The law of small numbers: unjustified conclusions based on insufficient information. Read on to test your logic skills by answering the hospital riddle and find out how charts can be misleading and what you can do to avoid losses when placing bets using statistical data.

The Riddle of Hospitals

In 1974, two psychologists Daniel Kahneman and Amos Tversky conducted an experiment in which subjects were described a situation and asked a question. Here is the situation. There are two hospitals in the same city. In a large hospital, about 45 babies are born every day, and in a small hospital, about 15 babies are born.

It is known that about 50% of all newborns are boys. However, the exact ratio varies from day to day. Sometimes more than 50% of boys are born, sometimes less. Within one year, both hospitals noted the days when the number of newborn boys exceeded 60%. Which hospital do you think has more days like this?

• In a big hospital.
• In a small hospital.
• Approximately the same (difference no more than 5%).

According to the binomial distribution theory, the number of days when at least 4-6 more boys were born than girls will be almost three times more in a small hospital only due to the more pronounced volatility of birth rates. A distribution in a large sample is likely to be less likely to deviate from 50%. However, only 22% of respondents gave the correct answer.

What is a heuristic?

Kahneman and Tversky explained that this misconception is due to people's belief in the law of small numbers. Generally speaking, conclusions drawn from data from small samples are often incorrectly considered to be representative of the larger population. For example, a small sample that appears to be randomly distributed will reinforce the belief that the larger population to which it belongs will also be randomly distributed.

The hospital conundrum: the distribution in a large sample is likely to be less likely to deviate from 50%. However, only 22% of respondents gave the correct answer.

On the other hand, a small sample that reveals seemingly obvious patterns (e.g. nine heads in a series of 10 coin tosses) would give the observer reason to believe that the same trend would be observed in the aggregate. In this case, we can assume that the coin is "biased", that is, the outcomes of its tosses cannot be considered fair. Perception, which is the ability to see patterns in random or meaningless data, is called apophenia.

Belief in the law of small numbers belongs to a broader group of mental tricks that people use when making decisions under uncertainty. Kahneman and Tversky called these techniques heuristics. Generalizations from small samples are an example of the representativeness heuristic, where people estimate the likelihood of an event based solely on generalizations from previous similar events that immediately come to mind.

Another example of the representativeness heuristic is the gambler's false inference. Indeed, this bias arises from belief in the law of small numbers. Kahneman and Tversky said the following:

"The essence of the gambler's problem of false inference lies in the misconception of the validity of the laws of chance." The player believes that in the case of a coin, the law of justice will operate in such a way that the deviation from the expectation of the fall of one side of the coin soon will be eliminated by deviating from the expectation of the other side of the coin coming up. People act like everyone element random sequence allows you to realistically evaluate true proportion aggregates; if the sequence deviated from the proportion of the population, a corrective bias in the opposite direction should be expected.

Reading Plots for Samples of Unequal Sizes

Sports bettors are especially prone to error in identifying patterns due to unwarranted belief in the law of small numbers. Misjudging profitability based on the analysis of a small sample of rates and taking it as a representative indicator of deviation from randomness and confirmation of predictive skills can lead to unpleasant financial consequences in the long run. Consider the graph below of the hypothetical profitability of 100 bets on NFL game score difference. All bets are made with odds of 1.95. Impressive, isn't it?

How would you react if you found out that this chart was compiled from the betting data of a well-known sports handicapper from the USA? Your gullibility is quite understandable, because the dynamics are quite good, and the income is 15%. But this, of course, is not true. In fact, the following 1000 bet chart gives you a better idea of ​​the situation.

In fact, long-term profitability was completely absent. The reason is that this data was obtained using a random number generator, which allowed us to determine that the probability of an individual winning is 50%, and the expectation of profit is -2.5%. The first chart simply represents the first 100 bets of the second chart.

But even in the second longer series of bets, the positive dynamics of profitability persisted for several hundred bets. In addition, despite the fact that there is a general unprofitability, the regularity inherent in the elements of this time sequence is not random and has a moderately stable wave-like dynamics.

However, as Kahneman and Tversky have recognized, people are much more likely to believe that sequences of similar outcomes are not random, even if there is no reason for this. Which of the two binary sequences below looks random and which doesn't?

0, 0, 0, 0 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1

0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1

Most people will choose the second sequence. In fact, the first sequence was randomly generated in Excel, and the second was specially formed in such a way that the segments with "1" and "0" were shorter. If you ask people to form random sequences that would look like the examples above, many will alternate "1" and "0" or vice versa if they feel that one of the numbers occurs too often.

Now consider the chart below for 1000 bets. All of them were randomly generated. The wide range of possible outcomes gives some idea of ​​how easy it is to be fooled by seemingly obvious patterns.


Don't forget that this series includes 1000, not 100 bets. Let's look at the average graph. It seems clear that the bet was made by a professional player or tipster: the return is 5% and a steady increase in profits is observed throughout the betting series - only the best handicappers are able to demonstrate such indicators for a long time. And yet it is the result of chance.

Using the binomial distribution, we can determine the probability of making a profit after several rounds of betting, even if the expectation is -2.5%.

Even though it's nothing more than a fluke, the odds of making a profit after a streak of 1,000 bets are still 1 in 5. If we were to place one handicap bet for every NFL game, it would take almost four season. It takes a long time for us to believe that only luck helped us.

How small should small numbers be?

The Law of Small Numbers is a cognitive bias in which people tend to believe that a relatively small number of observations accurately reflect the properties of the population. Also, as this exercise has shown, small things are sometimes quite large. This phenomenon exists because people prefer certainty, validity, causality, regularity, and skills (especially those focused on achieving personal goals) over uncertainty, ignorance, associativity, disorder, and randomness. Failure to realistically appreciate its value can be costly to sports bettors.

On the afternoon of April 18, 1775, a young boy working at a stable in Boston overheard a British officer saying something like this to another: "Tomorrow we will make hell for them." The guy immediately rushed to North Epd, Boston area, to break the news to Paul Revere, a silversmith. 11ol Revere listened to him in all seriousness: he was not the first person who had told him something like that that day. Prior to this, he had been informed of an unusual gathering of British officers, who looked like conspirators, on Boston's Long Wharf wharf. We also noticed a lot of British sailors in lifeboats along the sides of HMS Somerset and Boyne in the port of Boston. Several more British sailors were seen ashore this morning. They scurried around as if they were carrying out some important orders. By the end of the day, Paul Revere and his friend Joseph Warren were almost convinced that the British were about to take the drastic measures that had been talked about for so long. They are preparing to march on the city of Lexington, southeast of Boston, to arrest the colonist leaders John Hancock and Samuel Adams, and then attack the city of Concord and seize the depots of weapons and ammunition set up there by the people's militias.

What happened next has become part of the historical tradition, a legend that is told to all American schoolchildren. At ten that evening, Warren and Revere met. They decided that it was necessary to warn the neighboring cities of the impending attack, to raise volunteer militias to their feet and to meet the British properly. Paul Revere rushed to the Boston port, from there - to the ferry pier in Charleston.

At midnight he mounted his horse and galloped to Lexington. In two hours, he covered a distance of more than 20 kilometers. In every city that he met on the way - in Charleston, Medford, North Cambridge, Menothomi - he knocked on all the doors, reported the news of the British advance and asked to pass it on to the others. Church bells rang, drums beat. The news spread like a virus as those Revere told about it sent messengers of their own, and so on, until a disturbing message spread throughout the area. By 1:00 a.m., word had been heard in Lincoln, Massachusetts. By three in the morning - in Sadbury. By five in the morning - in Andover, a town 65 km northeast of Boston. And by nine in the morning the news reached Ashby, which is not far from Worcester, located 55 km west of Boston. When the British marched on Lexington on the morning of the 19th, they met, to their complete amazement, fierce and well-organized resistance already in its suburban areas. At Concord, the British were defeated by local militia units, and after that a military confrontation began, now known as the American Revolution.

The message spread by Paul Revere is perhaps the most striking example of a rumor epidemic in history. Exceptionally important news spread over a long distance in a very short time, forcing the entire area to take up arms. Of course, not all epidemics of rumor are so rampant. But it's safe to say that the spoken word, even in this age of media and multimillion-dollar advertising campaigns, remains

the most important form of communication. Recall, for example, the expensive restaurant you last visited, the expensive clothes you bought, the movie you watched. In how many cases did your choice of what to spend your money on be influenced by a verbal recommendation from a friend? Many people in the advertising industry believe that it is the intrusive omnipresence of today's advertising that has made word of mouth the only kind of persuasion that most of us still succumb to.

However, the origin of the rumor remains a mystery. People are constantly passing information to each other. But only in rare cases does such an exchange trigger a rumor epidemic. There is a small restaurant in my area, I like to sit in it and for about six months I kept telling my friends about it. But there are still few people there. My stories were clearly not enough to start a rumor epidemic, although there are restaurants no better than this, which opened only a few weeks ago, and they are not without customers. Why do some ideas, trends and messages cause an "explosion", while others do not?

In the case of Paul Revere, the answer seems obvious. Revere carried a sensational message: the British were coming. But if you get to know the events of that memorable night better, the explanation will no longer look so convincing and unambiguous. At the same time as 11ol Revere began his journey northwest of Boston, his associate, the tanner William Dose, set out on the same urgent message to Lexington through the cities located east of Boston. He carried exactly the same message, passed as many cities, covered the same distance as Revere. But after the message delivered by Doz, the counties did not take up arms. The commanders of the local militia units did not sound the alarm. One of the largest cities that lay in Dose's path was Waltham. But the next day, so few of its inhabitants fought the British that some historians later decided that the city was predominantly pro-British. However, this was not the case at all.

It's just that the people of Waltham found out too late that the British were coming. If the content of the message itself played a major role in the epidemic of word of mouth, Doz would now be as famous as Revere. But few people know about him. So why did Revere succeed where Doz failed?

The fact is that the emergence of a social epidemic of any kind depends to a large extent on the participation of people with a set of certain and rare communicative abilities. The message given by Revere set off a rumor epidemic, but that given by Doz did not, because they were two completely different people. This is where the law of small numbers comes into play, which I talked about briefly in the previous chapter. But there I cited as an example people who are promiscuous, sexually hyperactive, playing a decisive role in the spread of epidemics of sexually transmitted diseases. And this chapter is about the people who matter most to social epidemics and what separates Paul Revere from William Dose.

Such people are all around us. But we often do not notice the role they play in our lives. I call them Unifiers, Connoisseurs and Sellers.

In the late 1960s, psychologist Stanley Milgram conducted an experiment to answer what is commonly known as the "small world" problem. The crux of the problem is this: How are people related to each other? Do we belong to separate worlds living simultaneously but autonomously, so that there is very little connection between any two people on our planet? Or are we all entwined in a huge and complex web? My way Milgram

asked the same question with which this chapter began. How does an idea, or a trend, or a message (the Brits are coming!) spread among the people?

Milgram hoped to receive a response through a letter that is sent and transmitted through the chain, as "letters of chains." He chose 160 people living in Omaha, Nebraska, and sent a letter to each of them. The letter included the name and address of a stockbroker who worked in Boston but lived in Sharon, Massachusetts. Each recipient was asked to write their name on an envelope and send the package to a friend or acquaintance who could deliver the letter somewhere as close as possible to the broker. For example, if you live in Omaha, but you have a cousin near Boston, you can send a letter to him on the grounds that it will be easier for him to get to the broker in two, three, or four steps. The idea was that when the letter was finally delivered to the broker's house, Milgram would be able to look at the list of those in whose hands it had been before reaching its destination. On the basis of this, he wanted to establish how closely connected someone, taken at random and living in one part of the country, can be with someone from another part of it. Milgram learned that most of the letters reached the broker in five or six installments. Through this experiment, the concept of six handshakes was formulated.

Now many people know about it, and it is even difficult to imagine how amazing Milgram's discovery was in its time. Most of us don't have many friends. In one well-known study, a group of psychologists asked people living in the Diekman public housing complex in northern Manhattan to name close friends who lived there. It turned out that 88% of friends lived in the same building, half of them - on the same floor. In general, people chose friends close in age and of the same skin color. But if a friend lived in the neighborhood, then age and skin color no longer played such an important role. Spatial closeness overpowered personal resemblance. During another

A study conducted among students at the University of Utah found that if you ask someone why this person is friends with someone, the answer will be: because friends share the same outlook on life. But if you ask these two in detail about their views, it turns out that in fact friendship is based on joint activities. We make friends with people with whom we do things together, as well as with people who are similar to us. In other words, we are not looking for friends. We communicate with those who occupy the small physical space that we ourselves occupy. People from Omaha generally don't make friends with people across the country in Sharon, Massachusetts. “When I asked one of my learned friends how many stages he thought the packet would take from Nebraska to Sharon, he suggested that it would pass a hundred or more intermediate destinations,” Milgram wrote. “Many give roughly the same estimates and are very surprised when they find out that on average only five intermediaries are enough.” How did the package reach Sharon in just five steps?

The fact is that not all of these six handshakes are equivalent. When Milgram analyzed the results of the experiment, he found that many chains from Omaha to Sharon had the same asymmetric pattern. So, 24 letters reached the broker's house in Sharon, and 16 of them were handed to the addressee by the same person, Mr. Jacobs, a clothing salesman. The rest of the letters went to the broker's office, and most of them were transmitted through two people whom Milgram identified as Mr. Brown and Mr. Jones.

Thus, half of the letters that reached the broker were delivered by only three people. Think about it. Dozens of people, randomly selected in a large Midwestern city, sent letters independently. Some turned to their classmates. Others sent letters through relatives. Still others sent them through former work colleagues. Everyone had a different strategy. And in the end, when all these separate independent

the chains closed, half of the letters ended up in the hands of Jacobs, Jones, and Brown. The concept of six handshakes does not mean that someone is connected to someone in six steps. But it shows that a very small number of people are connected to us in several ways, and we are all connected to the rest of the world through these people.

There is an easy way to make sure this idea is correct. Let's say you've made a list of 40 people you can designate as your friends (not including family members and co-workers). In each case, try to remember the person who started the series of connections that eventually led to your friendship with someone. My oldest friend Bruce, for example, I met in my first year of high school, so I started that friendship myself. It's simple. I became friends with Nigel because he lived at the end of the hallway in the college dorm, but through my friend Tom, whom I met in my freshman year (he then invited me to play futsal). It was Tom who introduced me to Nigel. When you analyze all the connections, you will be surprised that the same names appear again and again. I have a friend named Amy, whom I met when her friend Katie took her to the restaurant where I was having dinner that night. I know Cathy because she is the best friend of my friend Larisa, whom I know because I was asked to meet her by our mutual friend Mike A., whom I knew because he went to school with another friend of mine, Mike X., who once worked for a political weekly with my friend Jacob. Similarly, I met my friend Sarah S. at my birthday party a year ago because she was a gambler with a writer named David who came to a party at the invitation of his agent Tina, whom I met through my friend Leslie, whom I I know because her sister Nina is a friend of my friend Ann, who I met through my old flatmate Maura, who was my roommate because she worked with a writer named Sarah L. who was a college friend

my friend Jacob. In fact, when I look at my list of forty friends, one way or another, I come back to Jacob. My social circle is not actually a circle. This is a pyramid. And at the top of that pyramid is one man, Jacob, who is responsible for most of my friendships. Not only is the social circle not a circle, but it is also not “mine”. It belongs to Jacob. It's more like a club he invited me to join. The people who connect us to the world, who bridge the gap between Omaha and Sharon, who bring us into their social circle, the people we depend on more than we realize, are the Unifiers, or people with a special gift for gathering.

What makes someone a Unifier? The first and most obvious criterion is: Unifiers know a lot of people. They know everyone and everyone. Each of us knows such a person. But I don't think we often think about how important such people are. We are not even sure that such a person who knows everyone actually knows everyone. But he really knows. And there is an easy way to demonstrate this. Below in this section is a list of 240 names taken at random from the Manhattan telephone directory. Go through the list and add yourself a point each time you see the last name of someone you know. (The word "acquaintances" in this case is interpreted quite broadly. For example, your fellow travelers on the train can be considered acquaintances if they gave their names, and you introduced yourself to them.) Duplicate names are considered, i.e. if it's a Johnson and you know three Johnsons, you get three points. The meaning of the test is to approximately determine how sociable you are. This is a very simple way to estimate how many friends and acquaintances you have.

Algazi, Alvarez, Alpern, Ametrano, Aran, Arnstein, Ashford, Bailey, Ballout, Bamberger, Baptista, Barr, Burrows, Baskerville, Bassiri, Butler, Bailey, Bell, Billy, Blau, Bok-geze, Bon, Borsuk, Bowen, Bravo, Brightman, Brandao, Brendle, Brook, Weinstein, Weisshaus, Waring, Wasillow, Weber, Wegimont, View, Villa, Water, Wong, Gardner, Garil, Hauptmann, Gelpy, Gilbert, Gladwell, Glascock, Glassman, Glazer, Gomendio, Gonzales, Horowitz, Goff, Grandfield, Greenbaum, Greenwood, Greenstein, Gruber, Guglielmo, Gourmet, Dagostino, Dali, Delacas, Dellamann, Gerard, Jerick, Diaz, Dillon, Dirar, Donahue, Dawson, Duncan, Eastman, Easton, Yara, Yon-son, Kavanau, Kalkaterra, Calle, Calleger, Kahn, Cantwell, Carrel, Cardboard, Couch, Keville, Keller, Keegan, Kiu, Kimbru, Kiesler, Kleine, Clark, Kozicki, Collas, Cohn, Korte, Co-soff, Cosser, Cohen, Crowley, Cook, Curbelo, Kuroda, Carey, Laber, Levin, Leibovitz, Leif, Leifer, Leonardi, Lin, Liu, Logrono, Lockwood, Locks, Long, Laurane, Lones, Lowet, Lund, Michaels, McLean, Marin, Maraudon , Marten, Matos, Mendoza, Murphy, Miranda, My, Muraki, Muir, Null, Neck, Needham, Noboa, O'Neill, Orlovsky, O'Flynn, Piper, Palma, Pao-lino, Perez, Perkins, Pons, Popper , 11ortocarerro, Potter, Pruska, Punvasi, Purple, Pierre, Raisman, Ramos, Rankin, Rastim, Reagan, Raider, Raze, Repe, Renkert, Ritter, Richardson, Roberts, Rosario, Rosenfeld, Roth, Rothbart, Rowan, Rose, Rus , Rutherford, Ray, Sadowski, Sutphen, Sigdel, Sears, Silverman, Silverton, Silverstein, Sklyar, Slotkin, Snow, Spencer, Speros, Stagoski, Steers, Stallman, Stopnik, Stone Hill, Stuart, Sirker, Theiss, Townshend, Temple , Tilni, Thorfield, Trimpin, Turchin, Fineman, Falkin, Farber, Fermin, Fialko, Filkenstein, Friedman, Famous, Habercorn, Hyman, Hardwick, Harrel, Hedges, Hemann, Henderson, Herbst, Hybara, Hogan, Hawkins, Hoskins , Hoffman, Hsu, Huber, Hussein, Chen, Chinlund, Chung, Shapiro, Shapirstein,

Swede, Sheehy, Schlee, Schonbrod, Steinkol, Edery, Elliot, Ellis, Andrews, Ashford, Jacobs, Yaroshi.

I gave this test to dozens of groups of people. One of them was freshmen from the Department of World Civilization at Manhattan City College. All students were about 20, many of them had recently immigrated to the United States and had a low income. The average score was 20.96, which means that each student knew about 21 people with the last name on my list. I also presented this test to a group of medical educators and professors at a conference in Princeton, New Jersey. The age of the participants in this group was 40-50 years. They were mostly white, highly educated, well-to-do people. Their average score was 39. Then I tested a group of my friends and acquaintances, mostly journalists, but also representatives of other professions in their 30s. The average score was 41. These results are not surprising. College students don't have as wide a circle of acquaintances as people in their 40s. Logically, between the ages of 20 and 40, the number of acquaintances should double, and professionals with high incomes should know more people than immigrants with small earnings. And in each group, there are those who get the lowest score and those who get the highest score. This is also logical. It is clear that real estate agents have more connections than computer hackers. It is surprising, however, how significant the gap can be. In college, the lowest score was 2 and the highest was 95. In my group of friends, the lowest score was 9 and the highest was 118. Even at the Princeton conference, where a rather homogeneous group of people gathered, the gap was huge. The lowest score was 16, the highest was 108. Overall, I tested about 400 people. Of these, about 20 scored less than 20, eight scored greater than 90, and four scored greater than 100. Another surprising thing is that I found people with high scores.

tatami in every social group with which he worked. On average, city college students scored lower than adults. But even in this group there were people whose social circle was four times wider than the others. In other words, there are people everywhere who have a truly extraordinary gift for making friends and acquaintances. These are the Combiners.

One of the highest scores I've ever met went to a man named Roger Horshaw, a successful Dallas businessman. Horchow founded the Horchow Collection, a large mail order company. In addition, he has achieved significant success on Broadway, producing successful productions such as The Humiliated, The Phantom of the Opera and the famous Gershwin musical Mad About You. I was introduced to him by his daughter (she is my friend). I went to meet him at his Manhattan apartment in a skyscraper on Fifth Avenue. Horshaw is thin, by nature he is a reserved person. He speaks slowly, in a Texan drawl. Charms the interlocutor with a slight self-irony, so that it is impossible not to succumb to his charm. If you happen to be near Roger Horshaw on a plane crossing the Atlantic, he will speak to you before takeoff. When the inscription “fasten your seat belts” appears, you will already be laughing with might and main, and when you land on the other side of the ocean, you will be surprised to note that time has flown by completely unnoticed. When I gave Horshaw the list from the Manhattan directory, he went through it very quickly, mumbling names and sliding his pencil along the lines. His result was 98 points. I suspect it could have been higher if I had given Roger another ten minutes to think.

Why was Horshaw so successful in his task? After meeting him, I was convinced that the ability to make acquaintances is a kind of talent that can be quite consciously developed. I asked Horshaw many times about how his many contacts helped him survive in the business world, because it seemed to me that there was a direct connection. However, this issue seems to

pissed off Roger. It's not that the connections don't help him. The fact is that he does not consider them as part of his business strategy. He treats communication as one of the aspects of his life. It's in his nature. Horshaw has an instinctive and natural gift for making friends. However, he does not show much zeal. He's not one of those overly talkative, back-slapping types who assert themselves by trying to appear very outgoing. By nature, he is more of an observer, always remaining a little aloof. He just really likes people. He finds the process of making acquaintances and communication endlessly interesting. When I met with Horshaw, he told me how he managed to get the rights to a new production of Gershwin's musicals Crazy Girl and Mad About You. The whole story took 20 minutes. That's just part of it. If you suddenly think that Horshaw is calculating, remember that this is not so. He told the whole story with his usual light self-irony. I even think that he intentionally stuck out some traits of his character. However, this story gives a complete picture of how his mind works, as well as what makes a person a Unifier:

“I have a friend in New York named Mickey Scheinen. Once he said to me: “I know you love George Gershwin. And here I am with his old girlfriend. Her name is Emily Paley. She is the sister of Ira Gershwin's wife, Lenore. She lives in the Village and invited us to dinner.” That's how I met Emily Paley and saw a painting of Gershwin. Her husband, Lou Paley, wrote music with Ira and George Gershwin at a time when Ira still called himself Arthur Francis. This is one thread...

And then I had lunch with Leopold Godowsky, son of Frances Gershwin, sister of George Gershwin. She married the composer Godowsky. Arthur's son was with us

Gershwin. His name is Mark Gershwin. And so they say, “Why should we give you the rights to Crazy Girl? Who are you? You have never been to the theatre.”

And then I began to remember my connections. Your aunt Emily Paley - I've been to her house. Her portrait in a scarlet shawl, have you seen this picture? I remembered the smallest details. Then we all went to Hollywood together and there we went to Mrs. Gershwin's house. I said that I was very happy to meet her, that I knew her sister and that I loved her husband's music. And then he mentioned my friend from Los Angeles. When I was working for Neiman Marcus, a lady wrote a cookbook. Her name was Mildred Knopf. Her husband Edwin Knopf is a film producer. He worked with Audrey Hepburn. And his brother was a publisher. We distributed the book in Dallas and I became friends with Mildred. She is an amazing person, and every time I get to Los Angeles, I definitely look at her. I always keep in touch. Well, it turned out that Edwin Knopf was Gershwin's closest friend. The Knopfs have his photos all over the house. Knopf was next to Gershwin in Asheville, just as he was writing Blues Rhapsody. Mr. Knopf is already dead, but Mildred is alive. She is now 98. And so, having come to visit Lee Gershwin, I immediately mentioned that we had just from Mildred Knopf. She exclaimed: “Do you know her? Ah, how have we not met before?” And then she gave me the rights."

Telling all the ego, Horshaw again and again rejoiced at how these life threads connected with each other. On his seventieth birthday, he tried to find Bobby Hunsiker, an elementary school friend he hadn't seen in 60 years. He sent a letter to every Bobby Hunsiker he could find in the directory. In the letter, he asked, "Are you the Hunsiker who lived at 4501 Perth Lane, Cincinnati?"

This behavior looks somewhat unusual. Horshaw collects people like someone else collects stamps. He remembers the boys he played with 60 years ago, the address of his long-grown best friend, the name of the man his college friend went crazy over when she was a freshman overseas. These details are very important to Horshow. He has a list of 1600 names and addresses on his computer, and under each entry there is a note under what circumstances he met this or that person. While we were talking, he took out a small pocket diary. “If I met you and liked you and if you told me your birthday, I will write it down here, and then you will receive a birthday card from Roger Horshaw. Look, Monday was Ginger Vroom's birthday, and the Wittenbergs had their first anniversary. And Alan Schwartz has a birthday on Friday, and our gardener has a birthday on Saturday.”

Most of us do not support casual dating. We have our own circle of friends, and we remain faithful to it. And keep everyone else at a distance. We don't send cards to people who aren't particularly important to us because we don't want to be obligated to dine with them, go to the movies with them, or visit them when they're sick. For the most part, we make acquaintances in order to assess whether we want to make this or that person our friend. It seems to us that we have neither the time nor the energy to maintain close contact with everyone.

Horshaw is completely different. The people whose names he enters into his computer or diary are just acquaintances (those with whom he can meet once a year, or even once every few years), and he does not shy away from the duties associated with maintaining all these contacts. . He mastered what sociologists call weak ties - friendly but irregular contacts. Moreover, he is ideally suited for these weak ties. After meeting with Horshaw, I even felt a little sad. I would like to get to know him better, but I am not sure that I will have such an opportunity.

ness. I don't think he shared that sadness with me. It seems to me that he is one of those who knows how to find joy in fleeting and random encounters.

Why is Horshaw so different from the rest of us? He doesn't know himself. He thinks it has something to do with the fact that he was the only child in the family, and his father often left home. But this is hardly the only reason. Perhaps the drive that drives the Unifier is best explained as just one of the many personality traits that distinguish one person from another.

Unifiers don't just know a lot of people - they know all sorts of people. Perhaps a better understanding of what is meant by this is the popular board game Six Steps to Kevin Bacon. The point of the game is to connect any actor or actress through the films they have played in with actor Kevin Bacon and do it in less than six steps. So, for example, O'Jay Simpson played The Naked Gun with Patricia 11resley. She starred in Ford Fairlane with Gilbert Gottfried. He played in Beverly Hills Cop 2 with Paul Reiser, who played in The Visitor with Kevin Bacon. Ego four steps. Mary Pickford played Screen Tests with Clark Gable and then played America Fights with Tony Romano. After 35 years, Romano starred in the movie "Starting Again" with Kevin Bacon, and it's three more steps away.

Recently, programmer Brett Tjaden of the University of Virginia calculated the average "Bacon number" for about a quarter of a million actors and actresses who played in television series or famous movies. He deduced a value of 2.8312 steps. In other words, anyone who has ever acted in a movie can be linked to Bacon in less than three steps, on average. Ego is impressive. However

Tjaden decided not to stop there and made a truly incredible calculation, calculating the average degree of contact of everyone who has ever played in Hollywood. For example, how many steps does it take to link someone to Robert De Niro or Shirley Temple or Adam Sandler? Arranging all the Hollywood actors in his list in order of their "contact", Tjaden found that Bacon was only in 669th place. Martin Sheen can be linked to any actor in just 2.63861 steps, which puts him 650 steps above Bacon. Elliot Gould can be linked to anyone even faster - in 2.63601 steps. Among the top 15 are Robert Mitchum, Gene Hackman, Donald Sutherland, Shelley Winters and Burges Meredith. And who is the most contact actor of all time? Rod Steiger.

Why is Kevin Bacon so far behind the leaders? One of the reasons is that he is younger than most of them and therefore has acted in fewer films. But there are actors who have played in a variety of films, who nevertheless do not have extensive connections. For example, John Wayne has appeared in 179 films over his 60-year film career, but is only in 116th place, 2.7173 steps away from Kevin Bacon. The problem is that more than half of John Wayne's films are westerns. That is, he played in the same type of films, along with the same actors, over and over again.

Now let's take someone like Steiger: he's been in some of the most famous films like the Oscar-winning On the Waterfront or the horror movie The Parking Lot. He won an Oscar for his role in the film "Stuffy Hot Night" and at the same time starred in the category "B" films, so worthless that they were immediately sent to video rentals. He played Mussolini, Napoleon, Pontius Pilate and Al Capone. He has appeared in 39 dramas, 12 detectives and comedies, 11 thrillers, eight action films, seven westerns, six war films, four documentaries, three horror films, two sci-fi films and one musical. Rod Steiger -

he is the most connected actor in the history of cinema because he has been able to move up and down, back and forth, through the different worlds, subcultures, niches and levels that the acting profession offers.

Here he is, the Unifier. This is Rod Steiger of everyday life. This is a person that we can connect with in just a few taps, because for one reason or another, he manages to be in many different worlds, subcultures and niches at the same time. Steiger owes his extensive connections to his versatile acting talent, as well as, to a certain extent, luck. But in the case of the Unifiers, their ability to build bridges across the most diverse worlds is a derivative of something inherent in their personality, a combination of curiosity, self-confidence, sociability and energy.

One day I met in Chicago with the classic Unifier - Lois Weisberg. She then worked as a commissioner for cultural affairs in the city administration. But this is only the latest of her many positions and professions. In the early 1950s, Weisberg directed a theater company in Chicago. In 1956, she decided to organize a festival in honor of the 100th anniversary of the birth of Bernard Shaw, then she began to publish a newspaper dedicated to Shaw, which eventually transformed into an alternative magazine The Paper. On Friday evenings, people from all over the city gathered for editorial meetings hosted by Weissberg. William Friedkin, who later directed the films The French Connection and The Exorcist, was a regular visitor here. The lawyer Elmer Hertz, who later became one of Nathan Leopold's lawyers, also came here. Dropped by Weisberg and some editors of the magazine Playboy whose building was on the same street. Stopping in the city, Art Farmer, Thelonious Monk, John Coltrane and Lenny Bruce came here. (Bruce actually lived at Weissberg's for a while. "My mother was on the verge of hysterics because of this. Especially one time, when she rang the doorbell and he opened it for her in a bath towel, -

Weissberg said. - We had a window on the terrace, but he didn't have a key. Therefore, the window was always kept open. The house was full of rooms and there were always a lot of people staying there. I didn't even know who was there. I couldn't stand his jokes. And I really didn't like the way he played. All these words of his infuriated me.") After The Paper closed, Lois took a job in the public relations department at the Post Traumatic Rehabilitation Institute. From there, she moved to a public interest law firm. The firm was called BPI. While working there, Lois became concerned about the deplorable state of Chicago's parks. Then she gathered a motley bunch of nature lovers, historians, community activists and housewives and founded the Friends of the Parks lobbying group. Then she became worried about the impending dismantling of the commuter railroad that ran along the south shore of Lake Michigan, from South Bend to Chicago. And Lois again gathered railroad enthusiasts, environmentalists, passengers of this line and founded the South Shore Restoration community group. And saved the railroad. Then she became the executive director of the Chicago Bar Council, led the election campaign of a local congressman. Then she received the position of director of the special events department under the first black mayor of Chicago, Harold Washington. Subsequently, she left the administration and opened a flea market stall, and later went to work for Mayor Richard Daley (she still works for him) as a cultural commissioner.

If you follow her story and count how many “worlds” Lois belonged to, it comes out to eight: actors, writers, doctors, lawyers, politicians, park lovers, railroad lovers, flea market regulars. When I asked Weisberg to make her own list, she got 10 because she added architects and people from the hospitality industry. But perhaps she was modest, because, if you take a closer look

look at Weisberg's life, you can pick out another 15 or 20 worlds from her connections. Although these are not separate worlds. The peculiarity of the Unifiers is that, being in so many different worlds, they bind them all together.

One day (it was sometime in the mid-1950s) Weisberg took the train to New York to attend a science fiction convention. Just. At the convention, she met the young author Arthur C. Clarke. And he liked it. The next time he was in Chicago, he called her. “He called from a pay phone,” Weisberg recalls, “and asked if there was anyone in Chicago he should meet. I told him to come to me."

She has a low, husky voice - from the fact that she has been smoking for half a century. She pauses between sentences to take a drag. And even when he doesn’t smoke, he still pauses, as if preparing for the moment when he smokes. “I called Bob Hughes. He was one of those who wrote for my The Paper. Pause. “I asked him if he knew anyone in Chicago who would be interested in talking to Arthur Clarke. He replied, “Yes, Isaac Asimov is in town right now. And this guy, Robert, Robert... Heinlein.” And they all came and gathered in my office.” Pause. "Then they told me, 'Lois - you...' I don't remember the word, they called me something, but the bottom line was that I'm a person who brings people together."

At first, she was drawn to someone who was not from her world: she was doing theater then, and Arthur C. Clarke was writing science fiction. Then, just as importantly, this man answered her. Many of us are drawn to someone who is different from us, more famous or more successful than us, but our gesture is not always accepted. And then Arthur Clark comes to Chicago and wants to contact someone, make contacts. And Weisberg brings him together with Isaac Asimov. She says that it is a happy accident that Asimov might not be in the city... But if it wasn't Asimov, then there would be someone else.

All the participants of those Friday evenings that Weissberg hosted in the 1950s remembered how easily they found a common language there. And it's not that nowhere else African Americans could communicate with whites from the North Side area. Such communication, although it was then a rarity, but still happened. The important thing is that in Chicago in the 1950s, if African Americans interacted with whites, it did not happen by chance. This happened only when a certain type of person did everything to ensure that such communication took place. This is what Asimov and Clarke meant when they said that Weisberg had the ability to bring people together.

“There is absolutely no snobbery about her,” says Wendy Willrich, who worked for Weisberg. - Once I went with her to a professional photo studio. People wrote letters to her, and she looked through them all. And so the owner of this photo studio invited her to his place, and she agreed. He photographed mostly weddings. She decided to see it all with her own eyes. I thought, “Oh my God, why are we dragging ourselves into this studio?” She was right next to the airport. Remember, this is the Commissioner for Cultural Affairs of the City of Chicago. But Lois found it all incredibly interesting."

Was this photographer all that interesting? Who can say? However, Lois found him interesting because, in one way or another, all people are interesting to her.

“Weisberg,” one of her friends told me, “always says: “Oh, I met an absolutely amazing woman. You will definitely like it.” And she is full of enthusiasm and delighted with this person - the same as from the person she met before. And you know what, she's always right." Helen Doria, another friend of hers, said that "Lois sees things in you that you don't see in yourself."

The same idea can be expressed this way: by some quirk of nature, Lois and people like her have some kind of instinct that helps them maintain relationships with those they wind-

tea on their life path. When Weisberg looks around, or when Roger Horshaw sits next to you on an airplane, they see a very different world from the one we all see. They see opportunities, and while most of us choose who they want to do business with, rejecting those who don't look right, or live too far away, or who haven't been seen in 65 years, Lois and Roger they all like it.

An excellent example of how Uniters work is provided by sociologist Mark Granovetter. In his classic 1974 study Getting a job, Granovetter described the stories of several hundred professionals and workers from the Newton suburb of Boston. He questioned them in detail about how they got their jobs. It turned out that 56% of respondents found their place thanks to personal connections. Another 18.8% looked for work through advertisements and through recruitment agencies, and about 20% applied directly to the employer. This is not surprising: the best way to get somewhere is to do it through personal contacts. However, it is curious that most of these contacts were weak ties. Of those who found a job through an acquaintance, 16.7% met this acquaintance "often" (as with a good friend), 55.6% - only "from time to time", and about 28% of respondents - "rarely" at all . That is, people did not find work with the help of close friends.

Why is that? Granovetter argues that when it comes to finding work (or information, or ideas), weak ties are always more important than close ones. After all, your friends revolve in the same circles as you. They may work with you or live next door, go to the same church, the same school, or the same parties. How much can they know of what you do not know?

And your casual acquaintances, by definition, occupy a different space. They are much more likely to know something that you don't. Granovetter called this apparent paradox the power of weak ties. In other words, acquaintances are the source of social strength, and the more acquaintances you have, the stronger you are. Unifiers like Lois Weisberg and Roger Horshaw, who are masters of weak ties, are exceptionally powerful. And we rely on them to access opportunities and worlds to which we ourselves do not belong.

This principle applies, of course, not only to job hunting, but also to restaurants, movies, fashion, and anything that depends on the spoken word. And the point is not that the one who is closer than others to the Unifier gains more power, wealth or opportunities. Could the Uniter be one of the links in the chain of reasons why Hush Puppies suddenly became mainstream? Somewhere along the path from the East Village to "single-story America," the Unifier or a group of Unifiers became enamored with these shoes and through their countless personal contacts, through endless threads of weak ties, using their presence in numerous worlds and subcultures, managed to spread the word about it simultaneously in thousands of directions. They provided her with a real breakthrough. Hush Puppies, one might say, got lucky. And, perhaps, many fashionable novelties never find themselves on the crest of popularity for one simple reason - because of ordinary bad luck. They do not meet the Unifier on their way.

Sally, Horshaw's daughter, told me how she once invited her father to a new Japanese restaurant where a friend of hers was the chef. Horshaw's cuisine was very good. When he returned home, he turned on his computer and sent letters to friends who lived nearby, where he announced a great new restaurant that he discovered for himself and which they should visit.

That's the power of the word! When I tell my friend about a new restaurant, and he tells another one, and he tells another one, this is

not at all. Word of mouth starts where someone in the thread talks about a new restaurant to someone like Roger Horshaw.

And here's an explanation for why Paul Revere's midnight ride started a rumor epidemic, and William Dose's trip ended in nothing. Paul Revere was the Roger Horshaw and Lois Weisberg of the time. He was the Unifier. Apparently, Paul was a talkative and extremely contact person. When he died, his funeral was attended by, as one newspaper in those days put it, "hordes of people." He was a fisherman and a hunter, a gambler and a theatergoer, a frequenter of bars and a successful businessman. He was active in the local Masonic lodge and a member of several select clubs. He was active, he was endowed, as David Fisher says in his book Paul Rever's Ride ("The Way of Paul Revere"), "with a supernatural gift to always be in the center of events." Fisher writes:

“When the first street lamps were brought into Boston in 1774, Paul Revere was asked to serve on the committee that handled it. When the Boston market demanded regulation, Paul Revere was appointed secretary of the board. After the revolution, during the epidemic, he was elected health inspector of Boston and coroner of Suffolk County. After a terrible fire in the old wooden town, Revere helped found a cooperative insurance company, and his name was the first in its charter. When the problem of poverty confronted the young republic, he called a meeting at which the Massachusetts Artisan Benevolent Association was founded. Revere was elected chairman of the association. And when there was controversy among the people of Boston over a sensational murder trial, Paul Revere was chosen foreman of the jury."

If Paul Revere had been given a list of 250 names taken at random from the 1775 Boston census, he would undoubtedly have scored well over 100.

After the Boston Tea Party of 1773, when the hatred of American colonists for their British rulers began to spill over, dozens of committees and congresses began to spring up all over New England like mushrooms after rain. They had neither a formal status nor established ways of interacting. But Paul Revere quickly assumed the role of link between all these widely separated centers of revolution. He went to Philadelphia, then to New York, then to New Hampshire, passing messages from one group to another. And in Boston itself, he played a special role. During the years of the revolution, there were seven revolutionary groups in the city, which included about 255 men. Most of them (more than 80%) were in only one group. No one was a member of all seven at the same time. And only two were part of five groups at once. One of the two was Paul Revere.

It is not surprising that when the British troops launched their secret campaign in 1774, planning to locate and destroy the stores of weapons and ammunition created by the revolutionaries, Revere became a kind of unofficial "communication center" of the anti-British forces. He knew everyone. To whom, if not to him, you should have turned if you were a guy from the stable and on that day, April 18, 1775, you heard two British officers talking about how they would make hell tomorrow? Not surprisingly, as Revere set off for Lexington that evening, he already knew how to spread the word as far as possible. When he met people along the way, being extremely sociable, he stopped them and told them the news. When he came to the city, he knew exactly whose door he needed to knock on, who was the commander of the local militia, who was the most influential person here. After all, he had already met most of these people before, and they knew and respected him.

What about William Dose? Fisher thinks it's unlikely that Doz could have driven all those 27 kilometers to Lexington and never said a word to anyone. But he obviously didn't have the same communication skills as Revere, because there's no evidence that anyone remembered him that night. “On Paul Revere's northern route, the city sergeants and company captains immediately issued an alarm,” Fisher writes. - On the southern route of William Dose, the reaction was belated, and in one city it was not at all. Doz did not wake up the city sergeants or militia commanders in Roxbury, Brooklyn, Watertown, and Waltham." Why? Because Roxbury, Brooklyn, Watertown and Waltham are not Boston. And Doz, in all likelihood, was a person with an ordinary social circle (like many of us). Once in a foreign city, he could not know which doors to knock on. Only one small community along Dose's path seemed to take the message - a few farmers in the Waltham Farms area. But warning a few families is not enough to sound the alarm. Rumor epidemics are the work of the Unifiers. And William Dose was just an ordinary man.

It would be a mistake, however, to think that the Unifiers are the only people who start a social epidemic. Roger Hareshaw sent out dozens of faxes recommending his daughter's friend's new restaurant. But he didn't find this restaurant. Someone else did it and told him. At some point in the Hush Puppies renaissance, they were noticed by the Uniters, who heralded the brand's return. But who first told the Uniters about Hush Puppies?

Perhaps the Unifiers get new information by accident, because they know so many people that they have access to the latest news as soon as it appears. But if you study social epidemics carefully, it becomes clear that there are people who

which we rely on when we need to contact other people, but besides them there are also people on whom we rely when we want to get fresh information. There are people specialists and there are information specialists.

Sometimes, of course, these two types of specialists are found in one person. For example, Paul Revere's influence is partly due to the fact that he was not only the organizer of contacts and not only the man with the thickest notebook in colonial Boston. He was also active in gathering information about the British. In the autumn of 1774, he organized a secret group that was supposed to track the movements of British troops. The members of the group met regularly at the Green Dragon Tavern. In December of that year, the group learned that the British intended to seize the Colonial Militia's secret munitions depot at the entrance to Portsmouth harbor, 80 kilometers north of Boston. On the cold morning of December 13, Revere rode north on horseback through deep snow to warn the local militia that the British were coming towards them. He helped to get information, and he also forwarded it. Paul Revere was the Unifier. But at the same time he was also an Expert - and this is the second type of people who influence the emergence of rumor epidemics.

Connoisseurs are those who accumulate knowledge. In recent years, economists have paid a lot of attention to the study of the Maven phenomenon for a very obvious reason: if markets depend on information, then the people with the most information should be the most influential. For example, when they want to increase sales of a product in a supermarket, they put up an advertising sign in front of it with something like this: “Every day the price is lower!” In fact, the price remains the same, but the product becomes more visible. Every time supermarkets do this, there is always a spike in sales of the item, as if it were actually on sale.

If you think about it, the situation seems rather alarming. The whole point of supermarket sales or promotions is that we consumers are very price sensitive and react accordingly: we buy more when prices go down and less when prices go up. But if we buy more even if the price doesn't go down, then what's to stop supermarkets from ever lowering their prices? What or who will prevent them from deceiving us with meaningless signs “every day the price is lower” every time we enter the store? The point is that while most of us don't watch prices, every retailer knows that there is a minority who do. And if these people discover something (for example, that sales promotion is actually missing), they will take action. If a store tries to pull off the sale stunt too often, such people will realize this and file a complaint with management, and then advise friends and acquaintances not to go to this store. These people guard the fair market. It has been ten years since they were first classified, and all this time economists have struggled to understand them. Their presence has been found in every walk of life and in every socio-economic group. One of their names is price watchers, another, more common, market experts.

Linda Price, professor of marketing at the University of Nebraska and a pioneer of research into the Maven phenomenon, videotaped interviews she conducted with several Mavens. In one of them, a well-dressed man talks very animatedly about how he goes to the store. Here is an excerpt from his story:

“As I watch financial news closely, I begin to see trends. The classic coffee example. When ten years ago the first

the coffee crisis, I followed the news about the frost in Brazil and how it could affect the price of coffee in the long run, and I said in advance that I was going to stock up on coffee.”

At this point in the interview, the man's face broke into a smile.

“I then accumulated about 40 cans of coffee. I bought them at those ridiculous prices when 1.5-kilogram cans cost $2.79 and $2.89. Today, such a jar costs about 6 dollars. It amused me."

Do you feel how passionate he is? He can remember the cost, down to the cent, of those cans of coffee he bought ten years ago.

The most important feature of Connoisseurs is that they are not just passive gatherers of information. Their interest is not in saving more on a can of coffee. As soon as they realize that they can save money on something, they immediately want to tell you about it. “A connoisseur is a person who has information about various goods, or prices, or places of sale. This person always goes to talk with other consumers and is ready to answer their questions - explains Price. - They like to help people in the market. They distribute coupons for discounts, take you shopping with them, go to the store instead of you. They know where the toilet is in retail outlets. That's the kind of knowledge they have." They are more than experts. “Experts,” says Price, “will talk about cars, for example, because they like cars. But they won't talk to you just because they like you and want to help you make a decision. The market expert will do just that. He's more socially motivated."

Price claims that a good half of Americans know such an Expert or someone who looks like him. She founded her own

concept on the example of a person whom she met while studying in graduate school. He was such a memorable character that his personality spawned an entire industry of marketing research.

“I was doing my PhD at the time at the University of Texas,” Linda told me. “I didn’t realize it at the time, but I met the perfect Connoisseur. He was Jewish. It was Easter and I asked him where I could buy ham. He replied that he was Jewish, but he still knew that I had better go to such and such a grocery store and buy ham at this price. Price laughed. - You should meet him. His name is Mark Alpert."

Mark Alpert is a short, energetic man in his early fifties. He has dark hair, a big nose, and small, burning, intelligent eyes. He speaks quickly, accurately and thoroughly. He is the kind of person who will never say that it was hot yesterday. He will say that the air temperature yesterday was 30.5 °C. He never goes up the stairs, he runs up them like a boy. It seems that absolutely everything is interesting to him, everything is curious, and that at his age, if you give him a children's chemical kit, he will immediately sit down at the table and create some new mixture.

Alpert grew up in the Midwest. His father opened the first chain of discount stores in northern Minnesota. Mark received his PhD from the University of Southern California and now teaches at the College of Business Administration at the University of Texas. However, there is no connection between his position and his status as an Expert. If Alpert were a plumber, he would still be just as precise and meticulous in everything related to the intricacies of the consumer market.

We met in Austin for lunch at a lakeside restaurant. I arrived first and chose a table. Soon Alpert appeared and killed

he made me transfer to another, saying that it would be better there. And so it turned out. I asked him how he buys anything and he started talking. He explained why he had cable TV and not a satellite dish, gave me all the details of the latest film review from Leonard Moltin, and named his man at the Park Central Hotel in Manhattan, who always gets a room at a good price. (“Malcolm, a hotel room is actually $99. And they're ripping off $189!”) He explained to me that there was a fixed but flexible retail price for a room. He pointed to my recorder and said, "I think you've run out of tape." Exactly. He told me why I shouldn't buy an Audi. (“These are the Germans and dealing with them is a headache. They will give you a guarantee for a while, but no more. The dealer network is not developed, so it is difficult to service the car. I like driving an Audi, but I don’t like own them.") He advised me the Mercury Mystic, because the car handles just as well as much more expensive European-made sedans. “It doesn't sell very well,” he said, “so you can get it for a very reasonable price. You need to go to a retailer. Go to him on the 25th of any month. Well, what am I going to tell you...” He then launched into an incredibly long, sometimes very funny description of how he bought a new TV for several months. If you or I were to go through this (returning a TV, endless comparisons of tiny electronic parts, comparing the fine print on a warranty document), I suspect we would think it was terrible. But Alpert seems to find it all amusing.

Connoisseurs, according to Price, are the type of people who read voraciously consumer reports(“Reviews of the consumer market”). Moreover, Connoisseurs write in consumer reports and correct their compilers.

“Once they said that the Audi 4000 was based on the Volkswagen Dasher. It was the end of the 1970s. But "audi 4000" is more

large car. I wrote them a letter. Then there was a mistake with the “Audi-5000”. consumer reports put this car on the "don't buy" list because of the sudden acceleration problem. But I looked through the literature and realized that this is not true ... Then I wrote to them and said that they needed to understand this better. They never answered me. This pissed me off. They should be above this.” After saying that, Alpert shook his head in displeasure. He does not like it when the commandments of the Experts are violated.

It should be noted that Alpert is not at all a nasty know-it-all. Although he could cross that line. He himself is aware of this. “I once stood in line at the supermarket for one guy. He had to show ID to buy cigarettes, Alpert told me. “I was tempted to tell him that I was diagnosed with lung cancer. This desire to serve and influence choices can go too far. You can start sticking your nose everywhere. I try to be a passive Connoisseur... We must remember that this is their decision. This is their life."

What saves him is that you never get the impression that he is drawing. There is something reflexive in his involvement in the problems of the market. This is not acting. This is very close to the social instinct of Horshaw and Weisberg. Mark Alpert told me about the complex pattern of using discount coupons when renting tapes from the Blockbuster video store. Then he stopped, as if realizing that he was too carried away, and burst into laughter: “See, you can save a whole dollar! In a year, I can probably collect for a bottle of wine.

Alpert is almost pathologically eager to help others. He is unable to contain himself. "A connoisseur is someone who wants to solve other people's problems, usually at the expense of their own," says Alpert. And this is true, although I suspect that the reverse is also true. The connoisseur solves his problems (satisfies his emotional needs) by solving the problems of others. Mark Alpert, deep down, was pleased that from now on I will be

bathe a TV or a car, or check into a New York hotel armed with the knowledge he gave me.

“Mark Alpert is an amazingly selfless person,” Lee Makalester, Alpert's colleague at the University of Texas, told me. - I must admit that he helped me save $15,000 when I came to Austin. First he helped me negotiate the price of the house because he knows how to buy and sell real estate. Then I needed a dishwasher and dryer and Alpert found them for me at the best price. Then I bought a car. I wanted to follow Mark's example and buy a Volvo, and then he showed me a site on the Internet that had all the prices for Volvos in the state of Texas. And went with me to buy. He helped me navigate the intricacies of the university pension system and made everything easier for me. He has everything systematized. This is Mark Alpert. This is a market expert. God bless him. He's the one who makes America great."

What makes people like Mark Alpert so important to starting an epidemic? Obviously they know things we don't. They read more magazines, more newspapers than we do, and they are the only ones who read junk mail. Mark Alpert is a connoisseur of electronic home appliances. If there is a breakthrough of new technologies in the production of televisions or camcorders, then his friends will be among the first to know about it. Connoisseurs have enough information and the art of communication to start a rumor epidemic. What distinguishes the Connoisseurs, however, is not the content of the information, but their ability to communicate it. The unselfish desire of Connoisseurs to help - simply because they love to help - invariably attracts the attention of others.

This partly explains why, on that memorable night, Paul Revere's message had such an effect. Message about the

While talking to Alpert, I mentioned that I would be in Los Angeles in a few weeks. “There is a place that I really like. It's in Westwood, he said quickly. — Century Wilshire. European style room and breakfast. They have excellent rooms, a heated pool, underground parking. The last time I stayed there (it was five years ago), single rooms cost from $70 and the cheapest apartments are $110. If you stay for a week, they will give you a discount. They have a toll-free phone number for inquiries."

Since he was a real Connoisseur, when I arrived in Los Angeles, I stopped at Century Wilshire, and everything was exactly as he said, and even better. A few weeks later,

As soon as I got home, I - quite contrary to my own habits - recommended Century Wilshire to two of my friends, and a month later to two more. Then I began to imagine how many of the people I told about the hotel also told someone about it. And how many people like me did Mark Alpert tell about the hotel. I suddenly realized that I was in the middle of a rumor epidemic launched by Mark Alpert. Alpert, of course, hardly knows as many people as a Unifier like Roger Horshaw, so he doesn't have such a huge distribution network. But if Roger Horshaw had spoken to you on the eve of your trip to Los Angeles, he would hardly have advised you where to stay. But Alpert will definitely advise. And if Horshaw advises, it is not certain that you will follow his advice. You will treat it in the same way as the advice of any other person you know. But if Mark Alpert gives advice, you will follow it. certainly. The combiner can tell ten of his friends where to stay in Los Angeles, and half of them can listen. A connoisseur can advise five people where to stay in Los Angeles, but will praise the hotel so passionately and convincingly that all five will do exactly as he said. Here you have different, doing things with different goals, individuals in action. But both have the ability to start a rumor epidemic.

One of the hallmarks of the Connoisseur is that he won't convince you. Alpert's motivation is to educate and help. He's not the type to twist your arms. During our conversation, there were several key moments where he seemed to be trying to worm information out of me, to pry what I knew to add to his formidable database. To be an Expert is to be a teacher. But at the same time it means being a student - and with no less fervor. Connoisseurs are a kind of information

brokers who accumulate knowledge and trade it. But for a social epidemic to start, for people to take some action, they have to be convinced.

For example, many young people who bought Hush Puppies for themselves would not even want to lie in a coffin at another time. Similarly, it can be imagined that after Paul Revere broke the news, members of the local militias got together and began to make plans to meet the British. But at the same time, some, perhaps, were eager to fight, while others doubted the wisdom of the action by the forces of home-grown formations against the trained army of the British. Still others, who did not know Revir personally, could even question the information he provided. But in the end, everyone fell under the influence of what we now call the influence of others. However, the influence of others is not always an automatic or unconscious process. This means that most often someone from those around them turns to a person and puts pressure on him. In a social epidemic, Connoisseurs play the role of a database. They provide information. The unifiers are the consolidators of society: they disseminate information. But there is another group of unique people - Sellers. They know how to convince us if we do not believe well what they tell us. And they're just as important in starting a rumor epidemic as the two previously introduced groups.

Who are the Sellers? And what makes them unsurpassed masters of their craft?

Let's meet Tom Gau from Torrance, a town just south of Los Angeles. His firm, Kavesh & Gau, is one of the country's largest financial planning firms. Tom earns a million dollars a year. Donald Moyne, a behavioral psychologist who has written extensively on the art of persuasion, advised me to meet Gau because, he says, he has a "charm." And it is true. By the will of fate, Tom Gau sells financial planning services. But if he wants, he can

sell anything. And if we want to understand what kind of people who can convince, then Gau is a great example.

Tom Gau is in his early forties. He has a pleasant appearance, but without a sugary charm. Average height, skinny. Slightly tousled dark hair, mustache. The facial expression is a little guilty. Give him a hat and a horse and he'll make a great cowboy. He looks like actor Sam Elliot. When we met, Gau shook my hand. But, as he told me later, when they meet, he usually hugs, and if it's a woman, he kisses her savoryly. As you would expect from a true Salesman, he radiates the joy of life.

“I love my clients. I'll go out of my way for them, - says Gau. - I call my clients my family. I tell them that I have two families. I have a wife, children and - you.

Gau speaks quickly, impetuously. His speech speeds up, then slows down a little. Sometimes, when he says lines along the way, he says them so quickly, as if he wants to enclose them in some kind of verbal brackets. He asks a lot of rhetorical questions.

"I love my job. I am a workaholic. I get up at six or seven in the morning and leave work at nine in the evening. I manage huge amounts of money. But I don't tell my clients about it. I'm not here for this. I'm here to help people. I love helping people. I don't have to work anymore. I am financially independent. So why am I working late? Because I love helping people. I like people. It's called an attitude."

Tom Gau points out that his firm offers a level of service and experience that can hardly be found anywhere else. Across the lobby, across from his office, is a subsidiary of Kayeb & Vai, a law firm that handles wills, insurance policies, and all sorts of other legal documents related to financial planning. Gau has insurance specialists, stockbrokers for investments, pension specialists for older clients. His

arguments are rational and consistent. Moyne, in collaboration with Gau, compiled what he called a financial planner scriptbook. What distinguishes a good Salesperson from a mediocre one, Moyne argues, is the quantity and quality of his responses to objections that a potential customer might raise. One day, Moin sat next to Gau, recorded all his answers on a dictaphone, and wrote a book on this material. At that time, Moyne and Gau calculated that the planner should be ready to answer about 20 questions or statements. For example: "I can do it myself." In response to this, the scenario book provides 50 possible answers. For example: “Doesn’t it bother you that you can do something wrong, and there will be no one around to help you?” Or: “I'm sure you're great with money. However, you should know that in most cases, wives outlive their husbands. Is not it? If something happened to you, would she be able to handle the money alone?”

I can imagine someone buying this script book and memorizing all the potential answers. I can also imagine the same person over time becoming so familiar with the material that they begin to get a good sense of what answers work best for different types of people. If you record this man's conversations with his clients, he will sound like Tom Gau because he will only use Tom Gau's words. By the standard measures by which we measure the power of persuasion (the logic and persuasiveness of the persuader), this will force people to use a collection of scripts as persuasive as Tom Gau himself. But will they succeed? What's interesting about Tom Gau is the extent to which he is persuasive even when he deviates slightly from his own words. There is an elusive character trait about him, something powerful, infectious and irresistible. Something beyond what comes out of his mouth. Something that makes those who meet this person always agree with him. This is energy. This is enthusiasm. It's charm. This is

sympathy. All this together and something else. At one point I asked him if he was happy and he almost jumped out of his chair.

“Very much,” Gau replied briskly. - I'm probably the biggest optimist you can imagine. Take the biggest optimist you know, multiply it by a hundred, and that's me. Because positive thinking can overcome anything. So many people with negativity! Someone will say: "You will not succeed." And I’ll say: “What do you mean, I won’t succeed?” Five years ago we moved to Oregon, to the city of Ashland. We found a house that we liked. But it was a bit pricey. And I told my wife that I would offer a ridiculous price for it. She said they would never agree. I said, “Maybe they won't agree. And what are we losing? The worst thing that can be is that they will say “no”. I'm not going to pressure them. I will only briefly explain to them why I am doing this. I will explain to them the essence of my proposal.”

And you know what? They agreed".

When Gau told me this story, I had no trouble imagining him in Ashland, somehow persuading the salesman to part with his wonderful house for a ridiculous price. “Thunder strike me,” Gau continued in the meantime. “If you don’t try, you won’t succeed.”

The question of what makes someone (or something) persuasive is not nearly as straightforward as it might seem. We recognize the ego at the first glance. But we are not always able to explain "this". Consider two examples taken from the psychological literature. The first is an experiment conducted during the 1984 presidential election campaign, when Ronald Reagan and Walter Mondale competed. For eight days before the election, a group of psychologists led by Brian Mullen of Syracuse University videotaped the evening news

on three national TV channels. Then, as now, they were hosted by Peter Jennings on ABC, Tom Brokaw on NBC, and Dan Reiser on CBS. Mullen analyzed the notes and highlighted all references to candidates. He ended up with 37 separate fragments, each about 2.5 seconds long. These fragments were then played silently to a randomly selected group of people who were asked to rate each speaker's facial expression. The subjects had no idea what kind of experiment they were participating in or what the announcers were reporting on the news program. They were only asked to rate the emotional content of the facial expressions of these three people on a 21-point scale, where the lowest score meant "extremely negative" and the highest meant "extremely positive."

The results were amazing. Dan Reiser scored 10.46, which stands for almost completely neutral when talking about Mondale, and 10.37 when talking about Reagan. He looked the same when he spoke of both Republicans and Democrats. The same was true for Brokaw, who scored 11.21 on Mondale and 11.50 on Reagan. But Peter Jennings with ABC is a completely different story. For Mondale, he earned 13.38 points. But when he talked about Reagan, his face lit up so much that he got a 17.44. Mullen and his colleagues struggled to find some neutral explanation for this. What if Jennings was just more expressive than his peers? But it seems that was not the case at all. The subjects were shown other fragments from the reports of the same three announcers, and the reports told about both sad and joyful events - about the funeral of Indira Gandhi, about a breakthrough in the treatment of an infectious disease. And this time around, Jennings didn't get more points for any of those posts than his peers. He was even less expressive than the others. On the "joyful" fragments included for comparison, he received 14.13 points, i.е. significantly less than Reiser and Brokaw. It turned out that the only

Another possible explanation is that Jennings put on "a slightly more pronounced facial expression" when talking about Reagan.

And then the study became even more interesting. Mullen and his colleagues called residents of different cities of the country - those who regularly watched the evening news on the main channels, and asked whom they voted for in the elections. In each case, those who watched ABC voted for Reagan far more than those who watched CBS or NBC. For example, in Cleveland, 75% of the ABC audience voted Republican, while only 61.9% of CBS or NBC viewers voted Republican. In Williamstown, Massachusetts, Reagan was supported by 71.4% of the ABC audience and 50% of the viewers of other channels. In Erie, Pennsylvania, the difference was 73.7% and 50%, respectively. The slight pro-Reagan accent on Jennings' face seemed to influence voters who watched ABC.

Of course, the ABC News program vehemently disputed the results of this study. (“As far as I'm concerned, I'm the only sociologist to get a highly ambiguous confession after Peter Jennings called me an idiot,” Mullen says.) Indeed, it's hard to believe that all of this is true. I think most of us are inclined to assume that it's just something else: Reagan supporters prefer to watch ABC because of Jennings' bias. But Mullen argues that this is not true. Indeed, on other, more obvious levels, say, in the selection of news, ABC has shown itself to be the television company most hostile to Reagan, so staunch Republicans should rather switch from ABC News to competing channels.

To answer the question of whether the result of the experiment was just random, four years later during the Michael Dukakis-George Bush election campaign, Mullen's group repeated their experiment and got a similar result. “Jennings smiled more often when talking about the Republican candidate than about the Democrat,” Mullen said. - And again, according to the result -

there, a telephone poll found that viewers who watched ABC were more likely to vote for Bush."

And here is another example of how many subtleties there are in the process of persuasion. A group of students were told they would take part in a market research study for a high-tech headphone company. Each was given a kit and told that the company wants to test how the headphones will work if the user is in motion: jumping in a dance or shaking his head. All the students listened to Linda Ronstadt and the Eagles and were then given a radio program urging them to raise their university tuition from $587 to $750. One-third of the students said they should vigorously nod their heads up and down while listening to the entire tape. Another third were asked to shake their heads from side to side. The last third served as the control group. They were asked not to move their heads. When the experiment was over, all students were given a short questionnaire with questions about the quality of the songs and the effect of shaking on the headphones. And at the end was the question that the experimenters really wanted to answer: “What do you think is a reasonable annual tuition fee?”

The answers to this question turned out to be as incredible as the results of the newscaster experiment. The students who did not move their heads remained indifferent to the radio broadcast. They found the $587 tuition fee to be normal. Those who shook their heads from side to side stubbornly objected to the proposed pay increase. They wanted tuition to drop to an average of $467 a year. And those students who were asked to nod their heads found the radio program very persuasive. They agreed that tuition fees would rise to an average of $646. A simple nod of the head for some reason was enough for them to agree to shell out more money from their own pocket. In to-

In the end, head nodding played the same role as Peter Jennings' smiles in the 1984 election.

These studies, it seems to me, provide very important clues to understanding what makes a person like Tom Gau, or any Salesperson we meet, so effective. First, small things are likely to lead to large-scale changes. In the headphone experiment, the radio program had no effect on those who did not move their heads. She was not particularly convincing to them. But as soon as the listener began to nod his head, the transmission acquired a tremendous power of persuasion. In Jennings's case, Mullen says, someone's cautious signals in favor of one politician or another usually don't matter. But given the special, "insecure" state in which people watch the news, a small gesture can have far-reaching consequences. “When people watch the news, they don't filter out this kind of 'information', they don't feel the need to object to the expression on the announcer's face,” Mullen explains. - We are not talking about the fact that someone confidently declares: this is a very good candidate who deserves your vote. This is not a direct verbal message that we automatically start to rebel against. It's much more subtle and, for that reason, more sophisticated, and so it's much harder for us to fence ourselves off from it."

The second conclusion that can be drawn from these studies is that non-verbal cues are just as important, if not more important than verbal cues. That, as we say, sometimes it means more than that, what We are speaking. After all, Jennings did not insert any pro-Ragen comments into the news. Moreover, according to independent observers, ABC was the most hostile TV station to Reagan. One of the conclusions drawn by the authors of the headphone experiment, Gary Wells of the University of Alberta and Richard Petty of the University of Missouri, was this: “Television advertising is most effective if there is repetitive vertical movement in the video sequence (for example, a bouncing ball)

and TV viewers nod their heads following this movement. Simple physical movements and observations can have a huge impact on how we feel and think.

The third (and perhaps the most important) result of the conducted research is that the power of persuasion often manifests itself in ways that are often incomprehensible to us. The point is not that smiles and nods are subliminal messages. They are straight and visible on the surface. The bottom line is that their impact is completely inexplicable. If you asked a third of the students why they agreed to a significant increase in tuition, no one would tell you that it was a matter of nodding their heads while listening to a program. They would say they found the show very thoughtful. They would back up their opinion with logical arguments. Similarly, ABC viewers who voted for Reagan will never, even in a thousand years, tell you that they made their choice because Peter Jennings smiled every time he mentioned the president's name. They will say that they liked Reagan's political program or that he did his job well. Yes, it would never occur to them that something as random and, at first glance, insignificant, like a smile or a nod from a news announcer, could influence their decision. If we are to understand what makes people like Tom Gau so persuasive, we need to see more in him than his sheer ability to speak beautifully. We need to see something elusive, secret and something that cannot be expressed in words.

What happens when two people communicate? In our case, this is the most important question, since we are talking about the main context in which all belief takes place. We know that people

speak in turn. They listen, interrupt each other, gesticulate. Tom Gau and I talked in his small office. I was sitting in an armchair pulled up next to his desk with my legs crossed. In the hands - a notebook and a pen. He is wearing a blue shirt, black pants and a black jacket. He sat at his desk in a high-backed chair. He wears blue suit trousers, a perfectly pressed white shirt and a scarlet tie. At some moments, he hung over the table and put his elbows forward. Then he leaned back in his chair and waved his arms. Between us, on the table, lay my voice recorder and recorded the conversation. This is what you would see if I played the video of this interview for you. But if you slowed down the playback of the recording until it turned into a sequence of fragments of video sequence of a fraction of a second, you would see something completely different. You would see that we are both participating in what can be defined as a complex dance with a clear pattern.

The pioneer of this kind of analysis - what is called the study of cultural microrhythms - was William Condon. In the 1960s, in one of his most famous research projects, he set out to decode a four-and-a-half-second movie sequence in which a woman says to a man and child at dinner, “You must come every night. We have not sat at the table so wonderfully for many months. Condon broke the episode into separate segments, each about 1/45th of a second long. And then he looked and looked. Here is how he describes it:

“In order to carefully study the construction and sequence of all this, a naturalistic or otological approach is indispensable. You just sit and watch

rub and sit and stare, for thousands of hours, until there is some order in the material. It's like sculpting... Long-term research reveals new logical forms. As I watched this tape over and over again, I had erroneous ideas about the order in which communication occurs between people.

It was sort of an established model. You send a message and someone sends a message back to you. Messages are sent back and forth and in all directions. But there was something about it that didn't make sense."

Condon devoted a year and a half to studying this short clip from the film, until finally he saw in peripheral vision what he had anticipated was: “the wife turns her head at the very moment when the husband raises his hand.” From that moment on, he began to distinguish other micro-movements, other patterns that appeared again and again, until the researcher realized that, in addition to speaking words and listening, three people at the table were involved in what he called "interactive synchrony". Their conversation had a rhythmic physical characteristic. Each person in a frame of 1/45, 2/45 or 3/45 of a second moved a shoulder, cheek, eyebrow or arm, delayed this movement, stopped it, changed direction and started all over again. Moreover, all these movements ideally coincided with the rhythm of the words that each of the interlocutors uttered, emphasizing, emphasizing and improving the process of articulation, so that the speaker actually danced to the rhythm of his own speech. At the same time, the rest of those present at the table danced along with the speaker, moving their face, shoulders, arms and body in the same rhythm. This does not mean that everyone danced the same way. People don't always move in unison, dancing to the same tune. But the bottom line is that the synchronicity of micro-movements of all interlocutors (shudders and vibrations of faces and bodies) were in absolute harmony.

In the course of subsequent studies, it was found that not only gestures, but also the rhythm of the conversation was in harmony. When two people talk to each other, the volume and timbre of their speech are mutually balanced. What linguists call the speed of speech (the number of speech signals spoken per second) equalizes. The same happens with what is denoted by the delay - the period of time that elapses between the moment when one interlocutor falls silent, and the moment when another begins to speak. Two people can start a conversation with very different speech patterns, but almost instantly they reach the same pattern. And this happens every time, always. Infants of one or two days of age synchronize the movements of the head, elbows, shoulders, hips and feet with the speech patterns of adults. Synchronicity is found even when humans interact with primates. This is one of the features of our nature.

When Tom Gau and I sat opposite each other in his office, we almost instantly reached physical and verbal harmony. We performed a dance. Even before trying to convince me with words, he had already established a connection with me with his gestures and manner of speech. But what made my conversation with him so special, so much more convincing than all the conversations that I have every day? It's not that Gau deliberately tried to establish harmony in communication with me. Some books on the art of selling recommend that persuaders try to copy the posture or manner of speaking of their customers in order to reach agreement more quickly. But this is too obvious and cheap trick.

We are talking here about a kind of overreflex, a fundamental physiological ability that we are barely aware of. And, as with all special human abilities, some people control this reflex better than others. Consequently, a person who has the power of persuasion, to a certain extent, can subordinate others to his own rhythm of communication and dictate his own terms. According to some studies, students with

high degree of synchronicity in communication with teachers, more satisfied with life, interested and good-natured. During a conversation with Gau, I felt that I was being seduced, of course, not in a sexual sense, but in a universal one. I felt the conversation was on his terms, not mine. I felt in sync with him.

"This feeling is familiar to experienced musicians and public speakers," says Joseph Capella, professor at the Annerberg School of Communication Engineering at the University of Pennsylvania. “They always know when the listeners are in the same rhythm with them.”

This is a strange feeling, because I did not want this at all, it happened against my will. But the amazing thing about Sellers is that at some level they are impossible to resist. “To build trust and reach agreement with the interlocutor, Tom needs five to ten minutes. For most people, this task will take at least half an hour,” says Donald Moyne about Tom Gau.

There is another important feature here. When two people talk, they have harmony not only on the verbal and physical levels. They are subject to what is called motor mimicry. If you show people a photo of a person with a smiling or frowning face, they will smile or frown in response, even with the slightest movements of the facial muscles that can only be recorded using electronic sensors. If I hit my finger with a hammer, most people who see it will grimace: they will imitate my emotional state. This is what, in a physical sense, is meant by empathy. We imitate each other's emotions, thus expressing support and care, and on a more elementary level, communicating with each other.

In their brilliant 1994 book Emotional Contagion, psychologists Elaine Hatfield and John Cacioppo and historian Richard Rapson argue that imitation is, among other things, one of the means

with which we infect each other with emotions. In other words, if I smile and you see it and smile back (even a micro smile no longer than a few milliseconds), this will mean not only that you imitate me and empathize. This may be the way I convey my happy state to you. Emotion is contagious. Some of this happens at the level of intuition. All of us, as a rule, are cheered up if there is someone nearby in a good mood. But if you think about it seriously, it becomes clear that this is an extremely important point. We are used to thinking that facial expressions are an external sign of our internal state. I am happy - and I smile. I'm sad - and I frown. Emotions come out from within. Emotional contagion, however, shows that there is a movement in the opposite direction. If I make you smile, you will feel happy. If I make you frown, you will be sad. That is, emotions are transmitted from outside to inside.

If we consider them from this point of view (from the outside - inward, and not from the inside - outward), then we can understand why some people are able to have a huge impact on others. Be that as it may, some of us are very good at expressing emotions and feelings, which means that they are much more emotionally contagious than the rest. Psychologists call such people transmitters. Transmitters have a special type of personality. They also differ in their psychological characteristics. Physiognomists claim that there are huge differences in the location of the facial muscles - in their shape and (which is quite surprising) in the predominance of their particular type. “The situation is very similar to an epidemic,” Cacioppo explains. - There are carriers, people who are emotionally very expressive, and there are very receptive people. Emotional contagion is not associated with illness, but its mechanism is exactly the same.

Howard Friedman, a psychologist at the University of California, Riverside, developed a research method he called

"emotional communication test". The test consists of thirteen questions. For example, can you sit still while listening to good dance music? Do you laugh out loud? Do you touch your friends while talking? How good are you at making eyes? Do you like being the center of attention? The highest potential test result is 117 points. And the average result, according to Friedman, is 71 points.

What does a high score mean? To answer this, Friedman performed an exciting experiment. He selected a few dozen people with very high scores on his test (over 90) and a few dozen people with the lowest scores (under 60) and asked them to complete a questionnaire that measured how they feel "in the moment." He then placed all the high-scoring participants in separate rooms and paired each of them with the two low-scoring participants. They were asked to sit together in the same room for two minutes. They could look at each other, but not talk. Friedman found that in just two minutes, without a single word being spoken, people with low scores picked up the mood of participants with high test scores. If the charismatic person was initially depressed, and the non-expressive person was satisfied with life, then after two minutes the non-expressive participant in the experiment also found himself in a depressed state. But not vice versa. Only a charismatic person could infect someone else in the room with their emotions.

Wasn't that the case with me and Tom Gow? During our meeting, I was most struck by his voice. He possessed the range of an opera singer. At times, his voice sounded dry (Tom's favorite phrase in this state: "Excuse me?"). Sometimes Tom drawled, lazily and calmly. At times he chuckled, and then his words melodiously echoed the laughter. In all these states, his face changed accordingly, moving from one expression to another - quickly and habitually. There was no uncertainty in his emotions.

Everything was clearly marked on his face. Of course, I did not see my face, but I can assume that it reflected the emotions of my interlocutor. It is interesting in this context to recall the experiment with nodding and headphones. This is an example of someone being persuaded from the outside, through an external gesture that influences an internal decision. Did I nod when Tom Howe nodded? Did I turn my head when Tom Howe turned his head? Later, I called Gau and asked him to take the Howard Friedman test of charisma. As he answered question after question, Tom began to grin. When he got to point 11 (“I’m very bad with pantomime, just like with solving charades”), he was already laughing with might and main: “And I’m doing it very well! I always win at charades!” Out of a possible 117 points, he got 116.

Early on the morning of April 19, 1775, the people of Lexington, Massachusetts began to gather in the town square. They were between 16 and 60 years old. Everyone armed themselves with what they could - muskets, swords, pistols. As the disturbing news spread, more and more of them - due to the militias who came from neighboring cities. Dedham sent four troops. From Lynn people went to Lexington on their own initiative. In the towns farther west, where word reached later, farmers were so eager to take part in the Battle of Lexington that they literally abandoned their plows in the fields. In many cities, almost the entire male population was mobilized. They didn't have a uniform, so they put on their usual clothes: jackets to keep out the cold and wide-brimmed hats.

While the colonists rushed to Lexington, British regulars were sent there in orderly ranks. At dawn, the fighters of the forward detachments saw the silhouettes of armed people walking through the surrounding fields and ahead of the British.

on their march to Lexington. When the regulars (as they were then called) approached the center of the city, they heard drumming nearby. Finally the British reached Lexington Square and the two sides came face to face: a few hundred British soldiers and about a hundred militiamen. In the first clash, the British overpowered the colonists, hitting seven militias in a short skirmish. But this was only the first of the battles that were to come that day. As the British advanced towards Concord to search for the armaments and ammunition depots they had been informed of, they again ran into the militia and this time were severely defeated. This was the beginning of the American Revolution, a war that claimed many lives and engulfed the entire American colony. When the American colonists declared independence the following year, it was enthusiastically received as a victory for the entire nation. But it didn't start out that big. It all began on a cold spring morning with a rumor epidemic that spread from a groom boy to all of New England, passed on by special people. There were very few of them: several sellers and one person with the talent of the Connoisseur and the Unifier.

• See Milgram S. Experiment in social psychology: Per. from English. - St. Petersburg: Peter, 2001. - 336 p.
• Subthreshold perception - subjectively unconscious, but influencing human behavior, the processes of perception, occurring as if "under the threshold" of consciousness. - Note. ed.
• Ethology is the science of the biological foundations and patterns of animal and human behavior. The main attention in ethology is paid to vidogynistic (genetically fixed) forms of behavior that are characteristic of all members of a given species. - Note. ed.

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BOROTH- BORT, Max (Max Borst), eminent pathologist. Genus. in 1869, graduated from medical school. faculty of the university in Würzburg in 1892. From 1893 to 1904 he was an assistant stalemate. in that Würzburg University, where he worked under the guidance of Rindfleisch, one ... Big Medical Encyclopedia

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NUMBERS THEORY, a branch of pure mathematics concerned with the study of the integers 0, ± 1, ± 2,... and the relationships between them. Sometimes number theory is called higher arithmetic. Separate calculations performed on specific numbers, for example, 9 + 16 = 25, are not of particular interest and are usually not included in the subject of number theory. On the other hand, the equality just written out becomes incomparably more interesting if we note that it is the simplest solution in integers (except for the trivial solutions x = z, y= 0) the Pythagorean equations x 2 + y 2 = z 2. From this point of view, the last equation leads directly to some genuine number-theoretic problems, for example, (1) does x 2 + y 2 = z 2 infinitely many or only a finite number of solutions in integers and how can they be found? (2) What integers can be represented as x 2 + y 2 , where x and y- whole numbers? (3) Are there solutions in integers of the analogous equation x n + y n = z n, where n is an integer greater than 2? One of the intriguing features of number theory is that these three questions, stated so easily and understandably, are in fact on completely different levels of complexity. Pythagoras and Plato, and perhaps much earlier Babylonian mathematicians, knew that the equation x 2 + y 2 = z 2 has infinitely many solutions in integers, and the ancient Greek mathematician Diophantus (c. 250 BC) knew that each such solution can be represented as x = r 2 – s 2 , y=2rs, z = r 2 + s 2 for suitable integers r and s and that for any two integers r and s corresponding values x, y and z form a solution. As for the second question, the structure of the set of integers representable as the sum of two squares was described by P. Fermat (1601–1665), the founder of number theory in its modern form. Fermat showed that the integer m representable as the sum of two squares if and only if the quotient of the number m by the largest square dividing the number m, does not contain a prime factor of the form 4 k + 3 (k is an integer). This result is much more subtle than the first one, and its proof is far from obvious, although it is not too difficult. The third question has remained unanswered, despite the most stubborn efforts of the most brilliant mathematical minds, for the past three centuries. Fermat wrote in the margins of one of his books around 1630 that the equation x n + y n = z n has no solutions in integers x, y and z, different from zero, when n greater than 2, but did not leave the proof itself. And only in 1994 E. Wiles from Princeton University managed to prove this theorem, which for several centuries has been called Fermat's Last Theorem.

Outside of mathematics itself, number theory has quite a few applications, and it developed not for the sake of solving applied problems, but as an art for the sake of art, with its own inner beauty, subtlety and difficulty. Nevertheless, number theory has had a great impact on mathematics, since certain branches of mathematics (including those that later found applications in physics) were originally created to solve especially difficult problems in number theory. MATHEMATICS.

multiplicative bases.

Let us agree to assume that in what follows all Latin letters will mean (unless otherwise stated) integers. We say that b is the divisor of the number a(or what b divides a) and denote it b|a if there is such an integer c, what a = bc. The numbers 1 and - 1 ("units"), the reciprocals of which are integers, are divisors of any integer. If ± 1 and ± a are the only divisors of a number a, then it is called simple; if there are other divisors, then the number a is called composite. (Prime numbers are, for example, 2, 3, 5, 7, 11, 13.) If a positive integer a composite, it can be represented as a = bc, where 1 b a and 1 c a; if either b, or c composite, then it can be further factorized. Continuing to factorize, we must eventually come to the representation of the number a as a product of a finite number of primes (not all of which are necessarily different); for example, 12 = 2x 2x 3, 13 = 1x1 3, 100 = 2x 2x 5x 5. Otherwise, the number a could be written as an arbitrarily large number of factors, each of which is at least 2, which is impossible. The uniqueness theorem for factorizations, one of the fundamental theorems of number theory, states that, up to obvious changes in the signs and order of the factors, any two factorizations of a number a match; for example, any decomposition of the number 12 into prime factors can be represented by three numbers - 2× 2× 3; 2H 3H 2; 3H 2H 2; other expansions are obtained by replacing any two factors with negative numbers equal in absolute value. The theorem on the uniqueness of factorization occurs in Euclid's Elements, where it is proved using the concept of the greatest common divisor (gcd). If a d> 0 - common divisor of numbers a and b and, in turn, is divisible by any other number that divides a and b, then d is called the greatest common divisor of numbers a and b, which is written like this: GCD( a, b) = d; for example, gcd (12, 18) = 6. If gcd ( a, b) = 1, then the numbers a and b are called coprime. Euclid showed that for any two numbers a and b, there is only one GCD, and proposed a systematic method resembling "division by an angle"; with gcd numbers a and b their least common multiple (LCM) is related - the smallest positive number that is divisible by each of the numbers a and b. The least common multiple is equal to the product of the numbers a and b, divided by their gcd, or | ab|/gcd ( a, b).

According to the theorem on the uniqueness of decomposition into prime factors, prime numbers are the "bricks" from which integers are built. Apart from ± 2, all other prime numbers are odd, since a number is called even only when it is divisible by 2. Euclid already knew that there are infinitely many prime numbers. He proved this by noting that the number N = (p 1 p 2 ...p n) + 1 (where p 1 , p 2 ,..., p n are all prime numbers) is not divisible by any prime number p 1 , p 2 ,..., p n and therefore either N, or one of its prime factors must be a prime number other than p 1 , p 2 ,..., p n. Hence, p 1 , p 2 ,..., p n cannot be a complete list of all prime numbers.

Let be m i 1 is some given integer. Any number a when dividing by m gives a remainder equal to one of the numbers 0, 1, ..., m– 1. (For example, when m= 13 and a, taking successively the values ​​29, 7, - 21, 65, we get: 29 = 2× 3 + 3, 7 = 0× 13 + 7, –21 = –2× 13 + 5, 65 = 5× 13 + 0, and the remainders are respectively 3 , 7, 5, 0.) If the numbers a and b when dividing by m give the same remainder, then in some cases they can be regarded as equivalent with respect to m. Mathematicians say in such cases that the numbers a and b comparable modulo m, which is written like this: a є b(mod m) and is called modulo comparison m. We are all familiar with modulo 12 in the case of hours: 17 o'clock means the same as 5 o'clock in the afternoon, since 17 є 5 (mod 12). This relation, called comparison, was introduced by K. Gauss (1777–1855). It is somewhat similar to equality in that comparisons modulo the same m can be added and multiplied as usual: if a є b(mod m) and c є d(mod m), then a + cє b + d(mod m), a-cє b-d(mod m), ah sє bh d(mod m) and ta є tb(mod m) for any integer t. Reduction by a common factor is, generally speaking, impossible, because 20 є 32 (mod 6), but 5 No. 8 (mod 6). However, if ta є tb(mod m) and ( t,m) = d, then aє b(mod( m/d)). At d= 1 this essentially reduces to a common factor reduction; for example, 28 є 40 (mod 3), and since the numbers 4 and 3 are coprime, we can divide both sides of the comparison by 4 and get 7 є 10 (mod 3). It can also be shown that if aє b(mod m), then the gcd of numbers a and m equal to gcd of numbers b and m. As an example, consider the comparison 6 є 10 (mod 4): gcd(6, 4) is 2, and gcd(10, 4) is also 2.

All integers comparable to any number form one deduction class. For each module m exist m deduction classes corresponding to m remainders 0, 1,..., m- one; each of the classes contains one of the numbers 0, 1,..., m– 1 together with all numbers comparable with this number modulo m. If two numbers a and b belong to the same class of residues, i.e. satisfy the relation aє b(mod m), then GCD ( a,m) = gcd ( b,m); therefore, either all elements of a given residue class are coprime with m, or neither is coprime. The number of "reduced" classes of residues, i.e. residue classes whose elements are relatively prime to m, denoted f(m). Thus, a function arises on the set of integers called f-Euler function in honor of L. Euler (1707–1783). At m= 6 there are six residue classes, each containing one of the numbers 0, 1,..., 5. With this m only the elements of the class containing the number 5 and the class containing the number 1 are coprime. Therefore, f (m) = 2.

As with equations, one can consider comparisons with one or more unknowns. The simplest is a linear comparison with one unknown axє b(mod m). It is performed only when m divides the number ( ax – b), or ax – b = my for some integer y. So this comparison is equivalent to the linear equation ax - my = b. Since its left side is necessarily divisible by GCD ( a, m), it cannot be executed for any integers x and y, if gcd ( a, m) does not divide the number b.

It can be shown that the comparison ax є b(mod m) is solvable if and only if the gcd ( a, m) divides the number b, and if this condition is satisfied, then there exists exactly gcd ( a, m) residue classes modulo m whose elements satisfy this comparison. For example equation 2 x + 6y= 5 is undecidable in integers, because gcd(2, 6) = 2, and the number 5 is not divisible by 2; equation 2 x + 3y= 5 is solvable, because gcd(2, 3) = 1; similarly, equation 2 x + 3y = b solvable for any integer b. Indeed, for any a and m, such that GCD ( a, m) = 1, equation ax - my = b allowed for anyone b.

The equation ax - my = b- this, apparently, is the simplest example of a "Diophantine equation", i.e. an equation with integer coefficients that needs to be solved in integers.

General quadratic comparison ax 2 + bx + cє 0 (mod m) can be analyzed quite comprehensively. Multiplying by 4 a, we get 4 a 2 x 2 + 4abx + 4acє 0 (mod 4 am), or 2 ax + b) 2 є ( b 2 – 4ac) (mod 4 am). Assuming 2 ax + b = u and b 2 – 4ac = r, we reduce the solution of the original comparison to the solution of the comparison u 2 є r(mod 4 am). In turn, the solutions of the last comparison, with the help of slightly more complicated reasoning, can be reduced to solving comparisons of the form u 2 є r(mod p), where p- Prime number. Therefore, all the difficulties and all the interest lie in this seemingly special case of a general quadratic comparison. If comparison u 2 є r(mod p) is solvable, then u called quadratic residue modulo p, otherwise quadratic non-residue. The "quadratic law of reciprocity", discovered empirically by Euler (c. 1772) and proven by Gauss (1801), states that if p and q are different odd primes, then each of them is either a quadratic residue modulo the other, or this is not true for any of them, except for the case when and p, and q look like 4 k+ 3 and when only one of these numbers is a quadratic residue modulo the other. Gauss's theorem, which he called the "golden theorem", serves as a powerful tool for number-theoretic research and allows one to answer the question of whether a given quadratic comparison is decidable.

Comparisons of higher degrees of a kind f (x) j 0 (mod m), where f(x) is a polynomial of degree higher than 2, are solved with great difficulty. According to J. Lagrange's theorem (1736–1813), the number of solutions (more precisely, the number of residue classes, each of whose elements is a solution) does not exceed the degree of the polynomial f(x) if the module is simple. There is a simple criterion for the solvability of the comparison x n є r(mod p) due to Euler, but it is not applicable to comparisons of general form, on the solvability of which under n> 2 little is known.

Diophantine equations.

Despite the fact that the study of Diophantine equations dates back to the beginning of mathematics, there is still no general theory of Diophantine equations. Instead, there is an extensive set of individual techniques, each of which is useful for solving only a limited class of problems. Starting the study of the Diophantine equation, we would like to obtain a description of all its integer solutions, as was done above for the equation x 2 + y 2 = z 2. In this sense, only a small class of equations has been completely solved, most of which are either linear or quadratic. Solution of an arbitrary system from m linear equations with n unknown when n > m, was obtained by G. Smith (1826–1883). The simplest quadratic equation is the so-called. Pell equation x 2 – Dy 2 = N(where D and N are any integers), which was completely solved by Lagrange (1766). Also known are solutions of various individual equations or systems of equations of the second degree with more than two unknowns, as well as a few equations of higher degrees. In the latter case, mostly negative results were obtained: the equation under consideration has no solutions or only a finite number of solutions. In particular, K. Siegel showed in 1929 that the only algebraic equations in two unknowns that have infinitely many integer solutions are linear equations, Pell's equations, and equations obtained from both using special transformations.

Forms.

form is called a homogeneous polynomial in two or more variables, i.e. a polynomial, all members of which have the same full degree in the totality of variables; For example, x 2 + xy + y 2 - form of degree 2, x 3 – x 2 y + 3xy 2 + y 3 - form of degree 3. One of the main questions is similar to the one formulated above for the form x 2 + y 2 , namely: what integers can be represented using the form (i.e., what integer values ​​can the form take) with integer values ​​of the variables? And this time the quadratic case was most fully considered. For simplicity, we restrict ourselves to only two variables, i.e. forms of the form f(x,y) = ax 2 + b.xy + cy 2. D value = 4 ac – b 2 called discriminant forms f(x,y); if the discriminant is zero, then the shape degenerates into the square of the linear shape. This case is usually not considered. Forms with a positive discriminant are called definite, because all values ​​accepted by the form f(x,y) in this case have the same sign as a; with positive a the form f(x,y) is always positive and is called positive definite. Forms with negative discriminant are called indefinite because f(x,y) takes both positive and negative values.

If in f(x,y) change the variables x = Au+Bv, y = Cu + Dv, where A, B, C, D are integers satisfying the condition AD-BC=± 1, then we get a new form g(u,v). Since any pair of integers x and y matches a pair of integers u and v, then every integer represented by the form f, represent the form g, and vice versa. Therefore, in this case, we say that f and g are equivalent. All forms equivalent to a given one form an equivalence class; the number of such classes for forms with a fixed discriminant D is finite.

It turns out that in the case of positive definite forms in each equivalence class there is a unique form ax 2 + b.xy + cy 2 with such coefficients a, b, c, either - a b J a c, or 0 J bЈ a = c. Such a form is called the reduced form of the given equivalence class. The above form is used as a standard representative of its class, and the information obtained about it is easily extended to other members of the equivalence class. One of the main problems, which in this simplest case is completely solved, is to find a reduced form that is equivalent to a given form; this process is called casting. In the case of indefinite forms, we cannot specify the inequalities that the coefficients of only one form from each class must satisfy. However, there are inequalities that are satisfied by some finite number of forms in each class, and all of them are called reduced forms.

Definite and indefinite forms also differ in that any definite form represents (if it represents) an integer in only a finite number of ways, while the number of representations of an integer by an indefinite form is always either zero or infinite. The point is that, unlike definite forms, indefinite ones have infinitely many "automorphisms", i.e. substitutions x = Au+ bv, y = Cu + dv leaving the form f (x,y) unchanged, so f (x,y) = f (u,v). These automorphisms can be fully described in terms of solutions to Pell's equation z 2+D w 2 = 4, where D is the shape discriminant f.

Some particular results related to the representation of integers by quadratic forms were known long before the appearance of the general theory just described, which was initiated by Lagrange in 1773 and developed in the works of Legendre (1798), Gauss (1801) and others. Fermat showed in 1654 that every prime number of the form 8 n+ 1 or 8 n+ 3 is represented by the form x 2 + 2y 2 , every prime number of the form 3 n+ 1 is representable by the form x 2 + 3y 2 and there is no prime number of the form 3 n– 1, represented by the form x 2 + 3y 2. He also established that any prime number of the form 4 n+ 1 is representable, and in a unique way, as the sum of two squares. Fermat left no proofs of these theorems (as well as almost all of his other results). Some of them were proved by Euler (1750-1760), and the proof of the last of these theorems required him seven years of intense effort. These theorems are now known as simple corollaries of the quadratic reciprocity law.

Similarly, one can define the equivalence of quadratic forms from n variables. There are similar reduction and representation theories, naturally more complex than in the case of two variables. By 1910 the development of theory had progressed as far as it was possible with the help of classical methods, and number theory remained in a dormant state until 1935, when Siegel gave it a new impetus, making mathematical analysis the main tool for research in this area.

One of the most amazing theorems of number theory was proved by Fermat and, apparently, was known even to Diophantus. It says that any whole number is the sum of four squares. A more general statement without proof was made by E. Waring (1734–1798): every positive integer is the sum of no more than nine cubes, no more than nineteen fourth powers, etc. The general statement that for every positive integer k there is an integer s, such that any positive integer can be represented as a sum of at most s k-th degrees, was finally proved by D. Gilbert (1862–1943) in 1909.

The geometry of numbers.

In general terms, we can say that the geometry of numbers includes all applications of geometric concepts and methods to number-theoretic problems. Separate considerations of this kind appeared in the 19th century. in the works of Gauss, P. Dirichlet, Sh. Hermite and G. Minkowski, in which their geometric interpretations were used to solve some inequalities or systems of inequalities in integers. Minkowski (1864–1909) systematized and unified everything that had been done in this area before him, and found important new applications, especially in the theory of linear and quadratic forms. He considered n unknown as coordinates in n-dimensional space. The set of points with integer coordinates is called a lattice. All points with coordinates satisfying the required inequalities were interpreted by Minkowski as the interior of some "body", and the task was to determine whether the given body contained any lattice points. Minkowski's fundamental theorem states that if a body is convex and symmetrical with respect to the origin, then it contains at least one lattice point other than the origin, provided that n-dimensional volume of the body (at n= 2 is the area) is greater than 2 n.

Many questions naturally lead to the theory of convex bodies, and it is this theory that Minkowski developed most fully. Then stagnation set in again for a long time, but since 1940, mainly due to the work of British mathematicians, there has been progress in the development of the theory of non-convex solids.

Diophantine approximations.

This term was introduced by Minkowski to describe problems in which some variable expression must be made as small as possible when the variable takes integer values ​​not exceeding some large number. N. Currently, the term "Diophantine approximations" is used in a broader sense to refer to a number of number-theoretic problems in which one or more given irrational numbers occur. (An irrational number is a number that cannot be represented as a ratio of two integers.) Almost all of these problems arose from the following fundamental question: if some irrational number is given q, then what are the best rational approximations to it, and how well do they approximate it? Of course, if we use sufficiently complex rational numbers, then the number q can be approximated arbitrarily accurately; therefore, the question only makes sense if the accuracy of the approximation is compared with the value of the numerator or denominator of the approximating number. For example, 22/7 is a good approximation to the number p in the sense that of all rational numbers with a denominator of 7, the fraction 22/7 is closest to the number p. Such good approximations can always be found by expanding the number q into a continued fraction. Such expansions, somewhat similar to decimal expansions, serve as a powerful research tool in modern number theory. With their help, for example, it is easy to verify that for each irrational number q there are infinitely many fractions y/x, such that the error | q – y/x| less than 1/ x 2 .

Number b called algebraic, if it satisfies some algebraic equation with integer coefficients a 0 b n + a 1 b n – 1 +... + a n= 0. Otherwise, the number b called transcendent. What little is known about transcendental numbers has been obtained using the methods of Diophantine approximations. Proofs usually come down to finding approximation properties of transcendental numbers that algebraic numbers do not have. An example is the theorem of J. Liouville (1844), according to which the number b is transcendental if for an arbitrarily large exponent n there is a fraction y/x, such that 0 b – y/x| x n . Developing the ideas of Hermite, F. Lindemann in 1882 proved that the number p transcendentally, and thus gave the final (negative) answer to the question posed by the ancient Greeks: is it possible to construct a square equal in area to a given circle with the help of a compass and a ruler? In 1934, A.O. Gelfond (1906–1968) and T. Schneider (b. 1911) independently proved that if an algebraic number a, other than 0 or 1, raise to an irrational algebraic power b, then the resulting number a b transcendent. For example, number is transcendental. The same can be said about ep(expression value i –2i).

Analytic number theory.

Mathematical analysis can be called the mathematics of continuously changing quantities; therefore, at first glance, it may seem strange that such mathematics can be useful in solving purely number-theoretic problems. The first who began to systematically use very powerful analytical methods in arithmetic was P. Dirichlet (1805–1859). Based on the properties of "Dirichlet series"

considered as functions of a variable s, he showed that if gcd ( a,m) = 1, then there are infinitely many primes of the form p є a(mod m) (thus, there are infinitely many primes of the form 4 k+ 1, as well as infinitely many primes of the form 4 k+ 3). A special case of the Dirichlet series 1 + 2 - s + 3 –s+... is called the Riemann zeta function z (s) in honor of B. Riemann (1826–1866), who studied its properties under complex s to analyze the distribution of primes. The task is as follows: if p (x) denotes the number of primes not exceeding x how big is the value p (x) for large values x? In 1798 A. Legendre suggested that the ratio p(x) to x/log x(where the logarithm is taken to the base e) is approximately equal to 1 and with increasing x tends to 1. A partial result was obtained in 1851 by P.L. "the prime number theorem", was proved only in 1896 using methods based on the work of Riemann (independently by J. Hadamard and Ch. de la Vallée Poussin). In the 20th century A lot has been done in the field of analytic number theory, but many seemingly easy questions about prime numbers still remain unanswered. For example, it is still unknown whether there are infinitely many "pairs of primes", i.e. pairs of consecutive primes, such as 101 and 103. There is another hitherto unproven Riemann conjecture that concerns the complex numbers that are the zeros of the zeta function and is so central to the whole theory that many proved and published theorems contain the words "If the Riemann Hypothesis is true, then...".

Analytic methods are also widely used in additive number theory, which deals with the representation of numbers as sums of a certain type. Analytic methods were essentially used by Hilbert in his solution of Waring's problem, which was mentioned above. Attempts to give a quantitative character to Hilbert's theorem by means of an estimate of the number k-th powers necessary to represent all integers, led in the 1920s and 1930s G. Hardy and J. Littlewood to create circular method, improved further by I.M. Vinogradov (1891–1983). These methods have found application in the additive theory of prime numbers, for example, in proving Vinogradov's theorem that every sufficiently large odd number can be represented as the sum of three primes.

Algebraic number theory.

To prove the law of reciprocity of fourth powers (analogous to the quadratic law of reciprocity for the relation x 4 є q(mod p)), Gauss in 1828 investigated the arithmetic of complex numbers a + bi, where a and b are ordinary integers, and . Divisibility, "units", prime numbers and GCD for "Gaussian numbers" are defined in the same way as for ordinary integers, and the theorem on the uniqueness of decomposition into prime numbers is also preserved. Trying to prove Fermat's Last Theorem (that the equation x n + y n = z n has no solutions in integers for n> 2), E. Kummer in 1851 switched to the study of the arithmetic of integers of a more general type, determined using the roots of unity. At first, Kummer believed that he had succeeded in finding a proof of Fermat's theorem, but he was mistaken, because, contrary to naive intuition, the theorem on the uniqueness of factorization does not hold for such numbers. In 1879 R. Dedekind introduced the general concept algebraic integer, i.e. an algebraic number that satisfies an algebraic equation with integer coefficients and the coefficient a 0 with the highest term equal to 1. To obtain a certain set of algebraic integers, similar to the set of ordinary integers, it is necessary to consider only such algebraic integers that belong to a fixed field of algebraic numbers. This is the set of all numbers that can be obtained from some given number and rational numbers by repeated application of addition, subtraction, multiplication and division; the field of algebraic numbers is analogous to the set of rational numbers. Algebraic integers from the given field, in turn, are subdivided into “units”, prime and composite numbers, but in the general case for two such numbers there is no uniquely defined GCD and the theorem on the uniqueness of decomposition into prime factors does not hold. The simplest examples of algebraic number fields, apart from the set of rational numbers, are algebraic number fields defined by algebraic numbers of degree 2, i.e. irrational numbers satisfying quadratic equations with rational coefficients. Such fields are called quadratic number fields.

Kummer owns the fundamental idea of ​​introducing new so-called. ideal numbers (1847), chosen in such a way that the theorem on the uniqueness of prime factorization is again satisfied in the extended set. For the same purpose, Dedekind in 1870 introduced a slightly different concept of ideals, and Kronecker in 1882 introduced a method for decomposing a polynomial with rational coefficients into irreducible factors over the field of rational numbers. The work of these three mathematicians not only laid the foundations for the arithmetic theory of algebraic numbers, but also marked the beginning of modern abstract algebra.

The question of whether a given field has a unique prime factorization is very difficult. The situation is clear only in one case: there are only a finite number of quadratic fields with this property, and all such fields, with the exception of one dubious case, are well known. With the "units" of the field, the situation is simpler: as Dirichlet showed, all "units" (of which, generally speaking, there are infinitely many) can be represented as products of powers of some finite set of "units". Consideration of such problems in connection with some particular field necessarily precedes deeper arithmetic studies within this field and applications to problems of classical number theory. There is another, more subtle theory, initiated in 1894 by Hilbert, in which all number fields with certain properties are simultaneously considered. It is called "the theory of class fields" and belongs to the most technically rigorous branches of mathematics. A significant contribution to its development was made by F. Furtwängler in 1902 and T. Takagi in 1920. In recent years, significant activity has been observed in this area of ​​mathematics.

My collaboration with Amos in the 1970s began with a discussion about the claim that people have an intuitive sense of statistics even if they have not been taught statistics. At the seminar, Amos told us about researchers at the University of Michigan who were generally optimistic about intuitive statistics. This topic worried me very much for personal reasons: shortly before that, I discovered that I was a poor intuitive statistician, and I could not believe that I was worse than others.
For a research psychologist, sample variability is not just an oddity, it is an inconvenience and a hindrance that comes at a cost, turning any study into a game of chance. Suppose you want to test the hypothesis that the vocabulary of six-year-old girls is, on average, larger than that of boys of the same age. In the volume of the entire population, the hypothesis is correct, girls at the age of six have, on average, a larger vocabulary. However, girls and boys are very different, and you can randomly choose a group where there is no significant difference, or even one where boys score more. If you are a researcher, this result will cost you dearly, because, having spent time and effort, you will not confirm the correctness of the hypothesis. The risk is reduced only by using a large enough sample, and those who work with small samples leave themselves to chance.
The risk of error in each experiment is estimated using a fairly simple operation, but psychologists do not use calculations to determine the size of the sample, but make decisions according to their own, often flawed, understanding. Shortly before the discussion with Amos, I read an article that perfectly illustrates the typical mistakes of researchers. The author noted that psychologists often use such small samples that they run the risk of not confirming the correct hypotheses with a probability of 50%! No reasonable researcher would take such a risk. A plausible explanation seemed to be that psychologists' decisions about sample sizes reflected prevailing intuitive misconceptions about the range of variability.
I was struck by the explanations contained in the article, which shed light on problems with my own research. Like most psychologists, I consistently used too small samples and often got nonsensical, bizarre results that turned out to be artifacts from the very method of my research. My mistakes were all the more embarrassing because I taught statistics and could calculate the sample size needed to reduce the risk of failure to an acceptable level. But I never did this when planning experiments and, like other researchers, I believed in tradition and my own intuition, without thinking seriously about the problem. By the time Amos attended my seminar, I had already realized that my intuition was not working, and during the seminar itself, we quickly came to the conclusion that the optimists from the University of Michigan were wrong too.
Amos and I set out to find out if there were any naive fools like me among researchers, and if scientists with mathematical knowledge make the same mistakes. We have developed a questionnaire describing realistic studies and successful experiments. Respondents were asked to determine sample sizes, evaluate the risks associated with these decisions, and provide advice to hypothetical graduate students planning a research project. At a conference of the Society for Mathematical Psychology, Amos conducted a survey of those present (including the authors of two textbooks on statistics). The results were clear: I was not alone. Almost all respondents repeated my mistakes. It turned out that even the experts are not attentive enough to the sample size.
The first article I co-authored with Amos was called Faith in the Law of Small Numbers. It jokingly explained that "... an intuitive estimate of the size of random samples seems to satisfy the law of small numbers, which says that the law of large numbers applies just as well to small numbers." We also included in the article a strong recommendation for researchers to treat their “statistical hunches with a grain of salt and replace impressions with calculations whenever possible.”