What makes math worth learning?

May 2026

This essay is about a question which has bothered me my entire adult life. We all go through a compulsory mathematics education in school. But what makes math worth learning?

I probably don’t need to justify that it’s worth it to you, a reader in 2026, to learn some amount of math. For some professional occupations, that need may extend to using specialized techniques for solving differential equations ; for others, it’s plenty useful to know arithmetic to get about your day. At an individual level, math is worth learning as a school student for an obvious reason: if you fail your math courses, you won’t get a degree and it’ll be harder to get a job.

Of course, today’s revolution in AI threatens to upend many of these norms. Just as digital calculators reduced the necessity of by-hand calculation skills, now AI is reducing the necessity of programming skills. Some think AI will replace humans in even creative thinking, as well as all of mathematics. It’s certainly raising some important questions about what mathematics and science even are as a human endeavor.

The question of usefulness is a little different from what I’m asking here though. Even if learning mathematics were useful to thriving in tomorrow’s society, that might be a quirk of how society is constructed. Perhaps our modern math education is an accident of the historical levers of power, rather than an optimal state of things. If we boil those factors away — take a 10,000 foot, idealistic view of how you wish to construct a moral, sustainable society, and chart a course to get there — then should people in that society be learning math? In what form, and why? If not, is mathematics an important thing to learn to get us there, or should we all be focusing our efforts elsewhere?

I have a lot of scattered thoughts on this question, which are evolving. This essay is a crystallization of some of them, weaving in a bit of my own story.

Three possible arguments in defense of mathematics

Here are a few justifications in favor of learning mathematics.

Mathematics forms a technical basis and a rigorous language for the sciences, i.e. the study of the natural and social world. Creating science is both a fundamental part of who we are, and a vehicle for thriving society. Number sense is also essential as an everyday life skill. Not all types of thinking demand numeracy, but the necessity of mathematics abounds in the modern world — so much so that it is worth it to study mathematics in its own right as a foundation. In other words, math is a useful tool.
The act of studying mathematics is an excellent arena for developing clarity of thought, and critical thinking — the ability to question, to methodically dissect arguments, to form hypotheses and reason. Math is materially extremely accessible: you need only a mind, a pen and paper (with the latter two being optional). Truth is exact and unforgiving, so one gets used to struggling and being incorrect. In other words, math is a practice which develops a keen mind, and this is good for the individual (and perhaps society as well).
Math has an intrinsic importance for its artistic and philosophical qualities, as well as its rich history. One studies math to be inspired by its beauty, and to cultivate a deeper ability to appreciate and share artistry. The history of math is intertwined with many cultural histories — it provides a deeper way to understand events in history and cultural ways of knowing. All of this creates a person who is more tempered, joyous, and experiences greater satisfaction. This is good for everyone.

I started thinking about this question around the time I turned 20. That’s ironic, considering that I spent so much of my pre-adult life doing mathematics, far beyond what was asked of me by school or my parents. I had access to math books and contests, and I was driven first by childlike curiosity and later by teenage competitiveness. It was fun and the adults approved of it, and that was enough for me.

It’s only when I started teaching to others that I was required to provide a motivation that was relatable, and that’s when I turned inwards. Of all the questions I was asked in a mathematics classroom, the hardest to answer was always “Why are we learning this? What’s the point?” At that age, I had no real life adult experience which would enable me to answer this meaningfully.

So I drew upon what I knew. The primary force responsible for my mathematical development at that time was through contests, whose culture is embodied by organizations like Art of Problem Solving or the IMO Foundation. These organizations emphasize the value of solving mathematical problems. The fruit is in the labor itself, as it trains one’s capacity to struggle, think hard, and exhibit creativity in new situations. This aligns most closely with argument #2 I listed above.

At that age, I was a student at MIT, a university which emphasizes the importance of learning to use. The motto of MIT is Mens et manus: “mind and hand”. The applications of mathematics — from partial differential equations in mechanical engineering, to statistics and linear algebra in AI — were apparent in a vast array of engineering projects, on display at all times as I walked around campus. It seemed apparent to me that the reason for developing a keen mind was to solve technical problems in the world, and my thinking shifted towards argument #1.

Certainly the idea of mathematics as a lifelong practice which cultivates temperance and creative inspiration isn’t something I gave weight to at this stage in my life. I saw benefit to society as primarily constructed through external transformation of the material world, rather than internal transformation of the human condition.

Returning to the original question, many professional mathematicians have written at length about the role of mathematics in society, and I figure that’s a good place to start finding answers.

Where does mathematics come from?

A little over a year ago, I read several papers by Vladimir Arnold, a Russian mathematician of the 1900s who takes the view of mathematics as intertwined with physics and geometry, emerging from the natural world. In the introduction of one paper¹ he asserts that all of mathematics can be traced genealogically to three applied domains:

All mathematics is divided into three parts: cryptography (paid for by CIA, KGB, and the like), hydrodynamics (supported by manufacturers of atomic submarines), and celestial mechanics (financed by military and by other institutions dealing with missiles, such as NASA).

Cryptography has generated number theory, algebraic geometry over finite fields, algebra, combinatorics, and computers. Hydrodynamics procreated complex analysis, partial derivative equations, Lie groups and algebra theory, cohomology theory, and scientific computing. Celestial mechanics is the origin of dynamical systems, linear algebra, topology, variational calculus and symplectic geometry.

The existence of mysterious relations between all these different domains is the most striking and delightful feature of mathematics (having no rational explanation).

One can nitpick at details here. For example, the comprehensive study of Diophantine equations (an early form of algebraic geometry) goes back to at least the ancient Greeks and perhaps even further — long before anyone was using much math in cryptographic applications. Another is the fact that this classification appears to gloss over the fields probability and stochastics which, as far as I can tell, evolved from an attempt to price market phenomena (e.g. insurance premiums for maritime ventures) and to win at games of chance.

Arnold was almost certainly being a little facetious with such a sweeping claim. But there are kernels of truth.

It’s a reminder that there is, and has always been, an interplay between abstract mathematics and real world strings of power. Mathematics doesn’t persist without mathematicians, and professional mathematicians wouldn’t exist without some way to get paid for it. In a few cases, research mathematics has come from non-professional mathematicians, but these are exceptions to the norm.

Historically, who have those funders been, and what professional services did they want in return? Well, maybe cryptographers, astronomers, mechanical and naval engineers — and people who could assemble a comprehensive theory to turn these practices into communicable bodies of knowledge, i.e. mathematicians. Today, the connection between the latter and the former is made invisible (a result of the professionalization of mathematics as apart from engineering or physics), but it exists.

Pure and applied: who are you benefiting?

When I started studying university-level math, I was told that math comes in two flavors: theoretical (also called “pure”) and applied. At that time I understood that one strain of mathematics is proudly abstract and artistic (yet disconnected from reality), while the other is concrete and material (and also happy to serve power).

I studied topology in graduate school — “pure” math through and through — and I loved it as an intellectual exercise. But a large part of me felt deeply dissatisfied. I wanted to be useful, to serve society in some way. The narrative of pure mathematics as enriching to the human soul simply didn’t hold weight for me, given that the majority of people didn’t seem to care about math, and the beautiful ideas were stuck behind highly technical formalism. Plus, you can’t eat a more enriched soul. I imagined my eventual pathway after finishing a PhD would be to transition into a more applied line of technical work, where I could build something real and material.

By about five years after the end of graduate school, I was employed in a major industry lab in the computing field, working on applied problems — a step in the direction I’d hoped while in graduate school. But I didn’t feel any closer to helping society than I had as an academic. The real issue is that I didn’t have a clear idea of what “society” means, what vision I had for change, and most importantly who my work was benefiting.

When I started reading more about the history of math, it firstly helped me understand the interplay between pure and applied. From an applied math perspective, the purpose of pure math is to spot the ethereal ideas which cross over from one applied domain to another — in fact, that’s what pure math is. So the two need each other.

It also helped me to see from where the applied problems come. As long as we use digital communication, there will be a need for cryptographers — and the deepest pockets are those of state surveillance agencies. As long as we keep building bridges, airplanes, or missiles, there will be a need for people versed in numerical methods for differential equations.

If I’d been placed back in that math classroom from my 20s, I could point to concrete applications of the math they were learning, and even who might pay them to do it. But now a new question had arisen: on the balance, I don’t know for sure if the downstream applications of math are more harmful or more beneficial for society. It’s made us more effective at feeding ourselves, transporting ourselves, entertaining ourselves, spreading information to more people, and also at killing each other and looting the earth.

Many have written about the purpose of a mathematics education, whether institutional or not. To draw on Arnold’s perspectives, a quality mathematics education program is a foundation for a country’s economic and military strength, and is also important to instill critical thinking in its population so they are not controlled by demagogues.²

The first point is widely acknowledged as true. There is a reason so much of the historical progress in applied mathematics (and that to which it’s applied — e.g. the natural sciences) has been stimulated by periods of war, conquest, and colonization, because these endeavors supplied problems which required mathematical thinking.

I also agree with the second point, though while I do believe mathematics helps train a keen and critical mind, experience has told me that’s nearly orthogonal to the task of building wisdom: the internal understanding of what to do with that skill in personal alignment. Nor does it effectively train kindness and ethical sense.³ A society full of people with a keen mind but lacking wisdom or ethics is a scary prospect.

My own personal journey has led me back to studying mathematics once again, no longer for the desire to be useful nor to prove myself, but just because I find it personally enriching once more. Moreover, it makes far more sense to me with an emphasis on its history. Foregrounding the why and the stories around the mathematics has brought much greater depth to the technical understanding.

In my experience, the dominant narrative about the value of a math education centers on arguments #1 and #2 above. I believe that the solutions to the most pressing problems of today (e.g. climate change) don’t lie in STEM training. But regardless of whether math is “useful” or not for external purposes, I believe that engaging with it deeply has great internal benefits.

My dream (which may sound foolishly idealistic!) is that if we taught mathematics in a different way — starting with de-compartmentalizing “STEM skills” from the more humanistic and community-centered aspects of education — it would steer us towards better uses.

¹“Polymathematics: is mathematics a single science or a set of arts?” https://math.ucr.edu/home/baez/Polymath.pdf

²See “Innumeracy and the Fires of the Inquisition”, https://www.pdmi.ras.ru/~arnsem/Arnold/arn-papers.html

³This isn’t a new realization, and I think it’s a major impetus behind programs such as SPARC, or Thought-Full.