Truths and illusions of mathematical thinking

Table of contents: The Kazakh-American 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 self-reflection (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 outward-subject, 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 (subjective-psychological 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, subject-objective 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 above-individual 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 multi-dimensional 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 so-called non-Euclidean 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 non-Euclidean.

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 self-reflection, 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 pre-experiment 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 "non-metrizable topological spaces. "If the space of a microcosm is really non-metrizable, 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 space-time 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 self-identity, the moment of inalterability i.e. timelessness. In other words, quietude of the thing, its self-identity 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 self-identity, 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 sphere-like 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 sphere-like", 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 sphere-like 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 sphere-shaped 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 pre-conscious 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 self-identity 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 non-spatial (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 self-identity 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 self-identical certainty, mathematics and turns into a point, a numerical form. But things in the form of their self-identity, 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 Kazakh-American Free University Academic Journal №7 - 2015

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