A few similes about mathematics

Mathematics is useful but what makes mathematics mathematics is not the same as what makes it useful.

Mathematics has its own intrinsic needs that have to be addressed for it to stay alive.

Let us compare mathematics with a cow. Cow is useful, it gives us milk, cream, butter, cheese — the list can be continued.

Applied mathematics can be compared with the cow’s udder — it produces milk.

Some branches of pure mathematics are best described as the cow’s immune system — they keep the cow alive.

Cow of course has other uses. To make a steak, it suffices to take a piece of cow and gently roast it to taste. What is a piece of cow? Mathematicians.

Financial industry, security sector, etc. are connoisseurs of good steak. NSA advertises itself as the biggest employer of mathematicians in the USA.

Some people claim that pure mathematicians’ focus on “useless” artificial problems “they invent for themselves”. But Let us look at geneticists’ obsession with a pretty useless creature: Drosophila melanogaster.

An article in Wiki devoted to it says:

The species is known generally as the common fruit fly or vinegar fly. Starting with Charles W. Woodworth’s proposal of the use of this species as a model organism, D. melanogaster continues to be widely used for biological research in studies of genetics, physiology, microbial pathogenesis and life history evolution. It is typically used because it is an animal species that is easy to care for, has four pairs of chromosomes, breed quickly, and lays many eggs”.

Very frequently, “famous” mathematics problems are means of concentrating effort of generations of mathematicians on development of methods of proof in particular areas of mathematics, they are drosophilas of mathematics. The Last Fermat Theorem is perhaps the most famous example. In some cases (and the Riemann Hypothesis is the archetypal case) they, however, have exceptional importance for mathematics as a whole.

Дети, любите корову – источник мяса!


Why were soviet mathematics/physics textbooks so insanely hardcore in comparison to US textbooks?

The most complete answer from a discussion in Quora:

Alex Sergeev, PhD in Physics

As I understand from reading comments, the OP means not school textbooks, but university textbooks, in particular Landau-Lifshitz was mentioned. In such case, I have to disagree with most answers presented.

Firstly, yes, they are indeed noticeably more hardcore than courses of a similar level in the US. Enough to compare two classic courses: Landau-Lifshitz and Feynman Lectures (which are, in turn, not really a walk in a park either, there are plenty of friendlier books). Same can be said about mathematical analysis books which I encountered. Soviet textbooks just go straight to the point and throw lots of definitions and formulas at you, without any preparation. The US textbooks try to explain simple things in more detail, and increase the complexity as they progress.

The reason for it, I think, is the difference in education systems. In the US, the point of education system is to teach students, as well as possible. In the USSR, the point was to get rid of weaker students and have only very good ones left, who would understand the subject no matter how hardcore the approach to it is. It might be more psychological rather than intentional, but in Soviet times it was a general sentiment: if you can’t do it straight-away, you are simply not good enough and should do something else. The US system tries to improve students and then select the best, the Soviet system tried to select the best and then improve them. The US system tries to make geniuses out of average students, the Soviet system tried to select geniuses disregarding average students. I might be a bit too categorical with this, but I don’t think it is too far from truth.

Another possible reason, stemming from the above is a lack of competition. In the US, the education system is adapting to students’ need, if the books are not teaching good enough they get replaced or amended. In the USSR, the textbooks were centrally selected and approved, and students had to adapt to whatever they were given.

Edit: I also have just recalled this phrase very widely circulated during Soviet times: “We don’t have irreplaceable people”. (It actually originated much earlier, and was used by Woodrow Wilson, but is widely assigned to Stalin, who in fact never said anything like that. I also believe that the connotation was intended to be different.) This phrase, however, well demonstrates the psychology of Soviet system. No one cared if you fail, there’ll be another person who’d take your place. In the US, if student is struggling, it is partially a teacher’s fault; in the USSR, it is 100% student’s fault.



A love poem without verbs

This appeared on Twitter:

In Arabic you can have a full sentence without a verb. It reflects our relaxed nature.

In Russian, it is possible to have a full love poem without a verb. This is a prominent  example:

Шепот, робкое дыханье.
Трели соловья,
Серебро и колыханье
Сонного ручья.

Свет ночной, ночные тени,
Тени без конца,
Ряд волшебных изменений
Милого лица,

В дымных тучках пурпур розы,
Отблеск янтаря,
И лобзания, и слезы,
И заря, заря!..

What should this reflect?


The Astronomer and The Anthropologist

I used this classical image by Vermeer as a frontispiece to my book Mathematics under the Microscope:

Astronomer by Jan Vermeer, 1632-1675. A portrait of Antonij van Leeuwenhoek?

Astronomer by Jan Vermeer, 1632-1675. A portrait of Antonij van Leeuwenhoek?


I even repeat, in the Preface to the book, speculations that this is a portrait of Antonij van Leeuwenhoek.  I write about the painting:

The portrayal of human thought has rarely been more powerful and convincing than in  Vermeer’s Astronomer. The painting creates the illusion that you see the movement of thought itself — as an embodied action, as a physical process taking place in  real space and time.

I use Astronomer as a visual metaphor for the principal aim of the present book. I attempt to write about mathematical thinking as an objective, real-world process, something which is actually moving and happening in our brains when we do mathematics. Of course, it is a challenging task; inevitably, I have to concentrate on the simplest, atomic activities involved in mathematical practice—hence “the microscope” of the title.

This is why I turned speechless when I discovered a photograph The Anthropologist by Tom Hunter from his magnificent series Persons Unknown.

The Anthropologist by  Tom Hunter, 1997

The Anthropologist by Tom Hunter, 1997

Perhaps it is time to switch from microbiology of mathematics to anthropology of mathematics.


Daniil Kharms: On Time, Space and Existence

I am happy to add an English translation to Daniil Kharms’ philosophical parody О времени, о пространстве, о существовании.


A note from the translator:

The text, as prepared for the Web by Serge Winitzki, is an imprecise rendition of the original work and may contain errors, despite proofreading. May be used for information purposes only. Distribute for free, but always together with this notice and in unchanged form.

On Time, Space and Existence

1. A world which is not can not be called existing, because it is not.

2. A world consisting of something unified, homogeneous and continuous can not be called existing, because in such a world there are no parts and, once there are no parts, there is no whole.

3. An existing world must be heterogeneous and have parts.

4. Every two parts are different, because one part will always be thus one and the other that one.

5. If only this one exists, then that one cannot exist, because, as we have said, only this exists. But such a this cannot exist, because if this exists it must be heterogeneous and have parts. And if it has parts that means it consists of this and that.

6. If this and that exist, this means that not this and not that exist, because if not this and not that did not exist, then this and that would be unified, homogeneous and continuous and consequently would also not exist.

7. We shall call the first part this and the second part that and the transition from one to the other we shall call neither this nor that.8.

We shall call neither this nor that ‘the impediment’.

9. Thus: the basis of existence comprises three elements: this, the impediment and that.

10. We shall express non-existence as zero or a unity. Therefore we shall have to express existence by the number three.

11. Thus: dividing a unitary void into two parts, we get the trinity of existence.

12. Or: a unitary void, experiencing a certain impediment, splits into parts, which make up the trinity of existence.

13. The impediment is that creator which creates ‘something’ out of ‘nothing’.

14. If this one, on its own, is ‘nothing’ or a non-existent ‘something’, then the ‘impediment’ is also ‘nothing’ or a non-existent ‘something’.

15. By this reckoning there must be two ‘nothings’ or nonexistent ‘somethings’.

16. If there are two ‘nothings’ or non-existent ‘somethings’, then one of them is the ‘impediment’ to the other, breaking it down into parts and becoming itself a part of the other.

17. In the same way the other, being the impediment to the first, splits it into parts and itself becomes a part of the first.

18. In this way are created, of their own accord, non-existent parts.

19. Three, of their own accord, non-existent parts create the three basic elements of existence.

20. The three, of their own accord, non-existent basic elements of existence, all three together, make up a certain existence.

21. If one of the three basic elements of existence should disappear, then the whole would disappear. So: should the ‘impediment’ disappear, then this one and that one would become unitary and continuous and would cease to exist.

22. The existence of our universe generates three ‘nothings’ or separately, on their own account, three non-existent ‘somethings’: space, time and something else which is neither time nor space.

23. Time, of its essence, is unitary, homogeneous and continuous and thereby does not exist.

24. Space, of its essence, is unitary, homogeneous and continuous and thereby does not exist.

25. But as soon as space and time enter into a certain mutual relationship they become the impediment, the one of the other, and begin to exist.

26. As they begin to exist, space and time become mutually parts, one of the other.

27. Time, experiencing the impediment of space, breaks down into parts, generating the trinity of existence.

28. A split down and existing, consists of the three basic elements of existence: the past, the present and the future.

29. The past, the present and the future, as basic elements of existence, always stood in inevitable dependence, each on the other. There cannot be a past without a present and a future, or a present without a past and a future, or a future without a past and a present.

30. Examining these three elements separately, we see that there is no past because it has already gone and here is no future because it has not yet come. That means that there remains only one thing — the ‘present’. But what is the ‘present’?

31. When we are pronouncing this word, the letters of this word which have been pronounced become past and the unpronounced letters still lie in the future. This means that only that sound which is being pronounced now is ‘present’.

32. But of course the process of pronouncing this sound possesses a certain length. Consequently, a certain part of this process is ‘present’, just as the other parts are either past or future. But the same thing too may be said of this part of the process which had seemed to us to be ‘the present’.

33. Reflecting in this manner, we see that there is no ‘present’.

34. The present is only the ‘impediment’ in the transition from past to future and past and future appear to us as the this and that of the existence of time.

35. Thus: the present is the ‘impediment’ in the existence of time and, as we said earlier, space serves as the impediment in the existence of time.

36. By this means: the ‘present’ of time is space.

37. There is no space in the past and the future, it being contained entirely in the ‘present’. And the present is space.

38. And since there is no present, neither is there any space.

39. We have explained the existence of time but space, of its own accord, does not yet exist.

40. In order to explain the existence of space, we must take that incidence when time performs as the impediment of space.

41. Experiencing the impediment of time, space splits into parts, generating the trinity of existence.

42. Broken down, existing space consists of three elements: there, here and there.

43. In the transition from one there to the other there, it is necessary to overcome the impediment here, because if it were not for the impediment here, then the one there and the other there would be unitary.

44. Here is the ‘impediment’ of existing space. And, as we said above, the impediment of existing space is time.

45. Therefore: the here of space is time.

46. The here of space and the ‘present’ of time are the points of intersection between time and space.

47. Examining space and time as basic elements in the existence of the universe, we would say: the universe expresses space, time and something else which is neither time nor space.

48. That ‘something’ which is neither time nor space is the ‘impediment’, which generates the existence of the universe.

49. This ‘something’ expresses the impediment between time and space.

50. Therefore this ‘something’ lies at the point of intersection of time and space.

51. Consequently this ‘something’ is to be found in time at the point of the ‘present’ and in space at the point of the ‘here’.

52. This ‘something’ which is to be found at the point of intersection of space and time generates a certain ‘impediment’, separating the ‘here’ from the ‘present’.

53. This ‘something’, generating the impediment and separating the ‘here’ from the ‘present’, creates a certain existence which we call matter or energy. (Henceforth we shall provisionally call this simply matter.)

54. Thus: the existence of the universe, as organised by space, time and their impediment, is expressed as matter.

55. Matter testifies to us of time.

56. Matter testifies to us of space.

57. By this means: the three basic elements of the existence of the universe are perceived by us as time, space and matter.

58. Time, space and matter, intersecting one with another at definite points and being basic elements in the existence of the universe, generate a certain node.

59. We shall call this node — the Node of the Universe.

60. When I say of myself: ‘I am’, I am placing myself within the Node of the Universe.


Daniil Karms on mathematics and dialectics

У Хармса были поразительно интересные взаимоотношения с математикой – и с диалектикой Гегеля. Если бы у меня было время, я бы написал про это статью.


Моя теория — что Хармс играл в “Cargo Cult” (в смысле Ричарда Фейнмана) игры с господствующей идеологией, созданной для потребления пришедшими к власти необразованными массами. Он пытался осознать, сам для себя, логические пружины, которые эту идеологию двигали. Поэтому он писал то, что выглядит ядовитыми пародиями на Гегеля, но что для него, мне кажется, было честными мысленными экспериментами.


Вот короткая хохмо-Гегелевская вещица, у Хармса были и много более развернутые.

“Не теперь”

Это есть Это.
То есть То.
Все либо то, либо не то.
Что не то и не это, то не это и не то.
Что то и это, то и себе Само.
Что себе Само, то может быть то,
да не это, либо это, да не то.

Это ушло в то, а то ушло в это.
Мы говорим: Бог дунул.
Это ушло в это, а то ушло в то,
и нам неоткуда выйти и некуда прийти.
Это ушло в это. Мы спросили: где?
Нам пропели: тут.
Это вышло из Тут. Что это? Это То.
Это есть то.
То есть это.
Тут есть это и то.
Тут ушло в это, это ушло в то,
а то ушло в тут.
Мы смотрели, но не видели.
А там стояли это и то.

Там не тут.
Там то.
Тут это.
Но теперь там и это и то.
Но теперь и тут это и то.
Мы тоскуем и думаем и томимся.

Где же теперь?
Теперь тут, а теперь там, а теперь тут,
а теперь тут и там.
Это быть то.
Тут быть там.
Это то тут там быть. Я. Мы. Бог.


Более развернутый текст выглядит еще более злобным. Скачки логики в нем (мне кажется, ясно осознававшиеся Хармсом) просто комичны:

“О времени, о пространстве, о существовании”

1. Мир, которого нет, не может быть назван существующим, потому что его нет.
2. Мир, состоящий из чего-то единого, однородного и непрерывного, не может быть назван существующим, потому что, в таком мире, нет частей, а, раз нет частей, то нет и целого.
3. Существующий мир должен быть неоднородным и иметь части.

[… я пропускаю середину –АБ …]

55. Материя свидетельствует нам о времени.
56. Материя свидетельствует нам и пространстве.
57. Таким образом: три основных элемента существования Вселенной, воспринимаются нами, как время, пространство и материя.
58. Время, пространство и материя, пересекаясь друг с другом н определенных точках и являясь основными элементами существования Вселенной, образуют некоторый узел.
59. Назовем этот узел — Узлом Вселенной.
60. Говоря о себе: “я есмь”, я помещаю себя в Узел Вселенной.


В математике, лучшая работа Хармса — это

Я вам хочу рассказать одно происшествие …”

Я вам хочу рассказать одно происшествие, случившееся с рыбой или даже
вернее не с рыбой, а с человеком Патрулевым, или даже еще вернее с дочерью Патрулева.

Начну с самого рождения. Кстати о рождении: у нас родились на полу…
Или хотя это мы потом расскажем.

Говорю прямо:

Дочь Патрулева родилась в субботу. Обозначим эту дочь латинской буквой

Обозначив эту дочь латинской буквой М, заметим, что:

1. Две руки, две ноги, посередке сапоги.
2. Уши обладают тем же, чем и глаза.
3. Бегать — глагол из под ног.
4. Щупать — глагол из под рук.
5. Усы могут быть только у сына.
6. Затылком нельзя рассмотреть, что висит на стене.
17. Обратите внимание, что после шестерки идет семнадцать.

Для того, чтобы раскрасить картинку, запомним эти семнадцать

Теперь обопремся рукой о пятый постулат и посмотрим, что из этого

[ … После чего идет длинный ряд абсурдистских хохм, которые гениально кончаются 100% тавтологическими выводами – АБ …]


Дочь Патрулева отца Патрулева дочь
Значит и дочь Патрулева отца Патрулева дочь.
Коли так то и дочь. Патрулева отца
Значит и дочь Патрулева отца.
Вот и дочь, а отец Патрулев
Дочь Патрулева, отец Патрулев
Значит отец Патрулевой дочери Патрулев
И никто не скажет что он Петухов
Это было бы противоестественно


И, как утверждают, Хармс был глубоко верующим православным, что, впрочем, ясно и из его стихов. Два стихотворения, что я сейчас привожу — они как будто из “Часослова” Райнера Мария Рильке, но только сразу написанные на русском языке.


Во имя Отца и Сына и Святаго Духа
вчера я сидел у окна выставив ухо
земля говорила дереву: произростай
дерево медленно росло — но всё же заметно глазу
то голым стояло то прятало ствол в зелёную вазу
на солнце читая значёк своей радости
планеты порой шевелились меж звёздами
а дерево гнулось махая птичьями гнёздами
семь радуг над деревом возносилось
я видел доски ангельских глаз
они глядели сверху на нас
читая годов добрые числа


Вот грянул дождь,
Остановилось время.
Часы беспомощно стучат.
Расти, трава, тебе не надо время.
Дух Божий, говори, тебе не надо слов.


How to Criticize with Kindness: Philosopher Daniel Dennett on the Four Steps to Arguing Intelligently

A quote from Maria Popova at Brain Pickings:

Dennett synthesizes the steps:

How to compose a successful critical commentary:

  1. You should attempt to re-express your target’s position so clearly, vividly, and fairly that your target says, “Thanks, I wish I’d thought of putting it that way.
  2. You should list any points of agreement (especially if they are not matters of general or widespread agreement).
  3. You should mention anything you have learned from your target.
  4. Only then are you permitted to say so much as a word of rebuttal or criticism.

Calling a Spade a Spade

A pdf file: Calling a spade a spade: mathematics in the new pattern of division of labour

Short URL: http://goo.gl/TT6ncO

The growing disconnection of the majority of the population from mathematics is
increasingly difficult to ignore.

This paper attempts to point to deeper roots of this cultural and social phenomenon. It concentrates on mathematics education, as the most important and better documented area of interaction of mathematics with the rest of human culture.

I argue that new patterns of division of labour have dramatically changed the nature and role of mathematical skills needed for the labour force and correspondingly changed the place of mathematics in popular culture and in the mainstream education. The forces that drive these changes come from the tension between the ever deepening specialisation of labour and ever increasing length of specialised training required for jobs at the increasingly sharp cutting edge of technology.

Unfortunately these deeper socio-economic origins of the current systemic crisis of mathematics education are not clearly spelt out, neither in cultural studies nor, even more worryingly, in the education policy discourse; at the best, they are only euphemistically hinted at.

This paper is an attempt to describe the socio-economic landscape of mathematics education without resorting to euphemisms.

Comments are warmly appreciated. Please email them at alexandre >>at<< borovik.net


If on a Winter’s Night a Traveller

The title of this wonderful novel by Italo Calvino is already a masterpiece. As soon I saw saw the book on s shop shelf, I bought it on impulse because of its title.  To the Russian ear, it had immediate connotations with the famous passage from Chekhov’s “Ionych”:

Then they all sat down in the drawing-room with very serious faces, and Vera Iosifovna read her novel. It began like this: “The frost was intense… .” The windows were wide open; from the kitchen came the clatter of knives and the smell of fried onions… . It was comfortable in the soft deep arm-chair; the lights had such a friendly twinkle in the twilight of the drawing-room, and at the moment on a summer evening when sounds of voices and laughter floated in from the street and whiffs of lilac from the yard, it was difficult to grasp that the frost was intense, and that the setting sun was lighting with its chilly rays a solitary wayfarer on the snowy plain. Vera Iosifovna read how a beautiful young countess founded a school, a hospital, a library, in her village, and fell in love with a wandering artist; she read of what never happens in real life, and yet it was pleasant to listen — it was comfortable, and such agreeable, serene thoughts kept coming into the mind, one had no desire to get up.

In Russia of “the period of stagnation”, the expression “The frost was intense” (“мороз крепчал”) became proverbial and was transformed into a less politically correct, but more politically charged, derivative.

And I was delighted to discover that my instinctive choice was correct and that, indeed, Calvino’s book “did exactly what it said on the tin“!

I mention this novel here because of its remarkably intensive built-in self-reflexivity.

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