What is the relationship between metamathematics and metaphysics

Principles of a meta-mathematics according to Aristotle


- Introduction

Nowhere did Aristotle leave behind a philosophy of mathematics, let alone a coherent doctrine such as has been handed down for physics, ethics, politics or poetics. From this some have concluded that he wanted to give mathematics far less importance than Plato or that he deliberately held back on questions of mathematics due to a lack of mathematical knowledge. At least since Albert Görland's thorough overview of Aristotle and Mathematics (1899), however, the diversity of Aristotle's mathematical ideas is known. So there is in the Celestial science (De Caelo), the two Analytics with numerous examples from arithmetic and geometry and on the edge in the Soul science (De anima) large passages on mathematics. The most widely discussed are the statements in physics about the concept of the continuum. Görland was a pupil of the Neo-Kantian Paul Natorp and wanted to show that Aristotle means a step backwards compared to Plato, since in his opinion he understood mathematics psychologically based on questions of the soul's perceptual and cognitive abilities. This contradicted the prevailing philosophical current in Germany around 1890, when philosophy with the criticism of psychologism by Husserl and the Conceptual writing von Frege went exactly the other way.

How stimulating Aristotle's ideas can still be for mathematicians today was discussed in 1936 by Max Dehn in two articles about Space, time, number in Aristotle from the mathematical point of view shown. Immediately afterwards, however, he was expelled from Germany as a Jew and was no longer adequately recognized even in American exile. As a result, his works fell into oblivion and are still difficult to access today.

The dissertation published in 1977 by Hans-Joachim Waschkies fared not much better From Eudoxus to Aristotlewho was one of the few who took up Max Dehn's contribution. Waschkies has shown how well Aristotle was familiar with the most important mathematical discussions of his time and possibly tried to inspire them with his own ideas. Instead of continuing this approach, numerous philological studies were published in Oxford in the 1980s and 1990s that analyzed, very close to the text, the ideas on mathematics handed down by Aristotle (finally John Cleary Aristotle & Mathematics, 1995). Since about 2000 there has been a group of philosophers in Sydney who want to revive the direction of Aristotle. An important article by Anne Newstead about Aristotle and Modern Mathematical Theories of the Continuumin which she confronts Cantor's mathematics with the ideas of Aristotle.

But the real difficulty for me is not in a more precise text-critical analysis of the individual passages in which Aristotle writes something about "mathematical objects", but in recognizing the systematic value of mathematics in relation to physics and metaphysics. With Aristotle there is only in physics II.2 a clear indication if he or she is about to become independent (chorizein) of the mathematical figures (schemes) speaks of the forms of natural bodies (Phys. II.2, 193b31). If the size and proportions of the shape are considered, then what is found in the shape is what can be analyzed mathematically independently of the material. There is no form without material, but every form has a size, and size has a mathematical figure that can be viewed independently of the material carrier.

In terms of language, however, shape, size and figure are used almost synonymously. Often the same thing is meant when the shape, size or figure of something is mentioned. To make the confusion perfect, form is referred to as morph and times as eidos Roger that.

Likewise, number and size are not clearly separated from each other, and neither is place and size (it is often said that something is the same size as the place it occupies).

Finally, the term independence (chorizein) has not become a clearly defined technical term in philosophy to this day. Plato spoke in Symposium 192c on the separation of men and women, im Phaedo 67c of chorizein the soul from the body, in Politeia 522b of the separation of pure science from music, commercial art and gymnastics, and has im Philebos 55e Arithmetic, geometry and weighing skills separated from the other skills (after Gottfried Martin Plato's theory of ideas, GoogleBooks, pp. 165f). in the New testament becomes chorizein Used both as parting in the sense of divorce (Mt. 19,6), but also as establishing, determining ("But of the disciples each one established something as the means allowed him", Acts 11:29).

All of these meanings of chorizein only partially meet what Aristotle means:

“This is what the mathematician is concerned with now, of course not in so far as all of this is a limitation of a natural body; nor does he consider the properties in so far as they apply to them as such; therefore becomes independent (chorizein) He too, because in thinking they can be separated from the general change in things, and that makes no difference at all, and nothing wrong arises if one separates them "(Phys. II.2, 193b)

When the mathematician separates the spatial form from the form, as with the separation of the soul from the body or science in contrast to the arts, this is a process that can only take place in thought. In contrast to these examples, however, it remains in a certain way on a sensually vivid level. In their imagination, everyone can almost immediately visualize what happens when the geometric figure is separated from the shape on which it was previously seen.

This understanding of mathematical independence is to be expanded in this commentary to an overarching approach: Just as Aristotle used the three principles of substance, form and lack of form as a basis in physics, the three principles of connection, size and in-between are to be developed from this Mathematics has been made completely independent from physics and constituted as a separate teaching. What Aristotle says about the "limitation of a natural body", i.e. about an aspect of form, should be transferred to matter and a lack of form.

And this independence is to be divided into two steps. Just like Aristotle's shape, size (megethos) and figure (scheme), a distinction should be made according to substance, context and dimension or lack of form, between and interchangeability.

This is sure to be unfamiliar and may seem arbitrary. Its justification will prove to be true if the fundamentals of mathematics are developed step by step in this sense. In order to distinguish the principles of size, relationship and between from mathematics, they are understood as principles of meta-mathematics, just as Aristotle distinguished between metaphysics and physics. This gives the overview:

materialContext (synecheia, Continuum)Dimension (slide topic, Expanse, potency)
shapeSize (megethos)Figure (scheme)
Form defect (steresis)Between (metaxy)Indistinguishability (interchangeability)

Dimension (expanse) and indistinguishability (interchangeability) are to be explained as equal principles of mathematics alongside the figure.

This novel approach means: In a figurative sense, the subject of mathematics is asked and the interplay (enantiodromia) of size and in-between (lack of size) in meta-mathematics or of figure and indistinguishability (lack of shape) in mathematics, as there is the interplay between form and lack of form in physics. Dimension, expanse, is understood as the subject of mathematics. (Schelling meets this understanding with his theory of powers.) Dimensions should not simply be understood as equal axes in the coordinate system, in three-dimensional space or generalized in the infinite-dimensional Hilbert space, but each dimension contains its own contribution to the subject of mathematics. With Aristotle, this understanding can be found especially in the Celestial science (De Caelo) Find suggestions. There he systematically uses the term expanse (slide topic). With a sure instinct, Görland has, contrary to tradition, place and distance (thesis, slide topic) and not size or figure (megethos, scheme) at the beginning of his investigation. He describes the distance in its Platonic terminology as the "concept of method" (Görland, p. 14).

These three principles lead to the "mother structures" of mathematics developed by Bourbaki: topology (theory of the continuum), lattice theory (theory of sizes), algebra (rules of interchangeability). It will be investigated in what way the category theory developed by mathematics has to do with the Aristotelian theory of categories.

The interplay of scheme and indistinguishability continues the interplay of form and lack of form. Schemas are the classic mathematical numbers and geometric figures. The interplay of scheme and indistinguishability using the example of numbers: All numbers lose a certain degree of distinguishability when they become independent from the form. Mathematicians put three dots ›...‹ at this point. With the introduction of the natural numbers, counting one, two, three begins until it ends in the indistinguishable ›...‹ or ›n‹. In a similar way, each new number class creates its own indistinguishability, most spectacularly the complex numbers with the overlapping of the Riemann number planes, but also the periodic fractions (1/3 = 0.333 ...), the irrational numbers (√2 ​​= 1.414213562 ...). Every mathematical system of axioms fixes a certain state in which number and indistinguishability can be brought together, and at the same time contains an inner explosive force that pushes for new numbers with a different kind of distinguishability. In other words: no system of axioms can avoid that a certain kind of indistinguishability is contained, out of which tasks arise which are insoluble within the given axiomatics.

Undoubtedly, the principle of interchangeability is far removed from today's discussion of the philosophical foundations of mathematics. Therefore, in another article it will be shown how precisely this principle leads to a new understanding of mathematics.

After all, there are two philosophers in modern French philosophy, Badiou and Deleuze, from whom ideas in this direction originated. Badiou goes to P.J. Cohen follows newly developed methods of axiomatic set theory and works out the meaning of the indistinguishable. He even gives it its own symbol, the sign of Venus. »The basic situation is part of every generic extension and the indistinguishable (Venus symbol ) is always one of its elements «(Badiou, p. 428). - Deleuze has a "small mathematics" (minor mathematics) compared to "Royal Mathematics" (royal mathematics) spoken of the axioms that nomadically transcends all the limits of mathematics. Here should the minor mathematics can be explained from the interplay of number and indistinguishability. In every axiom system (royal mathematics) something necessarily remains open and indefinite, making it susceptible to infiltration (minor mathematics) becomes.

There is something like a zero state for each of the three principles of mathematics: the dimensionless point, the empty quantity one, the absolute singularity (the "black hole") when producing a complete loss of information through unlimited entropy. The paradoxes of set theory can be developed from this. In contrast to Badiou, I therefore do not see the empty set as the principle of a new mathematics, but, conversely, its concept and significance can be explained from the three principles of mathematics mentioned.

This gives the second overview:

mathematicsMother structureZero state
Figure (scheme)Association theoryone
InterchangeabilityalgebraAbsolute indifference


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