What textbooks had the biggest impact on you as a mathematician?

My answer to a question in Quora: What textbooks had the biggest impact on you as a mathematician?

Of course, books which I read as a schoolboy, in the last 3 years of secondary school (Years 8–10 in the education system of my home country, that is, I was 14 to 16 years old). The list is bizarre and reflects a rather steep learning curve which was quite common in my generation.

Year 8

  • Vasilyev and Gutenmacher, Lines and Curves.
  • Gelfand, Glagoleva, Schnol, Functions and Graphs.
  • Kirillov, Limits.

(These three small books were among little cute booklets which were mailed to me from Mathematics Correspondence School; true masterpieces of mathematical didactic.)

  • Courant and Robbins, What is Mathematics? (Russian translation).
  • R. Hartshorn, Foundations of Projective Geometry (Russian translation). A little gem of abstraction.

Year 9

  • O. Yu. Schmidt, Abstract Group Theory (1933, republished with minor changes from its first edition of 1916).
  • R.Baire, Lecons sur les Fonctions Discontinues (1932, Russian translation of the French original of 1904). This was perhaps the first book of descriptive set theory. After reading it, I had never had trouble with functions of real variables.

Only later I realised that Schmidt and Baire helped me to put mathematics into a historic perspective and see it as something dynamic, developing.

Years 9 and 10

  • Birkhoff and MacLane, A Survey of Modern Algebra (in English)
  • Gorenstein, Finite Groups (in English) (a hardcore research monograph, it was the Bible of finite group theory for a couple of decades)

From Gorenstein on, my reading switched to research mode, highly selective, with tight focus on key ideas and arguments.

I was privileged to meet, in later years, Gelfand, Gorenstein, and Hartshorn in person and thank them for their books. Gelfand looked at me with suspicion and asked: ”And what have you learnt from “Functions and Graphs?” My answer: “The general principle: always look at the simplest possible example” delighted him. He immediately started to explain that this was his most important contribution to pedagogy of mathematics, and that in his famous Seminar he was always able to find a example simpler than the one given by the speaker.

The question about textbooks was asked to me by Michael Kheifetz; he also asked me:

What’s the most amazing conversation you have had about mathematics?

I can answer it now: quite a number of conversations with Israel Moiseevich Gelfand that followed this first encounter.

An interested reader can find comparative analysis of Schmidt and Gorenstein in my paper Mathematics discovered, invented, and inherited (Section 3).


What should we do to stop calculators hindering children’s math learning?

My answer to a question in Quora: What should we do to stop calculators hindering children’s math learning?

It is the best kept secret about mathematics: mathematics is done by the subconscious. Learning mathematics means learning to communicate with, engage one’s subconscious. I wrote about that in my answer to As a mathematician, how would you mentor your child and help her to learn, do and live mathematics in her free time as she is growing up?

Calculators do not help child to engage his/her subconscious. There is a class arithmetic problems that help: these are word problems. An easy example:

A girl said: “I have three more brothers than sisters”. How many more boys are there in her family than girls?

Is calculator needed? No. Is thinking needed? Yes.

If you think that word problems are so-o 20th century, let us add a twist more suitable for the 21st century. Why not give children a toy programming language, say Scratch, and ask them to produce their answers as a code in Scratch which solves all problems of that type.

For an example, let us consider a simpler problem:

A girl said: “I have two sisters”. How many girls are in the family?

On the screen, it will be child’s favorite sprite (a term from
Scratch , a programmable object, usually chosen to be some cute animated object, say, a cat, or dog – very easy to control and manipulate) who starts data input dialogue by asking

  • Are you a girl or a boy?

and then:

  • How many sisters do you have?

And returns: “aha, you have [output] girls in your family!”

Please notice – what we have here is (a) creating a model of a real life situation, and (b) implementing it as a computer algorithm. And it is doable in Scratch. And also notice: at no stage calculators are needed.


What do pure mathematicians do?

My answer to a question in Quora: What do pure mathematicians do?

I have already answered this question elsewhere: what do I do as a pure mathematician? I study symmetries. More precisely, I study multi-mirror symmetries, where mirrors get reflected in other mirrors, and reflections breed and multiply all over the place. I design ways to find safe paths in the maze of mirrors, and to tell real objects from sneering phantoms.

If you happen to be involved in a shoot-out in a Hall of Mirrors — like in the famous scene from Orson Wells’ The Lady from Shanghai, see a YouTube clip:

— I can provide a consultancy, at UCU recommended professorial rates.

This clip also gives an answer to a question I heard from a graduate student: why all calculations in simple algebraic groups end up in manipulations with root systems? Because a root system is just a convenient way to write down on paper the skeleton mirror system of the group, and calculation with roots is the final shoot-out at the end of the theorem.


Sir Michael Atiyah: Obituaries

Sir Michael Ariyah: One of the greatest British mathematicians since Isaac Newton, by Ian Stewart; The Guardian, 15 January 2019.

Sir Michael Atiya (1929 – 2019), by Nigel Hitchin; IMU-Net 93: January 2019, A Bimonthly Email Newsletter from the International Mathematical Union.

Sir Michael Atiyah died in Edinburgh, aged 89, on January 11th 2019. He was one of the giants of mathematics whose work influenced an enormous range of subjects. His most notable achievement, with Isadore Singer, is the Index Theorem which occupied him for over 20 years, generating results in topology, geometry and number theory using the analysis of elliptic differential operators. Then, in mid-life, he learned that theoretical physicists also needed the theorem and this opened the door to an interaction between the two disciplines which he pursued energetically till the end of his life. It led him not only to mathematical results on the Yang-Mills equations that the physicists needed but also to encouraging the importation of concepts from quantum field theory into pure mathematics.

Born of a Lebanese father and a Scottish mother, his early years were spent in English schools in the Middle East. He then followed the natural course for a budding mathematician in that environment by going to Cambridge where he ended up writing his thesis under William Hodge and becoming a Fellow at Trinity College where he started to pursue his research. But, attending the ICM in Amsterdam in 1954, his eyes were opened to the exciting work that was going on in the outside world and the opportunity then arose to spend a year at the Institute for Advanced Study in Princeton where he met his future collaborators and close friends Raoul Bott, Fritz Hirzebruch and Singer. The benefits of international collaboration which he valued so highly were made concrete when in 1957 Hirzebruch established in Bonn the annual Arbeitstagung where Michael was always the first speaker. In those years he and Hirzebruch developed topological K-theory, which subsequently became the natural vehicle for the index theorem.

A visit by Singer to Oxford in 1962 (where Atiyah had recently moved) began the actual work on the Index Theorem, which ultimately led to a Fields Medal in 1966 and, with Singer, the Abel Prize in 2004. Another visit in 1977 brought mathematical questions concerning gauge theory. Using quite sophisticated algebraic geometry and the novel work of Roger Penrose this yielded a precise
answer to the physicists’ questions: the so-called ADHM construction of instantons. The fact that mathematicians and physicists had common ground in a completely new context made a huge impression on Michael and he was energetic in the following years in facilitating this cooperation.

With a naturally effervescent personality he possessed, in Singer’s words, “speed, depth, power and energy”. His strong voice could be heard across many a departmental common room explaining some crucial point. Collaborations were all-important, bouncing ideas around, two or three people in front of the blackboard, exploring ideas, erasing them, sudden insights. This also held for his students — he needed continuous feedback and challenges. He had a natural talent for lecturing: leaving the lecture theatre you always had the feeling you had understood things, though trying to reproduce them later was a different matter. Beauty in mathematics was a feature he took seriously. It was in evidence in so many of his ideas and proofs and in his later years he actually instigated a neurological experiment to detect its presence.

Sir Michael received numerous awards and honours. He worked for the mathematical community in many ways. In particular, he was instrumental in founding the Isaac Newton Institute (where he insisted that it should be for the Mathematical Sciences) and the European Mathematical Society. He was also President of the Royal Society of London where he found himself in a situation where he could voice long-held views about science in general. He contributed to the IMU itself in many ways, including two terms on the Executive Committee. He will be greatly missed by all.

Nigel Hitchin (Oxford, UK)

The Polar Star and the Life Endgame (elegy for a departed friend), by Matilde Marcolli.


A sampler of mathJax

I am pleased that this random formula renders correctly and with high typographic quality in a default browser of my smartphone:

\[ x = \frac{2^{2n+1}}{\sqrt{7}+1} + \frac{\sqrt[3]{1+\sqrt[5]{1234567}}}{e^{x^2+1}+1}. \]


Anything goes

In absence of clear socio-economic criteria for mathematics education policy (I wrote about that in my papers  Calling a spade a spade: Mathematics in the new pattern of division of labour and  Mathematics for makers and mathematics for users), the mathematics education theory is open to the most bizarre proposals. This one beats everything.

Patricia Morgan and Dor Abrahamson, Applying Contemplative Practices to the Educational Design of Mathematics Content: Report from a Pioneering Workshop. The Journal of Contemplative Inquiry, 5 no. 1, 1-13 (2018).


Researchers in the field of mathematics education are beginning to appreciate the potential of contemplative practices such as mindfulness to alleviate students’ stress and increase their focus. What researchers do not yet know is whether, and if so how, bringing focused attention to somatic experience through a wide variety of contemplative–somatic practices (i.e., yoga, Feldenkrais, body–mind centering, and attending to bodily sensations in meditation) may support student learning of specific mathematical content. As a first step toward conceptualizing and ideating the pedagogical design and facilitation of content-oriented contemplative exercises, we convened a workshop to explore these ideas. Here we report on findings from this pioneering workshop, which brought together international scholars and practitioners interested in the relations between contemplative–somatic practice and mathematical reasoning and learning. This report elaborates on participants’ experiences and derived pedagogical insights to offer the field new horizons in the development of the theory and practice of contemplative mathematics.

At a practical level, I wonder what if bringing students’ “focused attention to somatic experience” will result in their responses “I am sick of it”.


What makes a proof beautiful?

My answer to a question on Quora: What makes a proof beautiful?

It is like the beauty of a human being, or a flower, or work of art: it is next to impossible to give a definition of any degree of precision, but we can make a list of attributes which make an object (a human being, or a flower) more beautiful than other objects of the same class.

Timothy Gowers, one of the leading mathematicians of our time, once said in one of his talks:

The following informal concepts of mathematical practice cry out to be explicated: beautiful, natural, deep, trivial, “right”, difficult, genuinely, explanatory.

Many of these informal (but instantly recognisable for every mathematician) words apply to proofs and may contribute to characterisation of a proof as “beautiful”:

natural, deep, “right”, explanatory, elegant, revealing, self-contained, concise, streamlined, well-structured, clear, unexpected, surprising, constructive, clean …

— this list (utterly random) can be easily continued.

For a human to be beautiful, it is expected that he/she has “clear eyes” (which is actually a sign of good health). The same with “clear” proofs: this is a good sign that the proof is in some sense “healthy”. On the other hand, some proofs (especially in their draft form) could be “suspicious” and “unreliable” — and even “fishy”; frequently it is lethal, and means that, under closer consideration, the proof is likely to collapse. And “clear” proof is not the same as “clean” proof —I feel the difference, but it will take some time for me to make it explicit even for myself.

In short, a book can be written on this topic — but, to the best of my knowledge, has never been.


SATs – are you passing the test?

Reposted from SATs – are you passing the test?on Let Our Kids Be Kids.

[…] Right now the education accountability system is a mess. Kids are suffering as a result and teachers are despairing at how they can protect children from a flawed standardised testing system.  Are you doing the easy thing or the hard thing to solve that problem?

If you are a primary headteacher we understand you have difficult ethical choices to make.  The hard thing right now is to say no to Baseline pilots taking place in your school – testing 4 year olds is nonsense and everyone knows it… the data will be unreliable and feed a system which is deeply flawed.  Every year you face the moral issues regarding SATs in both Year 2 and in Year 6 – the easy thing is to continue to feed the data machine, the hard thing is to work with other heads to try and fight the broken system that does so much harm.

If you are a parent watching your child prepare for SATs the hard thing right now is to ask difficult questions and challenge your school, your LEA, your MP – the right thing is to be part of the change you want to see by asking those difficult questions and challenging the system you know is failing your child.  The easy thing is to hope someone else will fight on your child’s behalf – as parents we are advocates for our children, we can not take the easy route.

If you are a secondary headteacher and can see how Progress 8 is damaged by over-inflated primary SAT scores, setting your children and your teachers up for failure, the hard thing right now is to work with primary colleagues towards a better solution. The right thing is to fight against SATs to protect yet more children entering a damaged system.

The easy thing is to think this is someone else’s problem, to maintain the status quo and allow this mess to continue.  The hard thing is the right thing, the thing that helps us sleep at night knowing we are part of the solution and not the problem.  If we all try to do the right thing, we can sort this mess out together. Ask yourself… are you passing the test?