Advice from Famous Mathematicians to Young Students
A collection of wisdom from some of the greatest mathematicians in history — practical advice, philosophical reflections, and encouragement for young people beginning their mathematical journeys.
Why Their Words Matter
Every great mathematician was once a student — uncertain, struggling, and wondering whether they had what it takes. Many of them have shared advice that is both practical and profound.
This post collects some of the best advice from mathematicians across different eras and areas of mathematics. Their words are relevant whether you are just beginning your studies or deep into a research career.
On Learning Mathematics
Terence Tao: Work Hard and Be Patient
Terence Tao, one of the most accomplished living mathematicians, has written extensively about learning and doing mathematics. His advice is refreshingly practical:
"The popular image of the lone (and possibly slightly mad) genius — making an ineffable leap of inspiration, creating a revolution, and single-handedly overturning the establishment — is a charming and romantic image, but also a wildly inaccurate one... The actual work of making progress in mathematics consists mostly of hard, unglamorous effort."
— Terence Tao, "Does one have to be a genius to do maths?"
Tao emphasizes that mathematical talent is developed through sustained effort, not bestowed at birth. His blog post "Does one have to be a genius to do maths?" should be required reading for every mathematics student.
Paul Halmos: Do Problems
Paul Halmos, the great expositor, was emphatic about the importance of active problem-solving:
"The only way to learn mathematics is to do mathematics."
— Paul Halmos
This seems obvious, but many students fall into the trap of passive learning — reading proofs without attempting them, watching lectures without working exercises. Halmos insists that mathematical understanding can only be built through active engagement.
Richard Feynman: Understand, Don't Memorize
Though primarily a physicist, Feynman's approach to mathematical thinking is deeply relevant:
"I learned very early the difference between knowing the name of something and knowing something."
— Richard Feynman, "What Do You Care What Other People Think?"
In mathematics, it is tempting to memorize definitions and theorems. But real understanding comes from knowing why a definition is the right one, why a theorem is true, and what happens when its hypotheses fail.
On Doing Research
Alexander Grothendieck: The Rising Sea
Grothendieck, who revolutionized algebraic geometry, described two approaches to solving a hard problem:
"I can illustrate the second approach with the same image of a nut to be opened. The first analogy that came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more flexible through weeks and months — when the time is ripe, hand pressure is enough; the shell opens like a perfectly ripened avocado!"
— Alexander Grothendieck, Récoltes et Semailles
Grothendieck's method was not to attack problems directly with clever tricks, but to build so much general theory that the problem dissolves naturally. This "rising sea" approach is not for everyone, but it reflects a deep truth: the right framework can make hard problems easy.
Andrew Wiles: Persistence in the Dark
Wiles, who proved Fermat's Last Theorem after seven years of secret work, described the experience memorably:
"Perhaps I could best describe my experience of doing mathematics in terms of entering a dark mansion. One goes into the first room, and it's dark, completely dark. One stumbles around bumping into the furniture. Gradually, you learn where each piece of furniture is. Finally, after six months or so, you find the light switch."
— Andrew Wiles
The message is clear: confusion and stumbling are not failures. They are the normal process of mathematical discovery.
William Thurston: Mathematics Is About Understanding
Thurston, who transformed topology and geometry, wrote one of the most important essays about mathematical research:
"Mathematics is not about numbers, equations, computations, or algorithms: it is about understanding."
— William Thurston, "On Proof and Progress in Mathematics"
Thurston argued that the goal of mathematics is not to produce proofs but to produce understanding. Proofs are a means to this end, not the end itself.
On Overcoming Difficulty
Maryam Mirzakhani: Embrace the Struggle
Mirzakhani, the first woman to win the Fields Medal, was candid about the difficulty of mathematical research:
"I don't have any particular recipe. It is the reason why doing research is challenging as well as attractive. It is like being lost in a jungle and trying to use all the knowledge that you can gather to come up with some new tricks, and with some luck you might find a way out."
— Maryam Mirzakhani
Michael Atiyah: Ask Questions
Atiyah, who won both the Fields Medal and the Abel Prize, emphasized curiosity:
"I think it is said that Gauss had ten different proofs for the law of quadratic reciprocity. Any good theorem should have several proofs, the more the better. For two reasons: usually, different proofs have different strengths and weaknesses, and they generalise in different directions — they are not just repetitions of each other."
— Michael Atiyah
The lesson: do not be satisfied with understanding something one way. Seek multiple perspectives on every important idea.
Pólya: Start with What You Know
George Pólya, whose book How to Solve It remains the most influential text on mathematical problem-solving, offered structured advice:
Pólya's Four Principles
- Understand the problem. What are you trying to find or prove? What are the given conditions?
- Devise a plan. Can you relate the problem to something you already know? Can you solve a simpler version?
- Carry out the plan. Execute your strategy step by step, checking each step.
- Look back. Can you check the result? Can you derive it differently? Can you use it for some other problem?
On the Mathematical Life
Bertrand Russell: The Austere Beauty
Russell, who co-authored Principia Mathematica, described the emotional experience of mathematics:
"Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture."
— Bertrand Russell
Paul Erdős: Collaboration and Joy
Erdős, the most prolific mathematician of the 20th century, embodied a joyful, collaborative approach to mathematics:
"A mathematician is a device for turning coffee into theorems."
— Paul Erdős (often attributed, via Alfréd Rényi)
Beyond the humor, Erdős's life demonstrated that mathematics is fundamentally a social activity. His willingness to work with anyone, anywhere, on any interesting problem, made him one of the most beloved figures in the history of mathematics.
Emmy Noether: Dedication to Ideas
Noether's dedication to mathematics persisted through extraordinary adversity. Driven from her position in Germany by the Nazi regime, she continued her work at Bryn Mawr College until her untimely death. Her student and colleague P.S. Alexandrov eulogized her:
"Her tireless energy, her selflessness, her devotion to science, and her friendliness to all those with whom she came into contact — all these qualities made her a model human being."
— Pavel Alexandrov, eulogy for Emmy Noether
On Being Wrong
Timothy Gowers: Mistakes Are Essential
Gowers, a Fields Medalist, has been unusually open about the role of mistakes in mathematics:
"One of the main skills of a mathematician is the ability to be wrong efficiently — that is, to make mistakes, recognise them quickly, and learn from them."
— Timothy Gowers
Henri Poincaré: The Value of Failed Attempts
Poincaré, one of the greatest mathematicians of the 19th and early 20th centuries, understood that unsuccessful attempts are part of the creative process:
"It is by logic that we prove, but by intuition that we discover."
— Henri Poincaré
On Advice Itself
John von Neumann: Humility Before Mathematics
Von Neumann, one of the most versatile intellects of the 20th century, offered a surprisingly humble perspective:
"In mathematics you don't understand things. You just get used to them."
— John von Neumann
While partly humorous, this captures something real: mathematical understanding often develops gradually, through repeated exposure and practice, rather than through sudden comprehension.
Ravi Vakil: Show Up
Vakil's advice to graduate students captures the importance of persistence:
"The way to get started is to show up. Go to seminars. Talk to people. Read. Work. And keep doing it."
— Ravi Vakil
Summary of Advice
If we distill the wisdom of these mathematicians into a few principles:
- Do mathematics actively. Solve problems. Write proofs. Do not just read.
- Be patient. Understanding takes time. Confusion is part of the process.
- Persist through difficulty. Being stuck is normal. Keep working.
- Seek understanding, not just results. Ask why, not just what.
- Collaborate and communicate. Mathematics is a community.
- Make mistakes freely. They are how you learn.
- Find beauty in the subject. It will sustain you through the hard times.
References
- Terence Tao, Does one have to be a genius to do maths?
- Paul Halmos, I Want to Be a Mathematician, Springer, 1985
- Alexander Grothendieck, Récoltes et Semailles, 1985–1986
- William Thurston, "On Proof and Progress in Mathematics," Bulletin of the AMS, Vol. 30, No. 2, 1994
- George Pólya, How to Solve It, Princeton University Press, 1945
- G.H. Hardy, A Mathematician's Apology, Cambridge University Press, 1940
- Paul Hoffman, The Man Who Loved Only Numbers: The Story of Paul Erdős, Hyperion, 1998
- Ravi Vakil, Advice for potential graduate students
- Maryam Mirzakhani, interview with the Clay Mathematics Institute, 2008