Mathematics and Existence
In the text, I have exposed my belief in the existence of Ideal World and in the orgin of live
in Time and Space. Naturally, I learned and adopted a lot from literature of Great Cultures,
the unique sours of human knowledge. I have no intention to convince or to preach others
about thoughts expressed in the text.
Radical Idealism. From the beginning of human life on Earth, the imperative thought on
Ideal World was present in spiritual life of People. Radical Idealism pronounced by
Pythagorean School (569-500 B.C.) and Plato’s Academy (429-348 B.C.) about Ideal Beings
existing beyond time and space in non-material form became fundamental in Philosophy.
The Material World existing in its Time and Space was understood as the fixed form of the
Spirit. There is continuous interaction between the two Worlds. The notion of existence has
been understood as to be in Ideal World. But the Material World is in its time and space
initiated from Ideal World.
The natural numbers 1,2,…; were considered as omnipotent Godly Beings which have
great influence on our daily life. In fact, there exists Godly ONE in the Ideal World named
one. Nothing is called zero means it does not exist and is not considered as a Natural
Number. Collections of omnipotent ONE are called two, three, etc. Through thousands of
years broad literature has been designed for Number Theory as the subject in Pure
Mathematics. Studying the subject, one can learn the power of numbers which exceeds
whole human imaginary. In computers, finite sequences of numbers are used as computer
words. Thus, communication with computers is based on sequences of numbers and
relations between them. Human knowledge in books of all libraries could be written in a
computer language as the ordered huge set of numbers saved in a disc memory. In fact, the
reach language of natural numbers 1,2,…, contains more than infinite number of words.
Also, sematic of the language is infinitely creative to express objectively phenomena which
can be seen by human mind.
Apart of numbers existing in the Ideal World, there exist ideal geometrical figures like
CIRCLE, TRIANGLE, SQUARE and perfect SPHERE, CUBE and CYLINDER. On Earth there are
only not perfect images of these Ideal Beings. To mention some like the shape of Earth, of
the Sun and Moon and repeating patters of trees leafs and flowers. All those shapes have
origin in the Ideal World and they have great influence on human civilization.
Richness of Ideal World is far beyond boundaries of human mind. The Godly Beings and
perfect relations between them truly exist with no beginning and no end. Meaning of
existence and uniqueness in Mathematics and in the Ideal World is independent of time and
space. For example, there exists and only one the number ONE, when multiplied by itself
remains the same ONE, everywhere forever. Also in Mathematics the statement “it does not
exist” is used as complementary to the statement “it exists”. For example, it does not exist a
number x such that x+1=x. In both cases the results “exists” or “not exists” are independent
of time and space. But the material things like rocks, stones threes and animals including
human they life begin and end in time and space. In fact life as an episode does not exist in
the ideal sense but it comes from the Ideal World and returns to its origin. Great discoveries
and innovations in human civilizations are about to see and use the images of Ideal Beings
and Laws. The Light of Ideal Beings is to create life.
Of course, awareness of the Ideal World is not necessary to spend life on Earth successfully,
but it opens within us hope for a great existence. For example elephants have been living in
Okavango Delta since millions of years, they do not need to be aware that penguins have
been living in the same time at northern pole in the temperature -50 degrees also for
millions of years. In temporary time people have discovered new stars, galaxies and infinitely
small particles like neutrino. However, before knowing these objects, people have been
living successful life.
Numbers and Probability. A prediction is also related with Natural Numbers. The
omnipotent presence and great power of service of Natural Numbers they create conditions
for a prediction. Only very small part, (infinitely small part), of the Natural Numbers are used
in human life. Even though, the small part of Natural Numbers satisfies more than enough all
needs. Dealing with large Natural Numbers probability of an event maybe established. It has
been observed that if an experiment is repeating many times then it outcomes have a
regular appearance. Like in the academic example when tossing a dice 1200 times, the
observed outcomes are closely to ˜200, with the frequency of appearance 1/6 on each face.
How does it happen ?. One can think that Natural Numbers as Godly Beings are equally
available and behave like these. Indeed, on the first face Natural Number named one is
equally available as Natural Numbers named two, three, four, six related with the second,
third, fourth and sixth faces. In fact, if tossing of the ideal dice would be made in the Ideal
World the outcomes could be the same equal to 200, even at first six tossing each face could
appear one time with frequency 1/6. Accepting this there is no need to speak about
probability when experiments are made in ideal conditions. However, in the Material World
tossing of a non-perfect dice is not the same at each time. The outcomes are not the same
but after large number of tossing the outcomes stabilize on fixed frequencies. It is generally
known that in the Material World all kind of measurements are always biased by errors. One
may conclude that probability is related with experiments made in the Material World when
perfect measurement is impossible.
Being in the Material World. The whole Material World known as Universe (or a number
of Universes) is in its time and space. By human mind, it can be seen as continuously moving
space. In Mathematics, each location point (t,x) in the Material World is determined by the
time - ∞< t <∞ and the point x=(x_1,x_2,…) for the coordinates ∞ < x_i< ∞, i=1,2,…: So
that, the space is not limited to three dimensions. Human mind has limited view to see the
great long distance material beings and the smallest ones. But the human view can be
extended by Mathematics to see large objects, billions light years away and small material
objects which are on the boundary of existence. For example, the “Black Wholes” have been
first seen by solving a Mathematical Model. In conclusion, if the unique solution exists of a
model, it means that the object must be, too. Also, for the smallest particles, an unseen
particle infinitely small neutrino was first discovered finding a unique solution of the
Mathematical Model. Later the “Black Wholes” as well as “Neutrino” were confirmed by
experiments.
The continuous movement of matter caused by gravitation or inertia forces is seen
through Mathematical Models. People in ancient time knew how to calculate the position of
Earth with respect to the Sun. First solar calendar was known since 4242 B.C. in ancient
Egypt. Kepler published in his book Harmony of the World in 1619 the Mathematical Model
of planetary motion around the Sun. At present time, one of the most extensive and
interesting area of research concerns motion in the Universe. Deterministic and Stochastic
Mathematical Models have been publishing in broad literature about motion of matter in
the Universe. Also to mention the satellites which have been sent in the space for research
and communication purposes.
Amazing possibilities to build by human mind great projects for construction of buildings,
roads, bridges, cars and airplanes, radio and television stations, computers and telephones
come from the Ideal World. It would be impossible to build these entire things if they Ideal
Forms do not exist in the Ideal World. Thus through Mathematics the Ideal World is seen and
transfered to the Material World.
In all kind of constructions physical forces are present. They impact on durability and
elasticity of a construction has been modeled by application of ordinary and partial
differential equations. As academic examples of Mathematical Model let us list the models in
civil engineering that is the beam equations and Laplace equation. Also, let us mention here
two great phenomena in the Material World, which are Diffusion and Waves propagation.
Diffusion and waves are other natural phenomena which have great influence in life on the
Earth. Like diffusion of air and liquid to support life in flora and fauna and diffusion of heat.
In order to create Mathematical Models of a diffusion phenomenon, parabolic differential
equations are appropriate to use. There are many options and possibilities to see Ideal
Diffusion by human mind and to build mathematical models for all kinds of diffusion in the
Material World. All those models have origin in the Ideal Laws of change. In human minds
the mathematical modules have to be written in terms of symbols used in Mathematics. As
the academic simplest form of a diffusion equation, let us write the heat equation
U_t(,x)=a^2 U_xx(t,x),
t ≥ 0, a ≤ x ≤ b.
There is infinite family of solutions of the equation. To choose one, the initial condition and
boundary conditions have to be put on the solution U(t,x). The rigorous proof of existence
and uniqueness of the solution under the initial boundary conditions is given in academic
hand books. The solution U(t,x) represents density in a material undergoing diffusion at
time t and point x, or in case of heat diffusion, it determines temperature t at the point x.
The Mathematical Models of a nuclear explosion to control diffusion of heat in space, when
nonlinear parabolic equations are involved. Of course, virtual nuclear explosion is not such
dangerous and costly as a material one.
The most impressive are waves of light which maintain life. On the Earth, light of the Sun
maintain life in flora and fauna. Waves are part of the environment like waves on water of
rivers and lakes, of seas and oceans, waves in communication including voice, radio,
television and telephone. Differential equations are used to explain dynamic phenomena in
the Material World like continuous moving of matter and waves. In particular, Partial
Differential Equations of hyperbolic type are used to build Mathematical Models of waves
phenomena. A correct model equation has to have a unique solution, that is, the solution
must exists and to be only one. Existence and uniqueness of solution of a model is subjected
to a proof. There are different methods of proving existence and uniqueness in Mathematics
even for the same model. Proofs in Mathematics are objective. They have completely
different character than proofs in administration of human law.
Mathematics in Society. Mathematics became the main tool exploring all aspects of
human life. In education, Mathematics is an important part of programs taught at schools
and universities. It is a great privilege to have the opportunity of intellectual development
studying Mathematics as a part of the Ideal World.
New highly sophisticated and interesting professions have been created in the area of
Mathematics and related subjects such as Computing, Modeling and Biomathematics. One
can observe that more young people wish to study science to get serious knowledge and to
have solid education for serving society better. The pure Mathematics and its applications
are inspiring subjects. To study the subjects successfully an enthusiasm and regular work are
needed.
Professor in Mathematics
Tadeusz STYṤ
February 5, 2012