Questions to Ask a Mathematician
For a student, a teacher or a curious adult who has a working mathematician in front of them, whether at a class visit, in a career interview or at the microphone after a talk. The list moves in the order such a conversation usually takes: what the research is and where its problems come from, how a proof is found and who checks it, the long stretches of getting nowhere, the big questions (a favorite theorem, infinity, whether any of it is useful), the route into the job, and what they would change about math class. One good question is enough for a Q&A, and a dozen fills an hour.
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The questions
Each question, and why to ask it
The work
What does a mathematician actually do all day?
Why ask it
Expect less calculating than you pictured and more reading, scribbling, talking at a board and teaching. Then ask how many hours of last week went to their own research, because the number is often smaller than they would like.
What are you working on, and can you give me the version for someone who stopped at high school algebra?
Why ask it
Give them a moment, since the plain version is harder to build than the technical one. If it arrives as an analogy, ask where the analogy breaks, which is usually where the mathematics begins.
What does research mean in a subject where the answers are supposed to be in the back of the book?
Why ask it
Many students assume mathematics was finished centuries ago. A good reply makes clear that a research problem is one nobody alive knows how to solve, and that choosing which of those to attempt is a skill of its own.
Where do your problems come from?
Why ask it
The sources vary: the last paragraph of someone's paper, a colleague's remark over coffee, a physicist or engineer who needs a thing to be true. Find out which it was for the problem on their desk now, and who else is after it.
What is your area called, and what is the simplest object in it that you could draw for me?
Why ask it
A field name such as algebraic topology or combinatorics tells an outsider nothing. A drawing of one knot, one graph or one curve gives you something to point at, and you can ask what they are trying to find out about it.
Which of your results are you proudest of, and what was known the day before you proved it?
Why ask it
The second half gives the result a size, since a theorem only means something against what came before. Pride often attaches to one idea inside the proof, so ask which step they remember finding.
Do you work with chalk, paper or a computer?
Why ask it
Blackboards are still loved in math departments, and plenty of mathematicians also write code to try examples before attempting a proof. The follow-up is what the computer is for in their case: experiments, checking or only typesetting.
Is mathematics as solitary as it looks from the outside?
Why ask it
The picture of a lone genius fits some people and misleads about many others. If they have co-authors, ask how it runs day to day: who talks, who writes it up, and what happens when two authors disagree about whether a step holds.
How long does one paper take, from the first idea to publication?
Why ask it
Listen for two clocks, the time to find the result and the time spent writing it up and waiting on referees. When the total runs to years, as it often does, you have the reason many researchers keep several projects going at once.
What problem in your field can a twelve-year-old understand that nobody has been able to solve?
Why ask it
Most areas have one, in the spirit of Goldbach's conjecture that every even number above two is a sum of two primes. Copy the statement down exactly, then ask for the best partial result so far, which shows how progress is made on a problem that will not fall.
Proof
What is a proof, and why is checking a million examples not enough?
Why ask it
Ask it even if you think you know, because this is the center of the subject. Request a pattern that held and then broke: a classic is the circle cut into regions by joining points on its edge, which counts 1, 2, 4, 8, 16 and then 31.
When you set out to prove something, where do you start?
Why ask it
Seldom at line one. You will hear about special cases, working backward from the goal and guessing the answer before justifying it. A finished proof hides the order in which it was found, so ask what came first in one of theirs.
How can you be convinced something is true before you have a proof?
Why ask it
Mathematicians talk about evidence more than outsiders expect: computed cases, an analogy with a solved problem, the way a statement would make other things fit. A student who thinks the subject is all certainty tends to be surprised by how much of it runs on informed guessing.
Who checks a new proof, and how long does that take?
Why ask it
The usual chain is colleagues and seminar audiences first, then one or more referees reading for a journal. How closely a referee is expected to read differs between journals and fields, so ask how it went for their last paper.
Have you ever found a hole in your own proof after you thought it was done?
Why ask it
Nearly everyone has, and the story ends in a patch, a weaker theorem or a withdrawn paper. It happened in public to Andrew Wiles, whose first announced proof of Fermat's Last Theorem had a gap that took about a year to repair.
What makes one proof more beautiful than another?
Why ask it
Mathematicians use the word without embarrassment, and the reasons they give tend to rhyme: short, surprising, or joining two areas that looked unrelated. Get an ugly one too, a proof nobody doubts and nobody enjoys, because the contrast is what gives the word a meaning.
Can a computer prove a theorem, and would you believe it?
Why ask it
Two different things hide in this. One is a machine grinding through cases, as in the four color theorem, and the other is proof assistant software that checks each logical step a person has written. Find out whether they have used either.
Has AI changed how you work yet?
Why ask it
Whatever you hear is a snapshot of this year, so treat it as one. Skip the forecast and ask what they have tried it on and where it went wrong, since a specific failure tells you more than a prediction.
Why bother proving something everyone already believes?
Why ask it
A fair challenge from a student, and it deserves two answers. Belief has been wrong before, and a proof says why a thing is true, which often hands over a method that works somewhere else.
How do you read another mathematician's proof?
Why ask it
Slowly, with a pencil, and far below the speed of ordinary reading. Many try to prove the statement themselves and look at the text only when they run out of ideas, a habit a student can copy with any textbook.
Stuck
What is the longest you have been stuck on a single problem?
Why ask it
Replies come in months or years, which resets a student's idea of struggle when twenty minutes on a homework problem feels long. Put it early in a class visit, since it makes every later answer about difficulty easier to believe.
What do you try first when you are stuck?
Why ask it
Collect the list, because most of it transfers straight to coursework: a smaller case, one condition dropped, a picture, explaining it aloud to someone. Then ask which one works most often for them.
How do you decide it is time to put a problem down?
Why ask it
No rule exists, so the reply shows their appetite for risk. Some keep one long shot running beside safer projects. Ask whether a problem they dropped ever came back years later with a new idea attached.
Where were you when your best idea arrived?
Why ask it
Often nowhere near a desk: a walk, a train, the edge of sleep. The weeks before matter more than the place, so ask how long they had been failing at the problem when the idea turned up.
After all these years, does it still frustrate you not to know the answer?
Why ask it
A personal one, better across a table than from the back of a lecture hall. A mathematician who admits to doubt does a lot for a student who reads their own frustration as evidence they are not built for the subject.
What happens in the first hour after something finally works?
Why ask it
Many describe a short burst of joy followed by a nervous reread of every line. How long they wait before telling anyone is a good measure of how often the first version has let them down.
Which of your wrong ideas turned out to be useful anyway?
Why ask it
A failed approach tends to leave something behind: a lemma, a counterexample, a clearer view of why the problem is hard. Their example is a working model for how to treat a wrong answer of your own.
Big questions
What is your favorite theorem, and can you tell me what it says?
Why ask it
Many have one ready, and it is often something they met as a student and not a result of their own. Have them state it in words before any symbols go on the board, then tell you what surprised them the first time they saw it.
Is infinity a number, and can one infinity be bigger than another?
Why ask it
The first half depends on what you agree to call a number, and the second has a firm yes. Georg Cantor showed in the nineteenth century that the decimals between zero and one cannot be paired off with the counting numbers, so there are more of them, and the argument fits on one board.
Is mathematics invented or discovered?
Why ask it
The argument is old and unsettled, and most working mathematicians hold a position loosely. Bring it down to their own work: which parts felt found and which felt made. Definitions and theorems often get different answers.
Why does mathematics fit the physical world as well as it does?
Why ask it
Nobody has a settled explanation, which makes it a good Q&A question because the speaker has to think aloud. If the reply is that it does not always fit, ask for a case where the mathematics was elegant and nature ignored it.
Which famous open problem would you most like to see settled while you are alive?
Why ask it
The Riemann hypothesis and P versus NP are frequent choices. What surprises most listeners is that a good deal has already been worked out on the assumption that some of these hold, so a proof might confirm more than it overturns.
Would you call yourself a pure or an applied mathematician, and how real is that line?
Why ask it
Some departments are divided this way and people fit the division less neatly. Notice how they describe the other camp, and whether any of their own papers would be hard to file on one side.
Is your work used for anything, and would it bother you if it never were?
Why ask it
Ask from curiosity and not as a dare. Some will name a use and others will say understanding is the point. Number theory was long held up as the useless branch until cryptography came to depend on it, which is why a careful mathematician rarely says never.
Where would I meet your kind of mathematics in an ordinary day without noticing?
Why ask it
Made for a class visit. Route planning, weather forecasts, medical scans and secure payments are frequent examples, and 'nowhere yet' is an honest reply. In that case ask which neighboring area does show up.
Career
Were you the fastest in your class at school, and did speed turn out to matter?
Why ask it
Quickness on a timed test and the patience research takes are different abilities, and plenty of researchers describe themselves as slow and stubborn. Worth asking in front of students who have ruled themselves out because of a clock.
What was the first piece of mathematics that made you want more of it?
Why ask it
You are after a specific thing: a puzzle book, the argument that there is no largest prime, a teacher's aside. Note how old they were, since the range runs from early childhood to the second year of a degree.
What were the steps between your first degree and the job you have now, and how does that route work here?
Why ask it
The outline is often a doctorate followed by temporary research posts, but the names, lengths and funding differ by country and institution, so have them describe it for where you live. Then ask at which step most people leave.
What did your doctorate consist of, week to week?
Why ask it
For anyone weighing one. The balance between courses, exams and a single problem with an advisor differs by country and program, so have them describe each phase as they lived it. How they picked the advisor, and whether they would pick the same way again, is the part to write down.
What jobs do people with math training do outside universities, among the ones you know personally?
Why ask it
Finance, software, insurance, security work, teaching and government agencies all hire them. Keeping it to people they know gets you real cases in place of a brochure list, along with what each person's mathematics is used for now.
What separates a mathematician from a statistician or a data scientist?
Why ask it
Students choosing a degree blur these together. The reply usually turns on what counts as an answer: a proof, an estimate with its uncertainty, or a model that predicts well. Say which jobs you have in mind and ask which training fits them.
How hard is it to get a permanent research position, as far as you can see from where you sit?
Why ask it
Their view is from one department in one country in one decade, and conditions differ widely, so treat it as a sample. The useful version is what they tell their own students who are considering the path.
Who pays for mathematical research, and what do they expect in return?
Why ask it
It may be a university paying for teaching, a grant body, a company or a government lab, and each comes with different strings. This varies by employer and country, so ask how it works in their post and how much of the year goes on applications.
What do you look for in a student who wants to do a research project with you?
Why ask it
Researchers tend to name persistence and the nerve to say 'I do not follow' ahead of grades. Ask what a student could put in a first email that would make them reply.
If you were eighteen today, what would you study alongside mathematics?
Why ask it
Programming, statistics, physics and writing are all plausible answers, and the one they pick shows where they think the subject is heading. It is also the most concrete course advice you will get out of the conversation.
When you tell a stranger you are a mathematician, what do they say next?
Why ask it
Often a confession about hating math at school, or a request to split the restaurant bill. What they say back to that stranger amounts to a one-sentence job description, which is handy for a write-up.
What is the hardest part of the job that has nothing to do with mathematics?
Why ask it
Moving cities for short contracts, grant writing, committee work, or going months without a result to show are all candidates. A career interview that skips this gives you half the picture, so ask it before the time runs out.
Teaching
What one thing would you change about how math is taught in school?
Why ask it
Complaints tend to cluster around speed, memorized procedure and the absence of proof or play. A teacher in the room should ask what the change would look like in a single lesson, since a principle with no lesson attached changes nothing.
What would you say to a student who has decided they are not a math person?
Why ask it
Pay attention to whether the reply is a slogan or a story. The better ones name something specific, a gap from an earlier year or a habit of stopping at the first confusion, that a student could act on next week.
Which topic do schools leave out that you think teenagers would enjoy?
Why ask it
Graph theory, probability puzzles, infinity, codes and logic come up often. Whatever they name is a ready-made reading list for a student who is bored by the syllabus, so get a starting point with it.
Is there a problem you would give our class that shows what real mathematics feels like?
Why ask it
Arrange this before a class visit if you can, so they bring one. The good ones need no formulas and take longer than a lesson. Have them leave it unsolved, and report back by email if the class gets anywhere.
How do you explain an idea to beginners when you have known it for twenty years?
Why ask it
The difficulty is real: once something is obvious it is hard to remember not knowing it. Listen for a habit, such as testing an explanation on a non-mathematician, and for what they do when a lecture room goes silent.
What book, video or puzzle would you hand to a curious beginner?
Why ask it
Hold out for one title and the level it is pitched at, and write it down in front of them. A recommendation aimed at you personally is worth more than a list, so say what you last enjoyed.
Do you think memorizing multiplication tables and formulas matters?
Why ask it
Mathematicians disagree, so you are collecting an opinion and not a ruling. Some see fluency with basics as what frees attention for ideas, and others recall very little and rebuild formulas when needed. Ask which they do.
What do you wish students asked you more often?
Why ask it
A closing question that hands them the floor. The reply is frequently 'why', or anything at all during a lecture. Finish by putting to them the question they just named.
How to talk to a mathematician about the work
Practical guidance for the conversation itself
Before you sit down with them
Look up their area and take one word from it
A mathematician's web page usually lists an area and a few paper titles. You will not understand the titles, and you do not need to. Pick one word from them and open with 'what is a ...', which starts the conversation on something they can explain.
Know what you came for
A class visit works when students leave with a sense of what research is. A career interview needs the route and the hard parts. A Q&A needs one good question. Choose from the matching groups and leave the rest.
Send the plain-language questions ahead
Explaining a research area without jargon takes preparation, and the first attempt on the spot is rarely the best one. Two or three questions sent a few days early, above all the ones that ask for an example or a problem for the class, come back with much better answers.
Put a board or paper within reach
Mathematics is explained by writing. Meet where there is a whiteboard, or put a pad and a pen between you, and ask before photographing whatever ends up on it.
Choosing for the room you are in
A class visit
Share the questions out so each student owns one, and start with the day-to-day ones in The work. Save ten minutes for a problem the visitor brings, because watching a mathematician think aloud does more than any account of it. Stuck and Teaching are the groups a class tends to remember.
A career interview
Go to Career early and ask for the route step by step, then check how it works in your own country with a university or a careers office. Add three from Stuck, since handling months without progress is a large part of what the job asks.
The Q&A after a talk
Ask one question, short, with no preamble. You do not need to have followed the talk: 'what was the first case you understood' or 'what made you believe it before you could prove it' are welcome from anyone, and the speaker can pitch the answer to the room.
Dinner, a train or a family gathering
Leave the research alone at first and take the wide ones from Big questions: a favorite theorem, invented or discovered, the open problem they would like settled. If they light up, ask what they are working on and settle in.
When the answer gets technical
Ask for the smallest case
'Can you show me the simplest example where that is not obvious' is the most useful sentence to bring. A great many theorems have a small case that fits on a napkin, and mathematicians reach for it themselves when they talk to each other.
Sort theorem from conjecture
A theorem has been proved, a conjecture is believed and unproved, and a mathematician keeps the two strictly apart. When you write up the conversation, check which word they used for each statement, because swapping them is the error they will notice first.
Stop at the first unknown word
Technical talk builds, so one undefined term makes the next five minutes unreadable. Interrupting to ask what a word means is normal in a math seminar, and it shows you were following up to that point.
Wait through the pauses
A mathematician will often go quiet before answering, sometimes for longer than feels polite. They are working it out. Resist filling the gap with a second question, and the answer that comes will be the considered one.
What tends to fall flat
Arithmetic party tricks
Asking for a large multiplication on the spot tests something most research mathematicians do not practice and may be no better at than you. Ask what they can do quickly instead: estimate, spot an error, see that two problems are the same one.
'What is it good for' as a dare
Asked with an edge, the usefulness question gets a rehearsed defense. Asked with curiosity, it gets the real reasons someone spends years on a problem. The words are the same and the tone decides which you receive.
Your own proof of a famous problem
Mathematicians are often sent amateur proofs of well-known conjectures and cannot referee one over coffee. If you have an idea, ask how someone outside the field could learn enough to test it, which they can answer kindly.
One career as the map
A single person's path through degrees and posts reflects their country, their decade and some luck. Take the shape of the story and verify every detail that matters to your own plans with the institutions you would apply to.