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I'm currently enrolled in an introductory real analysis course.

While I enjoy the beauty of pure mathematics, there are 2 problems I face.

1) Constructing proofs

2) Attempting to understand a proof without seeking help despite hours attempting to understand.

While I do in general understand many of the proof, I find it difficult/ impossible to construct the proof. Is this a matter of a lack of practice?

Should I be asking help in understanding the proof whenever I am facing a brick wall? Should I be expecting pure mathematics to be time-consuming?

Ivan Lerner
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Guest
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  • Check out the book "How to Prove It: A Structured Approach" by D. Velleman (ISBN-10: 0521675995). And yes, you can expect pure mathematics to be time-consuming if you really want to get under the skin of things. Have fun! – Moritz Mar 13 '15 at 07:08
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    My experience for proving is practice, practice and practice. Moreover, after reading a proof of a theorem, close the book and redo it again. And, after for a while, you will be confident with your skill. – Alexander Mar 13 '15 at 07:11
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    In addition to practicing a lot, make sure you don't start with proofs that are too advanced for your level. You should be reviewing easy proofs in the beginning, and when you are comfortable enough to prove such easy statements on your own, you can work your way up. Sometimes drawing pictures really help (though you can't always draw a picture). – layman Mar 13 '15 at 07:12
  • Best way to learn pure mathematics - As opposed to what ? Impure mathematics ? – Lucian Mar 13 '15 at 07:25
  • @Lucian Maybe as opposed to applied math? – layman Mar 13 '15 at 07:27
  • @user46944 Dear sir, I am applied Math student; and, I strongly disagree with your opinion. Why??? the reason is in our world, we have to write proof a lot, but the only difference with pure math is we can assume that our function is ideal, from that we can use any theorem without proving it. For example, let say "dirac delta function," in Riemann integral, you can't integrate it because Upper sum is not equal Lower sum, but in lebesgue integral, you can do it. It says that we just make things easier for our life not hard as pure math. – Alexander Mar 13 '15 at 07:45

3 Answers3

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  1. and 2. are hardly unique to your case. Literally no mathematician has ever avoided these two "problems" (I prefer to call them "challenges") at any stage of their careers.

"While I do in general understand many of the proof, I find it difficult/ impossible to construct the proof. Is this a matter of a lack of practice?"

Yes. It's hard for everyone, especially at first. Keep at it and you'll be astonished at how your mind adapts. "Mathematics is not a spectator sport."

"Should I be asking help in understanding the proof whenever I am facing a brick wall?"

In the early stages of your education, only after trying your absolute hardest to understand the proof, consulting other references if need be, should you perhaps ask for a hint about a proof or a solution. Be honest to yourself - only ask for help when you have actually exhausted all other options. You only hurt yourself by asking for help prematurely.

"Should I be expecting pure mathematics to be time-consuming?"

Yes. It's hard, and that makes it worth doing. Individual results in mathematics (pure or otherwise) are culminations of weeks to months to years to decades of thoughts and ideas. Centuries if you include your predecessors that added to the foundations of these ideas. If you are studying mathematics full-time and your problems do not take you very much time to solve, chances are you should be looking at harder problems.

Gyu Eun Lee
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  • I generally spend an entire day trying to break the brick wall, most times to no avail. I have a tendency not to ask for help although I eventually do. The issue with this is that, in University, time is an opportunity cost. What is a good estimate of time I should be spending on understanding the proofs before seeking help? Being an undergraduate doing a double major in Physics and Mathematics, I understand both activities to be time consuming but time is fairly limited and is an opportunity cost when considering the amount of workload involved. – Guest Mar 13 '15 at 07:18
  • Depends on the level of mathematics you're doing. You have to realize that there are no shortcuts to understanding mathematics - spending a long time and getting stuck is an essential part of the process. I would say it's reasonable to ask for a hint after you make a genuine effort at attempting the problem, but you're the one that has to judge what a "genuine effort" means. Then again, it's also important to see how others approach the problem, so I would look into working together with other students and querying your professor on how they would approach problems in general. – Gyu Eun Lee Mar 13 '15 at 07:24
  • (continued) But don't get stuck to the point of being left behind. It's equally important to manage your time well. If an obstacle is such that it's actually preventing you from progressing onto further topics or seeing the bigger picture/main ideas, that's a good indication that you should ask for some aid. Not every roadblock is equally important to overcome unaided. Again, use your best judgment. – Gyu Eun Lee Mar 13 '15 at 07:28
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If you are interested in pure mathematics, and would like to give a try, I recommend Linear Algebra (it is on its 4th edition now, i guess) written by Friedberg and other two authors. In fact, this was my first book studying in this field, and I found it profitable to me. Most of students encounter the two problems you mentioned, so don't be that depressed. One of the reason (and I believe, the reason of most time) of being unable to construct a proof is not understanding all the definitions, which let one be confuse with what should be proved. The book starts from the definition of vector space and also offers enough explanations with intuition. Moreover, I believe the problem sets it contain helps a lot too, especially the true-or-false problems. Try to give every details in the beginning to support your proofs, and you will improve.

The second book I learnt from is Baby Rudin, which is not easy at first glance, but I found it profitable to copy his proofs. You will learn from it a lot :)

user223261
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It is practice sir. Math is one of those things where, if you have a calm, relaxed mind, things will just come if you give it time. So take your hours to understand your proofs, you will find which way you learn best.

CaptainFantastic
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