Logical paradoxes. Monte Carlo false conclusion

Gambler's fallacy

O.I., or Monte Carlo fallacy, reflects a common misunderstanding of the randomness of events. Suppose a coin is tossed many times in a row. If there are 10 heads in a row, and if that coin is the “right” coin, it would seem intuitively obvious to most people that there is a delay in landing tails. However, this conclusion is false.

This error has received the name “negative recency effect” in the specialized literature and consists of a tendency to predict the imminent cessation of something that often occurs in Lately events. It is based on the belief in local representativeness (i.e., the belief that a sequence of randomly occurring events will have the characteristics of a random process even when it turns out to be short). Thus, according to this misconception, a generator of random events, such as tossing a coin, should lead to outcomes in which - even after a short time - there will not be a significant predominance of one or another of the possible outcomes. If a series of identical outcomes occurs, there is an expectation that the random sequence will correct itself in the near future, and a deviation in one direction will thereby be necessarily balanced by a deviation in the other. However, randomly generated sequences, especially if they are relatively short, turn out to be completely unrepresentative of the random process that produces them.

The gambler's fallacy is more than just a reflection of ordinary statistical ignorance, since it can be observed in the private lives of even statistically sophisticated people. It reflects two aspects of people. cognitive function: a) a strong and unconscious motivation of people to find order in everything that they observe around them, even if the sequence of outcomes they observe arises as a result of a random process, b) universal human. the tendency to ignore calculation-based estimates of probabilities in favor of intuition. Although logic may convince us that a random process does not control its outcomes, our intuitive reaction can be very strong and at times overwhelm logic. Reed, who explored the comparative power of logical and intuitive thinking, argues that the latter is often more compelling than the former, probably because such conclusions come to the mind suddenly, therefore do not lend themselves to logical analysis, and are often accompanied by a strong sense of being right. In contrast to the fundamental impossibility of tracing the process by which such intuitive “solutions” are found, the process of logical reasoning is open to analysis and criticism. That's why people rule logical thinking, and from intuitive thinking they simply get results, which fill the latter with a strong sense of rightness.

O. and. most common in situations where outcomes are generated purely by chance. If some skill factor is involved in the development of events, a positive recency effect is more often observed. An observer is likely to view a series of successes (eg a pool player) as evidence of his skill, and will base his predictions of subsequent outcomes in a positive rather than a negative direction. Even throwing dice can lead to a positive novelty effect to the extent that the individual is convinced that the outcome of the event is somehow influenced by the “skill” of the thrower.

See also Barnum Effect, Player Behavior, Statistical Inference

176 Gya. 1K paradox in basic probabilities

e) Literature

Vapas 5., Tagb1 A. "5nr 1a yesogproyshchop ye epepegpyy ye rogp1y ep ragpe gerres11nepgepg sopigpep1e", Rnny. Mvy., 6, 244 - 277, 11924)

51gorpegi K. "Tie Vapas - Tag21 ragajokh", Tlv Lgpsysvp May. Mopiny, 66, 161 - 160, 11979).

3. The paradox of the Monte Carlo method

a) History of paradox

The Monte Carlo method is a numerical method based on random sampling. When solving computational problems, you can often find a suitable probabilistic model that includes the unknown number you are looking for. Then, to solve the problem, the outcomes of random experiments included in the probabilistic model are observed many times so that the desired number can be estimated with a given accuracy (based on the observed values). Although the idea of ​​this method is quite old, its real application began only with the advent of computers, when E. Neumann, S. Ulam and E. Fermi used the Monte Carlo method to approximately solve difficult computational problems associated with nuclear reactions. The name of the method is explained by the fact that it uses the sequences random numbers, which could be the regularly announced results of games held in casinos, for example, in Monte Carlo. However, in practice, the random numbers needed for the method are generated by the computer itself. Consequently, the cute name 1was first used in 1949 by N. Metropolis and S. Ulam) is misleading 1method is unlikely to help you win in Monte Carlo). The idea of ​​the Monte Carlo method first appeared in 1777 in the work of Buffon 1cm. 1. 11), which outlined a method for estimating the number n by throwing a needle at random. Suppose that parallel lines are drawn on the table at a unit distance from each other, and a needle of length E (1) is thrown onto the table at random, while the angle between the straight lines and the needle and the distance from the middle of the needle to the nearest straight line are independent random variables, uniformly distributed respectively on 10.2p) and 1 - 1/2, 1/2). Then the needle will intersect some line with probability 2b/n. If the experiment is carried out many times, then the relative frequency of intersections will be very close to the theoretical probability 2b/n, and in this way the value of n can be calculated. This method of finding an approximate value has a purely theoretical value, since in order to obtain two exact decimal places it is necessary to perform several thousand throws. 1Using another method, you can determine the mil-

8. The paradox of the Monte Carlo method

lion digits of n, see the article by G. Mila.) Buffon's needle problem shows that the Monte Carlo method is not suitable for very accurate calculations. Even obtaining results accurate to two or three digits requires thousands or millions of experiments. Therefore, the Monte Carlo method is only applicable when the experiments are simulated by a computer. Instead of throwing a needle, two independent random numbers are given that determine the position of the hypothesized needle and whether it intersects with the hypothesized straight lines. Since a computer can produce several million numbers per minute, simulating millions of experiments will not take too long; without a computer, this would take a lifetime.

The theory of generating random numbers on computers has become an important area in mathematics. Instead of real random numbers (which arise during random physical processes, for example, during radioactive decay), pseudo-random numbers constructed using deterministic computational algorithms are becoming popular.

In connection with non-pseudorandom numbers, the following question arises. In what sense can they be considered random if they are obtained using deterministic (non-random) algorithms? Since von Mises's paper in 1919, several eminent mathematicians have explored this problem. 1The philosophical aspects of the problem were dealt with by P. Kirschenmann, P. McShane and others.)

b) Paradox

In 1965 - 1966 Kolmogorov and Martin-Löf presented the concept of randomness in a new light. They determined when a sequence of 0s and 1s could be considered random. The main idea is as follows. The more difficult it is to describe the sequence 1t. that is, the longer the “shortest” program that constructs this sequence), the more random it can be considered. The length of the "shortest" program naturally varies from computer to computer. For this reason, a standard machine called a Turing machine is chosen. A measure of the complexity of a sequence is the length of the shortest Turing machine program that generates the sequence. Complexity is a measure of irregularity. Sequences of length L1 are called random if their complexity is close to maximum. 1It can be shown that most sequences are like this.) Martin Löf proved that these sequences can be considered random, since they satisfy all statistical tests

This episode with the clever missionary is one of the paraphrases of the paradox of the ancient Greek philosophers Protagoras and Euathlus.

But every researcher who tried to strictly define all the concepts in his theory encountered a similar paradox of formal logic. No one has ever succeeded in this, since everything ultimately came down to a tautology like: “Motion is the movement of bodies in space, and movement is the movement of bodies in space.”

Another version of this paradox. Someone has committed a crime punishable by death penalty. At the trial he appears the last word. He must say one statement. If it turns out to be true, the criminal will be drowned. If it is false, the criminal will be hanged. What statement must he make to completely confuse the judge? Think for yourself.

Puzzled by this paradox, Protagoras devoted himself to this dispute with Euathlus special essay"Pay Litigation" Unfortunately, it, like most of what Protagoras wrote, has not reached us. The philosopher Protagoras immediately felt that behind this paradox there was hidden something essential that deserved special study.

Aporia of Zeno of Elea. According to the laws of formal logic, a flying arrow cannot fly. A flying arrow at every moment of time occupies an equal position, that is, it is at rest; since it is at rest at every moment of time, it is at rest at all moments of time, that is, there is no moment in time at which the arrow moves and does not occupy an equal place.

This aporia is a consequence of the idea of ​​discreteness of movement, that a moving body in discrete units of time passes discrete intervals of distance, and the distance is the sum of an infinite number of indivisible segments that the body passes. This aporia raises a deep question about the nature of space and time - about discreteness and continuity. If our world is discrete, then movement in it is impossible, and if it is continuous, then it is impossible to measure it with discrete units of length and discrete units of time.

Formal logic is based on the concept of discreteness of the world, the beginning of which should be sought in the teaching of Democritus about atoms and emptiness, and perhaps in earlier philosophical teachings ancient Greece. We do not think about the paradoxical nature of formal logic when we say that speed is the number of meters or kilometers traveled by a body, which it travels per second or per minute (physics teaches us that distance divided by time is speed). We measure distance in discrete units (meters, kilometers, versts, arshins, etc.), time - also in discrete units (minutes, seconds, hours, etc.). We have a standard distance - a meter, or another segment with which we compare the path. We measure time with the standard of time (essentially, also a segment). But distance and time are continuous. And if they are discontinuous (discrete), then what is at the junctions of their discrete parts? Otherworld? A parallel world? Hypotheses about parallel worlds are incorrect, because are based on reasoning according to the laws of formal logic, which assumes that the world is discrete. But if it were discrete, then movement would be impossible in it. This means that everything in such a world would be dead.

Indeed, this paradox is unsolvable in binary logic. But it is precisely this logic that underlies most of our reasoning. From this paradox it follows that a true judgment about something cannot be built within the framework of this something. To do this you need to go beyond it. This means that the Cretan Epimenides cannot objectively judge the Cretans and give them characteristics, since he himself is a Cretan.

In science, this means that it is impossible to understand and explain a system based on the elements of only this system, the properties of these elements and the processes occurring within this system. To do this, you should consider the system as part of something larger - external environment, a larger order system of which the system we are studying is a part. In other words: in order to understand the particular, one must rise to the more general.

The paradox of Plato and Socrates
Plato: “The following statement of Socrates will be false.”
Socrates: “What Plato said is true.”
That is, if we assume that Plato is telling the truth, that Socrates is lying, then Socrates is lying, that Plato is telling the truth, then Plato is lying. If Plato lies that Socrates is lying, then Socrates is telling the truth that Plato is right. And the chain of reasoning returns to the beginning.

This paradox is that within the framework of formal logic, a judgment can be both true and false. This statement, which constitutes the liar paradox, is neither provable nor refutable in formal logic. It is believed that this statement is not a logical statement at all. An attempt to resolve this paradox leads to triple logic, complex logic.

This paradox shows the imperfection of formal logic, simply - its inferiority.

This paradox suggests that in order to characterize the elements of a system by the elements of this system, it is required that the number of elements in this system be more than two. Thesis and antithesis are not enough to characterize an element. If a statement is not true, then it does not follow that it is false. Conversely, if a statement is not false, this does not mean that it is true. It is not easy for our minds to agree with this statement, because we use formal alternative logic. And the case with the statements of Plato and Socrates suggests that this is possible. Judge for yourself: they tell us: “The ball in the box is not black.” If we think that it is white, then we may be mistaken, since the ball may turn out to be blue, red, or yellow.

In the last two examples we see that paradoxes are born from the defectiveness of formal (binary) logic. Let us think about how the phrase should be constructed correctly: “History teaches a person, but he learns nothing from history.” In such a formulation, with such clarification, there is no longer any paradox. The last two paradoxes are not antinomies; they can be eliminated within the framework of the laws of formal logic by constructing the phrase correctly.

The barber does not shave himself; Russell's paradox forbids him to do so. Photo from the site: http://positivcheg.ru/foto/837-solidnye-dyadenki.html

Russell's Paradox: Does the set of all sets contain itself if the sets included in it do not contain themselves (are empty sets)? Russell popularized it in the form of the “barber paradox”: “Barbers only shave people who don’t shave themselves. Does he shave himself?

There is a paradox of definition here: We began to build a logical construction without defining what a set is. If the barber is part of the multitude of people whom he shaves, then he must also charge himself for shaving. So what is the definition? But scientists often operate with concepts that they do not define in any way, which is why they cannot understand each other and argue pointlessly.

The concept of "empty set" is absurd by definition. How can a set be empty, containing nothing? The barber is not one of the many people whom he shaves as a barber. After all, any man shaves himself not like a barber, but like a shaving man. And a man who shaves is not a barber, since he does not charge himself for it.

A paradox from the category of antinomies is generated by an error in reasoning, in the construction of a phrase. The following paradox also applies to antinomies.

In this case, we must remember that a person must learn to think, and not just remember. Learning as mechanical memorization has no great value. Approximately 85-90% of what a person remembers while studying at school and university, he forgets during the first 3-5 years. But if he was taught to think, then he has mastered this skill almost all his life. But what will happen to people if, during training, they are given to memorize only those 10% of information that they remember for a long time? Unfortunately, no one has ever conducted such an experiment. Although...

There was one man in our village who completed only the 4th grade of school in the early 30s. But in the 60s, he worked as the chief accountant of a collective farm and did a better job than the accountant with a secondary technical education who later replaced him.

But if a ship is defined as a system, the essence of which is determined by its properties as a whole: weight, displacement, speed, efficiency and other characteristics, then even when all parts are replaced with similar parts, the ship remains the same. The properties of the whole differ from the properties of its parts and cannot be reduced to the properties of these parts. Whole more than the amount its parts! Therefore, even at 50 years old, a person remains himself, although 95% of the atoms of his body have already been replaced many times during this time by others, and there are more atoms in his body than there were at the age of 10 years.

So the ancient philosopher was not entirely right when he said that you cannot enter the same river twice, since the water flows in it and all the time its molecules in the stream are replaced. In this case, it is implicitly postulated that the river is the sum of precisely these water molecules and no other water molecules. But this is not so, because we perceive a river not as a set of water molecules, but as a flow of a certain depth and width, with a certain flow speed, in a word, a river is a dynamic system, and not the sum of its parts.

Balding orangutan. Photo from the site: http://stayer.35photo.ru/photo_125775

Balding dandelion. Photo from the site: http://www.fotonostra.ru/4101.html

Often the answer to the question about baldness lies in a different plane than the one in which it was formulated. To answer such a question, one must move from one plane of reasoning and perception to a completely different one. For example, the publications of one scientist are cited 100 times a year, and another 1 time a year. Question: which of them is a brilliant scientist? There can be four different answers to this question: 1 - no one, 2 - both, 3 - the first, 4 - the second. And all four answers in in this case equally probable, since the number of citations, in principle, cannot be a sign of genius. The correct answer to this question can only be obtained in 100 years or a little less.

The absurdity in this case stems from the lack of a clear definition of the concept of “democracy”. If the social system (state) is to be democratic, then equal representation from voters should be achieved. Equal representation from states if their populations are different is not a principle of democracy, but something else. Equal representation from parties is something third, from religious denominations - fourth, etc.

But in politics, even formal logic is not held in high esteem, and often it is violated deliberately in order to fool the electorate. In the USA, “brain powdering” technologies are simply excellently developed. Their elections are not democratic, but majoritarian, but the Americans firmly believe that they have Democratic state and are ready to tear apart anyone who thinks differently about their social system. They manage to pass off the aristocratic form of government as democratic. Are democratic elections possible in principle?

But in practice, the Monte Carlo conclusion may be false for another reason. After all, the condition about the independence of elementary events when playing roulette may not be satisfied. And if elementary events are not independent, but “linked” to each other in both ways known to us and still unknown... then in this case it is better to bet on black rather than red.

It may turn out that there are other carriers of energy and information in the Universe, and not just oscillations of the electromagnetic field and flows of elementary particles. If at its core the Universe is not discrete (vacuum), but continuous, then this paradox is inappropriate. Then every part of the Universe is influenced by the rest of it, then every atom of the universe is connected and interacts with all other atoms, no matter how far they are from it. But in the infinite Universe there must be an infinite number of atoms... Stop! The brains are starting to boil again.

This paradox stems from our misunderstanding of what time is. If time is the flow of the world with many channels (as is often the case with a river), and the speed of flow in the channels is different, then a sliver that falls into a fast channel will then again fall into a slow channel, when the fast channel merges with the slow one in which another sliver is floating , with which they once sailed next. But now one sliver will be ahead of its “friend” and will no longer meet with her. To meet them, the lagging “friend” must get into another fast channel, and the one ahead must swim in a slow channel at the same time. It turns out that the twin brother, who flew away on a sublight ship, in principle cannot return to the past and meet his brother. The slow flow of time (sublight ship) delayed him in the time flow. During this time, his brother not only grew older, but he went into the future, and with him everything that surrounded him went into the future. So, in principle, a brother who has fallen behind in time will no longer be able to get into the future.

And if the river of time does not have channels at different speeds, then there can be no paradox. Maybe the theory of relativity is incorrect, and time is not relative, but absolute?

The paradox of the murdered grandfather: you travel back in time and kill your grandfather before he met your grandmother. Because of this, you will not be able to be born and, therefore, you will not be able to kill your grandfather.

This paradox proves that traveling into the past is impossible. In order to get into the past, a person needs to turn into a different entity - move into the five-dimensional space of time, in which the past, present and future exist together - fused together, he will have to be born, die and live, and all this in the form of some kind of consubstantial phenomenon when "to be born, to live and to die" are not separate from each other. Becoming such a creature for a person means certain death - disintegration into subatomic particles. In general, we live in a four-dimensional world, and the way to the fifth-dimensional world is barred to us.

And thank God! Therefore, the grandfather is not in danger of his grandson coming from the future and killing him. And today there are many such grandchildren who have smoked marijuana.

China's Central Bureau of Film, Radio and Television recently banned time travel films because they "show disrespect for history." Film critic Raymond Zhou Liming explained the reasons for the ban by saying that now time travel is a popular topic in TV series and movies, but the meaning of such works, as well as their presentation, is very questionable. “Most of them are completely fictitious, do not follow logic and do not correspond to historical realities. Producers and writers are taking the story too lightly, distorting it and pushing this image on the audience, and this should not be encouraged,” he added. Such works are not based on science, but use it as an excuse to comment on current events.

I believe that the Chinese hit the nail on the head when they realized the harm of such films. Fool people with nonsense, passing them off as science fiction, dangerous. The fact is that such films undermine people’s sense of reality, the boundaries of reality. And this Right way to schizophrenia.

Salvador Dali showed the absurdity of our ideas about time through painting. The current clock is not time yet. What is time? If there were no time, there would be no movement. Or maybe it would be more correct to say this: if there were no movement, then there would be no time? Or maybe time and movement are one and the same thing? No, rather, with the help of the categories time and space, we are trying to characterize and measure movement. In this case, time is something like an arshin malalan. To travel in time, you must stop being living (living) people and you must learn to move within the movement itself.

There is no time, there is movement, and movement is time. All paradoxes associated with time arise from the fact that the properties of space are attributed to time. But space is a scalar and time is a vector.

Past and present. If it were possible to connect the past with the present like this, then in the evenings we could go for a walk in the yard of our childhood and meet there with childhood friends, and our childhood friends would be children, and we would be adults. But this is impossible to do. Time is not a characteristic of any movement, but a characteristic of irreversible movement. Even if you start the movement in a circle - loop it, then each cycle will differ in some way from the previous one. Photo from the site: http://kluchikov.net/node/76

The turtle felt sorry for Achilles and stopped. Only then was the exhausted and aged Achilles able to catch up with her and finally rest. Picture from the site: http://ecolours.pl/life.php?q=zeno-of-elea&page=2

In mathematics, an attempt to break out of the captivity of formal logic was the creation of differential and integral calculus. Both presuppose a continuous change of some quantity depending on the continuous change of another quantity. Column diagrams depict the dependence of discrete phenomena and processes, and graphs (lines) depict continuous processes and phenomena. However, the transition from a diagram to a graph is a kind of sacrament - something like sacrilege. After all, all experimental data (results of specific measurements) are discrete. And the researcher takes and draws a graph instead of a diagram. What is this? If we approach strictly, then the situation here is like this: a graph is a transformation of a diagram into a graph that approximates this diagram. By constructing a graph in the form of a continuous line, we make a transition from the world of discrete phenomena and objects to the continuous world. This is an attempt to break out of the boundaries of formal logic and thereby avoid its paradoxes.

In philosophy, already in the 19th century, scientists realized the inferiority of formal logic, and some began to try to solve this problem. They started talking together about dialectics, about the triad (Hegel), about a different theory of knowledge. Philosophers understood earlier than scientists that formal logic leads knowledge to a dead end. The result of the introduction of dialectics into science was, for example, the doctrine of evolution (development). After all, if you strictly adhere to the positions of formal logic, then development is impossible in principle. Preformationism is a pathetic attempt by formal logic to explain the evolution occurring everywhere. Preformists argue that everything is predetermined in some program in embryo, and the observed development is only the implementation (deployment) of this program. Formal genetics was born from preformationism, but it could only explain the development of the organism in ontogenesis. But formal genetics could not explain the change in species and macroevolution. It was necessary to add a new building to that original formal genetics, which turned out to be several orders of magnitude larger than the building of classical genetics, even to the point of denying discrete genes. But even in this modified form, genetics could only explain microevolution, and macroevolution was too tough for it. And the attempts that geneticists make to explain macroevolution produce paradoxes similar to those discussed above.

But even today the positions of formal logic are very strong in the minds of scientists: biologists, biophysicists, geneticists, biochemists. Dialectics has difficulty making its way into this science.

The paradox says that someone omnipotent can create any situation, including one in which he will be unable to do anything. In a simplified version, it sounds like this: can God create a stone that he himself cannot lift? On the one hand, he is omnipotent and can create any stone he wants. On the other hand, if he cannot lift a stone he created himself, then he is not omnipotent!

In reality, this does not happen, since in the world there are no identical things, phenomena, bundles of hay, or equivalent types of execution. Even if the bundles of hay are the same taste qualities and size, then one of them may be a little further than the other, or one of the donkey’s eyes may be more keen than the other, etc. Unfortunately, formal logic does not take this into account, so it should be used carefully and not in all judgments, and it should not always be trusted.

People in life and in their activities (including economic activity) do not behave at all like “ideal” balls in theory. In addition to benefits, people strive for sustainability and comfort in in a broad sense this word. An unknown risk can be either less than or greater than the known one. You can, of course, win more and become richer. But you can lose more and become bankrupt. But non-poor people give money on loan; they have something to value, and they don’t want to end up homeless.

Let's say I took 100 rubles from a friend, went to the store and lost them. I met a friend and borrowed another 50 rubles from him. I bought a bottle of beer for 20 rubles, I had 30 rubles left, which I gave to my friend and I still owed her 70 rubles. And I owed my friend 50 rubles, a total of 120 rubles. Plus I have a bottle of beer for 20 rubles.
Total 140 rubles!
Where are the other 10 rubles?

Here is an example of a logical fallacy embedded in the reasoning. The error lies in the incorrect construction of the reasoning. If you “walk” in a given logical circle, then it is impossible to get out of it.

The theory of black holes has become very fashionable in cosmophysics today. According to this theory, huge stars in which thermonuclear fuel “burns” are compressed - collapse. At the same time, their density increases monstrously - so that electrons fall onto the nuclei and intra-atomic voids collapse. Such a collapsed super-dense extinct star has strong gravity and absorbs matter from outer space (like a vacuum cleaner). At the same time, such a neutron star becomes denser and heavier. Finally, her gravity becomes so powerful that even quanta of light cannot escape her. This is how a black hole is formed.

This paradox allows us to doubt physical theory black holes. It may turn out that they are not so black after all. They most likely have structure and therefore energy and information. Moreover, black holes cannot absorb matter and energy indefinitely. In the end, having eaten too much, they “burst” and throw out clumps of super-dense matter, which become the cores of stars and planets. It is no coincidence that black holes are found in the centers of galaxies, and in these centers there is the highest concentration of stars escaping from these centers.

Any contradiction in the theoretical dogmas of science should encourage scientists to change (improve) the theory. Such a large number of paradoxes in logic, mathematics, and physics shows that not everything is going well in these sciences with theoretical constructs.

In 1850, the German physicist R. Clausius came to the conclusion that heat passes only from a warm body to a cold one, and never vice versa, which is why the state of the Universe must change more and more in a certain direction. Physicist William Thomson argued that everything physical processes in the Universe are accompanied by the conversion of light energy into heat. Consequently, the Universe faces “thermal death” - i.e. cooling down to absolute zero-273 degrees Celsius. Therefore, the infinitely long existence of a “warm” Universe in time is impossible; it must cool down.

The theory of the heat death of the Universe is, in all likelihood, a beautiful theory, but false. Thermodynamics does not take something into account, since its postulates lead to such a conclusion. However, gentlemen physicists love this theory too much and do not want to give up on it or at least greatly limit its applicability.

Another revolution in physics is brewing. Someone brilliant will create a new theory in which energy can not only be dissipated in the Universe, but also collected. Or maybe it gathers in black holes? After all, if there is a mechanism for the dispersion of matter and energy, then there must necessarily be an opposite process of concentration of matter. The world is based on the unity and struggle of opposites.

Photo from the site: http://grainsoft.dpspa.org/referat/referat-teplovoy-smerti-vselennoy.html

Clausius wrote about it this way: “The work that can be produced by the forces of nature and is contained in existing movements celestial bodies, will gradually turn more and more into heat. Heat, constantly moving from a warmer to a colder body and thereby trying to equalize existing differences in temperature, will gradually receive a more and more uniform distribution and a certain equilibrium will also occur between the radiant heat present in the ether and the heat located in bodies. And finally, with regard to their molecular arrangement, the bodies will approach a certain state in which, as regards the prevailing temperature, the total dispersion will be greatest possible.” And further: “We must, therefore, draw the conclusion that in all natural phenomena the total value of entropy can always only increase and not decrease, and we therefore obtain as short expression Always and everywhere the process of transformation takes place, the following proposition is: the entropy of the Universe tends to a certain maximum. (http://msd.com.ua/vechnyj-dvigatel/teplovaya-smert-vselennoj-i-rrt-2/)

But everything goes fine until a production crisis occurs. And with a production crisis in the United States, the balance of payments deficit disappears. A lot of capital has accumulated in banks, but there is nowhere to invest it. Capital lives only through circulation through production. As they say: “Airplanes only live in flight.” And capital lives only in the processes of production and consumption. And without production and consumption, capital disappears - it turns into nothing (yesterday it was, but today it is not), this causes the balance of payments deficit to grow in the USA - the airbags of other countries in US banks have disappeared without a trace. The United States, having made the dollar an international currency, has put itself on the dollar needle. The global economic crisis is sharply aggravating the situation and the health of the dollar “addict”. In an effort to acquire the next “dose,” the addict goes to great lengths and becomes aggressive.

China is developing well under socialism. Not at all because there is little private property there, but more state property. It’s just that the Chinese began to determine the price of goods by the demand for them. And this is only possible in a market economy.

The paradox of thrift. If everyone saves money during an economic downturn, aggregate demand will fall and, as a result, the total savings of the population will decrease.

I would call this paradox the paradox of Angela Merkel and Sarkozy. By introducing budget austerity in the countries of United Europe, politicians sharply reduced the population's demand for goods and services. The reduction in demand led to a reduction in production, including in Germany and France themselves.

In order to cope with the crisis, Europe must stop saving and must come to terms with the inevitability of inflation. In this case, part of the capital will be lost, but production will be saved due to consumption.

Photo from the site: http://www.free-lance.ru/commune/?id=11&site=Topic&post=1031826

But inflation will inevitably lead to the loss of capital - savings that the population keeps in banks. They say that under the euro, the Greeks lived beyond their means; the Greek budget had a large deficit. But by receiving this money in the form of salaries and benefits, the Greeks bought goods produced in Germany and France and thereby stimulated production in these countries. Production began to collapse, and the number of unemployed increased. The crisis also worsened in countries that considered themselves donors to the European economy. But the economy is not only about production and its lending. It's also about consumption. Ignoring the laws of the system is the cause of this paradox.

Conclusion

Concluding this article, I would like to draw your attention to the fact that formal logic and mathematics are not perfect sciences and, boasting of their proofs and the rigor of their theorems, are based on axioms taken on faith as completely obvious things. But are these axioms of mathematics so obvious?

What is a point that has no length, width or thickness? And how does it happen that the totality of these “incorporeal” points, if they are lined up in a row, is a line, and if in one layer, then a plane? We take an infinite number of points that have no volume, line them up in a row, and get a line of infinite length. In my opinion, this is some kind of nonsense.

I asked my math teacher this question back in school. She was angry with me and said: “How stupid you are! It’s obvious.” Then I asked her: “How many points can be squeezed into a line between two adjacent points, and is it possible to do this?” After all, if an infinite number of points are brought close to each other without distances between them, then the result is not a line, but a point. To get a line or plane, you need to place the points in a row at some distance from each other. Such a line cannot even be called dotted, because dots have no area or volume. It’s as if they exist, but it’s as if they don’t exist at all, they are immaterial.

At school, I often wondered: do we do arithmetic operations, such as addition, correctly? In arithmetic, when adding, 1+1 = 2. But this may not always be the case. If you add another apple to one apple, you get 2 apples. But if we look at this differently and count not apples, but abstract sets, then by adding 2 sets, we get a third one, consisting of two sets. That is, in this case 1 + 1 = 3, or maybe 1 + 1 = 1 (two sets merged into one).

What is 1+1+1? In ordinary arithmetic it turns out to be 3. But what if we take into account all combinations of 3 elements, first by 2, and then by 3? Correct, in this case 1+1+1=6 (three combinations of 1 element, two combinations of 2 elements and 1 combination of 3 elements). Combinatorial arithmetic at first glance seems stupid, but this is only true out of habit. In chemistry, you have to count how many water molecules you get if you take 200 hydrogen atoms and 100 oxygen atoms. You will get 100 water molecules. What if we take 300 hydrogen atoms and 100 oxygen atoms? You will still get 100 water molecules and 100 hydrogen atoms remaining. So, we see that a different arithmetic finds application in chemistry. Similar problems occur in ecology. For example, Liebig’s rule is known that plants are influenced by chemical element in the soil, which is at a minimum. Even if all other elements are in large quantities, the plant will be able to absorb as much of them as the minimum element allows.

Mathematicians boast of their supposed independence from real world, their world is an abstract world. But if this is so, then why do we use the decimal counting system? And some tribes had a 20 system. It's very simple, those southern tribes those who did not wear shoes used the base-20 system - according to the number of fingers and toes, but those who lived in the north and wore shoes used only their fingers when counting. If we had three fingers on our hands, we would use the six-digit system. But if we descended from dinosaurs, we would have three fingers on each hand. So much for the independence of mathematics from the outside world.

Sometimes it seems to me that if mathematics were closer to nature (reality, experience), if it were less abstract, if it did not consider itself the queen of the sciences, but if it were their servant, it would develop much faster. And it turns out that the non-mathematician Pearson came up with the mathematical chi-square test, which is successfully used when comparing series of numbers (experimental data) in genetics, geology, and economics. If you take a closer look at mathematics, it turns out that everything fundamentally new was introduced into it by physicists, chemists, biologists, geologists, and mathematicians best case scenario this was developed - proven from the standpoint of formal logic.

Non-mathematical researchers constantly pulled mathematics out of the orthodoxy into which “pure” mathematicians tried to plunge it. For example, the theory of similarity and difference was created not by mathematicians, but by biologists, the theory of information by telegraph operators, and the theory of thermodynamics by thermal physicists. Mathematicians have always tried to prove theorems using formal logic. But some theorems are probably impossible to prove in principle using formal logic.

Sources of information used

Mathematical paradox. Access address: http://gadaika.ru/logic/matematicheskii-paradoks

Paradox. Access address: http://ru.wikipedia.org/wiki/%CF%E0%F0%E0%E4%EE%EA%F1

The paradox is logical. Access address: http://dic.academic.ru/dic.nsf/enc_philosophy/

Paradoxes of logic. Access address: http://free-math.ru/publ/zanimatelnaja_matematika/paradoksy_logiki/paradoksy_logiki/11-1-0-19

Khrapko R.I. Logical paradoxes in physics and mathematics. Access address:

Finally, my hands and other organs got around to the next article.

So, meet the next guest in our studio - Gambler's error or Monte Carlo false conclusion. The term was not invented by me, although it sounds somehow pop, without abstruse words, typical of high-brow guys. This distortion is very simple to understand, nevertheless, it lives everywhere, both in the thin gray substance of the lumpen, who have reached the letter E in studying the alphabet, and in the dense thickets of raisins, wise by experience with a lot of knowledge of gray-haired sages. Here's what Wiki says about this:

The gambler's fallacy, or Monte Carlo fallacy, reflects a common misunderstanding of the randomness of events. This is due to the fact that, as a rule, a person is not intuitively aware of the fact that the probability of the desired outcome does not depend on the previous outcomes of a random event.

For example, in the case of tossing a coin many times in a row, a situation may well occur that will result in 9 tails in a row. If the coin is “normal”, then for many people it seems obvious that the next toss will be more likely to show heads: it is difficult to believe that “tails” can come up tenth time in a row. However, this conclusion is erroneous. The probability of getting the next head or tail is still 1/2.

However, it is necessary to distinguish between the concepts: the probability of “heads” or “tails” falling out in each specific case and the probability of “tails” falling out ten times in a row. The latter will be equal to . However, the probability of getting any other fixed sequence of “heads” and “tails” with 10 coin tosses will be the same.

What does this mean translated into our Pihar-trader language?

The simplest and most well-known example is the classic catch-up with a flat. Those. Popan plucks TB 2.5 no matter what match with odds of +-2, merges, doubles the bet on another match TB 2.5 with odds of about two, merges, doubles the bet again, etc. Well, or Martingale, call it what you want, that’s not the point. And if you suggest to him in the third or fourth iteration to push the total less, he will probably be indignant with the mega-argument “Why, there were already 3 TMs, now the probability of TB is higher.” And it turns out to be absolutely right. But only in your imaginary universe, in reality everything is somewhat different. The probability of a future event, other things being equal, does not depend in any way on past ones, even one or even a million. Axiom.

About a million. Recently we had a conversation with Kent on this topic (¡Hola senor Alejandro!). At some point, a person who perceives this world absolutely adequately responds to a simple question: “Before this, heads came up a million times. What is the probability that tails come up?” He replies that it’s a little bit, but still higher. We quickly eliminated this issue, but the situation is indicative.

Got off topic. So what should a person who has gotten caught up in a catch-up (of which I am a fierce opponent) do? The most important thing is not to think red or black, total is more or total is less, fish or chicken, nothing depends on you. Just give a damn about any outcome and hope in front of the TV, or better yet, go in for sports, sex, fishing, emphasize what you need. This way you will burn fewer calories from the “wrong choice”, which, in fact, never happened. Now mathematics (gods, fortune, mastushka, call it what you want) has turned its face or ass towards you, and there is nothing you can do about it. There is no need to catch up with seven iterations of the total more, feel free to give the total less, this does not affect the result in any way. More precisely, the only effect is that catching up will ultimately put you on your back, you can’t fool mathematics, the margin will do everything for you. For many years I watched the tops of the pihars on the pump room; among those who were successful at a significant distance there was not a single catcher, but that’s not about that now.

Let's take another example. At one time I communicated online during trading sessions with a well-known horse trader, I will not mention his name. So, he too was caught in the web of this cognitive error. His train of thought followed the following course: 3 times in a row the favorite mare came first, which means the next race of the fava needs to be laid. She won - hsn, layim fava in the next race with double the fury, then tripled, etc. And this “system” gave a profit for a certain period of time. But at one crappy moment the inevitable happened: mathematics defeated him, he got into such a mess that he left our orderly, albeit unstable, ranks for a long time. He couldn’t believe that this was possible, it took him a long time to accept, understand and rethink it, he was so depressed that a massage by Australian koalas would not have helped him at that moment. I think this is not an isolated case.

I had a case when I myself got into something similar. I vaguely remember the details, it’s a long time ago. The long-standing Italian Championship is a sad spectacle, catenaccio, draws - frequent guests. In one of the rounds there was not a single draw, and my fragile brain tells me that the trend will return in the next round. Stupidly took draws in all matches and... mega-suck, again no draws. But I’m a tough guy, you can’t take me that easily, in the next round I again take draws with double the bet (hello Illusion of Control) - and only one draw in the entire round. According to the classics of the genre, I had to push and fight back, but now everything will definitely be fine. But reality hit me deeper, I stupidly ran out of money. I’ll answer your question: I don’t know what happened in the next round, I didn’t watch the cuts, I thought I’d go crazy if I saw an ocean of nothing. An expensive lesson, but as it turned out, very useful.

I'll finish at 3 am. I’ll make a riddle to consolidate, independently analyze and improve the absorption of the above. What is the probability that Barcelona will not win at home against, say, Malaga twice in a row? Odds on p1 - 1.2. And how soon can this happen? The first person to answer correctly will get me a small fee, say, I’ll write an article on the topic of his choice.

So, to summarize. Don't look at what happened before, it doesn't matter. If you look at it, don’t draw any conclusions, they are subjective. We have drawn conclusions - do not make predictions from them, they are unreliable. Still, you have made a prediction - be prepared to easily change it, do not cling to it as the only true one (one of my favorite cognitive errors, let's talk about it another time). If you grab hold of it and can’t let go, go to a factory, get a job in a taxi, as a pizza delivery person, choose any other pick, games with probabilities are, alas, not for you yet. But don’t despair, read, work on yourself, improve your understanding of the processes happening in your head, drill your brain. Having passed through the oil-bearing and coal layers, sooner or later you will drill to states of mind that are not so ossified and compressed, and someday, with some degree of probability, you will be able to again set foot on the ornate path of non-Kail dough.

Players are undoubtedly aware of the Monte Carlo fallacy. Some, however, will be surprised to learn that this is a false conclusion - they consider it a “Monte Carlo strategy.” Well, that's exactly what the dealers are counting on.

We all know that the roulette wheel has half black and half red sections, which means we have a 50% chance that when you turn the wheel, you will land on red. If we spin the wheel many times in a row - say, a thousand - and it is in good order and there are no tricks on it, then red will come out about 500 times. Accordingly, if we spin the wheel six times and black comes up all six times, we will have reason to think that by betting on red we will increase our chances of winning. After all, red should come out, right? No it is not true. On the seventh time, the probability that red will appear will be the same 50%, as well as every next time. This is true no matter how many times black appears in a row. So here's some very sound advice based on the Monte Carlo error.

If you have to fly on an airplane, for your own safety, take a bomb with you: after all, the likelihood that two guys with bombs will meet on the same flight is extremely small.

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