A Letter to My Younger Mathematical Self
A reflective letter to a younger self just beginning the study of mathematics — the things I wish I had known about learning, struggling, growing, and finding meaning in the subject.
Dear Younger Me,
You are about to begin studying mathematics seriously. You have chosen this path because something about the subject draws you in — the elegance of a proof, the satisfaction of solving a problem, the feeling that underneath the chaos of the world, there is a hidden order waiting to be understood.
I am writing to you from further down this road. I cannot spare you the difficulties ahead — they are part of the journey. But I can share some things I wish I had known.
It Gets Harder, and That Is Good
You are used to mathematics being something you are "good at." You solve problems quickly. You understand things the first time. Your grades are excellent.
This will change.
At some point — in real analysis, or abstract algebra, or when you encounter your first truly hard proof — you will hit a wall. The material will not yield to quick thinking. You will read a page five times and still not understand it. You will stare at a problem for hours with no progress.
This is not a sign that you have reached the limit of your abilities. It is a sign that you have reached the beginning of real mathematics.
"Everyone struggles in mathematics. The difference between those who succeed and those who don't is not the absence of struggle, but the willingness to persist through it."
— A truth known to every working mathematician
The struggle is not an obstacle to learning — it is the learning.
You Are Not an Imposter
There will come a time when you look around a lecture hall, a seminar, or a study group and think: everyone else understands this better than I do. They are smarter. I do not belong here.
This feeling has a name — imposter syndrome — and it is nearly universal among mathematics students. Even the most accomplished mathematicians experience it.
Terence Tao, who won a Fields Medal at 31, has said:
"I think what many students don't realize is that even the most successful researchers feel confused and uncertain much of the time."
— Terence Tao
The people you admire are not effortlessly brilliant. They have simply learned to keep working through the doubt.
Slow Down
You will be tempted to rush through material — to cover as many topics as possible, to get to the "advanced" stuff quickly. Resist this temptation.
Deep understanding takes time. It is better to understand one theorem thoroughly — to know why each hypothesis is necessary, to see what happens when a hypothesis is dropped, to work through three examples — than to skim ten theorems superficially.
Mathematics is not a race. The mathematicians who contribute most are not those who cover the most ground, but those who understand their territory most deeply.
A Principle Worth Remembering
Speed is not depth. Breadth is not understanding. Take the time to understand things properly. Your future self will thank you.
Ask for Help
You will sometimes refuse to ask for help because you think you should be able to figure everything out on your own. This is pride, not strength.
Asking a professor to explain something is not an admission of weakness. Going to office hours is not a sign of inadequacy. Working with classmates is not cheating.
Mathematics is a collaborative enterprise. The myth of the solitary genius is exactly that — a myth. In reality, mathematical understanding grows through conversation, questions, and shared effort.
Go to office hours. Form study groups. Ask "dumb" questions. The questions you think are dumb are almost always the ones that lead to the deepest understanding.
Write Things Down
You will be tempted to do everything in your head. You are clever enough to hold several steps of an argument in memory simultaneously.
Write things down anyway.
Writing forces precision. It exposes gaps in your reasoning that mental arithmetic conceals. It creates a record you can return to. And it trains you for the most important skill in professional mathematics: communicating ideas to others.
Start a mathematical journal or notebook. Write up your solutions carefully. Explain things to yourself on paper as if you were explaining them to a friend.
As the mathematician Paul Halmos wrote:
"The only way to learn mathematics is to do mathematics."
— Paul Halmos
And "doing" mathematics means writing it down.
Not Everyone Will Understand
At some point, your friends and family will stop being able to follow what you are studying. They will ask what you do, and your answer will not fit into a sentence they can easily parse.
This can be isolating. You may feel that the thing you care most about is incomprehensible to the people you love.
Learn to live with this gracefully. Find joy in explaining what you can. Accept that some things cannot be shared with people outside the field. And build relationships with people who do understand — your mathematical community will become one of the most important parts of your life.
It Is Okay to Change Direction
You will develop strong feelings about what area of mathematics you want to study. You will identify with a field — "I am an algebraist" or "I am an analyst" — and feel that changing direction is a failure.
It is not.
Many of the greatest mathematicians changed fields multiple times. Grothendieck moved from functional analysis to algebraic geometry. Atiyah moved between topology, geometry, and mathematical physics throughout his career.
Your interests will evolve as you learn more. Follow them. The most interesting mathematics often lives at the boundaries between fields.
Take Care of Your Body and Mind
You will be tempted to work all the time. You will feel guilty when you are not working. You will sacrifice sleep, exercise, and social connection in pursuit of mathematical understanding.
Do not do this.
Your brain is a biological organ. It requires sleep, exercise, nutrition, and rest to function well. The most productive mathematicians are not those who work the most hours — they are those who work the most effectively, which requires being well-rested and healthy.
Take walks. They are good for your mathematics and for your soul. Poincaré, Hadamard, and many others reported that their best ideas came during walks.
"At the moment when I put my foot on the step, the idea came to me, without anything in my former thoughts seeming to have paved the way for it."
— Henri Poincaré, describing a mathematical insight during a walk
Comparison Is Poison
You will meet people who are faster than you, who learn more easily, who seem to understand everything instantly. You will compare yourself to them and feel inadequate.
This comparison is destructive and misleading. You never see the hours of private struggle behind someone else's public competence. You never see the problems they cannot solve, the papers they do not understand, the doubts they do not share.
Compare yourself to who you were six months ago. If you have grown — if you understand things now that you did not understand then — you are on the right track.
Find Beauty
In the midst of problem sets, exams, and career anxiety, do not lose sight of why you came to mathematics in the first place.
There is genuine beauty here. A proof can be as elegant as a poem. A theorem can reveal hidden structure that takes your breath away. A connection between two distant areas of mathematics can fill you with wonder.
Seek out this beauty. Read proofs that are famous for their elegance. Explore areas outside your specialty. Let yourself be awed.
Bertrand Russell captured it:
"Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show."
— Bertrand Russell
Be Kind
The mathematical world is small. The people you study with today will be your colleagues, collaborators, and friends for decades.
Be generous with your knowledge. Help others when you can. Share ideas freely. Celebrate others' successes. Create the kind of mathematical community you want to live in.
The mathematicians who are remembered most fondly are not always the most brilliant — they are the ones who were kind, who mentored generously, who made others feel welcome.
It Will Be Worth It
There will be moments of profound frustration — weeks when nothing works, exams that go poorly, papers that are rejected, career uncertainties that keep you up at night.
There will also be moments of extraordinary joy — the first time you prove something no one has proved before, the moment a difficult concept suddenly clicks, the deep satisfaction of teaching someone else what you have learned.
The difficult moments make the joyful ones possible. You cannot have one without the other.
One More Thing
You do not have to become a famous mathematician for your mathematical life to have been worthwhile. If you learn to think clearly, to appreciate beauty, to persist through difficulty, and to communicate ideas to others — you will have gained something of immeasurable value, regardless of where your career takes you.
Mathematics is not just a profession. It is a way of seeing the world.
I am glad you chose it.
With respect and affection,
Your older self.
References
- G.H. Hardy, A Mathematician's Apology, Cambridge University Press, 1940
- Paul Halmos, I Want to Be a Mathematician, Springer, 1985
- Bertrand Russell, "The Study of Mathematics," in Mysticism and Logic, 1917
- Henri Poincaré, Science and Method, Thomas Nelson and Sons, 1914
- Terence Tao, Career advice
- Francis Su, Mathematics for Human Flourishing, Yale University Press, 2020
- William Thurston, "On Proof and Progress in Mathematics," Bulletin of the AMS, Vol. 30, No. 2, 1994