6 ms·

How to Read Proofs Faster: A Summary of Useful Advice

98 points by dlo 12y ago

chunky1994 12y ago
As far as I see it, every proof I have ever encountered has been of two types: A => B, or A <=> B. The first of which has the following structure: A: Given conditions (a,b,c,...) are true; B: We can say (..., x, y, z) are true. The second having a structure: A: Given conditions (a,b,c,...) are true; B: We can say (..., x, y, z) are true. AND B: Given conditions (..., x, y, z) are true; A: We can say (a,b,c,...) are true. The easiest (and quickest) way for me to parse proofs given the above representation is by ensuring I know what the conditions (a, b, c, ...) mean and how removing one of them changes the proof, therefore ensuring I know exactly how each condition contributes to the final result. (This is probably what he was talking about in his second point)
pavpanchekha 12y ago
This is great advice for reading proofs. In particular, the bit about never diving into a proof until you know why you expect it to be true is a big one. I find it very helpful, after I've understood a proof, to try to put together the informal version of why it works, and try to explain that informal version to someone else.
kimonos 12y ago
Great post! I find this very useful! Thanks!
zak_mc_kracken 12y ago
As far as research articles go, what I've noticed is that I never finish them on the first read. And I don't think anyone should expect to. Too often I see students start reading articles, getting lost after a page and just giving up. It's perfectly normal to stop understanding after a while and putting the article down. What you need to realize is that it takes time for your brain to process new abstract concepts. Give your mind some time to digest these new concepts. Put the article down and think about it on your off time in the following days. Then pick the article again and see if the part you already read looks more trivial and if you're able to go further this time around. Most of the time, you will. Repeat. There is no shame in taking multiple reads and months in order to read a research paper: keep in mind that it took the author(s) years to reach the level of expertise they are showing in that article. Cut yourself some slack. But don't give up on the first try.
deleted 12y ago
[deleted]
kenjackson 12y ago
Has anyone done rap genius for research papers? I think informal context associated with proofs would be a huge help in understanding. And even as a proof writer I'd put details in it that I wouldn't put in the formal proof.
- hackplus 12y ago
  Several years ago I started a website called WTFormula.com Unfortunately I was the only contributor, and since I'm not a mathematician my contributions were very few. I haven't been able to convince other people with helping writing content, and eventually I dropped it for lack in interest (the domain is also available again). I still believe it would be a very useful and nice tool to have, though. Currently I'm working on http://libflow.com http://libflow.com (on my spare time) and I have a lot of ideas to improve it (this "rap genius for research papers" was one of the possible improvements I had in mind). Unfortunately the project is on hold because I currently don't have the money to move it to a dedicated server and enough free time to make major changes to the site.
grovulent 12y ago
So I'm going to make a controversial claim and then try to defend it: There is a cultural problem in mathematical communication and pedagogy, which is keeping the discipline from having a wider accessibility and appeal. I come from a background where I ended up studying mathematical logic without having any mathematical skills to speak of at all. You'd be surprised how far you can go in the former without any training in the latter. I'm now teaching myself maths. The reason this is significant though is because of the notion of a proof as it is defined in mathematical logic is different to the proofs you see presented in mathematical papers and textbooks. In mathematical logic, what we care about is the precise mapping between the syntax and the semantics. The ideal (which for some systems is entirely met) is a perfect functional mapping between the two. And because we care about this - a proof is something which is defined carefully so as to ensure that no invalid WFF (well formed formula) is derivable from your axioms and rules of inference. Obviously maths doesn't want to prove falsehoods either, but the difference is in the level of explicitness. Every line in a proof of logic must have it's justification written beside it - whether it be by use of a rule of inference, or a previously derived theorem. You don't skip steps just because to you they seem 'obvious'. And my wonderful logic teacher at the time would take off marks whenever we did. Reading proofs in mathematical logic - even long and complex ones - is more often than not straightforward. Even the meta-logical results about the systems I studied were presented with greater clarity than your average proof in normal mathematics. Why is this? Well it seems to me that mathematicians un-apologetically skip steps in their proofs all the time. And it's extremely rare that they will state their justifications. To me this is as bad as un-commented, un-documented code with no unit tests. Any coder that inherited that sort of code would be rightfully annoyed! And this leads to situations that require this blog post. I have a friend who is a maths PhD (who has been a great help in my mathematics studies) - who defends this aspects of mathematics. His main argument is that of simple convenience. The problem, he says, is not the presentation but my lack of mathematical experience. If every proof had to be written for those with little mathematical experience, then no maths would ever get done. As I get more experienced with Maths - I can't disagree that reading proofs gets easier. But I feel his reply misses the point. These gappy proofs are endemic in pedagogical literature as well - not just research papers. And a pedagogical text shouldn't be about the convenience of the author. Different textbooks assume different levels of background knowledge and I find it to be a complete mess. Besides this, I generally have little sympathy for this reply because of my background in logic which trained me to be explicit. Whereas all these mathematicians are growing up thinking that it's normal to just assume everyone knows what you're on about. And, besides the teaching context, in the research context I've heard things aren't exactly great - you get this absurd situation, described to me by my friend, where mathematicians gather at conferences - present papers, but generally have little understanding of what anyone else is talking about. Mathematicians seem fine with this, apparently. Why? I believe this to be a cultural reality that could be changed - not something essential to the nature of the discipline. Don't get me wrong - there are some wonderful examples of mathematicians that are working really hard to change this. I found the Ohio State university calculus course on Coursera to really try to be innovative in their pedagogical approach (Their text book was a bit meh - but their online exercise application was brilliant). But in general, I just don't see a lot of this. Agree? Disagree?

98 points by dlo 12y ago

chunky1994 12y ago

As far as I see it, every proof I have ever encountered has been of two types: A => B, or A <=> B. The first of which has the following structure: A: Given conditions (a,b,c,...) are true; B: We can say (..., x, y, z) are true. The second having a structure: A: Given conditions (a,b,c,...) are true; B: We can say (..., x, y, z) are true. AND B: Given conditions (..., x, y, z) are true; A: We can say (a,b,c,...) are true. The easiest (and quickest) way for me to parse proofs given the above representation is by ensuring I know what the conditions (a, b, c, ...) mean and how removing one of them changes the proof, therefore ensuring I know exactly how each condition contributes to the final result. (This is probably what he was talking about in his second point)

pavpanchekha 12y ago

This is great advice for reading proofs. In particular, the bit about never diving into a proof until you know why you expect it to be true is a big one. I find it very helpful, after I've understood a proof, to try to put together the informal version of why it works, and try to explain that informal version to someone else.

kimonos 12y ago

Great post! I find this very useful! Thanks!

zak_mc_kracken 12y ago

As far as research articles go, what I've noticed is that I never finish them on the first read. And I don't think anyone should expect to. Too often I see students start reading articles, getting lost after a page and just giving up. It's perfectly normal to stop understanding after a while and putting the article down. What you need to realize is that it takes time for your brain to process new abstract concepts. Give your mind some time to digest these new concepts. Put the article down and think about it on your off time in the following days. Then pick the article again and see if the part you already read looks more trivial and if you're able to go further this time around. Most of the time, you will. Repeat. There is no shame in taking multiple reads and months in order to read a research paper: keep in mind that it took the author(s) years to reach the level of expertise they are showing in that article. Cut yourself some slack. But don't give up on the first try.

deleted 12y ago

[deleted]

kenjackson 12y ago

Has anyone done rap genius for research papers? I think informal context associated with proofs would be a huge help in understanding. And even as a proof writer I'd put details in it that I wouldn't put in the formal proof.

hackplus 12y ago
Several years ago I started a website called WTFormula.com Unfortunately I was the only contributor, and since I'm not a mathematician my contributions were very few. I haven't been able to convince other people with helping writing content, and eventually I dropped it for lack in interest (the domain is also available again). I still believe it would be a very useful and nice tool to have, though. Currently I'm working on http://libflow.com http://libflow.com (on my spare time) and I have a lot of ideas to improve it (this "rap genius for research papers" was one of the possible improvements I had in mind). Unfortunately the project is on hold because I currently don't have the money to move it to a dedicated server and enough free time to make major changes to the site.

grovulent 12y ago

So I'm going to make a controversial claim and then try to defend it: There is a cultural problem in mathematical communication and pedagogy, which is keeping the discipline from having a wider accessibility and appeal. I come from a background where I ended up studying mathematical logic without having any mathematical skills to speak of at all. You'd be surprised how far you can go in the former without any training in the latter. I'm now teaching myself maths. The reason this is significant though is because of the notion of a proof as it is defined in mathematical logic is different to the proofs you see presented in mathematical papers and textbooks. In mathematical logic, what we care about is the precise mapping between the syntax and the semantics. The ideal (which for some systems is entirely met) is a perfect functional mapping between the two. And because we care about this - a proof is something which is defined carefully so as to ensure that no invalid WFF (well formed formula) is derivable from your axioms and rules of inference. Obviously maths doesn't want to prove falsehoods either, but the difference is in the level of explicitness. Every line in a proof of logic must have it's justification written beside it - whether it be by use of a rule of inference, or a previously derived theorem. You don't skip steps just because to you they seem 'obvious'. And my wonderful logic teacher at the time would take off marks whenever we did. Reading proofs in mathematical logic - even long and complex ones - is more often than not straightforward. Even the meta-logical results about the systems I studied were presented with greater clarity than your average proof in normal mathematics. Why is this? Well it seems to me that mathematicians un-apologetically skip steps in their proofs all the time. And it's extremely rare that they will state their justifications. To me this is as bad as un-commented, un-documented code with no unit tests. Any coder that inherited that sort of code would be rightfully annoyed! And this leads to situations that require this blog post. I have a friend who is a maths PhD (who has been a great help in my mathematics studies) - who defends this aspects of mathematics. His main argument is that of simple convenience. The problem, he says, is not the presentation but my lack of mathematical experience. If every proof had to be written for those with little mathematical experience, then no maths would ever get done. As I get more experienced with Maths - I can't disagree that reading proofs gets easier. But I feel his reply misses the point. These gappy proofs are endemic in pedagogical literature as well - not just research papers. And a pedagogical text shouldn't be about the convenience of the author. Different textbooks assume different levels of background knowledge and I find it to be a complete mess. Besides this, I generally have little sympathy for this reply because of my background in logic which trained me to be explicit. Whereas all these mathematicians are growing up thinking that it's normal to just assume everyone knows what you're on about. And, besides the teaching context, in the research context I've heard things aren't exactly great - you get this absurd situation, described to me by my friend, where mathematicians gather at conferences - present papers, but generally have little understanding of what anyone else is talking about. Mathematicians seem fine with this, apparently. Why? I believe this to be a cultural reality that could be changed - not something essential to the nature of the discipline. Don't get me wrong - there are some wonderful examples of mathematicians that are working really hard to change this. I found the Ohio State university calculus course on Coursera to really try to be innovative in their pedagogical approach (Their text book was a bit meh - but their online exercise application was brilliant). But in general, I just don't see a lot of this. Agree? Disagree?