Mathematics is “a language that can neither be read nor understood without initiation” (Edward Rothstein, “Emblems of the Mind”)

The reading protocol is a set of strategies that the reader should use to take full advantage of reading the text. The set of strategies for poetry is different from fiction, and strategies for reading fiction are different from scientific articles. It would be ridiculous to read an art book and wonder what sources have allowed the author to claim that the protagonist is a tanned blond; but it will be wrong to read scientific literature and not ask such a question. This reading protocol extends to viewing and listening protocols.in painting and music. In fact, most of the introductory courses in literature, music, and art are devoted to the study of these protocols.

For mathematics, there is a special reading protocol. As we learn to read literature, so we must learn to read mathematics. Students should study the reading protocol for mathematics just as they learn the rules for reading a novel or poem, and learn to understand music and painting. The remarkable book Emblems of the Mind by Edward Rothstein reveals the relationship between mathematics and music, implicitly affecting reading protocols for mathematics.

When we read the novel, the plot and the characters absorb us. We are trying to follow different storylines and how each of them affects the development of heroes. The characters themselves must become real people for us - both those who delight us and those whom we despise. We do not stop at every word, but present the words as strokes in the picture. Even if the specific word is unknown, the overall picture is still clear. We rarely interrupt to reflect on a specific phrase or sentence. Instead, we let the work captivate us in its flow and quickly bring it to the end. This is a useful and relaxing activity, it provides food for thought.

Writers often characterize characters by involving them in carefully selected action episodes rather than describing them with carefully selected adjectives. They depict one aspect, then another, then again the first in a new light and so on. As this progresses, the overall picture grows and becomes more and more clear. This is a way to convey complex thoughts that cannot be precisely defined.

Mathematical ideas are inherently accurate and well defined, so you can give a clear and very brief description. Both the mathematical article and the fiction tell a certain story and develop complex ideas, but the mathematical article does this with a much smaller number of words and symbols than in the book. The beauty of the book lies in the aesthetic way of using language to arouse emotions and represent things that cannot be precisely defined. The beauty of a mathematical article lies in the elegance and effectiveness of a laconic expression of clear ideas of great complexity.

What are some common mistakes people make when trying to read math? How can these errors be fixed?

Don't lose the big picture

“Reading mathematics is not linear at all ... To understand the text, you need cross-references, quick scanning, pauses and re-reading” (from there)

Do not think that understanding each phrase individually will allow you to understand the whole idea. It’s like trying to see a portrait, looking at every square centimeter of the picture at point-blank range. You will see details, texture and color, but completely miss the portrait. A math article tells a story. Try to understand what this story is before you delve into the details. You can take a closer look later when you build the understanding framework. Do it the same way you could reread an art book.

Do not be a passive reader

“Three-line proof of a modest theorem is the quintessence of years of work. Reading mathematics ... involves a return to the thought process that occurred during its writing ”(from the same place)

Examine the examples for patterns. Try special occasions.

A mathematical article usually tells only a small part of a large and lengthy story. As a rule, the author wanders for months in the dark, trying to discover something new. In the end, he organizes his discovery in the form of an article where he hides all the errors (and their causes) and presents the finished idea in the form of a clean, neat stream. To truly understand the idea and recreate what the author has hidden, read between the lines.

Mathematicians are very concise, but much is hidden behind this laconicism. The reader must participate. At each stage, he / she must decide how clear the idea is. Ask yourself the following questions:

• Why is this idea true?
• Do I really believe in her?
• Can I convince someone that she is true?
• Why didn't the author use a different argument?
• Do I have a better argument or method explaining the idea?
• Why didn't the author explain it in a way that I understood?
• Is my method wrong?
• Did I really get the idea?
• Am I missing a nuance?
• Does the author miss a nuance?
• If I can’t understand the meaning, maybe a similar but simpler idea will be clearer to me?
• Is it really necessary to understand this idea?
• Can I accept this idea without understanding the details, why is it true?
• Will my understanding of the big picture suffer from the fact that I do not understand the evidence for this particular idea?

If you spend too little effort on such participation, then this is how to read a book without concentration. After half an hour, you suddenly realize that you simply turned the pages, thinking about your own, and did not remember anything from what you read.

Don't read too fast

Reading mathematics too fast leads to frustration. Half an hour of concentration on an art book will allow the average reader to read 20-60 pages with full understanding of the text, depending on the book and reading experience. The same half an hour over a mathematical article will give 0-10 lines, depending on the article and the experience of reading mathematics. Work and time cannot be replaced by anything. You can speed up math reading with practice, but be careful. As with any skill, trying to do too much too quickly can throw you back and destroy your motivation. Imagine that for an hour you are energetically engaged in physical exercises, if you have not been involved in this for two years. Maybe you will go through the first stage, but you are unlikely to continue further. Disappointment from constant observation,

For example, we take such a theorem from the treatise “The Case of the Calculator” by Levy bin Gershom, written in 1321.

“When you add consecutive numbers, starting from 1, and the number of terms is odd, the result is equal to the product of the average and the last.” For modern mathematics, it will be natural to write the theorem as follows:

$$

The reader will need about the same amount of time to unravel this small formula as he needs to understand two lines of the text version of the theorem. An example of Levy's theorem:

$$

Make your own idea

The best way to understand what you read is to make the idea your own. This means tracing the idea to its roots and reopening it on its own. Mathematicians often say that to understand something, you must first read it, then write it in your own words, and then teach someone else. Each has a different toolkit and a different level of “assimilation” of complex ideas. You need to adapt the idea to your own vision and experience.

“When I take a word, it means what I want”

(Humpty Dumpty from Lewis Carroll's Alice Through the Looking Glass)

“The meaning will rarely be fully understood, because each symbol or word already represents an extraordinary condensation of concept and reference” (from the same place)

A well-written mathematical text will neatly use the word in only one sense, making a distinction, say, between combination and permutation (or by arrangement ). A clear mathematical definition may imply that a "yellow rabid dog" and a "rabid yellow dog" are a different arrangement of words, but the same combination. Most English readers will not agree with this. Exceptional accuracy is completely alien to most of the fiction and poetry, where in good formthe use of a variety of words, synonyms and variable descriptions is considered.

The reader is expected to understand that the absolute value is not some arbitrary value that turned out to be absolute, and the function has nothing to do with anything functional.

A particularly notorious example is the use of the phrase “it follows easily from that ...” and similar constructions. They mean something like this:

Now we can verify the validity of the following statement. Verification will require a certain amount of essentially mechanical, although possibly hard work. As an author, I could have conducted it, but it will take up a lot of space and, possibly, will not lead to good results, since it is in your interests to do the calculations yourself and clarify what is happening here. I promise that no new ideas are involved. Although, of course, you may need to think a bit to find the right combination of good ideas to apply.

In other words, this phrase, when used correctly, is a signal to the author that something tedious and even complex, but not involving bright insights, is present here. Then the reader is free to decide how much he wants to understand all the details himself or if the author’s assurances are enough for him - and then we can say: “Okay, I take your word for it”.

Now, regardless of your opinion on the appropriateness of using such a phrase in a particular situation and on the correctness of its use by the author, you should understand what it should really mean. The phrase “It easily follows from this that ...” doesn’t mean.

If you don’t see it at once, you are a fool.

And it doesn’t mean.

It should not take more than two minutes

But a person unfamiliar with the mathematical dictionary may misunderstand it and will face disappointment. This is in addition to the question that the “tedious task” for one person can be a very difficult problem for another. Therefore, the author must correctly evaluate his audience.

Know yourself

Texts are written for readers of a certain level. Make sure that you are such a reader or want to do everything to join their number.

From Thomas Eliot, “Song of Simeon”:

Lord, the winter sun creeps between the snowy peaks,
Roman hyacinths flared in bowls;
Time froze - a stubborn dumb sovereign,
Waiting for the blow of death with a lightened soul,
Like a feather on an old hand. One.
Dust under the sun and the memory of the nooks and crannies of
Tea are cool that blows for mortal valleys.

Lord, the Roman hyacinths are blooming in bowls and
The winter sun creeps by the snow hills;
The stubborn season has made stand.
My life is light, waiting for the death wind,
Like a feather on the back of my hand.
Dust in sunlight and memory in corners
Wait for the wind that chills towards the dead land.

For example, Eliot’s poem largely suggests that readers either know who Simeon is or are eager to find out. It also assumes that the reader is either an experienced poetry reader or one who wants to gain such an experience. The author assumes that readers either know or understand allusions here. It goes beyond simple things like who Simeon was. For example, why are hyacinths “Roman”? Why is it important?

Eliot suggests that the reader will read slowly and pay attention to the pictures: the author compares dust and memory, compares old age with winter, compares the expectation of death with a feather on the back of his hand, etc. He assumes that the reader will perceive this as poetry; in a sense, he suggests that the reader is familiar with the entire poetic tradition. He should notice that the odd lines rhyme, and the rest do not, etc.

Most importantly, he assumes that the reader will connect not only his mind to reading, but also his emotions and imagination, which will draw the image of an old man, tired of life but forced to cling to it, waiting for some important event.

Most mathematical books are also written with some assumptions about the audience: that they know certain things, that they have reached a certain level of “mathematical maturity”, and so on. Before you start reading, make sure that the author expects from you.

Math notation example

To experiment with the principles presented here, I included in it a small mathematical fragment, often called the birthday paradox. The first part is an accurate mathematical article describing the problem and its solution. The second part is a fictional attempt by the Reader to understand the article using the appropriate reading protocol. The topic of the article is probability. It is accessible to the smart and motivated reader, even without a mathematical education.

Birthday paradox

A professor in a class of 30 random students often argues that at least two people in the class have the same birthday (month and day, optionally year). Will you accept the argument? And if there are fewer students in the class? Accept then the offer to argue?

Suppose birthdays$$people are evenly distributed over 365 days of the year (for simplicity, we do not take leap years into account). We prove that the probability of the same birthday (month and day) in at least two of them is equal to:

$$

What is the probability that among 30 random people in the room there will be at least two with the same birthday? For$$this probability will be 71%, that is, the professor will win the argument in 71 cases out of 100 over a long distance. As it turns out, with 23 students in the class, the probability is about 50%.

Here is the proof: Let$$will be the desired probability. Let be$$it will be likely that everyone has different birthdays. Now find$$by calculating the number of sets of $$ birthdays without duplicates and dividing it by the total number of possible sets of $$birthdays. Then find$$.

The total number of sets of$$ birthdays without duplicates:

$$

This is because 365 options are possible for the first DR, 364 for the second, and so on for $$human. Total amount$$ birthdays without any restrictions equals just $$because for each of $$birthdays there are 365 options. Thus,$$ equals

$$

Solution for $$ gives $$ and therefore our result.

Our reader's attempts to understand the birthday paradox

In this section, the naive Reader tries to understand the meaning of the last few paragraphs. Reader's remarks are a metaphorical expression of his thoughts out loud, and Professional comments are the research work that the Reader must do. The relevant protocols are highlighted in bold and pasted into the appropriate places of the narration.

It appears that the Reader seems to grasp things very quickly. But be sure that in reality a lot of time elapses between the reader’s comments and that I left out many of his comments that lead to a dead end. You can understand what he is experiencing only if you read between the lines and imagine the way he thinks. Thinking for him is part of your own efforts.

Know yourself

Reader (W): I don’t know anything about probabilities, can I figure this out?

Professional (P): Let's try it. You may have to make big digressions at every step.

W: What does the phrase “30 random (chaotic) students” mean?

“When I take a word, it means what I want, neither more nor
less”

P: Good question. The phrase does not mean that we have 30 crazy or sick people. It means that we must assume that the birthdays of these 30 people are independent of each other and that each birthday is equally likely for each person. A little further, the author more formally describes this: “Suppose that birthdays$$people are evenly distributed over 365 days of the year. ”

Ch: Isn't that obvious? Why specifically mention this?

P: Yes, the assumption seems obvious. The author simply establishes the main condition. This proposal ensures that everything is fine, and that the solution does not involve any imaginary bizarre science fiction.

W: What do you mean?

P: For example, the author is not looking for a solution like this: everyone lives in the Independence Country and was born on July 4th, so the chances of two or more people having their birthdays coincide are 100%. Mathematicians do not like solutions of this kind. By the way, the assumption also implies that we do not take leap years into account. That is, no one in this task was born on February 29th. Read on.

C: I do not understand this long formula, What is$$?

P: The author solves the problem for an arbitrary number of people, not just for 30. From this moment, the author calls the number of people$$.

C: I still do not understand. What reply?

Don't be a passive reader - try examples

P: Well, if you need an answer only for 30, just install $$.

W: Okay, but it's hard to calculate. Where is my calculator? Let's see: 365 × 364 × 363 × ... × 336. This is tiring, and the result does not even fit on the screen. It is written here:

$$

If even knowing the formula I cannot calculate the result, how can I understand where the formula came from?

P: You're right that the result is not the exact result, but if you continue further and divide, the answer will not be too far from the exact one.

C: All this is somehow uncomfortable. I would like to calculate the exact value. Is there another way to do the calculations?

P: How many factors do you have in the numerator? How many in the denominator?

W: Do you mean 365 as the first factor, 364 as the second? Then we get 30 factors at the top. Also 30 factors at the bottom (30 copies of 365).

P: Now can you calculate the result?

H: Ah, I see. I can pair each factor from the numerator with each factor of the denominator, so that is 365/365 for the first factor, then multiply by 364/365 and so on for all 30 factors. So the result will always fit in my calculator. (After a few minutes) ... So, I got 0.29368 if you round up to five characters.

P: What does this number mean?

Don't lose the big picture

C: I forgot what I was doing. So let's see. I was looking for an answer for$$. The number 0.29368 is all calculations except the subtraction from unity. If you continue, you get 0.70632. Now what does this mean?

P: It’s useful here to learn more about probabilities, but in a simple way this means that in a group of 30 people, two or more will have 70,632 birthdays out of 100,000 cases, that is, in about 71% of cases.

H: Interesting. I myself would not have guessed. Do you want to say that in my class of 30 students the probability is quite high that at least two people will have the same birthday?

P: Yes, right. You can take bets before you know their birth dates. Many people think that such a coincidence is unlikely. This is why some authors call it the birthday paradox .

Ch: So that's why I have to read math to earn a couple of dollars?

P: I understand that for you this can be a kind of incentive, but I hope that mathematics also inspires you without any monetary prospects.

W: I wonder what the result will be for other values.$$. I'll try to do some more calculations.

P: Good point. We can even compose a graphic image from all your calculations. You can make a graph of the number of people and the probability of coinciding birthdays, although this can be left another time.

W: Oh look, the author has done some calculations for me. He says for$$the answer is about 71%; I also got such a figure. And for$$it turns out about 50%. It makes sense? I think it does. The more people there are, the higher the probability of coinciding birthdays. Hey, I'm ahead of the author. Not bad. Okay, let's continue.

P: Well, now you will tell when to continue.

Don't read too fast

W: It seems that we got to the proof. It should explain why the formula works. What is this$$? Think that$$ means “probability”, but what does it mean $$?

P: The author introduces a new value. He uses$$ just because it's the next letter in the alphabet after $$but $$ - this is also a probability, and having a close relation to $$. It's time to take a moment to think. What$$ and why does it equal $$?

H:$$- This is the probability that no one in the room has the same birthday. Why does the author care about this question? Don't we need another chance that the birthdays are the same?

P: Good point. The author does not say this explicitly, but between the lines you can understand that he has no idea how to directly calculate$$. Instead, he introduces probability$$which is supposedly equal $$. Probably, after this, the author should show how it is calculated$$. By the way, when you finish the article, you may wonder how to calculate$$directly. This is a great continuation for the ideas presented here.

C: Everything has its time.

P: Okay. So now we know$$, what's next?

W: Then we can get$$. If$$then $$. Ok but why$$? Does the author think this is obvious?

P: Yes, he thinks so, and even worse, he doesn’t even tell us that this is obvious. Here is a rule of thumb: when the author explicitly says that this is true or obvious , then you should spend 15 minutes and convince yourself that it is so. If the author does not even bother to say this, but implies it, the process will take a little longer.

C: How do I understand that you need to stop and think?

P: Just be honest with yourself. If in doubt, stop and think. If too tired, go watch TV.

W: So why$$?

P: Let's imagine a special case. If the probability of coincidence of two or more birthdays is 1/3, then what is the probability of not getting a match?

R: This is 2/3, because the probability of the absence of an event is the inverse of the probability of an event.

Make your own idea

P: Well, you have to be careful when using words like the opposite , but you're right. In fact, you discovered one of the first rules that are studied in probability theory. Namely, the probability of the absence of an event is equal to one minus the probability of the occurrence of the event. Now move on to the next paragraph.

C: It seems to explain what equals$$- in a long and complicated-looking formula. I will never understand this.

P: Formula for$$hard to understand, and the author relies on your diligence, perseverance and / or existing knowledge to understand it.

W: It seems that he calculates all the probabilities of something and divides them by the total number of probabilities, whatever that means. I have no idea why he does it.

P: Maybe I can help you with some information on this subject. The probability of a particular outcome in mathematics is defined as follows: the total number of possible options for this outcome is divided by the total number of all outcome options. For example, the probability of throwing a four on a dice is 1/6. Since there is one four and six possible outcomes. What is the likelihood that you will throw the four or three?

W: Well, I think 2/6 (or 1/3), because the total number of possible outcomes is still six, but I have two options for a successful outcome.

P: Good. Now a more complex example. What about the probability of throwing four in total when you roll two dice? There are three options to get this amount (1-3, 2-2, 3-1), while the total number of combinations is 36. This is 3/36 or 1/12. Look at the following 6 × 6 table and see for yourself.

1-1, 1-2, 1-3, 1-4, 1-5, 1-6
2-1, 2-2, 2-3, 2-4, 2-5, 2-6
3-1, 3-2, 3-3, 3-4, 3-5, 3-6
4-1, 4-2, 4-3, 4-4, 4-5, 4-6
5-1, 5-2, 5-3, 5-4, 5-5, 5-6
6-1, 6-2, 6-3, 6-4, 6-5, 6-6

What about the probability of throwing seven in total?

W: Wait, what does 1-1 mean? Doesn't that equal 0?

P: Sorry, I'm to blame. I used the minus sign as a dash, just referring to a couple of numbers, so 1-1 means throwing the ones on both dice.

C: Could you come up with a better record?

P: Well, maybe I can or should do it, but the commas look worse, and the slash will look like division, and everything else can also be misleading. In any case, we are not going to publish this transcript.

W: Thank God. Well, now I know what you mean. I can get seven in total in six ways: 1-6, 2-5, 3-4, 4-3, 5-2, or 6-1. The total number of outcomes is still 36, so it turns out 6/36 or 1/6. It is strange why the probability of four falling out differs from the probability of seven falling out.

P: Because not every amount is equally likely. Here, the situation is different from a simple rotation of the wheel with numbers from 2 to 12 at equal intervals. In this case, each of the 11 digits has the same probability of falling 1/11.

C: Okay, now I'm an expert. Is probability calculated simply by counting?

P: Sometimes yes. Although it can be difficult to calculate.

C: I see, let's continue. By the way, did the author really expect me to know all this? My friend is taking a course of probability and statistics, but I'm not sure that he knows all these things.

P: A small section of mathematics contains a lot of information. Yes, the author expects the reader to know all this or that he will find this information and absorb it, as we did. If I were not here, you would have to ask these questions to yourself and find answers by reflection, in textbooks and reference books, or in consultation with a friend.

W: So the probability of coinciding birthdays in two people is the number of possible sets of$$ birthdays without duplicates divided by the total number of possible sets of $$birthdays.

P: Great resume.

W: I do not like the use$$so let's use 30. Maybe then it will be easier to learn the generalization $$.

P: Great thought. It is often useful to analyze a special case in order to understand the general case.

C: So how many sets of 30 birthdays are there at all? I can’t count. I think that I will have to further limit the conditions. Let's pretend we have only two people.

P: Good. Now you think like a mathematician. Consider$$. How many possible birthday combinations are there?

C: I count birthdays from 1 to 365, not including leap years. Then here are all the possible combinations:

1-1, 1-2, 1-3, ..., 1-365,
2-1, 2-2, 2-3, ..., 2-365,
...
365-1, 365-2, 365-3, ..., 365-365

P: When you write 1-1, you mean 1-1 = 0, how is subtraction?

Ch: Stop teasing me. You know exactly what I mean.

P: Yes, I know, and I can point out a good choice of recording method. Now how many pairs of birthdays are there?

W: For two people, 365 × 365 options are obtained.

P: And how many options, if you do not take into account the coinciding birthdays?

W: You can’t use 1-1, 2-2, 3-3 ... 365-365, so it turns out

1-2, 1-3, ..., 1-365,
2-1, 2-3, ..., 2-365,
...
365-1, 365-2, ..., 365-364

The total number of options comes out 365 × 364, because each line now has 364 pairs instead of 365.

P: Good. You're in a hurry here, but still 100% right. Now can you generalize to 30 people? What is the total number of possible sets of 30 birthdays? Try to guess. You are good at it.

W: Well, if you try to guess (although this is not really a guess, in the end, I already know the formula), then I would say that for 30 people you need to multiply 365 × 365 × ... × 365 30 times, for the total number of possible sets of DR.

P: Exactly. Mathematicians write 365 ³⁰ . And what is the total number of sets of 30 birthdays without repetition?

W: I know that the answer should be 365 × 364 × 363 × 362 × ... × 336 (that is, start at 365 and each time multiply by a number subtracting one 30 times), but I'm not sure I really understand why so. Perhaps you first need to consider the case of three people and find a way to increase to 30?

P: A brilliant thought. Let's finish for today. The big picture is clear to you. When you rest and you will have more time, you can return and fill the last gap in understanding.

C: Thank you very much; it was a good experience. See you later.