And now that it's obvious to you, it's suddenly really hard to explain to someone else.<p>My friend was TAing a freshman calculus class in college while working on his PhD. He clearly remembers that his (and everyone's) biggest problem with calculus was limits. Hands-down, limits were the most frustrating and un-intuitive part of the class. So he sat down to try to figure out a good, intuitive explanation of limits that would help his students avoid the frustration that he had.<p>He immediately ran into the problem that they weren't hard to understand. They were clear and intuitive. He could not for the life of him remember why he thought they were difficult when he was first learning them. He asked me for help, and I had the same response. I remember clearly banging my head against them and screaming in rage and frustration at those goddamn limits. But I look at them now, and I can't figure out how I could fail to understand something so obvious.<p>And thus, the cycle of pain continues, with limits remaining the eternal bane of freshman calculus students.
John von Neumann once told a confused young scientist, "Young man, in mathematics you don't understand things. You just get used to them."<p>I don't think that's quite true, but developing familiarity with a difficult concept is tantamount to coming to understand it, in my experience.<p>For me, the process seems to go like this:<p>1) assemble the disparate pieces of a concept, and try to wrap my head around the novel ones;<p>2) work out how Piece A relates to B, and B to C, and ... ;<p>3) forget what the hell Piece X is;<p>4) repeat steps 1–4 several times;<p>5) convince myself that all the pieces work together as they should.<p>And only later, after I've worked with this collection of pieces several times, do I stop seeing the pieces and start seeing something new, the emergent phenomenon of the new concept. And it's not until I've forgotten the details of how Piece X fits into Piece Y, and maybe why we need Piece Z in the first place, that I feel like I really grok this new concept.
Brilliant! Here's my similar, now animated take on it:<p>Obvious to you. Amazing to others.<p><a href="http://www.youtube.com/watch?v=-GCm-u_vlaQ" rel="nofollow">http://www.youtube.com/watch?v=-GCm-u_vlaQ</a>
I think this has to do with the structure of the brain. When you are learning something new, the brain is creating new routes and making new connections. It can even creates new neurons (something known as neuro-plasticity).<p>My guess is that when you finally understood the subject, your brain has completed the route for that particular problem. It's now clear how to solve the problem for the brain, since a specific route exists. It's obvious and very easy for the brain to do that with the route shortcut.<p>For you, the problem becomes obvious and simple. It's no longer complicated and requires less brain-processing-power to solve. You think you were stupid you didn't figure out that from the beginning, but you are not!
It would be lovely if this were true, but I think it's actually a fairly poisonous mindset. What the author's post seems actually to be saying, with which I heartily agree, is “obviousness is only in hindsight”; deep things become trivial only because of the time you've put into understanding them.<p>On the other hand, I think the things that become obvious once you've understood them <i>tend</i> to be (though are not always) the things that have already been fairly well digested by others, and so are presented to you in a smooth, flowing way to which you just have to accustom yourself.<p>My experience with (mathematical) research is that understanding has a roughly equal possibility of meaning that you find it trivial, or that you finally understand all the (apparently) irreducibly complex difficulties. Indeed, my feeling if anyone tells me that, say, Deligne–Lusztig representations are obvious is that he or she hasn't probably fully understood them (disclaimer: neither have I, not even close, which may mean that I'm just illustrating the author's point).<p>I don't mean at all by this that you shouldn't go on searching for the 'obvious' simplification—that way lies great insight. (As someone said much more elegantly, if things aren't already obvious in mathematics, we tend to make them obvious by changing the definitions.) What I mean is that you shouldn't drag yourself down by saying “I thought I understood that, but it's hard, not obvious!”
Corollary: if you are angry at something, it's because you don't understand it. Nobody gets angry at obvious things such as water being wet.<p>Another corollary: understanding brings peace.
This is precisely why I get nervous about offering to speak at professional groups / conferences. When I see a CFP or similar, everything cool that I do seems obvious and I don't think it'll be useful.<p>But then I get into conversations with my peers and realize that some of the stuff I do apparently <i>isn't</i> as obvious to everyone as I'd thought.
I think a better measure of understanding is how well you are able to explain it to a layperson. As Einstein once said, "If you can't explain it to a six year old, you don't understand it yourself."<p>As a researcher, who has also TAed classes, it takes a lot of effort not only to teach a particular topic, but also simultaneously present the crux of the idea as well as the right way to think about it and its place in the context of other information. And learning to think about everything with these in mind will make you understand things much much better. In particular, one of the most challenging things is not to explain breakthrough ideas to cutting-edge researchers but rather explaining breakthrough ideas to complete outsiders and laypeople.
This is partially why it can be frustrating to work with someone who isn't as good as you are. Things that come easily to you doesn't for others. You sometimes have to remind yourself that things that you consider obvious is only obvious for you.
Richard Feynman said you don't really understand something unless you can explain it to freshmen in a single lecture. Same concept, really.<p>The interesting consideration, to me, is how much we can get done with hardly any understanding at all. That's human nature. We do things, then we wonder how, and then we eventually come to understand what the hell we just did. It seems bass-ackwards, but that's the way it goes most of the time.
Thought: my linear algebra teach this past semester told me that some of the greatest mistakes in mathematics came from people assuming something was obvious without a proof. After building a huge intricate structure around the assumption, they would watch everything fall apart as it turned out the obvious fact wasn't true. Now, whenever I hear somebody say "it's obvious" I always get skeptical now.
There is a great line from Jonathan Ive in the documentary Objectified where he says: "A lot of what we seem to be doing is getting design out of the way. And I think when forms develop with that sort of reason, and they’re not just arbitrary shapes, it feels almost inevitable. It feels almost undesigned. It feels almost like, 'well, of course it’s that way. Why would it be any other way?'"
As Galileo said, "All truths are easy to understand once they are discovered; the point is to discover them." Insights are often so simple, so obviously true once you see them. But we are often blinded by our fixations, our hubris, our limited frames of reference -- our narrow perspectives -- which prevent us from seeing.
The intersection of "things I thought were obvious at one point in time" and "things I currently think are wrong" is a rather large set. It's hard for me to claim I "understood" those thing I now think I was wrong about.
Not sure about this. I understand the proofs that there is no largest prime number, and that the integers can't be put into a one-to-one correspondence with the reals, but I wouldn't say either is "obvious".
I guess the point can be summarized like<p>- When someone says that something is obvious, we cannot claim that he/she actually understands it well.<p>- But, if someone understands something well, it will seem obvious to him/her.