Have any major conceptual errors been found in math journals?

[andytoh]

I was wondering if anyone has ever spotted a mistake in any math journal before. Of course, I don't mean typographical errors, but conceptual errors like a flaw in a proof that had slipped by all the editors and referees. Since these works are all original, they have not been written off another source, so the accuracy of the proofs rest entirely on the specific author of the article.

 

[mathwonk]

i know of one instance, in a book, by camille jordan, classifying simple groups of small order. he missed the simple group of order 168, which was then treated by klein in a subsequent paper.

there is also a famous case of a theorem by lang in his algebra book, first edition, incorrectly generalizing a result of artin, which was corrected in a paper by ? i forget his name now (james cannon?) a famous topology student from utah.

another famous case is a false proof by someone that the 6 sphere has an almost complex structure.

fourier also apparently announced an incorrect result about existence of convergent Fourier expansions for overly arbitrary functions.

lefschetz is famous for, to exaggerate; "never having stated a false theorem nor given a correct proof". e.g. his proof of the "hard lefschetz theorem" is apparently nonsense. also hodge's proof of the hodge theorem is said to be inadequate.

fermat escaped this charge by usually not publishing his own arguments at all. and almost surely he deluded himself in thinking, and announcing, that he could prove his "last theorem".

zariski's book on algebraic surfaces is prefaced with the remark that he wrote it because he came to believe that many if not most of the basic results of the great Italian school of algebraic geometry of the the early 20th century, were incompletely proved.

zariski himself is credited with a famous "conjecture" later proved perhaps by harris, but which zariski had originally considered a theorem. this is also the status of a result once known as severi's conjecture.

andrew wiles' first proof of fermat was famously incorrect.

i know other cases as well. Usually the theorem is true and the proof is inadequate, but sometimes the statement is false.

yes people do make mistakes. notice too these are very fine mathematicians, most of whose work is correct.

so the moral is not to be afraid to make mistakes. the point is to make progress, not to always be right. as someone wise once said, it is not true that mathematics, historically speaking, is free from contradictions and errors. rather this is an ideal to be achieved, not a given fact which holds in advance.

 

[andytoh]

Thanks for your input Prof. Mathwonk. I noticed you mentioning false proofs in "books" and "editions" (implying "textbooks"). Did the theorems in these books you mention appear for the first time in these books? or are they theorems already proven by someone else but then another writer wrote his own proof that turned out to be flawed? I always thought that newly discovered theorems are first published in journals, and then they appear as theorems in textbooks only after they have shown to be useful. I know that many textbook writers write their own proofs to famous theorems and perhaps make an error in so doing, but that was not what I was asking about.

I was talking about a mathematician's original theorem, "proven" by him, then editted and refereed by other mathematicians, gained approval for journal entry, and then later on (after the proof has been published) somebody discovers that the proof is wrong. Or was that what you were talking about? I think most of what you mentioned are referring to original theorems (some of which turned out to be false altogether).

 

[andytoh]

After reading your examples again, I realize that the examples you've given are indeed orignal theorems with incorrect pushlished proofs. Which means that we cannot trust everything we read in those journals...

And how do these incorrect proofs escape the referees and editors of the journals? Is it because they read the proofs in a rush because they don't want to lose valuable time for their own research?

 

[robert Ihnot]

If you want to read about false proofs, try Dickson's, "History of the Theory of Numbers," and see how many mistakes were made on Fermat's Last Theorem.

I had a prof who complained that he did the work and solved the problem, but someone would get away with publishing a, "Cheap generalization," like the case n=0 had not been fully covered, or something similar.

 

[mathwonk]

all you have to do is look in journals for retractions, where people point out their errors and correct them if possible, or retract the results.

It is true as Robert Ihnot said, that errors by other people in journals are actually a source of rejoicing by some people, since then they get to correct them and thus obtain a publication.

As he implied, often also the corrected or augmented publication is less impressive, or less original, than the earlier flawed one.

I was talking about original theorems, not always in journals, as you noiced. If you consider incorrect proofs of old known thorems in books, these are legion. In fact I used to judge calculus books by whether or not they proved the fundamental theorem of calculus correctly, and many did not. I admit I was very critical, sometimes as petty as faulting them for not defining what they meant by an antiderivative thoroughly enough to cover closed intervals. The flaw usually involved not dealing with the endpoints of the interval properly.

Some of my best work, joint with Robert Varley, concerns finding and publishing correct versions of some falsely stated theorems by earlier workers. It was not so easy, since we had to find both the correct statements and the correct proofs. In fact my thesis dealt with giving a new argument for a result that had recently been retracted by someone. In this case also, the strict statement was false, and a weaker one needed. The most precise possible version of this result, the Torelli theorem for Prym varieties, is still conjectural after 30 more years.

As you know, Riemann's own proofs of many of his most famous results are considered "lacunary", such as the Riemann Singularities theorem, which was the subject of one of my early teacher's thesis at Columbia, and the general Riemann Roch theorem where he used the then unjustified Dirchlet principle, and of course the Riemann hypothesis remains unproved, but i am not sure if Riemanns own partial arguments there are considered flawed.

 

[mathwonk]

You have made a correct point, that one cannot fully trust results one does not understand. But in my experience flaws are usually discovered fairly soon, at least if the result is important enought to deserve close scrutiny. And mathematician's intuition is very good, so that the statements are more often correct than the proofs.

It is true that flaws may be found in the refereeing process, but refereeing is hard, unpaid, almost thankless work, and the responsibility for correctness still rests on the author.

I have found, usually not serious errors, but more often gaps, or insignificant errors, in papers I refereed, but only after months of close checking of every detail, something apparently no other referee had been willing to do before me. The author usually then quickly repairs it.

The problem is when the author says something like: "by the results of [26], we conclude that indeed all cases are dealt with." Then you look at [26] and find it is a 187 page difficult paper, with its own non specific references. Then you either write back and ask for a page reference, and wait another few weeks for an answer, or you just give him the benefit of the doubt and go on, or you could of course reject it for unclarity.

Just as authors make mistakes so do referees, and being anonymous, a referee has much less motivation to spend enormous amounts of valuable time on a paper. I have sometimes enjoyed spending large amounts of time and energy on a paper where someone does something I like and appreciate, but i could tell from the appreciative comments of the editor that my efforts are unusual.

I have often been helped by strong referees in the form of comments that help simplify a proof, or shorten an argument,a nd I also have tried to make comments which improve papers, but sometimes you just get it back with no useful comment, just a decision.

I have also had strict referees insist on our removing large parts of a paper as less interesting, only to have other authors years later reprove and publish precisely the removed parts, as especially interesting.

In many cases ideas and insights are more interesting and rare than correct proofs. Once someone has discovered an interesting, correct phenomenon, there are usually people found fairly quickly with the technical skill to prove it, but not always.

 

[andytoh]

If you consider incorrect proofs of old known thorems in books, these are legion.

Oh no. This was the last thing I wanted to hear! I thought conceptual errors would only be in math journals because the concepts are new. But old theorems proven incorrectly in textbooks? I never thought that would happen because they have been proven correctly over and over again by other mathematicians over the decades (centuries!), the textbooks editted carefully for publication, etc...

These are the following (famous) books that I've read every single word from (and did every single exercise in), and have practically memorized every single proof from them like a bible:

1. Linear Algebra, 2nd edition, by Friedberg, Insel, Spence
2. Calculus on Manifolds, 1st edition, by Spivak
3. Topology, 2nd edition, by Munkres
4. Differential Topology, by Gauld (not so famous)

I'm I filled with misinformation? I went over every single proof with a fine-tooth comb and believed every single proof I read.

 

[mathwonk]

well spivak is a geometer, not an analyst, and one of my analyst friends once complained to me about the rigor of his treatment of partitions of unity. also the second edition of spivak contains an enhancement of an argument he himself felt inadequate in the first edition.

but no, in my opinion you are not filled with misinformation from any of those books. besides, did you follow the proofs and understand them? if so you are probably fine.

but check out your linear algebra book and try to find where they prove that the key concept, linear dimension is well defined. i.e. do they really prove that all bases have the same number of elements?

many books do not prove this, and without this, the whole theory is unfounded.

dont memorize the proofs, criticize them, that's how you learn math. i had a student complain that my recent geometry course consisted merely of memorizing proofs, when i never want or encourage that, rather that was just the student refusing to think about the arguments and then blaming me.

 

[mathwonk]

missing proofs are a bigger flaw in elementary textbooks. look at thomas - finney 9th edition, page 314, where they admit that they will not prove the existence of the integral of a continuous function, not even in an appendix. this is the foundational result of the entire book.

in fact they do prove a more practical result on pages 321-322, as exercises 79-80, that every monotone function has an integral. it follows that every piecewise monotone bounded function has one, which suffices for essentially every application in the book. But they do not even point this out!

contrast this with the very clear discussion in the book, Calculus, the elements, by Michael Comenetz, where the theorem that uniformly continuous functions are integrable is correctly and clearly proved, along with the observation that in fact all continuous functions are uniformly continuous on closed intervals.

He also proves the monotone case, and cites Newton for the original argument. Good books try to give you a real feel for why the results are true, and choose reasonable special cases to prove when the general proof is more tedious.

 

[andytoh]

did you follow the proofs and understand them? if so you are probably fine.

but check out your linear algebra book and try to find where they prove that the key concept, linear dimension is well defined. i.e. do they really prove that all bases have the same number of elements?

many books do not prove this, and without this, the whole theory is unfounded.

I went over every proof with a fine-tooth comb and even wrote out subproofs were there were (intentional) gaps. My linear algebra textbook proves the equal cardinality of vector space bases only if the dimenions are finite. For infinite dimensions, I tried to fill in the void myself:
https://www.physicsforums.com/showthread.php?t=198854 (post #5).

 

[mathwonk]

i see you said you went over the proofs carefully. actually professionals recommend not reading proofs but giving your own, peeking when necessary.

and those authors you mention, like munkres and spivak, are especially careful and reliable.

by the way which proof did your linear algebra book give for constancy of cardinality of finite bases? "steinitz exchange lemma?" yes i see in your post that is the argument. did you know that argument is due to riemann? (in the case of homology bases for surfaces.)

 

[andytoh]

Ok, I won't be afraid to challenge any proof in a textbook that I find suspicious from now on. Thanks.