What are the qualities of math

Qualities and atoms

 

Basically, all of math is just symbols. The numbers and then also "placeholders" such as lower case letters, Greek letters and some special special characters should tell what is being used in the calculation. Signs such as +, -, √ (root), ∫ (integral) and so on determine how the calculation should be carried out. While the number signs stand for any numbered objects, i.e. represent reality in mathematics in the most general way, this is not so obvious with the arithmetic symbols. Rather, the question arises as to whether they arose out of convention, or whether they did not ultimately also refer to objects, the idea of ​​which is linked to the possibility of the respective arithmetic operations.

In other words: is it possible to imagine what is meant by '+' without thinking of any concrete objects that can be added? And then can't the '+' sign, similar to the number signs, create a reference to these objects? The sign '+' describes a common property of all objects that can be added. Can it not then be seen as a symbol for all these objects? - And which objects cannot be added: These are always objects that describe boundaries, be it the sky, the point of the Big Bang or the mathematical symbols for infinity. Every philosophy of mathematics quickly encounters these boundary objects to which the sign '+' is not applicable. The sign '+' can be clearly interpreted as a symbol of everything that is open to the outside world.

But normally nobody thinks about such questions in everyday arithmetic. If anything, it is argued that numerals and arithmetic symbols should ensure uniqueness and simplicity in arithmetic. There are 10 different digits because arithmetic has become established in the decimal system. The use of letters takes place with a view to better clarity (if e.g. the first letters like 'a' and 'b' are used to represent simple axioms like 'a + b = b + a', the last letters like 'x', 'y' and 'z' stand for unknowns in equations and 'i', 'j', 'k' for the indices in number sequences and matrix elements, e.g. 'ai',' aij'.).

Apparently there are two different basic ideas about the meaning of mathematical symbols. This becomes clearest with numbers: If symbols are understood as signs of qualities, then number symbols also have a qualitative meaning (number mysticism). The Pythagoreans assumed that the natural harmonies are symbolized by numbers. Each number has an individual character. The normative understanding, on the other hand, is exactly the opposite of understanding the numbers as abstractions of all qualitative properties and logically defining them without contradiction and avoiding the threat of circular conclusions, in order to be able to explain and secure their universal applicability in this way.

quality: If mathematical symbols are to be descriptions of qualities in nature, there must be something in nature that can be symbolized. These can be absolute dimensions, harmonies and patterns that create an inner unity of nature. One of the most vivid examples is the circle. In the most varied of sciences, circles are observed in nature (as circular disks, trajectories, cyclical movements, etc.). In algebra, its unique number measure is symbolized by π, an empirical quantity in mathematics, so to speak. Mathematical laws, in which π is included as a quantity, must agree with the proportions in real circles.

This approach can be generalized in such a way that mathematical laws reproduce natural properties. This basic idea is to be applied consistently except for the number concept. Even if it sounds paradoxical at first sight, it can be said that when a set is counted, the number of elements must correspond to the nature of the set. In other words: If, for example, a set can be broken down into 4 subsets, the number 4 of the set is by no means external. Rather, it shows an essential property of this set (the quaternity). As a result, it has an inner relationship to all other sets, which are also divided into 4 subsets. An arbitrary cut of a set into 4 subsets is not permitted. Each set must be examined for itself whether this division corresponds to its quality, whether it has the property of quaternity.

On the other hand, the first look, according to which the number of elements in most sets is completely random and arbitrary, such as the number of buttons on a shirt, the number of bicycles parked at a train station, etc. The qualitative understanding of mathematics goes against it on the principle that even with such amounts the numbers are never really arbitrary. The number of buttons is also selected on the basis of practical, cost-effective or aesthetic considerations. The number of bicycles parked is based on certain regularities, so that the bicycles are parked in such a way that they are easily accessible.

The qualitative understanding is of course not based on such everyday experiences, but on qualitative designs of the world. It is precisely here that it can prove itself, as, conversely, it attracts the strongest reproaches that this is nothing but speculation, arbitrariness and irrationalism. For example, the treatise "De Occulta Philosophia" by Agrippa von Nettesheim, published in 1531, contains separate chapters for all numbers up to 12 on their properties and their "ladder". In the ladder of the number 4, the 4 archangels, the 4 evangelists, the 4 elements, the 4 seasons, the 4 human temperaments, the 4 rivers of the underworld are listed. It is clear that the division into 4 areas is part of the qualitative property of the groups mentioned. Quaternity (and similarly duality, trinity, etc.) is by no means just a numerical value, but a property whose essence extends deep into religious and psychological considerations.

More mathematically speaking, a set can only be divided according to its symmetry. It may only be divided in such a way that the parts are similar to one another and that they repeat themselves periodically. An even star with 5 points can be cut in half, divided into 5 parts, but not into 3 parts, 4 parts etc. In linguistic terms it can perhaps be said: A set is split up according to its internal symmetry, otherwise it is cut up.

Counting must not be limited to what is immediately there, but must also feel when something is missing. This is the extreme criticism of the usual set theory: A set includes not only "that which is there", but possibly also "that which is missing", which belongs to it but is absent, repressed or hidden for some reason. The lost sheep is also part of the flock of sheep. Counting is a far more complex process than attendance checks. If, on the other hand, it is objected that such a counting process can become endless and will never lead to a conclusion, this only shows the lack of understanding for the qualitative concept of number.

For the qualitative understanding of the numbers, all questions remain interesting that have been explicitly excluded since Frege's work. For example, whether certain numbers can be assigned certain colors (for example: "which number is blue" or vice versa: "all blue objects also have related number properties").

And it is not just about the individual properties of the individual numbers, but between certain numbers there can be highlighted "elective affinities" that go beyond their arrangement on the number line. Since this has not yet been studied systematically, it should only be indicated here intuitively in terms of properties that are suggested by the shape of the numerals. What is meant here is not an internal relationship that exists between the numbers 10, 100, 1000, ..., i.e. the powers of ten. But if, for example, the sign '4' with the cross shows how something can be divided evenly into 4 areas and in this sense is similar to the '+' sign. This cannot be a coincidence for this understanding of the symbols. Apparently there is an inner relationship between the properties of quaternity ('4') and addability (i.e. objects that are not boundary objects).

And intuitively it is suggested that the '8' in the range of the smallest, finite numbers represents the infinity ∞. (For normal understanding this is of course an open contradiction: "How can the '8' be infinite when the '9' is already larger?" However, it will be shown that ultimately all mathematics is based on contradictions of this type .) Both symbols are composed of two touching circles, with the '8' completing the open '3', and thus show the limit of the qualitative number range opposite the '0', which in this view consists of the numbers from 1 to 7 and sets these numbers apart from the other natural numbers. (That is also the answer to Frege, who held up to the qualitative understanding of the numbers, what sensual meaning a number such as '753684' could have, whereby interestingly, in this arbitrary sequence of digits, on the one hand the 0, 1 and 2 and on the other side left out 9.) In many theories, the numbers 0 to 7 (or written in binary form 000 to 111) are the natural basis and 8, the third power of 2 (8 = 2 * 2 * 2 = 23), as the number of perfection.

The example of the characters '0', '3', '8' and ∞ shows how certain mathematical symbols such as constellations can be arranged in constellations within the apparently even sequence of numbers. These signs belong together through an inner relationship that is obviously at odds with their arrangement in the number line. To understand the '3' and the '8', it is not enough to characterize both by the common property of all natural numbers that they can be reached by taking equal steps from the '1' (i.e. 3 = 1 + 1 + 1 and 8 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1). It is conceivable that there are additional qualitative structures within the natural numbers that differ from the mere predecessor-successor relationship and are also not identical to the number classes defined by arithmetic calculation (e.g. all numbers divisible by 3).

standard: Much more common, however, is the opposite view that it is completely superfluous, if not leading away from the real goals of mathematics, to "speculate" about such qualitative relationships. Rather, mathematical thinking should be freed from such considerations and concentrate entirely on the further development and improvement of computing techniques. According to this understanding, symbols are used exclusively for an abbreviated and at the same time meaningful spelling of mathematical calculations and thus serve as an optimal means of communication for unambiguous scientific communication. Consciously independent of each quality, matching quantities are identified. The numbers are the first and most important example: 1, 2, 3, ... symbolize the number and should initially not say anything about the quality of things, except that they can be counted. Accordingly, symbols for equations, logical conclusions (e.g. =, ⇒), functions, etc. are set later, with which the proportions of any quality are to be represented and calculated. Further examples: y = f (x), (y is the function value of x), df (df is the derived function of f), TM (TM is the tangent space of the manifold M).

The normative understanding of symbols is based on the assumption that the question of an internal order of things is in principle unanswerable for human knowledge and that the order through symbols can relate exclusively to the natural science created by man. Nature is only assumed to be quantifiable. In principle, only passive properties are assigned to nature that ensure that it can serve as an object for human activity at all.

The normative understanding of symbols corresponds to an atomistic worldview. Ultimately, nature is thought of as the sum of simple atoms that are nothing more than the carriers of quantities. (In the picture of physics: charge carrier with elementary charge. Higher charges only occur in quantized form, i.e. as whole multiples of the elementary charge.)

This picture of nature is continued in the corresponding epistemology: Knowledge is the perception of individual, recordable facts in nature ("that which is the case", as Wittgenstein so vividly put it). The facts are the atoms of knowledge. Science orders and structures facts, recognizes their patterns and laws.

In this sense, mathematics is consequently built up atomically. For Euclidean geometry the points and the two elementary figures (straight line and circle) are the elementary units, for arithmetic they are one. Everything else can be constructed from them purely quantitatively. There is nothing beyond that.

A number 'n' is precisely defined in that it can be reached by taking 'n' simple steps from one:

n = 1 + 1 + ... + 1

This supports the inductive approach: With atomism it is set that everything is simple and equal to one another. What is shown for individual atoms therefore applies to all atoms and does not change when several identical atoms are added together. Only the quantity changes. Qualitatively, the numbers '1' and '1 + 1' and '1 + 1 + 1' and in general 'n' and 'n + 1' are identical to one another. Therefore the induction axiom was established for the natural numbers. (If a property holds for the '1' and it can be proven that it also holds for 'n + 1', if it holds for 'n', then it holds for all numbers.) The induction axiom will prove to be the actual basis and possibly prove to be the critical limit of today's dominant mathematics.

In the following, the history of mathematical symbols will be examined. Originally, the dominant idea was that mathematical symbols are the direct expression of qualities in nature, and that they are directly identical to them. When Euclidean geometry drew straight lines and circles, these were not regarded as symbols for "the straight line" or "the perfectly round", but were directly "the straight line" and "the completely round". Mathematical symbols were therefore not perceived as symbols at all. Depending on the worldview, mathematics was either natural science or humanities, in that its objects were understood as natural objects (comparable to physics) or as ideas of the spirit (e.g. with Plato).

The path to symbolic mathematics initially led to the opposite extreme. The quality side was completely disputed or at least took a back seat. Those were the centuries of the Enlightenment, in which the normative understanding of symbols prevailed. With the triumphant advance of the inductive method in modern natural science, empiricism has prevailed across the board. Only facts and techniques count, and everything that reminds of qualities or ideas is considered superficial talk. Although Aristotle had rejected the inductive method and Francis Bacon drew a clear dividing line with the "Novum Organum" at the beginning of the modern era, this attitude is considered Aristotelianism. Because he was the first critic of Plato and empirical natural science was founded in his school. And then it was Aristotelian thinking, coming from the Arab region, which attacked the rigid dogmatics of the Middle Ages for the first time and was experienced as liberation and opening to the world.

Since 1800, mathematics has been considered a purely quantitative science more than ever. After the development of calculation and proof techniques had stopped for centuries, the symbols were only accepted in their normalizing function in order to support the rapid development of new calculation methods. Qualitative properties and relationships between the numbers were of no interest as long as the calculation methods could be continuously improved. The occupation with mathematical symbols was therefore limited entirely to the arithmetic symbols. Starting with the symbols for the integral calculus and limit crossings, innovations are constantly being introduced for the increasingly complex technical apparatus (such as symbols for multidimensional integrals, tensors, Lie algebras). The analysis of the number symbols was relegated to pure logic, which since Frege has endeavored to derive the concept of number from a suitable concept of logic. When that in turn led to more and more technical logic calculations and the impression was never to be shaken off of getting into circularity dead ends, it was just one more proof for mathematics of how senseless it is to dwell on such questions.

The emancipation from Greek mathematics was only half successful.It is true that all notions of qualitative properties or number harmonies (which express natural harmonies or aesthetic harmonies) were "cleared up" and were therefore considered to have been overcome. But the claim of Greek mathematics to prove its scientific nature by being based on a transparent system of axioms was adopted and made the actual "renaissance" against all "dark" medieval thinking, which of course included texts like the "Occulta philosophia" by Agrippa. The textbook of Euclid with its exemplary axiomatics was next to the Bible the book with the highest circulation and the greatest impact. And no matter how much philosophy insisted on its prerogative of being able to make qualitative statements over mathematics, at the same time it admired mathematics for its system of axioms and tried to imitate the high scientific standard of mathematics with a comparable logical rigor "more geometrico" .

It was therefore all the more bitter for mathematics to finally have to give up all hope of an extended or more fundamentally set axiom system for the newly acquired symbols that would have complied with the axioms of Euclidean geometry and derived them as an application. From 1830 at the latest, when the preliminary character of Euclidean geometry became conscious, this problem came to the fore. For almost 100 years until 1920, new axiom systems were constantly being drafted and had to be dropped. It was not until the fundamental crisis of mathematics in the 20th century that an abrupt end point was set. What the problem of this crisis actually was is not asked within mathematics. In mathematics, there is a clear description of the axioms and formalisms involved in the fundamental crisis, but no agreement on what could lead to the fundamental crisis. Here it is interpreted as a general crisis of the natural science which appeals to Aristotle, that is, as a crisis of a natural science for which there are no leaps in nature or in one's own thinking.

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