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Easy as ABC
So for years, like all of you, I’ve been idly wondering if it’s true that For every ε > 0, there are only finitely many triples of coprime positive integers a + b = c such that c > d1+ε, where d denotes the product of the distinct prime factors of abc. I figure that’s a question that occurs to everyone in the small hours of the morning, now and again. Once, when I was stuck in the métro during a wildcat strike, I reckoned I’d figured out a marvellous little proof. But I scribbled it down on the back of a pack of Gitanes, shoved it into the Pile of Papers, and never found it again.
Anyway, in 2012, Shinichi Mochizuki had one of those restless nights, then quietly posted 2,000 pages of scribblings on his website:
In them, Mochizuki claimed to have solved the abc conjecture, a 27-year-old problem in number theory that no other mathematician had even come close to solving. If his proof was correct, it would be one of the most astounding achievements of mathematics this century and would completely revolutionize the study of equations with whole numbers.
But the thing is, like everyone else, I couldn’t make heads or tails of it. So I didn’t know whether to envision Mochizuki as one of the great geniuses in the history of mathematics or as a complete fraud and a lunatic (not that these categories are necessarily mutually exclusive).
The curious part is that it’s now late 2015 — and still, apparently, no one knows:
… Probably the first person to notice the papers was Akio Tamagawa, a colleague of Mochizuki’s at RIMS. He, like other researchers, knew that Mochizuki had been working on the conjecture for years and had been finalizing his work. That same day, Tamagawa e-mailed the news to one of his collaborators, number theorist Ivan Fesenko of the University of Nottingham, UK. Fesenko immediately downloaded the papers and started to read. But he soon became “bewildered”, he says. “It was impossible to understand them.”
Mochizuki speaks fluent English, but declines all invitations to discuss his work in English. He won’t speak to journalists. He’s generally described as surprisingly “articulate and friendly,” but I figure the suppressed qualifier is “for a mathematician.” So that’s probably code for, “at least his fingernails don’t look like Grisha Perelmen’s.”
Mochizuki has replied to e-mails from other mathematicians and been forthcoming to colleagues who have visited him, but his only public input has been sporadic posts on his website. In December 2014, he wrote that to understand his work, there was a “need for researchers to deactivate the thought patterns that they have installed in their brains and taken for granted for so many years.”
To mathematician Lieven Le Bruyn of the University of Antwerp in Belgium, Mochizuki’s attitude sounds defiant. “Is it just me,” he wrote on his blog earlier this year, “or is Mochizuki really sticking up his middle finger to the mathematical community?”
Recently, a few mathematicians have finished reading his proof — and they claim it’s solid. Fesenko finally got through it: “Fesenko has studied Mochizuki’s work in detail over the past year, visited him at RIMS again in the autumn of 2014 and says that he has now verified the proof … ‘We had mathematics before Mochizuki’s work — and now we have mathematics after Mochizuki’s work, Fesenko says.”
But here’s the really weird part:
… so far, the few who have understood the work have struggled to explain it to anyone else. “Everybody who I’m aware of who’s come close to this stuff is quite reasonable, but afterwards they become incapable of communicating it,” says one mathematician who did not want his name to be mentioned. The situation, he says, reminds him of the Monty Python skit about a writer who jots down the world’s funniest joke. Anyone who reads it dies from laughing and can never relate it to anyone else.
Mochizuki wrote a progress report last year. He notes, in a more-in-sorrow-than-in-anger tone, that “the most essential stumbling block” to appreciating his work
lies not so much in the need for the acquisition of new knowledge, but rather in the need for researchers (i.e., who encounter substantial difficulties in their study of IUTeich and related topics) to … start afresh, that is to say, to revert to a mindset that relies only on primitive logical reasoning, in the style of a student or a novice to a subject.
He hints in the paper that if you merely study it haphazardly, you’ll never be able to understand it. As his collaborator puts it,
… if you attempt to study IUTeich by skimming corners and “occasionally nibbling” on various portions of the theory, then you will not be able to understand the theory even in 10 years; on the other hand, if you study the theory systematically from the beginning, then you should be able to understand it in roughly half a year.
Roughly half a year? What do you think? Worth the investment? It’s at least plausible that this is one of the century’s great intellectual achievements, right? Six months — that’s not such a huge investment. I’ve spent longer waiting for a guy to call me back.
What about you? If you could take six months to do anything in the world you wanted to do, would you consider devoting the next six months to seeing whether this was worth reading?
Doesn’t it make you curious?
Published in General, Science & Technology
Not really. Was there any real doubt about the conjecture being true? While I know that isn’t the same thing as proof, when you have “pretty sure that it is true” a 2000 page proof isn’t nearly as impressive as a 20 page proof.
However, the article states that to do the proof he “invented a new branch of his discipline” – that is kind of impressive.
*shrug*
People act as if there’s no controversy in the foundations of mathematics, and never has been. But that’s the purest ray of <strike>BS</strike>revisionist history in mathematics. Some mathematicians, myself included, are not happy with the freewheeling approach to infinity in Cantor’s set theory. More generally, some logicians, myself included, are not happy with the “if it’s a contradiction for it not to exist, it exists” foundations of classical logic. A lot of heavy lifting is underway to refound mathematics on a constructivist basis. This is what I’m referring to when I say “computer science will save mathematics, whether mathematics wishes to be saved or not.”
So what bothers me about this story is phrases like “papers,” “read,” and “verified the proof.” I’m skeptical, because I remember the story of Fermat’s Last Theorem. I don’t understand why people are publishing formal mathematical results that have not been formalized and checked with a proof assistant, as the Four Color Map Theorem and Odd-Order Theorem were. We’re well past the point where human beings should be thought capable of correctly proving any interesting mathematics in a thousand or more pages, especially when there are challenges of a foundational nature in accepting the results. Formalize it on the computer, let the computer check it, then let the critics (if any) make the case that Coq’s, or Isabelle’s, or… logic is unsound (and good luck to them).
Not about that. I’m still working out the three-dimensional nature of time and the contradictory premise that time isn’t a fundamental aspect of the universe, but a fallout of other characteristics of it, beginning with Planck’s Length and the relationship among time, temperature, and motion.
Eric Hines
There are lots of things I’d like to spend 6 months studying, but none of them involve numbers. The photo attached to your post represents my number literacy. But then maybe that makes me a good candidate to study the proof–if I had the slightest interest that is. Oh, or aptitude.
It certainly isn’t as elegant or as beautiful. What’s odd about it is that usually mathematicians get all mystical about the stuff that’s beautiful, not stuff that’s all twisted and Daedalian, like this. But they’re giving off the telltale hints of having seen the sublime, aren’t they?
I don’t think I’d be too tempted to read through his proof, but math was always my weakest subject. (And, of course, I became an astrophysicist, so maybe I have a thing for challenges.) But I do enjoy math now. I wonder what the starting point would be for the knowledge needed to understand his proof. What’s the minimum you should know but not have to “relearn,” as he put it.
Wasn’t this the plot of Neal Stephenson’s Anathem?
Forgive me for even posting on this one – totally confusing – and I have no response, (complex math has made my head hurt since 9th grade), except that I was curious about the story because of the little primary-colored alphabets in the picture, which reminded me of the new name for you know who……and speaking of numbers, what does the name Google mean (before it was Alphabet)? So I will bail out on this story but hope someone at Ricochet can run with this next one……..
http://qz.com/520652/groundwork-eric-schmidt-startup-working-for-hillary-clinton-campaign/
But the author maintains that it is beautiful and elegant. He just insists the only way to apprehend it is from first principles.
Look at it this way: the logic of Russell and Whitehead’s Principia Mathematica is considered to be elegant and beautiful. It took them ~450 pages to prove 1 + 1 = 2. Here’s the proof script in Coq:
Coq < Goal 1 + 1 = 2.
1 subgoal
============================
1 + 1 = 2
Unnamed_thm < trivial.
No more subgoals.
Unnamed_thm < Qed.
trivial.
Unnamed_thm is defined
Coq < Print Unnamed_thm.
Unnamed_thm = eq_refl : 1 + 1 = 2
This is possible because, by default, Coq provides access to a boatload of definitions and theorems about the natural numbers, and provides proof tactics, like the slightly-snarkily named “trivial,” for discharging… well… trivial… proof obligations, and simplifying “1 + 1” according to the logical rules of Peano (well, Heyting, since Coq is constructivist) arithmetic, then using the fact that equality is reflexive is indeed trivial.
The author of the reported proof is offering the 450 pages in Principia Mathematica, claiming the standard mathematician’s tool belt doesn’t provide what he’d need to just say “eq_refl : 1 + 1 = 2.” It makes perfect sense to me.
GGG, I read somewhere Bertrand Russell had spent a very great deal of time trying to prove 1+1 = 2, never quite succeeded, and never quite managed to get past it.
True or urban legend?
Meanwhile in other mathematical news:
Given: Barney is a CUTE PURPLE DINOSAUR
Prove: Barney is Satan
The Romans had no letter ‘U’ and used ‘V’ instead for printing, meaning the Roman representation for Barney would be:
Extracting the Roman numerals, we have:
Decimal Equivalents are:
Adding those numbers produces: 666
666 is the number of the beast.
Therefore, Barney is Satan.
I was well along in the pursuit of a PHD in math some umpty-ump years back and decided the frontiers of mathematics were way too detached from what seemed to me to be useful knowledge, and quit – to go into fields where math had tremendous analytical power. This ‘problem’ is of a piece with the kind of mathematics I recoiled from. I to this day wonder if we aren’t wasting our tax dollars funding math research on such topics.
Urban legend. Principia Mathematica succeeded, if you accept the logical system it describes and employs. No one uses that system today—half because it left some surprising things unformalized, half just because, contrary to popular opinion, mathematical logic isn’t a static discipline, and the discipline has moved on since the mid-1930s. The proof assistant I have the most experience with, Coq, relies on a logic called the Calculus of Inductive Constructions. It’s a much simpler, and also more robust, logic than that used in the Principia Mathematica. As I alluded to before, Coq has been used to formalize and check the proof of the Four-Color Map Theorem as well as to do more practical things, like certify that the C source code the CompCert compiler compiles means the same thing as the machine code it generates (i.e. the compiler is bug-free). It’s the proof assistant I used to create my recent Formal Logic Undressed workshop and screencast.
Let me highly recommend you have a look at Davis Hestenes’ renewal of the appreciation of Clifford Algebra, which, following Clifford himself, Dr. Hestenes prefers to call “Geometric Algebra.” Dr. Hestenes has also done original work extending Geometric Algebra to Geometric Calculus. Because Dr. Hestenes is a physicist rather than a mathematician, he shares a thoroughgoing emphasis on applications. This page links to a number of excellent introductory resources, starting with Dr. Hestenes’ 2002 Oersted Medal Lecture.
To be clear, I have very little patience with pure mathematics myself. In particular, Zermelo-Frankel set theory with the Axiom of Choice is utter bollocks, pure or applied. But digging into that is, as they say, out of scope for this thread.
My favorite example of this is “Kepler’s Conjecture”, which was proposed by the astronomer Johannes Kepler in his 1611 treatise Strena seu de nive sexangula (“A New Year’s present; on hexagonal snow”). This little book seems to contain the first printed discussion on why snowflakes always have 6 sides–why not 5 or 7, if it happens by chance? Kepler thought that their regularity might stem from the geometrical arrangement of minute and equal brick-like units. Further reflection on this led him to think of close-packed spheres, and how they might be ordered in space. He describes the arrangement we now know as cubic close packing (e.g., the arrangement that occurs when cannon balls are stacked, or layers of oranges are placed in a box), and asserts: This arrangement will be the tightest possible, so that in no other arrangement could more pellets [or cannon balls, etc.] be stuffed into the same container.
This was intuitively reasonable, but the proof of it was difficult to come by. Carl Gauss (1831) proved that the cubic close packing is the densest arrangement of spheres in a regular lattice, but there always remained the possibility that some irregular arrangement might have a higher density. In 1998, Thomas C. Hales, a mathematician then at the University of Michigan at Ann Arbor, finally submitted a 250-page proof (with 3 GB of accompanying computer files) claiming that “Kepler’s Conjecture” is correct. It was a proof by exhaustion, and was published only after 4 years of checking by a panel of reviewers who were “99% sure” it was correct. He subsequently worked on a formal proof of the theorem (checked with computerized proof assistants); a description of it, in a 21 page paper, was published in January of 2015.
Six months to study something that has nothing to do with the price of tea in China, just to find out if it’s a fraud or not?
No thanks.
What does David Berlinski say about it?
I doubt this.
First of all, there are already several formal languages up to the task. I’ve mentioned Coq, but there’s also Isabelle, Mizar, HOL, HOL-Light… and several of these have already had a fair amount of mathematics, and other things, formalized with them.
In 2005, there was a comparison of 17 (!) proof assistants, their features and styles, and direct comparison of their formalizations of the irrationality of the square root of 2.
In other words, I don’t think the issues are technical. As usual, I think the issues are social, and insurmountable. Mathematicians just aren’t using proof assistants. And even if they want to, which one to use is a hard decision, and becoming expert at it can consume years of your life.
Minor quibble: no, it isn’t. :-)
It’s impossible with a predetermined, fixed axiom set (ZFC, for example). There’s nothing at all stopping anyone from offering axioms on a per-subject or even per-proof basis. (Of course, the user of the proof(s) has to accept the axioms.) One of the interesting things about tools like Coq and Isabelle (the two systems I’m most familiar with) is that they deliberately support the development of entire logics: first-order, higher-order, linear, modal… so the limit on developments in them is purely a matter of how much mental elbow grease you’re willing to apply.
Not sure the worth of this project. One of his other projects very much appeals to me though, his Computer Based Math (CBM instruction) initiative. If the Right wants to disentangle the State from everyday life, no better starting point exists than making computer aided instruction – from kindergarten up through college – accessible for peanuts to everyone in the US as a substitute for the public schools. We need philanthropists like the Koch brothers to finance something like this.
(Disclaimer: I use Wolfram’s Mathematica application extensively)
A great place to start would be financial support for SageMath Cloud, which loses its author money every month.
Next, someone needs to integrate Sage and MathBox2.
Yes, that’s the part that’s like catnip to me. Mind you, he’s the only one who says this — others are either coming away saying, “I don’t get it,” or so thunderstruck that they can’t explain it. Or saying other strange things about it, such as (admittedly in Mochizuki’s own words), “One aspect of these seminars with Sa¨ıdi that left an impression on me was the serenity of his demeanor as he emphasized on various occasions, with regard to the dissemination of IUTeich, the importance of maintaining a patient, long-term stance.” (He’s referring to visiting professor Mohamed Sa¨ıdi, from the University of Exeter, not the Persian mystic and poet, although it sounds a bit mystical and poetic, doesn’t it?)
The part that so tempts me, I think, is the idea that this could indeed be as beautiful and as elegant as he says, and that it can only be learned from first principles. In other words (in my fantasy; I truly have no idea), I would be at no disadvantage against the world’s great number theorists if I started trying to understand it now; or at least, they’d be as disadvantaged as I am. I’m asking myself why that matters, and I suppose the answer is that there are many things it’s just silly to begin studying at my age — it’s just too late to imagine, say, that I could take up the violin in the expectation of anything more modest than producing, at best, a sound that doesn’t painfully offend — but it seems the entire world’s going to have to start at the beginning with this. And if it’s as significant as it might be, wouldn’t it be quite amazing to be one of the first people ever to understand it?
Moot point, of course, because I can’t cloister myself off to study this for six months (or even a decade); and if I could, I’d put learning many other things ahead of it, given that life’s finite. But if I had world enough and time, figuring this one out might be on the top-fifty list.
No one is. That’s the mystery. And the only way to know is to learn it. Given his reputation, no one can write this off as mummery without at least trying to learn it. But given the strangeness of his claims about it, it seems to me there’s at least some chance he’s simply gone off the deep end. No one’s willing to say the emperor has no clothes, because who knows, he might be right, and maybe it’s just too hard. (I find an admirable modesty among all the people who are saying, “I just don’t understand it.” That’s quite atypical of our age.)
If it’s any comfort, the most accomplished arithmetic geometers in the world feel exactly the same way about this one. That’s exactly why I find the story so appealing. If he’s right (and if I’m understanding what he’s saying about it, and mind you, I may not be), they have to un-learn everything they thought they knew and start fresh.
He finds it fascinating too — but of course he isn’t among the four-odd people who claim to have fully understood it.
I’m not really good at emojis, but I reckon you wanted the one that means, “I’m spinning in my grave.”
Maybe like this? @@@@@@
I think my next book review needs to be of this, although (as always) Jaynes‘ broadsides are funnier.
GGofG,
0 < 1 < Aleph 0 < Aleph 1 < Theta < Rho < Omega < weeeeeeeee eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee!!!!!!!!!!!!!!!!
Sorry.
Regards,
Jim
1,2,3,…ω,ω+ 1,ω+ 2,. .. 2ω3ω4ωω^2
Then you keep going. 5ω^2 + 8ω + 96! And then much later you get to ω cubed! And then eventually ω to the fourth. You keep going and why stop there? This sequence goes on forever, but let’s put something after all of those. So what would that be? That would be obviously ω to the ω. This is starting to get interesting! Then you keep going and you have ω to the ω to the ω. This is a pretty far-out number already!
1, 2, 3, …ω,ω + 1,ω + 2, …2ω3ω4ωω^2ω^3ω^4ω^ω^ω^ω^ω
You can see why this is becoming theological. This is the mathematical equivalent of drug addiction.
Gregory J. Chaitin. Exploring Randomness (Kindle Locations 112-115). Kindle Edition.
No need to be sorry! You immediately reminded me of this great quote from Gregory Chaitin, so you’re in excellent company!
GGofG,
Didn’t you forget tetration…etc. Anyway, why do we need all those intermediate steps? I just like the big jumps. Well at least it relieves all the worry about the cost of the Many Worlds Hypothesis. I mean even if you use infinite many dimensions at say Aleph 1 per dimension, once you’ve driven the cost down by employing the higher transfinites you can have the whole thing for bupkiss.
Why Claire’s Macbook Air would cost more.
Regards,
Jim