the future ones will still play this role. Secondly, given the GENERAL definition, we mean, that besides time there are also other prototypes, which are not time at all. If we determined a certain unknown multitude, the legitimacy of the definition could be confirmed only in case when besides time, it would be possible to find others denotates for this nomination.

Areality is clearly visible during introduction of a measure on the axis of the real numbers. Actually, for the given axis it is naturally supposed, that the change of standard is possible: taking 2 for a new unity, we transform the old unity into 1/2 etc. In other words, the whole set of possible measures -standards is a typical areal multitude: if one of measures is taken - becomes real -, all others remain non-realised - so to say, "stay in unreality". Taking into consideration all unusual character of such estimations, the use of the definition "areal multitude" appears lawful here.

But the most remarkable thing is, that elementary areal ratio is nothing, but the logic law of the contradiction: either A, or non-A, the other way is impossible. That is, if A is real, NOT A is unreal. You see, this NOT A does not disappear. Without it this A is simply impossible, but we believe: if A exists, NON-A does NOT exist. That is, it exists imaginarily, but it exists somewhat "unreal". To put it briefly, A and NON-A together form areal multitude of the two elements.
Aristotle, and all the logic after him, constantly underlined, formulating the law of the contradiction: it cannot be A and NOT-A in the same ratio, in the same TIME. Now it is important to rearrange accents. We formulate the LOGIC RATIO, which models the time and we do not use the empirical time for a reinforcement of logic evidence.

Introduced the principle of AREALITY, we unexpectedly find out the special property in the empirical time itself. If we identify temporal continuum with areal multitude of standards on a numerical axis, it is necessary to make the strange conclusion: the temporal order is carried out in such a way, that the realisation of one of standards occurs only in the case when only one point is realised, - becomes an instant. The realisation of concrete standard can occur in time only through the realisation of one of its points, otherwise the whole multitude of points appropriate to the given standard should be real. In other words, in the given starting system any REALLY FINISHED interval of time is formed by points, each of which is a point of only one certain unique standard from the infinite multitude of those. If "the arrow of time" is linear, it is only because with each instant in unreality the infinite multitude of other instants is deduced, forming together with the data an ordinary linear continuum of material numbers.

The interpretation of this property will require inclusion multitudes of systems of readout into consideration, but here we shall limit ourselves by the above-said. At the given stage of CHRONOMETRICS of the elementary qualitative description, I believe, it would be enough.


IV. EMPIRICAL SPACE. SYNCRETICAL GEOMETRY

It is considered, that the geometry is a first science created by the human being consistently and logically. The geometry with its postulates, axioms and theorems became a paradygmal sample, by which all scientists: physics, and mathematics, and even philosophers were guided. To tell the truth, it remains not clear whether the idea applying for a scientific rank can be embodied in any other form.

When Decart imagined three interperpendicular straight lines crossing each other in one point, he imagined emptiness around us, in which multitudes of various material bodies move. They draw their trajectories in three-dimensional space, changing in each moment of time their co-ordinates, creating the abstraction of a line (according to Camile Jorden's definition). This notion of a ratio of the geometrical theory and the real (empirical) space was indisputable for a long time.

If Nikolai Lobachevsky still tried to answer a question: whether the real space corresponds "to its imaginary geometry", the non-euclidean successors came to an unequivocal conclusion: the space is only a formal model, a mathematical structure. So in the history of the science a very interesting chain of ideologies was built: at first they describe real space which surround us, then they understand, that "space" is something more general (space as a mathematical structure), and "the real space" appears to be just a particular case described by this model.

Let us think over the sense of the last statement. It appears that axiomatic mathematical structure, which is called "space" by us, is only a MODEL, which can describe many different situations, including that real emptiness, which surrounds us. Thus, the geometry ceased to be a science about "the empirical space". Describing space we use a MODEL, which is SUCH in itself, and not because the space is such at all. The wide applicability of spatial models has obviously revealed the division of a model and its prototype. In other words that SOMETHING, which is around us is completely not the same, that we have got used to express in a geometrical model of the 3 interperpendicular axes.

Maybe, the real "empirical space" that surrounds us, that space-like SOMETHING is, as a matter of fact, much more complex?

I am speaking here not about "the other thin worlds", and not about the physical vacuum or any imaginary ether. As Occam wrote: "It is not necessary to invent essence beyond necessity". It is necessary simply to see in the old "essence" what, as a matter of fact, is probably necessary now. It is necessary to develop new fundamental logic understanding, fundamental not in a less degree, than Euclidean points and straight lines. 2300 years ago Euclid formulated the notion that the axiomatic system of interrelations of such points and straight lines is space. Now we should find the same "simple idea" - fixing the essence of SOMETHING, that surrounds us.

When Richard Feynman declared: "The Theory, according to which the space is continuous, seems wrong to me", he rebelled not against geometry, but against the identification of geometrical models and the empirical space. It means that it is necessary to use available geometrical, mathematical and meta-mathematical models to create the uniform theory, which describes SOMETHING that surrounds us more adequately.

Conventialist Poincare did the naturalistic amendments to geometry. He tried to attract our attention to the so-called "the latent axiom" - empirical fact, which is disguised among Euclid's axioms as a postulate about drawing of a circle with the help of a compasses. The fact, that a turning half-straight line sooner or later coincides with its continuation, does not co-ordinate logically with the axioms about static points and straight lines, it does not result from them, and they do not mean it. This "empirical fact" itself is expressed in the concrete irrational number ?. The presence of such "empirical constants" in mathematics (they are p, natural logarithms е, and j  - the bound of the Fibonacci numbers, and other remarkable ratios) are the marks indicating a way to the uniform SYNCRETIC GEOMETRY. The Albert Einstein's idea that the world can be understood geometrically and by a speculative way seems too courageous to me.


Pavel Poluyan,
Russia, Krasnoyarsk-49, box 19589.
Tel. (3912) 27-50-77.
E-mail: poluyan@fromru.com


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