How to deceive, manipulate and present yourself in a favorable light in the greatness of mathematics?

At the beginning of November 2020, Mateusz Morawiecki referred to mathematicians from the Center for Mathematical Modeling that they showed that the Women's Strike caused an increase in infections by 5000. I have friends in this Center - they only learned that they had predicted this from a speech by Mr. - to Mateusz.

I would like to emphasize that, perhaps contrary to the title of the article, I will neither praise nor criticize the current prime minister. I think that mathematics is not his forte, but such an intellectual deficiency will not raise objections from most of you. And in general, wouldn’t a great mathematician be in a responsible position, but not wise in life and politics? I will also mention that Donald Tusk, in his former presidential campaign, said (as if jokingly): “you can’t write math exams without downloading.” You know, the math cloud is your man, just like me. Julian Tuwim was snobbish about his ignorance of mathematics. And they called me to the board. I will only note that we had a premiere in mathematics in Poland. It was (five times) Kazimierz Bartel, 1882-1941, rector of the Lviv Polytechnic, an excellent geometer. I cannot and do not attempt to judge his reign.

Wiping the mouth is versatile and old. Books, thin and thick, have been written about it. There are many ways, I will talk about some, I will start with those that are sewn with thick threads. Perhaps in the past there were even more such methods, because in the monumental and first of its kind Dictionary of the Polish Language Samuel Bogumil Linde (published in 1807-1814) we read:

Mathematician, mathematical mathematician, mathematical juggler.

We do not know the simplest actions, and we really want to prove ourselves. A few years ago, a journalist from Olsztyn wrote a long exposé about how manufacturers are deceiving us. For example: on a pack of butter it says “fat content 85 percent” - is it 85 percent in a cube or in a kilogram? All of Poland chirped. But only smart math teachers (that is, all math teachers!) noticed an error in the reasoning of one of our former prime ministers, Kazimir Martsinkevich, many years ago. I'll change the numbers a bit to make it easier to see. He said something like this: we spent 150 million zlotys on road construction, and received 50 million from Brussels, so we will spend only 100. We saved 50 percent. Well, 50/100 is 50 percent. Where is the mistake? And if we had 100 million, how much would we save? The mistake is subtle. Speaking of percentages, it is important to clarify where we get them from. This is a very common mistake teachers make. They say a percentage is one hundredth. This is not allowed! A hundred percent, but it's always something. If we spend 150 and spend 100, we save 50 out of 150, which is 33%. Prime Minister Martsinkevich was a physics teacher. Either he was such a bad teacher that he didn't understand percentages, or he deliberately manipulated them to get the best political effect. I would actually prefer the latter. Let me remind you of a very old, pre-war anecdote. “Dad, I saved 20 cents today!” "It's very good, son! How? “I didn’t ride the tram to school, I ran after it!” “Ah, son, run for a second time for a taxi - you will save 5 zlotys!”

Ideas, ideas! Most of the ideas of so-called creative accounting are based on legal loopholes (law written on the knee = crap) and stray from the notion of average. Here's an example: how can everyone's wages be raised while lowering the average wage? Simple: give a small raise to those who are already working, and in doing so, hire a lot of underpaid people. The average will fall… and in the context of the global wage bill, it was out of the question. Allegedly, until 1989, a certain director of a state-owned enterprise behaved this way.

You can fight directly, using the mathematical illiteracy of many circles of society and combining mathematics (??) with literature (??). Here is a demagogic but fictional text (albeit based on a real publication, before 2010 for attention).

The nurses will be better off. Two years ago, the average net salary of a nurse in Sochaczew county was PLN 1500. Last year, the government increased spending on healthcare by half a billion zlotys. This will be twice as much as in previous years. Hermenegilda Kotsyubinskaya, a nurse at the Central Clinical Hospital, says: last month my salary was PLN 4500. This means a huge, threefold increase in health care revenues.

Is there anyone to deceive? Even if the numbers are the same, you can see what we're comparing here. average salary in the provincial hospital with the salary of one person in a given month. Maybe Hermenegilda is the head of the nurses, maybe she had a lot of extra shifts this month, and besides, the CRH has a special salary scale? In addition, the mentioned PLN 1500 are net wages and it is not specified whether Ms. Kociubinska's wage is net or gross. Half a billion is a huge amount for an individual, but what does it mean at the national level? We note right away that “half a billion” sounds better propaganda than “500 million”. What 500 million zlotys went to is not reported. It is not known why 500 million zł twice as much.

How can I improve my learning outcomes? School X is criticized by education authorities for poor educational outcomes (i.e. a low GPA, although these are different things!). The headmaster finds a way to make things a little better. He transfers several students from class A to class B and achieves his goal: the average score in both classes has increased.

How is this possible? If there is a student in class A whose GPA is lower than the average in class A, but higher than the average in class C, then moving him to class B will have the same effect. Faith is based on this effect Mechislav Plague i Leshek Mazan, authors of the "Galician Encyclopedia" (publishing house "Anabasis", Krakow), that on the day when Sigismund III Vasa and his court moved to Warsaw, the average level of intelligence increased in both of these cities.

We tend to interpret data. This is the most common non-elementary stretch. I'll start with the most stupid, but reliable example. Many, many years ago, the now defunct Express Wieczorny reported that the average salary at the University of Warsaw would be 15000 24 złoty (then złoty). The rector was supposed to receive the highest salary, 6, the lowest novice assistant, 15. Average XNUMX!!! Manipulation the concept of the average is a topic for habilitation.

Here are two more examples. Do you know that the average person in Poland has less than two legs? Well, yes: there are those who have one, but no one has three! The second example is more subtle. Well, my wife and I have our own cars. My carrier consumes a lot of fuel, 12,5 liters per 100 km. This means that for 100 km I need 8 liters. My wife has a tiny Mitsubishi - it consumes 8 liters per 100 km. This is also a lot, but in order for the calculations to be simple, the data needs to be processed a little. We often ride the same one. Therefore, the average fuel consumption of our two cars is the arithmetic average of 8 and 12,5. Add up, divide by 2. It turns out 10,25 liters. Of course, it is important that we often ride the same way. So where is the scope for manipulation?

Oh, here. Did you know that US fuel consumption is calculated differently? They will answer: "I drive so many miles from one gallon." Let's leave the conversion of gallons to liters and miles to kilometers, but apply it to the aforementioned cars: mine and Our Marriage's Sole Review Board. I will only drive 8 km per liter (100 divided by 12,5), my wife 12,5 km (100 divided by 8). On average, one liter will take us ... the arithmetic mean of these figures. We have already calculated this once. It turns out 10 and a quarter - this time 10,25 kilometers.

Let's go back to European standards. If I drive 10,25 km on one liter, how many liters do you need for 100? Let's take a calculator: 100 divided by 10,25 is ... 9,76. The average consumption of our cars is 9,76 ... and before that it was 10,25. Where is the mistake? No! Actually, not in mathematics, but in the interpretation of the words “we travel equally often”. Careful analysis will show that in the first interpretation this means "we drive the same number of kilometers per month", and in the second "we use the same amount of gasoline." A third variable could be added: we spend the same amount of time driving (wife drives much faster)… and it would be different. If we are measuring something, we must have a measuring tape.

more subtle situations. Simpson's paradox. We explore what is better to remove dandruff: Coca-Cola or Pepsi-Cola. We test on women and men. Here is the data. Almost all calculations can be done in memory.

Please, Reader, sit down. Just to not fall out of the feeling. What is the best drink to remove dandruff in men? I've marked the larger numbers in red and the smaller ones in blue. 25 is more than 20, right? Gentlemen: buy Coke for dandruff! What about women? Probably the other way around? No, 60> 53. Ladies, have a Coke.

The company buys ads on television, where a happy couple (in the old fashioned way: a man and a woman) get rid of this mild ailment with the help of Coca-Cola. But there is a Pepsi ad. Well, because there were 250 people on the test both here and here, which means they were evenly divided. Coca-Cola helped 80 people (32%), Pepsi helped 100 people, 40%. On screen, the crowd is shedding their dandruff while a can of Pepsi rolls in front of the camera. “Our generation has already chosen!”

Where is the mistake? No. I mean, the math is fine. Or rather just arithmetic. To be mathematically correct, we must take comparable samples with the same proportion of M as K. Otherwise, the calculations do not make sense, as if we were calculating the average weight of a mosquito and an elephant. We can add and divide by two. What have we calculated? Well, the average weight of a mosquito and an elephant. What will it give us? A thread.

But let's take it to politics, to the US, of course. Supporters of one of the candidates, say Bump, would cry: we are better for both ladies and gentlemen. Vote for Jozef Podskok! Triden supporters would write on banners: We are the best in the world. Vote duck with 3 dens (Donald).

Okay, how is it really? This is the hardest part. What does "really" mean? We can say: "True is that which agrees with reality." However, another question arises: how to measure "correspondence to reality"? But this is no longer mathematics, and I would like to stick to it, because only here I feel confident.

About this paradox (called Simpson's paradox) is based on many, many others. It has been known in mathematics for a hundred years, but (relatively) recently the social sciences have taken an interest in it. It all started with the fact that at one of the American universities the rector noticed that girls were accepted much less than boys. She asked for reports from the deans... and it turned out that in every faculty the ratio of accepted to candidates was higher for girls than for boys - and quite the opposite. I recommend that the reader recast the example of Pepsi and Coca-Cola to the situation of university departments.

An even more subtle situation. Everyone in the mathematical world knows the "Nebraska example". Somewhere in Nebraska, a shop was ransacked and a cash register was robbed. Witnesses only remembered that this was done by a strange couple: a dark-skinned man with a beard and a woman with oriental features. They left (tires screeching like in the movie) in a yellow Toyota. A few hours later, the police detained ... a yellow Toyota, in which there was an African American with a beard, accompanied by an Asian woman. "It's you!". Handcuffs, court. An experienced mathematician calculated that such a set (Negro + Asian + yellow Toyota) is so unique that 99,999% of robbers are wanted. He threw memorized terms in the hall: elementary events, Bernoulli diagram, conjunction. The couple went to sit. However, they hired the best mathematician, who said in an appeal: “Good. Judge for yourself, my predecessor calculated that the probability that a randomly encountered car with two passengers will be a yellow Toyota with a black one and a Japanese woman is such and such. But here we need to solve another problem, the conditional probability. What is the probability of meeting another pair (or three, if you turn on the machine), if we know that such a one already exists. »

We do not know if the judge understood any of the arguments. Perhaps only that the answer depends on the choice of the situation. That was enough. He canceled the sentence.

A blow to the head with a pole. We have always treated such demagogy (1).

Bars are terrible: coal prices have doubled. Looking at the numbers is reassuring: they have indeed risen from PLN 161 per tonne to PLN 169 (exercise: by what percentage?). But since most people learn visually, they will remember the graph, not the numbers. Without going into political discussions, I must say that a similar method was used by the government (the one from the summer of 2020), imagining an increase in spending on cancer. This is not a criticism of this government. The next one will also use this method. It is safe and gives an immediate effect ("seen").

Let's wear masks. The laws of the spread of epidemics are simple and "in themselves" inexorable. The number of infected people is growing faster, the more of them there are already. This is how the avalanche goes. That's what the math says. There is, however, a big "but" - maybe more than one. First, it is so, while "nothing happens". When the avalanche in the forest is stopped, when the epidemic is slowed down by the wise behavior of all of us, then we will not so much “thank” mathematics as create a different model. Yes, a different mathematical model (as in the Nebraska store robbery example). Mathematics, a beautiful science, only helps to understand the world. So many, but only so many. Let's see: we jump almost six meters with a pole, without it we can't even jump 2,50. Then take the pole in your hand and jump. He's a hell of a nuisance, isn't he?

use mathematics in social sciences it is difficult, dangerous, and worse, tempting. The connoisseurs of the Tatras associate it with the Drege ravine: a gentle, grassy descent from Garnets to Chyorny Stav ... This is how it looks from above. Soon the ravine turns into a trap from which only TOPR, the Tatra Volunteer Rescue Service, can save us.

Mathematicians call this increase in avalanches and epidemics exponential growth. As I already wrote, this growth can be suppressed, but not again. However, let's look at two plots of the same curve (just on a different scale). Who will understand, I give the formula of this function: y = 2^xtwo to power. Please look at the charts. From what point does the rapid acceleration of growth occur? Everyone will indicate: it is more or less close to the point marked with a large dot. But on the first graph this value is close to 1,5, on the second it is more than 3, and on the third it is 4,5. If there will be some kind of street demonstrations then, then we can say: please, from the moment of the demonstration, the curve went up, went up sharply. In the glory of mathematics! And this is just a property of the exponential curve. The corresponding scale and point from which fast acceleration starts can be freely chosen (2).

Presidential elections ... in the US, of course. We still remember the farce of November 2020. The country, which is still the No. 1 power, has not coped with the page count. In the end it turned out that Joe Biden not only did he win more electoral votes, but he would have won if the decision had been taken by a simple majority. In the situation that I will describe, there is no mathematical manipulation - just an example of how the result of the elections can depend on the adopted resolution. If you know, it's hard to protest. A defender in football may consider the handball ban to be wrong, but if it is ignored, a penalty will be awarded.

Imagine that the following are running for the presidency of Greece: Apollonius, Euclid, Heron, Pythagoras i Such. Whoever voters choose will become president. There are 100 of them. They were elected by popular vote, and then the parties represented in Parliament, that is, the Circus Maximus, established the order of their preferences. Something is wrong because Circus Maximus is a Latin name, not a Greek one. But let's not argue with the sources.

Who will become president? Let's see how it depends on ordination. The preferences of the party should be understood in such a way that its voters vote for the first person from the list remaining in the elections after the next round.

If the ruling stipulates that the candidate who puts the most voters in first place wins, Pythagoras will win, because he will be elected by 25 + 9 = 34 voters. This is what happens at school when we choose, for example, the best student. In our place: Pythagoras is elected by the people!
In modern presidential elections, the second round system is most often used. We vote for one candidate, but if none of them exceeds 50 percent, a second round is held. The winner is the one who gains the absolute majority of votes, that is, simply more votes than his opponent. In this scenario, Pythagoras (34 votes) and Thales (20) will go to the second round. In the second round, voters distribute their votes according to their preferences. All but the Pythagoreans prefer Thales to Pythagoras. This is a common situation where a party has a tough electorate and is surrounded by general reluctance. So in extra time, Pythagoras will not receive a single vote. Result 66:34 in favor of Thales and a decisive victory. A similar situation occurred in 2001 in Slovakia, where a candidate who clearly won the first round lost in the second. It was similar in the presidential elections in Poland in 2005: the leader was defeated in the second after the first round. Long live Presidential tales!
In cycling, the so-called Australian system is used. After each lap of the track, the last one is eliminated. This version of the electoral law is called the "election of directors". Under this system, the first president of independent Poland, Gabriel Narutowicz, was elected. How would it look in our Greece?

The matter is more complicated. Please track. In the first round, Euclid received the fewest votes and dropped out (what a pity, such a good mathematician!). The party then votes in the second round for the second on its list: Tsaplya. In the second round Heron has 19 + 10 = 29 votes. Apollonius is eliminated (17 votes). Party, and then vote for Heron. In the third round Pythagoras (fixed electorate) has 34 votes, Thales 20 and Heron 29 + 17 = 46 votes. The stories are out. The Falesians (Party B) don't like the Pythagoreans either - they prefer heralds. Others too, except for stable parties A and E. In the final turn, Heron easily defeats Pythagoras 66:34. Vivat President Heron!

4. At the Eurovision Song Contest, 12 points were awarded for the first place in the list, 10 for the second place, 9 for the third, and so on. Let's assume about the same score 6-4-3-2-1. So points were awarded in three athletics matches (three teams, two players in each competition, in 1958 Poland won against the USA and Great Britain!). Our results will be as follows:

Euklides: 4+2+3+4+6+4=23.

Apoloniusz: 2+3+4+5+3+3=20.

Цапля: 1+4+6+3+4+1=19.

Сказки: 3+6+2+2+2+2=17.

Pitagoras: 6+1+1+1+1+6=16.

Greeks, here is your President Euclid!

5. Readers guess that we only need to count the votes so that it turns out that Apollonius is the best. Indeed, Apollonius is the best - because he is the best. Everyone loses to Apollonius! Why?

For how many electors placed Apollonius above Heron? Let's calculate: 25+17+9=51 means majority. Not much, but still.

How far is Apollonius ahead of Euclid? 20 + 19 + 17 = 56, most of them.

How many prefer Apollonius to Thales: 19+17+10+9=55>50.

Finally, Apollonius of Pythagoras prefers 20 + 19 + 17 + 10 = 66 electors out of 100.

Since then - the Greek people, able to think logically - since then, most of all, Apollonius prefers any other candidate; after all, it is he who should rule us for the next term! Come closer, Apollonius, our President-elect! You will be our 44.