Habits of people with a mathematical mindset

Hi, Geektimes! Recently, the developers of Wirex , a Fintech-start-up providing payment and money transfer services without banking intermediation, have come across very interesting material. His author analyzed some of the features inherent in people with a mathematical mindset, told what skills can really come in handy in life and outlined the benefits of a mathematical approach in assessing events. To ensure that this publication does not remain only in the field of view of the foreign media audience, we decided to make a translation of it, which we are in a hurry to share with all Geektimes users.

Next, we present an original translation of an article from the Medium blog platform devoted to the habits possessed by each mathematician.

One of the most popular questions that students ask teachers of mathematics is: "Where is it useful to me at all?" Few teachers can immediately give a reasonable answer that goes beyond the generally accepted point of view. Usually they give a standard explanation on the usefulness of the development of "critical thinking" and this ends with. At the same time, these same teachers should be able to calmly tell their students about the importance of knowing the arc cosine derivative.
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I offer you my list. In it, I included real, well-articulated skills that, being thoroughly mastered by students, will be useful to them in practice and will be useful in life beyond their mathematical activity. Some of them are of an applied nature: mathematicians use every day to talk about complex, multifaceted tasks. Others are socially useful and allow you to train your emotional intelligence, so necessary for everyone who wants to succeed in an area of ​​activity where almost all of their time has to be spent trying to understand something that really does not exist. All of them are studied in their purest form in the framework of mathematics.

Here is the list itself:

1. Ability to articulate definitions
2. Thinking examples and counterexamples
3. The ability to often make mistakes and admit their mistakes
4. Evaluation of the consequences of approval
5. Ability to consider the assumptions underlying the statement separately from each other
6. Method "stairs abstraction"

Ability to articulate definitions

The main skill that is developed by mathematicians in the course of their professional activities is flexibility and efficiency in working with the conceptual apparatus. And this skill is much more important than it may seem at first glance. By this I want to say that mathematicians are literally obsessed with finding the best and most useful meanings of each word they use. They need logical accuracy because they work in a world of concepts that can be unambiguously confirmed or refuted. And if a concept has a “semantic completeness”, then it must necessarily be defined.

Let me start with a mathematical example that has something to do with the real world. Let's talk about the "random". The concept of randomness has called off the eyes of mathematicians throughout almost the entire modern history of science, since it is quite difficult to give a precise definition of what event can be called random. Scientists-statistics solve this puzzle, considering as random not things, but processes and, accordingly, believing that it is possible to calculate the probability of an event, based on the results of the processes. So you can briefly describe the concept, which, despite its simplicity, underlies almost all statistics.

However, this is not the only definition of chance. Take, for example, the situation with a coin flip. The sequence of OROOOOOORROROOOOOOOR will seem quite random to us, whereas we will never want to recognize twenty identical “eagles” in a row as an accidental coincidence of circumstances. Mathematicians looked at this situation and decided that statistical determination of randomness was not enough and they invented a second definition called “Kolmogorov complexity”. Roughly speaking, an event is called “Kolmogorov random”, if the shortest computer program that reproduces it consists essentially of this event. Immediately, I note that the definition of “computer” is used here purely mathematical, that is, it is not about modern computers, but about the concept with which Alan Turing operated. Speaking in a simpler language, it is possible to imagine that an event that is random according to Kolmogorov requires that you describe it entirely in the source code of the computer program that reproduces it.

A separate remarkable area of ​​mathematics and computational theory has grown out of Kolmogorov complexity, but our story does not end there. Studying and developing this direction, mathematicians soon discovered that for many events the Kolmogorov complexity cannot be calculated, and therefore it is very difficult to use it for solving practical problems. A definition was needed that could describe numbers that would look random and were fairly random for practical use, even though it was actually non-random in the Kolmogorov sense. The result of these searches was the definition of cryptographically secure randomness used today.

A simplified definition of randomness from the point of view of cryptography assumes that no effective computer program that aims to distinguish between pseudo-random and truly random events (in statistical sense) will not have a significant advantage in this case compared to trying to guess the result with a probability of 50 by 50. This approach ensures that your sequence of numbers will be random enough so that your enemies are unable to determine which numbers you will use But because their attempts to make accurate calculations will be comparable in time with the duration of their life. This is the basis of modern cryptography, having adopted that, engineers have designed systems that support the security and privacy of our Internet communications today.

So, mathematicians spent a lot of time thinking about definitions, which ultimately influenced how we use mathematics in the real world. Nevertheless, I do not consider this an argument in favor of the need to teach mathematics to all.

So how can thinking about definitions help people in the real world? Let's look at specific examples. The first will be the case of Keith Devlin, a mathematician and consultant who helped the US defense agencies improve data analysis after the events of September 11. He begins the description of his first presentation by finding himself in a room with a large group of military contractors and starting his conversation by trying to figure out the definition of the word “context”. Next, I give you the main excerpts from his story.

I was preparing my PowerPoint project ... and I was sure that those present would stop me at half of the presentation, be asked to stop wasting their time and put me on the nearest plane to San Francisco.

Things did not go beyond one slide. But not because they let me out of the office. Just the rest of the session was held in discussing the contents of that slide ... As I was told later: “Only this slide alone justified your participation in the project.”

So what did I say? In my opinion, nothing special. My task was to find a way to analyze how the context influences data analysis and decision making in the extremely complex areas of activity that exist at the interface of military departments, politics and social factors. Well, I made a very obvious (for me) first step. I needed to write down as precise a mathematical definition of the concept of “context” as possible. It took me several days ... I can’t say that I was absolutely satisfied with the result ... Nevertheless, it was the best thing I could do, and this process, at least, gave me a solid basis for starting to develop some elementary mathematical ideas.

A rather large group of smart people, real academics, military contractors and senior personnel of the Ministry of Defense spent the entire remaining hour of the time allotted to me, discussing just this one definition. The discussion revealed that different experts had a different understanding of what context was, and this is the right path to disaster. From the very beginning I asked them the question: “What is context?” Each of those present in the room, not counting me, had a good working definition of this concept, but all definitions differed from each other. And none of the participants had previously proposed to write down a single formal definition. They are simply not used to doing this as part of their work. As soon as this was done, they had a common starting point, which allowed them to compare and oppose it first of all with their own ideas. Thanks to this, we managed to avoid a catastrophe.

As a mathematician, Devlin did not do anything unusual. In fact, the most common question that arises in a mathematician, faced with a new subject of discussion, is: “What exactly do you mean by this word?”

And even though the specific example given by Devlin of military intelligence counseling is very specific, the technique he used is universal. That is the basis of the so-called, but very vague, term “critical thinking”. Imagine a situation where an average citizen who sweeps away mathematical ideas listens to the news and hears the politician say: "We have strong evidence of the existence of weapons of mass destruction in Iraq." If the listener had a good mathematical education, he would have asked the question: “What exactly do you mean by“ weighty evidence ”and“ weapons of mass destruction ”?”. After all, in fact, the accuracy of these concepts plays a decisive role in determining whether the proposed response measure - a declaration of war - is legitimate. Without an understanding of the definitions, you can not make an informed decision and speak in favor or against. However, if you listen to news for fun or to feel like a part of a political flock, then the truth is the last thing that interests you.

Each of us has to deal with new definitions, whether it is a question of a new definition of marriage or gender, or of legal definitions of "intention", "rationality", "privacy". A sophisticated mathematician will immediately notice that the government cannot provide a single useful definition of such a thing as "religion." The ability to think critically, based on definitions, is the basis of any civilized dialogue.

The habit of thinking about definitions is developed by students of mathematics at an early stage of their studies at an institution of higher education and strengthened in the master's program and the subsequent stages of their research activities. Usually the mathematician is faced with new definitions every day and this happens in a variety of contexts. Well, the very ability to confidently understand the concepts and terms will be useful for everyone who masters it.

Thinking examples and counterexamples

Well, now I propose to practice a little work with definitions in an informal setting. By “counterexample” I understand an example that shows that something stops working or is incorrect. For example, the number 5 is a counter-example of the statement that 10 is a prime number, because 10 is divisible by 5 without a remainder.

Mathematicians spend a lot of time coming up with examples and counterexamples for a variety of statements. This item is very closely related to the previous one about definitions because:

1. Often, when inventing a new definition, a person keeps in mind a set of examples and counterexamples to which it should correspond. Thus, examples and counterexamples help create good definitions.
2. The first thing that every mathematician does when he encounters a definition that already exists for him is new, writes down examples and counterexamples that can help him to better understand it.

Be that as it may, examples and counterexamples are beyond the mere discussion of definitions. They help us evaluate statements and understand their meaning. Anyone who has studied mathematics is well aware of this approach, also known as "guess and proof."

And it is as follows. Working on a task, you study a certain mathematical object and write down the information about it that you want to prove. That is, you make a reasonable (or unreasonable) conjecture about a certain regularity that characterizes the object being studied. This is followed by evidence when you are trying to confirm or deny the assertion.

As a bad analogy, you can guess that the Earth is in the center of the universe. You reinforce this guess with the characteristics of the object that satisfy this statement. In our solar system, you could make a toy model showing an example of how, in your opinion, the model of the universe could look like with the Earth at its center, if the universe could be as simple as a toy. Or, on the contrary, you could perform some measurements, which include taking into account the characteristics of the Sun and the Moon, and obtain evidence that this statement is false, and in fact the Earth revolves around the Sun. So in the world of mathematics this “proof” is a counterexample and you can call it as such only if its truth is subject to unequivocal confirmation. “Proof” in mathematics often serves only as a temporary filler, until the truth is revealed. In spite of all this, however, there are some well-known problems, over which mathematicians have been fighting for hundreds of years, and so far they have not provided anything for them except “proofs”.

This analogy describes what happens in mathematics, even at the very microscopic level. When you immerse yourself in a project with your head, you make new small assumptions every few minutes, as a rule, as a result, refuting them, because later you realize that they were nothing more than unsubstantiated conjectures. This is a very intensive, "pumped" scientific process, consisting of the analysis of hundreds of false hypotheses, resulting in a pleasant result. The counterexamples that you find along the way act as road signs. Subsequently, they help your intuition, and as soon as they get firmly rooted in your head, the process of making or denying more difficult conjectures becomes relatively simple.

And here we come again to the fact that the ability to invent interesting and useful examples and counterexamples is one of the pillars of productive reasoning. If you have ever read the minutes of the Supreme Court hearing, for example, the case of discussing the legality of wearing prisoners for religious reasons, you will see that most of the arguments are test examples and counterexamples to verify the previously established legal definitions of “rationality”, “religion” and "Intentions" for strength. This approach has also found countless applications in physics, engineering, and computational theory.

There is another, much less obvious, but no less important point. Due to the fact that throughout their careers, mathematicians have to regularly make such a large number of incorrect, stupid and false guesses, they are immune to blindly accepting statements based on the strength of one's voice or cultural prejudice. If we recognize that in today's collective society, people have become too inclined to believe the voices of others (politicians, media “experts”, financial speakers), then studying mathematics is a great way to cultivate a healthy sense of skepticism in people. This skill will be equally useful for engineers and plumbers, nurses or garbage collectors.

The ability to often make mistakes and admit mistakes

Two mathematicians, Isabel and Griffin, discuss the mathematical statement at the blackboard. Isabel thinks the statement is true and ardently defending her point of view in a dispute with Griffin, who believes in the opposite. After 10 minutes, they change their points of view to the opposite and now Isabel already considers this statement to be false, while Griffin believes that they are true.

I observe such situations all the time, but only in the world of mathematics. The only reason why this can happen is that both mathematicians, regardless of which of them is actually right, are ready not only to accept their wrongful actions, but also willingly to change the side of the dispute, as soon as they feel in their arguments at least the slightest flaw.

Sometimes in a group of 4–5 people discussing a statement, I find myself the only one who disagrees with the majority opinion. If the argument I proposed is good enough, each of those present will immediately accept the fact that he was wrong, having done so without any regrets or negative emotions. More often, however, I find myself on the side of the majority and have to go back in my reasoning or revise and improve my views.

The habit of encouraging doubt, being wrong, recognizing it and starting all over again as often as possible - all this distinguishes the mathematical discussion from even the vaunted scientific discussion . Here you will not see any attempts to achieve the desired p-value or hidden lobbying. There is no place in mathematics for the desire to become famous, because almost everything you say usually does not leave the limits of a small group of participants in the discussion. The mathematician is completely absorbed in the process of searching for truth, and his professional habits allow him to discard personal glory or fear of shame for the sake of the main goal - insight into the essence of the problem.

Evaluation of the consequences of the statement

Scot Aaronson wrote a post about John Kennedy’s murder and conspiracy theories about it. In it, he examines the statement "John Kennedy's murder was a conspiracy, the scale of which is comparable to the size of the CIA" and gives him an estimate based on simple and understandable arguments, very similar in essence to the approach of mathematicians and computer scientists. Consider an example from his post:

10. Almost all conspiracy theories about John Fitzgerald Kennedy seem to be false simply because they all contradict each other. As soon as you understand this and begin to consider them on the assumption that at least one of them could be true, the insight will immediately fall upon you: you will understand that nothing prevents you from simply sweeping away all of them.

Another example:

12. If the organizers of the conspiracy were so powerful, then why did they confine themselves to the murder of the president only, without achieving any more impressive results? And why didn't the conspirators begin even earlier, with the election fraud, in order to prevent Kennedy from becoming president? In mathematics, you often find flaws in your argument thanks to the understanding that he alone gives you much more than you initially thought. Nevertheless, all the arguments in favor of conspiracy, with which I have become acquainted, appear to have the same drawback. For example, what happened to the conspirators after the successful implementation of their plans? Was their organization simply disbanded? Or did they continue to carry out plans for other murders and organize them? If this did not happen, what prevented them? Is the work of the world's secret puppeteers is not a perpetual activity? And where in general, if, of course, it is possible, the power of this organization ends?

In fact, the study of the limits of a statement is the daily bread for every mathematician. This is one of the simplest high-level tools available to everyone, allowing to evaluate the validity of an assertion before starting a detailed examination of the arguments. And this method can be used as a litmus test to determine which arguments should be considered in more detail.

Sometimes bringing this or that argument to the limits allows us to obtain an improved and more elegant theorem, which includes the initial statement. But more often, you just realize that you were wrong. Therefore, this habit is a less formal variation on the subject of frequent mistakes and inventing counterexamples.

Ability to consider the assumptions underlying the statement separately from each other

Mathematics has one, perhaps, annoying feature: it is full of ambiguities. We love to treat it as a kind of impersonation of steadfastness. And I am even ready to argue in favor of this idea. Be that as it may, the process of doing math - exploring existing ideas or inventing new ones - has much more in common with communication between two people than harsh and cold as ice steadfastness.

So, when a mathematician makes a statement, he, as a rule, tries to formulate the basic idea as simply as possible, in order to bring it to other people. This usually means that the meaning of the expressions used in the formulation may be unclear for other people, especially if the conversation takes place between two mathematicians who are familiar with the general context of the conversation, and you in this situation are a stranger trying to understand them.

When you find yourself in a similar situation in mathematics, you spend a lot of time getting back to basics. You ask questions like: “What do these words mean in this context?” And “What obvious attempts have already been made and rejected, and why?”. Trying to get deeper into the essence of the question, you ask: “Why are these questions so important?” And “Where does this line of research lead?”

These are the methods that the mathematician uses to gather information about the subject matter. A common leitmotif of this approach is to isolate every iota of information that is embarrassing you, every assumption underlying a particular belief or assertion. This approach is drastically different from any other type of debate seen in the world today.

Has anyone, for example, tried to thoroughly understand the worldview of Donald Trump during his preparation for the highly controversial presidential elections of this year? Most liberals hear only: “I will build a wall and make Mexico pay for it,” laughing at Trump and declaring him insane. Applying a mathematical approach to this statement, you first need to understand where it originates. What target audience does Trump appeal to? What alternative solutions to the immigration problem did he consider and rule out, and why? Why is immigration so important for its supporters, and what assumptions in its logic lead to such decisions? What does Trump understand and understand, what makes his election proposals so popular?

No, I am not trying to take this or that political position. I just want to draw your attention to the fact that if the mathematician finds himself in a highly ambiguous situation, a separate analysis of the assumptions underlying this or that statement will be part of the general scheme of his actions. The phenomenon of “liberal media underestimate Trump” owes its existence in many respects precisely to unwillingness to ask questions like the ones above and get answers to them. Instead, Trump's opponents just do tweets with quotes from his deluded and divorced supporters. However, if you believe the results of surveys, this approach does not bring tangible results ...

"Ladder of abstraction"

The last habit on my list is the concept of “ladder of abstraction,” which I borrowed from Bret Victor . Its essence lies in the fact that during the reasoning over the solution of the problem, you can disengage, look at it and think it over from different levels, by analogy with moving up and down the stairs, where a higher step means a higher level of abstraction. Victor gives an interactive example of developing a car driving algorithm. In it, you can view his work in the smallest detail, comparing the specific variation of the algorithm and the results of monitoring its behavior.

At a higher level (higher step), you can control various parameters of the algorithm (and time) using the slider, turning one variant of the algorithm into a whole family of derived algorithms, each of which can also be debugged. You can further generalize which parameters and behaviors can be debugged to expand the space of possible algorithms. So, in the course of the work, you are looking for generalized action patterns that can help you achieve the ultimate goal - developing a quality algorithm for driving a car from the point of view of the lowest level from which your work began.

Mathematicians regularly use this technique, especially at a later stage of graduate studies, when you need to learn how to handle a huge amount of research. There you do not have time to study in depth each part and each statement in this or that work, with the exception of the most important of them. Instead, you create a "ladder of abstraction", the lower level of which contains separate definitions, theorems and examples from the work, the next level is its generalized content, and a higher level considers how this work relates to other studies and fits into a wider mathematical context. . Even higher are the backbone trends for this area of ​​knowledge, what is considered important for it, fashionable, and so on.

You can start at the bottom of the ladder by examining and understanding a few examples of definitions and thus obtaining a reliable guideline, after which you jump over to the main work theorem and understand exactly what improvements it offers compared to previous work in this area. In the course of the reading, you may come across some technique from an unfamiliar area, invented in the 50s. Just use it as a ready-made solution, focusing on more useful proof of the main theorem for you, and, thus, go down one notch. After that, you can go to the chapters on unsolved problems, to see what else is left to be done in this area, and if they seem attractive enough, you can prepare yourself to work on them by carefully reading the rest of the work.

In fact, mathematicians have to exercise their "abstract muscles" whenever they talk about their own work. The audience at the lectures is different, and each student can evaluate the content of a mathematical idea at a different level of detail. Some theorems are best explained by the example of competitive games and their context, optimization problems are given by other examples, and in some cases it is appropriate to even give analogies from metallurgy .

Perhaps, it can be said that combining information from all the steps of a ladder into a single harmonious model that you can consider yourself and on the scale you need is one of the most common and difficult tasks in the world of mathematics. Victor tries to simplify this exercise for the mind by developing a functional user interface. Other mathematicians practice it using a variety of techniques that fall into their hands. Whatever the approach, the end result is always of great value.

Conclusion

In no case do I hint that the development of advanced mathematical habits is an exercise that is absolutely definitely useful. In the real world, many of these habits are a double-edged sword. Anyone who has received a university degree in mathematics knows a person (or was himself) who constantly makes remarks that the expression A does not always turn out to be true in a special case B, which no one was going to consider from the very beginning. To understand when a similar approach is productive, and when it infuriates others, it takes a lot of social maturity, which, in turn, is achieved beyond purely mathematical conversations.

Moreover, to get used to the need to “always be wrong” often requires the first few years of full-fledged work. Because of this, many students who do not have the support of comrades at the same stage of study or a good role model give up classes. Career math is really an emotional rollercoaster.

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