Does mathematics have a philosophical meaning?
Math and philosophy
1. Orientation with Kant
In view of the open questions in natural philosophy, Kant discusses the relationship between mathematics and philosophy. Does the world have a temporal and causal beginning? Does it have a spatial end? Does the matter consist of the smallest building blocks? Is there necessity in nature? Kant saw clearly that the prevailing popular, dogmatic natural philosophy asserts the existence of limit and necessity in everything. However, he could not have guessed what a boundless triumphant advance it would begin under the flags of materialism and anti-metaphysics, of all things. Today the time has come to say how many years have passed since the beginning of the world, how many kilometers the diameter of the world is. All elementary particles are considered to be combinations of 6 quarks, whose weight, diameter, etc. are measured. The development of all parts of nature is ordered in the theory of evolution in a single time series and postulated with more or less necessity. The logical conclusion is the evolutionary epistemology, according to which the end point of evolution is its own self-knowledge in the form of modern natural sciences. Something cannot be right there.
Kant wanted to avoid the dogmatic way as well as the opposite empirical way, which denies the existence of natural laws and limits itself to an in principle infinite number of individual analyzes. He sees two possible ways out in mathematics and philosophy. Mathematics finds it easier, and at first it is more successful.
It is not so tense in opposition to dogmatism and empiricism, since it can be guided by direct intuition. Without examining specific properties, you can use your hand with your five fingers to count as an illustration and for the geometry, the observation of simple figures such as triangles, circles, etc. In the following, it is limited entirely to the construction of concepts of size, such as numerical values, areas, lengths, Angles etc. This does not make it empirical, since all of its results are generally valid, and also not dogmatic, since it does not make any restrictive assumptions about its concepts of size. And yet, with the strict construction methods of arithmetic and geometry, she repeatedly succeeds in surprising findings that no philosopher would have expected, let alone prove, through mere thought.
Nevertheless, Kant's fascination with mathematics is broken. Historically, he follows the Platonic understanding of mathematics, according to which it has the eternal ideas of pure geometric objects as its object, but cannot break away from them either. In spite of all its successes, it is a frozen science that can only be seen on the outermost surface.
"We have two expressions: world and nature, which sometimes run into one another. The first means the mathematical whole of all phenomena and the totality of their synthesis, both large and small, that is, both in their progress through composition and through division but nature is called insofar as it is viewed as a dynamic whole, and one does not look at the aggregation in space or time in order to bring it about as a quantity, but at the unity in the existence of phenomena. " (Kant, Critique of Pure Reason, B 446 / A 418)
In the age of postmodernism and simulation, this may speak for mathematics. In EDP, too, only worlds are spoken of: Unix world, PC world, IBM world, etc. But of course that is not enough for Kant. Wittgenstein later put this in a nutshell:
"We feel that even when all sorts of scientific questions have been answered, our life problems are not even touched." (Wittgenstein, Werke Vol. 1, p. 85)
And Kant says that mathematics would immediately give up its business if it could solve the great questions of natural philosophy.
Thus in the mathematics of reason it is possible to achieve ever new successes in the knowledge of nature without dogmatism and empiricism, but it stops at mere quantity. Mathematics remains the theory of size and the art of measuring. Conversely, philosophy can formulate sublime ideas, but when they are solved it gets caught up in dogmatism, empiricism or skepticism. Kant sees the way out in the regulative use of philosophical ideas. Philosophy is not supposed to teach what nature is like. It should not interfere with natural science from outside and pretend that it can recognize nature better than it. Nor should it pretend that it can look into the heart of nature and unravel its inner secret, i.e., design a new theology or mythology. It is intended to provide the guideline by which natural science can orient itself. For orientation requires a science that is non-dogmatic and non-empirical at the same time, which, like mathematics, cannot rely on direct intuition. It should neither sink into the abundance of material, nor be constrained by the corset of dogmatic ideas or constrain nature. If the enlightened science fails to orientate itself, or if it loses it, then it inevitably gets caught up in a process that is just as immoderate as the sublime ideas about the whole of nature, which breaks down every hold and in the end appears to some as unleashed natural science, to others as the inescapable dialectic of the Enlightenment.
At first glance, however, the orientation turns out to be contradictory. Kant formulates this contradiction in the "Critique of Judgment": On the one hand, natural science should search for causal chains everywhere, on the other hand, it should gain a teleological understanding of nature. There are two different patterns of orientation in the models of the machine and the organism. It is too easy to resolve this contradiction by referring to two different forms of motion in nature, the machine to mechanical physics, the organism to biology. They refer to two different situations in Kant's guidance for orientation.
Teleological judgment is required to gain a basic understanding of nature. It is an interpretation and therefore cannot be expressed in natural laws or other mathematical sign systems. In the simplest case, teleological explanation of nature proceeds according to the question: How would I behave? In biology, of course, this is most obvious. When observing living beings, their behavior can be judged or compared with how people act. Be it in the communication of ants or in the energy balance of a plant. But teleological considerations also go into physics. For example, when looking at structurally stable systems in which small deviations are compensated for, or with the principle of action, according to which the path is taken with the least amount of effort. Physics is full of conservation and optimization laws. These laws themselves are not teleological explanations of nature, but they start with teleological interpretations such as balance, harmony, conservation, optimization.
Here teleology is by no means the only one for orientation, even if it occupies the most important place in Kant. It doesn't just have to be asked: How would I behave? Conversely, immersion can also take place in order to experience as much as possible how other behavior takes place in nature than ordinary human behavior. Instead of teleology, aesthetics can serve as orientation. In nature, certain relationships are emphasized by the fact that they are perceived as beautiful, sublime or, conversely, as threatening, frightening, peculiar. In all cases, the orientation leads to an order, assignment, and structuring of reality. But it is only about interpretation, only about becoming aware of how nature affects people, even if the results lead to practical success in experiments or industrial application. It's not about knowing the shape: the stone falls because it wants to do as little work as possible. It's not about your own will, your own sense of beauty in nature. Nothing can be said about this, unless man starts out from fantasies of omnipotence, in his consciousness the whole of nature becomes conscious of itself, his body is the highest organization of self-organizing nature, or else nature only exists in his consciousness. It is also not possible to see the gain in knowledge that statements about the inner drives of nature are supposed to bring. It's all about orientation, what belongs where, which parts form a common apparatus or a common organism, which tones sound beautifully together, etc. The only assumption about nature is that there is an objective basis for this interpretation in nature. Orientation is the attempt to find one's way in nature and only stick to how people experience nature. The cliffs of the dialectic of the Enlightenment are not reached at all if neither mythologically (e.g. through creation myths) nor through the sublime ideas of natural philosophy is attempted to jump out of nature and to judge it from the outside as a whole.
The orientation can be followed by a representation of nature in the most varied of sign systems. Orientation and presentation will mutually influence each other. In art, the aesthetic impression can be composed, visualized or expressed. In natural science a law is sought for the observations, the connection of which has already been recognized in the orientation. And here a different kind of orientation is required, to which Kant intends to give an answer in the "Critique of Pure Reason". This orientation is necessary, but by no means self-evident or unambiguous, as the history of mathematics shows. The Phythagoreans started out from aesthetic judgments in their music experiments (which sounds nice together), and then let themselves be guided by the idea that all relationships in nature can be expressed by numerical relationships (insofar the world consists of numbers). Kant wants to give a different orientation. Based on teleological judgments, the found connection breaks down into individual components, which can be represented in the form of causal chains as natural laws. For him, time is the orientation in which the laws of nature can be formulated mathematically.
In terms of content, this has reached the point where mathematics and philosophy touch each other in Kant's critical philosophy. Through an initial orientation, it is qualitatively recognized which interrelationships and internal relationships exist in nature. Then it is a matter of capturing these relationships quantitatively. In the simplest case, in the case of rigid natural structures, it is sufficient to first qualitatively determine what belongs to the structure and what does not, and then to apply a measuring stick, and the task of mathematics is to find a binding measure of size for the measurement and tasks of the kind to solve how area or volume, material density, weight, etc. can be calculated from external dimensions. With dynamic processes, the time is added, the clock serves as a measuring instrument. Mathematics works on an abstract space with 3 spatial axes and 1 time axis, the general properties of which are recognized regardless of which natural processes are represented in it. The size grid is examined purely quantitatively independently of specific natural processes. Conversely, in natural science it is examined in a purely qualitative manner which natural processes have the property that they can be represented in this grid. The philosophical orientation must on the one hand guide how qualitative properties can be found in nature, and on the other hand it must recognize which property it is in nature, where nature and the world of mathematics coincide, where nature can therefore be represented mathematically. For the Pythagoreans this was the numerical ratio, for Plato the ideal geometric figure, for Kant it is pure time (see the Schematism chapter in the "Critique of Pure Reason", there B 184 / A 145).
But here he is tangled up in contradictions. Orientation towards time is not clear. Kant describes 3 modes of time, which are the basic concepts of natural science: Substance as that which is preserved in time; Causality as that which necessarily follows one another in time; Interaction as that which necessarily occurs simultaneously (collectively). While substance and causality lie in the time series (as changeable or unchangeable), the interaction is perpendicular to time. It is recognized as a property of time, but then has no consequences. This is to be traced step by step.
I. The antithetics of the sublime ideas of natural philosophy is about the contradictions of an excessive, skimping observation of the time series when it is believed that it is possible for people to survey the time series as a whole and to make statements about it. Here, Kant expressly leaves aside properties of interaction. But all the big ideas appear in a completely different light when the possibility of interaction is taken into account. If many different things necessarily happen at the same time in every single moment of time, a causal chain can at best cover a cross-section. As a rule, many causal chains will run side by side at the same time. And there are intersections, branches, from where development can continue in different directions. The law of causality, according to which certain things necessarily follow one another in time, is retained. But the relationship between necessity and freedom has been re-established. And if the causal chains in the future direction and in every present have branching possibilities, this also applies to the natural history view back to the past, where every state once reached is part of an interaction and can be explained by various causal chains that lead to it. This concerns the question of the causal and temporal beginning of all things.
If interaction is taken into account, it can no longer be divided arbitrarily. Even aggregates like a lump of sand cannot be divided arbitrarily. First of all, dividing leads to smaller lumps, grains of sand, etc., Kant had this in mind. But the division comes to an end when the molecule is reached. If the molecule is divided, half molecules are not created, but this division is a chemical process that leads to other molecules and atoms with different chemical properties, because here the division tears apart a group of atoms that are interacting with one another. Kant envisaged division as a mechanical process, but this leads to the limit when a deeper form of movement of matter is reached, and so it goes on. At best, the quarks will also be a stopover. The antithetics is therefore to be resolved differently here. Quantum mechanics encountered this problem experimentally. In the Weizsäcker area, however, it was rashly concluded that causality and substance no longer exist where chance processes and indeterminacies were encountered in nature, which in turn are only an intermediate stage in the process of knowledge. In consideration of interaction, however, these phenomena are to be assessed quite differently, so that substance and causality retain their meaning.
II In every moment of time the entire space necessarily exists, which Kant consequently calls pure interaction, pure community. Therefore, when pure space is the subject of mathematics, interaction should be of fundamental importance. Kant avoids this by making the subject disappear behind the mathematical approach. For him, mathematics is ultimately counting, geometric construction and measurement. But all of this takes place step by step in the time series. The results are fixed, rigid sizes (the number or the figure). If Kant restricts mathematics to such an extent and does not see it, as, for example, his great role model Newton mathematically described the interaction of mechanical bodies through gravity, mathematics cannot do exactly what Kant assigns to it: the representation of dynamic processes in the time series.
III In the "Opus postumum" gravity is immediately the core question for the next problem. What is the relationship between the pure space of mathematics and the real space in which natural events take place? Kant clearly thought it to mean that pure space is the empty space in which natural things are placed, i.e. that the space between natural things is empty. Causality occurs when there is contact, when at least two things meet and change. According to Newton, however, gravity acts through empty space. First of all, this is confirmation of the existence of empty space. But for Kant a problem arises: "Newton's forces of attraction through empty space. How is empty space itself perceived, because the forces cannot be in themselves without physical reality." (Kant, Opus postumum, Vol. 1, p. 19).Kant decides that forces are not possible through empty space and that there is no empty space. But that breaks the connection between reality and the pure space of mathematics, a question that Kant no longer pursued.
The alternative is the filled room. And here Kant consults all the natural processes that were the subject of medieval natural research, such as bubble formation, droplet formation, elasticity, crystallization, malleability, brittleness, vibration, porosity, solidification, friction, affinity, wave oscillation, texture, warpage. As a common moving principle, he recognizes the flow of heat through which the most varied of solidification and liquid shapes are generated. A stream takes the place of the one-dimensional series, that is of fundamental importance. Because in the stream (as in the individual in the drop formation, bubble formation, warping etc.) the temporal movement of spatially extended natural things is considered, i.e. of communities, of interactions that develop over time. If the space is filled as the pure interaction, it must be so. "Through light and warmth, worlds establish themselves in community, as certain substances that fill the space are not only effective in it." (ibid., p. 105) This corresponds very much to the physical conception of particle fields.
IV With the empty space, however, the model of modern natural science, the machine, according to which nature is to be represented and processed, also coincides.
"The feeling of warmth (life), cold (death). Neither of the two are substances, but only relationships of forces. Air, light, warmth; positive and negative electricity." (ibid., p. 118) "Light, heat, electricity and nerve flow are the moving forces that bring about a life in the universe like galvanism." (ibid., p. 136)
These are all just notes, interspersed in "Opus postumum" between Kant's menus and excursion destinations, and they sometimes contradict each other from side to side (sometimes warmth is a substance, sometimes not). And yet the realization is gaining ground that a room filled with warmth is alive, that it is perceived by humans with all of their senses (not just with the eye), and that they cannot be a machine. How then can we explain that the movement of the planets in their regularity and rule still works like a machine, purely mechanically? Kant rethinks the relationship between machine and organism. "Organic bodies are those that act as machines under their own power." (ibid., p. 194). Here, the machine is a body in which various causal chains are systematically coordinated with one another and lead to certain results. In this respect, every living being can be viewed as a machine that processes food. Agriculture has always used this, biotechnology is supposed to push it further. These considerations lead Kant directly into the romanticism of nature.
"An immaterial moving principle in an organic body is its soul, and if one wants to think of it as a world soul, one can assume that it builds its body and even its housing (the world)." (Vol. 2, p. 97)
From here the question of why celestial mechanics appears as a machine is also resolved.
"Amphibolism (ambiguity, T.) of the reflection concepts with regard to physics, since what is made is thought a priori synthetically as a given." (ibid. 2, p. 285)
The celestial mechanics is found as something that has become. Everything fits together in it, so it appears as a machine, as a given. In truth, it has become historical, in which nature as an organic body has organized itself and continues to organize itself in this way. In place of empty space as a pure interaction, nature appears as a real interaction that is constantly changing and developing. The organism and the machine are both suitable as guiding ideas, in that either the resulting, planning, preparatory, controlling interaction is considered, or the interaction itself, as an overall structure of interrelated forces.
This must also have an impact on the pure space of mathematics. Kant assumed that pure space is Euclidean and that mathematics only constructs objects in this space. If, on the other hand, it is assumed that the space is filled, the first task of mathematics is to recognize the general properties of the space and then to construct them in this space. Examples of mathematically recognized qualitative properties of space are known from the further development of celestial mechanics, where studies are carried out to determine whether space is curved, whether it expands, etc. In a certain way, Kant has an inkling of where to look for an example for synthetic Method of geometry seeks.
"That the straight line between two points is the shortest is a synthetic proposition. Because my concept of the straight line contains nothing of size, only a quality. The concept of the shortest is therefore completely added and cannot be broken down from the concept must be drawn in the straight line. So intuition must be used here as an aid. " (Kant, Critique of Pure Reason, p. 816)
Here a qualitative property of the entire Euclidean space is recognized mathematically, its flatness, which distinguishes it qualitatively from other spaces such as the sphere or the curved universe. Mathematics is no longer a mere theory of sizes, but also investigates qualitative properties. However, this did not lead to her becoming independent of philosophy and history. Furthermore, it is to be determined philosophically which place mathematics occupies in the knowledge of nature and which bases it is based on.
2. Hegel's idea of a mathematics of nature
2.1. The infinity axiom in the differential calculus
For Hegel, the overwhelming achievement of philosophy is the design of the method. With it he can orientate himself everywhere and solve every problem. The relationship between mathematics and philosophy becomes a comparison of methods: axiomatic here, dialectical there. Euclidean geometry serves as a brilliant example of the successes of the axiomatic method, and Hegel rightly recognizes it in all natural sciences that imitate Euclidean geometry, like classical mechanics.
And yet in the "science of logic" mathematics does not merge with geometry. Hegel grants the differential calculus a special position. While Euclid's axiomatics had been in force for over 2000 years, a system of axioms was only systematically set up for the differential calculus in the time of Hegel, only in the 20th century.
(a) Measurements and geometric intuition
The differential calculus was preceded by geometrical investigations that went back to the question of the square of the circle. Curved curves were systematically approximated by straight tangents with the aim of expressing the circle by a tangent system, by a polygon. In the case of simple functions such as y = x², the derivation could be guessed using geometric methods by measuring at many points that the slope of the tangent at the point x₀ has the value 2x₀.
Figure 1 parabola
Functional progression of y = x², freehand drawing
The straight line that is the tangent at the point x₀ can also be specified: According to the measurement results, it must have the slope 2x₀, i.e. in the first approach:
(1) y = 2x₀x + C
The shift C on the y-axis must be selected so that exactly the desired function value is reached at the point x Stelle:
(2) y₀ = x₀² = 2x₀x
This results in the solution for the straight line that supplies the tangent at the point x₀:
(3) y = 2x₀x - x₀²
The calculation is of course more complex for more complex functions.
This geometric view is the basis of the derivation to this day. It was generalized to other spaces and multidimensional figures, for example, by a sphere defining a tangential surface at every point on its surface, which here approximates the spherical surface. In this way, Euclidean space also comes into its own, in that space can be approximated locally by Euclidean space. If movement curves are derived in real space, i.e. approximated by straight lines, then these lie in Euclidean or comparable space.
(b) Algebraic operations and pushing the boundaries of traditional calculus
In the next step, an attempt was made to calculate the derivative using arithmetic methods. The basic idea was that the slope of the tangent can be approximated by the ratio of the two sides in the slope triangle: y / x.
Figure 2 slope triangle
Here the function is to be derived at the point x₀, where y is the difference to a function value y₁ in the vicinity of y₀. The following approach applies:
|= x + 2x₀|
This is an arithmetic expression, and it can be solved by arithmetic calculation. The function y = x² serves as an example.
The function value y₁ is to be calculated a little above y₀.
(5) y₁ = y + y₀ = f (x + x₀) = f (x₁)
In this example, f (x) = x², therefore inserted:
(6) y + y₀ = (x + x₀) ²
Multiplied on the right side according to the binomial formula:
(7) y + y₀ = x² + 2xx₀ + x₀²
Since at the point x₀ the following applies: y₀ = x₀², the equation remains valid if y₀ is subtracted on the left and x₀² on the right with the result:
(8) y = x² + 2xx₀
Now all that remains is to divide the equation by x on both sides, and the arithmetic expression y / x you are looking for is found:
|(9)||=||=||+||= x + 2x₀|
From the geometrical investigations it was known that on the right-hand side with 2x die the solution had indeed been calculated. For this it can only be assumed that the difference is in the border process
x₁ - x₀ = x = 0 and with it the difference
y₁ - y₀ = y = 0 becomes with the searched end result:
|(10)||= 0 + 2x₀ = 2x₀|
Lagrange systematically completed the arithmetic method and showed that in a similar way all polynomials
(11) y = a₀ + a₁x + ... anxn
and also infinite rows like
(12) y = a₀ + a₁x + ... anxn + ...
can be derived. Since the trigonometric functions and the exponential function can be written as infinite series, they can also be derived.
Other functions cannot be derived (in any case there is no procedure for doing this). But it was proven: If somehow it can be shown that a function has the qualitative property of derivability (that is, that it can be approximated locally, i.e. on a small scale, by straight lines), then this function can be approximated piece by piece with arbitrary precision using polynomials (Theorem from Stones-Weierstrass).
(c) Boundary process and justification through an extended axiomatic of the number sets
Hegel and Marx based their studies on the differential calculus on the work of Lagrange. The arithmetic resultant of the form developed above
Marx describes as "provisionally derived", since here the arithmetic operation is completed and only the limit process has to be carried out, in which x = 0 and y = 0 are set.
The boundary process is in contrast to arithmetic, in which an expression of the form 0/0 cannot be calculated ("dividing by zero does not work"). Leibniz initially circumvented this problem with a new symbolism, in that he symbolically transformed the arithmetic expression of the "preliminary derived" into the border process
|(13)||= dx + 2x₀|
The transformation of the symbolism says that the calculation rules of arithmetic are abandoned and new rules apply. Lagrange adopts this symbolism. Leibniz formulated the differential calculus, which defines how these symbols can be used for calculations without contradictions occurring when dividing by zero.
But when Cauchy applied the differential calculus to other sets of numbers (the complex numbers), it became clear that the validity of the differential calculus is a property of the set of numbers in which it is derived and therefore belongs to the axiomatics of this set of numbers. For example, in the set of natural numbers it is not possible to derive, since the limit process is not possible if the distance between two neighboring natural numbers is always one. In the further development, this led to the fact that rooms were constructed that have precisely this property of derivability (standardized rooms). In particular it could be shown that mechanics can be derived in Euclidean space and in movement space. Conversely, there are normalized spaces that are not Euclidean. The idea of Kant and Newton that Euclidean space, the space of nature and the space in which one can deduce are identical, was finally gone. The axiomatic method first had the differentiating effect that different spaces could be constructed, in each of which certain axioms apply and others do not, from which different properties of these spaces then follow.
The course of axiomatization was remarkable. At the beginning there are well-known mathematical operations (such as geometrical construction and arithmetic calculation), which under certain circumstances can leave the field that was previously covered by axioms. This has led to the definition of new operations (arithmetic in differential calculus) for which the violated axiom does not apply. And from the new operations, conclusions were drawn about the axiomatics of the set of numbers and these were extended by axioms under which the new operations are permissible.
In its finished form, the axiomatic system consists of the definitions which determine the set of numbers or points, the axioms which record their basic properties, and the operations as to how the set can be constructed. So far it has always been shown that there is a tension between the definitions and axioms on the one hand and the operations on the other. Be it that it is unclear which operations are possible at all (e.g. in Euclidean geometry the question of whether the circle can be squared), or that, as in the case of derivation, the operations go beyond the scope of the axioms.
2.2. Dialectical method
To a certain extent we are already in the middle of the dialectical method by showing something like the dialectic of axiomatics. The axiomatic method keeps jumping over its limits, and when new axioms are sought, this search itself can no longer be called axiomatic. But in the end this leads to new axiomatics, and the axiomatic method cannot be canceled out in the dialectical method, as Hegel wished. That must be justified.
It is easy for Hegel to criticize the axiomatic method. The axioms stand side by side without any recognizable inner connection, they are not derived from the concept of the thing (e.g. the concept of numbers or the concept of space), and she does not seem to know such a concept at all. Hegel certifies that the theorems are synthetic in character, i.e. they lead to genuinely new results compared to the axioms and definitions, but they too are not developed from the concept of the thing, but result from a seemingly mechanical operation with the axioms. In view of the dogmatism of university science, his criticism has a lot to offer. The axioms are the basic law of the state building of the sciences, and anyone who doubts them as stupid or as a charlatan. Any relation to reality and practical work is triumphantly blocked; autonomous and free science is content with itself within the scope of its axioms.
In truth, the axioms are as a rule theorems from philosophy (mostly from logic), notes Hegel. The differential calculus led to the axiom of completeness: If a limit process is carried out, the limit value exists, the numbers are complete in this respect. With the development of the differential calculus, this axiom was re-established. It was of course nowhere found in this form, not even in philosophy. But there has been a long debate in philosophy since Aristotle about the continuum and various concepts of infinity. The concept of continuous movement was found in the physical experiments and observations. All this made it possible for mathematicians to find a suitable axiom for the differential calculus. One of the most influential axiomats, Cantor, made explicit reference to the scholastic controversy over the concept of infinity. Leibniz and Newton placed the differential calculus in the context of considerations of natural philosophy. The fact that axioms go back to doctrines in philosophy cannot at all be regarded as a defect. Unless axiomatic science is frozen in the dead letters of dogmatism, it thrives on the tension between scientific operations and axioms. It constantly urges new axioms. There arises from the course of natural science itself the necessity of clarifying its philosophical foundations.
The dialectical method is basically no different, albeit in the opposite direction. Let us take as an example how Hegel interprets the differential calculus. He has a philosophically motivated interest in this question in order to solve the "conundrum" of his logic (Hegel, Logic, vol.1, 169), like the infinite in the finite; The quantitative turns into the qualitative. He does not find the solution in philosophy - on the contrary, he wants to take philosophy a step further on this question - but in the differential calculus. Searching for the derivative leads to finding the slope of the tangent, and that is expressed by the size ratio y / x of the sides in the slope triangle. Once this specific size ratio has been found, there is a border crossing, which brings a new quality. In the boundary process, y and x are replaced by the new symbols dy and dx.
"dx, dy are no longer quanta, nor should they mean such, but only have a meaning in their relationship, sense just as moments. You are no more Something, something taken as a quantum, not finite differences; but also not nothing, not the indeterminate zero. Apart from their relationship, they are pure zeros, but they are only supposed to be moments of the relationship, as Provisions of the differential coefficient dx / dy be taken. In this concept of the infinite, the quantum is truly perfected to a qualitative existence; it is posited as really infinite; it is not only suspended as this or that quantum, but as a quantum in general. "(Hegel, Logic Vol. 1, p. 295f)
Hegel visibly reproduces the operation of the differential calculus: he tries to present it in the language of philosophy. He can by no means replace it, his philosophy does not provide any deductions and does not develop the differential calculus any further (as did his contemporary, the mathematician Cauchy). But in return he succeeds in exactly what mathematics does not succeed in: becoming aware of the inner connection between the various concepts used, the various operations and their results. As Kant demands of philosophy, he has an orienting effect, the result is the idea of a mathematics of nature.
Philosophy can only start from the material that is given to it by the natural sciences. She can interpret it. If it succeeds in clearing up new connections, opening up new horizons for science, it will push for this to become a program and be carried out. You can't do it yourself. Just as it accuses axiomatics of lacking concepts, it has to be said: The philosophers have only interpreted the world; what matters is to change it.
Similar to aesthetics and teleology, dialectics is given an orienting task, whereby these different approaches to nature cannot be traced back to one another; the result would be dogmatism. Kant correctly saw that philosophy cannot imitate the methods of mathematics. Hegel sees the characteristic achievement of dialectics as being able to track down an inner differentiation in the object of knowledge, to determine the typical inner contradiction. But not as a rigid paradox as with the pre-Socratic Greeks, but as a source of movement. The contradiction drives to cancellation, to a qualitative leap. In this way dialectic works analytically, where it differentiates the object, and synthetically, where in the process of dissolution it recognizes its relation to other qualities.
As in teleology, it leads nowhere to present the interpretation as an immediate natural property, that is, to say that nature is dialectical. This leads to the short circuit that every knowledge once gained receives the consecration, nature is like this and not different, and therefore cannot be further developed. Then the method blocks further orientation with its first findings. Following Hegel and Engels, this happened to dialectical materialism. Suddenly he was suspicious of every new scientific knowledge (such as the theory of relativity, quantum theory, cybernetics); it did not fit into the system. At some point their textbooks became just as boring and dull as those of the contested bourgeois science. The fatal result was that the dialectical method could not develop its orienting power, which contributed to the fact that the fundamental crisis of the natural sciences in this century ended in pragmatism.
2.3. Mathematics of nature
Even Hegel contributed strongly to this undesirable development with his natural philosophy. But in the "Science of Logic" the idea of a mathematics of nature is evidence of the orienting abilities of philosophy.
When quantity turns into quality, a measure is defined positively. With the derivation, a measure of the movement is found. Physically it is defined as speed and momentum, with two-fold derivative as acceleration and force, when integrating the momentum over time as energy. Where mathematics itself was only enriched by an axiom of infinity, Hegel recognizes that mathematics, with its ability to find dimensions, has completely new worlds open to it.
"In view of the absolute proportions, it should be remembered that the Mathematics of natureif it wants to be worthy of the name of science, it must essentially be the science of measures - a science for which empirically a lot, but actually scientific, i.e. i. philosophically little has been done. Mathematical principles of natural philosophy - how Be Newton Work - if they were to fulfill this determination in a deeper sense than he and the whole Baconian lineage of philosophy and science had, they would have to contain completely different things in order to bring a light into these still dark, but highly contemplative regions. - It is of great merit to get to know the empirical numbers of nature, e.g. distances of the planets from one another, but an infinitely greater one, to make the empirical quanta disappear and turn them into one general form of quantitative determinations so that they are moments of a Law or be measured; - immortal merits, e.g. Galileo in consideration of the case and Kepler in consideration of the movement of the heavenly bodies. They have the laws they found so proventhat they have shown that the range of details of perception corresponds to them. But there must be a higher one To prove of these laws are required, namely nothing other than that their quantitative determinations are recognized from the qualities or certain concepts that are related (such as time and space). "(Hegel, Logic Vol. 1, p. 406f)
The train of thought is unmistakably based on the representation of the differential calculus. (a) First, the gradients of individual tangents were measured (using geometric methods), which corresponds to the observation of individual empirical numbers, for example through astronomical measurements. (b) Then formulas for the derivation of certain functions were found (with arithmetic methods that led to the provisionally derived), which corresponds roughly to Kepler's laws of planetary motion. (c) Finally, the limit process and differential calculus were introduced, and this extended Hegel to the program of a mathematics of nature. After knowledge of further scientific research, it is to be defined more precisely.
Since the work of Poincaré, Kepler's laws in celestial mechanics have been explained from the qualitative properties of the mechanical space of motion. If a room has the property that the shortest connecting lines in it are straight lines, this means that this room neither grows nor shrinks. Under this condition, gravity acts in such a way that Kepler's laws are fulfilled. If the space has different qualitative properties, the laws of motion of the bodies also take on a different shape.
Quantum mechanics is more in a transition stage. A vast amount of empirical data and a whole series of laws are available, but an undisputed qualitative understanding of nature is not yet available. The properties of the space in which quantum mechanics applies are examined. This is the subject of quantum field and lattice theories. The quark theory is one possible solution.
Catastrophe theory also goes this way. In the beginning there were numerous empirical observations of turbulence, irregularities and chaotic situations. It was only very late that it was possible to discover laws in this tangle. Since the 1950s (and increasingly since Thom's work), research has been carried out to qualify the spaces in which catastrophes occur, i.e. folded spaces, broken spaces, spaces with singularities (points in which the axiom of completeness is violated: is approximating a singularity point in a limit process, the limit process gets into a vortex and falls into a space with qualitatively different properties).
Kant defined pure time and pure space as the point of contact between qualitative natural science and quantitative mathematics. If the room is full, the concept no longer works. According to Hegel's orientation, it is the proportions where science and mathematics come together. The dimensions can be found in natural science (such as the speed of light as the limit speed, Planck's quantum of action, the order of magnitude of the molecules where mechanical motion changes into chemical motion, etc.). Each of these measures is a challenge to mathematics to determine the perimeter where the measure is effective and how it can be calculated. This orientation is more in the Phytagorean tradition, which looked for the numerical proportions that express harmony.
1982 - 1984
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