Tag Archives: Self

CHAPTER 12. PARADOXES AND COMPLEX HIERARCHIES

Once, when I was talking about complex hierarchies, a listener said that this phrase caught her interest even before she knew its meaning. She said hierarchies remind her of patriarchy and power, but the term complex hierarchies has a liberating connotation. If you have the same intuition as she does, you should be prepared to explore the magical, puzzling world of paradoxes of language and paradoxes of logic. Can logic be paradoxical? Isn’t the power of logic the ability to resolve paradoxes?

As you approach the entrance to the Cave of Paradoxes, you encounter a creature of mythical proportions. You will immediately recognize him as the Sphinx. This Sphinx-like creature has a question for you, which you must answer correctly in order to gain the right to enter: what creature walks on four legs in the morning, two at noon, and three in the evening? You are momentarily confused. What kind of question is this? Perhaps your journey will be interrupted at the very beginning. You are just a newbie in this game of puzzles and paradoxes. Are you ready for what seems like a challenging riddle?

To your great relief, Sherlock Holmes comes to the aid of your Doctor Watson. “My name is Oedipus,” he introduces himself. “The Sphinx’s question is a riddle because it confuses logical types, right?”

This is true, you understand. It was helpful to learn about Boolean types before embarking on this exploration. But what now? Fortunately, Oedipus continues: “Some of the words of the phrase have lexical meaning, but others have contextual meanings of a higher logical type. It is the overlap between these two types that characterizes metaphors that confuses you.” He smiles encouragingly.

Right, right. The words “morning”, “noon” and “evening” should contextually relate to our lives – to our childhood, youth and old age. Indeed, in childhood we walk on all fours, in youth we walk on two legs, and three legs are a metaphor for two legs and a stick in old age. Fits! You approach the Sphinx and answer his question: “Man.” The door opens.

As you walk through the door, the thought comes to your mind: how did Oedipus, the mythical character from Ancient Greece, know such a modern term as logical types? But there is no time to think: a new task requires your attention. One man, pointing to another standing next to him, asks: “This man Epimenides is a Cretan who declares that all Cretans are liars. Is he telling the truth or lying? Okay, let’s see, you reason. If he is telling the truth, then all Cretans are liars, and therefore he is lying – this is a contradiction. Let’s go back to the beginning. If he is lying, then all Cretans are not liars, and he may be telling the truth – and this is also a contradiction. If you give the answer “yes”, it echoes “no”, and if you give the answer “no”, it echoes “yes”, and so on ad infinitum. How can you solve such a riddle?

“Okay, if you can’t solve a riddle, at least you can learn how to analyze it.” As if by magic, another assistant appears next to you. “I’m Gregory Bateson,” he introduces himself. “You are dealing with the famous liar paradox: Epimenides is a Cretan who says: “All Cretans are liars.” The first condition creates the context for the second condition. It qualifies him. If the second condition were ordinary, it would have no effect on its first condition, but no! It requalifies the first condition, its own context.”

Your face brightens: “Now I understand – this is a confusion of logical types.”

“Yes, but this is not an ordinary mixture. Look, the first one overrides the second one. If yes, then no, then yes, then no, ad infinitum. Norbert Wiener said that if you introduced this paradox into a computer, it would finish him off. The computer would print the sequence yes…no…yes…no…until it ran out of ink. It’s a tricky endless loop that logic can’t get out of.”

“Isn’t there any way to resolve the paradox?” – you ask sadly.

“Of course there is, because you are not a silicon computer,” says Bateson. – I’ll give you a hint. Suppose a merchant comes to your door with the following offer: “I have a beautiful fan for you for fifty bucks – that’s almost nothing. Will you pay by cash or check?” What would you do?

“I would have slammed the door on him!” You know the answer to this question. (You remember a friend whose favorite game was the question “Which would you choose?” – I will cut off your hand, or I will bite off your ear. Your relationship ended very quickly.)

“Exactly,” Bateson smiles. — The way out of the endless loop of paradox is to slam the door, jump out of the system. That gentleman over there has a good example.” Bateson points to a man sitting at a table with a sign that says, “Only two can play this game.”

The gentleman introduces himself as J. Spencer Brown. He claims he can actually show you how to get out of the game. However, to understand this, you have to look at the Liar Paradox in the form of a mathematical equation:

x = – 1/x.

If you plug in the +1 solution to the right, the equation gives – 1 ; you plug in -1, and the equation gives + 1. The solution oscillates between +1 and -1, exactly like the yes/no oscillation of the liar paradox.

Yes, you can understand it. “But what is the way out of this mad endless hesitation?”

Brown tells you that there is a well-known solution to this problem in mathematics. Let us define the quantity called i as √—1. Note that i ² = – 1. Dividing both sides of the expression i ² = – 1 by i gives

i= -1/i.

This is an alternative definition of i. Now let’s try to substitute the solution x = i into the left side of the equation

x = -1/x.

Now the right side gives -1/i, which by definition is equal to i – no contradiction. Thus i, which is called an imaginary number, overcomes the paradox.

“It’s amazing.” It takes your breath away. “You are a genius”.

“It takes two to play the game,” Brown winks.

Your attention is drawn to something distant: a tent with a large sign reading “Gödel, Escher, Bach.” As you approach the tent, you see a man with a cheerful face who waves at you invitingly. “My name is Dr. Geb,” he says. — I am spreading the idea of ​​Douglas Hofstadter. I assume you have read his book “Gödel, Escher, Bach.”

“Yes,” you mutter in some embarrassment, “but I didn’t understand everything about it.”

“Look, it’s actually very simple,” says Hofstadter’s messenger condescendingly. “All you need to understand are complex hierarchies.”

“Complicated what?”

“Not anything, but hierarchy, my friend. In a simple hierarchy, the lower level provides the higher one, and the higher one does not react in any way. In simple feedback, the layer above reacts, but you can still tell which is which. In complex hierarchies, these two levels are so mixed that you cannot define different logical levels.”

“But it’s just a label.” You shrug indifferently, still hesitant to accept Hofstadter’s idea.

“You don’t want to think. You missed a very important aspect of complex hierarchical systems. After all, I have been following your progress.”

“I trust you, in your wisdom, to tell me what I’m missing,” you say dryly.

“These systems—of which the liar paradox is the most important example—are autonomous in nature. They talk about themselves. Compare them with a common phrase such as “your face is red.” A common phrase refers to something outside of itself. But the complex phrase of the liar’s paradox refers to itself. That’s how you fall into her endless deception.”

You’re reluctant to admit it, but it’s a worthwhile guess.

“In other words,” continues Hofstadter’s messenger, “we are dealing with self-referential systems. A complex hierarchy is a way of achieving self-reference.”

You give in: “Dr. Geb, this is extremely interesting. I do have a certain interest in things that relate to the self, so please tell me more.” The person who spreads Hofstadter’s ideas does not need to be asked.

“The self arises as a result of the veil – a clear obstacle to our attempt to unravel the system logically. It is this lack of continuity – in the liar’s paradox, this endless fluctuation – that prevents us from seeing through the veil.”

“I’m not sure I understand this.”

Instead of explaining again, a Hofstadter supporter persuades you to look at a painting by the Dutch artist M. C. Escher. “In the Escher Museum, in that tent over there,” he says, leading you towards it. “The painting is called “Gallery of Engravings.” It’s quite strange, but it fits exactly with the essence of our discussion.”

Img. 32.
Escher’s painting “Gallery of Prints” is a complex hierarchy. The white spot in the middle shows discontinuity

Inside the tent you study the painting (Fig. 32). It shows a young man in an art gallery looking at a painting of a ship anchored in a city harbor. But what is it? There is an art gallery in the city in which a young man looks at a ship at anchor…

My God, this is a complex hierarchy, you exclaim. Having passed through all these buildings of the city, the painting returns to the starting point where it began to begin its cyclical movement again, thus prolonging the viewer’s attention to itself.

You turn to your guide with delight.

“You get the point.” Your guide smiles widely.

“Yes thank you”.

“Did you notice the white spot in the middle of the picture?” – Dr. Geb suddenly asks. You admit that you saw it, but did not attach much importance to it.

“The blank spot containing Escher’s signature shows how clearly he understood complex hierarchies. You see, Escher could not, so to speak, fold a picture back into itself without violating the generally accepted rules of drawing, so there had to be a discontinuity in it. The white spot reminds the observer of the discontinuity inherent in all complex hierarchies.”

“From discontinuity come veil and self-reference,” you cry.

“Right. — Dr. Geb is pleased. “But there is one more thing, one other aspect, which is best seen by considering the one-step self-referential phrase “I am a liar.” This phrase says that she is lying. This is the same system as the liar paradox you encountered earlier – only it removes the nonessential form of the condition within the condition. Do you understand?

“Yes”.

“But in this form something else begins to become clear. The self-reference of a phrase—the fact that a phrase speaks about itself—is not necessarily self-evident. For example, if you show this phrase to a child or a foreigner who is not very fluent in English, you might be asked, “Why are you a liar?” At first, he or she may not see that the phrase refers to itself. Thus, the self-reference of a phrase arises from our tacit, rather than precisely defined, knowledge of English. It’s as if the phrase is the tip of the iceberg. We call this the undisturbed level. Of course, it is unbroken from a systemic point of view. Take a look at another painting by Escher – it’s called “Drawing Hands” (Fig. 33).

Cassock. 33.
Escher’s painting “Drawing Hands”

In this painting, the left hand draws the right hand, which draws the left hand; they draw each other. This is self-creation, or autopoiesis. Moreover, it is a complex hierarchy. How does the system create itself? This illusion is created only if you remain logged in. From outside the system, where you are looking at it, you can see that the artist Escher drew both hands from an undisturbed level.

You excitedly tell Dr. Geb what you see in Escher’s painting. He nods approvingly and says with conviction: “Dr. Hofstadter is interested in complex hierarchies because he believes that the programs of our brain computer – what we call the mind – form a complex hierarchy, and from this complexity our glorious self arises.”

“But this is just a bold hypothesis, isn’t it?” You have always been suspicious of bold hypotheses. You have to be careful when scientists come up with crazy ideas.

“Well, you know, he thought about this problem a lot,” says a Hofstadter supporter wistfully. “And I’m sure that one day he will prove it by building a silicon computer with a conscious self.”

You’re impressed by Hofstadter’s dream—our society needs dreamers—but you feel the need to defend the logic. “I must admit that I’m a little wary of complex hierarchies,” you say. — When I studied logical types, I was told that they were invented to preserve the purity of logic. But you, that is, Dr. Hofstadter, intricately mix them not only in language, but also in real natural systems. How do we know that nature gives us such a right? After all, the paradoxes of language have a connotation of arbitrariness and artificiality.” You are very pleased that you can argue, if not with Hofstadter, then at least with his supporter, using what seems to you irrefutable logic.

But a Hofstadter supporter is ready for an argument.

“Who says we can keep logic pure? – he objects. — Or you haven’t heard anything about Gödel’s theorem. I thought you read Dr. Hofstadter’s book.”

“I told you I didn’t understand her. And it was Gödel’s theorem that became the final stumbling block for me.”

“It’s actually very simple. The theory of logical types was invented by two mathematicians – Bertrand Russell and Alfred Whitehead – in order, as you say, to preserve the purity of logic. However, another mathematician, Kurt Gödel, proved that any attempt to create a mathematical system free from paradoxes is doomed to failure if the system is sufficiently complex. He proved this by demonstrating that any sufficiently rich system is doomed to be incomplete. You can always find a statement in it that the system is not able to prove. Essentially, a system can be either complete but inconsistent, or consistent but incomplete, but it can never be both consistent and complete. Gödel proved this theorem using the so-called mixed logic of complex hierarchies. In doing so, he consigned several concepts to the scrapheap, including the possibility of a complete and consistent mathematical system like Russell and Whitehead’s theory of logical types. Do you understand everything?

You don’t dare ask any further questions. Mathematics seems like a hornet’s nest to you. The more you linger around it, the more you risk being stung. You hastily thank your interlocutor and head for the nearest exit.

But, of course, on the way to him I stop you. Seeing me, you are surprised. “What are you doing here?” – you ask.

“This is my book. “I can interfere whenever I want,” I tease. “Tell me, did you believe Hofstadter’s promise to build a self-aware silicon computer?”

“Not really, but it seemed like an interesting idea,” you reply.

“I know. The idea of ​​a complex hierarchy is fascinating. But has anyone explained how Hofstadter intends to create discontinuities in classical silicon computer programs, which by their very nature are continuous? It’s not so much that the programs are linked back to each other and become so intertwined that you can hardly trace their causal chain. That’s not the point at all. There really must be a discontinuity, a real leap beyond the system – an unbroken level. In other words, the question is how can our brain, considered as a classical system, have an undisturbed level? In the philosophy of material realism, on which classical systems are based, there is only one level of reality – the material level. Therefore, where is the place in it for an undisturbed level?

“Don’t ask,” you ask. – What do you propose?

“Let me tell you a story. One day someone saw the Sufi teacher Mullah Nasrudin, on his knees, adding yogurt to the water in a pond. “What are you doing, Nasreddin?” – asked this passerby.

“I’m trying to make yogurt,” answered the mullah.

“But it’s also impossible to make yogurt!”

“What if it works out,” the mullah said optimistically.”

You laugh. “Funny story. But stories don’t prove anything,” you object.

“Have you heard of Schrödinger’s cat?” – I ask in response.

“Yes,” you say, brightening a little.

“According to quantum mechanics, after an hour, the cat is half alive and half dead. Now suppose we set up a machine to observe whether a cat is alive or dead.”

You cannot help but say, “I know all this; the machine adopts the cat’s dichotomy. She is unable to give definite evidence until she is rescued by a conscious observer.”

“OK. But suppose we create a whole hierarchy of inanimate machines that successively observe the readings of each previous machine. Isn’t it logical to think that they will all acquire the quantum dichotomy of the cat state?”

You nod in agreement. This seems quite logical.

“So by having the cat wave function in the form of a quantum superposition, we are, in effect, opening up the possibility that all material objects in the universe can become infected with a quantum superposition. Quantum superposition has become universal. But this comes at a certain price. You understand?”

“No, I don’t understand.”

“The system is not closed.”

“Ah.”

“This openness or incompleteness becomes a logical necessity if, following Schrödinger, you give a quantum description to macroscopic systems. This is the real Gödel dilemma.”

“What are you implying?” – you ask, puzzled.

“To resolve the dilemma, we must be able to actually go beyond the boundaries of the system, and this means the existence of a quantum mechanism in our brain and a non-local consciousness collapsing it. Therefore, in order to have a truly complex hierarchy – discontinuity, unbroken level and all the rest – there must be a quantum system in our heads.”

“Indeed?”

But I end our conversation (jumping, using an undisturbed level). Everything that has a beginning must end somewhere in due time—even exciting ideas like the existence of a quantum system in our brains.

Okay, now you know what the complex hierarchy is, you agree that it only works for a quantum system within the overall idealistic framework, and you intuit that it may explain our own self-reference. Let’s see if this is true.

Return to Schrödinger’s Cat

To understand how complex hierarchy and self-reference arise in the brain-mind, let us return once again to Schrödinger’s cat.

According to quantum mechanics, after an hour, the cat’s state is half alive and half dead. If we set up a machine to measure whether a cat is alive or dead, then the machine becomes infected with the dichotomy of the cat. And if we install a whole series of irrational machines, in which each subsequent one measures the readings of the previous one, then the inevitable logical consequence of this is that they all acquire a quantum dichotomy.

It’s like the story of the islander and the missionary. The missionary explains how the earth is held up by gravity, etc. But the islander contradicts him, declaring: “I know who really holds the land. This is a turtle.”

The missionary smiles indulgently. “But then, my dear, who keeps the turtle?”

The islander remains unperturbed. “You will not deceive me,” he reproaches the missionary. “They’re all turtles, right down to the bottom.”

The essence of von Neumann’s chain, of course, is that the dichotomy of measuring instruments observing Schrödinger’s cat goes “all the way down.” The system is infinitely regressive. It doesn’t collapse on its own. We look in vain for collapse in the von Neumann chain, just as we look in vain for the truth value in the liar’s paradox. In both cases we come to infinities. We are dealing with complex hierarchies.

To resolve the dilemma, we must exit the system to an undisturbed level. According to the idealistic interpretation of quantum mechanics, non-local consciousness acts as the undisturbed level as it collapses the mind-brain from outside spacetime, thus ending the von Neumann chain. From this point of view, there is no Gödel’s dilemma.

However, from the point of view of the mind-brain, everything is different. Let’s build a rough model of the mind-brain’s response to a stimulus. The stimulus is processed by the sensory apparatus and presented to the dual system. The state of a quantum system propagates as a coherent superposition, and all classical measuring instruments connected to it also become coherent superpositions. However, there is no mental program that selects between different aspects of a coherent superposition; there is no program in the brain-mind that can be identified as a central processing unit. The subject is not a homunculus operating at the same level as the mind-brain programs.

Instead, there is a discontinuity, a disruption of causality in the process of choosing from possible alternatives in the pool of probability that is given by the quantum system. Choice is a discrete act in the transcendental sphere – the action of our non-local consciousness. No linear, cause-and-effect description in space-time is possible. This is the “blank spot” (as in Escher’s Gallery of Prints) in our picture of the complex hierarchy in the mind-brain. The result is self-reference. Consciousness collapses the general quantum state of the dual system, leading to the primary separation of subject and object. However, due to the complex hierarchy, consciousness is identified with the “I” of self-reference and experiences the primary awareness – I am.

The selfhood of our self-reference is conditioned by a complex hierarchy, but our consciousness is the consciousness of Being, which lies beyond the subject-object division. There is no other source of consciousness in the universe. The self of self-reference and the consciousness of primordial consciousness together create what we call self-consciousness.

The book “The Self-Aware Universe. How consciousness creates the material world.” Amit Goswami

Contents

PREFACE
PART I. The Union of Science and Spirituality
CHAPTER 1. THE CHAPTER AND THE BRIDGE
CHAPTER 2. OLD PHYSICS AND ITS PHILOSOPHICAL HERITAGE
CHAPTER 3. QUANTUM PHYSICS AND THE DEATH OF MATERIAL REALISM
CHAPTER 4. THE PHILOSOPHY OF MONISTIC IDEALISM
PART II. IDEALISM AND THE RESOLUTION OF QUANTUM PARADOXES
CHAPTER 5. OBJECTS IN TWO PLACES AT THE SAME TIME AND EFFECTS THAT PRECEDE THEIR CAUSES
CHAPTER 6. THE NINE LIVES OF SCHRODINGER’S CAT
CHAPTER 7. I CHOOSE WITH THEREFORE, I AM
CHAPTER 8. THE EINSTEIN-PODOLSKY-ROSEN PARADOX
CHAPTER 9. RECONCILIATION OF REALISM AND IDEALISM
PART III. SELF-REFERENCE: HOW ONE BECOMES MANY
CHAPTER 10. EXPLORING THE MIND-BODY PROBLEM
CHAPTER 11. IN SEARCH OF THE QUANTUM MIND
CHAPTER 12. PARADOXES AND COMPLEX HIERARCHIES
CHAPTER 13. “I” OF CONSCIOUSNESS
CHAPTER 14. UNIFICATION OF PSYCHOLOGIES
PART IV . RETURN OF CHARM
CHAPTER 15. WAR AND PEACE
CHAPTER 16. EXTERNAL AND INTERNAL CREATIVITY
CHAPTER 17. THE AWAKENING OF BUDDHA
CHAPTER 18. IDEALISMAL THEORY OF ETHICS
CHAPTER 19. SPIRITUAL JOY
GLOBAR OF TERMS