© P.V.Polyan




Author’s notes: During the International Mathematical Conference “Multidimensional Complex Analysis” (Krasnoyarsk, Russia, August 5-10, 2002) I made a poster report “Do the Hyperreal Numbers Exist in the Quantum-Relative Universe?” The report was devoted to the extensive theme “Non-Standard Analysis of Non-Classical Motion”, and in particular, was concerned with the question of building non-standard theoretical time model and application of non-standard mathematical approach to non-classical physics (http://res.krasu.ru/non-standard). The article offered below is the more detailed concretising of the results, presented there. The author expresses his gratitude towards the mathematicians and physicians, who gave personally or by e-mail their critical and constructive comments to the stated problem.


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.

And is it possible to speak of “metrical properties” of time, if its theoretical representation is a straight line? Metrical properties are the attributes of multidimensional space, where there appear linearly independent vectors. If we take “pure time”- numerical axis, where segments are put, the congruency of which is based upon the references to the periodicity of some natural processes, nothing could be done, except the linear operations with time segments. In connection with it is necessary to be more precise, I call such properties of “pure time” metrical, which cannot be limited to the special features of one-dimensional linear continuum of material numbers. In other words, we suppose beforehand that time is a more complicated object than an ordinary straight numerical axis. I want to remind that in 19th century William Hamilton formulated a prospective task: if there is geometry as a science of empty space, in analogy with it, we can imagine some science of “pure time”. Moreover, he thought that algebra was such a science, we just cannot see such specific temporal character in it, we do not understand how in REALITY in algebraic equation inner time properties are embodied. The fact that Hamilton’s discovery of non-commutative algebra came as a result of his attempts to model time in “The Theory of Algebraic Pairs of Numbers”.

In 1959 (J.L. Syng, “The New Scientist” 19th February, 1959,p. 40) Syng offered to create a special science about pure time named “Chronometry”- in analogy with geometry. But in Russian such a term is associated with the procedure of time measuring, that is why here I introduce other name “chronometrics” which includes the presence of special metrical properties.

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.

Voluntarily or involuntarily we attribute such properties to time which are considered characteristic of spatial relations.

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 anysotropy (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.

But before we begin to search, it is necessary to make a very important remark. (The necessity to make this remark was mentioned by Professor S.S. Kutateladze). The definition of areal multitude just introduced creates some specific object, which differs from that multitude, the notion of which is in the classical theory of multitudes. The unification of some elements into the one single multitude means a complete act, hence the notion of actual infinity. In our case an essential elaboration is made: areal multitude is actually the given total of elements, but its elements are such thanks to the fact that some other elements are not the elements of the given multitude. In other words, for the given areal multitude the presence of the possibility that some other elements CAN become its elements (on condition that its other elements are excluded from its staff).

I do not think that such a definition contradicts the logic, which forms the notion of multitude. On the contrary, we can see that here we find a point of view, where the traditional notion of multitude can be considered more detailed. And the main criterion, which I have at my disposal here is: constructivity of the approach – building such a model, which allows succeeding in the theoretical realisation of reality.

So, what examples of areality can be found? From the very beginning we exclude various empirical cases, which can be considered areal ratios (these are, for example, cases of populations from biology-domination of a particular phenotype in the given conditions, depending upon the presence of other possibilities, which are laid in the Genotype of this species). We shall concentrate our attention on the mathematical objects as more abstract and suitable for the precise analysis.

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.

Let us try to identify TIME as areal multitude with the just introduced areality of the multitude of linear continuum norms of real numbers.

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.

I remind that here we consider properties of “pure time”- multitude of instances, which are not equal to some events. And now it is discovered that any instant is not just a point on the axis, but multitudes of points, which with the regard to the given one go to the “past” and “future”. But the special feature is that all these points have already become “real” and never in future they will become instants, neither did they before.

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.

Non-Standard Analysis and Areal Multitudes.

Up to now I have tried to be within the generally accepted limits concerning the notion of areal multitude. All the above-mentioned ideas were based on the ordinary, well-known notions- what can be more ordinary than time division into Past-Present-Future! I suppose that critical readers could have taken the above-mentioned for some useless thinking, but I do not think that the notion of areality could have made them protest against it.

Now I shall try to use areality to make some moments more precise, these moments are concerned with the bases of mathematics. Here the author’s position is more vulnerable. Nevertheless, I shall try to state it.

Re-reading my preparatory notes, I caught myself that I felt as if I made tactlessness. And, of course, I can imagine readers’ reaction: arbitrary manipulations with mathematical notions lead to the thought that “there is something wrong” with the author. But I hope that the approach presented below will be taken at least for a curious thing, suitable as a reason for the philosophic-methodological discussions.

To base my approach somehow I must make the initial theoretical position more precise.

While we began with the time analysis, moreover, it was EMPIRICAL TIME, that was the object of the theoretical modelling, I suppose, not mathematical, but physical character of the article is evident to the readers.

Our main subject is: “Non-Standard Analysis of Non-Classical Motion”, that is the attempt to build up a model of mechanical Motion in its non-classical notion, typical of relative and quantum physics. The author’s idea is simple: if in the classical science the initial mathematical notion of derivative coincides with the initial physical notion of velocity - mechanical motion of a point along the trajectory, in non-classical science it is possible to build a model, where the close connection of the same type between mathematical notions and physical characteristics of motion will be discovered.

At the present moment mathematical motion modelling is of phenomenological descriptive character (at least, in quantum mechanics, in the theory of relativity time-space continuum plays more fundamental role). But no reason tells us that descriptive modelling is the way of mathematization.

Albert Einstein’s idea that the objective reality can be understood speculatively- with the help of mathematics does not seem good to the author. On the contrary, I am sure that in mathematical structures fundamental ratios, which are direct and precise, reflection of physical laws can be discovered. We shall not delve deeply into the philosophical details, let us leave them alone for another discussion. But it is doubtless: the basic mathematical notions are not enough for physics nowadays. To prove this idea I shall take 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 exist 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.

The fact that classical mathematical analysis with its idea of uninterruptedly - divisible continuum is not enough yet is quite evident. But it is not understood how this uninterrupted divisibility can be re-interpreted – what grounds do we have to do it?

Since 60-s of the last century Abraham Robinson’s non-standard conception of analysis has gained a firmer hold *, and if at the beginning the idea of actually infinitesimal and actually infinitely large hyperreal numbers were not treated very good, nowadays a definite ideology has been worked out where such numbers are considered admissible. But the extension of the field of real numbers thanks to hyperreal ones is of relative character – they (hyperreal numbers) are understood as Ideal “artificial” objects.

Hyperreal numbers appear in Abraham-Robinson’s model of analysis as a result of extension of the field of real numbers, if the breaking up of Eudocks-Archimedean axiom is admissible, but having built up the logical model, where such breaking up is admissible, we understand at the same time that the axiom itself is of more fundamental character, it is objectively dependent, and its negation is “relative”, artificial, subjective.

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.


* 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]

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.

The notion of time as areal multitude is the first step. It demonstrates that mechanical motion as such cannot be reproduced directly and precisely in the classical notion of ratio dt/dx, while time moments are not the points on the axis T, but the elements of some multitude of a different structure in comparison with an ordinary linear continuum. On the other hand, the introduction of ratio of areality lets us see such a continuum in a different way and discover some unexpected properties.

We shall begin with the simplest thing, one can say - a standard statement: “Gathering number series has final sum, but does not have the last member”. If we take this statement for some logical one, we shall see some signs of areal notion in it. Multitude of numbers – members of the gathering series- is created on condition that the succession has No last member. In fact, finiteness of the sum is actualisation of the multitude. We begin to add numbers with the first biggest one. The whole number of items is infinite. We can speak of actual calculating infiniteness of this multitude, but the main sign of the multitude elements remains: they EXIST, they form up in a particular order, we SUPPOSE that there is NO last member of the multitude. In other words, gathering succession of members of the series can be considered areal multitude, when the usual calculating infiniteness of elements of this multitude is added by some UNREAL element, which is the very last member that DOES NOT EXIST.

This strange statement does not seem to add anything important, serving just an artificial conjecture. But let us try to see what will happen if we take areal ratio for the base?

The first conclusion: Though this last member, excluded from the succession, is not the element of the multitude, but nevertheless, it EXISTS. That is, continuing areal logic, we must say that SUCH gathering succession of series numbers can be realised in other way. Really, such a notion of the given multitude must be realised when “the last member” EXISTS, but all the other members go to the non-reality, those that are larger than it beforehand. What is this then? it is nothing, but the sphere of hyperreal numbers in the sense of non-standard analysis.

Thus, within the limits of logic of areal relations we define mutual supplement of the sphere of real numbers (where members of gathering series are situated) and the sphere of hyperreal numbers, which are all less than “the least”. For hyperreal numbers Eudocks-Archimedean axiom does not work, because it works for the rest – real elements of this areal multitude.

The second conclusion: If areal multitude is something single, we cannot just “join” hyperreal numbers to the real ones, because we deal with a definite gathering succession of real numbers. In other words, in this areal multitude, taken as a whole, the general ratio of elements for all this succession must remain somehow. It is quite not clear how the law of agreement (the ratio of elements Ni And Ni+1) must go on in the hyperreal sphere!

Now we shall try to understand HOW IT HAPPENS, taking some concrete succession to help us. It is possible, that our consideration would look like some arbitrary thinking, but if the logic of AREALITY is accepted, those conclusions will appear with necessity. But before we do it, it is necessary to make some things clear.

It is clear that any gathering succession is an artificially taken fragment of series of numbers, connected by their ordinal ratio between Ni-1, Ni, and Ni+1. That is, such a series does not only have the last member, but also it does not have “the first”, to be more precise, we can begin with some N and build up some infinite gathering series with the finite sum of its members, but the same ratio, made in the direction of increase, of course, will not give us the finite sum, and the quantity of every next member of the series will increase unlimitedly. In other words, ratio of areality for such series is displacement of the both spheres of defining hyperreal numbers to the non-reality: either actually infinitesimal or actually endlessly large. Strictly speaking, having begun my concluding with the phrase: “there is no last member”, I just used the usual “school” definition to illustrate areal approach. Nevertheless, it was necessary, because the main thing is that, speaking of gathering series, we cannot describe it anyhow, but with the words: “This sum has no last item”.

Now another “school” definition will help us: series where the quantity of members of the series is constantly growing, there is still NO last member.

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 1.


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?

I answer these questions in the other part of my work “Non-standard Analysis of Non-Classical Motion”, when I describe motion with indefinite velocity. The fragment of this part is presented in the article “Do the Hyperreal Numbers Exist in the Quantum_Relative Universe?” You can get acquainted with it, the address is: http://res.krasu.ru/non-standard.


About the Applicability of Non-Standard Analysis in Physics

Thesis account of the material makes us do some logical shift from the one theme to the other. Now we shall turn our attention to physics. I shall try to demonstrate how non-standard approach allows us to join the spheres, between which there has been no connection before.

It was noted above that 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. (This reminds us of the addition of hyperreal numbers according to the non-archemedean principle that allows speaking that in non-standard analysis their sum cannot be more than one.) 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.

The main difference of mathematics from physics is that physical quantities are measured, in 4-dimensional pseudo-euclidean continuum of real temporal space imaginary unity is added by the co-efficient of proportionality, which is interpreted in physics as velocity of light. In classical physics the maximum limit of velocity was unlimited. Now the role of infiniteness is performed by velocity of light. In other words, all the beyond-infinite realisations of velocity appear to be displaced to the non-reality, and this fact makes us have some definite ideas. What if we try to apply here the technology, which we used dealing with infinite succession?

It would be especially interesting to see what happens in little- this is the sphere of quantum mechanics, and turning infiniteness to the series of numbers points to some likeness of the results.

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 reflaxions of the ray. When the value of the number of the reflaxions 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. As far as relative velocity addition is concerned, non-standard approach leads us to the hypothesis that as well as in the direction of increasing velocity the maximum quantity C – velocity of light - is discovered, in the direction of decreasing some minimum can be discovered. But such a hypothesis of “velocity of darkness” looks exotic, and the main thing is that it does not coincide with those conclusions, which can be made on the bases of the formal approach and its physical interpretation. The results, received this way, seem very interesting and physically interpreted to me.

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 2.

А. ®

¬ А.


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 3.

А. .В .С ®



* While the principle of relativity has more difficult interpretations, and that causes misunderstanding (the reviewer of “Nauchnaya Set”, for example, thought my interpretation to be mistaken), I want to quote Albert Einstein’s words from his work ‘What Is the Theory of Relativity”: “Co-ordinate system, moving uniformly and straightforwardly with regard to the inertial system, is inertial itself. The special principle of relativity is the consolidation of this statement, applied to all processes of nature: every universal natural law, which works with regard to some starting system C, must also work with regard to any other system C’, which moves uniformly and straightforwardly with regard to C”.(A. Einstein. Collection of Scientific Works, V1, M: “Nauka”, 1965, p. 679)


“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 its 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 BC distance for constant, then the former standard distance AB should be treated as changing.

Picture 4.

А.-const-.В .С®

А.® В.-const-.С


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: A comes closer to B, 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 begins 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?


What do we mean when we say: the given material point possesses the given velocity?

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 Ґ s/m (accordingly Ґ m/s and 0 s/m), are logically identified, but also 0 m/s and Ґ 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.

Let us sum up. It is known that in pseudo-euclidean 4-dimensional space-and-time continuum of Minkovski “single measure is put on the axis”, it corresponds to the spatial extent x[m], and transformation of the measure t[s] is realized with the help of co-efficient of proportionality C [m/s] – velocity of light and imaginary unity i. (In case of motion along a straight line it turns to an ordinary complex plane.) We have shown that the connection between x and t can be used in the same way to build up pseudo-euclidean continuum (complex plane), where a single measure will be put on the axis, which corresponds to the temporal periods t[s], and transformation of measure x[m] will be realized with the help of the co-efficient of transformation 1/v [s/m] and imaginary unity 1. In building of sucha kind there is nothing “mistaken”, though the approach is quite formal.

Nevertheless it was interesting to try, because no one has tried to build up such pseudo-euclidean continuum as applied to the physical quantities.

Having created it, we face the problem of interpretation, while “reverse velocity of light” possess measure [s/m] and cannot be velocity in the usual sense of the word. This strange quantity can be interpreted on the bases of the traditional principle of relativity as “velocity” of rotation along the single orbit, and co-efficient 1/v for the new type of continuum appears constant, which corresponds to the combination of constants e2/h in physics. It is unlikely to be coincidence. On the contrary, while in mathematical buildings, related to multi-dimensional complex analysis, all the quantities are dimensionless, and in physics they are connected with the concrete physical parameters, the mentioned dual character of pseudo-euclidean continuum of space-time has non-trivial sense. At least, this formal approach shows some mutual connection between notions and definitions of the theory of relativity and quantum-mechanical parameters.

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 here appears a question: why would mathematical dimensionless UNITY in physics somehow split, forming some sphere with non-archimedean addition of velocities (velocity appears here to be just co-efficient of proportionality between the axis of pseudo-euclidean continuum)? While in our buildings there were no dynamic physical quantities, it is clear that there are no answers to such a question. But I have no doubts, it is possible to explain the marked obscurities mathematically correctly and physically sensibly if we develop the offered approach.


I understand that I rouse quite natural negative reaction by offering this article for discussion. All this looks like some playing tricks of a dilettante with mathematical and physical notions- like the extraction of “n” from the Egyptian pyramid. I want to express hope that there will be readers to whom the offered approach will seem prospective. In the end, the only criterion of scientific character of an approach is its ability to give conclusions, which allow to see connections between the usual notions and events that have never been seen before.

Now the ideology that can be called model constructivism is accepted.

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...

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 and relative connection between velocity, mass and energy can be concluded theoretically - from logical bases, still being hideous and unrevealed I believe, that the matter is so.

Pavel Polyan,

Box 19589, Krasnoyarsk, 660049, Russia.

Tel. (3912) 27-50-77.