# Proof and Community Standards

May 10, 2013 2:51 PM Subscribe

In August of last year, mathematician Shinichi Mochizuki reported that he had solved one of the great puzzles of number theory: the ABC conjecture (previously on Metafilter). Almost a year later, no one else knows whether he has succeeded. No one can understand his proof.

This reminds me of software, in a way. There are genius programmers with idiosyncratic coding styles who write code that is inscrutable to all others, except for the fact that it works fantastically. This is fine —

Inscrutable mathematical proofs are even worse than inscrutable code, because at least one can run tests against the code and assert that it does what it claims. But proofs don't prove anything unless they're understood.

posted by savetheclocktower at 3:06 PM on May 10, 2013 [4 favorites]

*unless*others have to touch the same codebase. Excepting vanity projects, brilliance is useless unless it's also maintainable.Inscrutable mathematical proofs are even worse than inscrutable code, because at least one can run tests against the code and assert that it does what it claims. But proofs don't prove anything unless they're understood.

posted by savetheclocktower at 3:06 PM on May 10, 2013 [4 favorites]

*the goal is “to establish an arithmetic version of Teichmuller theory for number fields equipped with an elliptic curve…by applying the theory of semi-graphs of anabelioids, Frobenioids, the etale theta function, and log-shells.”*

timecube?

posted by The Bellman at 3:07 PM on May 10, 2013 [1 favorite]

*at least one can run tests against the code and assert that it does what it claims.*

That is not necessarily true. It can be difficult to

*prove*the code does what it claims.

posted by charlie don't surf at 3:31 PM on May 10, 2013 [2 favorites]

The twilight of genius, where dawn and dusk look alike.

posted by Jehan at 3:34 PM on May 10, 2013 [6 favorites]

posted by Jehan at 3:34 PM on May 10, 2013 [6 favorites]

*the theory of semi-graphs of anabelioids, Frobenioids, the etale theta function, and log-shells*

Four Mefi usernames that should exist but don't.

posted by miyabo at 3:47 PM on May 10, 2013 [14 favorites]

Does the proof contain non-trivial operations related to changing the universe?

posted by Obscure Reference at 3:59 PM on May 10, 2013 [2 favorites]

posted by Obscure Reference at 3:59 PM on May 10, 2013 [2 favorites]

And that's Metafilter's own escabeche interviewed in the article.

posted by needled at 4:11 PM on May 10, 2013 [1 favorite]

posted by needled at 4:11 PM on May 10, 2013 [1 favorite]

It's all understandable why mathematicians are excited and why it's created this controversy, but to ask a rude question: just how important is the ABC conjecture, from a

If yes, then it is just a matter of paying some mathematicians to invest the time to unpack this. There are plenty of people bright enough to follow any proof no matter how complex - it is a question of devoting the resources. How come nobody thinks it's worth devoting said resources? And that answers the question in the first paragraph, like it or not. And which mathematician is willing to devote his time - limited on this earth - for free, to unpacking this? Is there a burning desire? Not burning in words, but in action.

posted by VikingSword at 4:24 PM on May 10, 2013 [2 favorites]

*practical*point of view? To be facile, one can, of course say that since mathematics is essentially completely interrelated on*some*level, everything is of equal theoretical importance. But can you show a direct relation between getting this result and*economic*return? Like encryption - not on some theoretical level, but like in immediate dollars and cents return ("this encryption scheme is now obsolete"); the traveling salesman problem resulting in immense returns on optimizations of all kinds, etc.If yes, then it is just a matter of paying some mathematicians to invest the time to unpack this. There are plenty of people bright enough to follow any proof no matter how complex - it is a question of devoting the resources. How come nobody thinks it's worth devoting said resources? And that answers the question in the first paragraph, like it or not. And which mathematician is willing to devote his time - limited on this earth - for free, to unpacking this? Is there a burning desire? Not burning in words, but in action.

posted by VikingSword at 4:24 PM on May 10, 2013 [2 favorites]

*Inscrutable mathematical proofs are even worse than inscrutable code, because at least one can run tests against the code and assert that it does what it claims. But proofs don't prove anything unless they're understood.*

No, they are basically both equally bad.

Computer code is not a means for humans to communicate with computers. Computer code is a means for humans to communicate

*with other humans*. If it fails to do that, it's bad code, period.

Proofs are similar. The point of a proof is to convince. If you can't understand, you can't convince. If you can't convince, it isn't a proof.

posted by DU at 4:39 PM on May 10, 2013 [10 favorites]

VikingSword, my impression is that the work Mochizuki has done is so strange and complicated that nobody has the slightest idea what practical consequences it might have. Knowing whether the ABC conjecture is true or false is not particularly important, but the reasons why it's true or false could lead in all kinds of different directions. But understanding those reasons, even enough to know whether they might lead somewhere practical, requires a significant investment of time. Enough of an investment, it sounds like, that most mathematicians aren't interested, even though many agree there might be some great stuff in there.

posted by brianconn at 4:40 PM on May 10, 2013 [1 favorite]

posted by brianconn at 4:40 PM on May 10, 2013 [1 favorite]

VikingSword, I would guess the answer is: nobody knows for sure. I'm sure someone could give you a lower bound (possibly zero), but estimates beyond that might be difficult. That is the nature of research.

posted by Serf at 4:41 PM on May 10, 2013 [1 favorite]

posted by Serf at 4:41 PM on May 10, 2013 [1 favorite]

This is the MathOverflow thread referenced in the article, by the way. If you look at, say, his recent survey paper A Panoramic Overview of Inter-Universal Teichmuller Theory (which includes a sketch of his proof of the ABC conjecture) you will find 19 citations, 17 of which are to his own papers. (!) This is, I think, some indication of the daunting task that faces people who want to verify his proof.

I personally think this is never the right question to ask about pure math, since you will often see up to a century between the development of a mathematical theory and any potential application. (See, eg, number theory.) I think of pure math more as a wellspring for potential new ideas that lays foundations far in advance. Just speaking personally, the idea of economic return on my research is wildly outside the bounds of what I would ever concieve of, unless we're speaking in terms of grants or my getting tenure or something concrete like that.

This is not the case. There are only a handful of mathematicians in the world who would be truly qualified to vet all of the details of the proof. It's not a question of being bright enough; it's a question of (1) knowing enough math and (2) being willing to spend the huge amounts of time to learn all of the details of Mochizuki's theories. In particular, "knowing enough math" is a

posted by Frobenius Twist at 4:42 PM on May 10, 2013 [14 favorites]

*But can you show a direct relation between getting this result and economic return?*I personally think this is never the right question to ask about pure math, since you will often see up to a century between the development of a mathematical theory and any potential application. (See, eg, number theory.) I think of pure math more as a wellspring for potential new ideas that lays foundations far in advance. Just speaking personally, the idea of economic return on my research is wildly outside the bounds of what I would ever concieve of, unless we're speaking in terms of grants or my getting tenure or something concrete like that.

*If yes, then it is just a matter of paying some mathematicians to invest the time to unpack this. There are plenty of people bright enough to follow any proof no matter how complex - it is a question of devoting the resources.*This is not the case. There are only a handful of mathematicians in the world who would be truly qualified to vet all of the details of the proof. It's not a question of being bright enough; it's a question of (1) knowing enough math and (2) being willing to spend the huge amounts of time to learn all of the details of Mochizuki's theories. In particular, "knowing enough math" is a

*time*issue, not a*cleverness*issue: it involves (as far as I can tell, being a nonexpert in this field) heavy-duty knowledge of algebraic number theory and stacky algebraic geometry, both of which are seriously technical subjects. To get to the point where one would be comfortable enough with these fields to thoroughly vet the paper would require years of post-doctoral work. And the relatively small number of people who are in a position to do this apparently (and not so surprisingly) don't want to drop everything they're doing for the long time that it would take to unpack this.posted by Frobenius Twist at 4:42 PM on May 10, 2013 [14 favorites]

Much more than Timecube, it reminds me of Bitcoin: Japanese person (at least purportedly, in the case of Bitcoin) develops a highly complex system, dumps it in public for everyone to access, then withdraws and doesn't communicate about the system. Any chance that Mochizuki is related to "Nakamoto"?

posted by jiawen at 4:55 PM on May 10, 2013 [1 favorite]

posted by jiawen at 4:55 PM on May 10, 2013 [1 favorite]

**DU**:

*Proofs are similar. The point of a proof is to convince. If you can't understand, you can't convince. If you can't convince, it isn't a proof.*

A "proof" in the formal sense means more than just "I have convinced you". It describes a truth, nothing less (though

*preferably*more).

**If**correct, it means he has built a better hammer, that everyone

*else*can use to pound their own nails in - Whether or not they understand the particular alloys he used and the aerodynamics of the head,

*man*does that sucker sink nails well!

The problem with no one else

*understanding*his proof centers on my use of bold above - "If". If no one else can read it, we may not have the confidence to assume its conclusion. But that has no bearing whatsoever on the

*correctness*of the proof.

Or put another way - Rejecting a proof based solely on our inability to grasp it amounts to threatening the universe that we'll hold our breath until we turn blue if it doesn't start making sense to us.

posted by pla at 5:51 PM on May 10, 2013 [3 favorites]

Gosh, this inter-universality bit is such a huge rabbit hole. I will post back when (if) it all makes sense.

posted by the cydonian at 5:57 PM on May 10, 2013

posted by the cydonian at 5:57 PM on May 10, 2013

This is proof that we are all characters in a story written by:

posted by localroger at 6:18 PM on May 10, 2013 [1 favorite]

- Isaac Asimov
- Norman Spinrad
- Alfred Bester
- Philip K. Dick
- Stephen King
- D. F. Jones
- H. P. Lovecraft
- Neal Stephenson
- H. G. Wells
- localroger
- you

posted by localroger at 6:18 PM on May 10, 2013 [1 favorite]

*If correct, it means he has built a better hammer, that everyone else can use to pound their own nails in - Whether or not they understand the particular alloys he used and the aerodynamics of the head, man does that sucker sink nails well!*

This is false in at least two ways:

1) How are you supposed to use a proof you can't understand? How do you know you are using it right?

2) The purpose of proofs is not a merely a set of formally valid statements. It is understanding. Because otherwise, we eventually reach an point where #1 applies and we can go no further (or possibly we sit grunting in caves and manipulating symbols we don't understand and can never use in any way other than shuffling them around).

posted by DU at 6:32 PM on May 10, 2013

So, Gut feeling people, has he got it? I say no - its his refusal to explain - its like the mathematician who sailed on a boat and sent a postcard saying "solved Fermats last" as he figured god wouldn't let him die with such a massive lie on his lips. Or something.

posted by marienbad at 6:36 PM on May 10, 2013

posted by marienbad at 6:36 PM on May 10, 2013

*1) How are you supposed to use a proof you can't understand? How do you know you are using it right?*

You don't necessarily need to use the proof, if you're using the result. Yes, it's preferable if you actually understand the damn thing, but sometimes "These other people proved this thing I need using some stuff I don't know about" has to be good enough. You're trusting the people who are experts in that area to have read the proof and you're trusting the person who explained the general idea to you (if you in fact found someone who understood the proof). That's what the whole bit about 'community' in the article was about.

posted by hoyland at 6:50 PM on May 10, 2013 [1 favorite]

*Four Mefi usernames that should exist but don't.*

Sup.

Don't mind me, I will just sit here being vertically slim and totally aloof.

posted by theory of semi-graphs of anabelioids at 7:18 PM on May 10, 2013 [16 favorites]

So the theory of semi-graphs of anabeloids cost five bucks. Who knew.

posted by localroger at 7:21 PM on May 10, 2013

posted by localroger at 7:21 PM on May 10, 2013

If you want to read more, there's a really entertaining (long-ish) article by Barry Mazur linked in this comment. The first half of it is extremely accessible.

posted by benito.strauss at 7:35 PM on May 10, 2013 [2 favorites]

posted by benito.strauss at 7:35 PM on May 10, 2013 [2 favorites]

I have a genuine question for the mathematicians here, not snarking. I fully believe in the pursuit of knowledge for its own sake. But has any really complex math - so, say, anything you would generally need a PhD in mathematics to understand - been used in any applications apart from furthering the knowledge of mathematics? For example, did proving Fermat's last theorem give us some tools or knowledge we could use in some practical endeavor, or developing some new technology? Do ring theory or abstract algebra have any scientific applications outside of the mathematical sciences? Again not snarking, just genuinely curious.

posted by pravit at 7:51 PM on May 10, 2013 [1 favorite]

posted by pravit at 7:51 PM on May 10, 2013 [1 favorite]

It's possible to give a purely formal proof in a purely formal system. If you take a logic course or a mathematics course in college, you'll be learning a formal system. But a dirty little secret is that an awful lot of higher mathematics is

If an informal proof is insufficiently perspicuous to mathematicians, they'll claim it isn't a proof at all; you need to do enough work to show that each step-by-step transition in the proof is intuitive and clear. I find this ABC proof interesting because it's not clear whether it's a proof that no other mathematician can understand, or whether it's not a proof at all because no one else can understand it.

posted by painquale at 7:57 PM on May 10, 2013 [4 favorites]

*informal*. Mathematics papers are in English (or some other natural language). They don't always cite explicit transition rules that take you from one claim to another. Rather, they rely on their readers---other mathematical experts---just being able to "see" what follows from what. Sometimes, the agreed-upon intuitions of mathematicians end up packing in a lot of assumptions; these might get unpacked later on down the line, resulting in a new geometry or something like that. A lot of mathematical proofs would go out the window if you demanded that all proofs be formal.If an informal proof is insufficiently perspicuous to mathematicians, they'll claim it isn't a proof at all; you need to do enough work to show that each step-by-step transition in the proof is intuitive and clear. I find this ABC proof interesting because it's not clear whether it's a proof that no other mathematician can understand, or whether it's not a proof at all because no one else can understand it.

posted by painquale at 7:57 PM on May 10, 2013 [4 favorites]

> No, they are basically both equally bad.

DU, it goes deeper than that -- there's an elegant proof that proofs and programs are the same. ;)

I loved this article, and specially the candor the the mathematician who said, "It really is painful to read other people’s work."

posted by joeyh at 7:59 PM on May 10, 2013 [4 favorites]

DU, it goes deeper than that -- there's an elegant proof that proofs and programs are the same. ;)

I loved this article, and specially the candor the the mathematician who said, "It really is painful to read other people’s work."

posted by joeyh at 7:59 PM on May 10, 2013 [4 favorites]

pravit, math that seemed to have no application at the time -- literally complex math, the math of imaginary numbers which are the square roots of negative numbers -- made all of modern electrical distribution possible. Not electronics, I'm talking about generators and motors and transformers. Nicola Tesla assembled a bunch of previously free-floating symbols into a physical technology that's in use to this day, and at the time he seemed like a magician for being able to see it.

posted by localroger at 8:00 PM on May 10, 2013 [5 favorites]

posted by localroger at 8:00 PM on May 10, 2013 [5 favorites]

Has any really complex math - so, say, anything you would generally need a PhD in mathematics to understand - been used in any applications apart from furthering the knowledge of mathematics?The canonical answer here is cryptography, which is pretty necessary for modern society to function (for example, pretty much all finance depends on it).

Group theory turned out to be incredibly useful in low-level physics, which may or may not answer your question depending on how useful you think low-level physics is.

posted by dfan at 8:02 PM on May 10, 2013 [3 favorites]

*Do ring theory or abstract algebra have any scientific applications outside of the mathematical sciences?*

There are, believe it or not, industrial applications of algebraic geometry (or some of it anyway). What positions these things can be in turns into an algebraic geometry question.

There are some applications to biology as well. See the references to this article. (The article's actually about incorporating math into undergrad biology curricula, but sort of alludes to where algebra is popping up.)

Like dfan suggested, the canonical answer is RSA. But you see a proof of the Chinese Remainder Theorem in an undergrad algebra class, so that maybe doesn't meet your standard.

posted by hoyland at 8:36 PM on May 10, 2013 [2 favorites]

Has any really complex math - so, say, anything you would generally need a PhD in mathematics to understand - been used in any applications apart from furthering the knowledge of mathematics?

This is a tricky one because once a piece of PhD-level math becomes useful, we spend a generation of effort developing tools and tricks to teach it to the people who need it. See: calculus, fourier analysis, differential equations, RSA encryption, elliptic curves, network coding, graph theory, PDEs, error propagation, linear algebra, etc, etc.

Each of these things, prior to becoming widely agreed-upon as useful, needed a PhD in math to understand. Now that they are known to be useful, we teach them to undergrads!

posted by pmb at 8:57 PM on May 10, 2013 [8 favorites]

Oh, yeah, all those JPEGs you've seen on your computer? Fourier analysis.

posted by benito.strauss at 9:36 PM on May 10, 2013 [5 favorites]

posted by benito.strauss at 9:36 PM on May 10, 2013 [5 favorites]

*In particular, "knowing enough math" is a time issue, not a cleverness issue: it involves (as far as I can tell, being a nonexpert in this field) heavy-duty knowledge of algebraic number theory and stacky algebraic geometry, both of which are seriously technical subjects. To get to the point where one would be comfortable enough with these fields to thoroughly vet the paper would require years of post-doctoral work. And the relatively small number of people who are in a position to do this apparently (and not so surprisingly) don't want to drop everything they're doing for the long time that it would take to unpack this.*

Uhm, you're agreeing with me. That was exactly my point: cleverness is not the limiting factor (as there are plenty of clever mathematicians around), it is the other factors most important of which is time investment.

And how do you get people to invest the time? (a) pay them a ton of money or (b) it's important enough in its own right so that someone is willing to do it for free, sacrificing their own time.

For (a), only an entity that would immediately benefit financially would be willing to pay enormous amounts of money to essentially buy mathematicians' time. Clearly, it doesn't appear that anyone at present thinks there is such an immediate economic payoff - hence no money.

That leaves (b). Why would a mathematician drop everything and give of his time so massively if money is not at stake? Well s/he would, only if the importance of this proof was of earth-shattering importance to mathematics. The fact that nobody stepped up, answers the question quite adequately: "just how important is the ABC conjecture?"... clearly not important

*enough*(in the opinion of mathematicians today).

posted by VikingSword at 9:39 PM on May 10, 2013

At the point where you're a mathematician who is capable of figuring this out, you're probably not motivated by money. And you probably have too much ego and focus on your own work to give it up for the several years it would take to wade through this. It's not that the problem is unimportant but that this proof really is that impenetrable and in a sort of tragedy of the commons, no one wants to give up their calling to check it.

posted by matildatakesovertheworld at 10:31 PM on May 10, 2013 [1 favorite]

posted by matildatakesovertheworld at 10:31 PM on May 10, 2013 [1 favorite]

Well, let me put it this way. When I review a 20-page paper it takes at least a week of on-and-off work to read through the details repeatedly to make sure I understand the overall structure of the paper and then read all of the proofs multiple times to make sure I believe them. Then I need to carefully check for typos and then write up a letter to the journal editor carefully describing my conclusions. That is the work it takes for a 20-page paper in my field, which I know very well. And I do not begrudge this work, because peer review is done out of service to the community.

But now imagine reviewing 500+ pages,

And money has nothing to do with it. No amount of money is going to make someone peer review this if they aren't already interested. Believe me, nobody enters pure math for the money. If I was in it for the money I'd take my Ph.D. and 7 years of ongoing post-doctoral work and triple my pay elsewhere. I'd deeply mistrust any mathematician who only agreed to referee a difficult paper because they were paid to do so, since that runs counter to the ethos of the community on a fundamental level.

posted by Frobenius Twist at 10:43 PM on May 10, 2013 [8 favorites]

But now imagine reviewing 500+ pages,

*including an introduction to a field that essentially 1 person knows because he just created it.*And keep in mind that the complexity will increase exponentially with size. This could easily be the work of years, work that quite possibly would not pay off! You could spend a year away from your own research (which is what mathematicians love to do - it's why we are mathematicians) learning Mochizuki's theory only to find a mistake that invalidates it all. Altruistic service to math does not extend to losing years of one's own work and publications, nor should it. This is not about mathematicians not caring about the ABC conjecture -- it's a huge conjecture, after all.And money has nothing to do with it. No amount of money is going to make someone peer review this if they aren't already interested. Believe me, nobody enters pure math for the money. If I was in it for the money I'd take my Ph.D. and 7 years of ongoing post-doctoral work and triple my pay elsewhere. I'd deeply mistrust any mathematician who only agreed to referee a difficult paper because they were paid to do so, since that runs counter to the ethos of the community on a fundamental level.

posted by Frobenius Twist at 10:43 PM on May 10, 2013 [8 favorites]

[This is the most interesting conversation I've read on the blue in ages. As a simple performer, I have no stake in this game and couldn't comment on it, the ABC Conjecture, or much beyond algebra. But the conversation is fascinating and it's the reason so many people come here. It's like witnessing an argument between multiple Datas, Picards, and Umberto Ecos. I'm sure I speak for many when I say please don't let this conversation end!]

posted by artof.mulata at 12:48 AM on May 11, 2013 [5 favorites]

posted by artof.mulata at 12:48 AM on May 11, 2013 [5 favorites]

VikingSword 's view is extremely simplistic. The idea that all important works will be achieved by either a) capitalist economic incentives or b) philanthropic motives seems to disregard an entire realm of whats known (in economics) as Public Goods that are beneficial in some sense to everyone but no-one has a direct interest in performing/generating.

If it would take a person a year or two to verify the proof who should pay for their expenses for that year? What about their career, their need for income.

The cutthroat nature of academic funding allocations these days means that for a lot of mathematicians spending a year or so checking this proof could really derail your "career". If "we" (developed countries) funded mathematical research a bit more freely then perhaps someone would have already started on it. But instead we have decided to ask educational institutions to be more "economically viable".

posted by mary8nne at 3:24 AM on May 11, 2013 [4 favorites]

If it would take a person a year or two to verify the proof who should pay for their expenses for that year? What about their career, their need for income.

The cutthroat nature of academic funding allocations these days means that for a lot of mathematicians spending a year or so checking this proof could really derail your "career". If "we" (developed countries) funded mathematical research a bit more freely then perhaps someone would have already started on it. But instead we have decided to ask educational institutions to be more "economically viable".

posted by mary8nne at 3:24 AM on May 11, 2013 [4 favorites]

**matildatakesovertheworld**

*in a sort of tragedy of the commons, no one wants to give up their calling to check it.*

This is what doctoral students are for. I'm a bit out of touch with academic standards, but I would expect that verifying this particular proof would contain at least one PhD's worth of material.

posted by aeschenkarnos at 4:39 AM on May 11, 2013

Abstract algebra and group theory, particularly symmetry, are the theoretical underpinnings of the Standard Model, and have a lot of implications for materials physics. A lot of the new metamaterials coming out with seemingly impossible physical attributes are often the result of quirks of symmetry predicted by group theory.

posted by empath at 5:48 AM on May 11, 2013

posted by empath at 5:48 AM on May 11, 2013

**DU**: 1) How are you supposed to use a proof you can't understand? How do you know you are using it right?

Have you ever made use of the proof that 1+1=2? And even if you and I can (painfully) understand that ugly, ugly gem, do you suppose the average schoolkid or McDonald's cashier can?

*2) The purpose of proofs is not a merely a set of formally valid statements. It is understanding. Because otherwise, we eventually reach an point where #1 applies and we can go no further (or possibly we sit grunting in caves and manipulating symbols we don't understand and can never use in any way other than shuffling them around).*

While noble (or perhaps egotistical) to attribute a higher meaning to what we

*do*with them, you've just described the entirety of symbolic logic. And whether you live in a cave or in a 5th avenue penthouse, the computers we use to have this conversation "speak" that language natively, without the slightest "understanding" of their elegance in doing so.

posted by pla at 7:02 AM on May 11, 2013 [2 favorites]

On computer-aided proof and "understanding." With escabeche making another appearance!

There's an interesting analogy between Mochizuki's proof and computer-aided proofs: "

posted by painquale at 7:41 AM on May 11, 2013

There's an interesting analogy between Mochizuki's proof and computer-aided proofs: "

*Hales submitted his proof to the Annals of Mathematics, the field’s most prestigious journal, only to have the referees report four years later that they had not been able to verify the correctness of his computer code.*"posted by painquale at 7:41 AM on May 11, 2013

It really is hard not to be humbled by the epic nature of some of these modern proofs.

As something of an applied mathematician myself (heavy emphasis on applied) I feel pretty great when I work out a nice half page derivation in stochastic differential equations. I do hesitate to call myself a mathematician as such, but still, I think the level of stuff that I work with is not something that most aspiring mathematicians would start hacking until grad school or so (or at least the final year of undergrad).

Nevertheless, from what I can tell, the level that someone like Shinichi Mochizuki is working at is about as far removed from what I do as my work is from elementary school arithmitic. And I'm not even trying to exaggerate.

It does seem like some degree of automatic theorem proving is going to be necessary to allow mathematicians to push the limits over the 21st century.

Oh, and finally: Yah, the idea that academics are motivated by money is laughable. Fame and reputation and success, sure. Hence the fraud. But money? Pretty much all academics I know implicitly assume that if we put all our hard work into something else we could make quite a lot more money. And many do make that switch. But the ones who stay? They're not staying for the money.

posted by Alex404 at 8:08 AM on May 11, 2013 [2 favorites]

As something of an applied mathematician myself (heavy emphasis on applied) I feel pretty great when I work out a nice half page derivation in stochastic differential equations. I do hesitate to call myself a mathematician as such, but still, I think the level of stuff that I work with is not something that most aspiring mathematicians would start hacking until grad school or so (or at least the final year of undergrad).

Nevertheless, from what I can tell, the level that someone like Shinichi Mochizuki is working at is about as far removed from what I do as my work is from elementary school arithmitic. And I'm not even trying to exaggerate.

It does seem like some degree of automatic theorem proving is going to be necessary to allow mathematicians to push the limits over the 21st century.

Oh, and finally: Yah, the idea that academics are motivated by money is laughable. Fame and reputation and success, sure. Hence the fraud. But money? Pretty much all academics I know implicitly assume that if we put all our hard work into something else we could make quite a lot more money. And many do make that switch. But the ones who stay? They're not staying for the money.

posted by Alex404 at 8:08 AM on May 11, 2013 [2 favorites]

>

No kidding. I've been working for years now on a chronology of Russian prose literature that I started because I was wondering what else people were reading when

posted by languagehat at 8:48 AM on May 11, 2013 [1 favorite]

*VikingSword's view is extremely simplistic.*No kidding. I've been working for years now on a chronology of Russian prose literature that I started because I was wondering what else people were reading when

*War and Peace*came out. Am I expecting it to make me rich? No. Do I think it's "of earth-shattering importance"? No. But it makes me happy to do it. Does VikingSword not do anything for the sake of sheer pleasure?posted by languagehat at 8:48 AM on May 11, 2013 [1 favorite]

*But a dirty little secret is that an awful lot of higher mathematics is informal. ... Rather, they rely on their readers---other mathematical experts---just being able to "see" what follows from what. ... If an informal proof is insufficiently perspicuous to mathematicians, they'll claim it isn't a proof at all; you need to do enough work to show that each step-by-step transition in the proof is intuitive and clear.*

I think this is very slightly off. It's not, I think, that mathematicians read a logical step in an informal proof and "just see" that it's valid, or find it "intuitive" — it's that, due to their experience in the field, they're often able to come up with the missing formal details on the spot, or, sometimes, they can't come up with the details immediately, but they recognize the step as being the kind of thing that they would be able to prove with such-and-such techniques (again, by experience), and they move on because they trust that it will be possible to fill in the gap if they ever need to.

(Unless that's exactly what you meant by "just seeing" and "intuitive", but I don't think that's the usual meaning of those phrases.)

posted by stebulus at 9:15 AM on May 11, 2013

*I think this is very slightly off. It's not, I think, that mathematicians read a logical step in an informal proof and "just see" that it's valid, or find it "intuitive" — it's that, due to their experience in the field, they're often able to come up with the missing formal details on the spot...*

Agreed - Terence Tao calls this "post-rigorous" thinking in a really good blog post about mathematical intuition.

posted by Frobenius Twist at 9:20 AM on May 11, 2013

*VikingSword 's view is extremely simplistic. The idea that all important works will be achieved by either a) capitalist economic incentives or b) philanthropic motives seems to disregard an entire realm of whats known (in economics) as Public Goods that are beneficial in some sense to everyone but no-one has a direct interest in performing/generating.*

Nah, it's very realistic. In fact, one could say that this view:

*And money has nothing to do with it. No amount of money is going to make someone peer review this if they aren't already interested. Believe me, nobody enters pure math for the money. If I was in it for the money I'd take my Ph.D. and 7 years of ongoing post-doctoral work and triple my pay elsewhere.*

is simplistic in that it does not describe the actual options. Yes of course tons of mathematicians are not motivated by money and sometimes not even by fame/reputation (Perelman in rejecting the Fields medal "I'm not interested in money or fame, I don't want to be on display like an animal in a zoo."). The mistake here is in looking around at other academics and saying "well, nobody here is doing math for money", because while it may be the case that not a single academic would be motivated by money, this is a conclusion based on a self-selected group. As observed - these folks are pretty much defined by not being motivated by money - because if they were, there is a ton more to be made in various applied fields. And here is where the over-simplification occurs.

Because there are plenty of sufficiently clever mathematicians who are not academics and can be motivated by money. Those are the folks who end up on Wall Street and the like. Now, if you'd think that these are somehow folks who couldn't hack it in academia, you'd be sorely mistaken. When I was at the uni, the most brilliant student we had in the department, who everyone was certain would do foundational level of work in mathematical logic, ended up not even finishing his PhD, and working for a hedge fund. He's by no means unique. There is a level of arrogance among some academics, who assume that academia is a fail-safe method for attracting and retaining the best of the best; it is not so - life and human beings are more complicated than that.

The challenge is not the level of difficulty here, it is the time needed to delve into this, especially considering the one-of-a-kind little world Mochizuki built for himself. But unless he pulled the equivalent of "the margin of this book is too small" somewhere in the text, it can be done with enough time - and time mastering the field is part of the deal. And that's where money enters for those who are motivated by it - it's not 'capitalist motivation', it's human motivation, as I'm quite certain money motivated some during feudal times too - make the sum big enough and you'll have people - very, very smart people - step up to the plate. But there is no such sum being offered - and that's my point.

Now, again, if this was of foundational importance to the field, no doubt someone would step up to the challenge just for that reason, because it would be vital to the entire discipline. But clearly, nobody seems to think so, and hence nobody does it.

*Does VikingSword not do anything for the sake of sheer pleasure?*

VikingSword does, and understands this. But languagehat is incorrectly describing the boundaries of the sets of motivations I'm discussing. I am not claiming that there are no

*other*motivations beyond money and importance to the field - this is what languagehat seems, wrongly, to think I am claiming. I am not. I am claiming that money is not forthcoming and importance is lacking, and had we had either of those, this would not be an issue. That there may be people who would do it out of pure passion is also true, but not what I am discussing. So the fact remains: there are no people who think debunking/verifying someone else's proof of the ABC conjecture is important

*enough*.

Which leaves Mochizuki in the position of either being willing to make himself accessible and promoting and explaining his proof until acceptance, or seeing it languish... assuming he really thinks he's got the real thing here and it's not an elaborate obfuscation.

posted by VikingSword at 11:26 AM on May 11, 2013 [2 favorites]

*I think this is very slightly off. It's not, I think, that mathematicians read a logical step in an informal proof and "just see" that it's valid, or find it "intuitive" — it's that, due to their experience in the field, they're often able to come up with the missing formal details on the spot*

I think this might be an idealized conception of actual mathematical practice. But I'm no mathematician, so I could be wrong.

Jody Azzouni is excellent on this topic, I think: 1, 2, 3 (Part 2, esp. section 7.4).

posted by painquale at 12:44 PM on May 11, 2013

Sorry, I linked to the wrong book in (3) there. I meant to link to this one.

posted by painquale at 12:53 PM on May 11, 2013

posted by painquale at 12:53 PM on May 11, 2013

*So the fact remains: there are no people who think debunking/verifying someone else's proof of the ABC conjecture is important enough.*

Sure... but this is not at all what we gathered from your earlier comments. You seem to be assuming/arguing that if only mathematicians cared

*enough*or the abc conjecture were applicable

*enough*(because apparently math has no value beyond applications that result in someone acquiring money) that the proof would be checked in short order, while totally ignoring the scale of the task. The number of people qualified to start checking the thing is pretty small and you can't simply 'buy' more 'sufficiently smart' people, as it'd take years to train them. But that's not good enough for you--you seem to be saying it'd only be a significant result if it had already been checked.

As long as the abc conjecture isn't the most important thing in the universe (note that no one has said it is), there's a decent chance the handful of people who could slog their way through the proof will conclude their time is better spent on their own research. That doesn't make the abc conjecture not important. It simply makes it not pressing enough to drop everything else they're working on. Which is why people are hoping Mochizuki will explain it to someone (if not give talks)--having the rough idea beforehand greatly decreases the time and energy it takes to read the paper, meaning the people qualified to read the thing are more likely to be able to do it

*while*continuing to do the stuff they're already doing.

posted by hoyland at 12:53 PM on May 11, 2013

due to their experience in the field, they're often able to come up with the missing formal details on the spot

That's what we called "mathematical maturity", and it was (hopefully) first gained between sophomore and senior year. It's like when a Chilton car manual tells you to "tighten bolt X" and doesn't need to say "get a wrench", or tell you what kind of wrench or whether to turn clockwise or counter-clockwise.

posted by benito.strauss at 12:54 PM on May 11, 2013

That's what we called "mathematical maturity", and it was (hopefully) first gained between sophomore and senior year. It's like when a Chilton car manual tells you to "tighten bolt X" and doesn't need to say "get a wrench", or tell you what kind of wrench or whether to turn clockwise or counter-clockwise.

posted by benito.strauss at 12:54 PM on May 11, 2013

I actually read Mochizuki's paper and I found a serious error.

But I won't tell anybody where it is.

So there you go. Another problem solved.

posted by twoleftfeet at 1:32 PM on May 11, 2013

But I won't tell anybody where it is.

So there you go. Another problem solved.

posted by twoleftfeet at 1:32 PM on May 11, 2013

*I think this might be an idealized conception of actual mathematical practice. But I'm no mathematician, so I could be wrong.*

I'm (kind of) a mathematician, and I'll stipulate that it's idealized. I'll also stipulate that being (kind of) a mathematician doesn't necessarily mean I know what mathematical practice actually is.

*Jody Azzouni is excellent on this topic, I think: 1, 2, 3 (Part 2, esp. section 7.4).*

Thanks for this — it looks fascinating.

posted by stebulus at 3:41 PM on May 11, 2013

On much reflection I think I know what Mochizuki is up to here.

Everyone has hobbies, and sometimes mathematicians have hobbies in the form of little worlds they explore that have nothing to do with practical reality and are just their little playspaces. Kind of like what the Culture Minds in Iain M. Banks

So Mochizuki has been relaxing with his relaxingly meaningless fun time experiment for a decade or more, drawing it out into a whole Inception-like structure nobody else even knows about, and one day he realizes probably with a sense of horror that it has looped back in on something that exists IRL, namely the ABC problem. Being himself he cannot resist and he closes the loop, using his funtime theory to erect a proof of something he knows cannot be proved by more conventional methods.

What to do? He could of course bury his proof but that would be wrong in all sorts of ways.

So he releases it, with just enough explanation to make it possible to follow if you are really, really interested.

But he knows nobody else is going to be

So he releases it, knowing it will not be understood, and knowing that it will not be understood does not bother to take interviews or promote it or anything. It just is. It's now part of the record, itself a mystery like Fermat's Last Theorem, waiting not to be proved but to be made accessible to those who can judge whether it is in fact a proof.

I think he is expecting, most likely posthumously, that a machine will verify (or perhaps not) his proof. He knows he has no basis to ask his few fellows, all of whom might have fun times of their own, to spend enough time in his to follow the thread that leads to that conclusion.

But eventually a machine, not even necessarily an AI, will be able to follow his logic and tell his peers whether it's worth going down his rabbit hole or not.

posted by localroger at 4:34 PM on May 11, 2013 [4 favorites]

Everyone has hobbies, and sometimes mathematicians have hobbies in the form of little worlds they explore that have nothing to do with practical reality and are just their little playspaces. Kind of like what the Culture Minds in Iain M. Banks

*Excession*call "infinite fun time."So Mochizuki has been relaxing with his relaxingly meaningless fun time experiment for a decade or more, drawing it out into a whole Inception-like structure nobody else even knows about, and one day he realizes probably with a sense of horror that it has looped back in on something that exists IRL, namely the ABC problem. Being himself he cannot resist and he closes the loop, using his funtime theory to erect a proof of something he knows cannot be proved by more conventional methods.

What to do? He could of course bury his proof but that would be wrong in all sorts of ways.

So he releases it, with just enough explanation to make it possible to follow if you are really, really interested.

But he knows nobody else is going to be

**that**interested. He probably knows everyone in the world of mathematics who would be capable of following his proof and he knows he can't even begin to explain it to them. Nobody would sit still for it.So he releases it, knowing it will not be understood, and knowing that it will not be understood does not bother to take interviews or promote it or anything. It just is. It's now part of the record, itself a mystery like Fermat's Last Theorem, waiting not to be proved but to be made accessible to those who can judge whether it is in fact a proof.

I think he is expecting, most likely posthumously, that a machine will verify (or perhaps not) his proof. He knows he has no basis to ask his few fellows, all of whom might have fun times of their own, to spend enough time in his to follow the thread that leads to that conclusion.

But eventually a machine, not even necessarily an AI, will be able to follow his logic and tell his peers whether it's worth going down his rabbit hole or not.

posted by localroger at 4:34 PM on May 11, 2013 [4 favorites]

Just to give another example of a practical application of pure mathematics: do you ever use a GPS? The mathematics of general relativity is used in that. General relativity is a theory of physics, but it uses the language of pseudo-Riemannian geometry to model the interaction of matter and light and gravity.

posted by number9dream at 10:42 PM on May 11, 2013 [1 favorite]

posted by number9dream at 10:42 PM on May 11, 2013 [1 favorite]

The Foundations of Applied Mathematics - "Suppose we take 'applied mathematics' in an extremely broad sense that includes math developed for use in electrical engineering, population biology, epidemiology, chemistry, and many other fields. Suppose we look for mathematical structures that repeatedly appear in these diverse contexts — especially structures that aren't familiar to pure mathematicians. What do we find? The answers may give us some clues about the concepts that underlie the most applicable kinds of mathematics. We should not be surprised to find some category theory here."

also btw: "I don't know what these new Galois theories are good for, and I think Olivia said she doesn't either, since she just invented them earlier this year. But they subsume Grothendieck's Galois theory for algebraic varieties as a special case... and the whole idea seems pretty cool."

posted by kliuless at 6:00 AM on May 12, 2013

also btw: "I don't know what these new Galois theories are good for, and I think Olivia said she doesn't either, since she just invented them earlier this year. But they subsume Grothendieck's Galois theory for algebraic varieties as a special case... and the whole idea seems pretty cool."

posted by kliuless at 6:00 AM on May 12, 2013

*Any chance that Mochizuki is related to "Nakamoto"?*

As implausible at it is, how awesome would that be? Reclusive Japanese mathematical supergenius casually invents Bitcoin and proves the ABC conjecture. Add William Gibson to localroger's list.

It's totally irresistable to the credulous in the way of the best conspiracy theories. "wikipedia says he wrote papers about eliptical curves. The basics of encryption.... OMG", "5. Same number of Hiragana symbols in first and last names of the two identities."

posted by jjwiseman at 12:14 PM on May 12, 2013

jiawen:

Ted Nelson deduces Mochizuki is Nakamoto.

posted by bukvich at 2:17 PM on May 18, 2013 [1 favorite]

*Any chance that Mochizuki is related to "Nakamoto"?*Ted Nelson deduces Mochizuki is Nakamoto.

posted by bukvich at 2:17 PM on May 18, 2013 [1 favorite]

*5. Same number of Hiragana symbols in first and last names of the two identities.*

Okay.. that's probably true for about half of all Japanese names.

*Ted Nelson*

This is like a confluence of crazy.

posted by charlie don't surf at 3:11 PM on May 18, 2013

"I know who 'Satoshi Nakamoto' is, says Ted Nelson", from The Register:

Australian writer Stilgherrian told The Register that while it'd be easy to dismiss the claims, in spite of his eccentricities, Nelson "has the annoying habit of being right."posted by jjwiseman at 9:02 AM on May 19, 2013

Stilgherrian also noted that Nelson's long digression into Holmes-and-Watson may be a red herring with a nugget of data:

"He makes a big deal of Sherlock Holmes being able to come up with the right train of logic because the author already knows the answer he's working towards. Well, Nelson presents his logic, but does he already know the answer through other means? That certainly seems to be what he's implying."

Just watched Ted Nelson's video for the first time, and I have to say I agree both with the idea that he's a guy you blow off at your peril, and that he knows more than he's telling. I would guess that he at least attempted to contact Mochizuki and might have gotten a response crafted to let him know the truth without giving him a way to prove it to anyone else. His certainty that "Nakamoto" knows he will eventually be outed suggests more than a casual guess at work too.

Annoying as one might find Nelson (especially if it's your work he is calling crap because you're not using his methodology) he does in fact have a habit of being right. I've taken to giving people his the URL for Computers for Cynics whenever I'm asked why software is so generally crappy (which I get asked a lot, because I have a reputation for writing software that works in my little niche). He's certainly right that there is a problem, if not that he has a solution for it, and his observations on the rise of Apple and Microsoft are in particular spot-on.

posted by localroger at 4:29 PM on May 19, 2013

Annoying as one might find Nelson (especially if it's your work he is calling crap because you're not using his methodology) he does in fact have a habit of being right. I've taken to giving people his the URL for Computers for Cynics whenever I'm asked why software is so generally crappy (which I get asked a lot, because I have a reputation for writing software that works in my little niche). He's certainly right that there is a problem, if not that he has a solution for it, and his observations on the rise of Apple and Microsoft are in particular spot-on.

posted by localroger at 4:29 PM on May 19, 2013

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Inter-Universal Geometer.posted by klausman at 3:00 PM on May 10, 2013 [3 favorites]