I am reading the fascinating book Secrets of the First School by Tendai Huchu (available on Amazon), and have found remarkable words  on p. 161:

“What separates dark magic from normal magic is the practitioner’s intent. A Promethean fire spell can be used to warm the home on a cold night, or it can be used for arson. It’s the same magic, but with a different use.”

Mathematics is a kind of magic. Their difference is in the scale. Promethean fire spells of mathematics  can  be highly beneficial to the entire humankind, but also can destroy human civilisation as we know it. Practitioners of mathematics are sometimes facing serious moral choices  – this is discussed in  my paper  The Tool/Weapon Duality of Mathematics   that just appeared online at https://scholarship.claremont.edu/jhm/vol16/iss1/24.

My paper appeared online:

Alexandre Borovik, “The Tool/Weapon Duality of Mathematics,” Journal of Humanistic Mathematics, Volume 16 Issue 1 (January 2026), pages 365-392. Available at: https://scholarship.claremont.edu/jhm/vol16/iss1/24

Hopefully, it is sufficiently controversial. It raises issues which really deserve discussion. And some of them may equally apply to physics and computer science.

If you can, please spread the word and help distribute the link (or the paper itself) wider. The paper is covered by the Creative Commons Attribution licence CC BY 4.0.

I can add a comment that I have already received from Yuri Bazlov:

Thank you! Personally, I liked the conspicuous juxtaposition of the Isaiah and Joel quotes (exactly in the middle of the paper). Isaiah is overcited and honoured by the sculpture at the UN in New York. Yet Joel paints a picture that is no less valid, and resonates much more with the current reality.

And, by the way, waging war per se is not necessarily frowned upon by the Tanakh (Old Testament). Bad are spilling of innocent blood, sexual violence, extreme cruelty, unjustified plunder, and moving people out of their land to sell into captivity. This roughly correlates with the more modern concept of “crimes against humanity” – I would call them crimes against our civilisation.

I challenge you to give an example where mathematics was/is used not just for war but for such crimes.

Yet our civilisation – Western? Judeo-Christian? – is not defined just by the crimes against it. Commandment number 1 in the Old Testament is “Be fruitful and multiply”, which to me is a view that human creativity (a reflection of divine creativity) is not a process of redistributing a fixed stock, but of generating wealth. (The Bible uses the term “blessing” to avoid a narrow, materialistic view of wealth. Blessing is transmissible without depletion.)

In this sense, there is no duality to mathematics: essentially, it has always been a creative process, and so a means of generating wealth. The examples that you give do not at all convince the reader that mathematics can be used in some kind of zero-sum or negative-sum game: the long-term effect is always positive.

So, “mathematics is the one where people engaged have the highest degree of freedom…” reflects the creative essence of mathematics. But you seem to contradict this with “Mathematics, as we know it, was born as a weapon of subjugation and tyrannic control” – I doubt that that truly reflects the origin of mathematics. Perhaps one could say that human creativity, and in particular mathematics, found a way to flourish despite tyranny.

The way you showcase the views of James C. Scott does lend your paper an aura of (mild) controversy – his views are hardly mainstream. But then, all history and all ethics is controversial, unless merely an exercise in conformism.

If I need, or am asked, to do an “ethics review” of some research or a grant application, I will surely cite your paper!

Also, I discovered a website of The Deutsche Forschungsgemeinschaft a document The dual use problem in mathematics:

In many cases, basic research in both theoretical and applied mathematics has the potential to be significant to security-related research as well (possibly not until later). Just as a knife as a cutting tool can always be used as a weapon too, mathematical methods investigated in the field of control theory, for example, can often be used for both peaceful and military purposes. The same applies in the field of number theory with regard to its application in the fields of cryptography and coding theory. The answer to this cannot be a general ban on basic mathematical research of course.

However, aspects and issues regarding security-related research should be addressed in accordance with the applicable rules and laws when there is evidence of the potential concrete applicability of mathematical research in security-related areas. Indications might include the field of application targeted by the researchers themselves, or else their own previous research background or that of cooperation partners. In the case of cooperative projects, one point that should always be taken into consideration and indeed critically questioned is whether the participating institutions or their sponsors or funders aim to achieve or even require the utilisation of research results in a security-related context.

A further look at the Internet shows that many states see that a  serious security problem.

I do, on the basis of personal experience in mathematics.

I  retired after 50 years of teaching in universities in 4 different countries with different education systems and pedagogical traditions. This experience shaped my views on our profession. I firmly believe that

1. Teaching is not a science, it is an art, and should be treated as such.
2. Students are not customers (“persons who buy”) – they are clients (“persons who seek the advice of a professional man or woman”).
3. “Good learning experience” means mastering something new and advanced. To help his/her students, a university teacher has to be able to transform and restructure highly complex material from his/her subject area into a form suitable and accessible to the learners.
4. This cannot be achieved without teachers being experts in their disciplines.
5. Successful and inspirational teaching is a highly individual skill. The choice of teaching methods should reflect not only specifics of the target audience, but also the experience, teaching philosophy and individual psychophysiological characteristics of the teacher.
6. Structuring of the learning environment, choice of teaching and assessment methods have to be subject specific.
7. Values, standards, criteria of assessment in learning and teaching have to originate in, and be set by, the professional academic communities of their particular subject areas.
8. The role of managers is to create an environment which helps professional standards to be maintained; however, managers should not interfere in setting the standards.

This Manifesto was published in a an old blog. I wish to transfer from there a comment from one of the leaders of the math circle movement. It describes a creative learning environment flourishing in math circles but absent in universities.

At some point, I have compiled a short list of reasons why I get a lot of satisfaction from teaching a math circle. I love:

-the equality and feeling of mutual respect and attention that develops between me and math circle participants
-the democracy/lack of authority that shows us the “right answer”
-seeing the value alignment and deep intellectual friendship that develops among the participants
-sharing children’s excitement when they realize their own powers
-the feeling of freedom they develop when they get rid of their own mental blocks
-the intellectual stimulation of choosing the problems and personalizing and teaching them to a particular audience
-when children realize that they feel happy from doing a challenging job
-observing their self-discovery
-observing as children come up with amazing solutions and counter-intuitive discoveries
-getting a fresh view of the beauty and awesomeness of the world we observe and create – thus multiplying my own happiness

My answer to a question on Quora:

I do not remember seeing a mathematicians who was embarrassed by their conjectures disproved.

Why? Because making conjectures and refuting them is a normal cycle of mathematics. I think 90% of conjectures die on the same writing desk where they were born, being killed by the same mathematicians who formulated them. In mathematics, it is a daily routine. Refutations are as important as proofs. There is a famous book about the role of refutations in mathematics, Imre Lakatos’ Proofs and Refutations.

And the famous Lewis Carroll’s lines in Through the Looking-Glass capture the spirit:

“I can’t believe that!” said Alice.
“Can’t you?” the Queen said in a pitying tone. “Try again: draw a long breath, and shut your eyes.”
Alice laughed. “There’s no use trying,” she said: “one can’t believe impossible things.”
“I daresay you haven’t had much practice,” said the Queen. “When I was your age, I always did it for half-an-hour a day. Why, sometimes I’ve believed as many as six impossible things before breakfast.”

Proofs and refutations co-exist in the most natural way. Mathematical problems are conjectures. To solve a problem means to prove this conjecture or refute it.

Proofs are frequently done by constructing, in parallel, a counter-example: when a mathematician identifies obstacles for a proof, he/she may wish to try to use them to construct a counterexample; when this attempt at refutation encounters its own difficulties, a mathematician may try to isolate these difficulties and understand their nature – for use in the proof. In this zig-zag movement the aims — to prove a conjecture and refute it — alternate. In a happy outcome , the process converges on a definite answer: either proof or refutation.

But, if you look back at that zig-zag prowl in search of a kill, you may say that half of the time the mathematician believed impossible. Even worse, it is like lions in hunt: ten chases result in one kill; a mathematician normally solves about one problem out of ten that he or she tries.

There is one extreme case of the proof/refutation balance: the original proof of the Classification of finite simple groups. I quote Wikipedia:

The proof consists of tens of thousands of pages in several hundred journal articles written by about 100 authors, published mostly between 1955 and 2004.

As a rule, almost each of these “several hundred journal articles” contains a proof of a particular theorem, a special case or an intermediate step of the “global” statement. Since all that is about finite objects, proofs frequently use mathematical induction in a specific form: proof of non-existence of a minimal counterexample to the theorem. As a result, it makes thousands of pages of arguments about non-existent objects. At a first glance, it gives an answer to another question on Quora: What are some aspects of mathematics that are nonsense? But these arguments about eventually non-existent minimal counterexamples are not nonsense — for example, they can be re-used in proving theorems in other branches of group theory.

This text is based on my response to a question on Quora, slightly expanded:

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?

I write from the position of a mathematician about what a mathematician can do for her child.

First of all, a mathematician understands and can use the fact of life non-mathematicians are not aware of:

Mathematics is done by the subconscious.

Encourage in your child, and help her, to develop all kinds of intuition, guesswork (with subsequent checking, whenever possible, of the correctness of the guess). Help her to train her vision of the world, see [mathematical] relations in the world, identify mathematical structures present in the world.

As you can see, this immediately moves the discussion away from school and traditional approaches used in the traditional school education. In fact, this is more of an advice for home-schoolers, especially those who themselves have a sufficient mathematics background. The latter does not mean having a B.Sc. in mathematics; for example, reading economics at Cambridge, UK, suffices.

However, mathematical games briefly described here are not a replacement for a systematic course of mathematics.

But this in-family “protomathematics” could perhaps open to a child a door to mathematical thinking well above the demands of standard school curricula.

What follows are a few random examples, chosen from what I did myself with my (grand)children or had seen my colleagues doing with their (young, pre-school or primary school age) children.

Adult spends some quality time with Child, aged between 3 and 4, in a garden, watching insects and ants, and discussing with Child how the world looks from the ant’s viewpoint: that the tree trunk is like a street, and patches of algae and of moss on the bark are like lawns and bushes along the street. Child: “and these branches are like side streets”.

As you can see, a gentle introduction to mathematics can start just by watching the world, in this specific case insects and flowers in a garden.

A child needs to develop a habit of attentively looking at the world; to achieve that, it helps when an adult looks with him, and they discuss what they see.

Mathematics, in one of its many aspects, is a specific vision of the world.

Basic (but mathematically very deep) concepts of similarity and scaling are accessible to a 4 year old child.

The former is erased (with most of geometry) from school teaching, the latter was never properly taught. Euclid’s proof of the Pythagoras Theorem as given in Book 6 of Elements is an example of a scaling argument.

Let us look at one of the simplest examples of fixed point theorems (I have seen it in some popular books on mathematics):

If a (topographic or geographic) map is placed on another map of the same territory, but of larger scale, then it is possible to stick a pin in the both maps which goes through images of the same point (location) of the surface of Earth.

What I’ll give you now is not a proof but an immediately obvious explanation:

Imagine that you an ant and stand on the bigger map which is like a surface of the Earth for you, and you hold the smaller map in your hands; just mark your location on it.

Further development of the theme of maps:

Adult uses every opportunity to explain to Child the structure of actual streets in a big city: street signs, house numbers which go in progression and are odd on one side of the street and even on the other side. A year later, Child is able to confidently guide Adult across a unknown part of the city using a standard AZ map.

Observing an ant on a tree helped. This was an encouraging sign of mathematical development.

Of course, Child’s ability to read is useful. Street names, all kinds of shop signs provide an excellent material for reading and proof that reading gives information about the world.

In a supportive family environment, a child can learn to read by the age of 4 or 5—even in English with its inane orthography. Social class issues creep in here: this is realistic mostly in families of well educated middle class professionals. Sufficient disposable income for buying “quality time” with children (say, by inviting an au pair or hiring a caretaker of some kind for more routine tasks) definitely helps.

And the last episode in development of this theme of maps, a conversation on a street:

Adult: “Look at the name of this street. Does it tell you anything?”

Child: “Yes! Its end meets the street where Grandma lives, just over the corner!”

Adult: “And do you remember, what kind of house numbers there, large or

small?”

Child: “Small!”

Adult: “So, in what direction we have to go?”

Child looks at house numbers around and confidently points: “There!”

More about rearing and writing:

Adult and Child (aged 5) send to each other, from opposite corners of a sofa, small strips of paper with messages written in a substitution cipher: each letter is substituted by the next one (cyclically, z is substituted by a). Suddenly Child exclaims:

“And I invented my own cipher — each letter is replaced by the previous one!”

IMHO, this would later help Child to understand algebraic notation where numbers are substituted by letters. It is worth remembering that François Viète, the inventor of algebraic notation, was the first cryptographer and cryptanalyst known to us by name. He served to King Henry IV of France. Viète’s deciphering of intercepted diplomatic correspondence directly influenced European politics of his time. The King of Spain famously complained to the Pope that Viète sold his soul to the Devil.

Some time later, at the age about 7, Child can handle variables in Scratch, a toy programming language for kids.

Child is invited to guess weight of every household object she can handle by weighing it in hand and check the result by weighing on scales. The same with temperature of water in the bath, checked by a thermometer. Or temperature outside the house.

Further,

All kinds of estimates with subsequent checking: how many steps are in this staircase?

How many steps are to the end of the street?

How to estimate the number of cars in the parking lot without counting them all?

I can add cooking and baking, as a family activity, if Child is entrusted with control of numerical aspects of the recipe: how many spoons of sugar?

Lego:

Playing Lego (with child of 4). Adult encourages Child to pick correct bricks (say, 2 by 3 studs) without looking at them, by touch only. They together follow step-by-step instruction in the manual. And steps are numbered!

Building a symmetric model (say, a plane), Adult builds the left wing, Child builds the right wing by mirroring Adult.

Very soon Child starts picking details of correct orientation even before Adult touches his detail. (Orientation is an exceptionally important concept in geometry which is not even mentioned at school.)

Lego is a great propaedeutic for Scratch, a toy programming language for kids where computer programs are built on the screen from logic blocs which intentionally made looking as bricks in Lego.

And the real unadulterated fun, fun, fun:

Playing Snakes and Ladders with two dices. A player can pick one of the values or their sum. The catch is that, for winning the game, 100 has to be hit without overshooting—for otherwise the player gets back to the beginning, the path is circular, 97+6 = 3.

A fast, furious, and vicious game which trains tactical thinking.

Actually the rules of Snakes and Ladders can be changed in a variety of ways. Adult encourages Child to invent her own rules. Crucially, the new rules need to be agreed and written down before the start of the game.

A useful meta-activity: ask Child to compare two sets of rules: which one produces a more exciting game?

A jigsaw with a clear geometric structure could be very useful. The classical London Tube map is excellent, especially if Adult and Child work together and agree to ignore the white space. But perhaps the London Tube is best for kids who live in London.

I can continue this list, but, I hope, it already gives some idea.

And I have no idea how all that could be done at school.

Within the family—no problem. In small and friendly mathematical circles—it is also OK.

But in school?

And, finally, some general advice:

Never ever distract your child when he/she just sits and thinks.

This post was previously published on Substack.

This was what a colleague wrote today in some discussion of university life:

“I think the biggest problem is that the University effectively values academics time as zero.”

My response:

IMHO, this is indeed the core problem.

This is the Executive Summary of my paper arXiv:2103.04101v1 [math.HO] on 6 March 2021:

Executive Summary. This paper is written in lockdown and can serve as a testimony in support of the apparently self-evident, but largely ignored principle:

*The most important resource for a (pure) mathematician’s research is uninterrupted time for thinking.*

University administrators and research funding bodies systematically ignore it, and the bureaucratic burden imposed by them strangulates mathematics research. A short breathing space provided by lockdown made miracles.

And, in my paper, I provide evidence,  and, I think, pretty convincing (at least to mathematicians), in support of this my claim. I did some of the best work of my life because, thanks to lockdowns and retirement, I took back control of my time.

This is what mathematicians have to fight for: taking back control of our time.

This post is copied from my Substack   article.

Tony Gardiner died suddenly on 22 January 2024. He will be remembered as a national treasure, a man who made a unique contribution to the development of mathematics education in this country and internationally.

Tony set up, and made significant contributions to the work of, the UK Mathematics Trust, which runs problem-solving challenges taken by over half a million students every year. Tony was the Team Leader of the British IMO team in 1990–95 – and a mentor of many bright young mathematicians who are now the crème de la crème of British academia. For many years, he edited the Problem Solving Journal for Secondary Students, with a circulation over 5,000. He wrote and published more than 15 books on mathematical thinking and mathematical problem solving – as well as on teaching mathematics. He was consulted by several UK Ministers of State for Education, and acted as an advisor on mathematics education to the government of Singapore. More can be said about Tony’s contribution to this world, but there is no need to compete with Wikipedia where the article devoted to him, https://en.wikipedia.org/wiki/Tony_Gardiner, is being feverishly updated.

Tony started his work in mathematics in the 1970s. He was a PhD student of the legendary Bernd Fischer, who had just discovered his three sporadic finite simple groups. It was a very fruitful time, when group theoretic ideas were becoming widely applied in combinatorics. Tony’s further research was very successful and mostly straddled the two areas of combinatorics and permutation groups.

At the same time Tony started to forge an unusual path in combining research in mathematics with a commitment to high school and undergraduate mathematics. His instinct as a researcher led him to investigate the actual working of mathematics education as a system and look at the entire cycle of reproduction of mathematics: from preschool and primary school through all stages of school education to university to teacher training and then back to school as a teacher. This is also augmented by a smaller cycle: BSc – PhD studies and postdoctoral research, then back to university as a lecturer. This breadth of vision was supplemented by his attention to the socio-economic and political background of education and placed him in a very special position among British mathematics educationalists.

Tony had exceptional academic and intellectual integrity. He was very modest. And, above all – he was a very kind man always helping a talented student or a bright school child who needed help, and did so right up to the last days of his life.