The Chronicle. Definitions of
on fields of W. Heisenberg's
"Physics and philosophy"
3. Mechanics of time. Space of mathematics.
The law of prime numbers as the general field theory
Heisenberg wrote: "All things are numbers" is a sentence attributed
to Pyphagoras. The only mathematical
forms available at that time were such geometric forms as the regular
solids or the triangles which form their surface. In modern quantum theory
there can be no doubt that the elementary particles will finally also be
mathematical form, but of a mach more complicated nature. ... Modern science,
however, has from its
beginning in the sixteenth and seventeenth centuries started from the
dynamic problem. The constant
element in physics since Newton is not a configuration or a geometrical
form, but a dynamic low. The
equation of motion hold at all times, it is in this sense eternal,
whereas the geometrical forms, like the orbits,
are changing. Therefore, the mathematical forms that represent the
elementary particles will be solutions
of some eternal law of motion for matter. …
Actually this is a problem which has not yet been solved. The fundamental
law of motion for matter is not
yet known and therefore it is not yet possible to derive mathematically
the properties of the elementary
particles from such a law. But theoretical physics in its present state
seems to be not very far from this goal
and we can at least say what kind of law we have to expect. The final
equation of motion for matter will probably be some quantized nonlinear
wave equation for a wave field of operators that simply represents
matter, not any specified kind of waves or particles. This wave equation
will probably be equivalent
to rather complicated sets of integral equations, which have "Eigenvalues"
and "Eigensolutions", as mathematicians call it. These Eigensolutions will
finally represent the elementary particles; they are the mathematical forms
which shall replace the regular solids of the Pyphagoreans. We might mention
these "Eigensolutions" will follow from the fundamental equation for
matter by mach the same mathematical process by which the harmonic vibrations
of the Pyphagorean string follow from the differential
of the string.
If we follow the Pyphagorean line of thought we may hope that the fundamental
law of motion will turn out
as a mathematically simple law, so simple as it hoped to find as far
back as Plato. It is difficult to give any
good argument for this hope for simplicity - except the fact that it
has hitherto always been possible to write
the fundamental equations in physics in simple mathematical forms.
This fact fits in with the Pyphagorean
religion, and many physicists share their belief in this respect, but
no convincing argument has yet been given
to show that it must be so. (End of quotation.)
The chronicle uncovers the foundations of mathematics problem in the
following way: the mathematics is
the pure theory of space. If the physical time is uncovered as temperature
in its original physical sense, the
problem of measuring in the physics is uncovered consequently as the
creative function of space, i. e. as mathematics. The mathematics is the
measurement of physics. A fundamental problem of the Chronicle is
the problem of time measurement. The time measurement is a sole essence
of physical process, as far as
the process has mathematical form. The time measurement is physics
of formulas, strategy of universalization
of "divisions" of the modern physics. The space is a reality which
arises as the outcome of time measurement.
The time measurement is the dynamic and static function of time, a
function of time.
The time measurement is a theory of number. The time measurement -
Time the reversible, the recurring,
the multidirectional - is a theory of number. The classical thermodynamics,
keeping the problems of time in shade, has resulted in generating of nonequilibrium
thermodynamics, i. e. statistical physics. The statistical physics within
the framework of a reflection of physical knowledge program can be reduced
problems of the foundations of mathematics. The program of the Chronicle
brings us closely to the
development of the foundations of mathematics as to that field of knowledge
where the basic problems of
physical knowledge are solved.
More particularly, program of the reflection of mathematics is necessary.
The strategy of the present
program supposes re-definition, or, more clearly speaking, definition,
determination of number. The state
of mathematical knowledge is characterized by the same problems as
state of physical one. We notice here
the same forestalling development of "divisions" of mathematics in
relation to a state of development of the foundations of mathematics. Actually,
the present-day symbiosis of physics and mathematics also leads to radical
re-thinking of basic concepts used in mathematics, among which the most
fundamental value has
the concept of number.
The number is the basic subject of physical knowledge. Physics studies
number. Physics studies effects originating in connection with the existence
of number. The number is the form of existence of time in
nature, while temperature is the content of existence of time in nature.
The number is the actual subject of physics. Number is a wave and particle
simultaneously. The number is an actual object, the element of time;
a thing, subject of time which, from the point of view of the
researcher, is both wave and particle. The
number, thus, is a thing realizing oscillatory and undulatory processes.
The number radiates. It is the
problem of modern physics: the object which realizes the physical process
as such, physical process,
which explicitly and in detail is investigated in divisions of physics
has not been defined. The Chronicle
defines this object as a number, number of a numerical series, as an
object which concretely existing in
time, from time, by force of time; object which has all variety of
qualities uncovered in divisions of modern physics.
Basic foundation of the mechanics of time (chronicle) is the following:
the number is a concrete moment,
i. e. an outcome of making of a certain amount of space by certain
force of time. That is the formula of
number. The form of existence of a number is oscillation (harmonic
oscillations, mechanical harmonic
oscillations, free harmonic oscillations in an electrical oscillatory
circuit, relaxation and forced oscillations).
The real theory of number arises as radical criticism of quantum mechanics.
The quantum mechanics
develops as speculative physical knowledge around true, but not realized
subject of physical knowledge.
The quantum physics has closest approached than other divisions of
physics to true, but not explicit
object of modern physics, to number. But physics has stopped in front
of it and generates phantasmagoric constructions of "nominal" quarks.
Heisenberg's principle of uncertainty, concerned with a problem of
a measurability in the physics, on the assumption of sequential thought
means circumstance developed in this text: sole and basic matter of all
of modern physics is number as such, its reality, "behaviour",
law (laws) of its genesis and
structure, interaction among numbers. There is no other reality of
modern physics in the aggregate of its divisions, but number in itself.
The real object of the physics is number. The chronicle is a theory
of number. Mathematical aspect of
numbers corresponds to divisions of the physics. There is nothing in
concrete division of physics but the
study of concrete properties of a certain sort of numbers as chronosystems.
The natural numbers, integers,
prime numbers, fractions, positive and negative numbers, rational numbers
and irrational numbers are
types of unification of chronosystems.
The space consists of numbers. Exclusively actual infinity of numerical
series (denumerable infinity) as
such is space in itself. The relation of actual numbers is the definition
Solely actual infinity of a numerical series is "field". An infinite
numerical series is the consistence of
"nature", it is process of time as matter of any realization. The universal
and concrete number is a reality disappearing under the name "solid" in
the classical mechanics. There is only number. The interior relations
of a numerical series form transparent space of the physics. The quantum
physics (mechanics) disappears
in the theory of number. The new theory of number is the unified theory
of physical interactions in which distinctions of the divisions of physics
become substantive. The number is "velocity", "acceleration",
"impulse", "momentum", "energy", "thermal motion", "work", "fluctuation",
"electric field", "electric charge", "electric current", "dielectric",
"semi-conductor", "plasma", "magnetic field", "atom", "induction",
"oscillation", "wave", "radiation of heat", "photon", "radio-activity",
"fundamental interactions of ultimate particles". The number is a primary
subject of physics and coincides with the essence of mathematics. The number
is being of time in the form of space. "T" (linear time) of the physics'
formulas, is, undoubtedly, a turnover variable, but nevertheless should
be interpreted as real function, as outcome of transformations.
Even in case of the formula of a velocity as the relation of distance
to time it is necessary to understand that
just the time (even linear time) is produced in this process, and what
is required for this purpose is concrete velocity and distance which are
formed and become possible by virtue of necessity of making certain linear
amount of time.
All physical experiments are experiments "inside" of numerical series,
are experiments with concrete
numbers, experiments in the field of interaction of numbers, experiments
basing upon actual infinity of one
but really existing numerical series.
The general theory of number is the general field theory. The physical
field is an nfinite numerical series of
various types of numbers. For that simple reason, that there are various
kinds or qualities of numbers, i.e.
natural numbers, integers, prime numbers, fractions, positive and negative
numbers, rational numbers and irrational numbers. In other words, a numerical
series incorporates qualitative distinctions of physical
processes, is present by a transparence and interchangeability of all
physical processes. There is no nature investigated by physics, except
a numerical series as such, unique and infinite. Distinction of number
is a genuine physical reality of physical processes presented in divisions
of modern physics. The distinction
of number types is the actual form of distinction of physical interactions
and of kinds of physical substance.
The formula of a general field theory is the law of prime numbers.
The various types of numbers are that concrete reality which are situated
beyond "types of a substance" investigated in divisions of physics.
The prime numbers are a fundamental reality - foundation of physical
processes - fixing indestructible, indissoluble connection between time
and heat. The law of prime numbers is the plain fact of existence of
an infinite amount of prime numbers, of "solids as generators of gravitation"
which in traditional
mathematical sense is nothing but the Fermat theorem heuristic (clear)
proving. The mathematics speak
about the Great Fermat theorem in such a way: "There are no such four
numbers which would satisfy to the equation: ?n + ?n = zn ["n" here is
exponent], where: n > 2; x, y, z are positive integers. Trying to solve
the problem of the "great Fermat theorem", it is necessary to answer a
question: "And what do we, in fact,
search?" If the answer is casual numerical coincidences, then it is
necessary to calculate regularities of
power numerical series according to general numerical series. But such
approach can not give the crucial
answer to the problem: "Why is it impossible nevertheless to divide
a cube by other two cubes, and why
is it impossible in general to divide number in a power above second
by other two numbers of the same
power?" As it is well known, it is possible to seek out many pairs
of integers which sum total of second
power as well is some of integer's second power. You may find as much
such numbers as necessary. But
try to seek similar examples for the third power. You will not discover
any! You will experience the
complete failure in the search of an example for fourth, fifth and
other higher power. That is the "great proposition of Fermat". It states
that equation xn+yn=zn has no positive integer solutions".
The fundamental formula of the Chronicle is the formula of space: "The
great Fermat theorem is the physical formula of space".
The space as a concrete infinity of a numerical series is subordinated
to the Law of the Great Fermat
theorem. The concrete infinity of prime numbers of a numerical series
(the Law of prime numbers) is the
Great Fermat theorem proving. The Law of prime numbers is the physical
formula of time. Thus, the
fundamental formula of the Chronicle - the formula of a general field
theory - states: "The Great Fermat
theorem proving is the law of prime numbers". It is the fundamental
formula of the mechanics of time.
This formula is a mathematical equivalent of the concept of time -
"Heat is Time. Time is Heat".
The types of numbers reflect all variety of physical processes and
are the investigated form of this variety.
Types of numbers, instead of "nominal laws", mythical and archaic "forces"
of interactions, form a priori
structure of any physical research.
The great Fermat theorem uncovers structure of real infinite numerical
series, whose genesis out of time is uncovered in the fact of existence
of an infinite amount of prime numbers. The divisibility of number is a
concrete physical essence of physical process. Indivisible, prime number
is the last genuine object of
physics. The law of prime numbers - the Great Fermat theorem - is the
law of creation of the world, of
the nature in its pure, primordial aspect.