The author of the book “How not to be mistaken. The power of mathematical thinking ”Jordan Ellenberg believes that everyone can understand the existing arithmetic concepts, and believes that the information received can expand the horizons of anyone.
The numerical approach to life
Hardcover reveals logic and a non-standard approach that allows a lot to explain. The mathematics teacher and author of numerous publications and newspaper articles in The Washington Post in the New York Times presents the school subject to readers not as a boring list of rules, but as a central system on which everything is based. Mathematics is a chance to see the veiled microstructures of the universe, to understand the true meaning of the source data and to interpret it critically.
Everyday life consists of many questions that concern every inhabitant of the planet. Why are men and women of high stature born children who are completely different from their parents, what is the opinion that is considered public, who can win the next election, what is the percentage of cancer formation in a healthy person? Ellenberg in the book “How not to be mistaken. The Power of Mathematical Thinking ”presents to the reader a method of analyzing life questions, which describes phenomena and ideas (reaganomics, lottery schemes) that are explained in an accessible form. The author claims that this manual will help to understand the world deeper. Especially for those interested in math.
Little bit about jordan
Ellenberg was a child prodigy in childhood; he learned to read thanks to television programs. In eighth grade, he was offered to attend classes at the University of Maryland. At this stage of life, the author of the book “How not to be mistaken. The power of mathematical thinking ”Jordan Ellenberg is a teacher of this subject at the University of Wisconsin, and at the same time, is publishing articles on similar topics. He, by the means of this textbook, wants to show the reader that science is not limited to ordinary calculations, it is a skill of non-standard thinking, which helps to avoid frequently made mistakes.
To the question why the book “How not to be mistaken. The power of mathematical thinking, ”the author called it that way, and what was the idea, Jordan replied that the idea originated a long time ago, and, initially, he wanted to express his thoughts on the power of exact science. In his opinion, to tune in to living a life path without the wrong steps is too arrogant, but to make a plan that will help to avoid incorrect steps in advance is practical.
Math submits a large part of the life of an individual, but even humanities who do not like algorithms and numbers are successful. There is a difference between “being partial” to mathematics and understanding its essence. If a person can understand mathematical ideology, knowledge of this science will enrich him.
Statistics and military tasks
The book of Jordan Ellenberg “How not to be mistaken. The power of mathematical thinking ”is designed for everyone who wants to change their lives, who want to look at everything from this point of view. In his manual, he takes statistical studies of specialists and proves his theories with examples.
One vivid example is the solution of military tasks. The discussion raised the question of how much armor needed to be added to fighters, and what places should be strengthened so that it was harder to shoot down, but at the same time, their maneuverability was not affected. A table was compiled in which the damage to the aircraft was painted.
Abraham Wald argued that it was necessary to pay attention only to protecting the engine, due to the fact that aircraft with holes in certain places returned to the base, unlike those that received a bullet in the working engine. Why did Wald notice what the officers did not pay attention to? The reason, as Ellenberg is confident in “How not to be mistaken. The power of mathematical thinking ”is in the corresponding structure of Abraham’s thinking. A person whose life is based on numbers, when solving problems, poses the questions: “From what assumptions does this or that conclusion come from? What facts are they based on? ”
In this story, the military assumed that the returning aircraft was a random sample of the total, but when it comes to the erroneousness of this assumption, it becomes clear that it makes no sense to expect the objective survival of all aircraft regardless of which part of the object the bullet hits . Such a conclusion can be described by the term "survivor's systematic error."
Inaccuracies and Flaws
Miscalculations often arise in different situations. Similarly, it cannot be argued that dolphins push people leaving under water to land, because aquatic mammals only support the sinking afloat, pushing it in an arbitrary direction. But only those who survived were able to talk about it. The theory presented by Abraham in the middle of the last century made it possible to understand the meaning of "infinitely small increments", which, earlier, were considered absurd. His mathematical mindset allowed him to avoid unnecessary mistakes, and take a step in the right direction to solve the problem.
Jordan Ellenberg in How Not to Mistake. The Power of Mathematical Thinking ”touches upon the comparison of the development of Sweden and the USA, showing on the graphs a linear relationship between material well-being and the level of social benefits. The Swedes are leading their economies towards the free market, reducing social security, while America, on the contrary, is on the path to increasing its volume. The textbook presents graphs that reflect the difference in linearity and non-linearity between countries. The author notes that thinking nonlinearly is important because not all lines are straight.
Straight and curve
A linear thought process is everywhere. Author “How not to be mistaken. The power of mathematical thinking ”states that each of us thinks in this way, especially when he acts on the principle“ if you have something, then it is better to increase its number ”. Ellenberg does not understand how one can be sure that all lines are straight, when, in his opinion, the opposite is obvious. Newton said it is necessary to reduce the field of view until it becomes as small as possible, but not equal to zero.
Linear thinking is characteristic of every person, because an unconscious perception of time and movement is formed under the influence of external phenomena. Even before Newton’s discoveries, everyone subconsciously understood that everyone around was trying to move in a straight line, unless there was an opportunity or reason to move differently.
Linear regression on an example
Genre “How not to be mistaken. The Power of Mathematical Thinking ”is a popular scientific literature, where the author analyzes arithmetic structures using specific examples. In chapter 3, Ellenberg takes as his basis the article published in the journalistic publication, which suggested that all Americans in the future, by 2048, will be obese.
He immediately hurries to reassure American readers, assuring that this assumption cannot be true, since not all lines are straight, you cannot design this hypothesis for everyone, initially laying the same result. As mentioned above, each line is close to a straight line, and this idea underlies linear regression.
Good luck and broker from Baltimore
“How not to be mistaken. The power of mathematical thinking ”has a unique structure. The author in the prologue raises an important issue of concern to readers. It sounds like: "Why do I need math?" In subsequent chapters, he answers it, showing the vastness of the applicability of this science and its direct connection with real reality.
In the sixth chapter, Jordan proposes to get acquainted with the parable, thanks to which one can imagine the following situation: a person receives a letter from the Baltimore stock broker, which refers to the near increase in the price of certain shares. Over the course of the week, stocks did indeed rise in price. A week later, another notification arrives, where there is already information about the reduction in the value of shares, according to the broker. Indeed, after a few days, stocks fell. This happens 10 weeks in a row, and every week a person receives an informational letter from a broker with a correct prediction.
On the 11th week, an offer comes from the same broker to invest money through it for a commission. Initially, there is no doubt that such an investment is a good deal. But, if you analyze the situation in detail, confidence will go by the wayside. In “How not to be mistaken. The power of mathematical thinking ”Ellenberg makes you think, and argues for the pros and cons. It is undeniable that the stock broker from Baltimore understands something in the game on the stock exchange, because 10 amateur forecasts cannot be made by an amateur without knowledge of the market and stocks.
But in fact, everyone can calculate the chances of success: if a beginner gives the correct forecast with a probability of 50%, then the probability of getting ten correct predictions in a row will be (1/2) 10 = 1/1024 = 0.1%. If we describe the situation from the point of view of a stock broker, then in the first week he sent out 10,240 letters: 50% of them were with a forecast for stock growth, 50% - their decline. Half of the people who received letters (with the wrong prediction) no longer received forecasts, the rest, again received letters according to the same scheme.
Accordingly, already a quarter of the initial number - 2,560 people, received two correct predictions in a row. After the tenth week, there are 10 people who constantly received the correct forecasts, it is in their eyes that the broker looks like a genius. With these ten people, the broker plans to collect large commissions in the future, playing on their trust.
The book "How not to be mistaken. The Power of Mathematical Thinking ”distinguishes between present-day phenomena and random obstacles using generally accepted methods for analyzing results. Geometry and arithmetic, according to the author, correspond to our intuition. But, likelihood is the other side of the issue, as is the null hypothesis - the opinion that the effect under study does not have an end result.
The procedure for contesting the null hypothesis is based on the experiment: an assumption is put forward in which the null hypothesis is taken as true, and the probability is indicated by the symbol "p". When the value of "p" is minimal, then we can assume that the facts obtained are important. If the value is large, then the fact remains: the fallacy of the null hypothesis has not been proven.
In the ninth chapter “How not to be mistaken. The Power of Mathematical Thinking ”describes the story told by Shalizi. The author invites readers to present themselves as a haruspex - a fortuneteller who understands the nature of future incidents on the offal of a sheep, slaughtered for the sake of science. All available statistical results are sent to the publication on international haruspi, where the published data are tested for veracity, in accordance with statistical significance.
Ellenberg says he does not believe in pseudoscience and believes that animals can not predict anything, and true predictions are just a coincidence. Considering the haruspius to be a swindle, Jordan is sure that if the null hypothesis is always true, then only 1/20 of the results of the experiments can be made public, despite this, these statistics support the belief of a certain number of people in a strange science.
Formed opinion about the book
Reviews on "How not to be mistaken. The power of mathematical thinking ”is extremely positive. Readers note that it is practically impossible to write a popular publication about this science, but Ellenberg did it. Many discover the book as an interesting and exciting textbook on mathematical science. The author’s subtle sense of humor is often noted, which makes the book even more attractive.
The author of the book tried to write the text in an accessible form so that every reader would not have difficulty in understanding the thought. Ellenberg demonstrates graphs, tables, formulas, but all in an accessible, understandable form.