In my salad days, when I still thought of myself as a mathematician-in-training, I would have been fascinated by this (Caroline Chen writes about Shinichi Mochizuki’s alleged proof of “a famed, beguilingly simple number theory problem that had stumped mathematicians for decades”; nobody has any idea whether it’s correct because it would take months or years to understand it). As things stand, I’m mildly interested, but what really grabs me is the OED’s latest appeal to the public: “A number of quotations in the OED derive from a book with the title Meanderings of Memory. However, we have been unable to trace this title in library catalogues or text databases. All these quotations have a date of 1852, and some cite the author as ‘Nightlark’. The only evidence for this book’s existence that we have yet been able to find is a single entry in a bookseller’s catalogue…” If you’ve ever seen a copy of this book, please let them know! (New Yorker, Guardian, MetaFilter)
Also: Don’t make fun of renowned Dan Brown! (Warning: May cause uncontrollable laughter.)
Renowned author Dan Brown: Last time this author was discussed (perhaps by Geoff Pullum at LLog) it occurred to me that the reason why many people like his style is that they are used to the journalistic practice of packing as much information as possible into initial sentences and repeating some of the same information in subsequent paragraphs, while good fiction writers tend to omit details which do not contribute substantially to the story or to the picture of a character, or to dole them out little by little as they become more important.
That was a very funny piece, but why should anyone think there might be a connection between good writing and the number of books sold by a current author? Of course Dan Brown is crap. According to the accountants in charge of selling stuff, the public prefers crap. And I hate his stupid tweed sports jacket with the ugly pattern and wide lapels too,
I like how Caroline Chen’s example of “huge numbers” is 3,072 + 390,625 = 393,697.
Dan Brown is worse than Jeffrey Archer: discuss.
You’d need to read them to discuss it in detail, dearie, but unlike Lord Archer Dan Brown hasn’t done a stretch in prison yet.
The mention of Grothendieck in the item on the ABC conjecture is a warning flag for me…
http://en.wikipedia.org/wiki/Alexander_Grothendieck
Note the last sentence of the first section:
“He formally retired in 1988 and within a few years moved to the Pyrenees, where he currently lives in isolation from human society.”
On the other hand, more seriously, scanning the comments here:
http://quomodocumque.wordpress.com/2012/09/03/mochizuki-on-abc/
one sees that Terence Tao takes Mochizuki very seriously, and that’s a big deal.
There’s no way in which mentioning Grothendieck is a warning sign. Mochizuki had to learn his stuff because everyone who uses algebraic geometry has to learn his stuff. The fact that he’s a gigantic weirdo who eventually fled from human contact is neither here nor there.
@Walt
I was joking, and I can see where algebraic geometers would find the joke inappropriate. I’ll go back now to the comforts of the world of spherical cows.
I think Mochizuki may be, as Dan would put it, Patient Zero in renowned new field of psychomathematics. Perhaps we could say further that Ramanujan was Patient -1, like those guys who died of AIDS back in 1959.
marie-lucie is right — Geoff Pullum was the first to do the “renowned author” bit, on Language Log here and here. And it’s been ripped off before — see here.
Mochizuki’s caper has the advantage of having provoked a few mathematicians into making statements about their field that you don’t often get to hear. Proofs as social constructs … From the article by Chen:
Pretty reasonable, doesn’t one think ? I notice that the article’s author, and both mathematicians quoted, are women. Pretty irrelevant, doesn’t one suspect ?
Thanks for the Geoff Pullum links, Ben Zimmer. I’m disappointed he didn’t discuss “His irises were pink with dark red pupils”. Now I’m wondering if the illustrated edition shows any of Van Gogh’s irises.
Dan Brown is worse than Jeffrey Archer: discuss.
Surely Dan Brown took his prose style not from Jeffrey Archer himself but from the caricature version of Archer on the satirical puppet TV show Spitting Image. According to this website:
[Puppet Archer] often spoke in literary terms, starting a sentence and then adding “said the successful author/politician/…”.
Top Italian poet Dante!
I was indeed in a state in which tears were falling from my eyes- when I read in the renowned Telegraph about Dan Brown’s morning!
Thanks LanguageSteve
–EK
Do not taunt Angry Dan Brown. 🙂
The Dan Brown article is utterly hilarious. There’s a lot of snarky cleverness going on there (“renowned deity God,” “the 190 pound adult male human being nodded his head to indicate satisfaction”), but I burst into laughter at the quiet brilliance of:
“Thanks, John,” he thanked.
marie:
…while good fiction writers tend to omit details which do not contribute substantially to the story or to the picture of a character, or to dole them out little by little as they become more important.
Or — gasp! — leave them unstated, so their readers can — double gasp! — actually figure out the hints and allusions for themselves, and come to their — renowned gasp! — own conclusions.
Sorry, MattF, I missed that you are joking. To think, I delurked here solely to dishonor myself and my family.
GrumblyStu, that struck me as well. It’s easy to misconstrue, though. The intent is that proof is an objective standard, but in practice, proofs leave steps that are “routine”. This is necessarily subjective.
Walt: To think, I delurked here solely to dishonor myself and my family.
Haven’t we all!
There is no dishonor in making mistakes, and there is certainly no dishonor in failing to see jokes.
Walt: The intent is that proof is an objective standard, but in practice, proofs leave steps that are “routine”. This is necessarily subjective.
It does not appear, from the quoted accounts of Mochizuki’s publications, that people are complaining about the high percentage of routine steps that he left to his readers – whatever one’s take is on the meaning of routine in this context.
I’m rather puzzled by your appeal to “intent” and “necessarily subjective” features of what you describe as objective standards. I would suggest that if one refrains from reliance on the notions of objective and subjective, there is nothing left that needs intent to explain it away.
To say that proofs are social constructs is to make a very abbreviated statement about a complex matter. For one thing, the expression does not imply any claim that such constructs are “arbitrary”, although many people think that it must do so, and thus reject the idea as ridiculous.
Objective and subjective are notions tainted by centuries of fruitless philosophical squabbling. I find that something like “transpersonal, methodically reproducible results” is a more accurate description of what mathematical proofs should provide – as in all of science. (It is no accident that I do not list “consensual” here.)
In other words, many people seem to feel that Mochizuki’s latest work is unscientific.
I put “routine” in “scare quotes” to be “scary” about the “concept”. To take a specific example, after Perelman’s manuscript for the Poincare conjecture was circulated, the Poincare conjecture was not considered proven until other people had fleshed out the proof to the point that the gaps were “routine”. Mochizuki’s work seems to be hard to understand because he introduces a bunch of auxiliary concepts, which makes it hard to read, and hard for other people to see where the gaps are, and whether they can be filled.
I disagree with your philosophical point — a mathematical theorem is true if it follows from its assumptions using correct mathematical reasoning. This standard is sufficiently rigid that a fully-specified proof can be checked by computer. Many theorems, such as Godel’s Incompleteness Theorem, have been checked in such a way. It’s just it’s such a boring activity to specify a proof in such detail that almost no one ever does it.
I disagree with your philosophical point — a mathematical theorem is true if it follows from its assumptions using correct mathematical reasoning.
I said nothing about the “truth” of theorems, in fact I deliberately avoided the word. It would be nice if one could treat “true” as a superfluous synonym for “provable”, but of course the first incompleteness theorem put paid to that.
There’s nothing wrong with using a computer to check proofs. But who checks the programs, and whether the computer is functioning “correctly” ? The AI people. Who checks the AI people ? Other mathematicians. Adherence to methodological principles is what is verified, not “truth”.
I’m curious: what do you think is the cash value of the word “true” over and above that of the word “provable” ? Apart from the special meaning “true” has in model theory, where it is compatible with “not provable”.
Maybe Mochizuki should take a tip from Dan Brown and try to pack as much information into the first sentence as possible.
Listen, I think I’ve got it: multiple-choice Dan Brown.
Renowned
crossword compiler/mathematician/liquor salesman
Genevieve Lafontaine/Jamie Snodgrass/Bert Rump
staggered through the vaulted archway of the
museum’s/olympic stadium’s/local delicatessen’s
Grand Gallery. He
lunged/thrust/dived
for the nearest
painting/wedge of aged cheese/academic paper
he could see, a
ten-year-old Port Salut/Caravaggio/treatise on Boolean logic.
Grabbing the
gilded frame/sticky rind/stapled endpapers,
the
twenty/seventy/ninety
-six-year-old
man/woman/undisclosed
heaved the masterpiece toward himself until it tore from the
wall/desk/countertop
and
Lafontaine/Rump/Snodgrass
collapsed backward in a
heap/mound/jumble
underneath the
bar/buffet/taproom
of the
public house/nineteenth hole/after-hours club.
Renowned philosopher/logician/social critic Bert Rump Russell ?
I should have said “proven” instead of “true”, since the question about the difference between the two is somewhat besides the point I’m trying to make.
It’s true that we can’t guarantee absolutely that nobody is making a mistake anywhere, but proof checkers can be pretty short, and it’s not that hard to write your own.
In practice mathematicians write down arguments and other mathematicians read them and understand them, or not. If the arguments are such that nobody can understand them, then (if the author really has something good to say) they need to be rewritten.
The idea that by writing down every detail the proof could be made checkable by machine has a certain appeal, but it loses some of that appeal when you contemplate the difference between the relentlessly critical but fundamentally stupid way that a machine reads things and the way an intelligent person reads things.
My hunch (and this should be taken with tons of salt because I know nothing about Mochizuki’s work) is that this is great stuff, and that sooner or later someone will expound it in a way that makes it accessible to the rest of the mathematical world.
I think we are talking pretty much at cross-purposes. To be sure, we are both addressing the subject of showing mathematical proofs to be correct (or, in the case of Mochizuki, intelligible). However, you are rehearsing techniques for demonstrating correctness, whereas I am reflecting on the notions of technique and of demonstrating correctness.
You probably suspect that I am sneakily aiming to claim that “there is no such thing as correctness in mathematics”. Or: “people can make mistakes in proofs, so we never know for sure whether a proof is correct”. But no, I would never make such silly claims. The problem seems to be that we understand “correctness” and “mistake” in completely different ways.
If I understand your last comment, for the moment you no longer insist that mathematical theorems must be true, but only that they must be proved. This discussion started out from the idea that proofs are “social constructs”. But proofs are not merely social constructs, you may well object. They have been proven, so they’re reliable, valid and even true (that word again !).
If I now inquire about how proof techniques are proven to be valid, you will refer me to the proof techniques of mathematical logic and model theory. More techniques, more social constructs !
Is there circularity here ? Yes. Is that a bad thing ? No. All results of mathematical logic taken together are circular, because they are conditionals, i.e. relative statements: “If X is consistent, then X + Y is consistent”, or “We must be careful, because with this cardinal we are operating on the brink of inconsistency” (a remark I happen to remember by Sierpinski in his set theory book, of which I read a very small fraction). They cannot be anything but conditionals, as Tarski and Gödel showed. I wrote above that “true” and “not provable” are compatible: all I meant was that there are statements not provable in a language L, but provable in a metalanguage L* in which L has been embedded.
proof checkers can be pretty short, and it’s not that hard to write your own.
Someone should write a proof checker for Mochizuki’s latest publication. It shouldn’t be that hard, and would obviate discussions about what that publication means.
Of course, if you are a Martian mathematician (per Kripke), you are concerned with truth rather than proof. In that case you are more interested in what results you can deduce from Riemann’s Theorem rather than obsessing about the fact that you don’t have a proof for it yet.
Why Martians ? Sure are a lot of them running around. I thought Martians and zombies were the weird but thought-provoking bad guys in philosophical anecdotery.
“I am concerned with truth rather than proof” is one statement. An equivalent statement is “I am concerned with what would follow from a statement I assume to be provable”.
In scientific matters the notion of “true” can serve as a mere abbreviation of “proved, or likely to be proved”. Anything more than that drags in unjustifiable, contentious, overweening, tiresome old metaphysicalities that don’t never done nobody no good.
Dunno about Mochizuki, but the usual problem is, rewriting a proof to fit the strict Bourbakiesque metalanguage of a proof checker is hard. It’s only then that you are reminded how much hand-waving, how much “abuse of language” you are introducing in even the most trivial of your mathematical writings.
It’s only then that you are reminded how much hand-waving, how much “abuse of language” you are introducing in even the most trivial of your mathematical writings.
Not waving but drowning. Actually, hand-waving is not such a bad thing, provided somebody waves back, however timidly. And I just love to abuse language, especially when it deserves it. If I were English, I would be a notorious Nurse.
Stu: I don’t think you’re trying to sneak in anything. My point is rather that people yearn to misconstrue the idea that proof is a matter of social convention as trying to sneak something in. You can see that in the comments at MathBabe, where O’Neil makes the same point, where people misread it in exactly that way. (There used to be more such comments, but I think they got deleted for being so hostile.)
I don’t agree that “X is a theorem of this formal system” is a social construct — it’s an objective fact, in that I can provide a checkable formal proof that X is a theorem of the system. This is one of the great triumphs of late 19th century and early 20th century mathematics. Before then, people would ask things like “Is hyperbolic geometry real?” or “Are complex numbers real?” Now, hyperbolic geometry or complex numbers are just formal systems, and asking if they’re “real” is besides the point.
What’s a matter of social convention is “Have I provided enough evidence that you’ll take my word for it that X is a theorem of this system.” In theory, if you don’t believe a step in my proof, you can ask me for more detail. The convention is how much work I’m allowed to expect you to do.
Any proof checker would reject Mochizuki’s proof, because he has not provided a formal proof.
To save Stu the trouble: he doesn’t believe in “objective fact.”
Hat, it just seems that Stu doesn’t believe in objective fact.
empty: Hat, it just seems that Stu doesn’t believe in objective fact.
My subjective impression is that it is an objective fact that Hat thinks as he does. To be more objective, though, I might have to concede that Hat’s opinion is a subjective one.
So much for the notions of subjective and objective. Effing waste of time, folks. We might just as well discuss the attributes of God. A few hundred years ago, someone who refused to do so might be suspected of atheism – possibly unfairly, in some cases.
Many people cling for dear life to the notion of reality. They get tetchy and dismissive when someone suggests that it would profit us to just drop the idea and discuss something more worthwhile. But arguments against religious belief are seldom convincing. I think God died from lack of interest – people would rather talk about Desperate Housewives.
Walt: I don’t agree that “X is a theorem of this formal system” is a social construct — it’s an objective fact, in that I can provide a checkable formal proof that X is a theorem of the system.
If you insist on the word objective: is it your view that social constructs are not objective facts ?
Now, hyperbolic geometry or complex numbers are just formal systems, and asking if they’re “real” is besides the point.
Indeed: the notion of “formal system” arose as a way to avoid the notion of “real”. Similarly, the notions of provability and relative consistency serve to avoid the notion of “true” – in any sense other than “provable in a metalanguage”.
Huh, you may actually be misconstruing O’Neil’s comment in the way I was mentioning. I don’t think she was trying to endorse your philosophical position.
I’m not sure what you gain by not talking about ‘reality’. When we’re talking about reality we’re talking about something, and if we invent new terminology for it, we’ll just run into the same philosophical problems in the new language.
Marilyn vos Savant wrote a book on how Wiles’ proof of Fermat’s Last Theorem was wrong because it used hyperbolic geometry, which isn’t ‘real’. The reality of hyperbolic geometry doesn’t matter because there is a formal system that represents the theorems of hyperbolic geometry, and formal systems are real. Using hyperbolic geometry in a proof is unproblematic because it uses that formal system.
Whether or not there are mathematically true statements that are not provable is a subtle philosophical question, but at the very least provable statements are true.
I don’t know if the statements like “X is a social convention” are ‘objectively true’. Since any social convention is a vague generalization about a great mass of people, some of whom it won’t hold for, it would be hard to say. There’s probably a sufficiently precise statement that I would say that there’s an objective truth of the matter.
Grumbly:
My subjective impression is that it is an objective fact that Hat thinks as he does.
Yet it might be that he does not in fact think as he thinks he thinks. Thinking is mostly an unconscious process: “I think, therefore I have no access to the level where I sum.” (Douglas Hofstadter).
Walt: I don’t think there’s much doubt that social conventions are facts like other facts. Consider “Americans drive on the right side of two-way roads.” There is the occasional American who doesn’t, though typically not for long, and there are other similar carps like passing on a two-lane highway. But overall this is a perfectly testable scientific claim. Granted that it might have been otherwise, but that has the same status as saying that facts about humans aren’t really facts because they too are contingent, viz. on the evolution of humans at all.
For myself, I like best Henry Baker‘s revision of Kronecker: “Die Consen und Fixnumen hat der liebe McCarthy gemacht, alles andere ist Hackerenwerk.”
Let me just insert a teeny-weeny caveat: I use the expression “social construct” self-advisedly, rather than “social convention”. I didn’t invent it of course, but the German writers from whom I borrow it have good reasons for not saying “convention”, as it seems to me.
“Convention” (like its German counterparts) connotes arbitrariness and take-it-or-leave-it consensus. “Construct” primarily suggests a result of goal-directed behavior, but not necessarily anything consensual (conventional). Of course you can construe arbitrary constructs if you feel like it, but they won’t sell – I’m not buying, at any rate.
“Americans” is an imprecise way to express “people driving on roads in the U.S.” On occasions when I have driven in the U.K. (and certain former colonies thereof), I managed to *not* drive on the right at least a majority of the time, albeit probably not with quite as high a frequency as those who first learned to drive in a jurisdiction with the keep-left social convention. I imagine the same is true in reverse for most U.K.-licensed drivers let loose with rented cars on U.S. roads. (It is actually useful in such a situation to be driving a car with the steering wheel on the side you’re not used to, because that is itself sufficiently disorienting to make you more conscious of inappropriate-in-context social-convention habits that might otherwise be unconscious.)
To parody myself, anyone who drives in America is an American pro tempore, at least if they want to be. I myself don’t drive at all, which makes me a European manqué. (Note to m-l: this term is not derogatory in English.)
JC, You are a New Yorker. New Yorkers are different, at least with respect to driving.
I would not say that French manqué is derogatory. It can be, depending on the context. For instance, a “tomboy” is un garçon manqué, lit. ‘a boy that did not quite turn out right’, but this is usually affectionate rather than derogatory. Raté, a more colloquial word which means about the same literally, would be much more derogatory: un raté is the kind of man who showed promise as a child but was a failure as an adult (perhaps because parental expectations were too high, or he had no social skills to match his intellectual skills).
Thanks for the concept, m-l!
yours, le raté
Well, thank you for saying so, though real New Yorkers would say you have to be born here. Which I was not: I didn’t learn to drive for other reasons, a combination of bad eyesight and being too young for practical driver’s training in high school.
yours, le raté
Come on, “Minus”, you are overdoing it! There are no ratés that I know of among the Hatters.
Well, JC, regardless of the reason, you have obviously fitted well in New York.
m-l: Well, someone whose pseudonym means “absolute zero” clearly has a conceit of self-denigration.
JC: that’s my point.
JC: actually I had not caught the whole meaning. In French “un minus” is someone of very low intellect. But at the bottom there is hope: nowhere to go but UP.
Hmmm … You could say the same about someone whose pseudonym means “empty”.
Eager to learn renowned best-selling novelist Dan Brown’s secret from the horse’s mouth?
http://www.cbc.ca/books/2013/05/dan-brown-discusses-his-latest-book-inferno-on-q.html
I actually think of absolute zero as a place you can’t get to, an ideal, an ultimate, not rock-bottom at all.
Meanderings of Memory turns out to have its very own Wikipedia article, surprisingly (and a bit ridiculously) long and exhaustive; according to it, “Following the appeal to the public, another reference to Meanderings of Memory was found in an 1854 Sotheby’s catalogue, which rendered less likely the notion that the work might be a hoax by a nineteenth-century miscreant.” Strange that no copy has turned up.
Of course, if you are a Martian mathematician (per Kripke), you are concerned with truth rather than proof. In that case you are more interested in what results you can deduce from Riemann’s Theorem rather than obsessing about the fact that you don’t have a proof for it yet.
There’s actually a lot of theorems to the effect of “assuming Riemann’s Hypothesis, we derive…”, and many further ones doing the same with the generalized Riemann hypothesis, or some other inconveniently unproved conjecture, or even something like the continuum hypothesis that cannot be proven.
Arguably it just means “let’s add X to our system as an axiom and see what happens” (a few cases, such as large cardinal axioms, are specifically formulated that way).
I’ve vaguely heard of a case where someone who tried adding an unproven conjecture X to their system as an axiom proceeded to derive a contradiction, thereby disproving X.
I think Mochizuki may be, as Dan would put it, Patient Zero in renowned new field of psychomathematics. Perhaps we could say further that Ramanujan was Patient -1, like those guys who died of AIDS back in 1959.
Surely it goes back at least to Urquhart’s Trissotetras.
I actually think of absolute zero as a place you can’t get to, an ideal, an ultimate, not rock-bottom at all.
There’s an alternate formulation of temperature (I don’t recall its name offhand) that runs on k=1/T, where T is standard temperature; in that formulation, absolute zero comes out to positive infinity.
[EDIT: apparently it’s called thermodynamic beta.]
…I didn’t know there were known examples of Gödel’s discovery!
@January First-of-May, David Marjanović: The Continuum Hypothesis, since it is well known to be independent of the axioms of Zermelo-Fraenkel set theory, is frequently used as an axiom. However, I do not think it is actually an example of a statement that is indeterminate in Gödel’s sense. A Gödel sentence is one that is actually true of the standard integers, but it is unprovable in first-order logic, because it is false in some nonstandard model of Peano arithmetic. (Specific examples of these have been worked out, however.) The Continuum Hypothesis, concerned as it is with whether certain uncountable sets exist, is probably true or false depending on your model of second-order logic. (Second-order logic allows quantification over whole sets, which means that you have to have a universe of sets in mind from the very beginning.) Gödel himself actually proved (by finding a model in which they held, the Constructable Universe) that the Generalized Continuum Hypothesis and the Axiom of Choice were consistent with ZF. A few decades later, Paul Cohen developed the method of forcing to construct models of ZF in which they didn’t hold, thus demonstrating complete independence.
Regarding thermodynamic β, there is nothing special about which direction your temperature variable goes, so long as it is monotonic. kT and 1/kT = β both come up a lot. Frequently, the easiest way to calculate something is via a derivative with respect to β. Nobody actually measures temperatures using β, however; T does have the convenience in ideal gasses that it is directly proportional to the average kinetic energy. For systems at negative temperature (not colder than zero, but hotter than infinity), which can exist in systems with only discrete degrees of freedom, like spin systems, T may also be more convenient.
The name β comes from the fact that this quantity is introduced as the second of two Lagrange multipliers in one version of the calculation of the Maxwellian distribution of particle velocities in an ideal gas. There are many ways to find that distribution. You can show that it is the unique stationary (and thus equilibrium) solution of the Boltzmann transport equation. You can develop the canonical ensemble theory for a system exchanging heat with a bath at constant temperature. Or you can apply the microcanonical approach. This means finding the most probable distribution of velocities (and then showing that the most probable is actually overwhelmingly probable) for a gas with a fixed number of particles and fixed energy. This is a constrained maximization problem, so you introduce two Lagrange multipliers; α fixes the number of particles, and β fixes the temperature. When you ultimately makes the connection to thermodynamics, β turns out to be the inverse of the absolute temperature. α turns out to be β times the chemical potential. α is not so important for classical ideal gasses, but with quantum gasses with identical particles (for which the presence of one particle in a particular state affects the likelihood of another particle being in the same state), it is also needed in order to understand the thermodynamics.