Don’t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?(Paul Halmos, “I want to be a mathematician”)

When you learn mathematics, whether in books or in lectures, you generally only see the end product – very polished, clever and elegant presentations of a mathematical topic.

However, the process of discovering *new* mathematics is much messier, full of the pursuit of directions which were naïve, fruitless or uninteresting.

While it is tempting to just ignore all these “failed” lines of inquiry, actually they turn out to be essential to one’s deeper understanding of a topic, and (via the process of elimination) finally zeroing in on the correct way to proceed.

So one should be unafraid to ask “stupid” questions, challenging conventional wisdom on a subject; the answers to these questions will occasionally lead to a surprising conclusion, but more often will simply tell you why the conventional wisdom is there in the first place, which is well worth knowing.

For instance, given a standard lemma in a subject, you can ask what happens if you delete a hypothesis, or attempt to strengthen the conclusion; if a simple result is usually proven by method X, you can ask whether it can be proven by method Y instead; the new proof may be less elegant than the original, or may not work at all, but in either case it tends to illuminate the relative power of methods X and Y, which can be useful when the time comes to prove less standard lemmas.

It’s also acceptable, when listening to a seminar, to ask “dumb” but constructive questions to help clarify some basic issue in the talk (e.g. whether statement X implied statement Y in the argument, or vice versa; whether a terminology introduced by the speaker is related to a very similar sounding terminology that you already knew about; and so forth). If you don’t ask, you might be lost for the remainder of the talk; and usually speakers appreciate the feedback (it shows that at least one audience member is paying attention!) and the opportunity to explain things better, both to you and to the rest of the audience. However, questions which do not immediately enhance the flow of the talk are probably best left to after the end of the talk.

- Martin Schwartz, “The importance of stupidity in scientific research“,

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20 October, 2007 at 7:00 pm

AnonymousHello Terry Tao,

When I study something myself (either on my own or taking a reading course) instead of learning it from a lecture course, I found I learn them a lot worse than I think I could have if I take a formal class. But I spend probably just the same amount of time. Do you have any advice on something that I should particularly pay attention to when I learn things myself? Thanks.

21 October, 2007 at 5:55 pm

Terence TaoDear Anonymous,

Besides the advice already on these web pages, the one thing I can offer you is that when you are learning by yourself, it becomes very important to find ways to really test your knowledge of the subject, since you do not have homework, exams, or other feedback available. Doing exercises from the textbook is of course one way to test yourself, though you should resist the temptation to “cheat”, for instance by persuading yourself that you can do a problem without actually writing down all the details. But, as I already discuss in the above post, there are plenty of other usefully instructive tests you can make for yourself, for instance seeing whether you can somehow improve one of the lemmas in a text, or working through a special case of a theorem, etc.

16 February, 2011 at 10:44 am

AnonymousThis can be put into a post in the “career advise”, I think:-) Especially for those who has to learn mathematics himself/herself due to some reasons.

22 October, 2007 at 12:04 pm

AnonymousDear Professor ,

I am a doctoral student in the fourth year’s thesis; I have worked for three years on an difficult question in Penalization theory created by the authors “Roynette + Yor + Valois” :(http://arxiv.org/find/math/1/au:+Vallois_P/0/1/0/all/0/1), it involves several domains of mathematics that I maitraise shortly. Unfortunately, I dont get at any result! .

I do not know what I will do and I do not have any precise brojet for my thesis.

so i’m loking for advice for that.

Thanks in advance Professor

15 June, 2008 at 3:01 pm

这等牛人也在wordpress上写blog！ « Just For Fun[…] to mathematics, one does need to work hard, learn one’s field well, learn other fields and tools, ask questions, talk to other mathematicians, and think about the “big picture”. And yes, a reasonable amount […]

26 February, 2009 at 9:52 pm

RavishankarHello Terry Tao,

I have recently become obsessed with mathematics and proofs. I am already 24 years of age. Also whenever i see a problem, even the first example of a particular topic, i try to solve it on my own and when i am not able to solve it for some time and i look at the solution, i think why didn’t i think of this and wonder if i am not good enough. Also this seems to terrible slow down the speed at which i can study. I come from an engineering background but would like to do my graduate studies in maths. Would you still advice me to keep the same process going?(I must say that my problem solving ability has improved a lot courtesy this process). Thanks a lot.

2 March, 2009 at 11:54 am

Pacha NambiWhen I teach my classes I always tell my students that there are no such things as “dumb questions”. I emphasize to them they are in my class to learn and not sit passively but participate actively. I think this is very important, especially at the undergraduate level. I invite my students to discuss topics we cover in the class, always critically questioning what we read in textbooks. I think it is very important to not just “follow” what is in the textbook but find new ways to learn. I also learn from my students (especially from their probing questions) as much as they learn from me. I especially like students who ask the toughest questions!. I have enjoyed being a college teacher for the past 20 years – even though the pay is poor. I have also learned a lot from my students.

5 June, 2009 at 4:54 am

Essential Career Lessons[…] 8. Ask “stupid” questions So one should be unafraid to ask “stupid” questions, challenging conventional wisdom on a subject; the answers to these questions will occasionally lead to a surprising conclusion, but more often will simply tell you why the conventional wisdom is there in the first place, which is well worth knowing. (source) […]

10 January, 2010 at 10:22 pm

joe the ratThis is a really good post, thx. Sometimes I wish math books were made in such a way you can see these messy/failed lines of inquiries etc… and not only the way where you can see only the finished product. Sometimes I try to fill in these “failed lines of inquiry” but it is easy to get lost…

Are there any books where one can see the “failed lines of inquiry”? That would be great. Thx.

23 March, 2010 at 9:07 am

Solving mathematical problems – by Terrance Tao « Press4ward: faith, hope and love[…] I find that “playing” with a problem, even after you have solved it, is very helpful for understanding the underlying mechanism of the solution better. For instance, one can try removing some hypotheses, or trying to prove a stronger conclusion. See “ask yourself dumb questions“. […]

12 October, 2010 at 8:09 am

Depression and problem solving in mathematics: the art of staying upbeat « Republic of Mathematics[…] He also strongly advises that you ask yourself dumb questions – and answer them! […]

16 February, 2011 at 12:47 pm

petequinnIn a similar vein, Thomas Edison famously said: “We now know a thousand ways not to build a light bulb,” and “I never failed once. It just happened to be a 2000-step process.”

If you never try because you are afraid of failure, you will never succeed. Embrace your mistakes – that’s how you learn some things most effectively, by first discovering first hand the wrong way to do things. This can really help solidify the right way for you.

12 March, 2011 at 9:16 pm

Depression and problem solving in mathematics: the art of staying upbeat — Republic of Mathematics blog[…] He also strongly advises that you ask yourself dumb questions – and answer them! […]

16 March, 2011 at 5:09 am

Trivial reduction question is deep | Nanoexplanations[…] I learned a lot from that answer, and I bet others did too, given how dismissive we were of the question. This reminded me of perhaps my favorite research advice: don’t be afraid to ask dumb questions (and answer them). […]

17 July, 2011 at 8:17 pm

Career Advice From a Mathematician « I, Geek[…] Ask questions and connect with other people […]

28 July, 2011 at 3:54 am

raminhi professor

can you give some detailed examples of dump questions about an easy problem and then answer them? for example consider the series 1+2+3+…+n. or any other easy problem you wish.

many hard textbook problems get solved after a week of concentration and hard work but at the end i feel like there is more to learn. polya mentions asking dump questions too. but i have seen no real examples in his book “how to solve it”.

28 July, 2011 at 7:19 am

Terence TaoWell, the key point is that

youhave to come up with the questions; it doesn’t work nearly as well if someone else provides the questions for you.That said, there are plenty of questions around the standard exercise of evaluating that are worthwhile. Some examples: can the formula and its derivation be extended to the case where n is zero? negative? real? complex? What is the relationship between the sum and the integral ? What can one say more generally about (for k a natural number, integer, real, complex, etc.)? For instance, is it always a polynomial in n? Is there a way to assign a useful meaning to the infinite series , and how does this connect to the Riemann zeta function? (I discuss this latter point in this post.) The quantity resembles a binomial coefficient – can one find a generalisation of the identity that emphasises this viewpoint? Related to this, is there a combinatorial proof of the identity? Can it be categorified? Is there a geometric proof? Does it extend to higher dimensions? What if one replaces by another arithmetic progression? What happens to all of the above questions if one replaces addition with multiplication? Or works in another group or ring? The standard proof of the identity uses induction; what happens if one somehow deletes the axiom of induction from the number system? etc., etc.

Note that many of these questions are vague and open-ended, and thus quite distinct from the typical exercise one may find in textbooks. Unlike such exercises, the point is not actually to find definitive answers to these questions, but rather to get your brain to start following original lines of inquiry, and to develop the type of mindset that is needed for genuine mathematical research. (In many cases, the answers turn out to be degenerate or otherwise uninteresting, but the

processof arriving at such an answer is often quite instructive.)7 September, 2011 at 11:24 pm

Vinícius Machado VogtI’m writing just to say I really enjoyed this answer!

25 December, 2011 at 8:41 am

HupaledI love this answer. Many students actually ask a kind of questions that Pro.Tao mentions in the begining, but soon lost their interest in doing so because of formal education forcing them to find just the right answers.

This answer should be written in every math textbooks.

14 December, 2016 at 1:14 pm

JensenThis answer shows the very skill one should learn in his undergrads, but you are right, we do all get lost in the formalities of solving the problems, because that is the only way to get into exams, and then hopefully eventully graduate.

On the other hand, this is also the very reason I have been losing passion and motivation for pure maths, because :

1. No one gives you such good examples of genuine inquisitive thinking

2. From my experience, different than expected proofs are rated much lower than the standard ones which follow the lecture, because they require more time given by the tutors/assistants to understand where it’s coming from, for which they don’t get more money, so they would basically “punish” you for it with low marks and points.

3. To solve all weekly problems, you just give up almsot all of your time, which leaves almost no time for independent “research”, inquisitive analysis of the course and a deep understanding of it.

4. it is never clear what is expected of you, for so many times, I got less points than should have, for “style” issues and very subjective ideas of “proper format”, depending on the tutor.

5. The exams have little to do with what we’re spending almost all of our time doing: proofs! Since the exam is only 90 min, they can’t expect us to prove anything serious, so it usually involves advanced exercises from engineering maths, which is a skill developed by practicing a lot of recipe-type exercises, in order to solve them faster, which we never do, since we’re only working on very special, specific or at times even eccentric exceptions of the “rules”.

6. The mindset that is valuable in research is actually counterproductive in an undergrads course for lack of time, and fast paced jumping from one chapter to the other.

7. Besides the fact, but also the very idea that there is a 90% failure rate at most exams, feels you’re set up for failure before you even started and then when you do start, the lack of proper training to pass the exams they themselves designed.

8. The fact that Professors are not accesible to students, as they need to go through tutors first, then assistant and only then Professor, should the former ones had failed to come with an answer to your question. Since tutors are students themselves, their interest is not in teaching but earning money, so usually one gets deflected reply to the question at hand.

9. We, the students, never see a proper proof to the homeworks we get, so for the most part we are left with huge gaps in understanding of the course.

So my question is, how does one keep his morale throughout his/her studies?

Since the time is always so limited, we almost never get to rethink an entire proof and thus try to find more elegant solutions, which would be a great way to keep high spirits and optimism.

Since it’s undergrads studies, our courses are also mostly introductions into major fields, with no specific target or scope, which again would be a great motivation.

My wish is to persue research in pure maths, but as I go on I only feel more unadequate to understand maths. This utter formalism, though so immensely neat and organized, is yet not the way that maths came about, which was very organic and natural in a sense, as ever more questions demanded answers. It also isn’t the way in which modern research in math is conducted, there are no chapters to bound you from using any other techniques than the ones learned recently, nor sets of 90 min periods to solve problems,nor a one chance only to give in a solution, but quite the opposite. So I have to ask myself, what is my programme training me for?

Please DO correct me if I’m mistaken, and set me on a better train of thoughts.

many thanks for this article!

15 December, 2017 at 3:21 am

AlpxHi,

I am not sure if it helps to write my thoughts here, but I read your post and I certainly hear you. First, I don’t agree that the skills you need for research math and the skills you need for undergrad success are so mutually exclusive. Try to get a decent grade from your courses, but don’t let this eat up all your time. Spare time for the type of math you find interesting and get into it.

You may want to visit a prof from the area of math you find interesting, and ask for accessible papers to read? If the prof is eager to give you a small research project and guide you through, it is even better.

I think personal interactions with an active research mathematician will make you feel math is an organic product :)

31 July, 2011 at 10:50 am

raminthank you professor.

i understand that I should come up with the questions, i just wanted to have an example in mind, because sometimes i ask many dump questions, and i don’t know which i should follow.

some of my questions are like:

1.how on earth gauss came up with that solution?

2.or after solving a problem i ask myself why did it take me so long to see this obvious point?

3.or can i solve it without any tricks, just ordinary, mainstream deduction?

4.why is the solution working?

5.is the solution a special case of some thing bigger?(i always feel like it should be. it really bothers me to have a solution that i don’t understand. even if i have come up with it myself)

but if i understand correctly the focus of dump questions should be on mathematical knowledge not on problem solving or things of that sort. so maybe questions 1 and 2 are not that good. Am i correct?

do you think my other questions can be fruitful?

5 January, 2014 at 2:29 am

BastiI think that 1 is a good question. This is mainly because it leads you to look for different ways of proving the identity. Usually it is proved by induction on the already established equation, which, of course, can’t be the way the identity was originally found.

So, actually your question 1 explores into many interesting opportunities for research because you have to dig into the structure of the sequence.

(Gauss’ original proof is much more elegant and clear than the typical first semester induction exercise. Afaik, he first observed that adding the highest and lowest number in the progression always adds up to n+1 (1+n = n+1; 2 + (n-1) = n+1; …) and then he just observed that he can do this n/2 times until all the numbers have been exhausted, thus the sum equates to (n/2)(n+1).)

6 September, 2017 at 12:36 pm

AnonymousPlease note you can’t do this n/2 times if n is odd.

6 September, 2017 at 12:28 pm

AnonymousThe sum of any collection of numbers is the size of the collection times the arithmetic mean (that’s the definition of the arithmetic mean). When you’re given a collection from 1 to n, it’s fairly easy to know both the mean (which is (1+n)/2 for reasons of symmetry) and the size of the collection (obviously, n). Hence, we get the formula. While not a formal proof (what that symmetry is about, anyway…), still a useful trick to keep the formula in memory.

The poster didn’t invite any kind of dumb questions, just constructive ones. How to tell a constructive dumb question from a non-constructive one, I have no idea, but the name suggests such questions either construct something or request a construction (such that a smart person, apparently, would not request because “it’s obvious” or …).

17 September, 2011 at 9:52 am

Hacerse preguntas « alexmoqui[…] para aprender matemáticas. En este post de Terence Tao se habla sobre este tema. En concreto, en este comentario Tao da numerosos ejemplos de cuestiones que uno puede preguntarse a partir del sencillo problema […]

6 December, 2011 at 3:28 am

Odiare la matematica - Dueallamenouno - ComUnità - l'Unità[…] blog, tra le qualità che un matematico deve sviluppare c’è sicuramente la capacità di fare domande stupide. Oggi questo non è incoraggiato nella scuola, e forse è da lì che bisogna ricominciare. Togliere […]

14 December, 2011 at 2:52 pm

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9 December, 2012 at 2:25 am

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27 March, 2013 at 11:38 pm

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1 May, 2013 at 9:41 am

Bored now | Mathematical Field Notes[…] the questions inside their heads, if they try to iron out the conflicts in their own understanding. Terry Tao, the Fields Medallist, recommends asking yourself “dumb” questions as a great way of learning, even (/especially) for researchers, because if you can’t work out […]

21 May, 2013 at 3:35 am

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19 July, 2013 at 8:58 am

Does one have to be a genius to do maths? | Singapore Maths Tuition[…] to mathematics, one does need to work hard, learn one’s field well, learn other fields and tools, ask questions, talk to other mathematicians, and think about the “big picture”. And yes, a reasonable […]

10 December, 2013 at 9:33 pm

QuoraWhy do proofs in math textbooks so rarely make sense?I think this is a skill that one has to learn herself or himself. From my experience the best way to do so is to look at a proof and pick away the key components. From this one can learn the key machinery of a particular field and how those pieces of m…

3 February, 2014 at 6:39 pm

AntonioHello professor Terence Tao

I have read the quote above and I really like it, so I have related with one question I would like to ask you. How is the interactions between the process of creating mathematics, I mean, development of new mathematics, and the process of resolving or proofing theorems, lemmas or something like that uniquely ? because I sometimes ask myself about what should be more important, to resolve or proof things or create mathematics.

I’m undergraduate student and I’d really like what you think about this.

I must stress English is not my first language, so whether there are any grammatical mistake, I sorry for it.

thanks

16 February, 2014 at 8:30 am

AnonymousDear Prof. Tao

Today textbooks make it very hard for a student like me to tell what is the main point of a chapter and which parts are just details. so that we can ask questions related to that main point. and if I spend asking questions about every single proposition or definition in the textbook it will get a huge amount of time and I think one will get lost in all the details. what do you propose we do?

for example should we read the whole thing first and get the large idea then get back and ask questions or change the textbook or…??

6 February, 2015 at 5:27 pm

Career Advice by Prof Terence Tao, Mozart of Mathematics | MScMathematics[…] I find that “playing” with a problem, even after you have solved it, is very helpful for understanding the underlying mechanism of the solution better. For instance, one can try removing some hypotheses, or trying to prove a stronger conclusion. See “ask yourself dumb questions“. […]

16 April, 2015 at 10:22 pm

Ask yourself dumb questions – and answer them! | ressonliao[…] Ask yourself dumb questions – and answer them!. […]

11 July, 2015 at 5:58 am

译：解决数学问题 by 陶哲轩 | 万里风云[…] 我发现，“把玩”一个问题，甚至是当你解决它之后，是对理解问题答案的本质机理是有很大帮助的。比如，你可以试图移除一些假设的条件，或者试图证明一个更强的结论。可以读我的另一片博文 《问你自己‘傻’问题——然后回答它们！》。 […]

18 August, 2015 at 11:37 pm

Moses1am 17, am studying “number theory”.I download online PDF books on the subject,But i usually face ” blocks” because i dont have any tutor, Can you illuminate me on a way to become better in the subject@ Proffesor tao,I need your advice.

24 December, 2015 at 2:17 am

RonitHello sir….does the number of groups upto isomorphism of order 2^n always exceed that of a smaller naturan no?

25 February, 2017 at 11:58 pm

Paul CerneaLet n = 1. Then there is one group of order 2 up to isomorphism. But there is also one group of order 1 up to isomorphism.

7 May, 2016 at 5:13 am

Intuition – unimathverse[…] you are in vector space and every object around you is properties, operators in vectors space. Then ask yourself those dumb question that terry tao mentioned in his […]

22 June, 2016 at 1:10 am

Smart Imposter Syndrome – G-Notes[…] one does need to work hard, learn one’s field well, learn other fieldsand tools, ask questions, talk to other mathematicians, and think about the “big picture”. And yes, a reasonable […]

18 July, 2016 at 8:29 am

Solving mathematical problems | nguyen Huynh Huy's Blog[…] I find that “playing” with a problem, even after you have solved it, is very helpful for understanding the underlying mechanism of the solution better. For instance, one can try removing some hypotheses, or trying to prove a stronger conclusion. See “ask yourself dumb questions“. […]

13 August, 2016 at 4:18 am

Random Stuff « Econstudentlog[…] skills to test and refine these intuitions rather than discard them. One way to do this is to ask yourself dumb questions; another is to relearn your field.” (Terry Tao, There’s more to mathematics than rigour and […]

9 June, 2017 at 9:53 am

ANSHUMAANTo Mr.Tao

I am an undergrad student of mathematics . As you stated in this article my approach is very similar but sometimes i get stuck in loop of questions which argues with my philosophical insight. But i get more deep and getting more confused. I can’t state that feeling but it annoys me and my pace according to syllabus becomes very slow ……and i am very critical to myself.

Sorry for my English ….Its not that good

13 June, 2017 at 1:13 pm

Michael M. RossI know how to ask “stupid but interesting” questions. I just have the wrong education for someone interested in math – that is, almost none except the stuff I’ve taught myself. I think of myself as a “conceptual number empiricist”. I like recasting conjectures so they are stated in a way that makes you think differently about them. For amusement:

Twin Primes Conjecture

There are infinitely many semiprimes of the form n^2−1.

Legendre’s Conjecture

The unequal cardinality of the subsets of odd and even composites between perfect squares is deterministic.

Goldbach’s Conjecture

Every integer greater than 3 can be expressed as the average of two primes.

Collatz Conjecture

I can tell you what it’s not about: It’s not about multiplying by 3, adding 1, and dividing by 2. That is an extremely crude definition that misses the point entirely. This conjecture is not even about even numbers. I think it can be stated in three rules that contain the proof for both parts of the conjecture. (Don’t believe me? Ask me please.)

You see, I don’t know enough to know how little I know….

23 January, 2018 at 8:24 pm

Có Nhiều Hơn là Sự Chặt Chẽ và Chứng Minh trong Toán Học | 5[…] vì vứt bỏ chúng. Một cách để làm điều này là tự hỏi bản thân các câu hỏi ngớ ngẩn; một cách khác là học lại phân ngành của […]

14 July, 2018 at 4:14 am

AnonymousAn interesting “dumb question” which even a teenager (e.g. the young Einstein) first learning Newton’s laws might ask himself: According to the first law “a body under no influence of external force should continue its trajectory through space in a straight line and constant velocity” – but it is still not clear

1. In which coordinate frames the trajectory is a straight line? and is there any “objective” (e.g. by certain geometrical properties of space) to distinguish such “special” (inertial) frames from all other possible frames?

2. Is there any “objective” (e.g. by certain geometrical properties of space and time) to distinguish the particular “time flow” or “time parameter” under which the velocity is constant ?

Such “dumb thoughts” (based only on the “democratic” principle of not artificially distinguishing among observer frames) have the potential to lead such teenager to (eventually) develop the special and general theories of relativity !

2 November, 2018 at 12:01 pm

Anonymoushow to understand deeply and accurately mathematical concepts?

15 December, 2020 at 8:19 am

V__johHello terry tao,

Where can i find some unsolved problems so

That i can try my luck on them.How can i track down some advancement on the problem.

15 December, 2020 at 11:43 am

Anonymoushttps://www.claymath.org/millennium-problems

Good luck have fun

31 January, 2021 at 7:51 am

LarryIn my beginning of study of topology l was doing ok Now I am floundering like a fish out of water. My proofs are horrid when l am post them and others state really idiotic statements. Don’t know how to cope…

28 February, 2021 at 8:46 am

DanielI’m an Engineering student in my second year, but with a very avid interest in higher mathematics. Is there any more efficient method of self-study and also is it a good idea to restrict my study to certain fields only at this stage?

8 May, 2022 at 2:28 am

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12 January, 2023 at 6:47 am

AnonymousThank you so much for your advice.

I have one doubt( for self teaching students)

If we are asking such “dumb” questions, and after struggling for a long time, we don’t find the answer on our own, where to find the answers?

These questions are generally poorly received by most mathematics QNA forums, although they are entirely legitimate questions. Any advice would be highly appreciated

12 February, 2023 at 12:30 am

Easy Lebesgue integral of a non-horizontal line but by definition – Mathematics – Forum[…] a dumb question based on all the questions I've asked for the last decade, but what's the general way to do the […]

12 February, 2023 at 12:39 pm

AnonymousMany (if not most) scientific breakthroughs were due to new ideas which resulted from a a new ihterpretation of existing knowledge. Such new interpretations and new (i.e. “fresh”) points of view could result from “asking dumb questions”).