Post Number: 16
|Posted on Tuesday, 03 January, 2012 - 10:08 pm: |
My question is mathematics-related, but this is not a 'problem with a problem' per se.
I've written on this forum a couple of times before and I got amazing replies. For that I am truly grateful. I hope you can help me with this one, too.
My question is this:
If one is to 'ask questions' to a math text (or textbook) that you are studying, in order to understand the text better, what would you include as your 'expert questions'? These expert questions would be the best of the best questions to ask to reveal the mathematics that might otherwise remain obscure for the not (yet) so experienced mathematician, or simply to bring out the sometimes hidden meaning and help understand the subject better. Here are a couple that I have come up with:
- How can this proposition/definition/... be used? Does it relate to more than this particular area of mathematics?
- Can this theorem be proved in some other way?
- Are there other theorems that (almost) say the same as this one?
Do you have any additions? It would be very helpful to hear the insight from you guys with a lot of mathematical experience.
Post Number: 1736
|Posted on Wednesday, 04 January, 2012 - 06:46 am: |
I find that different mathematicians have different interests in different topics.
Some people may be studying a certain bit of maths purely to help with something else (in which case they might not be too interested in the topic itself, they just want the main results listed). Some people (mainly students), may be forced into studying the topic for their studies. Some may want to know everything about the topic but with/without seeing the bigger picture.
I guess it all depends on the type of mind you have. One thing I can say with certainty is that everyone (especially on this forum) will at some point have come across a textbook which is just too complicated to read and is written in a "don't care about the beauty of maths" style.
I don't think it is only about the experience of the reader, in my opinion I think that quite a lot of maths books are actually written in a bad style. When I read maths I like intuitions, links and to be told "this is a great result/topic, here is why".
This is what I try to achieve in my own writings. Unfortunately, quite a lot of maths books just tend to go linearly through a vast amount of results with no explanation. Because of this people often have to read between the lines and spend a while figuring out for themselves why things must be interesting. If books like this were converted into fiction they would not sell :p.
However, I do love reading maths texts and I find lots of books that fit my style, but more often than not I find that online notes are useful and give more insight. As I say, it all depends on what kind of mind you have.
Post Number: 1036
|Posted on Wednesday, 04 January, 2012 - 07:50 am: |
I'm not sure what level of maths you're at, but you're talking about definitions and theorems and propositions, so I'm going to assume it's around first year undergraduate level or higher. Tell me if that's an unfair assumption!
When I encounter a definition, I like to ask myself questions of the form "Why has the author defined this? Why do I care whether (for example) a ring is noetherian or not?". But bear in mind that those aren't necessarily questions you can answer immediately. Often you only care whether (for example) a ring is noetherian or not because of various theorems that will be given later in your book. You can't study group theory without knowing what a group is, but just given the definition, it's not necessarily obvious why that definition is good or useful or interesting. (You could contrast this with, for example, the real analysis definition of continuity of a function. When you meet this definition, you probably do have enough machinery and intuition available to you to work out why it's sensible and other definitions aren't.)
Sometimes an author will define something that seems very similar to something you've seen before. You should definitely try to work out why it's different (if it even is at all). I remember being very hung up on one of the definitions of a projective module: [inline]P[/inline] is projective iff there exists [inline]Q[/inline] such that [inline]P\oplus Q[/inline] is free. No matter how much I racked my brains that day, I couldn't seem to come up with an example of a projective module that wasn't just free. It turned out I was just thinking about modules in the wrong way.
When I encounter something labelled as a lemma, I don't tend to give it much thought, unless it seems interesting. But if the author rarely calls things "theorem" or "proposition", I try to make sure I understand why exactly those things are so much more important than the lemmas. It's often not entirely obvious until you have intuition about the area, but that intuition comes from knowing what the main theorems are and why they're important!
If you really can't work out why a theorem is interesting, for example because it's too general for you to get your hands on, try applying it to examples.
Theorems start off with hypotheses: "if G is a group which is cyclic and of odd order then...". Try to work out why these are necessary. (Actually, sometimes they're not at all, but the proof given doesn't work in other cases.) This is quite good exam technique, too - sometimes you're asked to prove a statement under some hypotheses, and if you haven't used one of the hypotheses, as long as you trust your exam setter, you might well be wrong!
And, of course, the age-old advice that you should never learn anything from just one source. As many different perspectives as possible are good. I often find even some people I teach say interesting things I hadn't thought of, or things I know but in a way I wouldn't have phrased them, or draw analogies I hadn't drawn before.
Post Number: 3272
|Posted on Wednesday, 04 January, 2012 - 08:22 am: |
This is a great question!
There are lots of good answers, and I'm sure that other people will offer their favourite suggestions, in addition to Daniel and Billy. Here are a couple.
- Do we need all the assumptions? Where did we use them in the proof? Can we drop/change any of the assumptions? If we added a stronger assumption, could we prove a stronger result? (This is closely related to Billy's penultimate paragraph.) Playing around like this can help you to understand both the statement and the proof.
- What examples illustrate this concept/result? What are the `boundary' cases (examples that only just work, or that only just fail)? What is a `typical' example? What is an `extreme' example? What are good examples that don't work? Finding your own examples to test a definition or result is a great way to understand the limitations and to get a feeling for what it really means.
Here's one place to look for a few top tips.