Kids These Days Don’t Know How Good They’ve Got It by Qiaochu Y. '12
and they need to get off my lawn
Once upon a time, when you wanted to know something, you had to find out about it by going to a place, probably not the place you currently were, called a "library." Just imagine. You had to actually travel somewhere in the physical world to obtain information. Information!
Fortunately, you live in more civilized times. Nowadays, when you want to know something and have access to the Internet, you can Google it, which will probably lead you to a Wikipedia article. Depending on the subject, you might subsequently find an online article, blog, or forum dedicated to the subject that will suggest further directions for investigation or provide an opportunity for you to shoot off questions to helpful strangers. And you can do all of this without ever leaving your room.
What are you going to do with all that power?
I can't answer that question for you. What I can do is give you an idea of what's possible using the resources that you, as a person reading this on the Internet, have access to. I can tell you what I did: I learned a lot of math! And I didn't even have to be at MIT to do it.
Art of Problem Solving
My first encounter with Internet mathematics occurred when my high school calculus teacher directed me to ArtOfProblemSolving.com (hereafter abbreviated AoPS). AoPS has some great resources, including collections of problems from a lot of competitions, but by far the most valuable resource for me was the AoPS forum. Some of the best and brightest students around the world gathered here to post and discuss problems from competitions of varying difficulty. It was very easy for me to find problems just a bit harder than what I was comfortable with, and I got a lot of practice in problem-solving and proof-writing.
But I got more than practice: I got feedback! If I did something wrong, someone would quickly point out my mistake. If I wrote something unclear, someone would ask me to clarify it. If there was a nicer way to get a certain conclusion, someone would post the appropriate proof. Often a problem would be too hard for me, but someone else would post a short solution using a clever method that I'd never seen before, and I picked up some very clever methods this way.
In total, I wrote about 12,000 posts on the AoPS forum. When people ask me how I learned mathematics, I want to say I used some clever trick, but the truth is that I just invested a lot of time into it. It certainly helped that posting on AoPS was a fun way to learn math. I never felt at any point that I was doing work; in fact, I used to log onto AoPS to procrastinate on my actual work.
About a year after I joined, I started the first incarnation of my blog, Annoying Precision, on AoPS. My initial goal was to share some interesting ideas I'd learned about at PROMYS, and more generally to share interesting techniques for solving problems that I often found I needed to refer to on the AoPS forum. Annoying Precision gradually grew into another valuable tool for my mathematical education: I kept writing posts about a subject and learning new things about it while writing, and I also found the process of writing valuable as a way to sharpen my thoughts. Annoying Precision was also a good way to let other people know what I was up to: Richard Rusczyk, the founder of AoPS, was impressed enough with what I'd been writing to offer me a job writing handouts for one of AoPS's online classes.
During my freshman year at MIT, I discovered a second amazing online math resource: other math blogs! I no longer remember how this happened, but at some point I heard about Fields Medalist Terence Tao's blog, What's new. Reading through his archives was a revelation. A blog post isn't anything like a paper or a textbook; due to the more informal setting, Tao could explain his intuitions and big-picture ideas instead of just writing down proofs. There's plenty of interesting stuff going on in a mathematician's head between the time that he starts thinking about a problem and the time that he writes a paper explaining his solution, but a lot of it never finds its way into print. Tao's blog posts contain all sorts of insight into the mathematical thought process that it would be difficult to find in any other medium, and all of it was freely available online.
I was entranced, so I clicked through Tao's blogroll looking for more blogs like his. I found plenty of fascinating stuff, but let me single out Tim Gowers' (also a Fields Medalist) and John Baez's blogs in particular. Like Tao, Gowers is also a great writer, and I also gained a great deal of insight into the mathematical thought process from reading his posts. John Baez is
actually a physicist, but his a mathematical physicist whose blog contains all sorts of interesting mathematics explained in a big-picture way, without a lot of details, and it greatly expanded my mental conception of what mathematics could be about.
Reading all of these amazing blogs inspired me to take my own blog more seriously, so I moved Annoying Precision to WordPress, where it's been ever since.
MathOverflow and math.StackExchange
During my sophomore year, something very exciting happened in the world of mathematics. A group of UC Berkeley graduate students and postdocs started a website called MathOverflow (MO) as part of StackExchange 1.0 (SE 1.0). A basic problem in mathematical research is that a question will come up which is probably not difficult, but which lies outside of your area of expertise. If you knew someone in the appropriate area in your department, you could ask them, but if you didn't, or if that didn't work, it might take months to figure out the answer on your own. MO is at least in part for asking these kinds of questions: post your question on MO and an expert will probably find it and post an answer that makes everything clear. The general idea is to harness cognitive surplus to accelerate the rate of mathematical research.
MathOverflow was an incredible boon for me. I had a lot of questions, and until the advent of MO I didn't have a good place to ask them. With MO around, I could get answers to my questions from world-class experts. Just to give one example that's particularly stuck with me, I was struggling to understand something I read in a textbook, so I posted about it on MO, and the author of the textbook answered me with an explanation!
math.StackExchange (math.SE), on the other hand, began as part of StackExchange 2.0 as a general-purpose Q&A site for mathematical questions at all levels. Around this time, I had stopped posting on AoPS, and math.SE became my new place for answering rather than asking questions. At least for my purposes, math.SE is a major improvement over AoPS for several reasons. First, it draws from a much wider audience (such as people from other SE sites) of people curious about math but lacking a good resource for addressing their questions, and these people sometimes ask much more interesting questions than anyone would ever ask on AoPS. Second, for reasons that I don't completely understand, SE sites are very visible on Google, so answering a question well on math.SE makes the answer available to a potentially large audience of future questioners curious about the same thing on Google. Finally, math.SE was boosted from the beginning by an influx of users from MO, mostly professional mathematicians, and it is very interesting to see their answers to even relatively elementary questions.
That's probably worth rephrasing: at math.SE, professional mathematicians might answer your (interesting) mathematical questions regardless of level. How amazing is that?
My activities on math.SE eventually got me noticed by StackExchange, and they offered me a summer internship which it would be going too far off-topic to describe now.
It's hard for me to overstate the impact that having access to such amazing mathematical resources has had on my life, and it's also hard for me to resist pointing out that MIT had nothing to do with it. It's great that MIT has a lot of amazing educational resources, but most of those resources (with the notable exception of OCW) are only available to people in the MIT community, and at the end of the day that's not a very large community. But even people who don't have access to elite institutions and only have access to the Internet still have access to amazing educational resources, even if they don't always know it.
One more time: what are you going to do with all that power?