Posted by: Alexandre Borovik | April 9, 2020

Poems for lockdown

Beat lockdown blues with viral poetry: send your favorite poem to random people from your address book (use bbc: or a mass mailing of your email client), and ask them do the same (but include you on the list). Here is some stuff that I sent and received.

Федор Тютчев


Молчи, скрывайся и таи
И чувства и мечты свои –
Пускай в душевной глубине
Встают и заходят оне
Безмолвно, как звезды в ночи,-
Любуйся ими – и молчи.

Как сердцу высказать себя?
Другому как понять тебя?
Поймёт ли он, чем ты живёшь?
Мысль изречённая есть ложь.
Взрывая, возмутишь ключи,-
Питайся ими – и молчи.

Лишь жить в себе самом умей –
Есть целый мир в душе твоей
Таинственно-волшебных дум;
Их оглушит наружный шум,
Дневные разгонят лучи,-
Внимай их пенью – и молчи!..

A translation by Vladimir Nabokov:

Speak not, lie hidden, and conceal
the way you dream, the things you feel.
Deep in your spirit let them rise
akin to stars in crystal skies
that set before the night is blurred:
delight in them and speak no word.
How can a heart expression find?
How should another know your mind?
Will he discern what quickens you?
A thought once uttered is untrue.
Dimmed is the fountainhead when stirred:
drink at the source and speak no word.
Live in your inner self alone
within your soul a world has grown,
the magic of veiled thoughts that might
be blinded by the outer light,
drowned in the noise of day, unheard…
take in their song and speak no word.
Posted by: Alexandre Borovik | January 5, 2020

Reuben Hersh ( Dec 9, 1927 – Jan 3, 2020)

Reuben Hersh was a mathematician, unorthodox philosopher of mathematics, and a very kind and friendly man.

Posted by: Alexandre Borovik | December 17, 2018

How people learn: The case of Dr Brian May

I am obsessed with stories of how people learn, and of their motivation for learning. 

This is Dr Brian May, and his personal story that appears to be unbelievable: the interesting bit is  2”07 – 3”32 of the BBC film. Aged 7, Brian May got obsessed with stereo photography and very soon started to produce his own stereopictures. 

By the time he joined Queen, he was doing PhD in Astrophysics (he formally defended his PhD only years later).

Well, the story is quite believable to me. Once upon a time I knew a boy who, at age 14, was repairing TV sets (primordial by modern standards, black and white, vacuum tube) for all his neighbours in a small provincial town. This job required an oscilloscope; he made one from his family’s TV set by adding an additional circuit and a switch between the two modes of operation: as a normal TV set and as an oscilloscope. In later life, he became a guru and wizard of the black art of fine-tuning of accelerators of elementary particles and was in charge of one of the biggest one in the world.

And, of course, there was Richard Feynman who, as a boy, famously “Fixed radios by thinking“.

Back to Brian May: his PhD thesis is published, and the preface contains this passage:

“I inherited a Fabry-Perot spectrometer and pulse-counting equipment from Prof. Ring, and spent 18 months entirely rebuilding and updating both the optics and electronics, in preparation for obtaining essentially first viable set of radial velocity measuremnents, all around the elcliptic, of the Zodiac Light. The writing of my thesis was virtually complete in 2006, but the submission was deferred due to various pressures.” 

It is easy to believe that May, as the lead guitarist of Queen, did not have the same issues with scales of measurement as Nigel Tufnel of Spinal Tap famously had: 

This goes to 11…  [watch from 1”16].

Posted by: Alexandre Borovik | November 5, 2018

My answer to a question in Quora: Who was a notable person that was originally evil, but eventually regretted their evil and became good later on?

I apologise if I missed some answers in this thread, but one of more obvious answers is St Paul the Apostle (or Saul, how he was known prior to his inversion on the road to Damascus). Acts 9:1–6 say:

[1] And Saul, yet breathing out threatenings and slaughter against the disciples of the Lord, went unto the high priest,
[2] And desired of him letters to Damascus to the synagogues, that if he found any of this way, whether they were men or women, he might bring them bound unto Jerusalem.
[3] And as he journeyed, he came near Damascus: and suddenly there shined round about him a light from heaven:
[4] And he fell to the earth, and heard a voice saying unto him, Saul, Saul, why persecutest thou me?
[5] And he said, Who art thou, Lord? And the Lord said, I am Jesus whom thou persecutest: it is hard for thee to kick against the pricks.
[6] And he trembling and astonished said, Lord, what wilt thou have me to do? And the Lord said unto him, Arise, and go into the city, and it shall be told thee what thou must do.

There are conflicting interpretaions of this episode, but, in any case, Paul was a changed person since then. An evil man would not write in 1 Corinthians 13:4-7 :

[ 4] Charity suffereth long, and is kind; charity envieth not; charity vaunteth not itself, is not puffed up,

[5] Doth not behave itself unseemly, seeketh not her own, is not easily provoked, thinketh no evil;

[6] Rejoiceth not in iniquity, but rejoiceth in the truth;

[7] Beareth all things, believeth all things, hopeth all things, endureth all things.

Posted by: Alexandre Borovik | November 4, 2018

Intuitively, what is a finite simple group?

My answer in Quora to the question: Intuitively, what is a finite simple group?

There are two ways to describe an object: how it is made and what it is doing. For example, a knife can be described as “an elongated flat piece of metal, sharpened on one edge, with a handle attached” (it is how it is made), or “a thing to cut bread” (how it is used).

I have not red every answer, but the discussion of groups in this thread so far appears to be restricted to the viewpoint of “how they are made”. But what do finite groups do?

They act. They act on sets of various nature; this sets are made of elements. The same group may have many different actions. For example, the group of symmetries of the cube can be seen as acting on

  1. the set of 8 vertices of the cube
  2. the set of 6 faces,
  3. the set of 8 edges seen as non-oriented segments
  4. the set of 16 oriented edges,
  5. the set of 4 non-oriented “main diagonals”,
  6. the set of 3 non-oriented axes passing through the centers of opposite faces.

Indeed, every symmetry of the cube is moving elements of each of this sets within that set (perhaps actually fixing some of them or even all of them).

Elements of the group, for the purpose of this discussion, can be called actors (I invented that name specifically for my post on Quora). What follows is a description, not a rigorous definition.

Each actor moves elements in the set in some way (and this could be an identity move, when nothing is actually changed in the set) . What follows are properties of actions of a group:

  • There is an actor which does nothing.
  • For every actor there is another actor, which reverses its moves.
  • For any two actors, there is an actor who is doing the combination of movements of the first two actors.

An action of a group is called trivial if every actor does not move anything.

An action of a group is called faithful if different actors do different movements. In the example with the group of symmetries of the cube, actions 1 to 4 are faithful, 5 and 6 are not faithful.

The key point: if an action of a group is not faithful, the same movements of elements can be achieved by an action on the same set of another group with smaller number of actors.

Definition: A finite group is simple if all its actions are trivial or faithful.

In short, a simple finite group cannot be replaced, in its non-trivial action, by a smaller group.

This is why finite simple groups are atoms of finite group theory, and why classification of finite simple groups has tremendous importance for combinatorics.

As we can see, the group of symmetries of the cube is not simple. Moreover, its action 6 above contain only movements of 3 elements which can be done by the symmetric group Sym_3 on three letters, or, which is the same, by the group of symmetries of an equilateral triangle (this triangle can be easily seen within the cube). By contrast, the groups of rotations (symmetries which do not change orientation) of the equilateral triangle or the regular pentagon are simple.

By the way, “how it is made” and “what it is doing” are called, in Hegelian dialectics, essence and phenomenon. Questions “intuitively, what is …” refer to phenomena. Intuitively, a knife is a thing to put butter on bread. Intuitively, a group is a mathematical object that acts, or which can be used to describe action. There are other mathematical objects that also can act, in their own way: rings and algebras, for example. On the other hand, you can use a spoon to put butter on bread.

Posted by: Alexandre Borovik | November 3, 2018

Open Book Publishers

In my opinion, they deserve attention: Open Book Publishers ; it is likely that I will soon submit to them a book for publication. What follows is an excerpt from their recent email. Please notice a useful list of links at the end.

OBP is trialing a new platform to engage readers, encourage discussion and to keep our books alive and thriving. We have currently implemented on our  title: Hanging on to the Edges by Daniel Nettle – please take part! To annotate this book, all you have to do is click on the HTML version and look for the ‘Annotate this book’ button below the cover image.
OBP recently weighed in on the dangers of participating in Knowledge Unlatched Open Funding and if you want to understand why OBP will not be participating, visit here.

Finally, for those interested, our new and updated Autumn 2018 catalogue is available to download here.

OA Week Blog Series: see an excerpt from An Academic’s Guide to Open Access, in which we explain why authors should choose to publish Open Access or, to read the whole series, visit here.

“Open Access means more readers. A printed monograph will sell 200-400 copies in its lifetime, primarily to university libraries. At OBP, on average our titles receive 400 views per month. UCL Press, an Open Access university publisher, achieved 1 million downloads in three years. Open Access book chapters on JSTOR are downloaded 20 times more than closed-access book chapters. More readers mean more citations; an increased profile for your writing and your discipline; and more colleagues, more students, more interested members of the public who are able to access your books and articles without hitting price barriers. It means your work will be doing more work in the world. Digital publication also allows new and exciting forms of research. You can add sound and moving images to the written word, embed archival materials, and engage directly with fellow scholars or students, thereby improving the quality of your work.”Lucy Barnes, OBP Editor

This blog series includes answers to the following questions:

Will I have to pay a fee? How much is it? What does it pay for?
Is the publisher for-profit or not-for-profit?
What peer-review systems do they have in place?
Do they create Open Access editions?
Do they insist on an embargo period?
Is their Open Access edition just a downloadable PDF?
Are the Open Access editions easily discoverable? How is the work distributed?
Do they let you keep your copyright?

Posted by: Alexandre Borovik | November 3, 2018

Back to basics

My good colleague allowed me to distribute these extracts from his emails. They are quite interesting, in my opinion.

I have now come to the conclusion that if you want first year students to learn how to write mathematics properly, it is necessary and fully sufficient to spend two hours face-to-face with them, in front of a blackboard, and have them write any form of silly proof such as: a uniformly continuous function is continuous.

But they must hold the stick of chalk and write, and you must correct real-time every single f*****g comma.

I can mark homework week after week and am stupidly dedicated at that; but homework, even good-willed, will not force the lost ones to make any progress, as opposed to the above. Just a face-to-face two hour session, correcting every move.

What a gain of time it would be to simply teach them the trick! How delightfully readable would their papers be afterwards! And what a side-benefit for the whole society!

Why do we then not do this?

As you see I completely gave up on conveying intuitions to 1st year students. Not to mention my own research for the time being.

This term my teaching duties are in a small branch the University has in *******. Classes of 20; students are dedicated, which is not unprecedented. But we can afford being dedicated to them, which is.

So unfair to the crowd in *****! [However] it is not impossible to implement.

Of course the professor who lectures in front of 100+ (30+ is already quite too much) cannot afford it.

Our TA’s should be assigned such — pleasant — duties, could we rely on all of them.

For all I request is two (2, TWO) hours per student, not per student per year. The cost is reasonable.

And the savings are huge; homework and exams become less of a pain: you only have to grade the contents, not return every-single-mark-my-word-bloody-time to the difference between “if… then…” and “therefore…” (a linguistic ability quite useful in everyday life). Which I do in written form, every-single-mark-my-word-bloody-time.

This, for the student who will spend in the average two or three years studying or trying to study mathematics, takes overall more than two hours.

I can report on this more in the future. One of my duties (in *****) will be to mark and comment on exams for people applying to become secondary school teachers. We actually run a full course entitled “How to write mathematics”. It is a year 5 class!

I have not suggested my idea there yet.

Posted by: Alexandre Borovik | October 14, 2018

Confident students do not cheat

This an abstract of a talk given by me at the Meeting “Mathematical Academic Malpractice in the Modern Age”, Manchester, Monday 21st May 2018.

Confident students do not cheat: how to build mathematical confidence in our students


I think it could be useful to address the question which, in my experience, is almost never asked: what pushes problem students to cheat by plagiarising work from their peers and, increasingly, from the Internet? Some answer can be found in Denizhan (2014):

“These students exhibit an inability to evaluate their own performances independent of external measurements.”

Plagiarism is one of the psychological defenses of a student who does not otherwise know whether his/her solution / answer is correct.

Mathematics provides a simple remedy: systematically teach students how they can check their solutions. This will boost their confidence in their answers – and in themselves.

I teach linear algebra; I have at least two dozen undergraduate linear algebra textbooks in my office — none of them provides systematic advice on these matters. The same applies, I think, to any other undergraduate subject.

In my view, the most efficient methods for checking answers in a particular class of problems usually provided by a more advanced point of view. For example,

  • all these elementary problems about systems of linear equations can be effectively checked if the concepts of the rank of a matrix is used;
  • the correctness of eigenvalues of a matrix can be checked by using the fact that the sum of eigenvalues is the trace of the matrix, and the product is its determinant, etc.

This retrospective reassessment of previous material can give students a chance to see how actually simple it is — and boost their mathematical confidence.

In my talk, I’ll discuss how to incorporate error-correcting aspects of mathematics into course design.

My answer to a question in Quora:

What are some common gaps in logic that students make in mathematical proofs that lead to inaccurate results?

I wish to comment on two specific flaws exhibited by students who encounter proofs first time in their lives.

The first one is

inability to accept the Identity Principle: “\(A\) is \(A\)”, and arguments related to it, as a valid ingredient of proofs.

For many students, a basic observation

For all sets \(A\), \(A \subseteq A\) (\(A\) is a subset of \(A\)) because every element of \(A\) is an element of \(A\)

is very hard to grasp because of the appearance of the same words about  the same set \(A\) twice in the sentence: “element of  \(A\)  is an element of  \(A\)”.  I have observed that many times and I think that students cannot overcome a mental block created by their

expectation that a proof should yield some new information about objects involved

– and this is the second fundamental flaw.

And, of course, reduction, removal of unnecessary information, is seen by many students as something deeply unnatural.

Every year, I hear from my Year 1 students the same objection:

How can we claim that 2 is less or equal than 3, that is, \( 2\leqslant 3\), if we already know that 2 is less than 3, \(2 < 3\)?

I think we encounter here a serious methodological (and perhaps philosophical) issue which I have never seen explicitly formulated in the literature on mathematics education:

a proof of a mathematical statement can illuminate and explain this statement, it may contain new knowledge about mathematics which goes far beyond the statement proved; but

  • elementary steps in proofs frequently do not produce any new information, moreover, sometimes they remove unnecessary information from consideration.

A proof can be compared with a living organism built from molecules which can hardly be seen as living entities — and even worse, from atoms which are definitely not living objects.

This is closely related to another issue which many students find difficult to grasp: statements of propositional logic have no meaning, they have only logical values (or truth values, as they are frequently called) TRUE or FALSE. Any two true statements are logically equivalent to each other because they are both true; moreover, the statement

if London is a capital of England then tea is ready

makes perfect sense, and can be true or false, even if constitution of the country has no relation to the physical state of my teapot.

When my students express their unhappiness about logic which ignores meaning (and I provoke them to express their emotions), I provide an eye-opening analogy: numbers also have no meaning. The statement

The Jupiter has more moons than I have children

compares two numbers, and this arithmetic statement makes perfect sense (and is true) even if Jupiter has no, and cannot have any, connections whatsoever with my family life. Numbers have no meaning; they have only numerical values. Arithmetic, the most ordinary, junior school, sort of arithmetic is already a huge and deep abstraction. We did not notice that because we are conditioned that way.

Learning proofs also involves some degree of cultural conditioning. As a side remark, I suspect (but have no firm evidence) that the role of family — presence of clear rational argumentation in everyday conversations within family — could be important.

Posted by: Alexandre Borovik | September 7, 2018

Mathematics for teachers of mathematics

My new paper at The De Morgan Gazette:

A. Borovik, Mathematics for teachers of mathematics, The De Morgan Gazette 10 no. 2 (2018), 11-25.


The paper contains a sketch of a BSc Hons degree programme Mathematics (for Mathematics Education). It can be seen as a comment on Gardiner (2018) where he suggests that the current dire state of mathematics education in England cannot be improved without an improved structure for the preparation and training of mathematics teachers:

Effective preparation and training requires a limited number of national institutional units, linked as part of a national effort, and subject to central guidance. For recruitment and provision to be efficient and effective, each unit should deal with a significant number of students in each area of specialism (say 20–100). In most systems the initial period of preparation tends to be either

  •  a “degree programme” of 4–5 years (e.g. for primary teachers), with substantial subject-specific elements, or
  • an initial specialist, subject-based degree (of 3+ years), followed by (usually 2 years) of pedagogical and didactical training, with some school experience.

This paper suggests possible content, and didactic principles, of

a new kind of “initial specialist, subject-based degree” designed for intending teachers.

This text is only a proof of concept; most details are omitted; those that are given demonstrate, I hope, that a new degree would provide a fresh and vibrant approach to education of future teachers of mathematics.

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