My answer to a question on Quora:

Any mathematics is hard, not only pure mathematics, but pure mathematics is special, and is perhaps is hardest of all. I love this motto coined by my colleague Rob Wilson:

Mathematics: solving tomorrow’s problems yesterday.

Of course, it is about pure mathematics — applied mathematics solves today’s problems, and solves today. But it was pure mathematics which had ready mathematical methods for physicists when they started to develop the general relativity theory and quantum mechanics, and for cryptographers when they needed computer based methods for signing and authenticating electronic documents, including financial documents and money transactions — the very existence of the global financial system now depends on this tools based on mathematics of yesteryear.

So, what type of people would study pure math? People who love precise, concise, very abstract thinking, people who value clarity of thought. Serious study of pure mathematics means choosing a specific lifestyle: first of all, you have to love the process of thinking, deep systematic thinking with full concentration of all your attention on it. Also, you have to have a sufficiently long attention span, ability to think, and do nothing else, for long periods of time.

So, the shortest answer to your question:

People who love to think.

As a comment, I added later:

My own experience of application of mathematics was just picking from a shelf something known (in ”pure” mathematics) for some time, and known for completely different reasons. The hard part was to understand what engineers want from you, ignoring their clumsy efforts to express that mathematically, and looking at the core of the *raw* problem.

Keith Devlin kindly commented in his traditional column   on my paper  The truth/weapon duality of mathematics. Strongly recommend his post as well his previous one, How do you respond when you are asked to use your expertise to help your nation?

A quote from Devlin:

I received an email from a reader the UK (we have met, and occasionally exchange ideas) alerting me to a recently published paper by Alexandre Borovik, a retired mathematician from the University of Manchester (where I taught briefly back in the 1970s), titled The Tool/Weapon Duality of Mathematics and published in the Journal of Humanistic Mathematics, Vol 16, Issue 1, January 2026.

Some of what Borovik writes is familiar to me, but he took his own reflections on working on projects with military relevance and extended them by researching the literature. It was his paper from which I took, not only the title for this post, but also the question quotation I began with. Those words were spoken by Russian President Vladimir Putin, speaking to journalists and media executives from BRICS countries on 18 October 2024.

[Given that source, the current US government’s assault on universities and their slashing of research funds (all of which the UK government did in the 1980s when I was a casualty) seems particularly unwise. But again, we get the administration the People elect.]

I urge you to read Borovik’s paper. It’s a mere 24 pages long, plus five pages of highly useful references. It’s all highly relevant to American-based mathematicians today, some of whom will (assuming we remain a democracy and there is a change of government) undoubtedly find themselves facing a decision like the one I did back in the first decade of the 21^st Century. Vladimir Putin was correct. And yes, I just wrote that. In an essay about mathematics.

I’ll end with a quotation from Borovik:

 Mathematics, as we know it, was born as a weapon of subjugation and tyrannic control.

He brings the historical receipts to back up that statement. Though I was peripherally familiar with the historical events it refers to (particularly the beginnings of arithmetic in Sumeria over five thousand years ago, which I have written about in some of my books), the starkness of his assertion made me jolt. Facts have a habit of doing that.

I was invited to contribute a chapter to a a Collective Volume: Artificial Intelligence and Society: Crossed Perspectives, Issues and Prospects, to be published by Springer.

On 2 March 2026 I asked Copilot Think Deeper:

“Write a paper for a Collective Volume: Artificial Intelligence and Society: Crossed Perspectives, Issues and Prospects, to be published by Springer. Prepare the manuscript in \LaTeX format using Springer’s class file”.

The result came in a few seconds, under the title Artificial Intelligence, data totality, and the reconfiguration of human agency: Societal prospects and crossed perspectives. I checked a few items in the bibliography: they look genuine, even if slightly outdated. A pdf file of the paper is available here. It is passable. But I preferred to write my own paper. I hope it does not look like an AI slop.

To make the question as close to the set theory as possible, let us restrict our attention to the case of a vector space V over the field F_2 of two elements. If my memory does not betray me, it is an old result by Paul Eklof that existence of a basis in an arbitrary vector space over F_2 is equivalent to the Axiom of Choice (it is easy in one direction: the Axiom of Choice, in the form of the Zorn Lemma, implies the existence of a basis). So, let us accept it, and let B be a basis in V, which means that every vector v in V is defined by its support in B, that is, by the (finite) set of elements in B which sum up to v. Therefore V has the same cardinality as the set of finite subsets of B; if B is infinite, than it is easy to prove that the set of all finite subsets of B has the same cardinality as B, hence V has the same cardinality as B.

Now let us look at the dual space V*, that is, the set of all linear functionals from V to F_2. Each such functional is uniquely determined by its support in B, that is, by the set of basis vectors where it takes value 1. Therefore V* is in one-to-one correspondence with the set 2^B of all subsets in B. But it is a classical result by Cantor that 2^B has larger cardinality than B and hence V* has larger cardinality than V. Of course, the cardinality of the bidual V** is even larger.

The case of an arbitrary field F can be handled in a similar way, but we will need to deal with the cardinality of the set F^B.

Is this intuitive? Well, it is intuitive for me because it was the first thought that crossed my mind. But I am not a set theorists and I have no idea whether the same can be proven without the Axiom of Choice. Also, the term “infinite dimensional vector space is ambigous: does it mean “a vector space without a finite basis” or “a vector space with an infinite basis”? And can anything that intimately involves the Axiom of Choice be called intuitive?

My old post on Quora, an answer to question

Why do people have to learn algebra? You have no use for it.

Now I would perhaps slightly modify it, see my Addition at the end of the present post.

It could be argued indeed that it is wrong that everyone is forced to learn algebra at school.

However, without (pretty basic, between us) school algebra further study mathematics, or statistics, or computer programming is impossible.

Not learning algebra dramatically narrows further educational choices. Catching up later is very difficult — if you have not learnt algebra as a child or, at the latest, in your teen years, to do this later is of course possible but requires a degree of determination and self-discipline not normally found in general population.

Therefore, without algebra, the education system is less democratic. This is already a heavy price to pay.

In general, mathematical education is increasingly the issue of democracy.

Proper mastery of mathematics gives a person the ability to create in the head mental images of complex systems and operate with them, see them. We are surrounded by complex systems, we drown in them — in stuff starting from smartphones to the Internet and social media . Very complex computer programs make more and more decisions on peoples’ lives – whether someone can be hired to a particular job, or given a bank credit, or sold a a particular insurance policy, and so on.

Properly mathematically educated people are sighted among the blind.

If you wish to stay blind – it is your choice. Or you may think that it is entirely your choice. Actually it could happen that it is already pre-determined by a dismally bad education which, most likely, had been offered to you so far — otherwise, perhaps, you would not ask the question “ Why do people have to learn algebra?”.

Added 18.04.26:

I think the question needs a reformulation and should be

Why at least some people have to learn algebra?

with the  answer

Because  the human race still has to use it.

It is more and more common in media, even in science-focused media:

The man who ruined mathematics, by Jacob Aron. The paper is built on false premises. Kurt Gödel didn’t ruin mathematics. And Gödel’s incompleteness theorem is mispresented. I know that, I had taught an undergraduate lecture course with a full proof of this remarkable result.

Recommend this article, The 5 stages of the ‘enshittification’ of academic publishing from The Conversation. I’ve seen all that in my work as an editor of mathematical journals.

My answer to a question on Quora:

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.