Where do math symbols come from

The mathematical and the mythical symbol


- with an afterword "Virtual and Multiple Mathematics"


Mathematical symbols are no less a matter of course in everyday life than letters or traffic signs. They are considered to be conventions that are easier to calculate and the results of mathematics can be better transferred to other areas. This article deals with the tension between mathematical and mythical symbols, which today seems almost forgotten. Do mythical symbols become obsolete and obsolete when mathematical symbols are found? Are mythical symbols placeholders that are only needed until the appropriate mathematical symbols take their place? Or is the success of the mathematical symbols explained by the fact that they retain the power of the previous mythical symbols?

Just as the rhetorical figures can be used best and most convincingly when no one perceives them as rhetorical figures but lets themselves be influenced by them unconsciously, this also applies to the symbols. Today they are no longer kept, venerated and worshiped in holy places, but brought into the formats of everyday representation (the diagrams and graphics, charts, statistics, formalized instructions and lists, ancillary calculations, logical conclusions with the use of "if - then", "and", "or" etc. up to the mathematicians discovered and further developed by Lacan, Deleuze and Guattari speak of abstract machines, Foucault of orders), in which they appear as mere tools. Nobody should perceive the way in which they already agree to a fundamental tendency of statements if they unhesitatingly engage in the use of the symbols used as a matter of course. At the same time, the enlightened person is incapacitated if he forgets the meaning of the symbols and is no longer able to use them appropriately. This deprives him of essential means of perception, knowledge and communication and makes him unable to express his own opinion.

In order to find an approach to understand this, one should go back to the emergence of the modern symbols.

On the threshold between antiquity, oriental wisdom, Christian philosophy and modern science, Neoplatonism found an understanding of the symbol on which modern science could be based. His own position in this area of ​​tension is quite surprising: for Plato and against Aristotle, but with the desire to take over as much as possible from Aristotle. For paganism up to the active cultivation of traditional cults and oracle interpretations, and against Christianity, which at first glance seems closer to Plato in its teaching of the revealed word. The conversion of the Christian Ammonias Saqqas to paganism was spectacular. He lived in Alexandria until 242 and is considered the founder of the Neoplatonic school. Plotinus and Origines were his students.

The Neo-Platonists saw the forerunner of the symbol (1) in Plato, when he abandoned dialectics in his mythical stories and created images that go beyond the logical thinking ability of humans. (2) Aristotle introduced proto-terms in his "Metaphysics" such as the proto-substance (proto hyle, prima materia). (3) Oriental wisdom and Greek paganism trusted the meaning and effect of divine symbols in nature.

Since modern times, scientific thinking has been inconceivable without the triumph of the mathematical symbol. A look back at the Neoplatonist's comprehensive concept of symbols should help to understand the fundamental questions of the symbolic science that arose in their successor: What is systematically lost in the suppression of mythical symbols, mathematical symbols and the sciences operating with them? In what way does science then give its symbols divine powers again and thus charge them with something that threatens to blow them apart from mythical symbols? Where are the limits of logical thinking? And how can mythical symbols break into a world shaped by science that has undergone a profound process of de-mythologization (Blumenberg calls this "work on myth"), as happened during National Socialism?

Mathematical and mythical symbols have an orienting task in two directions: to create symbols of the gods and symbols of chaos. Mathematics has to translate the symbols of the gods (the transcendent, the infinite, the excess) so that they can be reckoned with in the finite without losing the reference to infinity. In the other direction, the mathematical symbols have to act like crystallization nuclei in relation to the chaos, around which knowledge can develop (traces of form, I ton eidon, an expression from Plato Timaeus). As long as the mathematical symbols "work" no one asks about them. But the mythical symbols always appear on the horizon: on the one hand, as the power of the gods, who give the symbols their power, and on the other hand, the threat of chaos, which threatens to fragment everything. The mythological symbols protect the mathematical symbols so that neither the divine nor the unsettling chaotic prevail with them.

In a first step, this article should lead back to the questions posed by the Neo-Platonists, in particular with Jamblichos, Proklos and Simplikios. Their tradition remained unfinished. As for the mathematical symbols, they had to leave open the question of a clear distinction between number and size. To take up this point again is the aim of the following article in the narrower sense. From there, a new perspective on the fundamental crises of mathematics and physics in the 20th century is to be gained.

In the course of the following investigation, the importance of the terms connection (synecheia) and size (megethos). They will later turn out to be key concepts in Aristotelian physics and should be noted here. The foundations of Aristotelian philosophy were hidden more deeply in the thinking of the Neoplatonists than they were conscious or ready to express.

The three roots of the Neoplatonic symbol concept

Mostly the influences of oriental wisdom on Neoplatonism are neglected. In a way, the term "Neo-Platonism", which has only become common since 1900, is misleading. Its most important representatives such as Iamblichus, Plotinus, Proklos and Simplikios came from Lebanon, Egypt, Syria and the south-east of Turkey. Their forerunner was Philo of Alexandria. Iamblichus taught the Egyptian mysteries and Proclus commented on the Chaldean texts. From this tradition comes their understanding of the symbol, which differs from Plato and Aristotle, even if they rightly see forerunners in them. Today this is pushed into the background, as modern science is reluctant to admit the extent to which it goes back to the same origins as the "para-sciences" it despises. With the Neoplatonists it is to be learned how easily these different influences could be merged with one another.

Science is working with symbols, and the more so, the more mathematically it is oriented. The prototype is Euclidean geometry, which works with symbolic figures and clarifies their mutual relationships. Although the symbol term from Iamblichos to Proklos is mostly used only incidentally, but then always as a matter of course, they have done the decisive preliminary work. The idea of ​​creativity only had to be added from the Christian faith so that modern science can understand itself as the ability to create symbols of their own, to work with them and to develop models of the world. We can learn from Neoplatonic philosophy what the concept of symbol looked like immediately before it was linked to the idea of ​​creativity. It still has strong mythical echoes that have been displaced with the influence of Christianity.

Instead of speaking of symbols, the Neoplatonists often spoke of the proto-body and the proto-time. Here very different ideas of Plato, Aristotle and paganism were brought together in a completely new way.

(A) Plato had in his Timaeus a Pythagorean doctrine handed down, according to which the cosmos is the image accessible to man (eikon) a floor plan that can only be grasped for a higher than human understanding (paradeigma) is. Only God overlooks both, that paradeigma and the eikon.

The term paradeigma is difficult to translate. paradeigma must not be confused with the platonic, transcendent ideas. Literally means paradeigma the side-pointing, composed of para (next to) and deiknunai (show) (see Wikipedia). In the Latin translations the term paradeigma from the Timaeus translated both as (example) and as archetype (archetype). The word "Grundrisse" or "Grundlinien" seems to me to be the most suitable for a German translation: Man is able to recognize lines in nature that are an excerpt from a larger network of baselines that is entirely inaccessible to him. His picture, composed of lines, therefore stands next to the broader basic lines. He trusts that eikon and paradeigma are connected to one another in the sense of analogy thinking.

What is meant here can perhaps best be shown by a thought by Nikolaus von Kues (1401-1464): The infinitely long straight line is at the same time an infinitely large circle, a triangle with an infinitely long base line and an infinitely large sphere. Nikolaus von Kues thinks of a border crossing and approaches the idea of ​​a non-Euclidean geometry:

Figure 1 Infinite Line

Source: Nikolaus von Kues "On the science of ignorance", chap. 13 "Of the possible changes in the greatest and infinite line", in: "Philosophical and theological writings", p. 65

Nikolaus von Kues studied Proklos intensively. His thoughts on negative theology can therefore be used in the following for a better understanding of the Neoplatonists. He inspired Giordano Bruno and the development of calculus. Unfortunately, it remained largely unknown to the Enlightenment and was only rediscovered by Ernst Cassirer in the 20th century.

The three circle segments g-h, e-f, c-d symbolically stand for an infinite bundle of circle segments that continue to approach the straight line until the line a-b can be understood as a segment of an infinitely large circle at the border crossing. This illustration can also be interpreted in such a way that an observer looks at a sphere from above. The circle segments g-h, e-f, c-d are understood as the latitude of the sphere, the line a-b as the equator of an infinitely large sphere.

If it were possible for humans to take the standpoint of the infinite, then straight lines, circles, spheres and triangles would merge into one another in a proto-topology. However, human perception remains in the finite and therefore in Euclidean geometry or transcendental aesthetics in the sense of Kant. The power of imagination can indeed go beyond that, but its illustrative images are also bound to Euclidean geometry, for example if, as indicated here, the geometry a spherical surface is an example of a non-Euclidean geometry.

Man can only grasp the thought of crossing the border. Illustrations like these are images of basic lines that elude humans as such.

At the same time, the cosmos must not be confused with the sensually perceptible sky, the ouranos. The cosmos is neither a generalization of the celestial sphere nor the pattern according to which God created the world, but only a variant of this pattern accessible to man, which man can develop from his limited understanding of the divine blueprint and his sensual perception of heaven and nature can.

Correspondingly, the time accessible to human understanding is an image of the Aion (the Eternal that only God can overlook). Humans cannot grasp the higher unit of time, which lies before the division into past, present and future, or in modern imagination exceeds all physical ideas such as a big bang and represents the inner unity of all events, but only images Make of it: One picture is the flow of time with its inner connection, another picture is the chain of events immediately following each other. Each of the images is incomplete on its own and the two do not fit together without contradictions. This is what Zenon's paradoxes are based on. If a person tries to go beyond the flow of time and the time atoms in his thoughts, then he necessarily becomes entangled in the antinomies described by Kant. - Both images are intuitive and immediately catchy, but it is easy to understand that in nature neither a flow of time nor an atom of time can be perceived. Instead, concrete temporal processes are perceived in nature, such as the wandering shadows cast by sundials, the mechanism of clockworks, or the unique time of all living things and their inner pulse. The perceptible measure of all temporal processes are the orbits of the stars in the sky (ouranos).

The platonic one eikon already comes very close to the symbol. The flow of time and time atoms can only be understood symbolically: the thinking soul can operate with them and try to describe the temporal processes that can be observed in nature, but they are not the object of perception. They can be read from the perceptible appearances with the help of the mind. On the other hand, the divine creation cannot be understood as an event within the flow of time or as a primordial time atom from which all other time atoms arise. Modern physics gets entangled in unsolvable dilemmas when it tries to let time emerge from timeless states.

(B) Aristotle had in the metaphysics the concept of prima materia (proto hyle) introduced. Primary matter is the uniform substance of the sublunar world underlying the four elements earth, water, air and fire, in which the laws of physics, known to us from our own perception, apply. In this respect, Prima Materie was a materialistic criticism of the theory of ideas of Plato, who wanted to reduce the four elements not to an underlying material, but to transcendent, mathematical forms.

The term proto hyle Aristotle has an inner connection with the enigmatic superordinate concepts hyle noete (the thought hyle or the stuff of thoughts, the stuff of mathematics) (Met. VII 10, 1036a; VII 11, 1036b-1037a; VIII 6, 1045a; X 8, 1058a), and hyle topike (The local matter, the matter of local movement, the star matter of the celestial phenomena understood without body in the supralunar realm) (Met. VIII 1, 1042b). The proto hyle stands between these two substances on the one hand and substances known from nature on the other. Aristotle wanted with the proto hyle to find a term that does not derive physics from mathematics, as in Plato, but, conversely, enables higher concepts to be formed in thinking from the sensual experiences of the substances of nature (metaphysics).

The proto hyle is therefore not a primordial substance, as the alchemists of the European Renaissance misunderstood it in their search for prima matter and unsuccessfully wanted to produce it in the retort, but rather to be understood as a symbolic substance that represents all sensually perceptible substances in thought and with the help of which general properties of the substance can be recognized in the mind, for example the relationship of the substance to the form and the lack of form (steresis). This relationship, which can only be understood in thought, is the starting point of the Aristotelian physics. (For the double meaning of nature, see what it is [category] and what's missing [steresis], the comment "steresis - a doctrine of missing and saying no ", Link. Important sources are Aristotle physics B 1 and Heidegger's explanations.)

(C) Finally, the pagan tradition. Despite the prohibitions imposed by the Christian state religion, the Neoplatonic philosophers continued the pagan cults in secret, and the symbolic signs inspired by them (e.g. dream images, formations of the flight of birds, fibrous livers of sacrificed animals, constellations, patterns of the location of stones, etc. ) were understood as divination. Proclus wrote a text "On the Mythical Symbol", of which, however, not even fragments or quotations have survived. But Ficino handed down the "Opus Procli de sacrificio et magia" ("On Sacrifice and Magic"). Proklos represents a sympathy between heaven and earth: on earth nature is arranged in such patterns that allow the divine to be understood. Nature forms symbols of its own accord that man only has to learn to understand.

Peter Crome pursued the concept of symbols in Proklos: In the Timaeus Commentary, Proklos interprets the myths mentioned by Plato about the sinking Atlantis, the Egyptian temple of Sais and the youth of Greece as symbols with which the teaching presented below is introduced. For his part, Plato here explicitly refers to Egyptian teachings. In the Politeia commentary, however, Proclus criticizes Plato's rejection of all imitative art. On the contrary, he wants to show that Plato also created symbols with his myths. However, imitation does not simply have the character of an image, but is a symbolic representation in order to make something sensually tangible and understandable that cannot be directly recognized. (Gilles Deleuze began his philosophy with a similar criticism of the doctrine of the illusions of Plato, see the appendices in "Logic of Sense".)

An example from Iamblichos (245-325) may explain this. He interprets the strange sounding Pythagorean rule "plant mallow, but don't eat it":

"This is an enigmatic expression of the fact that such plants turn to the sun and manage to pay attention to this; however, it also says 'replanting', that is: turn your attention to their nature, striving for the sun and resonating with you, but be Do not be satisfied with it or stay with this phenomenon, but change the meaning and transplant it, so to speak, to the related plants and vegetables and further to the unrelated living beings, the stones, rivers and simply all natural phenomena. You will find that the sign of unity and the consistency of the cosmos is manifold, diverse and amazingly abundant if you start from the mallow as if from the root and starting point. So do not eat of it and do not let such observations go under, but on the contrary increase them and, like a gardener, produce an abundance of them. " (Jamblichos, "Call to Philosophy", p. 83)

Figure 2 Mallow, historical herbal illustration
Author: By Jan Kops - www.BioLib.de, public domain, link

The mallow is a sign of how the material turns towards the higher, and it shows how nature is permeated by a movement to understand the unity and the cosmos. The philosophy to which Iamblichos calls in this writing should start from the observation of nature, recognize this movement in nature and allow itself to be grasped by it. When she recognizes this striving in nature, the sensually perceptible appearances of nature acquire symbolic meaning. Many are passed down in the symbolic representations of the Greek gods, such as the ear, the lyre, the lightning and so on. This is a completely different understanding of the symbol than, for example, the cross in Christianity. (Yehuda Liebes suspects that the writings of Iamblichos flowed into the "Zohar", one of the main works of Kabbalah, written around 1280.) The Neoplatonists trusted the power of natural symbols and therefore held on to their belief in miracles and the interpretation of oracles.

The symbol and the soul, with their ability to form symbols, stand in the middle of two movements which, as a whole, form a cycle: starting from the basic lines of reason (spirit, nous) the soul forms proto-terms, which are the seeds of science, with the help of which man can recognize nature and understand it systematically. These germs, such as the circle, precede science and cannot be justified inside in the way that all conclusions can then be justified in the following. Euclidean geometry is the prototype of science. It starts with the definitions (oristic) of elementary figures (point, circle, straight line) and opens up a horizon with them, within which everything else can then be derived. Proclus wants to prove this in his Euclidean commentary. For his own philosophy, the task remains to understand a step earlier the formation of the elements that stand at the beginning of science. The title of his main work "Stoicheiôsis theologikê" therefore appears to me to be translated neither as a "basic course on unity" nor as "theological elementary theory". The term stoicheiôsis rather the working of the elements (stoicheia) describe in the soul the "elementary elements", such as the "counting number" or, with Heidegger, the timing of time can be spoken of.

At the same time, in the other direction, the symbols are obtained from the consideration of nature, as Iamblichos described it. This results in a circular movement, Proclus speaks of the "circular reality" (kyklike energeia, Proclus, Unity, §33). The outgoing Monas is dynamis both as a force, as a constancy, as well as a possibility from which the many emerges (prohodos ). And it is also the reason to which the many that have arisen from it refer back again (epistrophe). Only the cycle as a whole results in the abundance of the symbol. "The result alone would leave the object undefined, since every object only becomes its essence when it is turned back to its cause (ousia) determined. "(Gombocz, p. 216)

Each symbol is therefore in two orders (tas taxeis, Proclus, Unity, §63). One order goes back to the basic lines of the mind, the other to the perception of nature. The divine abundance has lost something in the transition into the symbols formed by the soul and therefore retains a reason that is secret, unknowable, and not even nameable for the soul. Proclus speaks of secret (kruphion), incomprehensible (alepton), unspeakable (arreton) and unrecognizable (agnoston) (Proclus, Unity, §§121-123). (This is what Nikolaus von Kues means with the ignorantia and based his negative theology on it). The other order arises from the observation of nature and lays knowledge into it that cannot be seen directly in nature, but can be read from it. Everything in nature can be counted, but nowhere can the numbers be seen as such in nature. These two orders threaten to isolate themselves from each other as mythical and mathematical symbols. The separation is complete when, in modern natural science, the mathematical symbols appear as creatures of the creative scientific mind, as mere conventions. Then science is no longer within the cycle described by Neoplatonism, but stands above nature and creates the symbols from itself in order to be able to describe nature scientifically.

What is special about mathematics is that it understands how to make the interrelationship between these two orders the subject of its research and in this way provides knowledge that is by no means self-explanatory from the start. Nowadays this appears systematically in the comparison of local and global orders (locally every space is "Euclidean", i.e. rectangular in which the Pythagorean theorem applies; globally it is a differentiable manifold). Euclid is about the relationship between a straight line and a circle. Proklos: "Right lined figures are proper to sensibles, but a circle to intelligibles." (Proklos, In Eucl., Taylor translation, 82) There is a tension between the two, from which the geometry of Euclid develops. It begins with simple constructions of straight figures and leads to complex helical lines (helix, spiral) (Proklos, In Eucl., 179-180, see Becker p. 103). (See Manchester "Syntax of Time" about the Archimedean spiral as a mathematical representation of the case [ekpipto] of intelligible time into the natural stream of time.)

For Proclus, the elements of Euclid's geometry are both mathematical and mythical symbols. The point is both the smallest element of geometry, without parts, without dimension, i.e. negatively defined, empty, and at the same time the divine power from which everything arises and where everything returns. Most unusual for today's mathematics is the assignment of angles to gods. The right angle is the symbol of the earth goddesses Rhea, Demeter and Hestia (Proklos, In eucl., 128-130 and 173-174). Every angle shows how lines flow out of the point and, conversely, lines collapse into one point. Every corner has a qualitative, divine power. The right angle stands for the construction principle of the earth. Here the expanding and contracting forces are in equilibrium. The right angle is identified by the goddess Hestia with the hearth fire, the power-giving angle of the house, and the pole of the celestial sphere. That may sound far-fetched and certainly requires further interpretation. The same picture returns when the mathematician Bernhard Riemann places the number ball with its lower pole on the right-angled coordinate cross of the complex plane in the 19th century to describe the complex numbers.

When, in the late version of the first volume of his "Science of Logic", Hegel finally drafted the idea of ​​a mathematics of nature, which for him is the science of measures, he was on the way to the center of Neoplatonic theory. More recently Deleuze has presented a different intuitive approach in his "Logic of Sense" with the "Five Characteristics of the Transcendental Field". He, too, systematically differentiates between two series and contrasts the "royal", rational, axiomatic mathematics, that is, the mathematics of the mind, with the "small", molecular, problem-raising mathematics, the mathematics that draws from perception. (Compare with Aristotle Met. VII, 10 1036a and De.An. III, 4 430a as well as Proklos, In Eucl. 51.)

Both sides are not rigidly opposed to each other, but mathematics is in a reciprocal movement between the principles of the spirit (Proclus understands this as the dialectic in the sense of Plato) and the fullness of nature. Hence Proclus speaks of the "numbers moving by themselves" and the "living" figures that precede the numbers and the "visible" figures in the soul (Proklos, In. Eucl. 16). The following considerations on the dimensions of nature are intended to lay the foundation for understanding the dynamics of the dimensions of nature based on them. Hegel was no longer able to elaborate on this. Today, thanks to Einstein's theory of relativity and quantum theory, there are entirely new ways of understanding them.

From the measures of nature (number, size, place, time) to the principles of proto-mathematics

The abrupt end of the Neoplatonic school fell in troubled times. From 375 onwards, under the pressure of the Huns, Europe had got caught up in the maelstrom of peoples. The dissolution of the Roman Empire could no longer be stopped. Even the elevation of Christianity to the Roman state church, which was enforced in 391/92, did not bring about the desired stabilization. Nevertheless, it hit the ancient philosophy hard, which was pursued at one with paganism. Pagan religious activity has been officially prohibited since then. In 402 Ravenna had become the capital of Western Rome, in 410 the Visigoths sacked Rome. In 415, the mathematician Hypathia (370-415) was murdered by an angry Christian crowd in Alexandria because she remained loyal to Neoplatonic paganism. The advance of the Huns was not broken until 451 with the battle of the Catalaunian fields. In 529 the Academy in Athens was finally closed by Emperor Justinian. She was accused of pagan teaching. The last representative of the Neo-Platonists therefore remained Simplikios (490-560). He was a student of Damascius (458-540), the last director of the Academy in Athens. Both had to leave Athens for a few years.

With Simplikios, the Neoplatonic tradition broke off. Two important innovations go back to him, which are the starting point here: (a) Following the example of proto-material (prima materia), he introduced the concept of proto-time in order to systematically distinguish between the paradigm of time in the nous, the symbols To be able to differentiate between the time in the soul and the temporal processes in nature. Proto-time has its origin in Aion, the higher unity of time in the spirit (nous). In the other direction it is in nature (physis) the overlapping of becoming and passing away and thus the measuring, "timed" time. The relationship between Aion in the Nous, proto-time in the soul and Chronos in nature becomes a pattern for all other terms in order to distinguish paradigm, eikon and the "natural", the sensually perceptible. (b) For Simplikios, size and location are different than for Damascios (megethos, topos) not one measure, but two measures (Simplikios, Zeit, 773-774).

This shows a grouping: there are four instead of three measures, and of them number and size are mathematical, place and time are physical (scientific). Mathematical symbols are understood in a narrower sense to mean the arithmetic number symbols, the geometric symbols such as point, straight line, circle, angle, sphere and the symbols of the infinite (unlimited, infinitely large). The natural sciences will look for symbols for places and times.

The understanding of size as a separate measure has remained open, since the Simplikios approach could not be continued. There are still no generally recognized symbols for size. The system of the measures of nature is therefore incomplete. As for mathematics in the narrower sense: if the measure of size can be clearly distinguished from location, it will also be possible to distinguish number and size from one another. This task has also remained open, and that is the deep cause of the paradoxes into which natural science has developed in the 20th century.

How far had the Neoplatonists succeeded in clarifying the mathematical measures of number and size with as rigor as the concept of time?

number: In a first approach, the number (1) must be distinguished from the natural numbers, (2) the proto-numbers of the soul and (3) the paradigm of the numbers in the nous. Gyburg Radke (Uhlmann) translated and interpreted a text by Syrian (400-450) in detail in her work "The Theory of Numbers in Platonism". In a commentary on Aristotle's criticism of Plato's theory of ideas and numbers (metaphysics Book XIII, Chapter 8) he asks about the material and the form of the numbers. This leads him to differentiate between admitted, demiurgic and eidetic numbers. They should be interpreted here as the natural numbers, proto-numbers and noetic numbers we are looking for.

(1) The numbers allowed are the numbers used to count in the physical world: three apples, 111 plants, a thousand degrees, etc. These are the numbers of physically present things that can be counted. These numbers are still referred to as the natural numbers to this day. All natural things are described by the number of their parts and characteristics: a person walks on two legs, there are 365 days in a year, the honeycomb has 6 sides ...

(2) The demiurgic numbers are the numbers with which the soul uses its images (eikon) designs. The idea of ​​demiurgic numbers follows on from that Timaeus from Plato on. Plato tells how the demiurge created the cosmos. Here something is no longer counted, but the soul looks for suitable numbers in order to get an idea of ​​which numbers the Demiurge has chosen. Hence Syrian speaks of demiurgic numbers. If, for example, 4 regions of the sky are mentioned, the regions of the sky are not counted with the result "the number of regions of the sky is four", as is the case in the physical area, but the four has a symbolic meaning (quaternity). Its symbol is the coordinate cross, which can still be read remotely today in the sign "4", in which a vertical and a horizontal cross each other. The sky is symbolically understood as a coordinate cross with the 4 areas west, north, east and south.

The demiurgic numbers are symbols. Where in the sky can north, south, west or east be perceived, observed and counted? Zodiac signs or stars can be counted, but not the cardinal points. Quaternity (fourfold) is a symbolic property of the sky that results from understanding the demiurge's blueprint and not from observing the sky. The coordinate cross is a picture of the baselines of how the cosmos was created.

When we speak of demiurgic numbers, it means that the demiurge worked with numbers in his plan. Humans can only incompletely understand these numbers. In nature they only occur in approximation. But nature tends to form shapes that want to go back to this origin. The soul has the ability to form an image of these numbers and to read them out from nature. The result are the symbolic (demiurgic) numbers.

The understanding of the symbolic meaning of the numerals has largely been lost. The circle of zero ("0") is reminiscent of the symbol of the unit, but is not assigned to the number 1. The "1", on the other hand, shows the sign of existence, symbolized as a line, that something is present once (the notch, the line on the beer mat, the Roman one). The serpentine line of the "2" is reminiscent of the Chinese yin-yang symbol. The "3" has moved far away from the triangle, earlier a symbol of matriarchy, the trinity of the phases of the moon, later a symbol of the Christian trinity. With a little imagination it can be read as a symbol of branching (three-way, trivium) when the two semicircles separate from each other in the middle upwards and downwards.The "4" shows the coordinate system, the "8" is symbolically related to the symbol of the infinite "".

Today the symbolic numbers are no longer separated from the natural numbers. There is no distinction between counting things using natural numbers and the symbolic understanding of properties of the demiurgic blueprint. Therefore it has never become possible to develop your own dynamic of numbers, but only the familiar structure of numbers, which leads from natural numbers to whole, rational, irrational and real numbers. The question of the dynamics of numbers will prove to be uncharted territory, as will the question of the symbols of sizes.

Today the natural numbers are not only a measure of numbers, but also a measure of size. Sizes are only differentiated according to their numerical value. The order relation of the natural numbers (e.g. "7> 3" or "n2 > n ", in words:› seven is larger than three ‹or› n square is larger than n for all natural numbers n ‹is equated with the measure of size. In order to be able to distinguish the measure of size from the measure of number, is therefore it is necessary to first differentiate between the natural numbers and the symbolic numbers for the numbers.

Perhaps David Foster Wallace was better able to put this thought into words. In his considerations of the infinite in set theory, he encounters a similar ambiguity: "The third distinction may seem finicky at first. It concerns us related words such as 'amount' and 'number'. These have a strange and confusing double meaning that can also be found in words like 'length' or 'gram'. A length of rope is a certain length, but is sometimes referred to as 'one length of rope'; a certain amount of medicine that weighs one gram is also called 'one gram of medicine'. ... Therefore a term like 'infinite' can be ambiguous: either it is meant predicatively ('There is an infinite number of prime numbers') or nominally (' Cantor's first infinite number is 0. '). "(Wallace," Discovery of the Infinite ", p. 46) When one speaks of" one gram of medicine ", size (here measured in grams) and number (here" one ") go together.

In summary: Demiurgic numbers, symbolic numbers and proto-numbers mean the same thing, but are to be differentiated from the natural numbers and from the symbolic quantities (the proto-quantities).

(3) If the demiurgic numbers are the proto-numbers, the eidetic number is that paradeigma of the symbolic numbers in the nous. Each symbolic number is the image of something with which the demiurge created the cosmos. As far as I understand Radke's representation of Syrian, he wanted to distinguish between the underlying principle and the image in the cosmos. The quaternity of the cardinal points can be recognized by humans as a property of the cosmos. It is the image of a higher principle of quaternity. Everything that can be described with the symbol of quaternity has part of a higher principle, which I would like to call tetras to distinguish it from quaternity. Unity, duality, trinity, quaternity etc. are images of basic lines that can be named with Monas, Dyas, Trias, Tetras, ..., Dekas, ... Unfortunately, this approach was also discontinued. In my opinion it would be more natural to speak of noetic numbers. I expect these questions can be better clarified when the dynamics of the numbers are understood. Then a distinction can be made which possibilities and which reality are connected with the respective numbers and from which principles they arise.

Syrian distinguishes the so-called eidetic numbers from the material numbers, with which he apparently combines the natural and demiurgic numbers. On the other hand - following Aristotle - I see the relationship between form and substance of numbers differently: (1) Numbers and thoughts are the substance of the nous, hyle noete. When this side of the numbers is considered, one speaks of noetic numbers. Man doesn't know what these numbers look like. They are inexhaustible, and people only know what has arisen from them in a way that people can grasp. (2) The soul can only form an image of them, these are the number symbols. They are read from the zodiac signs (constellations), which in all early cultures were regarded as symbols of the gods and as the archetype of numerals and characters. The stars consist of a symbolic substance (hyle topike). With him the demiurge created the cosmos. The noetic matter is the paradigm of the symbolic matter. The stars are disembodied and therefore consist only of symbolic material. Their movement is infinite, always the same and knows neither deceleration nor acceleration. It is a pure local movement. (Anyone who likes to smile at such "quarks" today should be reminded that today's physics with the virtual particles, Higgs particles, quarks, light particles without rest mass, systematically based their theories on the symmetrical states of the Lie groups in the fiber bundle builds up a kind of symbolic substance without giving an account of it.) (3) The bodies are created from the natural substance, which can be counted with the natural numbers.

In addition to the noetic numbers, proto-numbers (demiurgic numbers, symbolic numbers) and natural numbers considered so far, new numbers have been added in modern times: (a) Negative numbers, considered as systems in which there are balancing processes, e.g. debit and credit in economics, or weights suspended from a lever arm on both sides. (b) Irrational numbers, when the numerical ratios were found within the range of sizes, e.g. the ratio of the base and the diagonal in the square (root of 2), as a measure of squaring the circle, golden ratio. (c) Differentials as the internal measure of time and place. (d) Complex numbers for currents. These numbers have led to new mathematical symbols: minus signs or red coloring for negative numbers, radical symbols for irrational numbers, differential symbols and integral symbols, i for the imaginary axis of complex numbers. These symbols were used as conventions only. No more attempts were made to understand them as symbolic signs of the soul with the paradigms on which they are based. To see a real philosophical problem here at all is the main concern of this article.

size: Is there such a thing as (1) natural quantities, (2) proto-quantities with their symbols, and (3) the paradigms of proto-quantities? Ancient and Neoplatonic thought left this question open. There are only a few clues. In order to be able to approach this question, one must distinguish between the following: It does not mean how big something is, i.e. what size value can be measured, but what properties something must have in order to be able to speak of its size. This question stands at the beginning of all natural philosophy, since Thales came up, what it means to ask about the size of water. How big is water What is the water missing that makes it impossible to determine its size? What does it have that is why it goes beyond all sizes? The water-like will prove to be a symbolic quantity that is not of natural size.

(1) As always, Aristotle can be traced back to in his physics to find a better understanding of what the natural concept of something, in this case natural size, is: Something is natural size when it is connected on the inside and separated by boundaries on the outside. Aristotle developed an early form of topology in his physics when he systematically differentiated: what has a natural size is independent and separate (choris) from other, can be arranged in sequence with other (ephexes) until it touches another (apteesia). The transitions can be continuous (syneches). What has no size of its own is between others (metaxy). (Phys. E 3, 226b). Natural dimensions of size are not the units to measure the size value (e.g. meter), but they are the measure of how strong the inner relationship of size is, whether it is porous, loose, prone to breakage, and how clearly the boundaries are drawn can. With fractal mathematics, the possibility was created to mathematically capture the dimensions of the internal connection and the external limit.

To come back to the example of water: A certain amount of water with clear boundaries, such as water in a bucket or in a lake, has a size. The amount of water can be measured. It is more difficult with clouds, the boundaries of which cannot be clearly determined. How much air and how much water do they contain? The water as a whole has no limits and therefore no size.

The natural size of a tree is its size when it is fully grown according to its nature. Then it is internally coherent and not distorted to too small or too large a measure (excess) by external influences. The inner coherence is only fully established in the natural size, otherwise there is a risk of instability either from being too crowded together or from being loosened up too far, which can lead to the dissolution of the interior. - At the same time Aristotle starts from the natural life cycle of all physical (sublunar) bodies. Everything has its natural size in its respective phase of life. The concept of natural (physical) size must be developed on the basis of an understanding of the physique.

In order to understand the natural size one has to ask about the natural relationship and the natural distance. In nature there are natural distances that physical bodies have from one another in order to be able to develop freely.

(2) As symbols of the connection and the size, ie as proto-sizes, the geometrical elementary figures offer themselves first of all: The point symbolizes everything that has no internal connection, that is limited to the outside, and therefore also has no size . The line is continuous and connected in every section, but as a whole it has no limit and therefore no size. It is infinitely thin and infinitely long. Only when it joins together to form a circle is the simplest connected figure whose size can be measured (the diameter). Therefore, the circle is often considered to be the symbol of greatness. The circle is held together by the center. The Pythagoreans saw the symbol of the connection in the center of the circle and called it Hestia, the goddess of the hearth, who creates the center and connection of the house and the household community.

"Philolaos says there is a fire in the middle around the center, which he hears (hestia) of the universe and house of Zeus and mother of gods and altar and cohesion (synoch) and measure of nature. "(Diels Kranz, fragments of the pre-Socratics, 44 A 16)

With another approach, the natural (or real) numbers are viewed as symbols or orders of magnitude. (Symbolism and order are also often mixed up.) Today, size usually means the measured height: This high-rise is 130 m tall, I am 1.80 m tall.

A third direction is looking for symbols of size based on human perception: For example, the cubit as a symbolic size when the size of a cloth is measured by putting on the elbow. The sea or the desert as a symbol for sizes that are infinitely blurred to the human eye on the horizon (and derived from this the smooth and the notched as symbols for certain types of sizes in Deleuze and Guattari in "A Thousand Plateaus"). Aristotle saw the size of the moon's distance from the earth as a symbolic quantity. Humans can only perceive changes in things that are visible to them in the sublunar area. Planets and stars move in the sky, but appear unchangeable to his perception, only their spatial movement can be seen. It was not until much later that changes in stars such as supernovae were observed. Today's physics sees similar limits when it designates an atom or elementary particle, the interior of which cannot be perceived with the measuring instruments known today. Basically, none of these are size symbols, but natural sizes from the perspective of humans and their technical capabilities.

Another direction does not distinguish size from place: the size of something is identified with the place it occupies. A balloon is sometimes bigger or smaller, depending on how much it is inflated. The size of a lake is equated with the length of its bank.

After all, the size has to do with the physical state. Water can be a coherent liquid, it can form individual drops in a cloud, or it can dissolve into an air-like state when it evaporates. The same amount of water is always a different size. The same amount of water takes up a larger space when it forms a cloud instead of being collected in a puddle after rainfall.

All of these aspects each show something right. Just as in a first approximation duality (the "two-like"), trinity (the "three-like") led to the understanding of the proto-numbers, the particle-like, liquid-like (wave-like), gas- Nice first approximations to understand symbolic quantities (the proto-quantities). The symbolic sizes do not differ from one another in terms of the measured size: It is, for example, pointless to ask how big the crystal-like is or whether the crystal-like is larger than the liquid-like. Just as there are symbolic numbers, there are symbolic quantities. When Thales spoke of water, he meant it in terms of a symbolic size.

The particle-like is at the same time a symbol of the discrete quantities, the atoms. Plato has the particle-like in the Timaeus systematized by the solids with regular outer surfaces (the Platonic solids, the smallest building block of which is the tetrahedron). In the other direction, the wave-like symbol is the continuous quantities, the river, the current.

The unit under the numbers corresponds to the element under the sizes. The elementary is the symbol of everything that is great. The other symbolic quantities such as the particle-like are developed from the elementary in a similar way as the duality, trinity, etc. from the unity.

In the case of symbolic quantities, the close connection to mythical symbols can be recognized even in common parlance. Romantic poetry often plays with it when, for example, the crystal as a symbol of the particle-like is taken once as a natural crystal and at another time in a symbolic meaning as the hardening of an entire way of life. Or there is talk of elements as well as the fact that someone is in his element. It then moves freely in its natural size.

And just like the sign with numbers as a symbol of the infinite, which lies beyond all numbers, so here the ether as the infinitely great, which is at the same time infinitely fine and spins everything, the proto-medium of the wave propagation of the proto-liquid. Trying to prove the ether experimentally is just as impossible as the prima matter or symbolic numbers cannot be "discovered" in nature.

On the basis of the symbolic quantities, both the apparent paradoxes of relativity theory and those of quantum theory can be understood. Einstein did not make a clear distinction between ether as a symbolic quantity, natural space and the order relationships in space (differential geometry). When Heisenberg spoke of a wave-particle dualism, he mixed symbolic sizes (the wave-like and particle-like) and natural sizes of natural waves and particles inadmissibly. There can be just as little dualism between the wave-like and the particle-like as there can be between the Trinity and the Quaternity. What Heisenberg means is something different: when natural quantities are measured, they are always changed by the measuring process at the same time, also in terms of their size. If the efficiency of the measuring instrument is of a similar natural size as the measured object (wave or particle), then the change caused by the measurement is significant. But this is not because the wave-like meets the particle-like.

However, this interpretation of the uncertainty relation is controversial. Others argue that indeterminacy occurs when something has both local and global properties. If the local properties are examined more closely, then the view of the global becomes somewhat blurred in the horizon. If the global properties are highlighted, the local properties can only be estimated statistically. A doctrine of the dynamics of size is intended to unravel these questions. The question to be asked to Aristotle and Plato is whether their understanding of the mean, metaxy, here helps.

The symbolic sizes differ systematically through their respective inner context. The liquid-like has a different internal connection and therefore also different external boundaries than the crystal-like.If we look back again at the quotation from Philolaos, then Hestia, the hearth goddess of the cosmos, is both a symbol of the inner connection of the cosmos established by her and the symbolic greatness of the fire-like. The fire-like leads over to the light-like and thus to the paradigms of greatness.

In mathematics, the concept of connection was only introduced with fluid mechanics in the 19th century. As with the introduction of negative, irrational, real and complex numbers, numerous new mathematical symbols have been created here, such as the Christoffel symbols Γ for derivation in Riemannian manifolds, symbols for differential operators such as the Nabla operator ∇ for a uniform representation of gradient, Divergence and rotation in currents. The differential topology deals with the mathematical basis of these terms. The mathematician group Bourbaki has published "Elements of Mathematics" for these new mathematical developments based on Euclid since the 1930s, but so far there has been no philosophical interpretation of these terms in a way comparable to that which Proclus made for Euclid.

Are there starting points which symbols can be used for sizes? They may be supplied by IT. The sign # can be understood as a symbol of the discrete and particle-like (numberable, countable), the sign ~ as a symbol of the continuous and wave-like (the similar and approximate). Another possible source is the pagan symbols. The concentric circle with a point is considered a symbol of gold and the cohesion of the universe. The Celtic double spiral is the symbol of becoming, ~ is a simplification of it.

(3) What is the paradigm of sizes? According to the thought developed so far, this can only be heaviness and light.

The Romantics upgraded the paradigm of the night, although there are quite different approaches from Novalis, Hölderlin, Hegel and Schelling. For example, Schelling writes in his Würzburg System of 1804, expressly criticizing the mathematical principles of Newton's natural philosophy:

"The reason for the gravity is therefore the inexplicable depth of nature itself, that which can never itself come to light because it is through which everything else is born and sees the light of day, the mysterious night, the fate of all things Or also, because in it things are as in their ground, in which they receive and from which they are born, the maternal principle of things. " (Schelling, Würzburger System, p. 256f)

Only what has weight can have a connection. The weight is the paradigm of the inner relationships established in connection, which establish the uniformity of the straight line and the circle, i.e. their respective concrete forms of the connection represented in them.

The light shows the connection. If the solid or the ether are only symbols that cannot be proven as such in nature, then the heaviness and the light, light and dark have a "visible side" that can be perceived, but elude them as a whole human knowledge and are therefore in all creation myths at the beginning, before natural things were created. Humans can only make pictures of them, for example the rays of light, light waves, photons or the gravity and darkness of the caves in the earth or, in modern physics, the pictures of the black holes and the Big Bang. Everyone understands intuitively that these are only incomplete pictures of a paradigm that humans cannot overlook. Black hole, big bang, light waves and photons can only be proven indirectly in terms of their effects in the language of modern physics. Modern physics does not want to recognize that, similar to proto-time, for example, light waves and photons are only symbols that can be distinguished from natural light in one direction, such as daylight, and in the other of the unknowable paradigm of light. Often the sun is used as the paradigm of light. Light in the meaning of enlightenment, enlightenment, or in Heidegger's usage of light are different images to understand this paradigm.

Physics necessarily becomes entangled in paradoxes if it fails to distinguish between natural light, symbols of light and the paradigm of these symbols. In the same way, a distinction must be made between natural observations of what happens when stars "die", the black hole, which can only be understood symbolically, and the paradigm, the image of which is the black hole, in the way that Schelling described the reason for gravity. A careful distinction between these levels will allow one to understand a modern metaphysics that is completely different from the old metaphysics, from which modern science has turned away with good reason.

On the basis of the symbolic numbers and sizes, the program can be designed for a proto-mathematics, which is fundamentally different from metamathematics in the sense of Hilbert and which tries to re-approach the questions that remained open in Neoplatonic thinking. This is where the fundamental discussion of mathematics stopped in the 1920s to 1940s. Mathematicians such as Max Steck, who published a German translation of Proklos' Euclid's Commentary in 1945, Hugo Dingler and the philosopher and mathematics historian Oskar Becker were already getting close to this program. However, they wanted to justify National Socialist convictions. Instead of questioning this, this approach of proto-mathematics was no longer continued, but transformed into a purely formalizing approach in the 1950s, until it could no longer be distinguished from its earlier counter-position, metamathematics. It is no coincidence, however, that mathematics made it possible for a new mythology to break in at this crucial point, but must be understood. This shows that the tension between mathematical and mythical symbols has not yet been clarified. (In the next step, the same question with Nietzsche will therefore be considered from a different angle.)

Assumption: The proto-distance seems to me the appropriate term to understand what Michael Zednik sees as the distance between the markances. I understand the markances again as proto-places when a distinction is sought between (1) the physical places as Aristotle considered them (physics, Δ 1-9), (2) the proto-locations of the soul and (3) the locations in the nous, i.e. the paradeigma the proto-places. Therefore, the approach of Graf and Zednik, which goes back to a new interpretation of the special theory of relativity, can be understood as a first attempt to find a doctrine for proto-sizes and proto-locations that continues where Neoplatonism was interrupted.

In a different direction, Helmut Hansen took up Epstein's interpretation of the special theory of relativity and created a "physics of the mandala" in which the symbolic dimensions of the sun and the earth, circles and squares are intertwined. I suspect that the Epstein circle, with which Epstein distributes all movements to parts of space and time and thereby receives an amazing new interpretation of the length contraction and time dilation, can be expanded to a sphere in which the number and Size proportions are taken into account, ie all four dimensions of nature mentioned by Simplikios. Such a figure takes up Kepler's ideas of world harmony and leads to a new design for the architecture of mathematics. Because of the rotational symmetry of the circle and sphere, this is a multiple mathematics.

What is lacking in the basic lines of thought

The question of the paradigms and symbols of greatness touches on a fundamental problem: Are there systematic boundaries between mythical and mathematical symbols? The symbols of the alchemical substances look very similar to the mathematical symbols. Symbolic material evidently has a lot to do with symbolic quantities. Typical proto-substances are the salty , Sulphurous , Watery . They are used to designate elements that should not be confused with the corresponding natural elements. It was a misunderstanding when alchemy tried to identify and produce the proto-substances as natural substances in nature. Some of these symbols can also be understood as symbolic quantities: The symbol of gold or the symbol of the ore offer themselves as a symbol of the particle-like, the symbol of fire or pictographically of Aquarius as a symbol of the wave-like. Related to ore is the symbol of the diamond ◊, which is used today in axiomatic set theory (symbol of the combinatorial principle from which the continuum hypothesis can be derived).

The Neoplatonists did not look for paradigms of proto-substances in the nous. At the beginning of the modern age around 1600 there were ideas to take up their teachings again and to bring symbols of this kind together with mathematical symbols. Best known was John Dee (1527-1608), who wrote a foreword for the English translation of the elements of Euclid and designed the sign of Monas Hieroglyphica, with which he wanted to justify science and para-science in a uniform manner.


Figure 3 alchemical symbols

Examples of element symbols used by the alchemists 1 = tin, 2 = lead, 3 = gold, 4 = sulfur, 5 = mercury, 6 = silver, 7 = iron; Monas Hieroglyphica by John Dee, in which the symbols of the 7 planets, 4 elements and some zodiac signs can be detected;
Author 1: Von MaEr - Eigenes Werk, based on former ElementsAlchemisten.jpg onclick = "window.open (this.href); return false;" by Roland1952, CC BY-SA 3.0, Link
Author 2: From PRiis at Wikipedia in English - Transferred from en.wikipedia to Commons by Leyo using the CommonsHelper., Gemeinfrei, Link

I am convinced that it is only a matter of time before alchemical symbols are taken up and used in the border areas of mathematics or particle physics. Alain Badiou provided a first approach. He has the Venus symbol in "Being and the Event" Introduced as a symbol of the indistinguishable, a borderline concept of set theory and particle physics.

Can a difference between the mathematical and mythical symbols be determined systematically? The Neoplatonists gave a clear answer: the mathematical symbols are images of paradigms in the nous. Proclus writes in his treatise on unity in §57 that the nous has no defect (deprivation, privatio, steresis) and therefore does not know evil either, if evil is understood as the deprivation of good. In the nous, possibility and reality are one. In the nous - and thus for the paradigms of mathematical symbols - there is no distinction between what is only possible but not real, and therefore also no distinction between what is missing but could be present. There is therefore also no temporal conception of substance (permanent) and causality (temporally necessary subsequent) in the nous, because there is nothing that could decay, i.e. is not permanent, and nothing that could perish in favor of a successor. These speculative ideas become immediately apparent when viewed in mathematics: There are no possible or missing numbers. Numbers do not have a specific duration. In the case of proof, inferences are made, but the relationship between the statements that have been inferred cannot be understood as causality.

The math symbols are all images of paradeigmatathat belong to the nous. Hence the conclusion of the Neoplatonists can be summed up in one sentence: The mathematical symbols lack everything that is missing (steresis). They are perfect and complete in an empty way. This emptiness occurs somewhere in all mathematics (be it the paradoxical concept of two, which in Greek mathematics means both the other and the distance, or the prohibition to divide by zero, the disappearance of the differentials and finally the paradoxes of the empty set since Cantor), but they elude it. This is evident in the mixture of certainty and futility that characterizes all mathematics, and is right in the middle of what Deleuze and Guattari are in royal and minor mathematics wanted to separate from each other with the hope of starting out from a previously under-noticed minor mathematics to be able to re-establish mathematics.

The regulations (diakriseis) of nous and deprivation (steresis) remain independent of one another among the Neoplatonists. The diakriseis come from the nous and only capture parts, places, beings and temporal durations within complete wholes that are not lacking anything. When something is missing in the physical world, the missing is a missing part or a difference between the whole and the sum of the parts, a lack of time to get something done, etc. But there is no "proto-lack", no symbolic one, in the soul Absence, because the soul can not for the absence eikon since there is no in the nous for this paradeigma gives. The mathematical symbol of the empty set with its zero crossed out, is obviously an art symbol and not an image of a higher paradigm. The blanks in mathematics such as zero, the empty set or the indistinguishability considered by Badiou therefore hang in the air. They show the incompleteness (or positively formulated the openness) of any mathematics that is based on Platonic or Neoplatonic. With Lacan, they can be referred to as step points.

The soul has the ability to understand what is missing, but it cannot adequately express it in the mathematical symbols it has created so long as these are understood as images of the baselines of the nous. She therefore needs myths to understand what is missing. The symbolic myths about loss, abandonment, being torn and vice versa, happiness and success make it possible to understand what happens in "real life", in natural life, just as the proto-terms make it possible to measure sizes and distances and to count.

Mathematical and mythical symbols are different expressions of the soul's ability to create symbols. Through the concept of cohesion (synecheia) there is an overlap. The absence can be understood as destabilizing to the point of complete loss of cohesion, and at the same time the mathematical symbols are also based on cohesion. All of mathematics will go to nothing if the underlying cohesion is removed from it. This can be felt within mathematical research when the traditional symbols become paradoxical and the soul loses the ability to create new, appropriate symbols. Then, within mathematics, when operating with familiar symbols, undesirable feedback effects, resonances or combinatorial explosions occur, which call into question all mathematical calculations. The science of the 20th century provides an almost unlimited number of examples of this.

In such situations the soul has to create a founding myth. This establishment is far more than a physical event, it is a symbolic event. (With his genealogy Nietzsche asked about symbolic events that are to be distinguished from historical events.) In the founding myth, symbols are created that create unity and cohesion across the board. Examples of founding myths of science are Prometheus, the Fall of Man described in the Old Testament and the Pythagorean myth, which is passed down in the fragments of Philolaos, in which Hestia returns in science when she was absent from Olympus.

If the symbolic quantities are clarified, the relationship between the mathematical and mythical symbols will shift again. With the symbolic quantities it will be possible to mathematically represent what has so far only remained in the area of ​​symbolic substances. Certain properties that were previously associated with the symbolic substances (such as the inner cohesion of gold , the loose order of the sand , the dynamic of fire, symbolized in the sign of Aries spring ), will be mathematical symbols. Only when that has been successful is Simplikios' concern carried out.

The question still remains whether there will be separate mathematical symbols for the dynamics of the measures of nature. So far there are only art symbols introduced by Leibniz, such as the differential and integral symbol, and later the symbols of tangential spaces and bundles as well as symbols for better operation in field theory.

outlook. This seems to me to describe the scope of Platonic-inspired mathematics as far as possible. It has provided fascinating insights and is still a long way from reaching the end of its potential for development. On the contrary, a resumption of Simplikios' ideas could bring about great upheaval and completely unexpected results. Nevertheless, this is primarily intended to prepare the reading of Aristotle. With Plato, mathematical and mythical symbols necessarily fall apart. Does Aristotle offer an approach to transfer the principle of absence to mathematics? That would not call into question the individual findings of Platonic mathematics, but would fundamentally change the entire architecture of mathematics.

Epilogue: Virtual and Multiple Mathematics

In succession to Deleuze, a group of philosophers led by Manuel DeLanda and Simon Duffy advocate the thesis of virtual mathematics. The idea of ​​virtual mathematics takes up the movement from the diversity of nature via the soul to the unity of nous and being. However, while the Neoplatonic philosophers see the soul in a mediating role within a superordinate movement from nature back to the one, the task of virtual mathematics is seen as mathematically overarching the movement space for this movement, i.e. to find a virtual space that which contains nature, the soul and the nous. This virtual space would contain the natural things, the symbols and their outlines. It would also include the natural places, proto-places and their baselines.

Mathematics describes the most general space in which problems can be posed, analyzed and solved. Every problem can be understood as an initial constellation in this space, the solution is an end constellation. The space is defined in such a way that all problems, their solutions and the possible solutions can be described there. The task of virtual mathematics is therefore to recognize mathematical properties of the problems, their solutions and solutions as properties of this space. One can speak equally of a geometry of problem-solving and a logic of promptings, a logic which promptings must follow if they are to be successful. This is a generalization of the ideas of differential calculus and Galois' theory, which were used to rephrase mathematical problems in such a way that the problem was given a suitable space in which to solve it. Specifically, these were the multiplicity of tangential spaces as a space for the solution of the questions of the differential calculus and the group of body extensions as a space for the representation of the problems of Euclidean geometry and the arithmetic solution paths of Renaissance mathematics by Galois.

This fascinating idea takes the ideas of Hertz, Boltzmann and finally Wittgenstein that emerged at the end of the 19th century to the extreme. (The authors of "virtual mathematics" do not want to know anything about them, however, because they go back against Wittgenstein linguistic turn turn.)

Every problem-solving path is a concrete figure in this space and is referred to by Deleuze as individuation. The term individuation is intended to indicate that in the variety (as it were of the genus) of all possible solutions and errors that start from a problem, a single (individual) solution is ultimately distinguished.

At the same time, the concept of the diversity of beings is generalized to the concept of the multiple, in order to be able to describe the particular situation of a problem that is not simply a diversity, but rather something multiple from initially unfamiliar contexts. Mathematically, the term manifold, which virtual mathematics uses synonymously, is better suited.

I would like to distinguish the idea of ​​"multiple mathematics" from these stimulating thoughts. (i) On the one hand, with physics, multiplicity is grasped more concretely: multiples is not a further generalization of the concept of element in set theory (just as category theory wanted to generalize the concept of element to the topos), but refers to rotations, to multiples of movements, to things nested apart, that can be brought together by a polarization and has the property that something can be multiplied. Mathematically, this means that multiple mathematics basically describes something that contains at least two interconnected operations along the lines of addition and multiplication (generalization of the body concept). (ii) Multiple mathematics is systematically justified by a proto-mathematics, while the virtual mathematics also wants to contain the proto-terms and their paradigms and for them the question of a proto-mathematics is only a certain problem, for that with the virtual mathematics the appropriate space has to be found in which it can be represented and solved. (iii) Multiple mathematics does not just describe one process, but will in turn have to be designed in multiple ways. Multiple mathematics deliberately moves to the limit where mathematical symbols merge into mythical ones. Virtual mathematics, on the other hand, seems to want to continue along the path of previous science, replacing mythical symbols with mathematical ones.

The ideas for virtual and multiple mathematics are still in the early stages. In relation to virtual mathematics the question must be asked whether it is not inadmissible the ratio of paradeigma and eikon in their approach and thus takes a position that, from the point of view of the Neoplatonists, is only accessible to the divine spirit. Thus virtual mathematics stands in the tradition of Christian science, which assumes that man has been given the ability by God to be able to look at the whole of creation from the outside as his image.

The problem of virtual mathematics is very well met with the concept of individuation. In virtual mathematics, individuation seems to be describable from purely mathematical symbols, but it must receive its power from a principle that goes beyond it. It seems to me to be more promising to look at the space of possible solutions as a matter of principle. In principle, the critique of pure reason presented by Kant applies to this space as well.

If the suggestions of virtual mathematics are taken up, then multiple mathematics can possibly be developed as a method that operates in the space of solution paths. Multiplicity is then no longer just a generalization of multiplicity, but takes up the dynamics of the dimensions of nature, which is shown in rotations, flows and multiples. It can therefore well be that something in common emerges from both, if on the one hand virtual mathematics finds a relationship to proto-mathematics and, on the other hand, proto-mathematics achieves a better understanding of how it can act as a crystallization nucleus in the chaotic.

* * *

Deleuze suggested how the idea of ​​proto-mathematics can be taken up anew. Mathematics is not only a "royal science" bordering on divine reason, but also as a "minor mathematics" is close to disordered, chaotic movement. It can possibly act like a catalyst there, revealing internal relationships in the chaos, which at first just look like white noise. This has become a very practical problem when evaluating experimental observations, for example in particle accelerators.

"The chaos is pure many, pure disjunctive diversity, while the something is one one is, not already a unit, but rather the indefinite article which denotes any singularity. How is it going to be many to the one? A large sieve must intervene, like an elastic or shapeless membrane, like an electromagnetic field or like the container of the Timaeusto let something out of the chaos even if that something is very little different from it. In this sense, Leibniz was able to give several approximations to chaos. "(Deleuze, Falte, p. 126f)

By the sieve, Deleuze certainly means the transcendental field of his "logic of sense", which he also understands as a biological membrane (Deleuze, Logic of Sense, p. 136). It is a very clear and helpful idea to understand proto-mathematics as a sieve. The proto-mathematics should show which basic functions and precursors precedes mathematics,

* * *

The understanding of virtual and multiple mathematics presented here is based on the basic Platonic approach, in particular the notion of nous and proto-mathematics. It is intended to hold the position against which a new interpretation in the light of Aristotelian philosophy will later be measured.



Aristoteles: Physik, in: Schriften Vol. 6, Hamburg 1995 (Link)

Aristotle: About Becoming and Decaying (Link)

Alain Badiou: The Being and the Event, Berlin 2005

August Böckh: Philolaus the Pythagorean teachings, Berlin 1819

Peter Crome: Symbol and Inadequacy of Language, Munich 1970

Manuel DeLanda: Intensive Science and Virtual Philosophy, London, New York 2002

Gilles Deleuze: Logic of Sense, Frankfurt am Main 1993 [1969]

Simon Duffy (ed.): Virtual Mathematics, Manchester 2006

Benjamin Gleede: Plato and Aristotle in the Cosmology of Proklos, Tübingen 2009

Albert Görland: Aristotle and mathematics, Marburg 1899

Wolfgang Gombocz: The philosophy of late antiquity and the early Middle Ages, Munich 1997

Wilhelm Graf: From point to energy, typescript, Vienna 1952

Heinz Happ: Hyle, studies on the Aristotelian concept of matter, Berlin, New York 1971

Helmut Hansen: The Lines of the Old, Norderstedt 2009 (Link)

Georg Wilhelm Friedrich Hegel: Science of Logic, 2 vol, Frankfurt am Main 1969

Martin Heidegger: Phenomenological Interpretations of Selected Treatises by Aristotle on Ontology and Logic (1922) (GA 62), Frankfurt am Main 2005

Martin Heidegger: On the essence and concept of the physis. Aristotle, Physics B, 1 (1939),
in: ders .: Wegmarken (GA 9), Frankfurt am Main 1996

Ernst Hoffmann: Methexis and Metaxy at Platon
in: ders .: Three writings on Greek philosophy, Heidelberg 1964

Ernst Hoffmann: The prehistory of the Cusanian Coincidentia oppositorum
in: Nikolaus von Cues: About the Beryl, Leipzig 1938

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in: ders .: Philosophical and Theological Writings, Wiesbaden 2005

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Peter Manchester: The Syntax of Time, Leiden, Boston 2005

Otto Pöggeler: Dialectics and Topics
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the theses are available online without the comments, link

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German translation: Proclus Diadochus 410-485: Commentary on the first book of Euclid's "Elements", translated by Leander Schönberger, edited by Max Steck, Halle an der Saale 1945
important parts of this are taken over in: Oskar Becker: Fundamentals of Mathematics in Historical Development, Frankfurt 1975 [1954], pp. 99-105, 122-129

Pseudo-Aristotle: About indivisible lines
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Paul Ziche: Mathematical and scientific models in the philosophy of Schelling and Hegel, Stuttgart-Bad Cannstadt, 1996

Paul Ziche: Dimensions of space and the deduction of principles
in: Wolfgang Neuser, Vittorio Hösle (ed.): Logic, Mathematics and Natural Philosophy in Objective Idealism, Würzburg 2004


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