The expression mind is good and two is better means. A mind is good but two are better meaning of the proverb

Every joke has some humor in it.

The fairy tale is a lie, but there is a hint in it,
good fellows a lesson.

A.S. Pushkin

Task

In a class on probability theory, I work with students on a standard problem about the probabilities of two independent events. As an example – the operation of two devices. We made a table:

Everything is as usual: q=1–p, products of probabilities, overall reliability, ... In general, routine. All sorts of other examples are spinning in my head along the way. And then I remember the proverb - intelligence is good, but two are better! I'm starting to build an audience.

How to measure the mind?

If “the device is working,” then we can say that of its two states, one has been selected – working. We can assume that the working device has chosen the “correct option”. We denote the probability of success as p. This number can be chosen by the measure of intelligence. That is, in our model

intelligence is measured by the probability of choosing the correct option from two given ones

It is quite reasonable to believe that when p>½ we are dealing with a (comparatively) “smart” person - after all, he gives the correct answer more often. Accordingly, at p<½ приходится говорить о «глупом». Можно много времени потратить на уточнение границ, ввести диапазон для «среднего» ума и т.п. Ограничимся пока такой дихотомией.

The natural question is: what if we take two minds? After all, they say: intelligence is good, but two are better. What does this mean? After a short manipulation, I push the audience to the conclusion I need - the proverb states that the probability of choosing the right option is higher with two minds. Let's check.

Is two better?

We immediately see (situation No. 1 in the table) that for any p 1 and p 2 their product is less than any of the factors. The audience is shocked, I'm on a high. Finally, someone says hesitantly - but we must also take into account situations B and C, where one still chooses the correct option.

I praise the student, but there’s a reasonable question: how do you know which answer is correct? This is why we organize a “consultation for two”, to increase the chances of getting the correct answer. Expected stupor in the audience.

We come to the conclusion that the only available criterion of correctness in this case is only the agreement of both “minds” - after all, we do not know the correct answer. We are looking for the probability that both give the same answer:

P acc =P A +P D =p 1 ×p 2 +q 1 ×q 2 =1+2×p 1 ×p 2 –p 1 –p 2

My joyful remark that this is a simple hyperbolic paraboloid was met with disapproving silence from the audience. Here, fortunately for me, the lesson ended.

Rice. 1. The surface of “agreement” for two minds

On the way home, I quickly turned this surface around in my head and thought that (due to its symmetry) such a bi-system should have an equal chance of agreeing and disagreeing - 50 to 50. After constructing the surface P agree (p 1, p 2 ) this became obvious. The yellow cross at the level P agree =½ precisely divides the surface into symmetrical parts - the red area of ​​disagreement and the green area of ​​agreement. I imagined the “silence of the lambs” in the classroom - and realized that showing such a picture to students was cruel.

I note that in this model the problem of agreement for one mind does not arise - no matter what answer is chosen, it is always agreed with itself.

The Case of “Equal-Size” Minds

To simplify the problem, let’s assume that our minds are “equal in size” - their probabilities of giving the correct answer are the same and equal to p. Then

P acc. =f(p)=p 2 +(1–p) 2 =2×р 2 –2×р+1

This line is a section of the surface of agreement by the plane p 1 = p 2 (the edges of this plane and the line itself are indicated in blue).

Rice. 2. Agreement curve for two minds.

Here is a graph of this agreement function between two “equal minds”. The red diagonal line is the probability of the correct answer for one mind. And with some surprise we discover that for an intelligent person (after all, we assume p>½) the probability of making the right decision alone is higher than when we wait for the opinions of two equally intelligent people to coincide! Moreover, this difference reaches 1/8 at p = 3/4 (red arrow). That is, for a person with “above average” intelligence (for whom p = ½) to consult with another, even equally intelligent, is simply... harmful?

Why is “a mind is good, but two are better”?

But where then did such an “incorrect” saying come from? Let us recall the student’s timid proposal to consider those two events in which only one of the minds gives the correct answer. That is, we consider our pair of minds successful if at least one of them gives the correct answer! Or both will be right. The function is simple: f(p)=2×р–р 2 . And her graph is quite nice:

Rice. 3. The probability of the correct answer from at least one of two minds.

He pleases our soul, because the mind is good, but two are better. Especially with p=½, i.e., with stupid fortune telling. Then our chances increase by one and a half times - from ½ for one to ¾ for a couple (blue arrow).

Yes, but this is only good if someone knows the “truth” and can conduct such an analysis of the answers. For example, with a “team response” from a couple of students to a ticket question and a very favorable attitude from the teacher.

But what happens in reality when you don’t know what the “correct” answer is? After all, the chances that the answers will be opposite or the same are the same. So what then, which option to choose? What is the criterion for choosing a solution for a pair of minds? Again we return to the only criterion possible here - consistency of answers. Is it possible to understand the origin of our saying in the light of such a criterion?

Blue graph in Fig. 2 gives the answer to this question - probability consent never less than ½! It is curious that either very smart partners (with p close to 1) or very stupid ones (with p close to 0) agree more often. In this case, there is no need to even assume that they are “equal” – on the surface (see Fig. 1) we have the same thing.

If we assume that intelligence (as the probability of a correct answer) is distributed uniformly for people on the interval from 0 to 1, then it is easy to show that on average agreement will occur in 2 cases out of 3. I note that this is the upper limit, since very smart and very stupid yet they are found much less often than average minds.

Well, it has long been known - when consulting, we are not looking for the truth, but for justifications. And the theory of probability confirms this to us: no matter how smart or stupid we are (as long as we have “identical” minds, in our model), we will more often agree with each other than disagree. Whatever decision is made - right or wrong. For us it doesn’t matter anymore - after all, there is agreement! So people from the same social group, who have approximately the same “minds,” are more likely to agree with each other. And this leads to group unity. Have you received a rationale for conformity? Such a complex socio-psychological phenomenon - and just a theory of probability!? Hmm...

But if the minds are still different, for example, from different social groups? The answer “lies on the surface” (see the first picture) – they have equal chances to agree or disagree (assuming an even and independent distribution of minds in each group). If so, there is no point in talking to them! Justification for the separation of social groups?...

What about “think three of us”?

This step is completely natural - the group still wants to increase the likelihood of a correct decision. Agreement is agreement, but you have to eat. Let’s simplify the situation and draw up a similar table, immediately assuming the “equal size” of all three minds:

The principle of the majority in the troika gives nothing - there will always be no more than one dissenter! It turns out that it is generally impossible to assess the correctness of a decision made by voting on the basis of a majority? Indeed, in any distribution of votes, the correct decision is simply “assigned” here!

Rice. 4. The probability of agreement of three minds.

Well, it's not all bad. The probability of making the right decision based on the majority rule is easily calculated: f(р)=р 2 ×(3–2×р), graph in Fig. 4. As we see, there is some hope: when p>½, the troika still makes the right decision more often. True, at p<½ столь же часто принимает и неверные. Так что решению тройки можно доверять только тогда, когда мы уверены в квалификации её членов в данной области. Иначе почти наверняка будет хуже.

What if the correct answer is unknown? The majority of the three simply assign it at their discretion, but under the guise of choice. What are the chances of complete agreement, in which all three must choose the same solution? Then P acc =p 3 +q 3 =3×p 2 –3×p+1 (see graph).

Alas, we get disappointing conclusions for the trio:

• the smallest agreement is again achieved at p = ½, but its probability is already equal to ¼;
• the average probability of agreement (with a uniform distribution of intelligence) also decreases and is equal to ½;
• starting from p = 1/3, one person gets the correct decision more often than three people agree;
• the greatest difference between them (red arrow) is already equal to 1/3 and is achieved at a lower value p = 2/3.

It seems that those who claim are right - one is always smarter than a group of his own kind. In vain they think that this is purely humorous...

And you, friends, no matter how you sit down...

It is easy to generalize these findings to a larger number of “board members.” So, we have a council of k equally intelligent members, each of whom gives the correct answer with probability p. The decision criterion is unanimous. Whether it’s right or wrong is the tenth matter; no one knows the truth anyway. Here's a chart for the Magnificent Seven.

Rice. 5. Agreement curve for seven minds.

Oil painting:

• The minimum (blue circle) is 1 / (2(k–1)) .
• The average agreement is 2/(k+1).
• One is smarter than seven overall, starting with (red square) p≈0.2034 (numerical solution).

What is characteristic is that complete agreement is most likely either for the very stupid (small p) or for the very smart (large p). Accordingly, the left and right parts of the graph. It is clear that stupid people will agree on the wrong decision, and smart people will agree on the right one. But the tolerance for such consent is very narrow. A significant portion of the spectrum of like-minded minds (one thought for all?) will disagree with all the ensuing consequences. And as always, the “average smarts” will argue the most (p - in the range from ¼ to ¾).

This begs the conclusion: if a group of people argue all the time, then their average level of intelligence is very average?

What does the picture look like for the probability of a correct answer using the majority criterion? Yes, nothing new (see Fig. 6), only a further “exacerbation” of the noted dependence is planned - the advice of the stupid (p<½) почти гарантировано примет неверное решение. А где ж их набрать столько, умных-то? Получаем известный закон transition of quantity to lack of quality.

The most serious problem in human society is these are disputes and conflicts . We know how many people there are, so many opinions. There is nothing to be done about this, because each person is an independent person. The trouble is that the inevitable differences of opinion are constantly lead to mutual hostility , quarrels, and in extreme cases even wars.

What to do about it? To resolve disputes by peaceful means, humanity created a whole system of courts - at all levels, from the district magistrate court to international arbitration and the UN court. But practice shows that going to court is often also only partially solves the problem: the losing side often leaves the court irritated and angry, and relations between the parties become even more tense than before (we talked about this). And we see all this in various spheres of life: from sports to business, from conflicts between housemates to conflicts between states. The question becomes more and more acute: how to make a trial more perfect? How can we ensure that court decisions bring not only justice, but also peace to the parties to the conflict?

This is what the Mishnah says:

“Judge not alone, for none but the One can judge alone. And don’t say, “Accept my point of view,” because it’s their choice, not yours.”

The main idea is to make a judgment as collective as possible based on a wide range of opinions. This is logical: when the parties go to court, they thereby admit that they need someone else’s opinion on the problem in order to reconcile them. So let there be more than one opinion in court, and several! It is not without reason that it is said that “a mind is good, but two are better”: in our beit din, for example, there must be at least three dayan judges.

However, the question arises: if everything is resolved on a rational level, why does the Mishnah mention the One - that is, G‑d? Why G‑d can judge alone is understandable: He is the highest authority, He knows everything and does not need partners to judge objectively and fairly. In addition, the Mishnah specifically uses the word “One” to refer to G‑d here: by this it reminds that G‑d is everywhere, including within us, and His purpose is unite, ensure harmony of the world .

And from here two more conclusions follow. Firstly, when making a decision, judges must always remember the “truth of God” - that He is our father, He loves us and wants us to feel good. This means that when we undertake to judge other people, we must look for the good in everyone - after all, every person has a particle of G‑d. This requires the judge, first of all, to have personal modesty. He should not put himself above those people who came to him for judgment. And at the same time, he should not impose his own opinion on his colleagues: the task of a judge is to try bring together all the positive elements of different points of view.

On this score there is a story about the Besht court. On Yom Kippur there is one Jew in his synagogue. dropped the snuff box , scattered the tobacco and began to collect it during prayer. His neighbor was indignant: how can one tinker with tobacco on such a holy day, and even in a synagogue!.. The neighbor’s prayer reached the throne of the Most High, and Heaven pronounced a sentence: the sinner will not live to see next year! Of course, Besht learned about the verdict. He also learned that the verdict would be canceled if the one who convicted him changed his mind. Besht immediately went to that Jew and began to question him about how tobacco ended up in the synagogue. Gradually, his interlocutor realized that not everything is so simple: on Yom Kippur people fast, the neighbor needed tobacco just to stifle hunger so that he would have enough strength to pray. The Jew realized that he was wrong in condemning his neighbor, and thus saving his life .

As we understand it, the first lesson of the mishna is that to judge “in the righteousness of G‑d” means to judge positively. And the second lesson, in some ways even more important, is that our mishna talks not only about the judicial system , about resolving disputes between people. G‑d is not only in court - He is everywhere, and most importantly, He is in each of us. And we must profess the principles of “God’s truth” everywhere, in all deeds, words and thoughts .

It is not for nothing that the word “judge” has several meanings, in addition to the narrow legal one. A person “judges” - forms and expresses his opinion - about everything in the world . Now, in the age of computers and the Internet, there are the widest opportunities for obtaining information on any topic. Unfortunately, sometimes people confuse the verbs “to judge” and “to condemn” and talk about problems rather in a negative way. And the Mishnah warns against such an approach and demands find positive grain in different points of view . A person “judges” his own actions: he plans his steps and argues for them to himself. And here it is all the more important to hear what others think about you, because only the One knows everything, and only He alone can judge. 

Mind is good... Boris Sergeev.

Mind is good......is two better?

“A mind is good,” says an old Russian proverb, “but two are better.” The obviousness of folk wisdom, it would seem, is beyond doubt. However, do not rush to make a hasty conclusion. Two horses harnessed to one cart is quite normal. And two coachmen on the same box? Even with two horses, duplicating crew leadership is a harmful excess. And with one?

The question of the duality and contradictoriness of human nature has long been raised in the works of psychologists and psychiatrists, in the statements of philosophers, poets, and writers. There is probably no need to convince anyone of the validity of such a judgment. Surely each of us can illustrate this situation with examples from our own observations.

A strong argument in favor of the duality of the human psyche is the symmetry of the structure of our brain and the asymmetry of some of its functions, discovered at the beginning of the last century. Essentially, it was the discovery of differences in the activity of the cerebral hemispheres that was the first success in the study of the higher mental functions of the human brain and gave impetus to its systematic study. It was a serious blow to idealism and religion and helped many scientists believe in the knowability of the work of the brain, in the possibility of studying the mechanism of mental activity.

The entire subsequent course of studying the brain made it possible to reveal some of the mechanisms of its work and confirmed the strict specialization of the cerebral hemispheres. He put an end to the idea of ​​the soul as a special independent substance, which is the bearer of mental experiences and the cause of any vital manifestations of our body, but completely independent of it. The study of the human brain made it possible, with facts in hand, to answer the basic questions of philosophy about the knowability of the world, about the relationship of thinking to being, consciousness to matter.

Serious advances in understanding the functions of the human brain were made possible thanks to the joint efforts of a number of scientific disciplines, primarily anatomy, physiology, neurobiology, biochemistry, psychology, neurology, psychiatry, and linguistics. As in other branches of knowledge that arose precisely at the intersection of scientific disciplines, the development of neuropsychology is particularly intensive.

This book is a story about the formation and successes of neuropsychology, one of the recently formed areas of science that studies humans. A new scientific discipline was born at the intersection of psychology, neurophysiology and medicine. She studies the brain organization of various mental processes. It was neuropsychology that helped to understand the relationship between the two coachmen, invisibly sitting on the goats of our brain.

A significant contribution to the study of the brain was made by a glorious galaxy of representatives of Russian science - I. Sechenov, I. Pavlov, N. Vvedensky - who proclaimed a materialistic approach to the study of its functions and substantiated the reflex theory of its work. The scientific concepts of Sechenov and Pavlov had a decisive influence on the formation of materialistic psychology, which was greatly facilitated by the works of such outstanding scientists as L. Vygodsky, A. Leontiev and A. Luria.

Luria’s research, begun half a century ago, now continues to be successfully developed in the physiological laboratories of Tbilisi and Old Peterhof, the Institute of Evolutionary Physiology and Biochemistry named after I.M. Sechenov in Leningrad, in the laboratories of many scientific institutions of our state. This is the work of a huge army of Moscow psychologists, clinicians, physiologists and morphologists, as well as researchers from other cities of our country. All of them are associates, students or followers of Luria.

Thanks to their combined efforts, brain science has made such impressive strides today. The results of many years of research by Soviet scientists will be discussed on the pages of this book. The author dedicates his work to them

More often we hear another saying: “A head is good, but two are better,” but the meaning does not change. How to develop children's ingenuity, quick wit, switchability from one action to another - all that is formed in the everyday consciousness as “mind”? Everything is as usual - with the help of a variety of smart games and toys!

- one of mankind’s favorite pastimes since time immemorial. Wise rulers asked their courtiers intricate puzzles, the correct answer to which promised priceless riches, and a mistake could shorten the body by exactly a head. In those distant times, technology was not so developed, so puzzles were reduced to either labyrinths or intricate boxes that had to be opened without a key.

Modern puzzles combine knowledge accumulated over generations, as well as the achievements of scientific and technological progress. Complex tasks can be solved on the road, on the beach, at home near the fireplace or on an airplane - it all depends on their form and content.

So what puzzles should you pay attention to when purchasing? Let's divide them by age gradation.

Puzzles for kids (from 3 to 5 years old)

Puzzles for children and their parents (ages 5 and up)

The size of the puzzles is absolutely independent of their complexity; an example of this is a series of compact puzzles from the Professor Puzzle publishing house. When you first get acquainted with puzzles made of metal, the task seems elementary - to separate two simple parts. What's so difficult about this?! But this is only at first glance! Such puzzles can be purchased in the online stores “Igroved” and “OZON.ru” (price category from 500 to 1000 rubles, depending on the modification). Or, for example, Bambusler puzzles. The player needs to disassemble into component figures created from sticks fastened in a special way, which is not so difficult to do. Then you need to put everything together, and here there will be painstaking work... Mind games of this type are waiting for their owners in the online stores “Beautiful Game”, “Games4Brain”, “OZON.ru” (and here the price will depend on the modification and the number of puzzles in the set , on average from
400 rubles up to 1700 per set).

Puzzles for the road – you will definitely like this version of convenient, compact puzzles. You will be surprised to know that we spend about a tenth of our lives on the road. Original puzzles that can be solved, for example, while sitting in an airplane seat will help make the trip even more interesting. Travel versions of puzzle games must meet requirements that are not typical for regular games: all elements must be held in place even when shaking, and the playing field must also serve as a container for storing components. It is also desirable that the game parts are made of shockproof materials that are not afraid of falls and environmental influences.

This group can safely include puzzles that look like a compact book with a magnetic base - “Business Bugs”; “Cheese Loopholes” or “Magic Forest” - puzzles from this series fit into a neat handbag, and their bright design and modest price (within 500 rubles) make them an original gift. Such games are available for sale in the online stores “MosIgra”, “Tango and Cash”, “Mladenets.ru”.

“Advanced” parents know that the listed puzzles do not constitute the entire vast world of entertaining games that develop the mental abilities of children and their parents. Well, it’s up to you - follow the new products, choose and buy the best!

Based on site materials

A mind is good, but two is better(meaning) - Russian proverb meaning: 1. the best solution can be worked out by jointly discussing all the pros and cons. 2. Learning is a good thing. The more intelligence a person has, the better.

In the proverb, the phrase “two minds” means two people discussing a decision together.

- "One mind is half a mind; three minds are one and a half minds; two minds are a mind" (section " ").

The proverb is listed in the Big Explanatory and Phraseological Dictionary (1904):

A mind is good, but two is better (and three is better)

Examples

(1921 - 1997)

“Almost Seriously” (1976): “The insidious Pavel Borovikov had his own way of putting a drunk in his place. He approached him and said heartfeltly to the whole audience:
- Dear comrade! Thank you very much for helping me with my work. As the saying goes: " The mind is good, but... one and a half is better». "

(1883 - 1923)

"The Adventures of the Good Soldier Schweik" (1923, translation by P.G. Bogatyrev (1893 - 1971)): " Good mind - two is better. One will advise one thing, another another, “and the path is open to success,” as it is sung in our anthem.”

(1860 - 1904)

" " (1888), chapter IV - an old man says to a boy heading to study: - " A mind is good, but two is better. God gives one person one mind, and another two minds, and another three... Another three, that's true... One mind with which the mother gave birth, another from teaching, and the third from a good life. So, brother, it’s good if a person has three minds.”

(1821 - 1881)

"The Brothers Karamazov"

(1823 - 1886)

“Don’t sit in your own sleigh,” 1 Jan. 3: “It’s good that you do it, Ivanushka, that you go to your elders for advice. A mind is good, but two is better... Even though you’re a smart guy, listen to the old man.”

“What you go for is what you will find” (1861), map. 3, yavl. 1: “Can’t we sort it out together with Matryona. A good mind, they say, but two is better."