A Lecture About Teaching Mathematics to Non-mathematicians, Part II

By V.A. Rokhlin

I was talking about the limit theory. Why, when I look at this theory, do I want to give it such a rough treatment, to banish it from the course of mathematics to non-mathematicians? I don’t want to say that it actually should be banned from everywhere, no. I only want to say the following. The theory of limits at present works as not a tool to introduce the basic notions of calculus, but as a very high and difficult to negotiate barrier that one has to climb over in order to understand anything.

And this barrier is absolutely unnecessary! It is, as a rule, impossible to cross for the students who are non-mathematicians, and it is absolutely unnecessary.

Let me give you an example that will demonstrate it. I could have taken as an example the differential or integral calculus, but I will take the simplest of the two cases, or, it’s better say, subjects, the one that can be explained faster, which is the integral calculus.

You have to explain to the beginners what is the integral. Of course, you can begin, and it is probably the right way, simply with the area, as it is usually done.

But why not declare that integrals are simply the areas? Indeed, none of your audience doubt that these areas exist. You would have to spend a lot of time if you wanted to raise any doubts about it in your audience.

Of course, having the area at hand, you can easily construct the integral sums. You will say to your listeners that the area has the well-known properties. Here they are.

If one figure is contained in the other, then the area of the first figure is no more than the area of the second. It is difficult not to agree with that, and everybody will agree with you.

You can say that if you add two figures, put together two figures without any common interior points, then their areas will add, and everybody will agree with that too.

Then you will say that the area does not change if the figure moves around on the plane as a rigid body. They will agree.

And finally, you will say that the area of a unit square is 1.
These four properties, as you well know, uniquely define the area on a wide class of figures, on the class of all exhaustible figures.

The same properties, a little modified, determine the integral. They uniquely determine the integral on a wide class of functions. This way, starting with the area, you can define the integral by its properties that nobody doubts, because we are talking about the area.

Later on you point out that, if you construct the upper and the lower Riemann (or Lebesgue — no difference!) sums, then the area you are interested in will be between these two auxiliary areas. It follows directly from what I have just said. You write the very same inequality that is usually written in the integral calculus texts: the lower sum is no bigger than the integral that is no bigger than the upper sum. Simply put, it is the only number between the lower and the upper sums.

All this, formulated in the language of areas, is quite obvious, it doesn’t cause any doubts and is easy to digest. On the other hand, all this gives us a method to calculate areas. No limits are mentioned. If you want to demonstrate some identity, say, between two integrals or between an integral and a number, you simply notice that both numbers that you want to be equal are between the upper and the lower sums. Therefore they are equal, because the difference between the upper end the lower sum can be made less than any fixed positive number by taking an appropriate partition. This difference is non-negative, and therefore it is zero.

The question about the technique used in any particular case does not arise. All the technique is available. It is described in details in all the available courses, but in these course everything is tuned upside-down.

Very similarly, you can define the derivative in many ways. It will be better, of course, if you start with the intuitive meaning of the derivative, for example with the tangent line or the velocity, how it is usually done. But there is no need for limit theory here. It doesn’t mean that later, when you want, or when the curriculum requires it in a really reasonable way, and when your students really have to get familiar with limits, you can not explain that in fact the derivative is such-and-such limit. But at the beginning it is absolutely unnecessary, and many students don’t ever need it.

To put it shortly, I would suggest the following approach, that I will call for convenience naively-axiomatic. The essence of this approach is that you in fact define the notions you are interested in by axioms.

For example, for the integral the axioms are the following.

The integral of a constant is the product of this constant by the length of the interval of integration. Of course, you don’t pull this axiom out of the thin air. At the beginning you will talk about the area. Everything will be prepared. That’s the axiom number one.

Axiom number two: if one function is no bigger than the other at every point, then the integral of the first function will be no bigger than the integral of the second.

And axiom number three: if you integrate a function over the interval that is the union of two smaller [non-overlapping] intervals, the corresponding integral will be the sum of the integrals over these smaller parts.

This is it! It is difficult to think of a less complicated approach. Of course, we can say these things about the area right away, as everybody is used to the area, about integrals we can say it a little later, but, based on these properties, all the integrals can be calculated beautifully. In fact, that’s how they are calculated in all the textbooks.

Moreover, this approach immensely simplifies all the applications of the notion of integral in natural sciences and in mathematics itself. If you want, for example, to show that some volume is expressed as some integral, you simply check that all these tree properties are satisfied. You don’t have to prove anything. Because you have uniqueness, the volume turns out to be an integral, and you get the formula at once, whether you deal with the volume of a body of rotation or the volume in some other situation.

The same applies to the torques, calculations of the centers of gravity, the moments of inertia and all the other mechanical and physical magnitudes. There is no need in any “passing to the limit” and in general in that long and boring presentation that the text books for technical universities are full of, there is no need for it at all.

I am just giving you some examples, as you understand. It would be possible to give many more. But if we want to talk seriously, we have to admit that the course of mathematics for technical universities is simply not put together yet. Not in the sense that there are no curricula and no textbooks, there are. By the course I man something else.

To avoid any misunderstanding, I will tell briefly what I mean by the expression “to put together a course.” Imagine a text that is longer, of course, than the text of the curriculum, but shorter than a textbook, and from which a competent person can see in details what topics to present and how to treat them.

It is not necessary that the students could understand this text. It has to be understandable to their teachers.
Unfortunately, if such a text were written for high schools, it would not be understandable by the teachers. But such a text, written down or kept in one’s head is the course that is put together.

What I want to say is that a course in mathematics that is put together in this sense does not exist yet. Of course, there are many possible approaches to achieving it, I hope that such courses will be put together and the textbooks will be written.

What I have just said about the integral calculus, these few remarks about it, apply not only to it, of course. I think that even the university students of mathematics would benefit from taking a one semester preliminary course of introductory analysis in which the basic notions were introduced not from the point of view of mathematical hairsplitting, so to speak, but were described meaningfully, with the view towards applications, with a discussion of the geometric and physical meaning and with plenty of material for the exercises. After this, a more systematic and scrupulous treatment of the subject can be given to the students of mathematics.

Such experiments had been done. I don’t know what is the situation now here, say, at the department of mathematics and mechanics at Leningrad State University.

In any case, it looks to me that this naively-axiomatic approach could be useful at the beginning even in teaching professional mathematicians. In any case, I think it is absolutely necessary for teaching mathematics to non-mathematicians.

Now I will say a few words about the more advanced parts of the course. Indeed, some non-mathematicians are taught not only calculus and infinite series. They are taught, for example, about integrals over curves and surfaces, about change of variables in multiple integrals, etc.

These things already are not so straight forward and pleasant, they may present some technical difficulties. What to do about them? Here, of course, our naively-axiomatic method is not always sufficient. Here we have to accept some compromises.

Very recently I had discussed similar matters with one professor from Leningrad who had to explain to his students the change of variables in double integrals. How to do it? Being a mathematician, this teacher doesn’t want to fool anyone. He is ashamed to misinform anybody, including his students. He wants to give them a proof.

On the other hand, change of variables in double integrals is a rather complicated subject. Several different methods are suggested. We can present our integral as an iterated integral and use the formula for change of variables in a one-dimensional integral. This is one way, there are many other ways. Here, as it looks to me, one habit is at play, professional mathematicians has a habit to give proofs.

But really, what should we instill in a future engineer, in a future physicist, to say nothing about a future philosopher? First and foremost we have to instill understanding. They have to understand the subject. We have to, and this is probably the most important, to stop teaching such things (or teaching them in such ways) that our students can not understand. The same goes for the school children.

It is very common to teach students such things that they don’t understand. What is the use to prove (in a rather restricted sense, of course), the formula of the change of variables in double integrals to the students of a technical university? Isn’t it better if they will understand this formula, even intuitively?

For example, can’t you first explain to the students how the area behaves under a linear transformation? Here is the plane, and a linear transformation is applied to it. How does the area of a triangle behave, or the area of a polygon? You probably can explain that. You can also explain that the area of a small region behaves approximately the same under a smooth nonlinear transformation. So the appearance of the Jacobian will not be surprising to anybody. Of course one have to get used to the Jacobian first.

[Here the recording was interrupted, probably for a tape change. In the lost piece of the lecture there was a discussion of the implicit function theorem.]

… axioms that establish equivalence of many approaches to, say, defining a surface in space. A surface in space can be defined parametrically, by three equations, it can be defined –implicitly– by one equation, and finally, can be defined as a graph of a function. All these three approaches are equivalent, and this equivalence is established by the theory of implicit functions.

To my surprise, few people know about it, even among the students of mathematics, before they really need it, maybe in some other subject. In the analysis course it is very rarely discussed. I haven’t seen it in textbooks either. The connection of this stuff with the maps is discussed in a very few places. But maybe to the students of a technical university all this can be explained without hairsplitting, so to speak, in such a way that they could understand (by some understandable examples, by some general formulations). To summarize, it looks like, in addition to our naively-axiomatic approach that I talked about, we need also a great freedom of dealing with the subject matter when we teach mathematics to non-mathematicians.

Maybe the teacher should try to recall how he experienced all this material when he was learning. We usually forget such things. For example, I can say about myself that I absolutely don’t remember what and how I learned in school. Maybe I remembered when I was a university student, but then I forgot it little by little. Maybe it is necessary to study this problem also by involving the students in a discussion and by listening to what they have to say. We don’t do enough of that.

As an example, I will mention the following observation. Without any doubts, the students that take this or that class give grades to their professors. They don’t put these grades into professors’ grade books, but they do grade their teachers. More than that, each professor or teacher has a rather stable average grade. It’s like rating in chess, so to speak. Unlike the chess rating, this rating is not published and even usually kept secret. I will not discuss whether it is good or bad. I think everybody knows whether it is good or bad. But undoubtedly, the help of the students is invaluable here.

I think their help is also invaluable in putting together a course that they take. Of course, the generations change, some students will help us put together a course for their successors. Nevertheless, I think that this activity, undertaken with such arrogance by the professors, can not be successful without help of their students. The students should be consulted first and foremost.

I nave recently met a girl who goes to high school and lives next door, my next door neighbor. I had invited her to visit a while ago. She visited in the evening an said: “Here I am, you had invited me, and I have come. I can’t solve a problem.” (Cheers)

Well, I started taking a look at the problem and was horrified. First of all, it was impossible to solve this problem because you could not understand what the question was. The problem, professionally speaking, had been written incompetently. One could guess, of course, what was asked. We started guessing. But then I discovered, to my surprise, that I was guessing, but she had understood it all from the very beginning. (Cheers) But more than that, when I started solving the problem, it turned out that she knew very well what I was talking about. I thought first that she knew it all. But after we talked a bit more, it turned out that she didn’t understand anything. She just knew all the words. But it turned out that these words were quite enough for solving the problem. And she knew all these words.

This kind of experience is invaluable for a teacher and for people that put together a curriculum. Keep in mind that we are talking about universal education, not about teaching the children that go to special schools, who had heard mathematical language and speak it themselves. No, these children are not taught mathematics professionally and have no intentions to work in it. They hear all these words, they learn to pronounce them, but their interpretation of these words is somewhat strange; they either have none at all, or it is strange. In many cases you discover really surprising interpretations.

It turned out that this girl and I understood some words very differently. Well, clearly we should wonder how her teacher understands all these words. And that is the real root of the problem, of course.

Let me draw your attention to yet another peculiarity of teaching and learning mathematics. This peculiarity is the following. This teaching is always like magic, so to speak, teaching by osmosis, as one could put it.

Many years ago I saw a program of an entrance exam to some universities. That program had been approved and signed by some high officials, and in this program I saw something very strange. The topic was solving first degree equations with one unknown, followed by 2 types of equations:

is one, and the other is .

(Laughter) It turns out that these types are different! What was the matter? Some other girl helped me understand this one. From a conversation with her I understood that all the numbers are positive. (Laughter) The negative numbers don’t exist, and indeed, a negative number is the following. It is a positive number in front of which there, for everyone to see stands a minus sign. (Laughter) But if it is so, then, of course, these equations are of different types. Indeed, if you move b to the other side of the equation, you have to change the sign, but all the numbers must be positive! So, a must be positive,
b must be positive, clearly there are two types of equations.

But how had such stuff ended up the program? Well, it got there from the school program. There was such a requirement that the program of the entrance exams should not differ from the school program. If not, then what would happen? How would you take an exam? Now, from the school program it had migrated into the entrance exam program.

And how did it get into the school program? Well, it’s clear how. The school programs are developed by teaching specialists who know how to teach mathematics, and tell us how to do it. Well, they know how to teach mathematics, but they really think that all the numbers are positive.(Liveliness in the hall)

Of course no teacher — well, maybe there are very few, but no ordinary teacher will tell in the class that all the numbers are positive. It is not written in a textbook, so how can he say such a thing? But he thinks so! And if he does, so will do his students. How do they catch it? It’s an interesting question, but they do, beyond any doubts.

The understanding of the subject by the teacher is passed to the students. The understanding of the subject by the lecturer is passed to his audience. It is passed in a mysterious way, but very reliably. We have to keep it in mind. No extra education of the teacher, no proper treatment in the textbook will help if the teacher thinks differently.

The teacher, the instructor is in this sense the central, the decisive figure. I want to repeat again that I absolutely do not believe that you can somehow improve, or change teaching by improving the programs, the textbooks, but without changing, very seriously, the training of the teachers.

True, there are some methods of retraining, various continuous education programs and so on. How effective are these? I don’t have any factual data on that. I have only my personal experience and the personal experience of my friends. And here I have to express a rather grim prognosis. According to my data, any retraining or additional training of the individuals who had learned mathematics at some institution of higher education, such as a pedagogical university, leads to nothing. If they had learned nothing while being students, if during the long following period of teaching they have managed … I am afraid to say forget, it would have been for the better, if they have managed … well, o.k., forget what they had learned, then, of course, an additional training may lead only to cosmetic changes. They will get used to the new words, to the new teaching methods, but it will not change anything that matters.

It seems to me that the universal teaching of mathematics can be improved only one way. It is a slow and difficult way, but it may be the way that may work. We have to gradually increase the production of competent teachers.

There are absolutely enough capable people available for that. Unfortunately, we don’t teach them properly. The reason is understandable: there are not enough teachers to teach them well.

In good old times, when all the mathematics teachers in high school were the graduates of the top notch universities (it was long ago, before the universal education), the situation was better. We hear it often, read about it, that long ago the mathematics teachers were better. They knew their subject better and taught better. I don’t know if it is true, but if it is, the reason is that these teachers studied not in pedagogical universities, but in a few top quality universities. Now they study in pedagogical universities and many other mediocre universities.

Well, I would not want this lecture to be remembered as such a grim one. I want to say something optimistic at the end. I think that if we need a hundred years to prepare, gradually, layer by layer, enough competent mathematics teachers, and organize good mathematical education in our high schools and our institutions of higher education, if hundred years is enough for it — then it is good. (Liveliness in the hall)

A Lecture About Teaching Mathematics to Non-mathematicians, Part I.

Here is a (still rough) translation of the first half of the lecture, the Russian original is available from
В. А. Рохлин Преподавание математики нематематикам

I will hopefully finish the second half in a couple of days.

V.A. Rokhlin

Vladimir Abramovich Rokhlin (1919-1984), a remarkable mathematician, widely known by his ground-breaking work in theory of dynamical systems and topology, who had mentored several top-rated mathematicians, was also a brilliant lecturer. He also thought a lot about the problem of teaching mathematics. His own pedagogical experience was wide and diverse, as he taught at a technical university, at several pedagogical universities, and — during the last 20 years of his life — at Leningrad State University (further details can be found in the book “В.А. Рохлин. Избранные Работы. Воспоминания.” Изд. МЦНМО, 1999).

Here we are publishing some parts of one of the last lectures by V.A. Rokhlin at a meeting of Leningrad Mathematical Society. This lecture took place on November 20 of 1981 at the Leningrad Scientist Hall. This publication is based on the audio tape recording made by B.A. Lifschitz, then transcribed and edited by the efforts of O.Ya. Viro, B.A. Lifschitz and S.V. Ribin.

Unfortunately, the beginning of the lecture (about 15 minutes) has been lost. In these 15 minutes V.A. talked, in particular, about his experience of teaching mathematics to non-mathematics majors at the universities of Arkhangel’sk, Ivanovo and Kolomna in 1950s.

–A.M. Vershik

…At the elementary school children are, naturally, not presented with any serious proofs, but they are given some formulations. For example, they are taught how to divide a fraction by a fraction. There is a rule that is formulated, and they have to know this formulation.

Once I had been to a talk by a teaching specialist who explained how to teach children to divide a fraction by a fraction. He said that he formulated the rule and showed several examples of its application. Then children did some examples too, and then they had a written test. And in this written test almost all of them made the same typical mistake. In some cases –the same for all who made the mistake– they for some reason didn’t flip the divider fraction before multiplying it by the dividend; in some cases they did flip it, and in some cases they didn’t.

What was the reason?
The teaching specialist explained that the analysis of the examples had shown the reason for the mistake. The reason was that in the examples given before the test all the divider fractions were proper. The children figured out from the examples what to do, and they did just that.

So, the rule that was formulated in the beginning by this teacher, was not at all the formulation that these children had understood and had been guided by. The intended rule was simply dictated to them as a matter of routine. It had to be dictated, and it was dictated, but it was ignored by most of the class.

The fact that the children at that age had to learn how to understand the formulation of the rule and how to use this formulation, not just the examples shown to them, this fact had totally escaped the attention of that teaching specialist and his audience.

I consider the idea of giving a worked examples immediately after the formulation of the rule to be a harmful mistake. I have no doubts that the children should, by following the rule, calculate several examples by themselves first. Of course, they have to become fluent and do it automatically later, but the very valuable observation, the observation that the children, by working on some examples, can learn and should learn how to understand the formulation, was totally lost.

This is only one example. In fact the story that I just have told reflects the basic pedagogical principles. They are widespread.

Nowadays one can graduate from high school without in fact solving a single mathematical problem. Children can just emulate the templates and examples given to them to fulfill their school duties.

(*)Probably at the lost beginning of the lecture Rokhlin talked about the harm done by the universally accepted methods of teaching mathematics. Unfortunately only his criticism of the principle that any rule should be immediately illustrated by examples of application, survived. (a footnote by O.Ya. Viro)

When during the recitation sections the first year –and often not only the first year– university students are given a problem to work on, please observe what they do. Of course I am not talking about the math majors.

My subject is different, it is teaching mathematics to non-mathematicians. But, according to my experience, in a class of non-mathematical students only 2 or 3 people actually solve the given problems, the rest of them are sitting and waiting. They actually don’t understand what is wanted of them. They are waiting for the problem to be solved on the chalk board, or for somebody to tell them how to solve this kind of problems. They show no initiative, they are totally unfamiliar with this setup.

It sounds strange, because, if you ask any of these young people to buy some groceries and some medicine and also tell them when the grocery store and when the pharmacy are closed for a lunch break, they will solve this problem beautifully. They will pick the right times to go to the stores and will not go to the pharmacy when it’s closed. Well, there are some strange cases even here (liveliness in the hall).

The problems the students are given during recitations are far simpler than this problem I am talking about, with the grocery and the pharmacy. Nevertheless, the idea of beginning to solve these problems even doesn’t cross the minds of these children, and some of them are 17 or 18 years old. This astonishing situation is, of course, the result of how they were taught mathematics.

Similarly, the level of knowledge in the area of exact sciences, not only mathematics, is surprisingly low among the adult population, i.e., among people who are no longer students, who had graduated from high school or a university a while ago and are considered educated.

If you take the writers, the musicians, the actors, the film directors, many doctors (naturally, I am not talking about all the people educated in humanities), you will find absolutely surprising things.

They proudly say and write that they are bad at math or physics, they say it with a sneer, and, in general, don’t make much of a difference between the two. If you let me use a rather mundane expression, for them math is some area of technology or physics, something not very deserving of respect, but, in any case, something that should serve their needs.

With the doctors, of course, this situation is changing. These days they teach mathematics even to philosophy students and students in other humanities. All are taught by using the same template, without achieving any insight into or understanding of the subject.

If you read carefully the articles –some of them quite interesting– written by people educated in humanities, you will notice that they enjoy using the expressions from mathematics and physics. It is fashionable, it is modern, but God, what they are writing! Strikingly, they have less than a vague idea of what a factor is and what a divider is, what a degree is, what the positive is and what the negative is. From the texts that you can come across in newspapers and magazines you can see clearly that they studied all this, for example, the idea that the negative is somehow related to division, to the degree… Well, take, for example, this phrase: “something is negative and therefore it is not zero, but infinity!” (laughter).

I am not exaggerating, I can give you some references where you can read such things. All of them are written by some well-known, even famous people, it is published in the newspapers. These people memorized something in the elementary school and somehow they think that they remember it correctly.

What should be done about all this? Well, it’s a difficult question.

I don’t believe that this problem, that is the problem of getting a higher level of universal, general education in exact sciences (and mathematics in particular) can be solved fast. It is a difficult problem, and solving it will take a long time and huge efforts.

One may ask whether it is possible or useful to solve this problem. Up till now, till very recent times, people in all the civilized societies were educated in humanities. There were no human society in history where people would be universally educated in exact sciences.

The classical education was widespread, very widespread among the educated strata of the society, but the exact sciences were never made available even to the educated to any serious degree.

Many modern developing countries have long history, going back centuries and millennia. Many have their own intellectual elite but again, this elite is educated mostly in humanities. They like to study humanities, but the exact science they study with some reluctance, and not very successfully.

Concluding these preliminary remarks, I want to say that nobody really knows, and, of course, I don’t know what would be the result of a serious universal education in mathematics and exact sciences, even if it were possible, whether it would make things better or worse. I don’t know it, and nobody does. We don’t have any experience. Nevertheless, for some reason we want to achieve this goal. We are trying. Somehow we feel intuitively that it will be good if our children and grandchildren become familiar with the logical culture, with the mathematical culture, if they will understand the exact sciences better.

It is very likely that it will lead to some incredible upheaval, some unprecedented results, who knows? I don’t know.

Turning to the more narrow subject of my lecture, I want to say that teaching mathematics to the future mathematicians is infinitely easier than teaching mathematics to non-mathematicians.

After all, we can be earnest with the future mathematicians. Here we have the subject that we know, and we are trying to teach this subject to the future mathematicians. No matter how masterful or how mediocre we are in lecturing or leading the exercise sections, we, by knowing the subject, can transmit our knowledge to the interested people. But how should we deal with those who are not interested and think they have no ability, or just say so.

Very often people who say so are simply stuck in, so to speak, intellectual laziness. This intellectual laziness is a very common phenomenon, and sometimes you can easily find out. By talking to a person who claims to have no ability, that he is infinitely far from all this, by talking to him just a little, you discover that he beautifully understands all you say.

So, the question of ability in this field is a complicated question, you don’t have just to take the word for it when somebody claims to have no inclination to learn these subjects, claiming to be interested only in humanities.

Before I really turn my attention to teaching mathematics to non-mathematicians, I should first explain who I call a non-mathematician.  It’s  rather useless to discuss this question in general. I will simply say what I have in mind right now, when I am talking about this topic. I have in mind teaching mathematics to people who have no intention of working in mathematics, who study it either for applications or because they have a non-professional interest in it.

In fact, the presence of interest makes it easier for the teacher. But very often there is no such interest, instead there is a feeling of disgust. There are plenty of students now that have to learn some exact sciences, but have no interest in them. Nevertheless, they have to pass the exams, etc. How should we treat such people? What to do about the people who are interested in mathematics but are taught in such a way that makes it impossible to learn? There are many mathematical curricula for the students of technical universities, they vary in their length and their content.

But very often these curricula and the corresponding textbooks are not independent, they are simply watered-down courses for the students of mathematics. There is the same order of presentation, the same limits, the same derivative, the same integral, the same second degree curves, and so on and so forth.

The order is the same, but the treatment of the material is less clear. There are no proofs that would help to grasp the inner workings of the subject. The authors of these books have no talent for writing. It is boring, it is not understandable. The students are lucky if the lecturer gets them interested by explaining something outside of their textbook, so to speak.

This situation is very common, and not only in our country, as I said before. This situation is totally international, and, as I think, the reason is the following. Probably the technical universities, the pedagogical universities and high schools need their own, special, courses of mathematics. Each category of students (if it is big enough, of course) probably needs its own course, and totally different from the one for teaching mathematics students.

I think the main defect of these courses, the main reason for their failures is the following. None of the prominent mathematicians ever worked on putting together a course of mathematics for non-mathematicians.

(*)Maybe because they were too preoccupied with being prominent and had no time for making mathematics understandable by ordinary people? (a note by Misha Livshits)

I am talking now not about a high school course, but about a university course, and I mean the following.

Usually, before doing differential and integral calculus, the students are taught the theory of limits. It’s the same in high school now. Limits are taught there too.

Nevertheless — and it is a striking example of the current difficulties — the limits is the part of the course that is the most difficult to understand, and, most interestingly, absolutely unnecessary. All the differential calculus, all the integral calculus, and, in general, all the classical mathematics, to say nothing of the finite mathematics, can be treated beautifully without limits. Even more, they are not needed there. They are an absolutely extraneous phenomenon, extraneous subject that had been introduced into this area by the people who wanted to build a proper foundation of analysis.

Meanwhile, the purpose of treating the foundations is not achieved in a technical university course, it is not even a purpose of such a course. From this example we can see already that these courses are absolutely thoughtless and are just watered-down university courses in analysis (*)for mathematics students, a note by Misha Livshits).

Now I will explain my thoughts about limits in some detail. When I went to high school (maybe it’s still the same now), they explained to me that the area of a circle is some sort of a limit, then they wrote something, said something, and got some formula for the area of the circle. What was said was difficult to understand then, but when I became a mathematician, it became totally clear to me why it was so difficult to understand, the reason being that it was all nonsense.

Nobody of my fellow students had any doubts that he knew what the area of the circle is. Rather it looked strange to us that for the circle the area was defined for us, but somehow there was no definition for the other figures. It was strange to us that the area of a circle that was totally understandable to us, is defined by using some limits that we could not understand. We felt strange of course (and all the children who thought about it a little bit also feel strange) that some theorems about limits are needed to establish some very clear an simple things that we had never doubted.

But really, why should we define the area of a circle and to prove that it is ? Why not just say that the area of a circle is by definition, what is the difference? It looks like the difference, and a really serious one, is that not only circles have the area, that the area is a general notion, that the area is defined for a wide class of figures, that it has the properties, known and used by everyone, that make the area a useful notion.

So, the attitude towards the area adopted in the high school course was then (and maybe in many cases still remains now) absolutely bizarre.

But in teaching mathematics in technical universities the same attitude is adopted towards integrals, toward derivatives, towards volumes and masses, towards density, towards charges, towards the moments of inertia and in general towards all mathematical and physical quantities of such integral and differential nature.

From the point of view of a person who is not a professional mathematician, all these things exist, they don’t require a definition, they require a computation, and they have to be ready for applications. That’s what is needed.

The point of view that these properties should be defined is not appropriate in teaching here, at least in teaching such a person that doesn’t doubt the existence of area of all the figures, you don’t have to define the area to such a person. Such person has to study the properties of the area, to learn how to calculate this area.

The same is true for the other mathematical notions. Let’s take the notion of the integral. On this occasion I would like to ask a question of historical nature. Tell me, please, did Archimedes have a notion of the integral or not? There are different points of view on it. Some say he did, some say he didn’t. I will express my own opinion. I think that Archimedes, maybe the greatest mathematician of all times, did not have the notion of the integral. Here is why I think so.

Archimedes many times, by many different methods, calculated the integral

He calculated it when he studied the areas bounded by a segment of a straight line and a segment of a parabola. He calculated this integral when he calculated the volume of a ball, and in many other situations. Each time he used a special approach, very ingenious, brilliant. But it looks like he did not know that it was all the same. He probably felt that it was.

The reason for it is totally understandable. The Greeks did not have a notion of the real number. The volume and the area were totally different entities for them, they were the geometrical entities that they did not compare to each other. For example, Archimedes would have protested against such an expression as . He would have said that and can not be added, that it is the same as to add a line segment with a planar figure. But we do it. We have numbers. And it looks like Archimedes didn’t have the numerical notion of the integral. If he had, he undoubtedly would have not calculated the same integral many times.

On the other hand, Archimedes had left us the method of exhaustion, that may be very fitting for teaching mathematics to non-mathematicians. Because of this method, instead of all the theory of limits we need only one single fact, I will formulate it now. This fact is very simple. It says that if a non-negative number is less than any positive number, then it is zero. I repeat, if a non-negative number is less that any positive number, it is zero.

This fact is not difficult to prove, but maybe a proof is not even needed. After you are familiar with this fact, you can use it to prove all the formulas that are encountered in the differential and integral calculus and in its applications, and more generally in all the analysis, provided that don’t deal with the theorems of existence.

You don’t prove the existence. The limit theory is only needed to prove the existence theorems.

If you don’t have to prove the existence of the area, limits are not needed. If you don’t need to prove the existence of the integral, the limit theory is not needed, and so on. If you only need to calculate, you can get by without the limit theory. This makes the differential and integral calculus much easier right away.

[A BREAK]

What is so Geometric about the Geometric Series

Behold the picture:

Exhibit number 2, details on Wikipedia:

Exhibit number 3, calculation of the area under a power curve by Fermat, pilfered from Analysis by Its History, by E.Hairer and G.Wanner (Springer 1996), Sect. I.3, pp. 33-34:

From this picture it looks like the area we want to calculate is sandwiched between the sums of two geometric series:

After summing the series we get

At the upper and the lower bounds come together. Unfortunately, is undefined for since it reduces to . But fortunately Fermat knew how to make sense out of the undefined expressions like this, he knew how to differentiate, so he could figure out that .

Had he noticed this peculiar link between differentiation, i.e., making sense of , and integration, i.e., calculating areas? I have no doubt he had, Fermat was Fermat. Had he pointed it out to his high-esteemed correspondents? I don’t know. But it was up to Isaac Barrow to expose differentiation and integration as two operations inverse to each other, and up to Newton, Leibniz and Bernoulli brothers to put his remarkable observation to work.