Here's my issue I faced; I worked really hard studying Math, so because of that, I started to realised that I understand things better. However, that comes at a big cost: In the last few years, I had practically zero physical exercise, I've gained $30$ kg, I've spent countless hours studying at night, constantly had sleep deprivation, lost my social life, and developed health problems. My grades are quite good, but I feel as though I'm wasting my life.

I love mathematics when it's done my way, but that's hardly ever. I would very much like my career to be centered around mathematics (topology, algebra or something similar). I want to really understand things and I want the proofs to be done in a (reasonably) rigorous way. Before, I've been accused of being a formalist but I don't consider myself one at all. However, I admit that I am a perfectionist. For comparison, the answers of Theo, Arturo, Jim Belk, Mariano, etc. are absolutely rigorous enough for me. From my experience, $80$% or more mathematics in our school is done in a sketchy, "Hmm, probably true" kind of way (just like reading cooking recipes), which bugs the hell out of me. Most classmates adapt to it but, for some reason, I can't. I don't understand things unless I understand them (almost) completely. They learnt "how one should do things", but less often do they ask themselves WHY is this correct. I have two friend physicists, who have the exact same problem. One is at the doctorate level, constantly frustrated, while the other abandoned physics altogether after getting a diploma. Apart from one $8$, he had a perfect record, all are $10$s. He mentioned that he doesn't feel he understands physics well enough. From my experience, ALL his classmates understand less than he does, they just go with the flow and accept certain statements as true. Did you manage to study everything on time, AND sufficiently rigorous, that you were able to understand it?**


Frequently, I tend to be the only one who find serious issues in the proofs, the formulations of theorems, and the worked out exercises at classes. Either everyone else understands everything, most or doesn't understand and doesn't care the possible issues. Often, do I find holes in the proofs and that hypotheses are missing in the theorem. When I present them to the professor, he says that I'm right, and mentioned I'm very precise. How is this precise, when the theorem doesn't hold in its current state? Are we even supposed to understand proofs? Are the proofs actually really just sketches? How on earth is one then supposed to be able to discover mathematical truths? Is the study of Mathematics just one big joke and you're not supposed to take it too seriously?


I have a bunch of sports I like and used to do. Furthermore, I had a perfectly good social life before, so you don't need to give advice regarding that. I don't socialize and do sport because digesting proofs and trying to understand the ideas behind it all eats up all my time. If I go hiking, it will take away $2$ days, one to actually walk + one to rest and regenerate. If I go train MMA, I won't be focused for the whole day. I can't just switch from boxing to diagram chasing in a moment. Also, I can't just study for half an hour. The way I study is: I open the book, search up what I already know but forgot from the previous day, and then go from theorem to theorem, from proof to proof, correcting mistakes, adding clarifications, etc. etc. To add on, I have a bad habit of having difficulty starting things. However when I do start, I start 'my engine', and I have difficulty stopping, especially if it's going good. That's why I unintentionally spend an hour or two before studying just doing the most irrelevant stuff, just to avoid study. This happens especially when I had more math than I can shove down my throat which I have, for mental preparations to begin studying. But, as my engine really starts and studying goes well (proven a lot, understood a lot), it's hard for me to stop, so I often stay late at night, up to 4 a.m., 5 a.m. & 6 a.m. When the day of the exam arrives, I don't go to sleep at all, and the night and day are reversed. I go to sleep at 13h and wake at 21h... I know it's not good but I can't seem to break this habit. If I'm useless through the whole day, I feel a need (guilty conscience) to do at least something useful before I go to sleep. I know this isn't supposed to happen if one loves mathematics. However, when it's 'forced upon you' what and how much and in what amount of time you have to study, you start being put off by math. Mathematics stops being enjoyment/fun and becomes hard work that just needs to be done. enter image description here

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    A remark by Gian-Carlo Rota is apropos: "The facts of mathematics are verified and presented by the axiomatic method. One must guard, however, against confusing the *presentation* of mathematics with the *content* of mathematics. An axiomatic presentation of a mathematical fact differs from the fact that is being presented as medicine differs from food. It is true that this particular medicine is necessary to keep mathematicians at a safe distance from the self-delusions of the mind. Nonetheless, understanding mathematics means being able to forget the medicine and enjoy the food." – Bill Dubuque Jun 11 '11 at 05:36
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    The above is excerpted from [p.96 of Rota's *Indiscrete Mathematics*.](http://books.google.com/books?id=sahFH2CcpywC&pg=PA96) – Bill Dubuque Jun 11 '11 at 05:37
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    @Bill: Indiscrete thoughts. – Martin Sleziak Jun 11 '11 at 05:46
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    As far as "having a healthy lifestyle", I have found that [Parkinson's Law](http://en.wikipedia.org/wiki/Parkinson%27s_law) definitely applies. Just like I find the need to "make an appointment with my research" (set aside a certain amount of time on most days to think about my research, whether I am making progress or not), I also find that if I set aside the time to exercise/eat out/etc, I can use it for that, but if I don't set it aside, it's not like I have a bunch of time I'm not doing anything. Set aside the time. – Arturo Magidin Jun 11 '11 at 06:12
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    I'd try to get an overview of whatever I'm learning first. I'd like to think of it as a big canvas; fill in the details according to whatever piques your interest. You absolutely positively can't learn everything (not even close), but you can learn top-down instead of bottom-up. For your comments on rigour, you might find this interesting: http://www.cheng.staff.shef.ac.uk/morality/morality.pdf – Eivind Dahl Jun 11 '11 at 09:42
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    Note that the lack of sleep can make it harder to remember things. Try sleeping more (i.e., at least 8 hours a night), it'll make you more productive, and might even end up gaining you more time than that you invest in sleeping due to not forgetting so much. – Sverre Rabbelier Jun 11 '11 at 11:01
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    This really struck a nerve in the community here; no other questions produces 6 lengthy answers in 6 hours! So @Leon: Cherish the fact that you're not alone in this situation until you find a remedy. :) – Roar Stovner Jun 11 '11 at 11:54
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    Most likely the source of your problems is lack of sleep. I find it impossible to be efficient without being well rested. (That is, I need my 9 hours) Also, think about math problems right before bed, that way you might dream about them, and part your mind will still be at work. Just last week, while lying in bed trying to get to sleep, I came up with an idea to solve a problem I had been thinking about for about 6 hours earlier that day. – Eric Naslund Jun 11 '11 at 22:42
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    I found this article [A Mathematician's Survival Guide](http://www.math.missouri.edu/~pete/pdf/140-MAA.pdf) by Peter G. Casazza "intended as a survival guide for those students, teachers and mathematicians who are having trouble interpreting the mathematical experience." Maybe it might help you. – Américo Tavares Jun 12 '11 at 23:24
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    I find that going for a walk helps me to think and get stuff straight in my head. Walking helps me to solve problems. Also, it isn't exactly running a marathon but going for lots of walks does get me enough exercise! – user1729 Oct 24 '11 at 14:34
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    As to your memory, an Erdos anecdote might ease your mind: He would forget definitions on a day to day basis, and would ask, "what is housdorff..." and then it was explained to him and he proved the theorem at hand, the next day someone says: "Let $X$ be a housdorff space" and Erdos asks "what is housdorff...". Its not just memorizing everything, but more about internalizing the techniques and understanding the big picture. (And it always helps to be Erdos :P) – Robert Cardona Oct 01 '13 at 20:30
  • I don't remember writing this but I must have because you are describing me. I suffered from stress throughout college, some ways I got by were to go to sleep by midnight no matter how much homework I had (otherwise I am useless) and get a 15min walk, even if it was just to my car. I started a meditation practice of 8min a day, which seemed to help. And I worked at a library so I could do homework at work. My going through school did hurt my relationship with my girlfriend, but we put things back together and were married afterwards. My biggest piece of advice is don't stress grades, haha no? – kleineg Jan 15 '14 at 20:35
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    "How to study math . . . and have . . . free time?" Solution: do math in your free time. – imallett Oct 25 '14 at 07:09
  • Do math in free time and have relax orientation, its ok if you misses something. Simple hardwork won't work in maths – Anurag Nov 06 '16 at 07:51
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    The triangle is quite beautiful and truthful. – Simply Beautiful Art Jan 28 '17 at 15:42
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    I have an idea but I'm not sure if it will help. I even don't know that it won't make it worse so I will leave it up to you to judge whether you think it might make it worse. One idea is reading the answers to https://math.stackexchange.com/questions/1333206/how-to-begin-self-study-of-mathematics. Another idea is to come up with your own system of pure number theory. I think everyone is able to keep taking in new pieces of information and retaining them at a nonzero rate even if it's a very sluggish rate. To train your memory, you could keep figuring out more statements provable in that system – Timothy Jan 17 '19 at 01:18
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    in your head. I think that occasionally you will figure out one statement and you will have figured out the exact same statement and forgotten it so many times before that you will finally retain it this time. By not keeping a record of what you figured out, you might slowly get better at retaining new statements you figure out and figure out new statements that you will retain slightly more often than before. You might be like I can't figure out any statements that are actually useful and feel stuck. That doesn't matter. It's still better to figure out one more statement and retain it than no – Timothy Jan 17 '19 at 01:25
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    more. Maybe as statements you previously figured out that you felt were useless can be used to figure other statements, you will eventually figure out enough statements to understand so much and then realize that those statements weren't useless after all because they eventually lead to useful discoveries. If the system is simple enough that you instead get the opposite effect that you claim to have gained a full understanding of the system and it seems so simple, then you could extend the system to stronger system to describe the statement that all statements that system proves are true. – Timothy Jan 17 '19 at 01:32
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    Finally, my third idea relies on my uncertain hypothesis that Math Overflow is suitable for questions that prompt useful math research that's not too hard to do and is to move this question to MathOverflow to prompt research on how to create a course suited for people like you. If they won't accept that question, then maybe you could first ask a question on Academia Stack Exchange to learn more about why they teach the way they do or why people's brains learn in a certain way then maybe you will have the resources to ask that question on MathOverflow. If you're a researcher, maybe you can also – Timothy Jan 17 '19 at 01:37
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    ask it on ResearchGate. If not, maybe you could ask a question on Stack Exchange to gain information to help you figure out how you could become a researcher but I don't know which Stack Exchange website. – Timothy Jan 17 '19 at 01:40
  • I also think it's probably better not to get any aid from a Python like computer program to make calculations for you or Google search to find answers to questions. Maybe those answers are answers you don't actually need because you will always be able to mentally figure out one more statement and retain it but using them will get you into the habit of not figuring out any statements on your own. Python can't do creative thinking for you and give you a true mathematical statement equivalent to a written English answer that will teach you how to be really good at math. There may be a – Timothy Jan 17 '19 at 01:53
  • provable mathematical statement that once you learn it, you will be able to figure out a very artifically intelligent specific math technique but you're probably more likely to find it if you don't get help from Python than if you do. According to https://matheducators.stackexchange.com/questions/7718/effects-of-early-study-of-advanced-books, people who study advanced books at a young age are probably less likely to be able to figure out stuff on their own and be stuck when they have a question that they can't find a book or Google search result that answers. – Timothy Jan 17 '19 at 01:57

27 Answers27


In my view the central question that you should ask yourself is what is the end goal of your studies. As an example, American college life as depicted in film is hedonistic and certainly not centered on actual studies. Your example is the complete opposite - you describe yourself as an ascetic devoted to scholarship.

Many people consider it important to lead a balanced life. If such a person were confronted with your situation, they might look for some compromise, for example investing fewer time on studies in return for lower grades. If things don't work out, they might consider opting out of the entire enterprise. Your viewpoint might be different - for you the most important dimension is intellectual growth, and you are ready to sacrifice all for its sake.

It has been mentioned in another answer that leading a healthy lifestyle might contribute to your studies. People tend to "burn out" if they work too hard. I have known such people, and they had to periodically "cool off" in some far-off place. On the contrary, non-curricular activities can be invigorating and refreshing.

Another, similar aspect is that of "being busy". Some people find that by multitasking they become more productive in each of their individual "fronts". But that style of life is not for every one.

Returning to my original point, what do you expect to accomplish by being successful in school? Are you aiming at an academic career? Professional career? In North America higher education has become a rite of passage, which many graduates find very problematic for the cost it incurs. For them the issue is often economical - education is expensive in North America.

You might find out that having completed your studies, you must turn your life to some very different track. You may come to realize that you have wasted some best years of your life by studying hard to the exclusion of everything else, an effort which would eventually lead you nowhere. This is the worst-case scenario.

More concretely, I suggest that you plan ahead and consider whether the cost is worth it. That requires both an earnest assessment of your own worth, and some speculation of the future job market. You should also estimate how important you are going to consider these present studies in your future - both from the economical and the "cultural" perspective.

This all might sound discouraging, but your situation as you describe it is quite miserable. Not only are you not satisfied with it, but it also looks problematic for an outside observer. However, I suspect that you're exaggerating, viewing the situation from a romantic, heroic perspective. It's best therefore to talk to people who know you personally.

Even better, talk to people who're older than you and in the next stage of "life". They have a wider perspective on your situation, which they of their acquaintances have just still vividly recall. However, even their recommendations must be taken with a grain of salt, since their present worries are only part of the larger picture, the all-encompassing "life".

Finally, a few words more pertinent to the subject at hand.

First, learning strategy. I think the best way to learn is to solve challenging exercises. The advice given here, trying to "reconstruct" the textbook before reading it, seems very time consuming, and in my view, concentrating the effort at the wrong place

The same goes for memorizing theorems - sometimes one can only really "understand" the proof of a theorem by studying a more advanced topic. Even the researcher who originally came out with the proof probably didn't "really" understand it until a larger perspective was developed.

Memorizing theorems is not your choice but rather a necessity. I always disliked regurgitation and it is regrettable that this is forced unto you. I'm glad that my school would instead give us actual problems to solve - that's much closer to research anyway. Since you have to go through this lamentable process, try to come up with a method of memorization which has other benefits as well - perhaps aim at a better understanding of "what is going on" rather than the actual steps themselves. This is an important skill.

Second, one of the answers suggests trying to deduce as many theorems as possible as the "mathematical" thing that ought to be done after seeing a definition. I would suggest rather the opposite - first find out what the definition entails, and then try to understand why the concept was defined in the first place, and why in that particular way.

It is common in mathematics to start studying a subject with a long list of "important definitions", which have no import at all at that stage. You will have understood the subject when you can explain where these definitions are coming from, what objects they describe; and when you can "feel" these objects intuitively. This is a far cry from being able to deduce some facts that follow more-or-less directly from the definitions.

Yuval Filmus
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    @Yuval: Very well written (even had to look up some words), and above all, great advice, I am really grateful for your suggestions and your time. For me, the issue is certainly not economical, since studies here are free. "I always disliked regurgitation and it is regrettable that this is forced unto you." No no, we are certainly not forced to memorize things like in social sciences in any way, but with each new subject, it is assumed that we know most of the material from previous ones, which is a lot, and I constantly refresh and forget. – Leo Jun 15 '11 at 02:08
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    It is really frustrating that I understand a subject and in 6 months it's almost gone from my head. Even more so, when some classmates seem to have far better memory. Of course when I relearn, it goes smoother & faster, but the point is I have to relearn over and over. And since the amount of material only increases, this is becoming harder and harder. There are many many fields of mathematics, and the number of all theorems, lemmas, propositions and definitions is immense. Of course, intuition always helps, but it doesn't create miracles. Like in sports, without cardio, an athlete is doomed. – Leo Jun 15 '11 at 02:18
  • @Yuval: BTW, I almost didn't notice there was another (your) answer. I was told that using @ informs a person of another comment directed at him. Where/how do I notice these, without browsing through all my questions every time? – Leo Jun 15 '11 at 02:23
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    @Leon: Doing mathematical research entail amassing a vast store of knowledge and continually drawing upon it; not necessarily by calling upon an obscure lemma or even a moderately important theorem, but rather through analogies and through an intuitive understanding of the nature of some mathematical objects, and techniques to reason about them. If you honestly find that you keep forgetting old material - and by that, I mean the essentials, not advanced results - you are probably studying either too fast, or too much, or too nervously. – Yuval Filmus Jun 15 '11 at 04:23
  • As for your technical question, I believe that you do get notified - some special signal appears at the top of the page. And if you tick the correct box when asking the question, you are notified by e-mail whenever anyone answers your question. – Yuval Filmus Jun 15 '11 at 04:26
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    I think I mostly don't forget essentials, but to apply a certain lemma, one has to know the hypotheses. There are many pitfalls, many counterexamples, in topology, algebra, and analysis that clearly demonstrate how easy it is to make an incorrect conclusion. – Leo Jun 15 '11 at 04:29
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    It takes time to get used to a new form of knowledge, especially one as complicated as mathematics. At some point you will reach a new level of understanding - suddenly all the previous material, that used to seem so difficult, will become manifestly simple. You will start to remember concepts and basic results soon after solving some relevant exercises. It may seem incredulous, but if you persevere it will eventually happen. Be patient and relaxed. Everyone was once in your current situation. Things do get better eventually. – Yuval Filmus Jun 15 '11 at 04:36
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    This may not be the highest rated answer (yet) but it is certainly the most well-rounded in my opinion. I agreed with the observation that some of the other methods outlined on this page are time-consuming. I've tried them, and you end up learning a lot less than if you had, say, just covered some basic material and tried some exercises, gently increasing with difficulty each time. – Sputnik Jun 18 '11 at 16:21
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    What did @YuvalFilmus do between 2002 (end of MSc) and 2009 (beginning of PhD)? – Nike Dattani Jul 18 '20 at 19:01

Let me tell you that the only thing that I have been doing for the last four years of my life is mathematics. I have enjoyed the experience thoroughly but I have also had points where I was somewhat unsure as to how to approach my learning. I think that there is no one rule that works for everyone; however, let me answer some of your questions. I hope that I can help:

Question: How to study mathematics the right way?

Answer: I think that the best way to study mathematics is as follows. Let us assume that you have already chosen a mathematics book on a subject that you are really interested to learn. When you read the book, aim to actively think about the subject matter in different ways. For example, if a definition is presented, spend at least 30 minutes to think about the definition. If you are studying a book on linear algebra and the definition of a "nilpotent operator" is presented, you should try to discover some basic properties about nilpotent operators on your own without reading further. This can be difficult at first but ultimately an ability to do this effectively with as many definitions as possible is important in research mathematics.

Let us take the following example in elementary group theory. The author presents the definition of a maximal subgroup of a finite group $G$: a subgroup $M$ of $G$ is said to be a maximal subgroup if $M$ is a proper subgroup of $G$ and if there are no proper subgroups of $G$ strictly containing $M$. You should try to take the following steps:

(1) Find examples of maximal subgroups in finite groups and begin with the most trivial examples! For example, the trivial group can have no maximal subgroup. If you understand this, you have grasped one point of the definition. The next step is to consider the simplest cyclic groups. What are the maximal subgroup(s) of the cyclic group of order 2? What are the maximal subgroup(s) of the cyclic group of order 4? Think about basic examples such as this one. When you are ready, try to formulate a general theorem on your own which concerns maximal subgroups of a cyclic group of order $n$. You should arrive at the theorem that a subgroup $H$ of a cyclic group $G$ is maximal if and only if the number $\frac{\left|G\right|}{\left|H\right|}$ is prime.

Continue to find other examples of maximal subgroups in a finite group. The next step is to consider the Klein 4-group and the permutation groups of low orders. I hope at this point you are really fascinated by the concept of a maximal subgroup. At first, the definition might seem like something arbitrary; however, now that you have thought about it, you have started to gain a sense of "ownership" over the definition.

(2) It is now time to formulate and prove some theorems about maximal subgroups. Again, think of the easiest examples. One thing that can be discouraging for a beginner is to not be able to answer a question that looks easy over a long period of time. What is a good example of an easy theorem? You can study those finite groups which have exactly one maximal subgroup. What can you deduce about such a group? If you find that you are stuck, try to work back to the examples of maximal subgroups that you devised earlier. In fact, this question can be answered quite satisfactorily; a finite group with a unique maximal subgroup is cyclic of prime power order.

(3) The next step is to conjecture some more properties about maximal subgroups based on the examples you devised in (1). For example, you worked out that if $H$ is a maximal subgroup of a finite cyclic group $G$, then $\frac{\left|G\right|}{\left|H\right|}$ is a prime number. Is this true for all groups $G$? Can you think of groups $G$ for which this is true?

Notice how one can deconstruct a simple definition to arrive at a host of interesting questions? This is what a mathematician does all the time and is a very important skill. It might seem difficult at first but doing this will make mathematics all the more exciting and will give you a sense of "ownership" over the content. You worked out this piece of mathematics. This is the way I learn mathematics and I can tell you with confidence that if you practice this, it will soon become the norm.

What do you do after you look at the definition and have thought about it extensively? You continue reading the text. There is a good chance that you will notice the author stating some of the results that you discovered on your own. With luck, there will be results that the author has not stated. If this is the case, it could be a good idea to ask (on this website, for example) about the originality of the result.

However, you will encounter theorems concerning the definitions that you simply did not think about. You should resist the temptation to see the proofs of these theorems and rather you should try to prove these theorems on your own. Think about the theorem for at least a few hours before giving up. Note that theorems with quite short proofs can require highly original ideas and therefore you should not pressure yourself to prove the theorem in a small amount of time.

At first, you will take a long time to prove some theorems. There will be routine theorems and these should be proven fairly quickly. But there will also be difficult theorems. As you become experienced, your thinking will be faster and these theorems will come more easily to you. However, you should not expect this to be the case initially.

For example, you might encounter the following theorem in linear algebra: if $N$ is a nilpotent linear transformation from a vector space $V$ to itself and if the dimension of $V$ is $n$, then $N^n=0$. Working out how to prove this theorem on your own is a very valuable and rewarding experience. If you have not seen it already, I suggest that you try to prove it. It is not too difficult, however.

Question: How to avoid forgetting mathematics?

Answer: I used to forget mathematics too when I learnt it. I have talked to various mathematicians about this and they have said exactly the same thing. The point is that you just have to accept from the start that you will forget what you learn. However, there are ways to ensure that you keep this to a minimum.

For example, the best way to not worry too much about forgetting mathematics is to work out the mathematics on your own. For example, consider the steps that I suggested in the previous question. Even if you do this, you can still forget the mathematics, especially if the result in question was fairly easy to prove. (Note, however, that if the result is hard to prove, and you spend, let us assume, 10 hours to prove it, then you will probably never forget it for the rest of your life.)

The best method to take is to write down all the mathematics that you learn. Take copious notes. For example, when I read Walter Rudin's "Real and Complex Analysis" last year, I took down 3 entire books of notes. In fact, I wrote down 600 pages of mathematics when I only read 315 pages!

Write down every definition, every theorem, and every proof. The definitions and theorems should be produced verbatim from the book since it is important to ensure that your understanding of the rigor is correct. However, the proofs should be written in your own words.

Question: How to have a healthy lifestyle?

Answer: I am afraid I really do not have a good answer for this. In the four years that I have been studying mathematics, I have certainly not done anything else. Therefore, I cannot really give advice on how to manage one's time. If you are a serious student in mathematics, you will find yourself spending virtually your entire day doing the subject. This is inevitable. For example, I set myself goals every day of how much mathematics I wish to do and usually I end up doing mathematics non-stop. Nonetheless, I really enjoy this and I would not wish to have it any other way.

But I can offer one small piece of advice: try to wake up early, let us assume, at 6:00 AM. However, do ensure that you sleep for at least 8 hours; therefore, go to bed at 9:00 PM. Sleep is one of the most important points when it comes to studying. Over many years of doing mathematics, I have found that I am most productive and energetic before 12:00. If you can finish off most of your work before 12:00, then you will be in a really good position to do well each day. Also, try to avoid eating big meals. Big meals often cause you to lose your concentration and this can, in turn, lead to several wasted hours.

I think the most important point when you set out to achieve any goal in your life is to take it day by day, hour by hour, even minute by minute. Often you can complicate goals too much by thinking of what you would like to do over the next 1 year or even one month. If you work hard each and every day and set realistic goals, then anything should be possible.

I hope that I have helped! (I hope that my usage of bold text is not considered offensive; I simply used it to highlight some of the key points in my answer.)

Disclaimer (Dec. 25, 2013): This answer was written when I was 16 years old and does not necessarily represent my current views of mathematics. (Some points, e.g., "write down all of the mathematics you learn" is not something I would recommend to anyone today.) But I leave my answer here because I think it is overall reasonable advice and has clearly been useful to many people as is evidenced by the 77 upvotes.

Amitesh Datta
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    Bold text is always welcome, to distinguish important parts. I've been through the phase of learning how to read mathematical text rigorously enough and asking myself questions. Problem is, I ask myself too many questions, and I spend way too much time to undersdtand things. I'm asking how to learn math quickly AND rigorously. – Leo Jun 11 '11 at 08:46
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    "The best method to take is to write down all the mathematics that you learn. (...) Write down every definition, every theorem, and every proof." This has crossed my mind several times, and I have made attempts to it. I agree this is one of the best ways to master and memorize the material, provided the things you write down are in your own words and thoughts. But of course the big problem is, again: time. – wildildildlife Jun 11 '11 at 17:03
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    @wildildildlife I disagree with this to some extent. It is true that writing down all the mathematics that you learn takes time but I find that it takes even more time to relearn mathematics that you forgot and did not write down. For example, I learnt finite group theory three times because I forgot the material each time. (Of course, it takes less time to "relearn" the material because you have already seen it but the total amount of time it takes to cement the material in your long-term memory seems much more than the total amount of time it takes to write down the mathematics.) – Amitesh Datta Jun 11 '11 at 23:47
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    Well, I still agree with you. But it's a fact that during my courses I don't have the time to do this. Doing all exercises I have to do, and reading up to be able do to them, takes up nearly all time. The rest is filled with trying to construct and reading proofs. Typing up all this stuff in an understandable way would require 3 days in the week extra. Perhaps live-texing (during lectures) would help. Otherwise I'd have to do it during my precious holiday in the summer. – wildildildlife Jun 12 '11 at 10:17
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    Or perhaps we should terminate math.stachexhange, that'd free up some time :) – wildildildlife Jun 12 '11 at 10:24
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    I think that math shouldn't be learned as you say 'you learnt finite group theory three times'. In research it is better to know where to find the results than to know every result. Also, I think that trying to prove every theorem you encounter by yourself it is a bit of an overkill. No one can re-prove all that's been written before. Try Evans and Gariepy, Measure theory and fine properties of functions; most of the proofs are pages long. Sure, at some times after much reading, proofs come naturally, but not at a first time reading. – Beni Bogosel Apr 08 '12 at 21:03
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    @BeniBogosel I agree with you. I understand that maintaining all mathematical results in a certain collection of textbooks in one's mind is fairly pointless given that even that will only be a small proportion of current mathematical knowledge and that, alone, will not help one to do something that someone has never done before. Also, even if one does not prove every theorem on one's own, one can still break each theorem up into steps (or paragraphs) and prove each step (or paragraph) on one's own. I think this is more efficient and also helps one to understand the proof very well. – Amitesh Datta Apr 09 '12 at 06:43
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    "...the best way to not worry too much about forgetting mathematics is to work out the mathematics on your own." Definitely. It's hard to remember the view from the top of a mountain if you only saw it in a picture; it's easy if you climbed the mountain to see it. – Will Apr 26 '12 at 05:14
  • @AmiteshDatta, When you say "In fact, I wrote down 600 pages of mathematics when I only read 315 pages!", then wouldn't re-reading your notes actually waste more time than re-reading the book itself? – Pacerier Jun 22 '15 at 22:50
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    Hi @Pacerier, thanks for your comment. I've never actually gone back and re-read my notes (though I would be curious to do so at some point, even if only to see what I was thinking the first time I was learning the subject). If I do review a subject, then for the reason you mention, I would go back and re-read the relevant parts of the book (which would probably be better exposition then my rough notes, anyway). Nowadays, I don't adopt this learning style; it takes too much time and doesn't necessarily help me to remember/understand the subject better! – Amitesh Datta Jun 22 '15 at 23:34
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    @AmiteshDatta, You still keep those 600 pages of notes? Are they soft-copy? – Pacerier Jun 26 '15 at 23:24
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    Hi @Pacerier, I wrote them in several notebooks by hand, and they are probably somewhere in my house. Perhaps I will try to find them now that you mention it! – Amitesh Datta Jun 27 '15 at 02:15
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    @AmiteshDatta I am just beginning to learn Real Analysis on my own.I will follow your advice like think on definition for half an hour,deduce basic properties,prove theorem without seeing its proof .However do you think this approach for average beginner ? Thanks – Taylor Ted Jun 27 '15 at 21:36
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    Hi @JPG, thanks for your comment! It's hard for me to recommend anything without knowing your specific situation but I think the main thing is to think about mathematics in your own way, in addition to what you are learning. Of course, there are many ways of doing this and what I suggested is just one way. It's worth trying various approaches and then deciding which ones work for you. – Amitesh Datta Jun 28 '15 at 00:18
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    @AmiteshDatta, But time is a limited resource. I think that what he's trying to save time by learning from your mistakes. He's asking what are the stuff you have done that you have found are timewasters? Basically, what are the mistakes you've made? – Pacerier Jul 08 '15 at 10:42
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    Hi @Pacerier, thanks for your comment; that's a good point. I think it is fair to say that it wasn't the best approach for me to take so many notes, in hindsight. It didn't really help me to remember the mathematics any better; I think it is better not to worry too much about forgetting things, and just to be at peace with it. As one gets more mathematically experienced, the really important ideas will be clear as well as a broad(er) vision of what certain aspects of the subject are about (details of proofs etc. are sometimes best forgotten) ... – Amitesh Datta Jul 08 '15 at 12:07
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    Also, proving results on your own definitely helps to be actively involved with the subject, and is a great way (I think) to test one's understanding early on (it was a slight jump for me when I first started reading serious math textbooks after calculus, because solutions to exercises weren't present, so that's why I initially started taking the theorems as exercises with solutions). However, in the long run, it can be time consuming to prove everything (or even many non-trivial things) on one's own; it's a tricky balance. (Also, some things appear trivial if you have the right perspective.) – Amitesh Datta Jul 08 '15 at 12:07
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    Ultimately, how one learns math really depends on many factors such as how experienced one is, how mathematically mature one is, how much time one has, how one is able to learn new things etc. so it is hard to give a really specific answer. I mean one thing that worked well for me early one was not to necessarily learn things in order but rather as I saw interesting. For example, I studied topology (point-set and basic algebraic topology) without having really learnt linear algebra, real analysis and complex analysis; I sort of just looked up some things in real analysis as I needed them ... – Amitesh Datta Jul 08 '15 at 12:13
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    In the long run, I managed to cover the basic topics, and this suited me better than "systematically" working my way through mathematics. In principle, I wouldn't recommend this to a beginning mathematics student (in fact, I may not even recommend it to a younger version of myself) but that's the way I learnt, at least earlier on. The main suggestion that I can honestly give is to be as creative as you can and find your own way through mathematics - and then give advice to other people based on what you've learnt! Sometimes the only way to learn from some mistakes is to make them yourself. – Amitesh Datta Jul 08 '15 at 12:16
  • Hi @JPG, please see my above four comments where I provide a more elaborate response to your question. – Amitesh Datta Jul 08 '15 at 12:16
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    @AmiteshDatta Hmm Thanks – Taylor Ted Jul 08 '15 at 12:45
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    @AmiteshDatta So you were 16 and doing group theory? Man... –  Feb 11 '18 at 16:29
  • Hey @AmiteshDatta, I'm just seeing this post now. When you said " take it day by day, hour by hour,..." in the last few lines, does that mean you did math by just waking up and focusing on trying to understand some concept, for example, and woke up one day and noticed you finished 10 or so advanced topics? I ask because I often feel that I can understand what I'm currently working on but seeing the long term benefit is hard to understand. That's why I find it interesting that you seem to be saying that you never concerned yourself with the long term benefit/gain. Thanks! – john fowles Jul 22 '18 at 02:44
  • Hi @johnfowles, thanks for your comment! Yes, I guess I focussed on some concrete goal each day that I thought was achievable (but only by pushing myself). I think one of the things that helped to motivate me was to have a long term goal in mind (some big project, e.g., when I was learning, trying to read through an entire math book) and figure out a day by day plan to help me achieve that goal. I knew that if I missed a day, it would be very difficult for me to catch up so that forced me to work hard each day and I always had a long term goal in mind. – Amitesh Datta Jul 22 '18 at 12:48

Most people probably won't like this answer, but mathematics is a field where there's an unstable separation between those of genius caliber understanding and those who are just able to get by through dogged hard work. Way too many people want to do proofs and aesthetically pleasing artful mathematics for a career who are in the dogged-hard-work category. I am speaking as someone who has worked myself to death over the past 3 years to hack it in an Ivy League Ph.D. program in applied mathematics. Next to my peers, the only advantage I have is that I am able to work much harder, and to some extent I am much better at writing software. In terms of mathematical prowess, they all dominate me.

If, as you admit, you are average with a bad memory (aside from your obviously above average tolerance for difficult technical work), then you need to consider that an actual career in pure mathematics is not right for you. I want to be careful to avoid other-optimizing so please take my advice with a grain of salt. It may not be right for you, and surely all of the other commenters have insightful advice as well. But one thing that I think will never work for you is to just "try to exercise, eat right, and have a balanced life." Whatever others say, this will not happen for a pure mathematician who has really good taste in the aesthetic beauty of results, unless that mathematician really is at the genius level.

You have a limited talent supply and a limited time budget. Your personal forecast that you'll enjoy a career in abstract mathematics is almost surely incorrect; you seem to undervalue important things like salary, competitiveness for tenure, geographic preferences, etc.

For example, I have a close friend who studied very pure aspects of cryptographic number theory. He did two post-docs and earned practically no money at all, sacrificed personal relationships to try to get tenure track faculty positions, and ultimately found no jobs doing pure math. He took a job as a programmer for a company that makes cryptographic software. He thought that at least some of his time would go to researching new asbtract ideas in cryptography, but it turned out not to be true. Instead, he writes Java programs most of the time, learns about new applied cryptographic research, and writes very little (though he still dabbles in research in personal time and is, in my view, far more educated about cryptographic research than most people who currently publish in that field).

Is he unhappy in this situation? No! Actually, he discovered that to do software design properly, virtually everything is all about understanding the right abstraction, the right data encapsulation, the right design pattern, and this not only has great mathematical aesthetic value, but also delivers a better product to a client. After acclimating to professional software development, he now sees all sorts of parallels between his former work coming up with abstract math ideas and his current job coming up with abstract software solutions. His skills set now has a far higher economic demand, he isn't pressured to compete for tenured positions, and he's able to keep a very healthy work/life balance because of his company's regular work schedule.

I would say that, just as so many small businesses fail, far too many bright-eyed grad students see themselves as the next Godel, gung-ho for tenured positions and "living the life of the mind." They are especially prone to do what you are doing and to let the rest of their lives deteriorate in the hopes of being able to pursue what they currently (probably mistakenly) think is their own preference for abstract beauty that can only be satiated (also a mistake) by generalized math. Many more of these people should own up to the fact that they are not talented enough, and that universities that have to dedicate an ever slimming number of resources to hiring the best tenured faculty shouldn't really hire them.

Here are a few links to consider:

The economics also matter. Tenure is greatly diminishing in many parts of academia, and math faculty are especially notorious because pure math doesn't bring in grant money the way applied projects do. With the advent of online courses and open courseware, and sites like Stack Exchange, the need for highly specialized math teachers is diminishing at the university level. You should expect competition for tenured jobs to tighten, and that if you want a tenured job you'll have to go anywhere that offers them, even if this is a small regional university that is nowhere near any major city, has no real cultural atmosphere, and doesn't attract gifted students. It would be a big mistake to fail to take this into account.

My advice to you is this. Think hard about what it is specifically that you enjoy about mathematics. If you like the abstractions and geometrical thinking that are often part of advanced analysis and topology, then there are many applied mathematics / applied physics / engineering career routes that will offer you the chance to explore math questions, but will also put that geometrical abstraction ability to work writing software to solve actual problems. Your familiarity with pure mathematics may give you a career edge if you switch to a field like this. You might be situated to compete more effectively for grants and faculty jobs if that is what you want, and to the extent that you master programming skills, you'll have marketable skills to get different jobs if the need arises.

If you prefer the more abstract thinking that often accompanies algebra and number theory (that is, if you are a "problem solver" type of pure mathematician according to Timothy Gowers' definition, then I think you will find a lot to enjoy about software design. You may be better served by focusing on abstract problems in computer science and software engineering.

If you read a good math history book (e.g. Stillwell), you'll notice that (a) most good abstract math begins by being some sort of ad hoc, "it's probably true but I can't see the details" intuition anyway and it only gets refined later; and (b) most awesome stuff invented in mathematics was not invented by people who thought that mathematics was the way they needed to earn a living. People have been driving themselves mad over solving math problems for millennia, staying up late into the night, leading destructive romantic lives, falling into ill health. If you really love math, you'll never be happy doing it in the half-assed way that a healthy work-life balance requires, and very few people are truly capable of sustaining a career like that. Most ultimately stop trying hard on the math part and become dissatisfied with their careers.

Earning a living by being a pure mathematician is a very modern concept that arose largely because of the implications of Lebesgue integration in analysis and computability theory in computer science. And now that we have enough of a handle on those fields and their subsequent children, there just isn't enough stuff to support a lot of career mathematicians. Almost surely, significant mathematical advances in the next 50 years are going to come from highly intelligent, dedicated hobbyists, who solve problems at places like Stack Exchange or polymath.

And there's no reason why you can't find some niche problems that you like to work on, do so in your free time, and meanwhile have a fulfilling and economically sensible career that affords you a more comfortable life. For as obviously smart as you are, it would not be wise to fail to consider this sort of thing in your youth. Many more math students should do so as well.

In fact, the really egregiously unfair underfunding of students and inflation of post-doc positions largely comes about because naive youngsters who think they will automatically get tenure if they just try hard, and who think that nerdy love for aesthetic science is a good thing to base a career choice on, seem to unquestioningly accept underpaid and under-insured academic positions with no question. Trust me, you don't want to just be another one of those folks.

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    This is the only good advice here – Norbert Nov 11 '12 at 19:20
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    Seems to be based on personal experience – Akram Hassan Jan 25 '13 at 00:02
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    Some of it is. But the linked articles on the lack of a wage premium for PhD holders in computer science and math over Master's degree holders, and also the rise of post-docs as principal investigators (hence more concentration of academic employment at the lower levels and less at associate or tenured professor positions) are from outside sources that derive from peer-reviewed research. Other observations such as having a limited time budget and needing to do something economically productive are more or less default attributes of nature. So if a person cares about these, it's less anecdotal. – ely Jan 25 '13 at 18:36
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    The world needs more mathematicians - working in fields other than mathematics. – Little Endian Jul 03 '13 at 04:50
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    As a programmer who is learning math. I am not sure how I feel about this. – Surya Nov 21 '13 at 15:23
  • Can you elaborate? – ely Nov 21 '13 at 15:26
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    "Is he unhappy in this situation? No! Actually, he discovered that to do software design properly, virtually everything is all about understanding the right abstraction, the right data encapsulation, the right design pattern, and this not only has great mathematical aesthetic value, but also delivers a better product to a client" - no, this is just not true. "Software design" does not come nowhere close to mathematical research and is much closer to a repetitive manual labour. One may enjoy it, as a good carpenter enjoys making nice furniture, but it is not a real intellectual challenge. –  Feb 21 '15 at 13:42
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    @JohnDonn You've clearly never done software engineering. One of the core concepts in all of software engineering is to create abstractions that *explicitly prevent* manual implementation. Even apart from that, you've got things like functional programming with e.g. Haskell, which is essentially a manifestation of category theory and how to solve practical problems with it. – ely Feb 21 '15 at 15:43
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    @Mr.F (just in case someone comes to read this) I think I know enough about "software engineering" to state again my view that it is not an intellectual challenge which gets anywhere close to mathematical research. –  Feb 22 '15 at 17:20
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    @Mr.F: In my opinion, John Donn is right in the sense that what he says accurately describes software engineering in general. There will always be exceptions where purity and elegance is highly regarded but most of the time software engineering is there to feed the dollar-eyed. Many well-known companies are consistently delivering software as early as possible to get the money even though thousands of bugs are already known to pervade it. The industry is almost entirely driven by money. – user21820 Mar 18 '15 at 11:36
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    @user21820 But if that's the standard we want to use, then the same is true for mathematics or statistics as well. Most of that work takes place in companies, defense labs, and R&D consulting firms who are notorious for only going after low-risk demo-ware. As a machine learning professional, I've seen this both with software and with formal math time and again, especially in mathematical finance. Why try fancier learning algorithms if simple OLS will get something (crappy) out the door right now? So if *that* is the criticism, then it applies just as well to math too. – ely Mar 18 '15 at 11:45
  • @Mr.F: Yes I do indeed mean that it applies everywhere. So sad, but true right? Also, I'm also not advocating doing only pure mathematics that one has absolutely no confidence will have relevance to solving problems in the real-world. Anyway, just my opinion. =) – user21820 Mar 18 '15 at 12:03
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    I may be misreading this post, but there seems to be a suggestion that it isn't worth pursuing an academic career in pure mathematics if one is mediocre or second-rate. I'm not sure how I feel about that suggestion. There's also no mention of that other part of being a professor, namely teaching. That omission makes me uneasy: there are reasons _other than research_ to consider academia. – Jesse Madnick Jun 03 '15 at 04:15
  • @JesseMadnick When I was a grad student, I was *actively discouraged* from teaching. When I spent extra time preparing recitation notes and interactive examples for a probability course for which I was service as teaching assistant, my advisor actually took me aside and admonished me for not spending more time on research. It was the same story I heard from my peers, and from my many contacts and friends in academia at other universities and in many different disciplines (not just math or science). – ely Jun 03 '15 at 13:55
  • When you combine this with articles like The Economist's "The Disposable Academic" and you look at statistics about the rate of growth of adjunct staff who are expected to take over the bulk of teaching duties, while being paid far less, given less comprehensive benefits packages, and almost zero opportunity for career advancement, I do indeed think it's very unwise to pursue academia for the sake of getting into university-level teaching. It's almost like a lottery. Maybe 1/1000 of the top PhD grads goes on to have a fulfilling teaching career. The rest publish or perish & go to industry. – ely Jun 03 '15 at 13:57
  • @EMS, What exactly does " unstable separation" mean? – Pacerier Jun 22 '15 at 22:52
  • Here I mean casually that nobody "is close to" the boundary. If you're good enough at math such that accomplishing very difficult career mathematics doesn't significantly prevent you from having a good work/life balance, then your math skill is much, much higher than even an average math Ph.D. graduate from top universities. On the other hand, if you don't have math prowess that extreme, but you still doggedly work as hard as necessary to do advanced math for a career, then you almost surely do not have good work/life balance. Of course there can be exceptions; I claim they are highly rare. – ely Jun 22 '15 at 23:45
  • I apologize for the late bump, and I hope this is not too off topic, but do you believe this advice also applies to many would-be physicists? – Striker Mar 06 '16 at 01:07
  • Are you saying bad memory will not allow you to study mathematics? – N.S.JOHN Apr 02 '16 at 14:31
  • Simple hardwork in maths won't work. – Anurag Nov 06 '16 at 07:54
  • I must tell you I completely changed my impression on you after reading your post here. An excellent one. – Nobody Apr 08 '18 at 13:19
  • Somewhat dangerous answer, as too many people give up on Math due to psychological issues and math phobia. It is true that there are always geniuses like Abel or Galois, but then if other fields also start getting elitist like this, world would have been worse. – senseiwu Feb 08 '22 at 19:54

Some very basic non mathematical advice and I'm sorry if I sound like your mother. If you feel like your memory is bad and you're not finding enough time to socialise, perhaps you're not finding enough time to eat well. Eating plenty of fresh fruit, fresh vegetables, fresh fish, olive oil and cereals will give your body the building blocks to do its best . Oily fish in particular are known to be good for the brain. http://www.newscientist.com/article/mg20827801.300-mental-muscle-six-ways-to-boost-your-brain.html

I find that the time I spend cooking / washing up is pleasurable mentally relaxing down time during which some of my best ideas come. Perhaps you could combine this with a social aspect and invite people round to tea if you're cooking something nice. Avoid the alcohol that usually goes with these situations if you're intending to get back to work after.

Giles Roberts
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    I really like this answer. The importance of proper diet (and not just a diet of stimulants) cannot be overstated. Fruit is especially important, of course, because the brain runs on sugars, and fruit's full of them. +1 – Jack Henahan Jun 18 '11 at 02:22
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    Candy is also full of sugar! (falls into hypoglycemic coma...) – The Chaz 2.0 Oct 24 '11 at 20:36

Regarding the "and have a healthy lifestyle" thing. Well, you also have to learn when to stop.

Sometimes you figure out you don't understand something and you don't have time right then to understand it. Try to "box" that as much as possible. Figure out the general form of the kind of thing you don't understand. How does it function? What context is this idea used in? What kinds of "inputs" does it have? Does something appear as if by magic? What? Have you ever seen anything like that before? If you keep these vague ideas in mind, you may very well figure it out in your sleep, in a conversation with someone, maybe weeks later, maybe years. It depends on the particular thing.

Sometimes you don't understand something because that thing is complete nonsense. Profs sometimes say nonsense -- they're human beings and make mistakes. Books make mistakes. I remember spending a lot of time trying to complete a proof for a homework problem and everything I tried failed. An hour before the homework was due I was talking with someone else in class. He showed me an example he cooked up to demonstrate the theorem and I reinterpreted it as an example that disproved the theorem. These things happen. Similarly, plenty of textbooks have subtle "lazy" errors in them. If you're not particularly confident in yourself you may spend hours being frustrated on such a problem. Talk with people.

Ryan Budney
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    You're completely right about "learn when to stop", but when I read notes, I place myself in the position that none of those things are known to be true, I'm the one who is supposed to verify it, and then be able to reproduce the proof to others. I'm guessing this is precisely the kind of work researchers do. And in researching, a single mistake unnoticed can cause the whole theory to fall apart like dominoes. **If I can't understand most of the proofs now, when the results are known and presented to me, how will I be able to discover new ones?** – Leo Jun 11 '11 at 06:28
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    It helps to build up "modules" that you're comfortable and confident with. In many introductory manifolds classes, many of the proofs boil down to some kind of understanding of the implicit or inverse function theorem. So having those theorems "hard coded" is key to making the transition to manifold theory more of a modest step. So your motivation in some sense shouldn't be getting to the point that you can explain a proof to someone else, it should be getting to the point where the proof seems natural to you. In many (most) subjects there are only a few key ideas. Getting a sense for – Ryan Budney Jun 11 '11 at 06:36
  • them and making the natural extrapolations is key. In that regard I find homework very valuable. – Ryan Budney Jun 11 '11 at 06:37
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    +1 for "If you're not particularly confident in yourself you may spend hours being frustrated on such a problem. Talk with people." – Jesse Madnick Jun 11 '11 at 07:09
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    @LeonLampret: Every proof form was new at some point and was discovered (probably multiple times, before it caught on) by someone who did not know it and had never had it presented to them. The discovery of new proofs is all about a skill that is not "understanding proofs that are presented to me". It is not your job to reproduce the proof to others, that is the job of publications. Finally, the publication of incomplete work does produce results that fall apart like dominoes. You have been exposed to the well-trodden paths. At the frontier, mistakes are common. – Eric Towers Feb 10 '14 at 10:24
  • +1: 'Try to "box" that as much as possible.' – Our May 24 '19 at 11:53

On studying math:

Your time on this rock is finite, the amount of mathematical knowledge is infinite. You must choose wisely about what you want to spend your time learning. Decide if you want to be a "jack of all, master of none" or "a master of one, jack of none" type.

Personally, I'd rather know a lot about everything than everything about one thing, thus I'd say don't waste your time learning every single detail. Appreciate the high level material, move on to the next field until you run out of fields. Then, work your way down to lower level material as your time allows.

Your learning doesn't stop once you complete your degree(s). Learn to pace yourself.

On the healthy lifestyle side of things:

The mind can only be as sharp as the body. Make the time to exercise, eat right and get enough sleep. You will find that you think more clearly, you retain information more effectively and that you are happier.

Balance is critically important. You get but one life to live and there is far more to it than mathematics. Take the time to explore other interests, discover new ones and become more well rounded. The greater your overall knowledge, the better you will be at math.

Live life. Socialize, fall in love, run a marathon, get into a fight, go to the ballet, paint a masterpiece, go fishing, explore the world; do the things that make you more than just a mathematician, do the things that make you a person.

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    Get into a fight. Like [Evariste Galois](https://en.wikipedia.org/wiki/%C3%89variste_Galois#Final_days), right? :-) – emallove Jul 31 '14 at 01:07

I can see that this question is a couple of month old, but I would like to add some remarks:

1) Most research mathematicians have a better memory and are quicker than what you describe. There are notable exceptions, but you have to understand that it will be hard to compete. You will always have to work harder than most of your peers. If it takes you three times as long to correct exams, then this time will be missing from your research even though you might be just as talented for actual research. On the other hand, don't trust your fellow students when they just say that they understand things and are quick. In many places, it is cool to claim to have aced the exam with little study time. In the long run, you might overtake some of the people who know how to learn just for an exam.

2) Of your peer group, a very small percentage will become researchers. There is no point in comparing yourself to people who efficiently pass the exams if you want to become a researcher. Seek out good, ambitious students and socialise with them. If they are quicker and have a better memory than you, then ask them what is wrong with Lemma 3.4 whose proof seems somehow strange to you. There is no point in finding all stupid errors yourself. Ask your peers, ask your professors, ask here. You are wasting time if it takes you three hours to find out that the professor wrote "c" instead of "e".

3) If you concentrate too much on details, you have to train summaries. Can you explain to a very talented beginner student what they will learn in linear algebra and analysis? In 10 sentences? In a couple of minutes? In a couple of hours? Without paper? When I need a result from a lecture that I heard as a student 15 years ago, I don't need to remember the conditions in the theorem. I need to realize that this theorem is probably applicable to my problem, in which lecture or book I saw it and then I can look it up to check whether there was some technical condition I forgot. To do so, I have to remember the gist of the theorem and the proof, not the details. Also, if you don't understand something during learning, preliminarily accept the result, continue and return to the result later, don't brood on one thing indefinitely.

4) Usually, the gist is something professors like to hear during an oral exam. They will check the details here or there, but they don't need to hear the gritty details all the time. Are you sure that you are acutally speaking at the expected level of detail during your oral exams? Or are you just assuming that the professor wants to hear all details and start right away at the epsilon level? Have you ever tried to sit in on other students exams?

5) Seek out younger students and help them preparing for their exams (or answer questions here). Helping others is the best method to keep your acquired knowledge fresh. This will not be wasted time.

6) You should absolutely not sacrifice your physical and mental health. Sleep, food, exercise, social life and hobbies are important and should not be neglected for an extended time period. Is it not possible to just take fewer lectures per semester? Who will care later if it takes you a year longer to finish? Something has to give, and it seems to me that the easiest thing for you is to just spread the work over more time. (And yes, I do realise that even without tuition there are high opportunity costs, but you seem to need more time now.)

J. W. Tanner
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Being in good physical shape, makes your head clearer. Besides, you only need one hour of exercise a day to keep in decent shape. Do some multitasking! Walking and thinking can be done at the same time, you train your memory, AND your stamina.

Solving math problems in your head is a good mental exercise.

Per Alexandersson
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    One of my professors had an acquaintance in grad school that took this to one extreme. The two of them were sitting together getting coffee when my professor started writing something out on a napkin when his buddy interrupted saying, "Real mathematicians don't use paper". I guess he was a bright fellow and wasn't joking! Of course this is a fringe case. Good luck with your studies OP – ttt Jun 28 '11 at 01:20
  • also see [BDNF][1] [1]: http://www.ncbi.nlm.nih.gov/pubmed/21198979 – T. Webster Mar 08 '13 at 08:24
  • I definitely agree +1. For the OP, though, I recommend doing that Krypto card game with house numbers. –  Nov 05 '15 at 05:39

If you choose to study mathematics so hard, as you describe it, then it means that you should like it, and I guess you do. Still, saying that it eats up all your time worries me. Math should be a pleasure; at least that's how it is for me. Math shouldn't take all your time, because your brain needs to rest for you to process things more easily. Here are a few tips:

  • when studying, choose a degree of detail or depth you wish to go through. Do not work out all the proofs from a book, when studying it. Choose what really interests you and what you need for your course. Usually a course does not cover an entire book, and for getting good grades you don't need to now way more than what has been taught in the course. Some things will get clear only with time and experience; you will learn them for the exam, but you'll understand the whole picture in a larger period of time, maybe years.

  • learn to relax everyday. Maybe I'm a lazy type, but I always find time for a walk, a bike ride, for playing the piano, for watching a movie. For example, I relax when solving problems on this site, or on my own, problems which do not have anything to do with what I'm studying at the moment. Take at least 8 hours of sleep per night. When relaxing you leave the brain a chance to put all the things in order. Many mathematicians had revealing ideas while doing ordinary things. A walk in the park can help you understand a key point in a proof, or a solution to a problem might pop up when doing some sport or some chores around the house (it happened to me more than once; the funniest one was that I solved a problem given to a team selection test for the IMO in my head, without any pen and paper while cutting the grass in the field with my father and grandfather).

  • usually you can focus better if you have a better goal than 'finishing the book'. For example, take an article in the field you're studying (maybe a teacher can help you with that) and try and understand that article in detail. Study only the theorems and proofs which are related to that. Mathematics has developed enormously in the past years. Trying to keep the pace with everything is impossible. Focusing on a narrower scope is usually easier, and in research this is really needed.

  • do not ever worry about memorizing everything. You will forget many things no matter how many times you learn them, but the essential thing to do is to remember where to look for the things you forgot. For example: theorem X with examples and counterexamples is presented in book Y, subject Z can be found in the book T, and so on. Try and split your proofs into steps you can remember. Do not memorize calculations. Remember only key points, and trust yourself that you can fill in the blanks.

  • find time to spend with friends or colleagues. Having someone to share an idea with, even a mathematical one, can be of great help.

  • find someone you can tutor ( at highschool or university level, in a lower year ). This can be of great help financially and you'll notice how good you understand things when you try and explain them to someone who doesn't know them at all. This has been of great help to me.

Good luck.

Beni Bogosel
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I am an undergraduate at an American university going through almost exactly what you're describing... the lack of sleep, lack of social life, weak memory for proofs, and a perfectionist insistence that mathematics be presented "my way." Like I say, I'm currently going through this, so I can't offer any answers. I do, however, have some suggestions from experience.

Recently I found myself in the unfortunate situation of having to memorize many proofs the day before an exam. Suddenly, it no longer mattered if I knew only a few proofs in great detail. What I needed was to know all of the proofs, but only in enough detail to warrant sufficient partial credit. To do this, I skimmed the proofs in my textbook one by one, writing little summaries of each in my own words.

My point is that this (for me) was a very effective method of grasping the big ideas of the proofs without getting hung up on the details. In writing my own summaries, I was also able to boil down entire proofs to a couple of sentences, which then served as mnemonics for memorization.

But as for the questions you actually asked...

Should you skip reading the proofs? Ideally, you'd read and understand all of them, but if you're crunched for time (as you seem to be), then you have to be efficient. Ryan Budney is right: you have to learn when to stop. Learn what you think is relevant to doing well in the class. Then, when the course is over, you can take the time to understand the details or less-important proofs or whatever you want, should you so desire.

Should you try less hard, get worse grades, but "have a life"? I don't think anyone can answer that but you, I'm afraid.

I will say, though, that efficiency really matters, and that you might be able to find ways to balance academics with a social life if you look for them. You know, somehow we're all pretty efficient when exam time comes around, managing to cram large amounts of information in a very short amount of time. We have no choice but to be efficient. So while I'm not saying that you should treat every day like it's the day before an exam, I do think that you can find ways of increasing efficiency if you look for them.

I should point out that all of this is meant to be practical advice rather than sage advice. For sage advice, I also recommend Terrence Tao's career advice, as well as talking to your professors and advisers.

Finally, I should mention that it is my understanding that -- although I am by no means a professional mathematician just yet -- that at the end of the day, discipline and hard work matters just as much as natural talent, if not more.

So if you're worried about being able to produce research-level math, then my advice would be to stop worrying about it. If you haven't actually tried your hand at research yet, then there's no reason to worry yourself about it prematurely. At least, this is what my adviser told me when I presented him with these concerns last year, and really, it's been some of the best advice I've ever received.

Jesse Madnick
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    Among Terrence Tao's Career Advice, I found his description of the pre-rigorous, rigorous, and post-rigorous stages of mathematics education to be particularly relevant for me. It sounds relevant to you, too: http://terrytao.wordpress.com/career-advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs/ – Jesse Madnick Jun 11 '11 at 07:03
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    And I've really found it to be true, too. I've been making a conscious effort recently to leave the "rigorous" stage and move into the "post-rigorous" stage. And I have to say -- again like Ryan Budney mentions -- I'm starting to see certain patterns arise in the proofs. I think this kind of pattern recognition happens to most people eventually, but they have to have been exposed to enough material first! – Jesse Madnick Jun 11 '11 at 07:05
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    Very interesting, glad I'm not alone on this, and also good advice. I've wondered for the last couple of days, if I were somehow (don't laugh, this is serious) permanently transported back in time to the ancient greeks, how much mathematics would I be able to teach them. Especially if they weren't 'buying' my proofs. I might've been disappointed if I found out... – Leo Jun 11 '11 at 07:37

A friend send a link to this post and I find it very interesting. One thing that is particularly interesting is your conviction that you do not have a talent. How do you know if you have a talent or not? What is talent anyway? Why are you so sure that your peers who get a proof faster or remember it longer have more talent than you?

It is very difficult to explain what mathematical talent is. Most math problems that are worth solving and most theorems that are worth proving take years to solve and proof, so speed or memory will not come so handy while solving or proving these theorems. I believe most humans have enough memory to store all the necessary information in order to work on worthwhile problems for few years (they have enough time to do that). Speed in a three or four year project is hardly ever useful (of course there are notable exceptions but in average I would say this is irrelevant). What actually comes much more handy is diligence, which you seem to have. If you can stay with a problem after few months of failure then you have the right qualities, I think.

One thing people bring up often is imagination. This you can never know if you have or not until you actually start working on problems. You can learn a language quickly but after doing that you may never become a poet. The undergrad math, in fact most math taught in standard classes, is just developing a language which some people learn faster than others and some remember more than others, but what will they do with it isn't something that is taught in those classes.

I would say that mathematical talent is in fact this imagination. To say someone is mathematically gifted is just to say that the person sees more mathematical connections and relations among mathematical objects than most people. A mathematically gifted person has a strong intuition about which line of thought will lead to beautiful new theories and new discoveries. Good memory and speed of performing mathematical computations and logical operations is often mistaken for mathematical talent. It is of course a talent, a very useful one, but I wouldn't call it a mathematical talent.

At any rate, you seem to be far away from a kind of place in your life where you can actually figure out if you are mathematically gifted or not. You can speed up your journey towards getting there by singing up for research projects rather than taking math courses, which are the most deceiving indicators of mathematical talent. You can also take reading courses and sign up for more higher level, like graduate level, courses. There are also summer research programs that you can sign up for.

As for healthy life style, I don't know. I actually think about math when I do my exercise. That is the great thing about the profession but you have to teach things to yourself. For instance, you can teach yourself to think about math when you are doing routine daily things like washing the dishes. But it took me a while to get to this kind of state of mind, as a college student I had an unhealthy life style as well and didn't exercise much thinking its waste of time. But its not! You can teach yourself to think about math while doing it and also it is a way to rejuvenate yourself and deal with the stress.

The social life part is hard. As a research mathematician you do need significant amount of time to yourself. I doubt that there are research mathematicians out there with huge social circles. But with some effort you can have enough people around you and these people, in most cases I know, are usually very interesting, intelligent and motivating people.

As to how to learn math, well, you never know. We all I guess put a lot of emphasize on learning the proofs and the details of it but when we go a higher level we realize that we didn't really understand the proof. So in a sense its actually a pointless activity to really learn the proof of a theorem you have seen first time. Probably doing some kind of circular thing where you learn things and as you go on you come back to earlier things and you re-learn them is better.

But what is more important I think is for you to figure out why you want to learn these proofs so well. There are many more proofs to learn and you will not know all of them by heart ever. In fact what is it that you want to gain out of math? Its best to concentrate on finding what area of math you like and learning that subject with a good professor in a circular fashion where the subject gets more and more sophisticated and important ideas that have been left behind get revisited from time to time.

In short, there is nothing to be afraid of and I am sure you will figure these things out for yourself.

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I think Terence Tao's Career advice can answer your question. I would strongly recommend you to read it.

Edit: And also Kevin Houston's How to Think Like a Mathematician: A Companion to Undergraduate Mathematics.

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    hmm, haven't found anything that relates to my actual problem: how to study so that I understand math, have leisure activities, and finish on time. Because it's just been one big suffering and hard work so far... – Leo Jun 11 '11 at 05:07

I was (and am still to an extent) going through much a similar phase some time back. I am not particularly good in all the main subject areas of mathematics and am precise about details myself and there occur times when it all sorts to overwhelm me a little. At such times I either indulge in fun-maths, and just try to prove results for fun which attract me, no matter how much time it takes, or how trivial they seem. This keeps me attached to mathematics while also relaxing me. Also I take time out daily for non-mathematical activities because if I dont, too much mental activity invariably gives me a headache.

I think doing maths is much like playing music. It is hard work, but occasionally you can play whatever tunes relax you. All said and done, I think it is important to remember why exactly we do maths: because it is fun!

Added after seeing the comment:

I see. I can sort of relate to my graduate days with that. I got through them somehow with a lot of angst, and what I learnt from that was this: Its important that you study the proper way and that proper way is unique to everyone. In my second semester I remember really struggling through Complex Analysis by Ahlfors (I still am a little apprehensive towards it) and the reason was that that book was not geared towards my way of studying, and there was no time to painstakingly give arguments for everything "assumed to be clear" in the book. Later on, I read another book on Complex Analysis(Brown & Churchill) and what I know of the subject is largely due to that. This is because the second book was more geared to my internal understanding process then Ahlfors. Perhaps you too will go a lot of quicker with the grades if you read from books that appeal to you intuitively and not from prescribed ones.

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    Thanks for the advice, but I think you don't understand the situation: there is no time for fun math, no time books or relaxation, etc. Each year, I have 12 courses, each has notes of about 100 pages, which I have to digest somehow, and be able to reproduce on the exams. This is just horrible. I'd just be happy to 'survive' with good grades and some free time/physical activity. I know what I want to do, if i had time. It's the courses in mathematics that eat up all of my time. If I don't allow this to happen and TAKE some free time, the consequences will be that I'll understand even less. – Leo Jun 11 '11 at 08:12

I suggest transferring to a program where you can make your interests coincide as much as possible with the things you're required to study.

Mid-way through my 1st year as an undergraduate I stumbled upon the honours mathematics program at the University of Alberta and I was pretty much hooked. The honours program emphasized rigor, understanding, technique, visualization, precision, basically just a really solid foundation.

In my 2nd year as an undergraduate I had a (required) rather unfortunate introductory differential equations course where it was all crank-the-formula. Proofs and ideas were nowhere to be seen. That course was quite frustrating for me -- it seemed like such a waste of an opportunity to start connecting the various threads we had been developing in analysis, linear algebra, algebra and so on.

I found as an undergraduate I usually had plenty of time to do everything I was required to do. There were moments when I got in situations that were close to being over my head but it all worked out for the best. If you never push yourself too hard you'll never know what "too hard" is. So it's a good thing to discover. I think it helps when the things you have to do are the things you want to do. If they're not, you can end up wasting time being bored out of your skull. It's a good lesson to learn how to accomplish boring things, but hopefully there's not too many of them in your undergraduate education!

Ryan Budney
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    Transferring to a different program would be problematic. In my country, there are 2 universities, and I'm at the larger one, attending the pure mathematics program. I expect things to be even worse at the other one university. Going abroad presents even bigger difficulties it seems. Especially if I want to eventually live here. – Leo Jun 11 '11 at 05:29
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    Not pushing myself isn't the problem, as you can read from my initial post. The problem is that I've been pushing myself for several years, to compensate the lack of talent and to really understand things, and I've reduced the quality of my life considerably. Efficiency is what I'm lacking. I strongly suspect, that most classmates don't read proofs, or just glance them, to get the feeling. Is this the way to do it? – Leo Jun 11 '11 at 05:34
  • Why not talk with some Slovenian professors, and ask them if they had similar problems, and how they dealt with it? You *can* go to universities outside of Slovenia and eventually get a job in Slovenia -- I know at least one prof in your department that did just that. – Ryan Budney Jun 11 '11 at 05:39
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    As an undergraduate I didn't dwell on every proof we were given in class. I did the ones required in the homework, and paid a lot of attention to the ones I found difficult or somehow informative in class -- for example, I remember my faith in the axiom of choice being shaken by the proof of the Urysohn Lemma in point-set topology. Even though the proof technically made sense it still felt *wrong* to me. – Ryan Budney Jun 11 '11 at 05:42
  • Did you go through all the proofs at all the courses? We have 12 subjects per year, each has on average 80-100 pages (40-50 sheets of A4) in handwritten form. If many of the notes are 'shaky' with many errors, both lapsus-like and deeper ones, how is one supposed to handle this, without sacrificing huge amounts of time on it? – Leo Jun 11 '11 at 05:43
  • As a student (and still to some extent) I rarely take detailed notes in class. My strategy was to condense the presentation down to a 2 or 3 line summary. If there was a really novel idea, I would write it up in detail but usually I'd just try to get the main idea from the presentation. But we had textbooks, or print-outs of lecture notes so there was little reason to make detailed notes. I know plenty of very successful people that write everything down and go over everything in detail -- we're all different that way. – Ryan Budney Jun 11 '11 at 05:46
  • "...ones I found difficult or somehow informative in class..." **How can I know which proofs are informative, without reading and understanding them?** I often don't understand something, then dwell on it, until it is clarified. Most of the time, it's just a proof, but sometimes I find out that it was crucial to the whole construction/proof. It was the whole point of it all, and if I hadn't read it, I would've missed the point completely. – Leo Jun 11 '11 at 05:49
  • I'm not sure if this is helpful short-term. But once you see the "elemental" aspects of mathematics in rigorous detail and have absorbed it, here I'm thinking about single-variable calculus starting from an axiomatic definition of the real numbers (like in the Apostol book), and basic sets, relations, group theory, basics of rings, basic linear algebra, representing linear maps w.r.t. a basis. Once you've got that stuff down it starts getting easier to notice the outlines of proofs and think more top-down rather than bottom-up, following every step linearly. – Ryan Budney Jun 11 '11 at 05:53
  • Hmm, I'm not sure what the theory (oral) exams look like at your university, but here, at many classes depending on the professor, we are asked to prove this and that, in fair detail. I have absorbed the undergraduate stuff, but still, I haven't found any shortcuts/easy ways around digesting the proofs one by one. – Leo Jun 11 '11 at 06:01
  • What kind of proofs are you talking about? Are you a graduate student? I had earlier assumed you were an undergraduate. – Ryan Budney Jun 11 '11 at 06:07
  • Umm, we have three stages: 1st=3yrs(diploma), 2nd=2yrs, 3rd=3yrs(doctorate). I'm in the 2nd stage, first year, but I've been studying for 5 years, not 4, because I switched from 'applied mathematics' program to 'mathematics' program. How can I go to an oral exam without understanding most of the proofs? Over here, everyone is taking notes, and there are (mostly) **no print-outs of lecture notes**. Therefore I copy the notes from others, since I can't really follow what the professor is explaining AND write it correctly in my notebook. – Leo Jun 11 '11 at 06:16
  • Oh, and I was talking about serious (at least from my viewpoint) proofs: abstract algebra, algebraic topology, analysis on manifolds, graph theory, etc. – Leo Jun 11 '11 at 06:23
  • Okay, I think everyone works differently when it comes to these types of things. I really like taking oral exams, being put on the spot. My strategy is to not study the fine details of the proofs too much, but to go over the broad outline, sometimes over-and-over again until I create the details myself. I try to avoid as much as possible reading other people's fussy arguments, since it's so easy for someone else's proof to look like clutter to me. So perhaps something like that could be a strategy -- try to not follow other people's proofs too closely. Do them yourself. – Ryan Budney Jun 11 '11 at 06:30
  • "try to not follow other people's proofs too closely. Do them yourself." I absolutely like this idea BUT it only applies to simple proofs. There is no way I will be able to just figure out the proof of Seifert van Kampen theorem, or Gauss-Bonnet theorem. I have to read through the professors proof, which sometimes is 3 pages, and there are numerous details unclear to me. – Leo Jun 11 '11 at 06:34
  • SvKT can seem non-intuitive at first but once you've really digested it you'll find it's really quite simple. It's just a lot of "accounting" and keeping track of relatively small details that bring in an explosion of notation and all the real technical detail. Perhaps look at other proofs -- Hatcher's and Peter May's are great starts. For GB, perhaps look at Guillemin and Pollack's proof (originally this one is due to Hopf). – Ryan Budney Jun 11 '11 at 06:40
  • example: we did an exercise at algebraic topology course: compute $\pi_1$ of the diadic solenoid. We have a solid torus $T$ in $\mathbb{R}^3$ and a homeomorphism $h:\mathbb{R}^3\rightarrow\mathbb{R}^3$ such that $h$ maps the torus into its interior, twice wrapped. Now $A_0:=T$ and $A_n:=h(T)$. Why is $\cap_{i\in\mathbb{N}}A_n$ ($A_0\supset A_1\supseteq A_2\supseteq\ldots$) even non-empty?? Maybe it's a dumb question, but I can't seem to answer it right away. Nobody asks these types of questions, but I have them in hundreds... – Leo Jun 11 '11 at 06:44
  • "SvKT can seem non-intuitive at first but once you've really digested it..." That's my point, it takes a huge amount of effort and time to digest most of the proofs at my faculty. I suspect many classmates are not going through the proofs and they're doing fine, but I just can't understand things until I've spent enough time thinking about it and digested most of the proofs. That's the problem. I can't understand things, until I've spent serious amount of time digesting it. Then only do I get a feeling of the techniques and methods of the field. That's why I have a horrible lifestyle... – Leo Jun 11 '11 at 06:49
  • That question is too vague to be a mathematical question. "twice wrapped" is too vague, for example. Any why isn't $h : T \to T$? – Ryan Budney Jun 11 '11 at 06:53
  • I know that it's vague, but that's the way we did it in our class. I don't blame the assistant, a more rigorous approach would probably lead to serious problems and technicalities. Nevertheless, is an intersection of a sequence of closed subsets $A_1\subset A_2\subset A_3\subset\ldots\subset\mathbb{R}^n$ always nonempty? – Leo Jun 11 '11 at 06:58
  • Anyway, I'm really not trying to nitpick at anything, but I imagine that when researching/discovering mathematical truths, these kinds of details can be crucial. – Leo Jun 11 '11 at 07:00
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    @Leon: that the intersection of a nested collection of non-empty closed subsets of $\mathbb R^n$ is a non-empty closed set, this is one of those fundamental theorems in analysis that follows directly from completeness. This is one of those theorems that was heavily covered in the 1st and 2nd year of my undergraduate education. Think about Cauchy sequences. – Ryan Budney Jun 11 '11 at 07:06
  • Since $\mathbb{R}^n$ is complete, every cauchy sequence has a limit. The limit in included in the intersection, since it's closed. I'm guessing we chose a point from every $A_n$ to get a sequence $a_n$. How do we know that it is cauchy? – Leo Jun 11 '11 at 07:17
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    Oh, I had the wrong assumption. The sets need to be nested, non-empty and *compact*. So they're contained in a big box. If you substitute closed for compact, you could take $A_n$ to be the complement of the open ball of radius $n$ centred at the origin. – Ryan Budney Jun 11 '11 at 07:33
  • Yeah, then it works. Hmm, this works also for the diadic solenoid case, since a homeomorphic image of a solid torus is closed. Thanks! That's probably why we need $h$ to map $\mathbb{R}^3\rightarrow\mathbb{R}^3$. Anyways, these kinds of questions are in the hundreds, when I read stuff, and nobody seems to be adressing them. Especially in the. That's why **I spend a lot of time trying to correct the proofs, to 'make them work'**, which is why I'm left with barely any time... – Leo Jun 11 '11 at 07:42
  • Perhaps you're over-complicating things? Looking for high-tech proofs when techniques from your 1st year as an undergraduate fit more directly? It's hard for me to say from this distance. – Ryan Budney Jun 11 '11 at 08:00
  • +1: 'If you never push yourself too hard you'll never know what "too hard" is.' – Our May 24 '19 at 11:52

This answer will attempt to only address the first part of your question. When I was doing undergrad work, I gained a lot of weight since my main way of doing my homework was just sitting down and eating chips or something while putting my nose to the grindstone. This, in combination with needing to constantly study, was really bad for my body and also caused some anxiety problems down the line. Around my 4th year, I started making daily to-do lists which included little bits and pieces of things which were not math-y: I found short (10-15 minute) exercise videos on youtube that I knew I had time to commit to, and I did that "100 Push up Challenge" which you can probably find via google (I didn't quite get there, but I had a lot of fun along the way!).

To this day, I set aside time for at least 20 minutes of exercise each night (you'd be surprised at how focused you are afterwards) and I feel significantly better, physically. Once you've been doing it for a month or so, it just becomes natural.

As far as the social problem, different people do different things. I ride my bike to school, so I joined a cyclist group. I also found a number of other social clubs in the city (Chicago at the time) that did things I liked. I was surprised at the number of people just looking for other people to talk to, and not all of it was science-y!

It may be the case that this does not work for you, but I wanted to share my experience just in case someone found it even a little bit helpful.

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    Thanks. Actually, I didn't gain weight because of such a habit, but rather because of lack of exercise, and because eating gives me the energy to stay awake at night and lessens nervousness. I used to be really sporty, training MMA and all, and now it's all gone to dust. – Leo Jun 11 '11 at 08:00
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    Also, I have a lot of interests and activities that I was and would have been doing now, if I had more time. And I had a perfectly fine social life before faculty. I'm just caught up in the horrible stress and hectic pace of life, from which I can't be able to escape, without sacrificing my understanding and grades. And to top it all, there is a distinct possibility that I'll not be able to have a career in (academic) mathematics, since the opportunities are very limited and I have no connections. So everything seems to be work in vain, which further decreases my motivation. – Leo Jun 11 '11 at 08:18
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    I completely understand your last points here; I've been called a mediocre mathematician by a number of individuals (including one famous one!) and this tended to get me down. In the last few years, I've decided that even if there is no room in academia for me, there's certainly room other places and this doesn't mean I ever have to stop doing math! That kind of thinking keeps me going, I think. I don't know what else you could do besides just try to schedule your days or take on a lesser workload. I'm not sure if either of these work for you, though. –  Jun 11 '11 at 08:35

You might consider distinguishing the understanding of mathematics from the requirements of your classes. If you need to memorize a proof for a test, in order to remain in school, then do it... but don't confuse it with understanding, or with working out proofs for yourself. You might find that breaking a proof into the parts you do understand and the parts you don't understand will simplify the process of coping with the proof.

Who knows? Maybe having memorized a proof you don't understand, you'll find that you can think about it while you're doing something else, and perhaps understand it in a flash of insight.

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I know there is one fellow here that wants to discourage you from doing your mathematics... He is absolutely right, graduate school is very difficult, post-docs are a nightmare... What is disturbing is that he tries to set you up for a comfortable life... Perhaps this is not what you want. One thing I know for sure is that any human being is "the captain of his soul". If you want to do something, and if you don't give-up you will get there. Nothing worth doing comes easily. This is why there is such a competition for professions such as mathematician or cook. Because the people that want to do it absolutely love it and for them there is nothing better they could be doing. Yes, you will not be paid well, yes some idiot is going to ride a better car than you. The ultimate question however is, if you want to follow your passion to the very final destination where it leads you, perhaps to the abyss, or if you want to never try, and be pushed around by every kinds of people that will tell what they think life is and how they think life should be lived. I know you don't listen to these people anyway because you question even mathematical objects, you ask "why?" which is all you need, coupled with a practical life-approach, for a career in mathematics. I didn't listen either. The same people told me I would get nowhere, that mathematics is too competitive, that there is no room for me there.

On the practical level, it is not normal for mathematics to cause a gain of weight; there could be external factors you should look into. Also mathematics should not cause a loss of social life, perhaps your friends were not the best fit for you?

Concerning academia, inform yourself about the practical means of staying and surviving in academia. You will need a game-plan and a well thought-off strategy always present for the next 3-5 years at every moment in your career until tenure-track. Otherwise you might end up with no job. The most important "currency" in academia are publications, their quantity and their quality. You should incorporate these in your plan very soon. Prestige of the academic institution and of the professors that you've worked with is important in the early beginnings. Look at successful mathematicians (adjust to the level where you want to be in 10 years but always try to do slightly better than they did), their papers, their career progression, etc. to get an idea what is necessary. I wish you the best of luck, no matter what you decide is the correct choice for you.

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The point not raised I think in the above varied comments is: what is the nature of mathematics, and how should one go about doing it? I believe that for any human activity one needs to discuss methodology, but there is not so much of a literature on this. Here is an article on this, for you to agree or disagree with. You can also look at the Prefaces to my book Topology and Groupoids, which you can browse in look inside on amazon.

One also has to explore the nature of ones own talents in relation to the subject. One professor advised his students to do whatever they found easiest!

Some people are fascinated by problems, and by combinatorics. I find myself more interested in the questions I have thought up myself, partly in the hope that it may satisfy three criteria:

  1. No one else has thought of it.

  2. The answer is not technically difficult.

  3. The answer is important.

All this may sound unlikely to find, but if you do not look you do not find. One of my students, Derek Waller, said that he liked to have a hundred ideas. If 10% of your ideas are good, that gives you ten good ideas! Try one damn thing aFter another!

The main theme of my research since 1965 or so, namely on groupoids, came about by writing a book on topology. Writing mathematics in order to make things clear and elegant, and so writing again and again, may eventually make you see that there might be another way of doing things.

The composer Ravel said that you should copy. If you have some originality, this will show. If not, never mind! In fact, an original idea may occur only after copying several times, as the idea has got into the brain.

Thus we have advised students that a thesis is supposed to have a "thesis". So you should start by writing up the background that thesis. As you write it up, you may come across not quite satisfactory bits. That is a start.

I have been fortunate in coming across this broad and flexible programme of higher dimensional group theory, which consisted in testing out uses of higher dimensional groupoids, in the spirit of group theory, and particularly in relation to homotopy theory. I suspect/know many "authorities" regarded it as nonsense, which has been a disadvantage. On the other hand, it kept down the competition.

I suspect my talent is a feel for mathematical structure. I like the comment of Philip Hall: "One should try and develop the algebra appropriate to the geometry, and not try to force the algebraic expression of the geometry into a particular mode, simply because that is available."

I have met "researchers" who either find it difficult to write mathematics, or do not do it all. My advice is the opposite: keep writing. For a paper, you write at the top of a piece of paper: Title. Author. Introduction. That says what you intend to do. Of course you may have difficulty with Section 1 on basic definitions, but you can always come back to it later!

The old army outline of methodology was to write: 1. Overall aims. 2. Present situation. 3. Immediate objective. 4. Method.

You do not start with "Method"!

Ronnie Brown
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I didn't see this answer so I figure I'd try.

“Young man, in mathematics you don't understand things. You just get used to them.” John von Neumann

I always strove to understand maths while I learned it, but I know when to stop and move on to learning something else. I don't think anyone could completely understand math, or physics for that matter, all great mathematicians and physicist still have problems with comprehension. It seems like the comprehension level you are looking for is too metaphysical. Some of the best mathematicians had an amazing memory, this undoubtedly helped them because they could use it to develop further analysis, etc

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"Also, my problem is having weak memory. I forget a lot."

Have you tried using a flashcards program? There is a program where you can add questions. If you answered a question correctly, the program will ask you the question again in 2 days. If you answer correctly in 2 days, it will ask you again in 4 days. After that, the interval will grow to 8, 16, and so on. I found it helpful. You can also set a custom interval. The program accepts mathematical symbols.

Also, don't add EVERYTHING you want to remember to the program. You'll remember a lot of things anyway. The trick is to figure out which ones you need to add.

Regarding how you can lose the temptation of doing math all the time, I'd like to mention the concept of marginal utility. The marginal utility of anything (say, an ounce of water) is how much acquiring one additional unit benefits you. If you're dehydrated, the marginal utility of an ounce of water is very high. As you drink more and more water, its marginal utility diminishes. The same thing happens with mathematical knowledge. Even if you've been consuming new math for years and are anxious to get more of it, there will be a point where its marginal utility will diminish, and you'll be able to focus on other things easily. You just don't know yet what that point is, and maybe you won't reach it even in years, but based on my experience, I expect that you'll reach it at some point.

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    Actually over-hydration can be fatal, if you drink too much you will damage your body perhaps to the point of death. To drive the marginal utility point home, if you do too much mathematics you might end up not actually learning anything at all. – Asaf Karagila Apr 09 '12 at 19:25

Something could be useful when Special Force Soldiers are trained, named Devil Training.

It seems the good reasons are related to transcend extreme limit of capacity to quickly extend the comfort-zone and equipped with massively difficult practices in very complicated environments NOT easy ones.

Some parts the same as good mathematical training, one could absorb both broad and important knowledge quickly, and move on quickly, intensively practices to be done in advanced and complicated 'environment' to train both elementary and advanced skills at the same time. Never stay in easy level but just move on even if some skills/knowledge are not absorbed well, since it'll be better as we go further. There's only a fine line between Wisdom and Foolish, wise persons absorb and move on fast with playing around with knowledge like a swimming fish and keep on deepening the understanding; while fool persons only suffer from the pain passively and repeat everything mechanically without thinking.

While some parts are apparently opposite, in Military Training it's obey the orders, no question asked, but in mathematics(natural science) it's break the orders, endlessly ask questions. Plus some passion, curiosity, self-automatic willingness to explore and some luck, it can generate a very beautiful road in mathematical world.

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As a CPA who is now returning to school to take a Masters in Computer Science (and thereby a few math courses along the way), I can indirectly relate with your problem; however, I do believe my perspective is relevant and may very well be of help to you.

I was like you in my undergraduate studies in accounting. I literally spent every waking moment of my day during the last two years of my studies - seven days a week - preparing for my classes. Like yourself, I am driven by the "why"; not just the "how". It also holds true in the business arena that the majority of students do not ask why a solution is true; they merely accept that it is true and memorize the steps to work out the corresponding solution.

Unlike you, I do not consider myself only marginally talented or mediocre at best; and, in fact, I encourage you to truly consider an earlier poster's comment who challenged your self-deprecating opinion by saying you may indeed be talented and to therefore not prematurely make this judgment concerning yourself. Proverbially: How can I build a house if I do not believe I am able too? It would not be a waste of time to consider how your internalized belief about yourself may be significantly affecting A) your academic performance, and B) your level of personal anxiety. Please take this to heart.

Having now made such statements, I would also like to say that from a practical point of view I totally "get" where you are right now. You want tomorrow's perspective, today, and thereby to make sure you do not make a foolish choice today that will cost you tomorrow. My advice is to follow the deepest passion of your heart today and trust "tomorrow" to work out the other details.

For example, if doing mathematics is truly the supreme passion of your life, then commit yourself to this path irrespective of whether you become a tenured professor or instead find yourself in some area of applied mathematics ... or even software development. Everyone must do their best and strive for the highest achievement. You will always be happiest striving to do your very best while at the same time following the deepest passion of the heart.

Of course, there is more to life than "X" (e.g., mathematics, in your case), so we must also nurture the other parts of our being. Even our base desires teach us this in their insistence every twenty-four hours to feed, rest, clean and relieve. And how about our, higher, emotional and spiritual needs? Is it mere vanity to seek humanity (companionship)? As with the assumptions inherent in mathematics, our carnal and psychological desires also provide self-evident assumptions to aid in the building process of our existence. Listen to yours and act accordingly, but understand that not every house is of the same shape, size, and symmetry; therefore, you may require very little, or very much, of such "y" variables while solving the "x" factor of your life.

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You are only as good as your weakest link. Alot of students and ppl who work in math underestimate the power of being in good shape. Your span of concentration increase and your sleep is often better when you exercise and eat healthy. I got so many insights after running a few miles while catching my breath after being stuck on something.

As for social life ; family is important, close friends aswell. I never ever attended any "social" events at the univeristy its just destructive imo, there is nothing to gain by getting wasted and staying up all night. Maybe a dinner with a few students and have a few beers and just chill out.

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Set your priorities. Get a calendar. Put exercise in it for a half hour 3 times a week. This is better than nothing. Don't do anything that's going to absolutely wear you out. Schedule time to hang out with people. Schedule other priorities like sleep. Then, whatever time is left over, which would probably be quite a bit still, do math. If you still feel the same way, schedule more exercise/sleep/hang out time/whatever.

Lots of people forget math. I wanted to study algebra stuff. I had Real Analysis 1 and 2 (graduate level) one year and spent so much time on those that I had no time for algebra. I couldn't remember the algebra stuff that was actually important. The real analysis isn't even important to me. I should have prioritized better and spent less time on real analysis and more time on algebraic stuff.

I spent so much time away from my family that year also, which was terrible. I now basically work 8-5 and go home. And, maybe Saturday I work for a few hours. Sometimes, I try to get in some more work. But, my family is more important than this math so I don't care if it takes longer to graduate or I don't do quite as well.

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    I'm not trying to be rude or anything, but often it does not help t simply say an imperative like "schedule time for exercise." Usually people are asking *how* to schedule such time. – ely Apr 09 '12 at 00:26
  • @EMS Not to be rude, but looking at the context, you can see that what I said made sense. He is saying that he puts all his time into math. I was telling him to schedule times where he isn't doing math and do math the rest of the time. Then, he gets things done that he didn't before. – GeoffDS Apr 09 '12 at 01:20
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    Right, but that doesn't help explain *how* to put time into other things. The poster wouldn't have even asked the question unless he or she already knew that they *should* put time to these other things. But that's very different than an actual course of action that gives implementational details about how to get one's self to actually commit to this stuff. More relevant advice would be to study the [state of the art of self help](http://lesswrong.com/lw/3nn/scientific_selfhelp_the_state_of_our_knowledge), things like the procrastination equation or pre-commitment. – ely Apr 09 '12 at 01:25
  • @EMS If you have such a great answer, then answer the question yourself. – GeoffDS Apr 09 '12 at 12:58
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    I did before even commenting on your answer ([link](http://math.stackexchange.com/a/129430/26927)). I'm sorry if my comments offended you. It was merely meant to be constructive, polite criticism. – ely Apr 09 '12 at 14:56

Perhaps one of the best pieces of advice I've heard regarding this type of dilemma (not only in math, but for all practices), is that if it's not fun then you need to change it up.

Your math interests sound like they've become more of an obsession than fun. Math clearly fits the way you think, and you strive for satisfaction in understanding it, but you've let it go too far.

Two points:

  1. You will not understand everything
  2. It's ok to not understand everything

Math is a tool, and rarely a lifestyle. You might find it odd to wrap your entire life around understanding and designing a perfect set of wrenches. While in doing so you might come up with a better set, you leave out all the fixing, building, and other productive things you can do with the set you've got.

Go use your math, and allow the understanding you get from its use to propel further understanding along the practices you find interesting. If it's fun then do it, and if it's not fun then change things up. Make it all mathematical along the way, just don't get stuck in the math.

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The agony regarding "How to study maths?" is not as real as posed. Before going to study a topic, one should devote some ten percent duration in making strategy of how to complete the Chapter. Never take anything for granted in Math. Do not rely on the proof of a theorem given in the books with too many "clearlys". Do not rely on Professor's authority. They are rarely ideal to follow. Simply try to discover your own proof, putting every definition and previous results before your eyes written on a broad sheet of paper. Never rely on memory regarding definition or statement of a complicated theorem when using somewhere. They mostly deceive a novice. Simply see it without shame. In this way you will enjoy and form a habit of proving anything on its own. Only in emergency read some parts of the proof. Do not go mad for solving Problems. They are rarely challenging for a student who has gone through the procedure I have described in completing a Chapter. Preserve the "proofs discovered by you" for future need. You will be amazed to observe after studying this way you are becoming bolder everyday. Even then, if you fall into trouble, you should devote more time in making your prerequisite vibrant. Abandon the idea of abandoning maths. Only fortunate people spend their life in Maths. Young people do not as much need exercise as old. Your priority is Math, not anything else. If you leave the table without completing the work, you will have to spend double the time in recapitulating the next time you sit. In this way you will be a great loser in the long run. So leave the study table only after completing the task at hand.

Omar Shaaban
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R K Sinha
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For the weight gain part: Read some Arthur De Vany's writings, and check out his diet. People eat constantly and expect not to gain any weight. De Vany suggests, some days, after a healthy, big breakfast it's okay not to eat anything, and going to sleep hungry. "Going to sleep hungry" part gets big objections from people, but our bodies have not evolved from their hunter gatherer structure just yet. When humans hunted, they could go days without eating much. Our metabolism is geared towards feasting (overeating) after a good catch, and going around hungry until that catch. Constant eating is not our way. Try this and you will happier, your body when it is hungry will be relieved even, because there will be one less thing it needs to work on. On days when you are not eating (after breakfast), it is also advised to set that day aside for physical activity, walking, etc. (yes, while hungry), but your mind will rest in the meantime.

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