Truths and illusions of mathematical thinking
Table of contents: The KazakhAmerican Free University Academic Journal №7  2015
Author: Lobastov Gennadij, National Research University of Electronic Technology “MIET”, Russia
Chair of a Philosophical Society “Dialectics and
Culture” (Moscow), Doctor of Philosophy, Professor, Chief of the Sense Genesis
Laboratory of L.S. Vygotskiy Psychology Institute, Professor of Philosophy and
Sociology Department
Mathematical science not only causes
fearful interest with a wide loop of mystic "intuitions" among the
average people, it has been differently interpreted over the development of
history. Its special status presumably associated with its subject field is
discussed not only by the scientists but also science managers in different
classifications: it does not fall into the category of natural or social
sciences. It was perceived as the incarnate mind, designed by this mind on the
basis of some ontological assumptions or based or on a priori structures of the
mind itself. That is why there is no definite answer to the question “what does
mathematics study”. There is something similar to the convention of chess,
which also deals with the secrets of mental activity. But if the conventional
rules of chess is an obvious thing, the mathematics in its selfreflection (in
the philosophy of mathematics) for ages has been trying to find a bases of its
own actions. Mathematics sees conditionality of seemingly conventional forms of
its work and looks for a solution of this problem in some "ontological"
intuitions and in constructivist abilities of the human mind. In its rules, the
game of chess game is closed and easy to master. Its simple logic and certainty
of the ultimate goal determine its goals, which are achieved on the chessboard
under changing circumstances. In mathematics like on the board, there is also a
certain goal, which appears in the process of moving towards the ultimate goal.
But the ultimate goal causes some confusion. Because there is no clarity: if it
is a science, not a game, what kind of truth does it look for? What does it
study?
Or, maybe, as some believe, it only creates the rules by which all other
sciences recreate space and time of objective phenomena of their subject field
in order to determine quality characteristics of objects under study using
quantity characteristics? Or, does it simply develop a universal language of
science? Treated like this it is perceived as a logic of a “special type, thus
revealing its fullness with the mind.
On the other hand, the emergence and development of logic as the science
of thought, performing a methodological role in cognition, naturally relates it
to mathematics. We cannot say that the algebra of logic is identical with the
logic of algebra, but the objective correlation of mathematics and mathematical
logic does not just clears up, it becomes philosophical and methodological
problem. The problem of the relationship of
mathematics to the concepts of objective reality is clearly a philosophical
problem, and it remains permanently relevant as long as mathematical thinking
retains its thinking character. Therefore, this problem arises not just from a
philosophical curiosity, but as a problem of practical importance, since
mathematics, in fact, reveals itself as a special theoretical tool in all kinds
of not only scientific but also practical activities. This, of course, once
again points to the universality of its "logic", its
"language". Kant clearly showed that in the formal forms of
"school of logic" (the logic that today turned into mathematical
logic) there are forms characterized by universality and necessity, providing a
real connection of thinking (logical) definitions and the categories. In other
words, the actual thinking, while comprehending the subject, is realized in an
entirely different way than it is thought from the point of view of traditional
formal logic. Mathematics in its actual work is consciously oriented on this
external form of thinking (formal logic) and uses it more clearly than any
other science. That is why earlier than other sciences it realizes the need for
reflection of ways of its own development. This reflection constitutes the
basis for work of mathematical mind related to the various philosophical
concepts. This forms different "methodology" position, explaining and
at the same time guiding the mathematical thinking (intuitionism, logicism,
formalism). But this, of course, is a question of philosophy. However,
philosophy itself finds its grounds different. And, of course, among
philosophical principles, mathematicians prefer those, that are closest to them
 by schematic nature of professional
thinking and by common sense of everyday life. After all, by and large, the
traditional formal logic has grown and mastered this everyday experience. In
this regard, formal logic is a purely empirical science: formal forms of
thought are abstracted from the everyday content of sensible things. This fact
can be explained, and formally  logical science does this  however, beyond
the content of its theory.
Similarly, mathematics seeks to explain its reasons and its certainty
with facts when it is not mathematics per se, but philosophy of mathematics,
its methodology. Explaining its own grounds is not what mathematics deals with;
it starts doing so when it has certain problems in its direct field of study.
However, we should say that these philosophical problems arise in any science,
as any science is only a special way of understanding the specific objective
reality.
This, of course, is a banal but curious fact: no science explains its
driving forces, it only explores its subject.
In both of these fields, there is no distinct clarity, and when there is
clarity, the questions disappear. Then, the science degenerates from research
into technological process  from an anonymous author and without thinking
about the subject. In the context of the division of labor (in this case, of
scientific labor)  it is very much a natural process, social in its nature and
with a wide range of negative consequences. Technologically wasted forms within
science are embodied into outwardsubject, social and pedagogical technologies
 and this (under certain social conditions) is seen as the essence see the
meaning and purpose of science. Truths, revealed by science and deprived of
their personal character by this science, come to life. This is the sign of its
practical efficiency, and philosophy of science deduces from it external causes
of science development.
Worked out in science "technological forms" as the embodiment
of the truth become stable forms of development of knowledge, operations and
actions with a hidden meaning and of hidden origin. Today, they are associated
with go to the computer technologies, with a special objectified world, and,
within the human subjectivity in personality psychology into prejudicial forms.
Human psyche is full of these forms, which generates a range of secondary
prejudices and pseudo problems both within the science and in everyday
consciousness. It is not surprising therefore, that inside the science of
mathematics, schematisms, fused with the methodology of formal logic and common
sense of ordinary consciousness, raise so many philosophical problems, often
leading it into mysticism. From the Pythagorean times one of the actively
discussed issues is the problem of a number  one of the fundamental concepts
of mathematics. It is naturally, that these "technological forms" of
science also become the main concepts of the educational process. And,
unfortunately, not only the content but also the form of the pedagogical
activity. And nonsense spreads. "The school system  says the famous physicist  forces the teacher, who
has a good idea about how to teach children to read, to do it in a different
way, and even confuses the teacher head so that the poor ting starts believing
to feel that her method is not good at all. A mother who ones punished her
hooligan boys in some way, then throughout her life feels guilty because
"the experts" convinced her that she acted "incorrectly"
(Richard Feynman. Surely you are joking, Mr. Feynman. M., "Kolibri",
2008, 468 p.). No wonder, therefore, that such formal logic cannot explain
creative development of mathematics. That is why people start searching for its
deep hidden grounds, which cannot be further explained: intuition (subjectivepsychological
basis), construction (creative ability of the subject), an empirically formal
rigor of rational thought. And again  it is not just a problem of mathematics
but science in general, in mathematics it is just seen more clearly, rather
than, say, in biology or chemistry, where the objects can be put into actual
experimental conditions. In mathematics, as well as in philosophy, the subject
content is not so explicit. And it is not easy to understand a banal truth that
this content is generated a transforming activity of a man. Like the subject of
biology and chemistry. Like any objectness. However, not any objectness is as
obvious as the actual existence of everyday life. But escape beyond the limits
of ordinary life involves entering the inner, the hidden and not obvious, the
content of this very ordinary life, unfolding its content and turning it into
the semantic field of consciousness, which historically differentiates itself
as a subject and thereby constitutes different sciences.
Each of these sciences develops its own methods, its "operational
technologies" and checks them for the truth. Mathematics seems to do it
more carefully, as if making it the subject of its own  and therefore creates
the illusion of logic "research." The accuracy of this science is the
subject of envy by many and taking its accuracy as a norm, people start to
classify sciences on this basis. If we proceed from the assumption that
consciousness reflects, reproduces the being, then we have to admit the
existence of specific objective content in both mathematics and philosophy. In
mathematics, however, there is an understanding that mathematical form of
thinking is generated by mathematics itself and there is nothing outside it
that could act as its basis or its object.
The position of science is to understand this through its nature and
unfold this nature through the development of shaping’s of this nature. This is
the theoretical knowledge of the thing, i.e. the reflection of the thing in its
development, in its natural dimension, or  in the forms of the universal and
the necessary. That is what should be called scientific knowledge. It is clear
that this knowledge is the true knowledge i.e. corresponding to the general
nature of the things, and a method of generating every special things inside
this general nature. Only then the thing is understood, and only then a man can
take the position of this thing, and coordinate their actions in accordance
with its logic. It is because the thing becomes "transparent" for the
thinking of a man and can be practically used. Philosophy is interested in the
nature of the concept, the form of subjective activity, the objective law of
its development. All this is one and the same. And there is basically no
difference what kind of material is used to explore these things. The adequate
to itself form of philosophy is its work with the general definitions isolates
from the structure of objective cultural and historical existence. The same
happens in mathematics: its adequate form of movement is realized only in pure
abstractions, generated historically, in the study of pure quantitative,
spatial and temporal relations with no load of sensuality. This circumstance
creates the illusion in mathematicians, for sensuality, subjectobjective
content in it is not explicit. The theoretical form of science is inaccessible
for the ordinary consciousness, it requires sensuous visualization and visual
authenticity, thus, clarity and sensual visual authenticity  thus it only guesses its essence,
but doesn’t understand it. But behind every thing there is essence, given by
the thing, but also hidden by this thing. It generates this thing and manages
it. Anticipation of this situation is given to each consciousness. Clarifying
this situation is the task of philosophy. But what about mathematics, which has
lost its direct connection with reality?! Mathematics deals with identifying pure spatial
and quantitative forms of existence, independent of any circumstances. Images
and schemes of spatial activity created by it in their quantitative expression
immerse into the cultural aspect of history – and at the same rime into the
culture of mathematics itself.
And that is why mathematics as any other theoretical science exists as
an aboveindividual form, which contrary to the views existing within
mathematical culture doesn’t depend in its universally required definitions on
a man or his consciousness. The human consciousness can either adequately
express them or distort. And, most interestingly, in both cases it is not always
capable to realize whether it is in the position of the truth or the error.
Therefore it is not surprising that it often cannot answer the question what
determines mathematical forms, what is their nature, and whether they are
created by consciousness or weather consciousness just reflects some objective
content, given in spatial quantitative forms.
And today, when mathematics loses spatial representations and the
concept of quantity is transformed in such a way that it almost disappears
along with concept of number, loss of an objective foundation of mathematics is
becoming a fact of its own identity. The opposite of this presentation, growing
in consciousness, lies in the fact that the mathematical relations reflect the
deepest universal ties, lying even outside of this world, but defining it. This
representation is not only logically possible but  as any logical possibility
 lives quietly in the mind, since a mathematical form is a form, which from
the very beginning is cleared from any empirical content, abstract, abstracted
from it, and therefore seemingly not related to it. This representation dates
back to the Pythagoreans, who believed the number was in the foundation of the
world based on the fact that the empirical world obeys mathematical numeric
relations. And if it was not so, then why would we need math? The world is
“inferior to” science, because the world is expressed in science. The world is
subordinate to the insane action of humanity  because the world allows this
action. And not just because the world forms
can be broken by the insane interference with their existence, but because the
world is full of variety, order and chaos are just moments of the existence,
pure and mixed forms are just an accidental discovery of one’s own essential
moments. With the immersion of science in the structure of the matter we
discover new properties, which, with its of its paradox nature, actualizes the
problem of space and time. And, thus, setting new challenges for mathematics.
Microphysics, for example, now allows the presence of such unusual properties
of the material world as a multidimensional space and time reversibility. In
the interpretation of the phenomena of the microworld, however, there is
another extreme  rejection of the concepts of space and time in general. Such
views, of course, are related to the mathematical moves of science. The problem
of relationships of mathematical "spaces" to the real space arises in
connection with emergence of the socalled nonEuclidean geometries. How are
these various geometric images of related to the space of the objective
external world? Poincare, for example, believed that all mathematical
(geometric) spaces are equal, none of them has any advantage over the other.
All of them are abstract models that exist only in the mind. And he didn’t even
set the question of their relation of the objective world, but considered that
in the description of physical phenomena mathematical spaces are more
convenient than others. Therefore, there is no sense asking which geometry is
characteristic of the real space  Euclidean and nonEuclidean.
If these geometries are consistent, then, from the point of view of
mathematics, they are acceptable. Such representation does not derive directly
from the mathematical concepts they are clearly of philosophical and
methodological nature. Therefore, we cannot conclude that behind any
mathematical concept or phenomena there is a certain physical reality. This
suggests that the interpretation of the mathematical provisions  is not an
easy matter, it involves the transformation of all ideas related to the sphere
of phenomena, and, most importantly, with the experimental verification.
One of the proofs of God's existence contains a conclusion from the
presence of the concept of God to his existence. The lack of a developed
ability of logical selfreflection, even in science can easily lead to all
sorts of "ontological" findings (the truth of which is not justified). Kant's analysis of the categories of
pure reason showed that the concepts of space and time cannot be derived from
experience, and therefore we cannot conclude that they belong to the reality
itself. But they exist within the structure of human subjectivity as
preexperiment forms of sensuous contemplation. In other words, space and time
belong only to the subject of cognition as its own forms, which it organizes
into various material, which we feel with different senses. Indeed, which kind
of the experience do the concepts of mathematics derived from? The science that
studies space, its structure, is geometry. Modern physics readily admits that
in the microcosm the space is characterized by a specific metrics and topology
(i.e. by a special dimensions and special quality characteristics). Metric
relations here may have a qualitatively different character than in our
ordinary space and time. In mathematics there are "mathematical
spaces" in which there is no concept of distance. Such spaces are called
"nonmetrizable topological spaces. "If the space of a microcosm is
really nonmetrizable, then it will be characterized by only topological
relations, and any metric relations will be missing. These facts cannot but
change and the structure of the mathematical thought. Rational understanding of
metric relations is possible only if certain standards of length and time are
present. In other words, if they are not measurable, they are not a part of
rational thinking; Physics, for example, rejects the absolute space of Newton
on the grounds that it is of no use, it is not measurable: the rational sense
of space appears only in relation to extended bodies. In a microcosm, lattices
and atomic vibrations serve as such standards. But on a very small scale or
under certain physical conditions (e.g., at ultrahigh density or infinite
speed), these standards cannot exist.
And if a measure is lost, then length and time lose their meaning. At
infinite speeds zero speed and infinite speed coincide, there are no numbers;
there is no scale for determining distances. Both of these are possible to express
only through a relationship to the values of energy, and space, its topology
and metric only is interpreted only through these values. The space of the
thing is determined by the nature of the thing itself. And if the thing exists
in the space of the other things, it means that its own spatiality is
compatible with and aligned to the space the other thing. Hence, there is an
illusion that the space is just an empty receptacle.
And what is the determinacy of the thing with its nature? This is the
determinacy of the relation of this thing with other things (its coordination
with other things). But pure space, studied by geometry (mathematics) has no
other definition other than the geometric ones, i.e. arising out of its
universal nature. Mathematics, preserving the universality of the definitions
of pure forms of space and time (quantity), seeks and finds ways to specify
peculiarities of spacetime dimensions of any reality. No thing is conceivable outside its relations
with other things, and space is the form of this relationship (Aristotle,
Leibniz).Time is an appreciative attitude of a thing to itself, to its own
discriminated forms and simultaneously the detection of its selfidentity, the
moment of inalterability i.e. timelessness. In other words, quietude of the
thing, its selfidentity is found only under the condition of its own change,
therefore, only in relation to itself. That is why quietude is the removal of
the time, the absolute identity of the thing to itself; super temporal
existence of the thing appears only as its spatial difference, as its spatial
existence. At the time of its spatial and temporal integrity and selfidentity,
space and time disappear. And appear again in its own motion. In all this,
however, there is a problem, of the universal character. Correlating geometric
sphere with a spherical body, I easily seen, express contradiction. This is a
contradiction between the shape formed by the science, and the shape of a real
thing. The subject here is opposed to the
object. Because the sphere, no matter how it is conceived in geometry, is a
shape of a subject. But the mind requires to relating a thing, as we have seen,
to its own universal form. The question arises: is a universal shape introduced
by the science related to the universal form of the thing itself. This question
is further divided into two other questions: how does math gets "its"
universal form, for example, the sphere? And is there in the objective reality
a universal shape – the shape that claims to be common for all spherelike
objects? And the science of logic, if it cannot resolve these contradictions,
and has no right to claim for universality. Because it is not enough to say
that" a stone is spherelike", we need to demonstrate a relation
between them, and this relation exists in reality, not just in our imagination.
Philosophy proved long ago that no provision of a theoretical science
reflects the reality directly, that there is no exact coincidence. For any pure
form, for example, a mathematical sphere must be aligned by the human ability
with the actual empirical form, with the empirical content of reality, must
allow seeing in this empirical reality things, that cannot be grasped by
sensory perception. Knowing this form, we recognize it in things, and in these things
we see what cannot be aligned with them, what is opposed to them. Here I align
a sphere with a stone ball, a mathematical formula with a specific problem of
my case. Here school mathematics, which taught me to express sensuous
correlations of a real situation through the formulas of mathematical theory,
at least indicates the direction of my efforts, and I know that I should
express the reality of being abstract logical and mathematical ratios. The
sphere itself is necessary to me only as the ideal condition of my practical
activity, among and by means of objective and geometric shapes, subjective and
objective.
However, can’t this
shape that emerged as a necessary condition of cognition belong to the essence
of the thing, to be its own internal definition? Do you remember a famous
statement that any content has a form? So what is the form of a stone? After
all, in order to judge about a thing I need to express something, which appears
to be this thing through its own form. Relate the thing with itself and not
with what I have in my head. Then the judgment about the thing will express its
nature. But where does this form come from? It seems clear that it is
impossible to abstract it as a а feature common
for a group of things. There is no sphere in the real life, as well as there is
no right line in the real life. Besides, we look for the “examples in nature”
for kids in school and for ourselves, smart people of science. A geometric
sphere and spherelike spatial existence of a stone  what is the connection
between them? These are quite different things! A geometric sphere is from the
abstract science, geometry, which I can be good at or can ignore, while a stone
is real and can be used in many functions. But do I need the sphere itself?
Do you remember how long they "constructed", having found no
examples in nature, the Lobachevsky space? A school teacher demonstrates a
sphereshaped stone and a ball as a model of a geometric sphere. But the
geometric sphere is modeled only by thinking, through the form of the concept,
through the synthesis of diverse abstract definitions in their logical sequence
that define the formation of this geometric sphere. For without any science
(i.e., before school) I carry this image in my heart and can make use of it.
That is why I compare the sun with the sphere, and not the sphere with the sun.
I carry all measures available in my existence and use this ideal forms to
measure the world. That is why a man is "the measure of all things"
(Protagoras).
But a man has to create any measure as well as his own ability through
his own transformational objective activity revealing is the dimensions of the
thing itself. That's math geometry appear. And having created this miracle of
science, a man starts bothering his head with what it is and why does he need
it. After all, the measures do not seem to be difficult, they are used to
measure external things. But what measures do we use to measure me a human
soul? Some people to use math even here. The desire to present an object
outside of consciousness, science is almost natural. Such representation occurs
within the everyday experience, which opposes consciousness to being.
Mathematics, which doesn’t deal with the objective reality tries, falling to
philosophy, to link its mathematical representations with reality  either
deriving the reality from its concepts, or trying to find the ontological
prerequisites for it actions. But in both cases it seeks for their
authentication. Space is a form of correlation of things, the original form and
condition of their particular connection in the mind. Absolute space is a pure
form of identity. Here things
are registered by consciousness as not related to each other. At the same time
this is the form of authentication of things. Consciousness is a differentiation
between the particular and the common. It's a contradiction. Consciousness
fixes the difference of things unrelated to each other, preserving the form of
identity, i.e., size, space.
To be more exact, it doesn’t fix the mind, because in order to fix it,
the mind should be present and it is possible only through form of and
contradictions. Therefore, a contradiction should reveal itself before
consciousness. It is found
with the real movement of the subject in the forms of its activity. The
identification of the various in the process of human activities leads to the
category of number, i.e. to recording simultaneous existence of the various,
and in this variety, the existence of identical things  outside their
qualitative determination.
But if differentiated moments represent an activity, then the activity
itself and the process of differentiation contain the form of their
correlation, unity and equivalence. The simplest, qualitative, mathematical,
yet preconscious equivalence, but existing in external sensuous activity.
Consciousness itself grows out of this activity contradiction. The form of
thinking lies in the forms of activity and with the help of activity is turned
into consciousness, which can turn itself into a subject. And it is easy to
understand that the forms of these activities represent the form of spatiality,
all necessary coexistence and all its necessary elements. Therefore, space and
quantity are expressed as identical forms. "Space  says Hegel  is a pure
quantity. It is a pure quantity not only as a logical definition, but also as a
direct heart of the matter" (Hegel. Encyclopedia of Philosophy. V. 2.
Philosophy of Nature. M., Mysl, 1975p. 45). Advancement of activity into
qualitative certainty of things reveals new forms of identity
"series", a new quantitative certainty and a new measure (and hence a
new form of a number). The form of identity is quietude, the form of the
undifferentiated. It is, therefore, not characteristic of the consciousness and
the subject.
The point, we should note, is the absolute selfidentity of the
existence, beyond any differences, therefore, outside of any dimension, it is,
therefore, outside qualitative and quantitative determination. That is why it
is because nonspatial (let us remind you that quantity and space are
identical). And that is why it allows mathematics to enter microcosm where
dimensions cannot be found. This purely logical fact allows thinking of any
thing as of a point. Here things are abstracted from quantity and quality. This
looks like a purely subjective process.
However, here we deal with inexorable objective logic of the active
existence, which presents its definition as a definition of external reality,
or vice versa  external determination as tits own. This happens because the actual process of
activities is carried out as the unity and the identity of the subjective and
the objective, detailed distinction between which are realized through the
development of consciousness and knowledge. Including, of course, and the
mathematical consciousness. Separation of a point  is being in two different
points simultaneously. Simultaneous fixation of two points in space gives an
image of the spatial differences, the distance, moreover, that of the line, the
first measurement of space. For the second dimension to emerge on objective
change of motion is necessary and fixing this fact, i.e. three points, as a
unity – a plane.
Third dimension emerges in a similar way. The change in direction
happens in reality not in accordance with the form assumingly inherent to
consciousness, hiding in the depths of mathematical thinking, but in accordance
with the objective forms of the external objective circumstances. This is
explained by the rigidity of the nature of things involved in the activity. A
house mouse in a psychological experiment, entering the room starts to move in
accordance with the objectively important characteristics of the space, it
doesn't care about, first "measuring" the skirting, and when it
reaches the corner it feels the necessity to change direction and starts moving
along the hypotenuse to the point of the entry into the room. Satisfying the
need for food is postponed until the muse creates a "subjective"
condition for its secure objective activity as if overcoming the Kantian a
priori spatial form. This is the emergence of a psychological image of space,
revealing the form of the location, objective conditions of real life. Geometry
does the same thing, in an abstract, pure form, in a distraction from the real
substantive work, revealing a purely spatial, indifferent to the things
conditions  dependence of the elements of the pure space on each other, their
relationship and interconnections. Thus, geometry provides a practically acting
man with the means of active transformation of real the space, i.e., conversion
of real spatial relations. Creating space by separation of the point gives the
image of a space as a set of points, as a set of coordinated with activity
(Leibniz) points (things), indifferent to each other.
Point as a form of selfidentity of the thing, which this definition
makes insignificant for the existence of other things (points), makes the
discreteness of space and time infinite. Everything that can be defined as a selfidentical
certainty, mathematics and turns into a point, a numerical form. But things in
the form of their selfidentity, i.e., quietude and indifference, are different
from each other in terms of space (quantity), which makes them essential for
the activity. The space is meaningless, there is just a condition of
representation (think ability) of an object (material or ideal). In geometry
(mathematics) the motion of a point, forming a line t coincides with the
activities of an ego, geometrician, but a space there is presented as an
objective and logical condition of activity, and is not generated by activity
of ego, its active movement.
The subject, a man, identifies himself in space, finds himself acting
and feeling. Reflection of things existing in one location as subject of
circumstances of real sensuous activity of the subject separates abstracts,
this location as space  as a kind of indifferent relationship of things,
changed by un indifferent interest of an acting subject. Here is the foundation
of mathematics, with its complicated categories of the finite and infinite, the
discrete and continuous, measure and number, axioms and methods of work with definitions of value and quantity and the amount (space and
time). The space in this way acts as an indifferent necessary condition of
the activity of the subject. Its flexibility is not absolute, but its
relativity and resistibility has a particular content, which may be seen as a
special form of quantitative ordering of objects as a special space. This leads
to the development of concepts of mathematics in all its representations.
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Table of contents: The KazakhAmerican Free University Academic Journal №7  2015

