03/20/23

Mathematicians wish Azat a Happy Birthday

Azat Miftakhov celebrates his 30th birthday, the fifth in prison, on March 21. On this occasion, many mathematicians from all over the world sent him their best wishes and expressed their solidarity.

Warmest birthday greetings Azat; may you have success in all things

Professor Barry Mazur, Harvard

02/27/23

A renewed call to the international mathematical community for the support of Azat Miftakhov, who is facing a likely new criminal prosecution of fabricated charges

Reposted from caseazatmiftakhov

The Azat Miftakhov Committee renews its appeal to the international mathematical and scientific community for the support of Azat Miftakhov. This call is made more urgent by the recently confirmed information that the FSB is currently in the process of fabricating a new criminal case against Azat, which may result in an additional lengthy prison sentence for him.

Azat Miftakhov, an opposition pro-democracy activist in Russia, was originally arrested by the FSB on February 1, 2019 in Moscow. Except for a brief few hours release on February 7, 2019, he has remained imprisoned since then. At the time of his arrest Azat Miftakhov was a mathematics graduate student in the Faculty of Mathematics and Mechanics of the Moscow State University. Azat was charged in relation to the January 2018 political protest at an office of the United Russia party in Moscow that came to be known as the “broken window” case. While in pretrial detention, Azat was extensively tortured by the guards in the attempt to extract a false confession. The premier Russian human rights organization “Memorial” recognized Azat as a political prisoner, and his case attracted widespread attention, both in Russia and internationally. Azat has always maintained his innocence of the false charges against him.

From the time of his February 2019 arrest, numerous mathematicians, scholars, and mathematical societies have expressed support for Azat and called for his freedom.

After multiple delays, Azat’s court trial took place in Moscow in the Fall of 2020. The trial was marked by numerous human rights violations, including the prosecution’s use of “secret” government witnesses and of testimony of a prosecution witness who died in 2020 while in prison. In a grotesque abuse of justice, on January 18, 2021 the court in Moscow found Azat guilty of “hooliganism” and sentenced him to six years imprisonment in a penal labor colony. Azat is currently serving that sentence at a colony in Omutninsk.

While imprisoned, Azat became a key symbol of the struggle against the brutal oppression of Putin’s regime and for democracy and freedom in Russia. Azat’s bravery and steadfast determination, in the face of torture, intimidation and unjust imprisonment on false charges, continue to inspire many. That role became even more vital after Putin unleashed a bloody fratricidal war on Ukraine in February 2022. Since the start of the war, the Russian government’s suppression of any type of dissent and opposition in the country has grown ever more brutal and complete. People like Azat, who continue to stand up to this oppression, provide hope and inspiration to all those fighting for a free, peaceful and democratic Russia.

For that reason the Russian authorities continue to view Azat as a threat. Azat is currently scheduled to be released on September 23, 2023. However, as we recently reported, it appeared that the FSB was preparing to fabricate a new criminal case against Azat. Unfortunately, this information has now been confirmed, according to multiple news reports that directly quote the FSB itself. Even though Azat has been imprisoned for over four years, the FSB is apparently planning to accuse him of being a member of the so-called “Network” anti-government group, an umbrella charge that the Russian government has been using to persecute numerous opposition activists. Reports indicate that the FSB has extracted coerced false testimony against Azat from at least one other prisoner whom they were interrogating at the infamous Lefortovo prison in Moscow. If convicted on these new fabricated charges, Azat would face a much longer additional prison sentence, likely in even harsher conditions. According to Azat’s family, Azat is aware of these developments and is preparing to face them.

Azat Miftakhov is our mathematical colleague and a courageous and admirable young man. Even while being imprisoned, he continued conducting mathematical research and published a new mathematical research article in 2021. Azat’s conviction in January 2021 on false charges represented a blatant case of political persecution and abuse of justice by the Russian authorities. Any additional prosecution of Azat would be even more outrageous and intolerable. At this fraught and difficult moment, we call on the international mathematical and scientific community to renew their support for Azat Miftakhov. Azat needs and deserves our help and solidarity.

We continue to demand Azat’s immediate and unconditional release.

The Azat Miftakhov Committee

11/12/22

Are mathematicians gifted people?

My answer on Quora: Are mathematicians gifted people?

I do not know for what my sins Quora bombards me with questions about giftedness, IQ, etc. For several years I tried to ignore them, but finally I realised that I have to formulate my position.

Yes, professional mathematicians possess some mental traits and skills which majority of population do not have. But these traits are not what is called “gift”, “talent”, “ability” in the mass culture; they remained unnoticed, unregistered in the public discourse about mathematics and mathematics education. However, my mathematician friends, when we discuss this topic, know what I am talking about.

IQ is mostly irrelevant to discussion of mathematical “ability”; specific traits of mathematical thinking belong to a much higher cognitive level than skills tested in IQ tests.

A simple example: I had seen once how an eight years old boy was solving some standard puzzle (not of IQ type), with some pattern of hexagons which had to be filled with integers from 0 to 9 so that certain sums were equal — you perhaps had seen this boring stuff . At some point he paused and commented: “Hmm, I have to somehow move information from this corner to that corner”. Moreover, after some thought he had successfully moved the information. This was meta-thinking, ability to reflect on one’s thinking, ability to look at the problem from above. This boy now is quite a successful student in one of the best university mathematics departments in the world, in a pipeline to becoming a professional research mathematician.

Perhaps you have heard this definition:

“Mathematics is the science of patterns”.

IQ tests pay much attention to the speed of pattern recognition. It is a useful skill, but it is not a sign of mathematical abilities. In my life, I had a chance to see a lot of children and teenagers who had an instinct (or maybe it was a trait absorbed in the family?), to look deeper and try to detect the structure behind the pattern — and the boy mentioned above was one of them. Indeed, the simplest description of mathematics is

“Mathematics is the science of structures behind patterns”.

Perhaps my personal experience is outdatet, but I was privileged to go through a viciously academically selective system of mathematics education — see my paper “Free Maths Schools”: some international parallels. Aged 14, at a Summer School which was the final step of selection to the specialist boarding school described in the paper, I and my friends were subjected to a battery of IQ tests — which, however, had no relation to admission to the school.

We were tested by professional experimental psychologists who were commissioned by the Soviet Army to study and assess reliability of the IQ tests used by the US Army for assignment of conscripts to particular duties (you see how long ago it was). The psychologists translated real American IQ tests into Russian and tried them on various groups of population. They were excited to discover that our performance refuted a claim that apparently was universally accepted at that time: that practicing IT tests could not improve results.

Indeed our results were quickly improving beyond applicability of tables for conversion of counts of correct answers into IQ scores. Why? Because we did not practice IQ tests — we had access only to tests which we have already taken — but, after every test, we spent hours classifying test questions, analysing them, inventing our own questions and challenging each other to solve them, and we did that in a collective discussion, in brain storming sessions, attacking problems like a pack of enthusiastic young wolves. Perhaps we had already had some specific habits of mathematicians; but there was nothing special about that, even some 8 year old kids might have them, as I have already said.

As I explain in my paper that I mentioned above, in the selection process for my mathematics boarding school, and in the school itself, the use of words gifted, talented, able was explicitly forbidden — they were seen as misleading and divisive.

I am a staunch believer that majority (maybe even all) pre-school kids have strong potential for understanding and mastering mathematics. Unfortunately, their mathematical traits are systematically suppressed in the mainstream school mathematics education — mostly because many teacher have no idea what it is about.

You may wish to take a look at my papers, they say more:

A. V. Borovik, Mathematics for makers and mathematics for users, in Humanizing Mathematics and its Philosophy: Essays Celebrating the 90th Birthday of Reuben Hersh (B. Sriraman ed.), Birkhauser, 2017, pp. 309–327. bit.ly/2qYHtst

A. V . Borovik and A. D. Gardiner, Mathematical abilities and mathematical skills, The De Morgan Journal 2 no. 2 (2012) 75-86. bit.ly/2jTYy4r

11/3/22

Geoffrey Howson died on 1 November 2022, aged 91

Tony Gardiner writes:

The mathematical “house” is fortunate in having “many mansions”, inhabited by a remarkable variety of workers. By any account Geoffrey Howson – who died on 1 November 2022, aged 91 – was a significant player throughout much of the period 1950-2000. However, like so many other workers, he operated effectively, but quietly, so may not have been noticed. Nevertheless his life offers interesting insights into how UK society has changed since 1931 (when he was born as the seventh in a family of seven children), and into how UK mathematics and mathematics education worldwide have evolved since the 1950s.

Geoffrey belonged to the generation, who emerged in significant numbers (perhaps for the first time) in the 1940s. Their families had never been to secondary school – let alone university. Yet – thanks to structural changes and committed teachers – they somehow emerged in small numbers at age 18, ready to take on whatever challenges the post-war world might present.

Geoffrey always remained faithful to his roots (a solid Yorkshireman, from a deprived, but proud, mining community). Yet he came to excel in mathematics, in university politics, and in international mathematics education – as well as in the world of opera, Bauhaus design and embroidery, and medieval church architecture.

Geoffrey went to Castleford Grammar School (founded 1906), and was probably the first from that school to study mathematics. He went on to Max Newman’s department in Manchester, where his teachers included: Max, Walter Ledermann, J.W.S. Cassels, Bernhard Neumann, Graham Higman, Kurt Mahler, Arthur Stone, James Lighthill (MA President 1970), M.B. Glauert, Charles Illingworth, and Bernard Lovell. He was Graham Higman’s second PhD student (proving that the intersection of two finitely generated subgroups in a free group is finitely generated). He also attended Turing’s lectures on morphology – interrupted only by Turing’s death.

Invitations from Reinhold Baer (Illinois) and Saunders Maclane (Chicago) were put aside in order to complete National Service (when he taught RAF trainees about guided missiles). He then moved to the Royal Naval College in Greenwich in 1957 (where he taught the new generation of future naval commanders about similar things).

In 1962 he went to Southampton to manage the School Mathematics Project (SMP). This was the UK equivalent of “new math”, but much more humane and less abstract. At its height SMP materials were “used” (in some sense) in 60% or so of UK secondary schools. But SMP remained a Teachers’ Cooperative, with no government support. Geoffrey’s job was officially to edit and to manage the program of new textbooks. In practice, he had to coordinate the writing (planned and completed by a remarkable group of full-time teachers); the production of draft materials; the revision process; and to deal with the publishers and the exam boards (since no project could survive if there was not a corresponding tailored public examination at age 16 and 18).

Geoffrey became a representative spokesperson for “modern maths” developments in the UK, and so came to interact with those similarly placed in other countries – in both East and West – producing many reports, and editing collections published in the 1960s, 70s, and 80s. He published and edited a huge variety of books and papers – all written in a thoughtful style. His goal was to inform and enlighten, rather than to engage in “theoretical research”. He became a leader in Mathematics Education internationally, but was never really appreciated by the new breed of “Maths Education” researchers. His contributions were mostly pragmatic comparisons, surveys, and analyses, designed to inform and to allow improved judgements to be made. He was also very active in supporting teachers’ colleges and those working in polytechnics.

He helped to salvage ICMI/ICME after it came unstuck around 1980. And it is a mark of the man that he managed this (with Jean-Pierre Kahane) while remaining great friends with those who had been part of the previous regime. He was recognised in other countries but not much within the UK.

He was Head of Department and Dean 1990-92 and may have helped in building up parts of what is now a very strong mathematics department. He also Chaired the LMS/IMA/RSS committee that produced the report “Tackling the mathematics problem”: this was a rare instance of the three scholarly societies acting together on a matter of mutual concern, and then having a significant impact on subsequent policy-making.

They don’t make them like that any more.

10/11/22

Which are the procedures you are following to memorize or learn so fast?

My answer to a question on Quora:  Which are the procedures you are following to memorize or learn so fast?

One my favorite procedure is called “understanding”. Prior to memorising, try to understand. Very frequently, the need to memorise disappears – you simply understand.

Another procedure that I try to popularise is called “interiorisation”: making something external a part of yourself.

I remember my conversation with my grandson, at that time 5 years old. I asked him: “What’s new at your new school”. He: “I do not like the school. Too many rules. Hard to remember.” I: “Why should you remember the rules? You simply have to follow them. I’ve seen in the lunch room at your school a poster:”Do not mess with food”. What is so difficult about it?”

My grandson fell in deep thought. We walked along a street on our way to his home, in silence. I didn’t distract him. It should be a law of the land: never distract young children when they think.

10/1/22

Do the children of math teachers always pass algebra?

My answer on Quora: Do the children of math teachers always pass algebra?

My passed.

I do not wish to generalise, and I do not know statistics on this specific issue, but I see some reasons for children of mathematically educated people be a bit more confident in their mathematical studies at school. For example, they have never seen emotions of fear, or dislike of, mathematics in their parents. It is not about genetics, it is about inheriting certain social / cultural capital: values, habits, motivation. I intentionally avoid the word “intellect”, this is not about intellect either.

10/1/22

How does category theory relate to other branches of mathematics?

My answer on Quora: How does category theory relate to other branches of mathematics?

An excellent question. Category theory is important on its own, and has important applications in a number of other mathematical theories; however, the crucial and the most fundamental impact of category theory is invisible and under-reported, it is of cultural nature. It can be compared with the influence of set theory: 99% of mathematicians use only a modicum of naive set theory, ignoring deeply penetrating and frequently very hard results of set theory as the live research discipline which continues to develop and flourish.

There is a telling example: the theory of games of chance was created in the 17th century and gave birth to probability theory; the latter was already quite developed by the time when, in the 20th century, the concept of a deterministic game had finally crystallized – in a paper, of all people, by Zermelo, who proved that chess was a deterministic game: for one of the players, there is a strategy, that is, a function from the set of permissible position to the set of moves, which achieves at least a draw. His paper was published in 1913 (see Zermelo’s theorem (game theory) – Wikipedia). Why did this happen so late? The word “set” in the definition of a strategy came to use only in the second half of the 19th century.

The same is happening with category theory: the vast majority of mathematicians use its ideas and terminology in a very rudimentary and naive form, frequently even without realisation that they are doing so. In the work that I am doing, it had happened to be very important to remember that an algebraic group was a functor from the category of unital commutative rings to the category of groups. Some my colleagues who work in the same theory continue to insist that a group is a fixed set with some operations on it. When my co-author and I recently solved a certain problem which was open since 1999, we were able to do that only because for us a group in question was a functor – not much deeper than that. Why it was not solved by someone else earlier? Because for them a group was just a set.

09/27/22

Is it natural for a mathematician to forget specifics of some parts of math?

My answer to a question on Quora:  Is it natural for a mathematician to forget specifics of some parts of math?

I think it is natural. I love to surprise my students by saying that I hardly remember any trigonometric formula beyond $$\sin^2x+\cos^2x=1$$– but I can deduce most standard trigonometric formulae on the spot. On a number of occasions I offered my students a game: give me a wrong formula, say, for $$\sin(x+y)$$ — and I will instantly explain you why it is wrong. I have always won. I am not special; I believe that the vast majority of my mathematician colleagues can recover statements of L’Hopital theorems even if they do not remember them exactly.

And there is one more aspect of mathematical memory. The great mathematician Andrew Wiles said in a recent interview Andrew Wiles: what does it feel like to do maths?:

I really think it’s bad to have too good a memory if you want to be a mathematician. You need a slightly bad memory because you need to forget the way you approached [a problem] the previous time because it’s a bit like evolution, DNA. You need to make a little mistake in the way you did it before so that you do something slightly different and then that’s what actually enables you to get round [the problem].

So if you remembered all the failed attempts before, you wouldn’t try them again. But because I have a slightly bad memory I’ll probably try essentially the same thing again and then I realise I was just missing this one little thing I needed to do.

09/22/22

Is it normal that one doesn’t understand a math procedure unless demonstrated by a teacher?

Today I attended a seminar of mathematics education experts. They are seeing inability of many students to adequately understand written/printed texts as a serious and growing problem facing school mathematics education. Mathematical texts are difficult. What are other (that is, non-mathematical) difficult texts that you have read in your life? For example, have you ever tried to read fine print in an insurance policy? By complexity, this could easily beat maths textbooks. Ability to read and understand a difficult text is a useful skill — a skill for life.

09/20/22

Why do I need to learn conics in math?

My answer to a question on Quora:  Why do I need to learn conics in math?

This is the second best kept secret of mathematics education: it does not matter what students are taught, what matters is how deep their learning, and how efficient is the network of connections between mathematical facts that grows in their minds in their minds: in how many steps they can get form Fact A to Fact B? Conics could be included in high school courses of mathematics, and could be omitted, it does not really matter. What matters is whether students develop specific mental skills of mathematical thinking.

In one of the best mathematics high schools in the world, conics are used as a training ground of mathematics problem solving, that is, solving problems not seen by students ever before — simply because the theory of conics, if taken seriously, is rich, and because it is really hard to find in the literature good books on advanced level but elementary theory of conics. Students, I was told, really have to work from scratch.

I can offer you one problem; I think it was used at one of the International Mathematical Olympiads of yesteryear.

Assume you are given a sheet of paper with a parabola printed on it. Using only straightedge and compasses, construct the Cartesian coordinate system in which this parabola has equation y=x^2.

People who can solve this problem have reasonably deep understanding of conics.