Understanding mathematics imprecisely

For a long time, it has been a complete mystery to me how any of my peers understood any math at all with anything short of filling in every detail, being careful about every set theoretic detail down to the axioms. That's a slight exaggeration, but I certainly did much worse in courses where I attempted to replicate this myself by not reading every proof to save time.

It's only recently, and only within the field of probability theory, that I've developed the ability to do this myself. I am currently following Grimmett's book on Percolation theory. There are far too few details for someone of my level to fill it in completely, but I am getting more than nothing out of it.

Question 1: I would like to learn how to get even more out of such "incomplete" studying.

What tends to happen even now, and more before, is that as soon as I don't understand something, I lose focus and everything just flies over my head. I imagine this is partly psychological, since from a logical perspective if I have to accept proposition $P$ to derive $Q$, I could just think of myself as having proven merely that $P$ implies $Q$, and then no "acceptances" are being made.

Most of the time, professors simply stare blankly at me, wondering how I could persist like this, and all they say is to stop. But it's not that simple, because it appears my intuition is also primarily symbolic. Sure, I think of some geometric pictures when they're called for, but most of my problem-solving creativity comes from pattern matching methods and tricks with situations.

Question 2: How does one distill out the important ideas of a mathematical reading, such as a proof or paper?

Grimmett's book is very helpful in this regard. He will always tell me what's important, and as long as I'm willing to believe him, then I don't have to do anything. But what if I need things that are different than he emphasizes? I always worry that by not understanding everything, I will eventually reach some point in my life when I need to use some fact/method I glossed over and forgot, and that it could be framed in such a way that I would not even be aware of what is missing. That way I wouldn't even be able to do a huge review to rescue the fact from the depths of my ignorance. My current way of thinking about this subproblem of question 2 is that mathematicians always take this risk by not studying everything. So it's a risk-minimization game with time as the constraint. If so, how do I make smart choices with regard to this game?

Question 3: With my recent ability to learn imprecisely in probability, I've started to see many connections, even with outside fields. Many of them are probably fictitious. Many of the questions that I think are highly-motivated might actually be not really worth answering. How does one decide what questions are interesting? As a graduate student who has barely popped out above what the traditional classroom has to offer, I am very lost in this regard.

Question 4: The revelations that enabled me to understand math imprecisely came all at once. A similar comment about the abruptness of my coming upon the ability to proof-check without significant error could be made 2 years ago. Most of my peers seem to learn rather continuously, but the evolutions of my way of thinking seem to come all at once. Is there anything bad or good about this? If so, how do I minimize the bad and maximize the good?

As always, answers to subsets are appreciated.

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What do you think of when you hear the phrase "mathematical intuition?" –  Alexander Gruber May 29 '13 at 19:42
I used to just tune out everything following such a phrase because I would develop my own intuition, typically only after full understanding. Lately, I have even requested specifically that people teaching me in some specific cases say why something is intuitive. I also seek it out myself pictorially, since in analysis, sometimes pictures help more than symbol manipulation. –  Jeff May 29 '13 at 19:57
Terry Tao has a blog post There is more to mathematics than rigour and proofs which seems relevant. –  Martin May 29 '13 at 20:26
I mean this post was mostly about how to become a better mathematician in the sense of what most people think mathematics means, which is less about rigor. So the last comment was more of an excursion into my own personal opinion than something totally on topic. –  Jeff May 29 '13 at 20:40
This is not to suggest I think rigor is everything. I only behaved that way in the past because it was the only frequency I could perceive. But even now, I still think that some of the beauty of mathematics is its rectilinear logical perfection. –  Jeff May 29 '13 at 20:46

4 Answers

Full understanding is illusory. If you pursue it, you will find yourself trying to say what a number is, or a set, and digressing into the problem of making language, which for math is a meta-language, precise. And, of course, that can't be done.

So regarding your first question, it might help to observe how futile that innate wish of yours is, and how much you understand without full understanding (or compunction) in all other aspects of your life.

Imagine trying to learn biology and studying the chemical processes in the body, then asking "what is a chemical". You are given an answer that has to do with molecules, a term which you then inspect for precision's sake. Atoms come up, then electrons. Eventually you are learning quantum physics when all you wanted to do was understand how allergens work, or some such thing.

You must operate at the appropriate level for a specific problem. It's no use to reinvent the wheel and do everything from first principles. That would be like writing every program in machine code.

One day, our brains may be augmented with enhancements that allow us to have enough knowledge to understand everything down to our "axioms". Until then, it is a matter of becoming comfortable with our limitations and trying to work with what we have to be awesome.

In terms of knowing which questions are interesting, I think that is one of the harder parts of research. One almost has to be prescient.

And as for getting the important ideas of a proof, my first answer is that sometimes you can't really. Some proofs are just a confluence of numerical estimations and limit results and don't give any real insight into what is going on. Since you seem to be a probabilist, I would point to the proof that a random walk in dimension $n$ is recurrent for $n=1,2$ and transient otherwise. One feels there should be an intuitively understandable reason, but all one gets is Stirling's formula.

For other proofs it is a matter of becoming comfortable enough with the terminology and techniques used in the proof (by re-reading) to see the forest for the trees. In Kung Fu one talks about "learning to forget". You learn the movements carefully so you can perform them without thinking about them when the time comes. You do the same when you learn to integrate or differentiate - you don't want to be doing this from the limit definition when crunch time comes (exam say).

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I've observed how my closest colleague did it back in the day of my Phd and postdocs and I took over his technique. What he did was to sit down, read the paper superficially and then try to work out simple stuff he understood on his own. Then he would try to build his own version of what he got from the paper, often without fully understanding what had been going on. But he just had a general idea of the gist of the paper and tried to rebuild the idea in his own words, math, etc...

I remember I then proceeded to do the same later when studying some ecological model that we were trying to pour into mathematical formulas. I felt that the work that had been done was not very rigorous or even incomplete. So I rebuilt the model for myself superficially imitating others at first but gradually abandoning their approach for my own. And this without ever fully learning the necessary techniques of Markov processes, stochastic equations, etc... I feel that by doing this work, my understanding of the material is much deeper than it would have been if I had read a book about it or followed a standard course.

What also helped was the countless conversations and presentations I had to do about my work that forced me to put my thoughts into words understandable to others. They might not have gotten much out of it, but it has been very beneficial to myself for sure.

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I'm not sure I have a strong answer to your questions, but here's my take on it.

I do work in "foundations". In particular, my interests are in computer proof assistants and type theory. The great thing about doing proofs on a computer is that you can tell with certainty stronger than any mere mortal that your reasoning is correct (assuming the consistency of your logic, the correctness of the program, the integrity of the hardware, blah, blah, blah). If your code typechecks, it's virtually guaranteed to be correct.

Doing this, you notice something right away: even the simplest proofs often require a lot of machinery. In written mathematics, we almost always handwave the tedious details. "It's trivial", as they say. But this is analogous to a software developer who only bothered to write the interesting parts of his code, but the "boring obvious stuff", he skipped entirely. (Such a developer wouldn't hold a job for too long).

But at the very least, you can quarantine the "trivial" bits. The unwritten bits of your proofs end up working almost identically to axioms. (You have asserted they are true without proof). But you end up with an awful lot of them floating around your proof.

There are some interesting lessons you can learn from a computer proof assistant, though, in terms of logical intuition. Perhaps most importantly, you end up seeing the logical structure of proofs. (At least, this is true in term-based systems, such as Agda or Epigram. It's not as true when you work with tactic-based systems, like Coq). To make a simple example, I worked through the first Sylow theorem from group theory this morning:

Sylow I: Let p be a prime, k a natural number, and G a finite group.
If p^k divides |G|, then there is a subgroup H of G that |G| = p^k.

If k = 0, then H is the trivial group.
Else if k = 1, then H is G itself.
Else if k > 1,
We can show |G| = mp for some integer m > 1.
Does G have a subgroup H where [G : H] is coprime to p?
Yes ->
Then by lagrange, |G| = p^k = [G : H] * |H|
We know that p^k doesn't divide [G : H], so by the division lemma,
p^k must divide |H|.
H is what we need, so that becomes our answer.
No ->
In this case, we know that there are no subgroups H where [G : H] is coprime to p.
Equivalently, for *all* subgroups H, p divides [G : H]
By the class formula, |G| = p^k = |Z(G)| + Σ[G : H_i]
So modularly, 0 ≡ |Z(G)| mod p
(Note the [G : H_i] drop out, because each [G : H_i] is divisible by p).
By the abelian version of Cauchy's Theorem,
there exists a z in Z(G) where |z| = p.
<z> is a subgroup of the center, so <z> is normal.
We appeal to the induction hypothesis, letting G = G/<z> and k = k-1.
This gives us a subgroup H of G/<z> where |H| = p^{k-1}.
We appeal to a minor lemma (I don't know if this has a name) which lets us show:
H = P / <z> for some subgroup P of G.
And we note that
|P| = [P/<z>] * |<z>|
|P| = |H| * |z|
|P| = p^k * p
|P| = p^{k+1}
So P is our subgroup of G, and it has the required order.


You can see the structure very cleanly here. I can imagine being able to port this to Agda in a fairly straightforward way (although it might take a while getting all the definitions laid out first). The proof is by induction on G and k. The induction is a strong one, as I recurse with arbitrary smaller groups each time. My induction has three cases. There's a rather major case split on line 5 when I ask about the existence of a subgroup H where [G : H] being coprime to p. (Constructively, I have to worry about whether or not this procedure is effective. In classical mathematics, I can appeal to the Law of Excluded Middle). I see there is some basic divisibility theorems required. And I need to know about generated groups, quotients, the class formula, and a few other things.

I can also spot the flaws. I can see a few places where I'm not 100% convinced my logic is air tight. Especially, my use of recursion is really awkward, and the "minor lemma" I refer to, I don't really know if it's true. (My book just asserted it, and I accepted it for now).

But in the book I have, the argument is written in prose (as most mathematics is). It's not clear where the induction occurs (I think the author made a minor error, mentioning the inductive hypothesis twice). And instead of indenting on the case splits, you simply see the author saying "it's trivial, so let's assume blah".

But I guess my point is, if held at gunpoint for about a week, I am highly confident I could write this out on a computer in a suitable dependently-typed programming language. I don't know how other mathematicians do it, but this is perhaps my favorite tool in my toolbox.

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Sometimes I take solace in:

"Young man, in mathematics you don't understand things. You just get used to them."

- John von Neumann

It seems to me that some of the art is "if-this-then-that" kind of stuff, but there's a whole bunch more that basically comes from the intuition you get from basically just solving problems.

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