๐๏ธ alreadyfinishedit ยท 279 points ยท Posted at 12:38:35 on December 27, 2016 ยท (Permalink)
I am just wondering if anyone remembers the story of the professor who said the proof was trivial but when asked further he couldn't solve it. He searched for it in papers and found it but again the proof was left as an exercise for the reader and it was he who had written the paper? I can't remember the story.
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chebushka ยท 178 points ยท Posted at 13:25:41 on December 27, 2016 ยท (Permalink)
Supposedly it was Kakutani. See http://curiosamathematica.tumblr.com/post/122398968526/obvious
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One day Shizuo Kakutani was teaching a class at Yale. He wrote down a lemma on the blackboard and announced that the proof was obvious. One student timidly raised his hand and said that it wasnโt obvious to him. Could Kakutani explain?
After several momentsโ thought, Kakutani realized that he could not himself prove the lemma. He apologized, and said that he would report back at their next class meeting.
After class, Kakutani, went straight to his office. He labored for quite a time and found that he could not prove the pesky lemma. He skipped lunch and went to the library to track down the lemma. After much work, he finally found the original paper. The lemma was stated clearly and succinctly. For the proof, the author had written โExercise for the readerโ.
The author of this 1941 paper was Kakutani.
From Mathematical Apocrypha by Steven Krantz.
[deleted] ยท 134 points ยท Posted at 13:37:59 on December 27, 2016 ยท (Permalink)
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[deleted] ยท 86 points ยท Posted at 18:36:05 on December 27, 2016 ยท (Permalink)
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turnipheadscarecrow ยท 14 points ยท Posted at 00:32:12 on December 28, 2016 ยท (Permalink)
That's essentially how I write my mathematics. I usually only emphasise whatever point took me a long time to figure out the first time, and I can usually recreate my steps just by the signposts I've left behind. I do the same with my marginalia when reading other people's work: just a quick jot as a hint here and there explaining a particular point that took me a long time to figure out.
dr_entropy ยท 12 points ยท Posted at 03:06:31 on December 28, 2016 ยท (Permalink)*
When I was young I thought that the best code was concise, compressing the most information into the fewest characters.
Now I realize that "language" is truly an apt description for code. The primary purpose of code is to be as expressive and intelligible as possible. Comments can help, but it's the structure of the code: the order concepts are introduced, how they're named, and which parts are abbreviated (functions/classes) that are important.
DoWhile ยท 17 points ยท Posted at 02:32:45 on December 28, 2016 ยท (Permalink)
//this is trivial
notmike_ ยท 4 points ยท Posted at 20:33:31 on December 30, 2016 ยท (Permalink)
//trivial also
HyperSpline ยท 2 points ยท Posted at 06:49:33 on December 28, 2016 ยท (Permalink)
Either that or the lemma is false (or the story is fictional).
[deleted] ยท 32 points ยท Posted at 15:02:13 on December 27, 2016 ยท (Permalink)
Sorry if this is a stupid question but was the lemma actually true and who proved it?
chebushka ยท 91 points ยท Posted at 16:07:57 on December 27, 2016 ยท (Permalink)*
In the story by Krantz at the link it is not called a lemma (it could still be one, of course). His story ends by saying it was in a paper in 1941. Kakutani's most famous 1941 paper is the one on his fixed point theorem. Just three pages long, no proof is omitted with a comment that it is trivial or left to the reader. He does say a couple of times that something is obvious, but not in order to omit the proof of an officially stated result.
On MathSciNet there are 4 other papers by Kakutani from 1941, all in the Annals of Mathematics vol. 42. By page and issue number they are on pages 188--228 (No. 1), 523--537 (No. 2), 994--1024 (No. 4), 1025--1028 (No. 4). I skimmed over these papers, and while Kakutani says "it is easy to see" and "well known" a few times, I don't find this being done in order to omit the proof of a result he needs. Maybe I missed something. You're welcome to look yourself.
helnott ยท 9 points ยท Posted at 13:28:29 on December 27, 2016 ยท (Permalink)
The irony.
๐๏ธ alreadyfinishedit ยท 7 points ยท Posted at 13:33:54 on December 27, 2016 ยท (Permalink)
Yes thank you this is the one I was looking for!
mnemoniker ยท 153 points ยท Posted at 16:26:44 on December 27, 2016 ยท (Permalink)
This reminds me of a joke I once heard: "A professor is at the whiteboard in the middle of a lecture. He goes 'So, it's trivially true that--' Then he thinks for a moment. Suddenly he rushes out of the room. Finally, 30 minutes later he comes back. 'So, it's trivially true that...'".
TheCannonMan ยท 89 points ยท Posted at 18:24:25 on December 27, 2016 ยท (Permalink)
Ive heard a similar quip, where th prof is proving something and a student asks about some step "But isn't that obvious?" So the professor starts thinking about it and she paces up and down the hall for 45 min, walks back in and says "yes it's obvious.."
auxiliary-character ยท 19 points ยท Posted at 21:38:19 on December 27, 2016 ยท (Permalink)
Is there a rigorous definition of obvious?
Dmasik ยท 26 points ยท Posted at 23:47:55 on December 27, 2016 ยท (Permalink)
this
alx3m ยท 2 points ยท Posted at 13:31:02 on December 28, 2016 ยท (Permalink)
I love that page.
auxiliary-character ยท 1 points ยท Posted at 00:00:08 on December 28, 2016 ยท (Permalink)
Accurate.
[deleted] ยท 14 points ยท Posted at 02:18:06 on December 28, 2016 ยท (Permalink)
Yes. It means someone who has a proper understanding of the subject can work out the proof without needing to be told any additional insights or tricks. The issue is that it takes a level of mathematical maturity to even understand what that means, and the vast majority of e.g. this sub don't have it.
AceCream ยท 19 points ยท Posted at 22:39:27 on December 27, 2016 ยท (Permalink)
Von Neumann and Norbert Weiner were both the subject of many dotty professor stories. Von Neumann supposedly had the habit of simply writing answers to homework assignments on the board (the method of solution being, of course, obvious) when he was asked how to solve problems. One time one of his students tried to get more helpful information by asking if there was another way to solve the problem. Von Neumann looked blank for a moment, thought, and then answered, "Yes".
http://www.infiltec.com/j-logic.htm
mnemoniker ยท 2 points ยท Posted at 18:31:40 on December 27, 2016 ยท (Permalink)
Heh, I like that version too!
akjoltoy ยท 1 points ยท Posted at 23:17:50 on December 27, 2016 ยท (Permalink)
I really don't think this is an ironic thing either. There are many obvious things in math that can be initially opaque to someone because their first train of thought, by luck, was way off base.
CreatrixAnima ยท 165 points ยท Posted at 16:20:15 on December 27, 2016 ยท (Permalink)*
I have a prof who always says stuff is obvious. It usually isn't to me, so I have no problem asking for further explanation. My favorite was the time he stood back, looked at the board and said "Well, I don't remember why it's obvious, but it is."
I don't think that word means what he thinks it means.
(Actually, he's one of the best profs I've had. LOVE his classes!)
[deleted] ยท 85 points ยท Posted at 17:11:24 on December 27, 2016 ยท (Permalink)
I guess professors just get in the habit of saying "the proof is trivial/obvious" and then overtime they forget what the actual proof is.
It's very easy in mathematics to simply forget the details of something. You know it's true, you know you once knew it, you might even be able to explain it, but you just forget how to derive or construct it and it's obviousness is not immediately obvious....obviously.
[deleted] ยท 45 points ยท Posted at 17:57:51 on December 27, 2016 ยท (Permalink)
Yeah but I feel if the proof is something that they have to remember it's not obvious.
philly_fan_in_chi ยท 28 points ยท Posted at 18:33:29 on December 27, 2016 ยท (Permalink)
Something can be obvious when viewed through the correct perspective but until you regain that perspective, it remains elusive. The correct thing to do there is say "it follows from", but when you're in the weeds, that concept might be taken as a primitive.
china999 ยท -10 points ยท Posted at 18:41:39 on December 27, 2016 ยท (Permalink)
I think the correct thing to do is give English the and respect people want of maths. If you're put into a thought coma when asked to explain something you just said was obvious you need to consider your use of language
VioletCrow ยท 30 points ยท Posted at 21:14:25 on December 27, 2016 ยท (Permalink)
Be the change you want to see in the world.
akjoltoy ยท -1 points ยท Posted at 23:20:11 on December 27, 2016 ยท (Permalink)
irony. and you're wrong too.
[deleted] ยท 9 points ยท Posted at 19:22:17 on December 27, 2016 ยท (Permalink)
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mangodrunk ยท 5 points ยท Posted at 19:48:01 on December 27, 2016 ยท (Permalink)
Well, it sounds like those puzzles that might be "obvious" after you are told the answer, but to get there requires a clever idea that isn't obvious. So, I think we're better off not using words like "obvious" and "trivial".
[deleted] ยท 3 points ยท Posted at 22:43:24 on December 27, 2016 ยท (Permalink)
Thankfully "we" mathematicians don't give a flying f-ck what r/math thinks. This thread is a prime example of how people who know just a little can mistakenly think they know more than the experts.
mangodrunk ยท 4 points ยท Posted at 00:25:37 on December 28, 2016 ยท (Permalink)
This thread is about people learning math, so I think the non-expert opinion is actually valid here. Also, given how poor math education is, and how low math literacy is, I think we can stand to change a few things. Also, it would be a good idea for you mathematicians to be explicit for yourselves, since something that is obvious now probably won't be later on, to yourself.
You can say "fuck" by the way, no need to say "f-ck". Unless you meant another word.
[deleted] ยท -6 points ยท Posted at 00:28:37 on December 28, 2016 ยท (Permalink)*
The post specifically mentions papers, so it's not about people learning math, it is about research level mathematics or at the very least graduate level courses.
Please don't use the word we in a way that implies you are a mathematician or teacher of it, as you clearly are not.
Edit: ok r/math, same deal as before: get my completely correct comment contradicting someone who doesn'tโ know what they're talking about to minus 10 and I'll leave.
mangodrunk ยท 3 points ยท Posted at 01:08:35 on December 28, 2016 ยท (Permalink)
You're really rude. Look, you weren't able to understand in the few words I used that I didn't say that I was a mathematician, which was pretty OBVIOUS, and I said it would be good for mathematicians themselves. You don't seem like you want to have a civilized discussion, so good bye.
[deleted] ยท -7 points ยท Posted at 01:22:37 on December 28, 2016 ยท (Permalink)*
Edit: so what we were you referring to there?
Edit 2: just fyi, rude would be me pointing out that your complaints with the use of the word obvious are the same as those complaining that closed does not mean not open, aka complaints of people who are not capable of abstract logical thought.
Xujhan ยท 4 points ยท Posted at 02:31:09 on December 28, 2016 ยท (Permalink)
As someone who is a teacher of mathematics: please stop making us look like assholes.
[deleted] ยท -1 points ยท Posted at 02:34:43 on December 28, 2016 ยท (Permalink)
Lol, ok. When someone know has no idea what they're talking about says something acting like they do, I will point it out. But if you are actually a math teacher then you should realize the importance of that. And btw, this thread is not about people learning math.
unborn0 ยท 0 points ยท Posted at 19:16:34 on December 28, 2016 ยท (Permalink)
It's also not about being an asshole
[deleted] ยท 0 points ยท Posted at 19:29:02 on December 28, 2016 ยท (Permalink)
I disagree. When someone spouts off nonsense trying to sound like an expert, the only response that is appropriate is a harsh and dismissive one.
unborn0 ยท -1 points ยท Posted at 22:41:35 on December 28, 2016 ยท (Permalink)
This is the internet. No one cares about your harsh and dismissive tone, especially when the original commenter clearly said he wasn't an expert.
[deleted] ยท 0 points ยท Posted at 23:40:48 on December 28, 2016 ยท (Permalink)
You appear to care. And, no, they did not do such a thing until after I called them out.
unborn0 ยท 1 points ยท Posted at 02:16:28 on December 29, 2016 ยท (Permalink)
Let me rephrase. No one is going to change their mind just because you were harsh and dismissive on the Internet any more than they would have if you hadn't acted like an asshole.
Furthermore, you called him out on Jack squat. You said he was a case of someone thinking he knew more than the experts when all he did was state an opinion and nothing that even remotely made him sound like an expert. Who cares if he said it after, you still acted like a child then, too.
[deleted] ยท 0 points ยท Posted at 02:21:40 on December 29, 2016 ยท (Permalink)
You are also missing the point. This is not a question of opinion. There is an accepted use of the words trivial and obvious in mathematical writing. And yes, using the word "we" as they did was an intentional attempt to make an uninformed "opinion" sound authoritative, despite what you may think.
I don't think I've acted like a child at any point, but that is a matter of opinion so I'm not going to argue that point with you. For the record, we mathematicians still don't give a flying f-ck what r/math thinks and will continue to use the words trivial and obvious in our writing as before.
Edit: to make this as clear as possible, I will quote the comment I initially responded to:
What do you think the word "we" there was trying to refer to? This thread is a discussion of mathematical writing by research mathematicians. Any suggestion that "we" ought to do something implies that the speaker is one of said "we". I seriously feel like the average intelligence of this sub drops in proportion to the number of votes a post has.
[deleted] ยท -1 points ยท Posted at 00:25:36 on December 28, 2016 ยท (Permalink)
Love it when r/math downvotes correct statements, always entertaining. Also, yes, well put.
[deleted] ยท -1 points ยท Posted at 04:52:51 on December 28, 2016 ยท (Permalink)
Who the fuck are you to speak for all mathematicians. Also the whole point of math is to prove theorems to other people. A proof is pretty useless outside a social context, it's just a fact about the structure of the universe. If you leave lemmas hanging, and not explain them when asked, you're doing a disservice to the future of mathematics.
[deleted] ยท 1 points ยท Posted at 17:07:48 on December 28, 2016 ยท (Permalink)*
The whole point of math is to prove theorems and write them so that people with the requisite knowledge can understand them, not to spoonfeed undergrads.
[deleted] ยท 2 points ยท Posted at 23:02:46 on December 28, 2016 ยท (Permalink)
"Who the"? Anyhow, math is an edifice that took centuries to build. Expecting people(even grad students) to get every detail of a proof without someone asking stupid questions about obvious stuff is just naรฏve. And it's been shown(by neuroscience) that with dedication, and proper guidance, even not so bright students can learn this stuff.
[deleted] ยท 1 points ยท Posted at 23:43:02 on December 28, 2016 ยท (Permalink)
Not what was said. Obvious means obvious to experts, if a student reads that something is obvious and doesn't understand it then yes they should ask about it. But papers are written for other experts and if something is obvious we say as much and move on.
I'm sorry that the kids feel so entitled, but I don't particularly care.
Zophike1 ยท 0 points ยท Posted at 02:15:38 on December 28, 2016 ยท (Permalink)
For something to be obvious you should get or "see" it without any thought what so ever.
[deleted] ยท 4 points ยท Posted at 03:04:40 on December 28, 2016 ยท (Permalink)
Context is important. Obvious to experts is what obvious means in research articles. Obvious to students who fully grasp the material is what is meant by obvious in textbooks.
If I say it's obviously daytime since I can see the sun is out, that doesn't mean it's obvious to a blind person. But that doesn't make me incorrect.
If you're reading something and it says X is obvious but you don't find it to be, most likely that means you don't have the prerequisite knowledge/familiarity with the subject that is assumed of the reader by the author.
Zophike1 ยท 1 points ยท Posted at 20:30:10 on December 28, 2016 ยท (Permalink)
That is true, is just that the word obvious has become so ambiguous these days in modern mathematics.
[deleted] ยท 1 points ยท Posted at 20:35:45 on December 28, 2016 ยท (Permalink)
I disagree. I think the only perceived ambiguity is on the part of those unfamiliar with how it's used. Ask any mathematician and you'll get an answer very similar to mine (you can actually see who in this thread is a mathematician and who is not from the comments).
Zophike1 ยท 1 points ยท Posted at 20:37:34 on December 28, 2016 ยท (Permalink)
Ahh I understand something being obvious is from the point of view of the individual.
[deleted] ยท 2 points ยท Posted at 20:45:26 on December 28, 2016 ยท (Permalink)
As with many other words, the mathematical usage does not match that closely with the common usage. Obvious and trivial may not have precise mathematical definitions, but there is very much an accepted meaning of the terms in mathematical writing that has little to do with their dictionary definitions.
The issue, as I said elsewhere, is that it takes a level of mathematical maturity to understand whether or not something is mathematically trivial. If you see something mathematical described as trivial or obvious but don't understand it, it means that you don't have the prerequisite knowledge the author expected. That's not a failing on your part nor on theirs, it's an indication that you have some work in front of you if you want to master the material.
Zophike1 ยท 1 points ยท Posted at 21:16:02 on December 28, 2016 ยท (Permalink)
Ahh now I understand.
buo ยท 47 points ยท Posted at 18:56:35 on December 27, 2016 ยท (Permalink)
In technical papers, "obvious", "trivial", "can easily be shown that", et al are codewords that indicate to the reader that what follows can be derived without special tricks or references to arcane theories. It doesn't necessarily mean that it's easy, or even that it's actually obvious in the normal meaning of the word.
I think the problem is then that professors use those terms in class, as if students understood that language, when they don't because nobody has explained it to them.
mangodrunk ยท 5 points ยท Posted at 19:49:40 on December 27, 2016 ยท (Permalink)
I don't think that makes it much better. Who knows if it's "obvious' or "trivial" if it's not actually done. Seems like it just adds more room for miscommunication on something already not "trivial".
Syrdon ยท 6 points ยท Posted at 21:01:05 on December 27, 2016 ยท (Permalink)
I've always read papers by doing the proofs as I read the paper. A step that is "obvious" must still be shown in my notebook. Even when I think that it is obvious in the usual English definition and not the math one.
On the other hand, that shit gets tedious if I'm explaining it to someone else. If they want to make sure they understand the proof by going through it themselves, more power to them. But I'm not going to spend time talking about it unless everyone there specifically requests that.
mangodrunk ยท -1 points ยท Posted at 00:21:38 on December 28, 2016 ยท (Permalink)
Fair point. When discussing it with a person, you will have to cater to your audience, but keep in mind even if no one speaks up, you might lose a few people from understanding what you're talking about.
Syrdon ยท 5 points ยท Posted at 00:34:46 on December 28, 2016 ยท (Permalink)
That's what asking afterwards is for. If you really care, you'll ask. If you don't, how much did you really care.
Then again, it's been suggested that I do t really have the temperament for things that boil down to customer service - and teaching kinda does.
[deleted] ยท 4 points ยท Posted at 01:29:53 on December 28, 2016 ยท (Permalink)
If you're teaching a proofs based class (other than an introductory one), using obvious and trivial with their usual mathematical meaning is fine, especially at grad level. The students need to get used to the cues about when they should be able to work it out themselves. And these sorts of things don't really come up in lower level courses (I mean, e.g. all of linear algebra is trivial from a research level pov).
dr_entropy ยท 1 points ยท Posted at 03:14:51 on December 28, 2016 ยท (Permalink)
I think trivial is a pretty good description, as in literally the realm of trivia. You could labor to reproduce the proof, but there isn't much there of relevance to the rest of the matter at hand.
Obvious should almost always get a footnote, with the relevant bits to trigger an a-ha moment. What is obvious from one domain of expertise is obtuse to another.
ghan-buri-ghan ยท 9 points ยท Posted at 23:58:44 on December 27, 2016 ยท (Permalink)
When you know the field well, your intuition is very good, especially for course-level stuff. So it entirely possible for a professor to know (with very high confidence) that something is easy to prove even without a proof in mind. And on an off day and in the moment not be able to find it.
LuxDeorum ยท 5 points ยท Posted at 00:33:12 on December 28, 2016 ยท (Permalink)
It's also strangely difficult for most people to actively problem solve in the middle of giving a lecture. I know more than once I've been caught off guard by drawing a blank on something I was certain I knew before beginning class.
[deleted] ยท 2 points ยท Posted at 02:20:48 on December 28, 2016 ยท (Permalink)
A good lecturer will essentially put themselves back in the mindset of someone learning the material for the first time. In doing so, you lose the ability to quickly "skip steps" as you would while doing research.
BennyPendentes ยท 5 points ยท Posted at 21:02:42 on December 27, 2016 ยท (Permalink)
My DiffEq prof was always saying things were 'trivial'. I had the deep impression that he was using that word in a way that didn't correspond to any dictionary definition, and that his own definition was fairly fluid.
By the end of the course, I'd come up with a working definition that made sense every time he said it:
A problem is 'trivial' if
It just means somebody somewhere solved it somewhen, so from the professor's POV there is no point in wasting time discussing it when you can just go look up that existing solution.
[deleted] ยท 5 points ยท Posted at 22:47:28 on December 27, 2016 ยท (Permalink)
If it's known that it can be solved exactly via computer then it is mathematically trivial.
akjoltoy ยท 2 points ยท Posted at 23:22:27 on December 27, 2016 ยท (Permalink)
yep
dr_entropy ยท 2 points ยท Posted at 03:17:57 on December 28, 2016 ยท (Permalink)
Trivial just means, "not interesting in a way that's relevant to what we're talking about'. Like, if you did the proof you may win some perverse math trivia game, but you won't do any better on the exam (and may not even learn much). It's more like, "don't sweat the proof, just accept this works."
Or maybe, "have fun with the proof, you'll never get that weekend back."
mattsains ยท 1 points ยท Posted at 05:14:42 on December 28, 2016 ยท (Permalink)
You could use the same logic to skip lessons in calculus entirely
[deleted] ยท 4 points ยท Posted at 19:26:34 on December 27, 2016 ยท (Permalink)
[deleted]
CreatrixAnima ยท 2 points ยท Posted at 00:21:36 on December 28, 2016 ยท (Permalink)
Could be the same guy. I have classmates who can't stand him! But honestly, I like classes that kick my ass, so I'm a bit weird that way.
mattsains ยท -2 points ยท Posted at 05:13:16 on December 28, 2016 ยท (Permalink)
When I was in uni, I was dealing with so much other shit I had no patience for lazy lecturers that forced me to pick up the pieces later
MadPat ยท 39 points ยท Posted at 17:44:53 on December 27, 2016 ยท (Permalink)
I remember reading a preprint of a paper where there was a line that said: 'By a boring but easy calculation we see that...' Then there was a resulting calculation.
I tried and tried to make the calculation but I couldn't do it. Finally, I decided to go to the library where I knew the paper had the published version. There I saw:'By a boring calculation we see that...'
That's when I gave up on the paper.
SardonicTRex ยท 22 points ยท Posted at 22:10:36 on December 27, 2016 ยท (Permalink)
There were some lecture notes on general relativity I was skimming that would use phrases like 'after an orgy of algebra...'
MadPat ยท 16 points ยท Posted at 23:04:03 on December 27, 2016 ยท (Permalink)
I have statistics book that refers to 'algebraic heroics.'
Gwinbar ยท 5 points ยท Posted at 02:42:18 on December 28, 2016 ยท (Permalink)
I always liked "index gymnastics". Very picturesque.
SOberhoff ยท 14 points ยท Posted at 21:35:10 on December 27, 2016 ยท (Permalink)
While OP seems to have found his anecdote, there is a similar story about G.H. Hardy that is related in Prime Obsession by John Derbyshire:
linusrauling ยท 19 points ยท Posted at 14:34:42 on December 27, 2016 ยท (Permalink)
A friend's paper had a similar genesis. She was working with two other people on a project and had a question about a step in the proof. The other people were both very certain the step was correct as they had seen the result published. My friend went to the reference they provided and wasn't convinced it applied. Eventually they determined that in fact it didn't apply and they had to come up with a new result.
[deleted] ยท -16 points ยท Posted at 23:23:54 on December 27, 2016 ยท (Permalink)
[removed]
Wild_Bill567 ยท 9 points ยท Posted at 20:59:48 on December 27, 2016 ยท (Permalink)
So calling things obvious or trivial is one of those pieces of math-jargon that I've had to get used to. Usually now when I hear that, it is referencing something that can be proven by simply 'unraveling definitions' or direct computation, where no 'tricks' are used.
Of course, what is nearly impossible from an elementary point of view may be obvious from a more advanced point of view, or simply a different point of view. 'Obviousness' has a lot to do with context.
[deleted] ยท 4 points ยท Posted at 22:48:50 on December 27, 2016 ยท (Permalink)
To blind people, it's not obvious if it's day or night. That doesn't make it incorrect to say it's obviously daytime since I can see the sun.
[deleted] ยท 32 points ยท Posted at 12:45:20 on December 27, 2016 ยท (Permalink)
[deleted]
Bromskloss ยท 9 points ยท Posted at 16:21:40 on December 27, 2016 ยท (Permalink)
Can you link to it?
The_Alpacapocalypse ยท 34 points ยท Posted at 20:53:31 on December 27, 2016 ยท (Permalink)
It can be linked...
[deleted] ยท 2 points ยท Posted at 16:42:40 on December 27, 2016 ยท (Permalink)
[deleted]
Bromskloss ยท 8 points ยท Posted at 16:46:29 on December 27, 2016 ยท (Permalink)
Still, it could be interesting to read, if you don't mind.
Yajirobe404 ยท 10 points ยท Posted at 18:52:20 on December 27, 2016 ยท (Permalink)
sorry I do mind
Bromskloss ยท 2 points ยท Posted at 00:11:11 on December 28, 2016 ยท (Permalink)
Never mind!
SimplyUnknown ยท 8 points ยท Posted at 18:12:21 on December 27, 2016 ยท (Permalink)
A prof of mine has a hand annotated version of the perceptron book by Minsky and Papert and somewhere it says "It can easily be shown that..." where the annotation says "Was harder than I expected" and I believe it is still unsolved as of today. Can't remember what is was about though...
ajgraven ยท 5 points ยท Posted at 16:34:23 on December 27, 2016 ยท (Permalink)
I know that there was a similar case for FLT, Fermat basically wrote that the proof should be trivial... It wasn't.
philly_fan_in_chi ยท 13 points ยท Posted at 18:39:09 on December 27, 2016 ยท (Permalink)
I had a problem set once where I invoked Fermat + "this is left as an exercise to the grader" because I ran out of time. I got a big ol' smiley face on that one.
LeepySham ยท 6 points ยท Posted at 23:16:46 on December 27, 2016 ยท (Permalink)
I would argue the opposite. He said the proof was marvelous and too large to fit in the margin, both of which suggest that the proof is nontrivial.
WormRabbit ยท 2 points ยท Posted at 06:14:28 on December 29, 2016 ยท (Permalink)
He clearly didn't consider it inportant enough for a separate paper.
Asddsa76 ยท 5 points ยท Posted at 23:17:48 on December 27, 2016 ยท (Permalink)
ajgraven ยท 4 points ยท Posted at 00:15:27 on December 28, 2016 ยท (Permalink)
He did not, however, imply that the proof would be over 150 pages long.
ChemicalRascal ยท 1 points ยท Posted at 05:15:29 on December 28, 2016 ยท (Permalink)
And rely on instead proving another, at-that-time undiscovered, branch of mathematics entirely, as well. (Personally, I'm inclined to suggest that his proof was flawed in some way he wasn't aware of.)
WormRabbit ยท 1 points ยท Posted at 06:15:47 on December 29, 2016 ยท (Permalink)
Well, the proof is indeed quite simple. I mean, by the modularity of the Frey curve...
[deleted] ยท 5 points ยท Posted at 22:42:00 on December 27, 2016 ยท (Permalink)*
ITT people who don't know mathematics.
Edit: watching this comment's score oscillate is also entertaining. Kinda proves the point though.
[deleted] ยท 10 points ยท Posted at 05:25:07 on December 28, 2016 ยท (Permalink)
[deleted]
[deleted] ยท 4 points ยท Posted at 17:09:35 on December 28, 2016 ยท (Permalink)
I am only rude to people who act like they know what they're talking about when they don't. Such behavior is dangerous in general and disastrous in math.
[deleted] ยท 2 points ยท Posted at 17:26:40 on December 27, 2016 ยท (Permalink)
Sounds like all my professors
[deleted] ยท 1 points ยท Posted at 20:38:47 on December 29, 2016 ยท (Permalink)
It was in the context of a lecture, when a professor said something was obvious. But it's fine if you don't care, just don't assume nobody else cares.
Server969 ยท -6 points ยท Posted at 17:19:53 on December 27, 2016 ยท (Permalink)
That sounds like the setup to fermats last theorem where he writes in the margin of his book that he could prove integer solutions for ax+bx=cx however it was to great to be contained in the margin.
[deleted] ยท -2 points ยท Posted at 08:14:34 on December 28, 2016 ยท (Permalink)
"Trivial" is code for "I cant do it but I am too snobby to admit it"
odh_moyer ยท -69 points ยท Posted at 13:55:21 on December 27, 2016 ยท (Permalink)
Trivia does not mean easy or even imply capability. It simply means that it is not important.
chebushka ยท 58 points ยท Posted at 14:10:24 on December 27, 2016 ยท (Permalink)
First of all, the word was trivial rather than trivia. Second of all, the word trivial is not used in math with that meaning. The proof of Schur's lemma is trivial, but that result itself is definitely not unimportant (and the proof is important because it is how we know Schur's lemma is valid).
[deleted] ยท 5 points ยท Posted at 14:25:48 on December 27, 2016 ยท (Permalink)*
Can you explain how the proof of Schur's lemma is trivial?
Edit:Oh, I thought you were talking about the linear algebra Schur's lemma. I guess you're talking about this one, which I'm not familiar with: https://en.wikipedia.org/wiki/Schur's_lemma
zornthewise ยท 17 points ยท Posted at 14:47:31 on December 27, 2016 ยท (Permalink)
Well it's usually a couple of lines and the idea is easy to come by once you have been working with simple things.
With_a_G ยท 6 points ยท Posted at 17:11:13 on December 27, 2016 ยท (Permalink)
Well if you don't know, I'm certainly not going to tell you.
Narbas ยท 5 points ยท Posted at 15:57:45 on December 27, 2016 ยท (Permalink)
I am not making any statements about the triviality of the proof, but James & Liebecks "Representations and Characters of Groups" has a short proof.
[deleted] ยท 2 points ยท Posted at 22:52:28 on December 27, 2016 ยท (Permalink)
It's literally just the definition of irreducible. It is trivial the same way that every exercise in an undergrad abstract algebra text is trivial.