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Articles and Publication Modeling NON-STANDARD ANALYSIS OF NON-CLASSICAL MOTION TIME AND CHRONOMETRICS. AREAL MULTITUDES.
NON-STANDARD
ANALYSIS OF NON-CLASSICAL MOTION
TIME
AND CHRONOMETRICS. AREAL MULTITUDES.
© Pavel Polyan,
Contact: Box 19589, Krasnoyarsk,
660049, Russia.
Tel. (3912) 27-50-77.
E-mail: polyan2002@mail.ru
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.
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.
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 ÖÃÀËÈ 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 "à" 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:
====================================================================
- These exist principal divergence between
modern physical notions of motion and classical notion of analysis.
- 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.Ãàðäíåð,
Ìàòåìàòè÷åñêèå íîâåëëû. Ì: “Ìèð”, 1974, ñ. 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.
À. .Ñ
®
¬ À.
.Ñ
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.
À. . .Ñ
®
-----------------------------------
*
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 14.
À.-const-. .Ñ®
À.® Â.-const-.Ñ
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. Ò. 1. M.: "Ìûñëü" 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"
(Ò. 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+Â) [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 å.
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.
Conclusion.
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.
Publishing date: October 7, 2002
Source: SciTecLibrary.ru
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