Topological Invariance of Thought

A cockroach perceives a human being as evil and treacherous wind that brings death and can overtake if you run away from it in a straight line. A cockroach has feathers-cilia in the back, which feel the strength and direction of the slightest air movement. This property gives the cockroach an advantage — a chance to escape from the killer wind. A cockroach uses chaos to trick the evil intelligence of the wind.

Running away from the wind means for the cockroach to run in a completely random direction, completely unpredictable for the wind, but gradually moving away from its source. Therefore, the first movement of the cockroach in response to the wind may not be from, but towards its source. A group of cockroaches will splash in completely different directions from the boot brought over it. At the same time, all strategies of cockroaches for running away from the wind will be topologically invariant in the space of phase states of the wind and cockroach system. I will try to explain what it is and how it is possible.

Chaos in everyday language is synonymous with disorder. For mathematicians and other scientists who study it, chaos is complete order, but so complex that it cannot be distinguished from complete disorder, where real randomness rules. The adjective “real” in this case is important because, in various programs and algorithms that simulate randomness, we are dealing not with real randomness, but with pseudorandomness, which has only limited complexity.

For example, we cannot toss a coin an infinite number of times to generate an infinite series of random numbers. But a finite row, no matter how long it is, must necessarily be repeated sooner or later. For an ordinary person, there is no significant difference, but for mathematicians who constantly deal with infinity, it is huge. The brilliant “Martian” mathematician John von Neumann, who built the first computer, laid down the mathematical foundation of quantum mechanics, created game theory and the theory of cellular automata, suggested seriously take this factor into account. And I suggest taking his opinion into account because I trust his competency.

Some opponents urged me not to refer to well-known scientists in discussions. But what can we do without referring to them? Should we be competent in everything? No, darlings, the question is not about refraining from referring to big names in science but choosing the right names to be trusted. After reading this chapter, you will hopefully understand how this can be done. All the scientists to which I refer are selected according to the principle of the topological invariance of their way of thinking. But we will return to this a bit later.

Mathematicians have long drawn attention to the strange behavior of simple (for them, I myself have not solved a single one in my life) logarithmic equations as the number of their variables increases. The outstanding French mathematician Henri Poincare, no less genius than von Neumann, at the end of the 19th century decided to accurately calculate the orbits of three celestial bodies relative to each other. Newton managed to do it with the orbits of two bodies three centuries earlier. Decades and hundreds of pages of formulas later, after the beginning of his work, Poincare was forced to admit defeat. The formulas remained simple and unambiguous (invariant in other words), but the complexity of the solution grew so rapidly that Poincare realized that he simply would not have enough lifetime left to even come close to solving the problem.

Since then, computers and new ways of solving the problem with the orbits of three and more celestial bodies have appeared, but all of them gave and keep giving an approximate, but not absolutely exact solution to the problem posed by Poincare. Moreover, observations in the real world show that the orbits of the planets really vibrate. Planets do not move like trains on rails, that is, invariantly. Their orbits vary, but only within a certain corridor of values, not just anyhow. Invariant values define the corridor boundaries. By and large, this is an example of topological invariance in the real, physical world.

The geometry of the phase states of systems made it possible to visualize topologically any process. If we take, for example, three variables: air temperature, its humidity, and CO2 concentration in it — and begin to measure them at regular intervals, then each measurement will correspond to a phase state of atmospheric parameters determined at a given time. Each phase can be entered into a table in the form of a column of numbers, or you can build a three-dimensional space with coordinate axes: temperature, humidity, CO2 concentration, and mark each phase state with a dot in this space. Then we can see the orbit along which the system moves, passing from one phase state to another.

If phase states of a system change cyclically, then its orbit in the phase space will be cyclic, similar to the orbits of planets. Moreover, if the measurement data in each cycle is not absolutely identical, then the orbit will fluctuate — vary. If the variations of the orbit will always fit within the corridor defined by the boundary parameters, then all the passages of the system along the orbit of its phase states will be topologically invariant. Thus, topological invariance gives an infinite number of variations, infinite variability, but within a corridor limited by invariant values.

This is only one of the complexity levels that scientists have to live with. They coined the terms metastability and almost periodic systems to describe this phenomenon. In my opinion, everything is simple so far.

Another thing is even more surprising: not only real-world systems, in which real randomness can be assumed, behave in such a messy way, but also mathematical models in which each step of a system in the phase space is clearly described by unambiguous mathematical equations. Such systems with three or more variables are called high dimensional. One-dimensional or two-dimensional — low dimensional systems, unlike high dimensional ones, are absolutely invariant. A complete order reigns in them.

Chaos reigns in high-dimensional systems. But, attention, please, in the topologically invariant systems bounded by no more than two parameters, there is a possibility of transition from the level of multidimensional chaos to the level of a full low dimensional order. Topological invariance allows one to simply neglect the variability of fluctuations within the channel. To understand this, it is enough to imagine a solid immovable thread, which under the microscope consists entirely of tiny fibers constantly moving like worms.

Thus, the topological invariance of meaning in Avicenna-Bayes-Nalimov’s creative or probabilistic syllogism is bounded by the categorical invariance of meaning in Aristotle’s syllogism. “You cannot enter the same river twice!” — this is exactly what Heraclitus spoke of 3,500 years ago, according to Plato.

So far, everything is logical, in my opinion. But here a different level of complexity intervenes. It was discovered by chance by meteorologist Edward Lorenz, who was one of the first to use computer modeling to predict the weather. Once Lorenz fed his computer with the measurement data with an accuracy of six decimal places. Later he decided to reproduce the result with the same data, but the computer rounded the data to the accuracy of four decimal places. Lorenz got a result that had nothing to do with the previous result. A small error in the initial parameters led to a glaring difference in the phase states of the system after a relatively short period of time. I repeat, it was a computer model consisting of simple logarithmic equations, but not the real weather. And this model ran awry entirely as a result of the most insignificant deviation of the initial parameters.

That way peaceful chaos broke out of the shackles of topological invariance, showed its wild face, and returned us to the questions that the ancient Greeks already asked themselves.

“There is no movement,’’ the bearded wise man said.

Another remained silent and began to walk in front of him.

He could not object more strongly;

All praised the intricate answer.

But gentlemen, this funny incident

Reminds another story to me:

After all, every day the sun walks above us,

However, the stubborn Galileo is right.”

The genius Russian poet Alexander Pushkin very accurately described the situation with the aporias of Zeno. Hercules will never catch up with the turtle, he said, because at first he will have to cross half the distance to the turtle, and before that half of the half, and even earlier half of half, of the half, and so on to infinity. In fact, Zeno went on, Hercules and will not be able to even move from his place because before he takes the first step, he needs to take a half step, and even earlier half of the half a step, and before that a half of half, of a half a step and so on to infinity. Yet we are somehow moving. See, we can walk, take steps. But how?

The ancient Greeks invented an atom, the smallest indivisible particle, to explain this. Before the atom, everything, according to them, was made of chaos. Chaos was just nothing until Eros fell in love with it and broke its symmetry. When the symmetry of nothing was broken, geometry emerged. Only geometry of nothing existed but nothing else. The matter was nowhere around. Geometry was continuous: it had no end either beginning and for that reason it was elusive. To catch it the ancient Greeks chopped it into pieces up to the smallest and indivisible — to the atom. Now we can structure chaos with atoms and it will turn into complete order. Computer models will cease to run away to the unknown, but they will accurately and invariantly calculate the future of every atom in the Universe, they probably proudly claimed. But it was too early to be proud.

No, of course, the ancient Greeks did not have computer models. And the atom, upon closer examination, turned out to be divisible. Zenon was laughing to tears on a cloud near Olympus, watching how the theory of his opponents collapsed. No problem there is a quantum, the smallest indivisible measure of the Universe! — Max Planck exclaimed and took his famous constant out of chaos in the same way as a magician takes a white rabbit out of his black cylinder, which used to be empty a moment before.

Now we have the smallest dots-grains, the indivisible measure of a phase trajectory, from which one solid and straight thread should go to the future, last century scientists had proudly claimed. But it was a bit early to be proud. When they took a closer look at how indivisible quanta actually behave they were stunned. One cannot say otherwise. It turned out that quanta, the tiny bitches, behaved exactly like those wriggling villi-worms on the phase state diagram of a complex system. In fact, they behaved even worse than that. Quanta behaves so weirdly that scientists till now can’t find enough adequate conceptual metaphors to explain its behavior so that we, the laymen, can imagine and understand it. The Schrodinger’s cat alone is a telling example of that weirdness but the genius polymath John von Neumann even proposed that a quantum can freeze in place only if a conscious creature looks at it. Turn away and it will begin to wobble like hell. Here it goes, quanta of matter in tomato sauce.

But let us return to the topological invariance of thinking, which helps to unambiguously separate real scientists from, perhaps, even very smart and well-read, but false ones. Let me end with a couple of descriptions of how real scientists think that the real scientists themselves left behind.

“The most beautiful thing we can experience is the mysterious. It is the source of all true art and science. He to whom this emotion is a stranger, who can no longer pause to wonder and stand rapt in awe, is as good as dead: his eyes are closed.”

Albert Einstein

“What sometimes enrages me and always disappoints and grieves me is the preference of great schools of learning for the derivative as opposed to the original, for the conventional and thin which can be duplicated in many copies rather than the new and powerful, and for arid correctness and limitation of scope and method rather than for universal newness and beauty, wherever it may be seen.”

Norbert Wiener

“And in fact, compared with immeasurably rich, ever young Nature, advanced as man may be in scientific knowledge and insight, he must forever remain the wondering child and must constantly be prepared for new surprises.”

Max Plank

PS: The topological invariance of the semantic space interests me, and not the topology of the phase states of all systems. In geometry, there is the established term “topological invariance.” In terms of meaning, it is similar to what I am writing about, but it is very special and therefore it is necessary to draw a line. The similarity ends with the example of the topological invariance of all simple polygons. They are all topologically invariant of a circle. They all satisfy two conditions (boundary parameters). They are all closed lines (1) that do not cross themselves (2).

PPS: I’ve stolen the idea of the topological invariance of thought from George Lakoff, a co-author of the theory of conceptual metaphor. He formulated the Invariance Principle in the following way, that “all metaphors are invariant with respect to their cognitive topology, that is, each metaphorical mapping preserves image-schema structure”.

His most powerful idea leads us to the domain of image-based intelligence, “If the Invariance Principle is correct, it has a remarkable consequence, namely that: Abstract reasoning is a special case of image-based reasoning. Image-based reasoning is fundamental and abstract reasoning is image-based reasoning under metaphorical projections to abstract domains.”