How do I explain Russell's paradox
Russell's Antinomy and the Paradoxes of Thought
Contribution to the Philosophical Colloquium of the Akademie 55plus Darmstadt on July 4, 2016, extended version
Russell's antinomy has occupied mathematics, logic, and philosophy since its publication in 1903. Since then, the trust in the security of logic has been shaken from the ground up, with effects on all sciences and arts, including ethics and questions of faith, which had also relied on the irrevocability of scholastic logic. Russell belonged to a larger group of intellectuals in England who came from a wide variety of backgrounds and shared the feeling that they were at the dawn of a new era. These were the Cambridge Apostles and the Bloomsbury Circle, including T.S. Eliot, Eric Hobsbawm, Aldous Huxley, John Maynard Keynes, John McTaggart, Katherine Mansfield, Vita Sackville-West, Ludwig Wittgenstein, Virginia Woolf. Is there a ground in sight from which thinking can reinvent itself? Can it lie elsewhere than again in logic? A look at the previously known antinomies and their relationships to one another should show how Russell's antinomy comes about and how it is related. It may provide a new perspective for the further development of logic.
What Russell means is easiest to understand in a newer version: »All mayors are not allowed to live in their own town, but have to move to the mayor town of Bümstädt, which has been set up especially for this purpose. Where does the mayor of Bümstädt now live? «(Duden useless language knowledge 2012, quoted from Wikipedia, here as accessed on June 5, 2016). For Ulrich Blau paradoxes of this kind occur in all cultures. He freely quotes the Buddhist teacher Nagarjuna, who lived in the 2nd century: “Buddha never spoke a true word. Anyone who understands my words as a theory is incurable. "(Blau, p. 445)
One of the most stimulating works on this topic for me is the book published in 1984 The productivity of the antinomy by Thomas Kesselring (hereinafter cited as K). He draws a connection to the developmental psychology of Jean Piaget. Piaget lived from 1896 to 1980, was a biologist by nature and worked in Geneva. His work was groundbreaking for mathematics education. He tried to show the age at which children and young people become receptive to mathematical ideas. Instead of confronting them too early with material that they have not yet been able to absorb, the respective skills should be promoted at every age group, something which mathematics lessons are still far from today. That would allow a clearer view of mathematics and, above that, logic and philosophy. He hadn't read Hegel, but is said to have shown great interest when Kesselring spoke to him about his plan shortly before his death. Hegel phenomenology of the Spirit is based on a comparable development model.
Kesselring advocates the thesis: The four stages of the development of thought distinguished by Piaget each come up against an antinomy that can only be resolved at the next stage. With Russell's antinomy the fourth stage of formal thinking is reached and with it - according to Kesselring - also dialectics. In the following, the 4 phases and their antinomies are presented until it becomes clear that Russell's antinomy is followed by other antinomies: the liar's paradox (›everything I say is wrong‹), the parable of Achilles and the turtle and the Cantor's antinomy. They can no longer be solved in the four stages distinguished by Piaget, but only in subsequent stages, in which thought develops a concept of infinity. In terms of content, I am referring to the two concepts of infinity introduced by Cantor, the transfinite numbers and the continuum, in Levels V and VI, as well as to those of Hegel in his Science of logic executed antinomies of infinity. Finally, there is an outlook on the further development of the logic.
Russell's antinomy has become known as the barber's paradox: "A barber can be defined as one who shaves all those and only those who do not shave themselves" (Wikipedia). It is not immediately obvious that this definition is contradicting itself. Where is there a contradiction when considering the set of all barbers who do not bar themselves? The crux of the matter lies in the tightening "which shaves all those and only those". The mentioned Wikipedia entry explains: “Assuming the barber shaves himself, then he is one of those whom he does not shave according to the definition, which contradicts the assumption. Assuming the opposite is true and the barber does not shave himself, then he himself fulfills the quality of those he shaves, contrary to the assumption. "
The first four stages of Piaget's cognitive development
Piaget does not speak of contradiction or antinomy, but of resistance to leaving the respective stage. At each level, thinking follows a certain pattern. If this schema for its part becomes the content, and if the schema is used to answer the question of when the schema does not apply, this leads to an antinomy. The child has got used to the scheme and wants to stick to it. This leads to a development crisis that is resolved with the next stage.
I: Sensorimotor, age: 0-2 years, infancy
Nobody can remember what happened in the first two years of life. It falls under infantile amnesia and works on a level below consciousness. Immediately after birth, the baby reacts to its environment with innate reflexes. These are motor processes that each follow a specific behavioral pattern (scheme). Examples: sucking and swallowing reflex, grasping reflex, reflex-like focusing of the eyes on a highlighted object. - Reflexes of this kind can be seen through later, but cannot be avoided. The Libet experiment demonstrates how adults react spontaneously to certain stimuli before they become aware of the stimulus and the response to it. Far-reaching conclusions have been drawn from this about the limits of human free will and delusions of consciousness. For me they only show the area in which the schemes of level I remain valid for life.
Originally, perception selects from all stimuli those for which there is an innate reaction scheme. "It is estimated that a person per second takes in 10,000 exteroceptive and proprioceptive sensory perceptions." (Watzlawick 1967, quoted K, p. 385, see definition of terms exteroception and propioception) Most of this is not consciously perceived and does not trigger any reaction. Leibniz describes the totality of all sensual stimuli as small perceptions, of which only a part penetrates into consciousness. Nomadic thinking is the ability to still be able to react to stimuli of this kind below the level of consciousness. Metzinger suspects that there is a second higher-order process that follows the process of selecting the stimuli and hides the awareness of how much has been filtered out (Metzinger The ego tunnel, Munich, Zurich 2014, p. 71f).
At this early stage, the infant can only distinguish between two situations: abundance or deficiency. If anything is missing, he is completely helpless. His bottomless despair can deceive others who do not understand what exactly is wrong with him. And vice versa: when all needs are met, he is timelessly happy and cannot imagine that it could ever be otherwise. He only feels how the state changes: something disappears, something comes. When something disappears, he cannot imagine that it will ever come back, and when something is there it seems to be there forever.
In this earliest development, the child gradually learns to differentiate between certain sensory images and, for example, can recognize a human face. But here, too, the following applies: If something disappears from the field of vision, it is gone for good. When the sun goes down it will never come again, when it shines it will never go down. Great the picture in the dusk von Baudelaire: As a rule, the "exhausted souls" look forward to the evening, "oh refreshing darkness! For me you are the announcement of an inner festival ", but there are mentally ill people in an asylum who then scream loudly and" consider the coming of night to be the announcement of a witches' Sabbath "(Baudelaire, Parisian whimsy, XXII). They still behave as the infant does when it loses fullness. In this first phase the pattern of all antinomies can already be recognized logically: the disappearance of the disappearance. When something disappears, it is gone and no longer disappearing. If it doesn't go away, it's not going away anymore.
In the course of further development, individual situations can gradually be more clearly differentiated from one another. The child must learn that their innate reaction behavior does not always lead to the expected satisfaction of needs. Example: He sucks on everything that corresponds to the sensory image of the mother's breast and experiences that sometimes there is no nourishment. The child learns disappointment and joy. From this the first antinomy arises for Kesselring: "It is the way it is not." (K, p.192) Something seems to be a source of food, but it is not.
The child learns that what can be seen in the sensory image may only be an appearance. The child has the experience that pictures can be deceiving, but pictures can see is not an appearance. What is the constancy of vision? The child will find an answer to this at the next stage of development.
The child has to learn to distinguish between appearances and non-appearances. It looks for new identifying features. It experiences situations for which no scheme is yet available. These are no longer the stimuli that are filtered out because there is no innate answer to them, but the stimuli with which the child is no longer sure whether they are sham or not. The child can find himself in situations of indifference in which he cannot react schematically. Which scheme then prevails? Is it spontaneously cautious, defensive, fearful, open, attentive, depressed? That depends on both his personal characteristics and the support of his environment. I suspect that his specific habitus develops in these situations. Every person can be recognized by their habitus, their specific and at the same time schematized gestures and behavioral patterns, and they experience their habitus from within as the shadow over which they are unable to jump.
If the child begins to exceed this level, despite all previous experience of their own helplessness, it is experienced as a loss of falling out of the security of a paradise and is associated with corresponding resistance. Piaget sees substitute gratifications appear here for the first time when the child learns - usually at the age of 4 to 5 years - to make use of disappointments and, for example, get used to reacting to all uncomfortable situations by sucking their thumbs or biting their nails. - The habitus arises from a deficiency (a disappointment), but it does not have to lead to a regression, but conversely, all positive properties of thinking such as intentionality, spirituality, openness etc. can only emerge from a habitus when the first level is exceeded .
II: Preoperational (symbolic or pre-conceptual), age: 2 - 7 years, kindergarten and preschool
With the experience of disappointment and joy, the child learns the difference between objects and symbols. It learns that not everything is a lactating mother's breast that feels that way and that it can suck on, but that the mother's breast is a certain object that is different from other objects as well as from the appearance (the picture) of a mother's breast be. The child learns in a uniform process how the individual objects can be recognized and differentiated by their respective properties, and how they can name them linguistically. Language acquisition takes place at this age. The linguistic symbols are different from what they say. They are of a different nature from what they designate. The word "yellow" is not yellow. The word "evening star" is not the evening star. The first understanding of concepts, judgments and numbers also falls in this phase. (K, p. 201) A number is not what is counted with it, a judgment is not what is said with it.
The previous antinomy is resolved: The appearance has permanence because, unlike the content it represents, it is a symbol that differs from its content. The content can come and go, but the icon remains. This insight is the basis of the earliest philosophical systems (see Plato's realm of ideas which, unlike what they represent, are not transient, and Hegel on the emergence of language in the first stage of development phenomenology of the Spirit, the sensual certainty).
Strangely enough, it is much more difficult to go back to this phase than to the previous phase of stimuli and sensations. At this stage the child does not yet learn in a systematic way as it does later in school. Children of this age like to hide and dress up and are enthusiastic about the circus and fairytale characters like Struwwelpeter. The III. Phase later becomes so dominant that everything that happened before is looked down on. At preschool age, the child is in a phase of magical thinking, as Piaget wanted to show with many exercises.
Transfer task: If a liquid is poured from a wide pot into a narrower pot and the level is higher there, the child believes that the amount has increased, although he has seen that nothing has been added. He can be shown how the same liquid can be poured back, the experiment can be repeated any number of times, but the child will remain convinced that the amount will change. - If the pearls are unthreaded from a pearl necklace and placed in a glass, they have become new pearls for the child. If the sister wears an unfamiliar dress, she is someone else until she dresses as usual again and is the old one.
This thinking is considered magical because there is a hidden transformative power for the child: when the properties of things change, things also change. The child must learn to distinguish which changes in an object are due to internal changes and which are due to changes in the environment that leave the object itself unchanged.
No substitute: If at this age a child is shown three balls with the list "one, two, three" and the child is then asked to give three balls, there are not all three balls, but the third, the one in the list had been labeled "three".
Inability to see something from a different perspective. Is that your brother? Yes. Does he have a brother too? No.
It is difficult to put into words the difficulty and antinomy that arises at this stage. The child learns that some things can change and some things cannot. Fort-there game: the ball can fall to the ground, and still remains a ball. Its place has changed, but its essence as a ball has not changed.
In retrospect, this phase may seem like an imperfection, and those who hold on to it or occasionally fall back on it are considered messed up. But it has its own magic and poetry. The writers are sometimes better at remembering it than the philosophers and pedagogues. That wasn't just the romantics' concern. It is also to be thought of transformation from Kafka or to Musil. He describes a man who is confused in his feelings towards a woman. This carries over to his perception of the everyday things in the room he is currently in, the wallpaper, the doors, the velvet chair. “It was true of them that the belief in them had to be there earlier than they themselves; if one does not look at the world with the eyes of the world and has it already in view, it breaks down into senseless details. «(Robert Musil Three women, quoted in Kulenkampff, p. 9 fn. *, text available from Projekt Gutenberg). But nobody hit the mood of this phase better for me than Katherine Mansfield (1888-1923), who had known Russell well since 1916 and who recalls her childhood in her stories.
The underlying question is still controversial in philosophy today. For Kant that which remains unchangeably as a point of reference for all changed appearances is the "thing in itself." It even precedes the appearances in space and time and, when viewed impartially, has a magical quality, as can best be seen from the perspective of the child in the second stage.Just one example: if the mother drives away for a long time, she gives the child a symbol (a fetish) so that the child can continue to sense her presence, be it a doll, a photo or another memento. As long as the child can see how the fetish is "doing well," he is reassured, but if it breaks or is lost, it is desperate, as if the mother had died. The fetish is only an appearance of the mother, but the child believes in it, and the "real" mother and the appearance are based on something that Kant can call a "thing in itself," outside of space and time. With the experience of the III. Level no one will want to see it that way anymore, and everyone will only have to admit to themselves with an honest view how often magic is still trusted and used in adult life. Psychologists like C.G. Jung described this phenomenon as synchronicity and used it in their practice, for example in the treatment of the physicist Wolfgang Pauli, who was in a psychological crisis at the time of his groundbreaking physical knowledge.
For this, Hegel found the abstract sounding concept of "other in himself" in a new, astonishing phrase. This means that, for example, a flower growing in a barren environment can still bloom naturally, even if it is not able to do so in the given adverse circumstances. In a grand thought, Hegel understands idealism to be the ability to recognize in a person or in a thing that which they potentially contain in themselves according to their nature or inner determination, even if they cannot actually realize it. This is the basic idea of humanity, also in people who in their misery give a sad form, to see their true form and to be able to give them the affection they need to break away from it.
The child learns to distinguish what something is as it is and what it can be according to its destiny. This leads to an antinomy, if it is negated in turn: Is there something whose purpose it is not to be what it is? This can be understood as a purely logical antinomy: the concept of contradiction. The contradiction is destined not to be what it is. The determination to prove a contradiction is to resolve the contradiction. It is the determination of the contradiction to perish, i.e. to show the way (the way out), which reason the contradiction is based on and which in turn is no longer a contradiction. This is almost inevitably associated with moral values. At this age, the child not only learns to differentiate between objects and to talk about them, but also begins to perceive how different people behave differently, how the mother or father in certain situations no longer behaves as expected of the child and how they are seen by other people in a completely different way than by their children. The child feels how it affects other people differently and triggers sympathy in some and aggression in others. It gets into double-bind situations when faced with conflicting expectations, and it learns to play other people off against each other. With the question of the something whose purpose it is not to be what it is, the question of evil is posed. Is there something whose essence it is to destroy other things until it destroys itself in negative self-reference? How do the negation of negation and the antinomy differ from this, which also negate something, but do not annihilate themselves in the negative self-referentiality, but are canceled in a higher level? The answer will lie in the concept of the border, which does not lock something in a cage and starve there, but in the essence of which lies the crossing of the border.
As in the first phase, exceeding the second phase is associated with separation fears and resistance. When the child learns to see through the disguises (Santa Claus) and the fetish as a deception, it is suddenly confronted with the harshness of the adult world and the unbearable power of the real and experiences this as disillusionment and disenchantment. Inside remains the secret wish: I never want to grow up.
Are there possibilities to cancel something from the II. Phase without making the transition to the III. Block phase? In the 1920s, art movements such as magical realism and surrealism emerged, which have made this their business and, particularly in literature, have continued to the present day. Effects on logic were almost completely absent, apart from a few exceptions in the context of Ronald Laing or Jacques Lacan, which, however, hardly got beyond the first approaches. Abolishing magical thinking is no less important than nomadic thinking rooted in Stage I. Those with a one-sided dominance of the III. The catastrophes connected with the fourth and fourth stage clearly demonstrate this.
III: Operational, age: 7-12 years, elementary school age
With school enrollment, the child begins to learn systematically, to acquire knowledge and, to a certain extent, to put the previous difficult questions aside (not to say: to suppress them). Freud speaks of latency in another context. The child gets to know objects and facts in all areas and to assign their properties to them. It acquires factual knowledge to a great extent. It understands the relationships and their order. It is enthusiastic about a world in which everything is understandable and has its clear place. It is receptive to new ideas and experiences like never before in life. It's from an original democracy of all people and things convinced: Compared to the truth, everything is of equal value in a positive sense and of the same validity. It feels secure in a belief in a higher being who has arranged everything in such a way that it merges into harmony.
The previous antinomy is resolved: the objects and their names, the counted and the numbers are not of the same nature and cannot be confused because it is possible to organize them uniformly in an overarching system without contradictions. The objects and the names, the individuals and the properties, the counted and the numbers are each assigned their own predicates, whereby they can be distinguished from one another and do not come into conflict with one another. The predication function (recognizing and assigning predicates to subjects in judgments of the form ›S is p‹) is the general principle. If inconsistencies arise, they can be seen through as deceptions, tricks or careless mistakes and can be easily rectified. Those who cannot know or learn everything on their own ask the others and come to an understanding with them. This is the world of the Enlightenment, of the empiricism of Aristotle and of English experimental philosophy, of the French encyclopedists, of logicians like Frege or Carnap, of analytical philosophy and American optimism, of social democracy and of trust in a communal communicative reason in Habermas. Who is not lucky to have taken part in this project at least once in their life?
If it weren't for the recurring doubts! That begins with Descartes, whether not possibly everything is a delusion and culminates in paradoxes in Russell. Frege declared his project a failure when he found out about it. Piaget names typical paradoxical statements from children of this age who do not perceive their own inner contradiction and say with honest conviction:
"I don't like onions, and I'm glad about them, because if I wanted them, I would keep eating onions when I absolutely don't like onions."
"I am not proud, because I believe that I am not half as intelligent as I am in reality." (Quoted from K, p. 244)
Sentences like this have to be read twice to understand. Anyone who experiences being able to form sentences of this kind for the first time in their life feels seized by an oceanic feeling of being in complete freedom above themselves and the world and of participating in overarching wisdom. The greater the disappointment and deep uncertainty when others (have to) laugh about it. A world collapses and there is a risk of becoming a cynic.
What is happening here? The previous stage came up with the question of how a thing's natural determination can be recognized, even if it is currently not realized. This antinomy can be resolved if two different properties can be distinguished in a thing: On the one hand, it is such as it has emerged under the pressure of external influences. The nature of something shows the traces of how it has been promoted and oppressed. In an abstract thought, Hegel can say that the quality is "the side open to the other" (Hegel, WL A, p. 70). Because things are not theoretical constructs, but rather emerged from a physical process of conception, birth and development, they have a physical quality that gives them stability and vulnerability. Hegel distinguishes the particular quality of something, as it has developed in its concrete environment, from the concept of quality: When it is said in a paradox-sounding way that something is made to be made in such a way that it has the property of quality This means that it has the property of being able to be shaped by others in its nature. It is changeable to the extent that it is subject to change. - At the same time, however, every something in itself does not contain its determination as a transcendent thing that lies somewhere far outside in a distant hereafter, but it is made in such a way that it can change according to its determination. Its quality includes inner restlessness and the ability to become what it is not yet, but should be according to its purpose. »The other determination is the unrest of something in its limit, in which it is immanent Contradiction to be who sends it beyond himself. "(Hegel Werke, HW 5.138) The determination cannot be taught from outside (" be as you are not "," finally be yourself "). It is only possible on the path of a new antinomy: "First of all, it is the quality that changes in such a way that it only becomes a different quality" (Hegel, Logic, 1st edition, p. 71). The quality is like this to repent yourself and become a different quality.
This antinomy can be more precisely understood as the contradiction of the inner limit. If something in its constitution contains both that which is exposed to external influences and that which it sends beyond its current state, then there must be an inner boundary between the two. For this, Hegel found the appropriate formulation of "Stopping another on him«. (HW 5.135) This can be understood in two ways: If the other influences the something and possibly presses it, there is still an inner limit in the something at which the influence becomes impotent. And at the same time the something contains its purpose at so that it ends in him and is there as potential. Whoever wants to become what he already is according to his destiny does not have to become someone else, but rather cross the inner limit that has hitherto prevented him from doing it.
With this distinction, however, the concept of order of every system dissolves. In a system, an element can be divided into two types if it has a property that occurs in two or more different forms. If there are red and yellow roses, then there are species that are all yellow and red, respectively, within their species. Here now the quality of something appears in two expressions, which are separated by a boundary: on the one hand the quality how something has become according to its biography in its uniqueness, and on the other hand the quality whereby it belongs to the next higher genus according to its determination . Each person can be seen in his uniqueness, which distinguishes him from all others, and yet remains a being of the human species according to his destiny. If these two expressions of the quality are isolated, each of them brings the system down for itself: (+) If the one quality is isolated through which something becomes individual and unique, then it falls completely out of the system. Adorno radicalized this idea with the negative dialectic of the non-identical. For him, the whole is untrue and every something has its dignity in that it does not fit into any system. The system is falling apart. (-) If, on the other hand, the other property is isolated, through which something belongs to the superordinate genus, then a definition of something arises that only refers to the next higher order. These are the impredicative judgments examined by Poincaré, Russell, Carnap, and Gödel. "A term is called impredicative (defined / definable) if it is defined (or: can only be defined) by means of an entity to which it belongs." (Roland Hagebücher and Volker Geyer The paradox, Würzburg 2002, p. 120). For example, the element ’largest city’ is defined imprecisely: It relates to the entirety of all cities and denotes the largest element there. With finite sets this is problem-free, but it leads to indeterminacy with infinite sets. For example, everyone understands what is meant by the phrase "greatest prime", but there is no such thing as a greatest prime. Impredicative definitions can reach into the void and undermine an order.
With the impredicative definitions, Russell's antinomy is achieved. It is an impredicative definition that also relates negatively to itself. Kesselring has put Russell's antinomy into a more general formulation: "The term›... does not fall under itself‹Means the same in relation to terms as the term›... does not fulfill itself‹." (K, p. 212) If the term 'fulfills-not-itself' is analyzed, it turns out to be an antinomy: If the non-fulfillment is fulfilled, it means that it is fulfilled, in contradiction to If it is not fulfilled, it is no longer a fulfillment. It can be clearly seen how the antinomy of the disappearance of the disappearance repeats itself and the liar paradox ('this sentence is wrong') already implies.
Russell's antinomy shows how the III. Level constitutive system thought leads out of itself into an antinomy. If this antinomy cannot be resolved, empiricism turns into dogmatism. Each individual is dogmatically imposed in his uniqueness as to how it should fit into the system. Russell's antinomy is to be corrected by a prohibition: An axiom is set up with which negative, self-referential, impredicative judgments are to be excluded. - In an empirically dominated science, dogmas of this kind arise everywhere today. Physics teaches us: The big bang is the limit and the principle of the cosmos, beyond which one cannot question further. Neuroscience wants to reduce all thinking to the finite system of neurons and their synapses. Every child at this age comes across dogmas and paradoxes of this kind with their unbiased questions. Mostly they are told: »Learn this by heart! You will understand later. ”Enlightenment turns into authority thinking. The same thing goes through every math and physics student today who is expected at some point to believe what he is learning. Half of all students therefore drop out of these fields of study. Few could dare to say openly "that nobody understands quantum mechanics" (Richard Feynman, founder of quantum electrodynamics and one of the most influential physicists of the 20th century).
With empiricism and dogmatism, the III. Level to their limit. Anyone who initially took part in the acquisition of knowledge and the Enlightenment project with great enthusiasm sees himself and his friends and allies at some point imperceptibly falling on the path of empiricism or dogmatism and do not really know how this could happen. At this point the wish arises to free oneself from it into something "higher", and if that succeeds with the fourth stage, the III. Opposite level, only scorn and ridicule left. The III. Level then only counts as "common sense" or simply "the mind", the philistine of education or the talker about the regulars' table. This condescension conceals the disappointment that the III. Stage could not be achieved. Everyone longs, unspoken, to be able to see the world again with open eyes like the "first time", impartially and in the naive conviction that infinity can be grasped free of all antinomies.
Hardly anyone has the III. Level associated stupidity and self-righteousness hit as well as Flaubert (Madame Bovary), whereby the almost encyclopedic attention to detail and diversity of his works also show him as a secret representative of this level. He was responsible for the captain's task, which sums up the "arithmetic arithmetic in clothes" of this school age, which is so unpopular with every student:
"A cotton-laden ship of 200 register tons coming from Boston sails to LeHavre, there is a cabin boy on the forecastle, twelve passengers are on board, the wind is east-northeast, the ship's clock shows a quarter past three in the afternoon and it is May ... How old is the captain? " (Letter from Flaubert to his sister Caroline, 1843)
What looks like a joke has been presented to elementary school students in a simplified form as an exercise since 1980, with the amazing result that they forget how to solve tasks of this kind while attending school. The proportion of correct answers fell from preschool age to 4th grade from 90 to 30% (Holger Dambeck School math absurd in: Spiegel Online from January 17, 2012). The knowledge acquisition of the III. Level has the consequence of relying entirely on schemes and losing the unbiased view. What better shows the loss than the simple fairy tale of The Emperor's New Clothes: The king is naked.
IV: Formal thinking, age: from 12 years, adolescence
And yet there is a solution to the III antinomy. Level: The formal systems. Even if it is impossible to consistently classify everything that can be found empirically in nature and everything that can be written down or said as text, and even if every order proves to be ephemeral, after a paradigm shift by a new one Order is replaced, but the idea remains that there is a higher system of finitely many rules with which new sentences can be spoken and works of art can be created.
The most obvious example is language. It will never be possible to describe all the sentences that can be spoken. But there are a finite number of letters and a finite number of grammatical rules that make up all sentences. Similarly, the infinite variety of musical melodies can be traced back to a finite supply of notes and a finite number of intervals, scales and modes with which one can compose or sing spontaneously. This suggests that the same should be possible for mathematics and logic.
So argued Hilbert and Carnap. Nobody will doubt that mathematical knowledge never reaches an absolute limit, but aren't there overarching rules that all mathematicians work with? That was the idea of a metamathematics from Hilbert, who, following the example of Euclidean geometry, was looking for a finite set of rules and methods to which all mathematical proofs can be based. And despite Russell's antinomy, Carnap wanted to rehabilitate the impredicative judgments by going back from the content (semantics) of the language, in which contradicting statements like Russell's, to its syntax. Even if it is possible to say something nonsensical or paradoxical, it still follows the syntax. If you think from this broader perspective, Carnap sees no difficulties with impredicative clauses: they refer to the next higher order (the genus), but they are not - open or hidden - dependent on the individual properties of individual elements, and they make possible it is to build a system with a finite number of proof steps (Carnap, p. 116). The most important example are the real numbers. Every real number is the element of an overarching set. It can only be defined imprecatively by properties of the set to which it belongs. In this set, it must be possible to cross the border leading to the real number. A typical example of impredicative determinations of real numbers: »For every real number x you can therefore specify an open environment with a compact closure. «(Wikipedia). Here is every open environment of x a lot that x contains.
Formal systems of this kind are learned from the age of 12 and are the most important material in the upper school from the 11th grade. Piaget gives a typical example of the formal, hypothetical thinking ability that develops at this age:
“John is thinner than Bill; John is bigger than Sam; who is the fattest of the three? Children under the age of 11 or 12 have great difficulty doing such tasks unless they are objects they can see. The reason is that solving the problem requires propositional thinking; H. Thinking about hypothetical statements. "(Guy R. Lefrancois, Psychology of learning, Berlin et al.1994, p. 138, quoted from psychologie.de, accessed on June 5, 2016)
The young person recognizes the abstract structure of the individual statements and can draw general conclusions. He frees himself from “calculating blindly” (Dambeck in the cited article in Mirror online) the III. Step.
You understand the distinction between object and meta level. There is a difference between saying "The snow is white" and "The claim 'the snow is white' is true". Were the children in III. On the naive level, assuming that everything they see, experience and learn is true, from the age of 12 they develop an understanding between the predication function (the assignment of predicates to subjects) and the truth function (the assertion that a judgment is true or is wrong). The truth function is the elementary example for a meta level: It is checked from the outside whether the statements of the object level are true.
Russell's antinomy can be resolved according to this model: Statements on the object level (set of all barbers who do not bar themselves) and on the meta level (the question of whether there is such a set or not) can be separated. Russell designed a system of meta-levels building on one another, numbering them in the levels 1st, 2nd, 3rd order etc. (or equivalent of type 1, 2, 3, ...) and demanded that every judgment must remain within its order .
With that everything seemed clear. Instead of continuing to work on a rigid, dogmatic system that can never meet its encyclopedic claims, the search was on for a manageable and in practice successful collection of rules that in turn finally followed many rules. This attitude still determines the self-image of science to this day. For Piaget, cognitive development is completed at this stage, and Kesselring, too, is convinced of Hegel that "the genetic location of his dialectic also falls into the fourth stage" (K, p. 316). That would mean that the dialectic, comparable to metamathematics or axiomatic set theory, has a method that can be carried out in a finite number of steps, with which it can see through and understand everything. (Hegel would have criticized such an interpretation as "formal idealism", cf. HW 5.173.)
Kesselring sees how the fourth stage also becomes an antinomy. On the one hand, it invokes the finite number of individual rules and methods that it has at its disposal, but on the other hand, it cannot avoid that its distinction between object and meta levels leads to an infinite progression of constantly new meta levels. But for him the knowledge and description of antinomies takes place again within a formal method, which takes place with a finite number of steps, and is therefore conclusive for him. In contrast to Popper, for him the dialectical method is not an impetus or annoyance for scientific thinking, but on the contrary, to a certain extent, its crowning glory.
Kurt Gödel (1906-1978) had shown in 1931, however, that the distinction between object and meta levels leads to an inner, indissoluble antinomy. He was able to prove that the liar's paradox ›This sentence is wrong‹ cannot be excluded by the distinction between meta and object levels. With this paradox there is again a typical negative self-reference. If the sentence is correct, then it asserts its own untruth. If the sentence does not apply, then it is true and contradicts its own statement. Gödel was able to show how it is possible to formulate this sentence within any formal system and thus to show a contradiction in this system. This antinomy goes much further than Russell's. It not only formulates a contradictory sentence, but it also shows that it is impossible to find a consistent system.
Everyone can imagine how unpopular Godel made himself with this knowledge. It became completely unforgivable when he and his friend Einstein proved that this is not an out-of-the-way construct by logicians who are unfamiliar with the world, but can be reproduced within the most advanced physical systems and thus also undermined them (paradoxes of cosmology).
There is still great resistance to leaving Stage IV. To this day, science cannot come to terms with it and is looking for ways out within the fourth stage. Carnap's program is to be redeemed with the help of model theory, the theory of formal languages and set theory. Ulrich Blau has dealt with it for decades and has put together an impressive list of rules with which the liar's paradox should sometimes be excluded and sometimes included. He is convinced that they have all turned out to be dead ends (Blau, pp. 451-454).
L1 Self-referential sentences are excluded by hierarchies.
L2 Such sentences are allowed, but labeled as false.
L3 You get your own third truth value.
L4 You get several additional truth values.
L5 Sentences without a truth value are allowed.
L6 Paradoxical sentences are allowed that are both true and false at the same time.
L7 Any number (even uncountable many) truth values are allowed.
L8 Paraconsistent or dialectical logic.
»One introduces a non-truth-functional, possibly partially truth-functional, negation and allows sentences like (1) (› This sentence is not true ‹, t.) To be true at the same time as their negation, or perhaps both to be false. This happens e.g. in systems of so-called 'dialectical' or 'paraconsistent' logic (e.g. Da Costa 1974 and Routley 1979). None of these systems have semantics worthy of the name. One would like to know which sentences keep their truth value when negated, which do not, and above all why. The answer is - well, an index for possible worlds or the like with a few formal properties. To call such semantics is bold, but unfortunately still quite common. "(Blau, p. 453)
L9 Bivalence is relativized »to situations, contexts or whatever. Perhaps the liar is true in some situations1 and wrong in other situations s2. «(Blau, p. 453)
L10 Truth values are completely dispensed with.
The V. and VI. Stage - Antinomies of Infinity
The formal systems of the fourth stage inevitably lead to an infinite progression. Each meta level is followed by another. Every system contains an open point or blind spot where it needs to be expanded into a new level. If the resistance to the insight is given up that no overarching, consistent, formal systems can be found, then conversely the question arises, what actually happens at the transition into infinity. For Hegel the philosophy begins here: "The basic concept of philosophy (is) the truly infinite" (HW 8.203, Enz. § 95).
V: The infinite in general, the transfinite numbers
Where is the boundary between the finite and the infinite and how do both determine each other? The finite drives beyond itself, and if ‘the infinite’ is regarded as an inner unity, it is finite as a unity in turn. Hegel distinguishes (a) the transition from the finite to the infinite and (b) the infinite progression from the finite to the infinite, their change determination. If the infinite progression is related negatively to itself, from my point of view (c) a new antinomy arises, the paradox of Achilles and the turtle set up by Zenon.
(a) A change occurs when the finite reaches a certain limit beyond which the infinite is located. Primitive peoples count "one, two, three, many" and see the envelope at the "three". In the Greek alphabet, the last letter Ω is used as an envelope and was therefore adopted by Cantor as the name for the transfinite numbers. In the Hebrew alphabet, every letter has an immediate numerical value. The last letter ‘Taw’ with the symbol has the numerical value 400. For the Egyptians the greatest number was reached with the million, represented with a hieroglyph for Heh, the god of infinity: . While the Omega and Taw symbols represent the arch of heaven, to where the finitude reaches and beyond which the infinity lies, the Egyptian hieroglyph shows the worship of open infinity. Today the Bremermann limit applies, with which the greatest physically possible speed is estimated for calculating numbers with computers. That is 1.35639 · 1050 bit / kilogram / second. Larger numbers, which cannot be represented technically and which therefore cannot be calculated, can be viewed as meaningless.
Within today's sciences based on the fourth level, the question of the transition into infinity arises as an induction problem. Is it possible, after a finite number of cognitions, to infer an overarching cognition inductively? The negation of the axiom of induction is a more precise version of the non-identity of the unique, demanded by Adorno. The induction axiom says that after a finite number of steps a rule can be inferred that can be inductively extended to an infinite set. Every induction changes from the finite to the infinite. The mathematician Manjul Bhargava (* 1974) showed one of the most astonishing recent discoveries in this area: Together with Jonathan Hanke, he was able to prove that for questions such as ›Can every natural number be represented as a sum of 4 square numbers‹ »is a test with exactly 29 numbers is enough to answer the question. [...] Here are the 29 numbers: 1, 2, 3, 5, 6, 7, 10, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 34 , 35, 37, 42, 58, 93, 110, 145, 203, 290 «(Spiegel-Online from August 15, 2014).
(b) When there is a transition from the finite to the infinite, there must be an internal measure through which the transition is mediated. There has to be something about which it can be said why it goes on in infinity as in the previously finite number of observations. If it has been experienced often enough that a new day dawns each morning, it is inductively deduced that there is an overarching, eternal regularity. The "Progress to infinity«(HW 5.155) does not stop at the limit of the infinite, but is continued in the infinite. To describe this mathematically, the mathematician Georg Cantor (1845-1918) introduced the transfinite numbers a few decades after Hegel. The decisive argument was the proof that there are uncountably many numbers, including numbers that do not belong to the natural numbers, but are still numbers. Therefore Cantor developed the transfinite numbers in order to continue counting with them beyond the natural numbers. This can be seen as the first approach to find an inner commonality between the finite and infinite numbers, even if the representation does not look spectacular at first glance:
1, 2, 3,… ω, ω + 1, ω + 2,… 2 · ω, 2 · ω + 1, ...
Here there is no longer a clear transition from the finite to the infinite, but an interplay of counting 1, 2, 3, ... on the one hand, and the introduction of transfinite numbers such as ω - which encompass the entirety of infinite counting processes - on the other . In my opinion, this is exactly what Hegel meant for the alternation of the finite and the infinite. It is noticeably running into an antinomy. With the infinite progress, every limit is crossed again and again, but it never succeeds in reaching a conclusion. It would have to "go beyond this [...] itself" (HW 5.155, quoted from K, p. 309).
As early as 1873, and in a new way in 1877, Cantor was able to prove with his second diagonal argument that not all numbers can be achieved with this method. There always remains a non-empty remainder, which he called uncountable numbers. They lie beyond the transfinite numbers and cannot be reached even with the alternation of the finite and the infinite. In order to determine it, a new method is necessary: the border crossing that came with the VI. Stage is introduced.
(c) How does the concept of infinity gained with the fifth stage solve the antinomy of the fourth stage? As on the previous levels, this is achieved through a new antinomy. The V.Level has led to infinite progression, and this has to be related negatively to itself: Instead of counting on and on, as is known from the natural numbers, the counting has to be turned against itself, so to speak. This was achieved in a very clear way with the paradox of Achilles and the turtle. Achilles can never catch up with the turtle, because every time he gets to where it was a moment before, it has run a short distance in turn. Everyone can feel what is happening here: The number of ways Achilles continually approaches the turtle continues to count, but he never reaches it because the distances observed become shorter at the same time. Two processes diverge: the counting of the respective distances considered, and the shortening of these distances. If the framework of the infinite progression is kept, this paradox cannot be resolved.
But with this paradox the previous antinomy begins to dissolve. Even if it remains open how it can be formulated mathematically that Achilles exceeds the infinitely many, infinitely smaller distances to the turtle in a finite time and overtakes it, this is an example, »how the infinite comes out of itself and into finitude«(HW 5.168f). Achilles runs through an infinite number of distances until he overtakes the turtle in a finite time. Infinity can be incorporated into a finite calculus without contradiction if it is possible to relate the infinite progression negatively to itself in the manner described with the example of Achilles and the turtle. The question of how the infinite can become finite, however, was the "conundrum" (HW 5.169) with which, from the point of view of the fourth stage, every philosophy should be made impossible in the beginning. For formal systems of the fourth level there is no infinity, but only finitely many rules and ultimately only finitely many models that can be formed from these rules. If, on the other hand, the example of Achilles and the turtle shows how an infinite process of rapprochement becomes finite at the moment of overtaking, it is ex negativo shown that the liar's paradox is only valid within a logic that has no infinity. Further steps will be needed to show how the liar's paradox can also be resolved in terms of content with the aid of the thought of infinity.
If the negative self-referentiality of the transfinite counting of continuously smaller amounts is considered separately, this leads to the continuum hypothesis: The set of uncountable numbers corresponds to the set of real numbers.
VI: The true infinity, the continuum of real numbers
Aristotle found the answer. The path of the tortoise and Achilles is not made up of steps that each have a constant step length. While counting assumes that the distance from a number ‘n’ to its successor ‘n + 1’ is always 1, the real distance from Achilles to the turtle is getting shorter and shorter. The numerical values of the continuously decreasing distances are in a different number class than the natural numbers. This other class of numbers has one property that distinguishes it from the natural numbers: that is its »context«(HW 5.161), the continuity. With the context, a new term has been found that goes beyond the previously introduced terms. (This is formulated in a criticism of Kesselring. Since for him the dialectical method belongs to the fourth stage, he sees nothing new on the following stages, but only a retrospective reflection. Consequently, he objects "critically against Hegel" that Hegel on the following stages "has not introduced any new terms into the dialectic" [K, p. 315]. For me, on the other hand, the previous distinction between envelope and progression and the following distinction between continuity, process and transition result in new terms within the IV. Stage were not yet attainable.)
To resolve the antinomy of Achilles and the turtle, it is not enough to point out the continuously decreasing distances. At the same time, there is something that transcends these distances with its own internal standard: the movement of Achilles. He does not brake and does not come to a stop when he approaches the turtle, but he keeps the momentum of his movement so that he easily overtakes it (see also Hegel in his Lectures on the history of philosophy, Vol. 1, HW 18.310).
»Anyone who makes a continuous movement has incidentally (symbebêkos) also go through infinity, but not in the real sense. It is only incidentally true of the line to be an infinite number of half-pieces, its essential being is something completely different. "(Aristotle, physics, VIII.8, 263b)
Without commenting on this passage in detail (see the corresponding Aristotle comment), two things should be emphasized: The movement is considered in its uniformity. In a subtle distinction it can be said that it does not consist of an infinite number of points, but rather contains an infinite number of points which are connected to one another by their continuity. Movement is described as being (literally as the subject, with Aristotle as the first category, ousia) and the infinity of their points as a property (second category, symbebêkos) that she wears. At the same time, Achilles only touches the infinite number of points "in passing" in his course. They become blurred in his conscious perception. Their multitude and infinity is an example of the small perceptions that are touched in their fullness, but do not penetrate into consciousness as perceived.
If the movement is viewed in isolation, it is a »transition« (HW 5.166). In simple examples, transition can mean overtaking. Furthermore, the process can be meant how the individual points are casually "passed over" in the continuous movement. And finally, transition means changing from one category to the next. Hegel only uses the expression "transition" in the strict sense when it comes to the transition from one category to the next. The logic of being is complete when all categories have passed through transitions. At the end of the VI. Level, there is not only a transition from a second category for the first time (symbebêkos) to the next, but self-referring to the transition to the transition.
The VI. Stage leads to an antinomy that Cantor found in 1897. Cantor was looking for a term to separate the various levels of infinity from one another: that is power. Finite sets, the set of natural numbers, and the set of continuous real numbers each have their own cardinality. If the question is asked whether the totality of the thicknesses in turn forms a set, this leads into Cantor's antinomy: the totality of the thicknesses cannot form a set. The antinomy is solved in mathematics by giving sets of this type their own name and calling them "classes". Classes are not examined with set theory, but with the allusion to Aristotle with category theory.
Outlook: the movement of the term and its objectivity
For me, the solution to Cantor's antinomy with a new name ((class ’instead of quantity’) is unsatisfactory. But it points to the more general question by means of which a solution can be found: the formation of terms and names. In conceptual logic, questions are no longer asked about the nature of the objects of thought, their modes of access (categories) and the transitions from one category to the next, but rather about the nature of thinking.
With this question a change of perspective is carried out again as in the solution of the paradox of Achilles and the turtle. Just as it was a matter of recognizing the movement-in-itself as the subject and the continuity as its property, here thinking-in-and-for-itself is to be determined as the subject and the rules of its formal inference as its property . While thinking at the fourth level is content with setting up a system of formal rules, Hegel asks about the subject that appears as a whole in these rules. This no longer means the sentence subjects, i.e. the individual contents that are described with concepts, judgments and conclusions, but a subject that shows itself in thinking and its rules. This corresponds to Wittgenstein's approach, who also asks about something that shows itself in language instead of asking about what can be expressed and shown with language.
In order to work this out, the sentences for their part are divided into two moments in a similar way as in III. Phase the two manifestations of the nature of something were distinguished: the literal statement (semantics) and the immediate use (pragmatics), how and why something is said on the one hand, and the trace of something else that shows itself in this statement (in a figurative sense the determination or the own poetry of a sentence, the shock and revelation emanating from a sentence, which goes far beyond the literal content of what is said) on the other hand. That is how I understand Wittgenstein. He criticized both the type theory of Russell and the syntax theory of Carnap, since they stop at the literal meaning of a sentence and seek the formal structures for it. For him, on the other hand, it is inconceivable that language says what it is in its own words, but can only show itself in language what transcends language, the other of language, just as the continuity of movement transcends individual steps which it is composed of. If, however, an attempt is made to formulate the provision hidden in the sentence as a sentence, the result can be referred to as a "speculative sentence" using Hegel. It shows the nature of the sentence. "Formally, what is said can be expressed in such a way that the nature of the judgment or sentence in general, which includes the difference between subject and predicate, is destroyed by the speculative sentence, and the identical sentence to which the former becomes the counter-attack to the former Contains proportions. " (HW 3.59, quoted in Kulenkampff, p. 1). Only with the speculative sentence does it seem possible to me to find a comprehensive solution to Russell's antinomy and the liar's paradox, whereby the question immediately arises whether 'speculative' is again nothing more than a new name to fill with content is (see introductory Kulenkampff Antinomy and Dialectics).
This gives rise to further questions that are only to be mentioned at this point as theses. They will be explored further using the example of the logic of Spencer-Brown and his interpreters.
(1) Within the one-dimensionality of a string of characters, memory can only be achieved by passing on a trace.
(2) How can a trace be recognized as a trace and not as a random collection of objects? Example: How can the traces on the earliest works of art or clay tablets be recognized as language and not as random scribbles or rubbing of stones against one another? What can be inferred from traces about the thinking and the liveliness of the beings who left this trace?
(3) Re-entry and sewing up a track. What is the medium (the reason) in which not only the trace but also the revelation passed down with it can be entered?
Conclusion Can a conclusion be drawn? For me it consists in the insight into the unity of thought, which disintegrates when individual levels are absolutized. The levels are separated from one another by antinomies. It is true that each new stage can resolve the antinomy of the preceding one, but it lives on in a new antinomy. After a phase of dominance of the III. and IV. level the others are to be upgraded. This cannot be done through some kind of decision according to the model Be spontaneous!, but only through openness and attention to the most varied of developments in all areas.
HW = Georg Wilhelm Friedrich Hegel: Works in 20 volumes. Re-edited on the basis of works from 1832-1845. Ed. E. Moldenhauer and K. M. Michel. Frankfurt / M. 1969-1971 (cited as HW); link
K = Thomas Kesselring: The Productivity of the Antinomy, Frankfurt am Main 1984 (cited as K)
WL A = Georg Wilhelm Friedrich Hegel: Science of Logic, Volume One: The Objective Logic. First book, 1st edition, Nuremberg 1812; link
Aristoteles: Physik, in: Schriften Vol. 6, Hamburg 1995 (Link)
Ulrich Blau: The Logic of Indeterminacies and Paradoxes, Heidelberg 2008
Rudolf Carnap: Logical Syntax of Language, Vienna, New York 1968 
Kurt Gödel: Russell's mathematical logic
in: Kurt Gödel: Collected Works II, Oxford 1990, pp. 119-141 
Kurt Gödel: Is mathematics syntax of language
in: Kurt Gödel: Collected Works, Volume III, Oxford et al. 1995, pp. 334-362 [1953-1959]
Heiko Knoll, Jürgen Ritsert: The Principle of Dialectics, Münster 2006
Arend Kulenkampff: Antinomy and Dialectics, Stuttgart 1970
psychologie.de: The development stage model according to Piaget; link
Urs Richli: Critical remarks on Thomas Kesselring's reconstruction of the Hegelian dialectic in the light of genetic epistemology and formal logic
in: Philosophisches Jahrbuch, 1988 1st half volume, Freiburg / Munich 1988, pp. 131-143
Dieter Wandschneider: Basic features of a theory of dialectics, Stuttgart 1995
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