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.

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.