TITLE: Struggling with "technique-based" mathematics, can people relate to this? And what, if anything, can be done about it?
QUESTION [46 upvotes]: I'm a 3rd year undergraduate, majoring in pure mathematics. I've done well in the "proof-based" subjects I've taken, and I think that's because I understand the "rules of the game." That is, predicate logic + how to write a coherent proof. Furthermore, people are explicit about what they mean, stating their premises, quantifying explicitly (for all $x$, there exists $y$ such that...), distinguishing between "A implies B" and "A iff B", etc. This obviously really helps.
Recently, however, I've been finding that "technique-based" (as opposed to "proof-based") subjects like complex analysis, vector calculus, differential equations etc. are beginning to frustrate me, and I'm starting to get bad marks, too. It's like when I'm sitting in these lectures, the "logic" of math suddenly becomes opaque. I can never tell what the premises are. I often don't know whether we're trying to show that "A implies B", or whether we're trying to show that "A iff B". Stuff is happening on the board, but the "rules of the game" just aren't clear to me.
Does anyone else have a similar problem with "technique-based" math? And if so, what can be done about it?
Let me give an example. Below, I've copied some of this problem from Wikipedia, and I have inserted my own thoughts in italics.

A separable linear ordinary differential equation of the first order must be homogeneous and has the general form
$$(1)\qquad \frac{dy}{dt}+f(t)y=0.$$
where $f(t)$ is some known function.
I can't tell if (1) is being taken as a premise or not.
We may solve this by separation of variables (moving the $y$ terms to one side and the $t$ terms to the other side).
$$(2)\qquad\frac{dy}{y}=-f(t)dt$$
Are you asserting that (2) follows from (1), or are you saying they're logically equivalent? And I still don't know whether equation (1) is a premise, or what our premises are.
Since the separation of variables in this case involves dividing by $y$, we must check if the constant function $y=0$ is a solution of the original equation. Trivially, if $y=0$ then $y'=0$, so $y=0$ is actually a solution of the original equation. We note that $y=0$ is not allowed in the transformed equation.
Clearly, if $y$ is everywhere zero, then equation (1) holds. But what's all this "we must check" nonsense? Are you trying to say that the statement "the function $y$ is everywhere zero, or equation (2) holds" is logically equivalent to the statement that "equation (1) holds?" If that's what you're meaning, why don't you just say so? If the argument was just laid out in a coherent fashion, nonsense like "we must check" simply wouldn't appear.
We solve the transformed equation with the variables already separated by integrating,
$$(3) \qquad \mathrm{ln} \,y = \left(-\int f(t)dt\right)+C$$
where $C$ is an arbitrary constant.
Are you trying to say that if (2) holds, then there exists $C$ such that (3) holds? Then why don't you just say so? Or maybe you're trying to say that for all $C$, (3) holds iff (2) holds. I honestly can't tell.

Well you get the general gist. So my question is, can other people relate to this, and what can be done about it?

REPLY [4 votes]: First of all, here's a somehow relevant blog post by Timothy Gowers.
Now, I understand the situation you describe very well. It's not just analysis, mind you, the problem may arise in every math topic.
The way I understand it, and you might agree by now (a year after the question was posted), it doesn't have to do with "techniques" at all (every math topic features techniques, after all), but with the assumed shared intuition among the participants. You can hear proof theorists talking to each other about their research, and you might again not see how they use the rules of the game. You can hear mathematicians talk about something using plain old slang, and slang feeds on linguistic unclarity (I really mean slang, not math jargon), and yet, they manage to communicate perfectly. How do they do it?
One can put it this way: "we tolerate linguistic unclarity when the intuition is clear". Or this way: "we don't need linguistic nitpicking when the intuition is clear". When the intuition is clear, formality hinders. When it's not clear, formality helps bring it home. One can compare this to talking about chess: how does one explain a certain family of, say, openings to a chess student, and how do grand masters talk about it? Certainly, the latter don't need to specify the squares of each move as often as the former.
Analysis has an old and very strong tradition among math disciplines, and its communication, in particular its teaching, in my experience, is usually based on the intended, mostly physical, intuition; it seems that people teaching it somehow decide that the intuition should be clear to everybody (it isn't so, of course). On the other hand, the teaching of algebraic topics, again in my experience, usually suffers from stressing the rules of the game too much. It's as if it should be clear to everybody that formal reasoning is important and necessary (again, no, it's not clear to everybody).
There are good technical reasons behind this situation. There's an abundance of most basic notions in analysis which are highly complex from a logical viewpoint. One can check for example the alternating quantifiers in the various notions of continuity. To be formal about such proofs would mean to spend forty five minutes on each of them; well, this is something a student who still misses the intuition could do on his or her own. It's also something that a professor should do at least once. On the other hand, it is easy for a course in, say, group theory, to start by stating the axioms for a group; their complexity is low enough. But why groups, and how come "groups"?... A professor should talk about it, and a student, as always, could chase the connections alone, starting from the ones that are usually cryptically stated as "examples" after each definition in the lecture notes, without any mention of the intuitive significance of each in the corresponding mathematical field. And later, when an adequate level of intuition is attained, and groups as objects are clearly justified in themselves, one would be able to talk about them using language that resembles the wikipedia excerpt of the original post.
(Beside this technical reality, there are probably also good historical reasons for this situation. The development of analysis was mostly driven by physics, and it has even led to formal mathematics and given rise to algebraic, more formalist thinking, and so on. Under this light, it's almost understandable that people would assume that the intuition behind it is familiar. It's also almost understandable that algebraic teaching would stress formality as an attempt to distinguish the topic from analysis by showing how it can be consistently exact. I'm not pretending to be good at history, actually, so take this paragraph with a grain of salt; I'm just trying to explain (to myself) my otherwise strong belief, that the teaching of various math topics is often based on some deep but questionable assumptions and stale traditions.)
So, what should one do about it? Well, people have answered well enough before me. Personally, when I don't understand a vaguely stated argument, I try and decipher the rules that must have been followed in a trial-and-error fashion. With enough patience and time, I usually get it, and it gets quicker and quicker each time, since it helps build intuition. When I'm trying to explain something to somebody who doesn't share my intuition (it being significantly lower or significantly higher than theirs), I do try and stress the rules of the game. When I'm talking to someone who more or less does share my intuition, I'm talking to them as if I were talking to a buddy about which bar to go next: not by giving the exact route, but just by handwaving.