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Possible Proof of ABC Conjecture
submeta writes "Shinichi Mochizuki of Kyoto University has released a paper which claims to prove the decades-old ABC conjecture, which involves the relationship between prime numbers, addition, and multiplication. His solution involves thinking of numbers not as members of sets (the standard interpretation), but instead as objects which exist in 'new, conceptual universes.' As one would expect, the proof is extremely dense and difficult to understand, even for experts in the field, so it may take a while to verify. However, Mochizuki has a strong reputation, so this is likely to get attention. Proof of the conjecture could potentially lead to a revolution in number theory, including a greatly simplified proof of Fermat's Last Theorem."
Assuming the paper is correct and as impenetrable as the summary claims, this won't simplify the proof of FLT. It'd be a massive rug that the hard parts of of FLT would be swept under.
LMWTFY: http://en.wikipedia.org/wiki/Abc_conjecture#Formulations.
That is precisely the point of the proof, to determine under which conditions the sum of 2 integers is less than the product of the prime divisors of the 3 original numbers. I hope that is less vague :P
...and solved. I think it was the early (19)70's. A researcher named Jackson
(with the help of his brothers) came to the conclusion that it was simple as 1-2-3.
Additional verification shown that do-re-mi fit the bill as well. At the time, people
were sing all about it - I'm surprised this has come up again.
"Rarely much smaller than"? What kind of mathematical statement is that? Are we to assume that most of the time, d is somewhat smaller than c? Are there conditions where d is larger than c? How are you supposed to get anything done with vague statements like "rarely much smaller than"?
There exists mathematical statements which sounds rather "unmathematical" at first, as an example, "almost everywhere" has a precise meaning in measure theory.
http://en.wikipedia.org/wiki/Almost_everywhere
Nobody's measuring anyone's penis--the truth is a lot more boring (and reasonable) than that. Wikipedia is a fantastic first reference for working mathematicians or grad students--I'm sure nearly all math article editors are in these groups--who just want to quickly find out e.g. what the hell an "ultrafilter" is. And so the articles are written in a way that makes them most useful to the people who donate their time to produce them. It's not that any (non-douchebag) mathematician gets off on throwing around smart-sounding jargon. It's just that you can't actually do anything with "intuitive" descriptions.
Yo dawg, I heard you like the Ackermann function, so OH GOD OH GOD OH GOD
Peter had a pretty good first glance reaction to the paper: http://www.math.columbia.edu/~woit/wordpress/?p=5104
I haven't seen any good discussions of the actual math content of the paper yet though.
Which makes it even more non-sensical to post it here, on slashdot, a general-interest geek site, where only very few are working mathemeticians or grad students.
A page like this: http://abcathome.com/conjecture.php would have been more apropos. No reaching for the jargon, and an actual mini-tutorial on what an ABC triple is and what the conjecture is.
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BMO
Well WP math articles aren't designed so that every concept comes with a layman's introduction; that would involve massive duplication and bloat. And so, yes, the link you posted would be more appropriate here than a WP link. But I really don't see how you get from there to accusing the volunteer WP math editors of having a big willy contest. There's a reason those articles are written the way they are, and it's not just to make you personally feel stupid. They don't give a shit how smart you think they are.
Yo dawg, I heard you like the Ackermann function, so OH GOD OH GOD OH GOD
Actually after reading a bit more, it turns out not to be as hyperbolic as it sounds. The author has come up with a whole constellation of new mathematical constructions to support his claimed proof. As the article points out, this means it'll take quite some time for mathematicians to understand these constructions before they'll be able to judge the correctness of the proof. This kind of thing would be dismissed out of hand if it came from Joe Nobody, but Shinichi Mochizuki's reputation in this case should ensure that it gets a good look. And before the crackpots hop on, no, that's not because of any ivory-tower prejudice, but simply because no sane (and busy) professional would judge that such a large personal time investment is likely be worthwhile, without some very strong past performance.
Yo dawg, I heard you like the Ackermann function, so OH GOD OH GOD OH GOD
No, you can't actually learn abstract mathematical ideas by basing your understanding on intuitive descriptions. If you think you have learned a concept that way, I can almost guarantee that your understanding is faulty. (I've learned this the hard way: I happen to be a mathematician who is particularly adept at providing comfortable metaphors that cause non-mathematicians to believe they've understood something when they really haven't.)
For anyone with a suitable background (say, a first-year graduate student or better), Wikipedia's math articles are generally the best, most accurate and most comprehensive free source of basic mathematics information available. If you don't have that background, no article of any kind is going to be explain to what a "scheme" is, for example. To think so is as naive as believing that you can understand all the nuances of Baudelaire's poetry without learning French; you may think you learn something from a translation into your language, but you actually don't.
See http://mathoverflow.net/questions/106560/philosophy-behind-mochizukis-work-on-the-abc-conjecture for a discussion on the mathematical content by experts.
Because if you look in all other commercial encyclopedias (encyclopediae?), you get a more english (well, natural language) translation of the concepts of a math article. But not even that, Wikipedia on this subject fails even at the post-secondary textbook level. I don't count myself among the dumbest of the population, but when I go to a Wikipedia page for something that is on my level for math, the articles on things like cycloids and such are much better explained by Machinery's Handbook or any other source, really, than there.
I am not saying that Wikipedia should dumb its articles down to the point where even the most innumerate among us would understand all of them, but the "spam equations on the wall with little explanation" model doesn't work very well unless you are immersed in the subject. For example, concepts covered in Algebra I and II in high school should be written for that level.^1 Also, this "write for the grad-student and mathemetician for everything" model does little to help people who use applied mathematics. Indeed, this whole focus on grad-student and up writing in the math articles is at odds with the rest of the Wikipedia.
As a result, anyone wishing to *learn* anything about math is better off using anything but Wikipedia.
Your response to me that the articles are written by grad students and mathemeticians (not all mathemeticians are jerks, btw) for grad students and mathemeticians reinforces the fact that it certainly seems like a giant circle jerk.
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BMO
Footnotes:
1. I had to explain to a high school student that she should not be using Wikipedia for help in her Algebra II class. Because all it did was confuse her. I mentioned that Wikipedia math pages are a "dick measuring contest for experts on the subject" and the light went on behind her eyes and she laughed and agreed. There are far better resources and I suggested she ask her teacher for them.
For anyone with a suitable background ..., Wikipedia's math articles are generally the best, most accurate and most comprehensive free source of basic mathematics information available. If you don't have that background, no article of any kind is going to be explain to what a "scheme" is, for example. To think so is as naive as believing that you can understand all the nuances of Baudelaire's poetry without learning French; you may think you learn something from a translation into your language, but you actually don't.
Goethe's comment is relevant here:
Mathematicians are like Frenchmen: whatever you say to them they translate into their own language, and it immediately becomes something entirely different.
Good question. Why don't you devote twenty years or so to becoming competent to judge, then spend all your time reading every crackpot's theory on trisecting angles or why pi isn't really transcendental, and let us know what you find out?
Yo dawg, I heard you like the Ackermann function, so OH GOD OH GOD OH GOD
Probably extremely few.
A friend of mine knew Shin (as he was known then) when he was an undergraduate. The guy was obscene insane-clown-level genius prodigy. Not the prodigy in the sense of the people who can shoot the lights out of the Putnam Competition but even far deeper than that, and jumping into very difficult and profound concepts by age 17 or 18. He did a small stint doing independent research with Ed Witten before moving up to pure mathematics. By 2nd or 3rd year undergrad (age 17 or so), he was already at an advanced graduate level.
I think he may be a different species.
Oh yeah and for fun he learned ancient Sanscrit.
Because if you look in all other commercial encyclopedias (encyclopediae?), you get a more english (well, natural language) translation of the concepts of a math article. But not even that, Wikipedia on this subject fails even at the post-secondary textbook level. I don't count myself among the dumbest of the population, but when I go to a Wikipedia page for something that is on my level for math, the articles on things like cycloids and such are much better explained by Machinery's Handbook or any other source, really, than there.
I am not saying that Wikipedia should dumb its articles down to the point where even the most innumerate among us would understand all of them, but the "spam equations on the wall with little explanation" model doesn't work very well unless you are immersed in the subject. For example, concepts covered in Algebra I and II in high school should be written for that level.^1 Also, this "write for the grad-student and mathemetician for everything" model does little to help people who use applied mathematics. Indeed, this whole focus on grad-student and up writing in the math articles is at odds with the rest of the Wikipedia.
As a result, anyone wishing to *learn* anything about math is better off using anything but Wikipedia.
Your response to me that the articles are written by grad students and mathemeticians (not all mathemeticians are jerks, btw) for grad students and mathemeticians reinforces the fact that it certainly seems like a giant circle jerk.
The problem is that these topics aren't what you'd see in high school algebra. In fact, upper level undergraduate courses would probably just touch on these. So yes, encyclopedias would have more easily understood articles but they almost certainly don't cover theorems like the ABC theorem or topology in any depth. In fact, most articles in encyclopedias will probably give you a very cursory explanation. To make an analogy it'd be like explaining people as living things with 2 legs, 2 arms and which breath air. It's not useful for any in depth topic and when you really want to understand, you'll need to go into details. And in math, those details come in the form of definitions and equations explaining how the definitions interact together.
"When you sit with a nice girl for two hours, it seems like two minutes. When you sit on a hot stove for two minutes, it
I agree with exploder. Wikipedia math articles are surprisingly useful.
>The problem is that these topics aren't what you'd see in high school algebra.
But the fact is that I was able to pull up a *better* explanation of what ABC triplets are and what this conjecture is by linking to a project dealing directly with this problem run by an actual mathemetician. And it was in terms that anyone in algebra I or pre-algebra, if they slowed down and took it step-by-step, could comprehend and it was accurate.
And if you clicked through to the other pages, on the site, you found clear common-english explanations as to why it's important.
Really, go look at the other page I linked.
http://abcathome.com/conjecture.php
And nothing anyone said defending the Wikipedia math pages contradicts my initial claim that you shouldn't link those pages to explain math, especially to a bunch of non-mathemeticians on Slashdot.
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BMO
The best way I can describe this is to say this:
When you see a cube, you define it's boundaries in this universe by it's sides and edges,
In this theory primes, q 1 make up the "dimensions" if you will.
Want an easier explanation? The abc conjecture is a universal equation through which (seemingly) all other equations can be refactored to make them comparable and translatable. Great for number theorists and programmers, not sure who else will use it. Maybe physicists.
Where genius and insanity become confused true wisdom is found
The Penis Measuring Armageddon will be fought on Pen Island.
http://www.penisland.net/
--
BMO
Your response to me that the articles are written by grad students and mathemeticians (not all mathemeticians are jerks, btw) for grad students and mathemeticians reinforces the fact that it certainly seems like a giant circle jerk.
I wrote a couple of the original pages on wikipedia dealing with some comp-sci type topics not usually taught in a 4 year program (back in ~2000). I thought they were fairly clear and understandable, complete with short pieces of pseudo code, and algorithmic explanations. I gave up on those pages because of the idiots editing them. I couple years ago I looked at them, and frankly was shocked, its like some grad students have been trying to out do each other on writing the most esoteric mathematical description of the problem/solution. In fact the Wikipedia pages probably should just be wiped and replaced with references to the original authors (of the algorithms/mathematics) works, because they are a far more accessible resource.
Basically, I would call the Wikipedia pages a complete failure on any metric other than a competition to obfuscate with math something that is fairly accessible.
And nothing anyone said defending the Wikipedia math pages contradicts my initial claim that you shouldn't link those pages to explain math, especially to a bunch of non-mathemeticians on Slashdot.
Nobody's really disputing that. You kept reiterating that Wikipedia Math articles are written by jerks. They're most unhappy about that. To support your point, you basically compared the Wikipedia article to a webpage that's simple enough to suit curious junior high pupils, but it is not up to your own opinion whether one shall cater Wikipedia to this level of pedagogy. It is wrong to say something's absolutely bad just because something else is better (for a different purpose). For example, I can easily find other people who can talk much more politely in a discussion; does that make you a jerk?
I am not a mathemetician. I know my way aronud a reasonable bit of maths as part of my day job (engineering of sorts), but I never studies maths apart from in my engineering course and what I have had to pick up from books since.
Ocasionally random bits of maths interest me (like involution matrices) so I read about them. I find wikipedia a very useful resource.
Because if you look in all other commercial encyclopedias (encyclopediae?), you get a more english (well, natural language) translation of the concepts of a math article.
If it exists at all. Which it usually doesn't.
Also, when it comes to maths, natural languages peter out quite fast. It turns out that maths can be quite difficult and you have to sit down and actually think hard about it in order to gain understanding.
But not even that, Wikipedia on this subject fails even at the post-secondary textbook level.
It's an encyclopedia, not a textbook, but...
but when I go to a Wikipedia page for something that is on my level for math, the articles on things like cycloids and such are much better explained by Machinery's Handbook or any other source, really, than there
Well, it's wikipedia. Quit complaining and write a better article! If you can't be bothered... then why complain that noone else can be bothered either.
For example, concepts covered in Algebra I and II in high school should be written for that level.^
Please bear in mind that wikipedia is very much an international site. There is no "Algebra I" or "Algebre II" outside of some specific educational syllabus. Maths tents to be entertainingly interconnected and even the most unlikely things turn out to be related. Keeping articles dumbed down to the level of the syllabus of a particular education system would be a shame.
Also, this "write for the grad-student and mathemetician for everything" model does little to help people who use applied mathematics. Indeed, this whole focus on grad-student and up writing in the math articles is at odds with the rest of the Wikipedia.
Not really. the whole point of wikipedia is that anyone can write it. It starts off with someone splatting something down and then others refine it. It is not terribly surprising that the initial articles on maths are written by working mathemeticians.
As a result, anyone wishing to *learn* anything about math is better off using anything but Wikipedia.
Well, of course a good text book is going to be a better place for learning than encyclopediea. It should be full of explanations, worked examples, exercises designed to stretch knowledge etc. But to claim wikipedia is worse than everything ele is, frankly, absurd.
And yes, I have learned stuff from wikipedia.
Your response to me that the articles are written by grad students and mathemeticians (not all mathemeticians are jerks, btw) for grad students and mathemeticians reinforces the fact that it certainly seems like a giant circle jerk.
And it's that comment that makes you sound like a grade A whiny entitled asshole. So a bunch of people working hard to provide free encyclopedic content are in a "circle jerk" because the level they have written (and to which you have not contributed" does noe exactly match the level you want out of it?
SJW n. One who posts facts.
I mentioned that Wikipedia math pages are a "dick measuring contest for experts on the subject"
Please check out the comment above by exploder (should be easy to find - it is rated +5 Insightful). In particular:
the articles are written in a way that makes them most useful to the people who donate their time to produce them
I just want to briefly provide an example as to why this is a good thing. I'm a math/stats guy. For me, the free and easily accessible Wikipedia pages are always my first port of call when looking into a new topic/method.
On the other side of the coin is my best mate. He is a med science guy. He avoids Wikipedia like it has the plague and instead uses a resource that is behind a paywall. Why? Because the Wikipedia med science topics are not written for guys like him. They're written to be more accessible. Unfortunately, this makes them of little use to researchers in the field, so they don't bother contributing.
So, what would you prefer? Personally, I think it is better to put up with a little jargon if it ensures a free and open resource that is constantly being peer-reviewed and updated by the top players in a given field. Surely this is preferable to a system where those top players instead choose to contribute to a resource that is behind a paywall?