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I've just started a second year course in real analysis. This is my first proof-oriented course. Last year, our maths curriculum was introductory tertiary calculus and algebra. When I practised problems, getting feedback was as simple as looking at the answers at the back of the book; if my answers consistently agreed with the authors', then I must be doing it correctly.

For this new and proof based course, there isn't such a simple way of finding out whether or not I know what I'm doing. How do you get feedback? I just want a way of verifying that my knowledge isn't fiction.

How do I know if I'm doing real analysis correctly?

Yai0Phah
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Daniel Levin
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    The ultimate way to know if you are doing real analysis correctly, is to make a prediction and test it with an experiment (like in physics, financial mathematics and any other field that uses it) :-P More seriously, what about asking someone if your proof is correct (it could be a teacher, or perhaps just ask here, at math.SE)? – dtldarek Feb 22 '14 at 09:13
  • @dtldarek The problem is that, even if the idea is right, the proof might be somewhere fauty. It's hard for one to evade from these tough though minor problems, especially when one is dealing with real analysis related to countability and measure theory. For example, we usually take $a_n>\alpha_n-2^{-n}\epsilon$ then sum up. We usually ignore the issue that $\alpha_n=+\infty$ for some $n$, for example. Such mistakes even appears in books, and then, in lectures of professors. – Yai0Phah Feb 22 '14 at 09:26
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    @FrankScience You can't know for sure, there is a non-zero probability, that all the people that studied mathematics since the Greeks, happened to have some kind of universal mental mathematical aberration which produced the same insane mistakes in everyone. Hence, all of the mathematics could be false and we wouldn't know about it, our own brain blocking and reinterpreting the signals for us. Peer review frequently fails. Still, I don't know any better than this or simply asking a person (maybe more than one if being paranoid) we trust to scrutinize what we have crafted. – dtldarek Feb 22 '14 at 09:41
  • @dtldarek I didn't say that peer review is useless. I just said that people aren't good at formal checking. As Thurston said in On proof and progress in mathematics, Our system is quite good at producing reliable theorems that can be solidly backed up. It’s just that the reliability does not primarily come from mathematicians formally checking formal arguments; it comes from mathematicians thinking carefully and critically about mathematical ideas. So if we want to produce a nearly absolutely correct proof, then computer assistant proof is a choice. – Yai0Phah Feb 22 '14 at 10:01
  • @dtldarek Thus it is not easy for newbies, just like me, to be managed to make the idea precise and to be managed to control those mistakes not to impact the key idea. I did ask teachers, professors or online to check my ideas and proofs. Incidentally, even if the probability is zero, it doesn't mean that we're safe. It's not a classical probabilistic model, I think. – Yai0Phah Feb 22 '14 at 10:04
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    @FrankScience I would say that in this case asking someone and spending more time on understanding the mathematical ideas involved is a more effective use of time (considering both the writer and the checker) than trying to encode your proof in a proof assistant. – dtldarek Feb 22 '14 at 10:17
  • @dtldarek That's it. But I found that MSE isn't, as far as I've experienced, a good place for peer checking (The word peer review is preserved for mathematicians). – Yai0Phah Feb 22 '14 at 10:30
  • @FrankScience You could have always put a bounty on it. – dtldarek Feb 22 '14 at 10:40
  • @dtldarek I posted this in meta. – Yai0Phah Feb 22 '14 at 10:55
  • @FrankScience Good idea. – dtldarek Feb 22 '14 at 10:57

3 Answers3

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General rules which could prescribe in detail the most useful discipline of thought are not known to us. Even if such rules could be formulated, they would not be very useful. Rather than knowing the correct rules of thought theoretically, one must have them assimilated into one's flesh and blood ready for instant and instinctive use. Therefore, for the schooling of one's powers of thought only the practice of thinking is really useful. The independent solving of challenging problems will aid the reader far more than the aphorism') which follow, although as a start these can do him no harm. One should try to understand everything: isolated facts by collating them with related facts, the newly discovered through its connection with the already assimilated. the unfamiliar by analogy with the accustomed, special results through generalization, general results by means of suitable specialization, complex situations by dissecting them into their constituent parts, and details by comprehending them within a total picture.

There is a similarity between knowing one's way about a town and mastering a field of knowledge; from any given point one should be able to reach any other point 1. One is even better informed if one can immediately take the most convenient and quickest path from the one point to the other. If one is very well informed indeed, one can even execute special feats, for example, to carry out a journey by systematically avoiding certain forbidden paths which are customary-such things happen in certain axiomatic investigations.

There is an analogy between the task of constructing a well-integrated body of knowledge from acquaintance with isolated truths and the building of a wall out of unhewn stones. One must turn each new insight and each new stone over and over, view it from all sides, attempt to join it on to the 'edifice at all possible points, until the new finds its suitable place in the already established, in such a way that the areas of contact will be as large as possible and the gaps as small as possible, until the whole forms one firm structure.

"Problems and Theorems in Analysis" (Excerpt of the) Preface to the first German edition, Polya & Szego.

Pedro
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The usual answer (and I think the best answer): Take a course with an instructor, where you do proofs as homework and the instructor grades the homework and criticizes it.

Some things are best learned with an instructor in person, as opposed to reading the textbook and viewing video lectures, however good these are.

Maybe in 10 (or 50?) years, there will be computer programs good enough to play the instructor role for this. But I think not yet.

GEdgar
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Check out books on writing proofs, they will give you general techniques. See for example The book of proof. Read some good introductory texts on the subject (you'll probably find many lecture notes on the web, for a variety of subjects I like the ones by William Chen). Check the proofs carefully. See why each step was taken, look for alternative proofs, try to come up with your own parts.

Much of reading proofs is like debugging: look for potential counterexamples, consider extreme cases, work through a few examples. It is rough travel at first; but as always, practice makes the master. Good luck!

vonbrand
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