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You don't understand something until you think it's obvious.
- hrabago 15y agoThis 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.
- adamdecaf 15y agoThis reminds me of a story from Feynman where he was with a math friend and they were trying to multi-task. Yet, they ended up going about the two things (counting for a minute while reading or speaking) in completely different ways. (Different enough that one could only count and talk while the other could only count and read.) http://www.youtube.com/watch?v=Cj4y0EUlU-Y&feature=player_detailpage#t=140s http://www.youtube.com/watch?v=Cj4y0EUlU-Y&feature=playe...
- boredguy8 15y agoThe 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.
- polynomial 15y agoI was going to point out that, ironically, many people who think something is "just obvious" completely fail to understand it at all. So yes it cuts both ways.
- lmkg 15y agoAnd now that it's obvious to you, it's suddenly really hard to explain to someone else. 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. 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. And thus, the cycle of pain continues, with limits remaining the eternal bane of freshman calculus students.
- haberman 15y agoThat's what makes guys like Sal Khan special. And why his videos are so valuable. Sal's "Introduction to Limits": http://www.khanacademy.org/video/introduction-to-limits http://www.khanacademy.org/video/introduction-to-limits
- espeed 15y ago"If you can't explain it simply, you don't understand it well enough." -- Albert Einstein
- waqf 15y agoAnd the tragedy is that nobody else understands the simple explanation once you understand enough to give it.
- espeed 15y agoYes, true insight requires a level of understanding usually obtained through a focused effort to understand all the components that make up the idea, several levels deep. Explaining the end result -- the epiphany -- to someone else is often difficult because most will have gaps in their understanding that prevents them from seeing it they way you do and therefore they will have limited ability to verify and understand the significance of what you what you are saying. See http://jamesthornton.com/blog/how-to-get-to-genius http://jamesthornton.com/blog/how-to-get-to-genius
- Swannie 15y agoWell that sounds like the problem of "I learnt it wrong the first time". Looking back at issues I've had, learning it right the first time has been very important. If I've skimmed over some concept, and learned enough to apply it simply, when it comes to applying it more generally, I'm struggling. Going back and relearning it correctly takes time, and generally has to be done more than once. Teachers should be careful to explain things simply, but ensure that the understanding really is correct. This can't really be assessed by exercise sheets, but discussion. (Well it can by exercise sheets, by asking students to apply the concept to a different problem, but personal experience is only a few people make the jump without help. And when they get the help, that short circuits the learning that should have happened to allow them to do it themselves.)
- csomar 15y agoI 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). 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. 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!
- pash 15y agoJohn von Neumann once told a confused young scientist, "Young man, in mathematics you don't understand things. You just get used to them." 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. For me, the process seems to go like this: 1) assemble the disparate pieces of a concept, and try to wrap my head around the novel ones; 2) work out how Piece A relates to B, and B to C, and ... ; 3) forget what the hell Piece X is; 4) repeat steps 1–4 several times; 5) convince myself that all the pieces work together as they should. 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.
- mechnik 15y agoI recall a section in Halliday and Resnick freshman Physics text where the authors advise to 'do exercises until you convince yourself that the solution method is correct'.
- Lost_BiomedE 15y agoI think this way as well. I was told that I am a slow processor of information, by a credible source. Also, that I would have a harder time learning on the fly but would have a deeper understanding due to making more cognitive links to each idea, and this would allow for tackling harder problems and long projects. It has turned out to be spot on, and if I would have to guess, I would say about a quarter of engineers that I know seem to think this way. In the general population it seems much lower.
- bdhe 15y agoI 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." 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.
- qq66 15y agoEinstein may have said that but it's just not true. How do you explain the Higgs boson to a six-year-old? If you succeed, it's because you made an analogy so simple that it has no meaning.
- chrismsnz 15y agoMaybe that's because we don't understand it fully?
- qq66 15y agoThere are plenty of things we understand fully (e.g. translation lookaside buffers) that are impossible to explain to a six-year-old.
- aristus 15y agoOff the cuff, I would come up with something involving parking valets and the pegboard for keys. It would illustrate the important bits of the mechanism and its purpose. I don't know if you would dismiss that as meaningless analogy, but explaining TLB (or Traveling Salesman, or functional decomposition, etc) to kids is not impossible.
- espeed 15y agoAs 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.
- JadeNB 15y agoIt 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. On the other hand, I think the things that become obvious once you've understood them tend 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. 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). 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!”
- ZackOfAllTrades 15y agoThought: 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.
- motxilo 15y agoYou are just describing what motivated Bertrand Russell to write Principia Mathematica.
- orangecat 15y agoNot 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".
- daniel-cussen 15y agoThe second proof you mention is the most controversial proof in the history of mathematics. Some mathematicians refuse to believe it.
- impendia 15y agoI am a mathematics postdoc at Stanford, constantly going to conferences and meeting other mathematicians, and I have not met anyone who refuses to believe this. Indeed, I have met no one who refuses to believe any proof that is widely accepted by the mathematical community. You could (presumably) construct axiom systems under which this is not true, and I believe that some logicians do this sort of thing, but this is rather far removed from mainstream research mathematics.
- xyzzyz 15y agoYou could apply the Lowenheim-Skolem theorem to the ZFC theory to obtain a countable model of set theory, but the object representing the set of real numbers in this model would still be "uncountable" inside of the model, altough clearly countable for us outside of it.
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- jeffcoat 15y agoDiagonalization is a poor example -- though I don't think Cantor's logic was widely accepted until the early 20th century -- but I think there are examples of proofs where mathematicians will accept the logic, but argue with it none the less: The Banach-Tarski paradox comes to mind first. I don't think anyone argues that the proof is wrong, but you'll easily find people to argue that that very fact means that the full axiom of choice should be held in deep suspicion. The 4-color map coloring theorem is more interesting in this thread, though; I think it's the best known example of a proof that offers little to no insight into why the theorem is true. I don't think any solution that requires breaking a problem into 1,936 special cases, and then mechanically checking each one of those cases, will ever lead to an understanding that makes the theorem "obvious".
- kylemaxwell 15y agoThis 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. But then I get into conversations with my peers and realize that some of the stuff I do apparently isn't as obvious to everyone as I'd thought.
- sivers 15y agoBrilliant! Here's my similar, now animated take on it: Obvious to you. Amazing to others. http://www.youtube.com/watch?v=-GCm-u_vlaQ http://www.youtube.com/watch?v=-GCm-u_vlaQ
- szany 15y agoI don't necessarily disagree, but it's worth pointing out that this might not always be the case: "Every mathematician worthy of the name has experienced the state of lucid exaltation in which one thought succeeds another as if miraculously. This feeling may last for hours at a time, even for days. Once you have experienced it, you are eager to repeat it but unable to do it at will, unless perhaps by dogged work." -André Weil
- bluekeybox 15y agoCorollary: 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. Another corollary: understanding brings peace.
- MatthewPhillips 15y agoYes, this is something I try to remind myself whenever I find myself getting angry. I find it helps tremendously; I quickly realize that it's something beyond my control or something that I don't completely understand and I find that calming. I think I've gotten better about just skipping over the part where you worry about things that are outside of your control (not that it doesn't still happen).
- petegrif 15y agoThis is why so many people think patents are unfair - because they are so obvious. :)
- petegrif 15y agoThis is why so many people think patents are unfair - because they are so obvious. :)
- drcube 15y agoRichard Feynman said you don't really understand something unless you can explain it to freshmen in a single lecture. Same concept, really. 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.
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- yhlasx 15y agoI guess the point can be summarized like - When someone says that something is obvious, we cannot claim that he/she actually understands it well. - But, if someone understands something well, it will seem obvious to him/her.
- codyguy 15y agoYet, we might not understand the things that seem obvious. Gravity, light etc can be "obvious" to most people, but not really understood.
- redtwo 15y agoYou don't understand something until you can explain it to your grand mother.
- Resonemang 15y agoIn that case I don't even understand English... ;)
- protomyth 15y agoThere 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?'"