Thinking as a function and its decomposition into a Taylor series

Strigin Mikhail PhD in Physics and Mathematics Director, "Mitrial" LLC 454004, Russia, Chelyabinskaya oblast', g. Chelyabinsk, ul. Akademika Koroleva, 4, of. 6 strigin69@rambler.ru Other publications by this author

One of the main tools of modern science is the Taylor series, which allows you to approximate a function describing an arbitrary entity using an infinite and zero-tending sequence of signs. Such an approximation is used everywhere: in physics, in mathematics, in chemistry, etc., because a daily function describing a certain phenomenon has a very complex form. Investigating the behavior of an arbitrary object, the scientist builds a model and further describes it in the form of some function, for an approximate description of which the specified series is used (for example, the sun in the first approximation is a ball, in the second, a flattened ball, in the third, an uneven flattened ball, the surface of which, due to prominences, deviates from the ideal view). It is easy to notice that the description of finer details requires more characters. Thus, the first geometric impression of the ball is complemented by the dynamic component of the prominences. Even Coulomb's law, which is considered canonical, describing the interaction of two charges, is actually an approximation showing the dependence of the interaction only on the coordinate (which demonstrates the initially geometric approach). A more accurate approximation takes into account the dependence on velocity and is called Lorentz's law. We will try to show that the Taylor series is actually a natural and basic tool of thinking (not only in science, but also in other spheres of life, up to everyday life) in the search for a hierarchy of cause-and—effect relationships - first geometric, then dynamic. Then we can try to extend the tool of the Taylor series to the humanities, which will allow us to apply physical approaches to describe the constitution of spiritual processes. As in mathematics, the Taylor series allows you to extrapolate the function of a phenomenon to a large area of definition, and in the humanities, such a sequence will allow you to extend some meaning to a large semantic area.

But, unlike mathematics, in the humanities it is more difficult to define the concept of a function, which in the traditional mathematical representation corresponds one set to another, for example, determines the dependence of the maximum deviation of the bridge from the horizontal y coordinate along it, . Then the Taylor series consistently approximates such a correspondence to empirical data, since even a minimal error in such a question can lead to tragedy. Similarly, the subject builds a correspondence between the real and the conceivable with the help of sentences, which are a special case of symbolic forms. The size of the latter varies from simple judgments to large texts equal to the volume of the book. Such symbolic forms, as has been established by many researchers, for example, Ernst Kassirer ^[8], are spiritual functions. Obviously, a mathematical function is a special case of a symbolic form. Spiritual functions, like any other, correspond one set to another, in our case they display the immanent in the semantic space and are explicated in the form of verbal constructions. Just as a physicist tries to expand any discovered function describing a certain phenomenon and having a domain of definition to a large area, so in the humanities, a researcher, having discovered a symbolic form whose meaning can be determined for some area A of the semantic space, will try to expand it to new areas of knowledge. Since symbolic forms are spiritual functions and have the same definition as functions in mathematics, they, like any function, can be decomposed into a Taylor series ^[6] in the vicinity of some meaning A:, where in our view means semantic removal from point A. Thanks to such a series, the meaning of A can be expanded to a large area of semantic space.

The sequence of symbolic forms constituting the representation of an object and constructed by the subject over a long temporal period can be called a function of thinking. Even in the simplest case of everyday use, this sequence is a series similar to the Taylor series, in which the first term carries the basic meaning, it is supplemented by secondary forms, tertiary, etc., which determine the dynamic aspects of meaning. Also, as in the case of the usual Taylor series, the first term is a constant equal to the value of the function at some point A and which can be correlated with the geometric description of the object, the second term is linear, determined by the first derivative of the function and describing the rate of change of the phenomenon, the third one is correlated with the quadratic term, in other words, with acceleration, that allows you to understand the features of the function on a longer temporal interval. Each subsequent derivative determines the change of the previous one, which forms an iterative sequence, just as the prominences determine the change in the ovality of the ball. In addition, for such a decomposition, as is known from mathematics, the thinking function must be a smooth function in the vicinity of some semantic point A, which arises together with the direction of the subject's intention to the real point A. The term smooth function applied to the symbolic form means that it has no breaks, in other words, if a stone is thrown into the sky, it cannot disappear at one point and appear at another, just as the existential function of caring should not be instantly replaced by indifference. Similarly, if an object in the process of apperception appeared initially in the form of a cube, then it can be successively refined and become an elephant or a hippopotamus, but not a snake. (Although there are exceptions to any rule: a snake can curl up in the form of a cube).

Through symbolic forms, the subject constitutes the meaning of the object, compacting the semantic space, which is somewhat conditionally located between the subject and the object. Indeed, the fact that a person in phenomenological research studies not the object, but the method of its comprehension, which can be called the function of intention, or the function of transcendental apperception, or the function of thinking, was discovered in his "Copernican revolution" by Kant. In other words, it is not exactly the object that is being studied, but what is between the subject and the object, and therefore no experience is possible in its pure form. The very concept of experience already introduces the idea of the subject into it. The temporal unfolding of an intention does not proceed continuously, but rather is a sequence of refinements (an iterative series) that can lead to drastic changes in the representation of the object. For example, Pluto in 2006 lost the status of a planet as a result of a sequence of astronomical studies that took into account not only the external parameters of this celestial body, but also its dynamic characteristics.

The researcher, producing, according to Kant, a transcendental apperception of the manifold, within which the synthesis of the point now takes place, what can be called the study of the object, uses a set of ready-made symbolic forms, like a master using an instrument in his professional activity. Developing the image of the master, we can add that the researcher, faced with a complex malfunction, first uses the existing tool of existing forms, and then, if they are not enough, comes up with his own devices – new forms. Thus, the temporal sequence of apperceptions forms a series of increasingly small refinements that form a whole idea of the object.

Since symbolic forms do not belong to the object, but are located between the subject and the object, it is obvious that the mechanisms of the psyche are involved in their construction. Therefore, for a better understanding of the evolution of symbolic forms, we will use the psychoanalytic concept of Jacques Lacan, who discovered the registers of the psychic in the subject: real, imaginary and symbolic. Such a representation increased the dimension of the intention. If the Cartesian "I think, therefore I exist" reveals its one-dimensional representation, then the psyche of the Lacanian individual is already essentially three-dimensional. In Descartes' view of the subject, there is a clear asymmetry towards the subject, since, articulating the self, he finds himself in the center of his own constitution. The subject turns out to be, like a positive charge, a divergent unit that radiates the field of speech and verifies the environment through doubt. Descartes, asserting the existence of the subject, does not give any description of it. "I only indicates the subject of the act of utterance, but does not mean it" ^{[10, p. 209]}. In the Lacanian description, the relation of subject and object is also asymmetric, and the subject resembles a negative charge more, since it absorbs the symbolic of others. It can be added that in the history of philosophy there have been regular shifts in the position of the function of thinking along the straight line connecting the subject and the object – Spinoza turned the concept of Descartes, identifying the subject with nature, in the language of projective geometry, Spinoza inverted the function of thinking. Such uncertainty of the placement of the function of thinking relative to the subject and object led to different concepts: "in relation to the spiritual being, the being that he knows is never "outside", but always just "here"" ^{[14, p. 95]}. The uncertainty of the position of the thinking function is primarily due to the fact that the spiritual boundaries of the individual do not coincide with his physical boundaries, which allow dividing the space into subjective and objective. In other words, if Descartes and Lacan begin the decomposition of the function of thinking with the main term – the constant – of their own identification or the geometric definition of their own Self in relation to the object, then Spinoza turned immediately to the next members of the series, starting to write it out in reverse order, namely, the connection of the object with the surrounding context, with nature.

The mapping, or reflection, of the immanent into the semantic is somewhat more correct to consider as the stratification of the immanent into the real and symbolic, produced by the individual. On the one hand, the symbolic, like any layer (for example, bark separated from a tree), after stratification begins to live its own life quite in the spirit of nominalism, on the other hand, it bears the imprint of the real (as the bark bears the imprint of the trunk). And the Taylor series helps to explicate the ever deeper connections of the object and context, and the hierarchy of meaning. The evolution of the symbolic is determined by the register of consciousness that Lacan called imaginary.

According to Lacan, the symbolic of the individual is formed by others, in other words, the subject is a set of symbolic forms that he has adopted from others, interacting with them throughout his life (with mom, dad, grandmother, colleagues). Figuratively speaking, a person absorbs others as he communicates with them in one way or another, acquiring skills in possession of symbolic forms, as a master learns to use a tool. In other words, the subject builds a Taylor series in his own intention, consistently using the symbolic forms discovered earlier, where he substitutes the perceptual data of the object to which the intention is directed. But such a constitution has a subjective color, since the individual, through the register of the imaginary, puts the symbolic of others in their own order, which creates his individual function of thinking. "The imaginary "imaginaire" is a complex of identifications and representations of a person about himself" ^{[10, p.210]}. Thus, the function of thinking depends on the imaginary, which is like a vessel, the shape of which determines how symbolic forms will be placed there. Using the image of a vessel, it is possible to supplement the idea of the Taylor series – first large symbolic forms (constants) are placed in the vessel, then small ones (related to the dynamics of the object) and, finally, the smallest ones that allow it to be densely filled.

Despite the asymmetry of the Lacanian subject's intention, his function of thinking is much richer. The register of the real, something unconscious and unformed, mixed up in perceptions, makes its significant contribution to the construction of the Taylor series of the function of thinking. In the realm of the real there is an irrational component of the subject, which makes us search for the noumenal, trying to know the truth. In response to a request for truth, the imaginary offers answers, creating them from symbolic forms that the subject already possesses. "It is naming, the name represents the function of symbolic identification" ^{[10, p. 211]}. It is possible to figuratively represent the work of the imaginary and symbolic as a work of form and content that defines the real subject, correlating, in turn, with the phenomenal, generating an endless series of perceptions and apperceptions. Indeed, such a correlation generates, according to Hegel, a shortage (the phenomenal never comes into exact correspondence with the real), which forces a number of refinements. Such a shortage can be compared with the difference between the real and reality, where the real is an infinite series that approximates reality, but which never reaches a complete coincidence with it, just as in mathematics the difference between a certain function and its approximation by an infinite series always has a very small, but quite finite value.

Even in everyday use, the Taylor series of the function of thinking, as already mentioned, explicates the hierarchy of cause-and-effect relationships. Most often, when detecting some phenomenon A (child smoking), we associate it with cause B (street influence), without noticing the deeper causes C and D, etc., as shown in the Taylor series diagram.

Here an attempt is made to demonstrate that the causes of C and D are hidden and do not lie on the A-B line, figuratively speaking, they "act from around the corner". Any thinker, encountering a new thing, tries to define it based on logic and some model, which is the sum of his a posteriori and a priori representations ^[7]. And if the causes C and D are not represented by symbolic forms, then the researcher is not able to detect them. Such phenomena, like neutrinos, "pass through" without interacting with the researcher. Thus, on the one hand, the significance of secondary, tertiary, etc. causes is usually belittled as insignificant, on the other hand, they may be invisible to the researcher. But, as Prigozhin discovered ^[11], at the points of bifurcation (family scandal), where relationships undergo topological changes, the whole hierarchy of causes becomes equivalent (street, bad example of the father, etc.).

One of the clearest examples of such a hierarchy defining the Taylor series is the correlation of the concepts of desire and demand in psychoanalysis. If the demand, as Lacan defined it, is always transparent and acts in a straight line, then the desire, on the contrary, is "hidden around the corner" ^[3]. If the first has rather geometric grounds, since it is always connected with the intention to get something concrete, then the second is more dynamic - the desire is much more difficult to articulate.

A similar picture can be found in the study of the social: first of all, direct cause-and-effect relationships are observed, and then, based on them, a theory is built. This is how Machiavelli built his theory of social, which justified the behavior of individuals based on the passions inherent in each of them. This description is a consequence of first-order logic, which is the first member of the Taylor series of the thinking function. The same order of social coherence in Freud's description: he attributed to each subject the concept of libido, which determines the attraction of an individual towards the leader of a social group (without fully defining the concept of libido) ^[15]. Both are obviously right, if we take into account only the first member of the series of the function of thinking. But, for example, the idea of group solidarity, called Ibn Khaldun asabiya, can hardly be described based only on the individual behavior of the subject (for example, libido). Obviously, this process has a higher level of complexity, for the description of which it is necessary to take into account not only the individual and his immediate environment, but also the entire social hierarchy, which again forces us to take into account the dynamic aspects of the phenomenon. And each subsequent member of the Taylor series makes its own, albeit small, contribution to the existence of the individual, which can no longer be considered as a unit (as a constant defining the first member of the series), but as part of a coherent social that has become a unit at a higher level.

Since the representation of the Taylor series uses the concept of the distance between the points of the space in which the function under study is defined, in order to find the decomposition of the thinking function in such a series, it is necessary to introduce a metric in the semantic space. But such a metric, although mathematically incorrect, already exists – the geometry of knowledge (the relation of sciences to each other) is described by epistemology, dynamic characteristics are set by epistemology. For a clearer definition, we can turn to descriptive linguistics ^[5], where the "distance" between words is defined as the frequency of their joint use. On the one hand, a very specific metric appears, on the other hand, it takes into account only statistical aspects and weakly takes into account semantic aspects. In addition, such a metric is obviously evolving – if the semantic "distance" between the words time and money, according to descriptive linguistics, was large before the 20th century, since these words were hardly used together, then at the present stage the appearance of the saying "time is money" has significantly reduced the distance between them.

Let's try to imagine the function of thinking, which can also be called a function now, in the form of a decreasing series of a sequence of perceptions and acts of imagination. , where T is a transcendental apperception, P1, P2, etc. is a series of consecutive perceptions, f is an apperception function. In an ordinary situation, the subject's intention fixes its attention on some point A of the real, followed by primary perception, which can be designated as P1(A). The primary apperception f(P1(A)), which correlates with a member of the Taylor series f(A), is a consequence of the strongest primary perception. The next step is the imaginary and secondary perceptions, generating secondary apperceptions, complete the real to some objective, which is indicated by the following term f(P2), correlating with the second term of the Taylor series, then tertiary apperceptions f(P3) – etc. The process of transcendental apperception can be represented figuratively by the example of the sudden discovery of an elephant in the jungle: then it will denote an abstract dark object-a cube (the shape of an elephant), the following members of the series will give contrast to the cube, define it as alive, add details in the form of ears, trunk, color palette, chewing grass, etc. (Provided that the ideas about ears, trunk, color palette and chewing grass are already present a priori or a posteriori). After that, the abstract object-cube turns into a certain elephant – each subsequent, albeit already small member of such a series of apperceptions adds uniqueness to the object of intention. And the larger the series, the larger the semantic area will be able to mean, the more significant will be the area of semantic definition.

Let us clarify that the mathematical term "smooth function" in our context means that the constitution of the symbolic form of the image of an object through a series of perceptual refinements of the object occurs sequentially, while simultaneously expanding the scope of the semantic representation of the object. For example, if after detecting a snake, as a result of the next iterative perceptual step, it is determined that the snake has paws, then the image of the snake will singularly move to another part of the semantics, where the snake will turn out to be a lizard. In this example, the function of transcendental apperception will not be smooth.

Since the concept of meaning is close to the concept of information, it is not surprising that such an idea of the evolution of symbolic forms correlates with the image recognition algorithm proposed by Herman Haken ^{[16, p. 211]}. If the algorithm of traditional recognition is based on the analysis of the image, its division into components, recognition of each part separately and subsequent reassembly into the desired image, then in its variant "at the first stage, the image is perceived at a global level, at which transitions from the initial state to several attractors are possible. Then the sensor system is activated, which allows taking into account additional features of the image and thereby selecting a more finely detailed set of attractors" ^{[16, p. 47]}. The understanding of the attracting attractor by Haken and us is identical – it is a certain image (idea) to which the entity is drawn. And the image, as in the previous example, can follow the path of a snake or a lizard, because they are similar, and only their detail will determine which attractor – snakes or lizards – is actually observed. Subsequent (already smoother) refinements will determine the type of a particular snake or lizard (for example, a viper or a copperhead).

In the last example, the constitutive series of apperceptions means the definition of subsequent, more subtle derivatives, which, as already mentioned above, characterize the change of the previous ones (just as prominences change the shape of the surface of the Sun). Thus, the sequence of apperceptions at point A has a cognitive meaning – the expansion of the semantic field of perception of the world allows, in turn, to expand the perceptual capabilities of the subject. This view is consistent with the concepts of anthropological evolution, intuition and its long-range effects, which will be described in more detail below. One of the main evolutionary steps in anthropological development was the appearance of symbolic forms expressed in mathematical formulas. Substituting the corresponding perceptual data into these formulas, the quality of which increases due to the use of sensors more sensitive than the sensors provided by nature, allows us to expand the hierarchy of transcendental apperception, increase the number of members of the Taylor series, covering not only the geometry of the object, but also its dynamic aspects (the paradox that the weather forecast with all modern hardware is valid for a maximum of a week, whereas experienced people in ancient times could predict the weather for up to six months, resolved by a conflict of intelligence and intuition).

The hierarchy of perceptions is connected with the hierarchies of the narrative (causal hierarchy) in which the object under study is located, and the internal hierarchy of the object itself. For example, in the image of an elephant, the contrast of skin pigmentation and the presence of ears and trunk refers to the hierarchy of perceptions of the object itself, but the process of chewing grass through an act of imagination connects us with the place where he could pick it and, accordingly, with the influence of the external environment on the object. It can be noted that, despite the general smoothness of apperception (not counting examples like snake-lizard transitions), it has a quantum nature of the constitution of thinking – the function of transcendental apperception from the addition of the next perceptual member of the series always changes abruptly. Another perception, albeit insignificant in its informativeness, nevertheless adds a very specific amount of information that can start an irreversible process, by analogy with a bridge, exceeding the deviation of which leads to its destruction.

Hegel also noticed that the different members of the Taylor series, which he called moments, have a qualitatively different nature, indeed – this is length, – this is area, – this is volume. The philosopher in ^{[4, p. 257]} paid great attention to the ontology of differential calculus and the problem of discarding higher derivatives. Hegel wrote that various explanations for such neglect of the next members of the series are not credible, and also that each subsequent member carries the ontological meaning of the subsequent increase in dimension, and, accordingly, the members of the series are not comparable. "In mechanics, the terms of a series in which a function of some motion is decomposed are given a certain value, so that the first term or the first function correlates with the moment of velocity, the second with the acceleration force, and the third with the resistance of forces. Therefore, the members of the series should be considered here not only as parts of a certain sum, but as qualitative moments of a certain concept as a whole. Because of this, discarding the remaining members belonging to a badly infinite series has a meaning completely different from discarding them on the basis of their relative smallness" ^{[4, p.282]}. In other words, the discarded, supposedly small members of the series can equally base the value of the function. Prigozhin and Haken called such moments bifurcation points.

It is necessary to pay attention to another aspect, clearly visible from the diagram of the approximation of reality given below. If Hegel pointed out the different nature of the multipliers and the impossibility of their comparison, then it is necessary to point out that the multipliers have a different scale of impact and are responsible for the hierarchy of long-range effects.In other words, another member of the Taylor series brings the objective closer to the real on a larger scale (in the case of an elephant, this scale will be the biological hierarchy: family, genus, species, and the hierarchy of the narrative: the area of the encompassing jungle). The diagram is an approximation of reality, demonstrates a change in the range of action, taking into account an increasing number of members of the series, a kind of "look beyond the horizon":

Here, the circle figuratively represents reality, while the colored curves are an attempt to approximate it, in other words, symbolic forms explicated by the subject. The functions approximating the circle are shown in different colors, from which different numbers of terms of the Taylor series are taken. The first and simplest level of approximation is a point, in this case we believe that the world merges with the subject, it is stationary, there is no movement, and the entire circle degenerates into a point, this is a figurative representation of solipsism. The next level of approximation is a straight line, with which you can describe a small neighborhood of point A on a circle. The nonlinear approximation level is shown in yellow, which takes into account the quadratic term, such a curve will be a parabola. The approximation that takes into account the third derivative is shown in red. It can be seen that as additional members of the series that constitute the approximated function appear, it becomes closer to the real circle, it presses closer and longer to it. The range of the point is now increasing, and the transcendental apperception captures an increasing Kantian manifold, structuring it, as shown by the sequence of green, yellow and red arrows.

In life, we are constantly engaged in approximating the "now" function (more precisely, we are only doing it, since approximation is the result of transcendental apperception), using a set of a priori, a posteriori knowledge and logic (more on this later) – will I have time to work if I get on this minibus, will he agree a child has porridge if I promise him candy, will she get married if I propose to her? Recalling the example of the elephant, it can be argued that the entire biological taxonomy is also an approximation and the allocation of the common (often external similarity leads to genealogical errors). And it depends on our a priori and a posteriori experience and our logic how close our approximation of reality will be to reality, or, as it will be seen from the following, how close reality and the product of the imaginary and symbolic are to each other, and the key point is how well we have taken into account the facts of the second plan, the third, etc., which constitute now, and which are usually overlooked (the butterfly effect). Indeed, when solving multicomponent problems (and there are no others in reality), very often (at moments of bifurcations) reality "jumps out" from under its approximation, showing its own "temper".

Man's representation of the universe is built on several levels. Firstly, at the level of rational judgments based on empirical data, in other words, experience, and in this area is science, including physics. All our practical daily experience is in the same area: taking such and such a medicine will lead to such and such a result. Secondly, it is a level of intelligence capable of observing broader connections, since ancient times called wisdom. Philosophy itself is in this area. At this level, we make generalizations: if we give up medications and exercise, we will be healthier. Obviously, for such a broader observation of the world, it is necessary to rise higher and look further. And finally, the third level is the level of speculation. Not just to look into the distance, but to look beyond. At this level, the synthesis of previous states is carried out. It is obvious that the concept of speculation correlates with the concept of scientific intuition, which Galileo used in a thought experiment, discovering the law of relativity: all processes in a uniformly moving system occur in the same way as in a stationary one. Galileo, as it were, observed all these systems from above. Such a consideration of several levels of thinking is the decomposition of the context of the "now" point created by transcendental apperception into a Taylor series. And here one can feel that symbolic forms belonging to different levels of thinking have, in accordance with Hegel, a different nature.

Just as in mathematics there is a Taylor spatial series that decomposes a mathematical function into a sequence of terms that depend on coordinates, so is the Taylor time series, where the decomposition occurs along a time coordinate. And here we again discover the presence of the Taylor series in the formation of symbolic forms, since the temporal resolution is associated with the concept of intuition and its range, which allows us to push the limits of pretence. In connection with the different types of judgments and the range of intuition, one can recall again the image of epistemology as the geometry of knowledge, and epistemology as its temporal evolution, like the Taylor spatial series, which shows how the function spreads in space, and its temporal counterpart, demonstrating dependence on time. Bergson in ^[2] also correlated intelligence with spatial thinking, in other words, with geometry, whereas intuition, according to him, is rather an instrument of time and a temporal anthropological characteristic. The spiritual edge of intuition, in turn, is conscience, which is able to guide an individual along some optimal trajectory: "if, however, we ask ourselves why conscience necessarily functions irrationally, then we need to take into account the following fact: consciousness is open to existence, conscience is not open to existence, but rather, on the contrary, then that does not yet exist, but only should exist" ^{[14, p. 97]}. According to the diagram, due can be separated from the point now by various temporal segments – from the near future to looking beyond the horizon of many centuries (or "many centuries ahead"), which is determined by the number of time terms of the Taylor series in the function of transcendental apperception, which an individual is able to explicate. In other words, conscience is synonymous with far from the first temporal members of the Taylor series of the thinking function.

For a better understanding of the role played by intuition in the construction of the Taylor series of the function of thinking, let's try to depict intuition as the range of temporal penetration of thinking into the transcendental, in a diagram that can be called a diagram of three types of intuition, where the numbers 1,2,3 indicate a different temporal range. A similar structure was discovered by Poincare ^{[1, p. 201]}.

Three types of intuition can be represented by epistemological "bursts" of different temporal scales on the surface of semantic space. The first type of "surge", depicted in green and indicated in the figure by the number 1, is a local deductive advance based entirely on the known, on the understandable, when an extrapolation of the immanent into the future is built on several points already determined as a result of the analysis. Such thinking can be called Cartesian, cleansed of subjectivity, since such penetration into the future is most often done mathematically (the project of the future bridge). Such a rational kind of epistemology is the slowest, and it can be represented, using an image from physics, as a "diffusion of knowledge".

The second type, indicated by the number two and yellow, is already an aggressive advance into the transcendental, based on some idea, which is later justified with the help of some metaphor. (The similarity explicated by the metaphor of the existing epistemological domain and the new one is depicted in the figure in the form of shaded rectangles). Metaphor helps to embed one theory into another; in other words, and using the topological term maps, it detects the intersection of various semantic maps (a map is a description of some semantic variety, as geographical maps describe the diversity of the globe). It is important to understand that the second kind of intuition does not just expand the immanent into the transcendental, but it kind of glues one area of semantics with another, putting one area into another, thus compacting semantics. This can be represented as a "crumpling" of the fabric of semantics, explicated by metaphors. The metaphor provides almost instantaneous temporal advancement.

And finally, the third view shown in red is an unprecedented leap deep into the transcendental, which has no justification. This is how the Indian mathematician Ramanujan composed his formulas, who claimed to communicate with the deity.

It can be seen from the diagram that each level of intuition has its own temporal range, its own time scale, which, within the framework of the Taylor series concept, means the use of more of its members. Ramanujan's contemporaries believed that he was several centuries ahead of his time.

In conclusion, we show that the sequence of paradigms of thinking itself makes up the Taylor series of the thinking function. This sequence is called Foucault epistems. Although Foucault himself identified only a number of epistems that were characteristic of three periods, starting with the Renaissance. All of them were characterized by their own sense of language (for example, in the Renaissance era, the sensory component was more important and there was still no strict correspondence between language and reality). We will try to trace how epistems relate to our model of the Taylor series in millennial periods.

The world in ancient philosophy seemed unchangeable or cyclical, which refers us to the concept of a constant or the first term of the Taylor series of the function of thinking. This consideration of time "without time" is characteristic of the Hellenes. "The world is perceived and experienced by the ancient Greeks not in terms of change and development, but as being at rest or rotating in a great circle" ^{[9, p. 64]}. At the next stage, Christianity broke the circle of time, turning it into a straight line, the concept of evolution and the idea of human transformation as a necessary task for getting to paradise appeared: "Christian time in the worldview of a medieval European became linear and irreversible, but also in a very limited sense" ^{[9, p. 66]}. This stage was completed by Galileo, who introduced into the paradigm of thinking the idea of speed, of the equivalence of reference systems moving relative to each other at a constant speed, which completed the formation of the second, linear term of the Taylor series of the thinking function.

Together with Newton, the next episteme included the idea of acceleration, or the third term of the series. At the same time, it became possible to study objects from the point of view of force interaction, which gave impetus to the emergence of various symbolic forms associated with force. In the nineteenth century, such important concepts as Hamiltonian and Lagrangian appeared. If ancient mechanics considered static objects: levers, suspensions, etc., then since Newton dynamic problems have been actively solved, which required the development of algebra and differential calculus. The concepts of the Hamiltonian H and the Lagrangian L unify solutions to problems of mechanics. The space in which the problems are solved doubles, now it is called phase space and consists of 3N coordinates and 3N pulses, thus connecting the previous epistems. An important axiom of such a synthesis is that the movement of any object is completely determined by setting a set of its coordinates and impulses – their sum is necessary and sufficient to describe evolution. Through this statement, Zeno's aporia about a resting arrow is revealed – the reduction of a six-dimensional phase space to an ordinary three-dimensional one leads to a dimensional paradox, explicated by Frankl ^[14]. Acceleration is not explicitly included in the functions L(r,v) and H(r,p) as determining evolution. "It is not difficult to continue this induction and put forward a working hypothesis that this axiom is outdated, and that it is also necessary to take into account the dynamics of accelerations or, in other words, the third derivative of coordinates in time. Such an examination should also reveal the next layer of philosophical problems, since then the traditional scheme of linearity loses its heuristic value" ^{[13, p. 75]}. Thus, we should expect the arrival of the fourth member paradigm in the near future. So far, physicists are only groping for its definition and have not even given it a name yet.

Based on the described evolution of the thinking function, it is possible to verify the anthropological evolution and assume that the thinking function evolves in accordance with the concept of the Taylor series, in other words, we find three or four members of the Taylor series, since they correspond a priori to modern thinking. If initially symbolic forms represented the real, correlating with the first, geometric member of the sequence, now we observe more complex symbolic constructions that derived semantics from direct correspondence, which allowed symbolic forms to increase prognosticality. Thus, it can be considered verified that the Taylor series is a natural way of thinking.