The new issue of the Notices of the American Mathematical Society turned up in my mailbox today. It features an interesting, if slightly disturbing, editorial (PDF format) by CUNY mathematician Melvyn Nathanson. He wonders about how confident we can really be regarding the proofs that appear in our research journals:
But why the delay? Surely, any competent person can check a proof. It's either right or wrong. Why wait two years?
The reason is that many great and important theorems don't actually have proofs. They have sketches of proofs, outlines of arguments, hints and intuitions that were obvious to the author (at least, at the time of writing) and that, hopefully, are understood and believed by some part of the mathematical community.
But the community itself is tiny. In most fields of mathematics there are few experts. Indeed, there are very few active research mathematicians in the world, and many important problems, so the ratio of the number of mathematicians to the number of problems is small. In every field, there are “bosses” who proclaim the correctness or incorrectness of a new result, and its importance or unimportance. Sometimes they disagree, like gang leaders fighting over turf. In any case, there is a web of semi-proved theorems throughout mathematics. Our knowledge of the truth of a theorem depends on the correctness of its proof and on the correctness of all of the theorems used in its proof. It is a shaky foundation.
He goes on to write:
Part of the problem is refereeing. Many (I think most) papers in most refereed journals are not refereed. There is a presumptive referee who looks at the paper, reads the introduction and the statements of the results, glances at the proofs, and, if everything seems okay, recommends publication. Some referees do check proofs line-by-line, but many do not. When I read a journal article, I often find mistakes. Whether I can fix them is irrelevant. The literature is unreliable.
How do we recognize mathematical truth? If a theorem has a short complete proof, we can check it. But if the proof is deep, difficult, and already fills 100 journal pages, if no one has the time and energy to fill in the details, if a “complete” proof would be 100,000 pages long, then we rely on the judgments of the bosses in the field. In mathematics, a theorem is true, or it's not a theorem. But even in mathematics, truth can be political.
Interesting stuff, and largely accurate, alas. I think Nathanson overdoes it a bit in likening the authorities in a field to political bosses or gang leaders, but otherwise I agree with every word of his editorial.
His remarks about peer review are especially on the money. Refereeing a research paper in mathematics is an awful, time-consuming task. The style among research mathematicians is to make brevity the highest of virtues, and anyone who writes three consecutive sentences of jargon-free exposition will immediately be accused of excessive wordiness. When reviewing a paper you can reasonably expect to be stopped dead several times, banging your head in frustration and asking yourself, “How the heck does this sentence follow from anything that came before?”
There is a practical reason for this, and also a sinister one. The practical reason is that paper is expensive. (On the other hand, I have not noticed that papers published in electronic journals are any clearer than their low-tech counterparts.) The sinister reason is that crystal clear writing would immediately make apparent the triviality of most research papers.
It is also true that there is a large element of faith in modern mathematics. As Nathanson writes, if the bosses say the result is valid, that is good enough for just about everyone. In my own field of algebraic graph theory, there is an article that I have cited in several of my research papers. It was authored by one of the true giants of the field, appeared in a prestigious journal, and its main result is certainly one that feels right given everything else I know about the subject. The fact remains that despite several attempts, both on my own and with my research collaborator, I can not make heads or tails out of the proof. It starts off sensibly enough, but then quickly fades to black as I go farther into it. But I cite the result anyway.
There are very few mathematicians who actually enjoy picking up a research paper and reading it. Even if the paper comes right from your own little corner of the mathematical world, it is still going to be difficult to read. What is fun, on the other hand, is going to a conference, meeting the author of the paper, and asking him to explain what, exactly, he did. It's quite rare for the explanation to take more than a few minutes.
I often tell my students that math is easy, it's math classes that are hard. Mathematics deals in clever ideas and moments of clarity. There is nothing quite so satisfying as looking into darkness and mystery, and, using nothing more formidable than your own brainpower, creating light. Math classes, alas, deal in tedious symbol manipulation, excessive abstraction, and in doing everything possible to make the subject appear to the world as a conspiracy against the laity. As Nathanson points out, it's not just students that have that feeling. Even professionals find themselves at the mercy of those who know more.
Is mathematical complexity then an excellent cover for the most intricate of deceptions?
I'm guessing the reason that electronic papers are no better is that people have gotten into the habit of writing for paper journals. On the plus side, the internet allows all sorts of really good exposition to get around, which helps GREATLY. If it weren't for lecture notes floating all over the place from lots of different people, I wouldn't understand half of what I do.
I do, however, think that part of the problem of trusting the literature would be solved if we had people whose JOBS it was to referee, as opposed to being volunteer labor for profitable publishing companies. I mean, CREATING mathematics and READING mathematics are two very different things. One of which I know I can do (reading), and one that I'm not sure (creating...I'm a grad student, after all, haven't made any of my own theorems yet). But I find errors and skipped steps in proofs that take days/weeks to figure out how to get around often enough that I think that graduate students and people who had to drop out at a Master's could continue studying and get jobs refereeing, even if they couldn't get jobs as research professors.
The style among research mathematicians is to make brevity the highest of virtues, and anyone who writes three consecutive sentences of jargon-free exposition will immediately be accused of excessive wordiness.
This is true, and as a physicist standing on the sidelines trying to understand some of this stuff, it is very frustrating to me. The impenetrability of much of the material slows its uptake and limits its scope. I think physicists like to have a bit of intuition thrown in with the mix, and I appreciate that. Perhaps the mathmos make the mistake of thinking that only other mathematicians are their audience. I can't imagine that being good for anyone.
I wouldn't say “deception” since that would imply a conscious element of fraud. The point is simply that honest mistakes happen routinely, and some of the conventions of the profession can make these mistakes unlikely to be discovered.
I think you're right about the electronic journals. The trouble is that a lot of mathematicians really do think that brevity is terribly important. I've heard many of my colleagues complain that this or that textbook was too wordy for having too much explanation, and enough Theorem-Proof. Too wordy for whom, one wonders. I've never heard a student complain about a textbook being too clear.
And good luck with your studies! Where are you doing yout graduate work.
Mathematicians definitely tend to be a bit insular. Another thing I've noticed is that math journals are, with almost no exceptions, just collections of papers placed between covers. No editorials, book reviews, perspective articles, research summaries, or anything else. Journals in other branches of science seem not to be so extreme. One thing I really like about journals like Science and Nature are the high-level summaries they provide for many of their research papers. I can usually understand the summaries even though the research papers themselves would be impenetrable.
It would be a hell of a job, but a computer program could theoretically take care of this.
Jason, as to requiring a conscious element of fraud, that's what I meant by "excellent cover."
Heretofore it was assumed (at least by me) that mathematical proofs were essentially unmanipulatable for that purpose, but clearly I was wrong.
As Nathanson mentions in the beginning of the article, proofs can be checked by a computer. I don't know why this is not a high priority objective for the mathematics community.
It doesn't have to replace human review, but the community would feel much better about proofs that they can't check manually. It would also help the author of a paper as he develops his work.
Could you generalize this a little further*, and publish it?
"Numbers these days just aren't as reliable as we used to think!"
*Further = far enough to cover my butt if/when the IRS gives me another audit.
I've found this generally true for a lot of computer science and engineering (esp EE) papers too. Bio tends to be a bit clearer, but at the cost of often leaving out very important details (or just referencing 'standard' methods which often have never really been described in intricate detail).
I'm actually using 'algorithmic descriptions' (essentially pseudo-code with descriptive variable names) instead of math in many places in my dissertation. My target audience is engineers, cs folks, and field biologists... so I consider being very clear critically important, otherwise I'll loose people in jargon that is outside their field.
Oh, and I keep running across papers (signal processing stuff mostly) which make completely unrealistic 'standard' assumptions and then derive some nice formula which is pretty but utterly useless in solving the engineering problem the paper is supposedly about. I really don't understand the fetish for an analytical solution/proof.
Koray: Proofs typically cannot be checked by computer; most mathematics does not lend itself to verification by algorithm. Moreover, even in those rare cases when calculation is used critically (eg Hale's sphere packing proof, the 4-color problem) the issue of whether the code is correct (and the computer processing correctly: remember the Pentium?) would need to be verified by hand, which would not be feasible.
In my view things are not as serious as the article makes it seem. 95% of all published Mathematics (and probably more in the other sciences) is forgotten a couple years after publication, and important work becomes important partly because it is pored over for decades, in which flaws eventually become apparent. The 95% typically serve as support to the large breakthroughs, exploring consequences and clarifying/simplifying/expanding the important results.
In my experience, most of the great articles are well written, and mathematicians usually become better writers as they get wiser and exchange the urge to impress for the need to communicate clearly.
I think the more serious problem is that the "publish or perish" mentality, employment/tenure stress, the overly competitive drive for recognition and credit, excessive influence of fashion, etc. leads to the writing of too many bad articles, overwhelming referees with work, and hence a sloppiness enters the writing and refereeing process. But compared to the other sciences we have very little to worry about.
I'm at the University of Pennsylvania, studying algebraic geometry, and hopefully will pass orals in the Fall.
I've actually complained about a book or two, if I remember right, for being too wordy. But it was mostly that they spent a lot of time going over and over on the basics of something and then burned through the hard stuff like it was trivial.
This overlaps a posting in the Good Math, Bad Math scienceblog of 4 July 2008.
One of the best articles on the crisis from a mathematical and computational viewpoint is "Whither Mathematics?", Brian Davies, Notices of the AMS, Vol 52, No. 11, Dec 2005, pp.1350-1356.
I just an hour ago [on 4 July 2008] put a lengthy quotation from that on the "Michael Polanyi and Personal Knowledge" thread of the n-Category Cafe blog, with a reference to Greg Egan fiction and a glimpse at the year 2075.
Before we address the crises in Astronomy (the Inflation theory tottering, Dark Energy, and the like), Biology (what is a "gene" now that the old paradigm has fallen in a deluge of genomic data?), and Planetary Science (now that comparative planetology has covered much of this solar system and something of 200+ others), we look back at Math.
The triple crisis, as explained at length in the Brian Davies article may be summarized:
#1: Kurt Godel demolishes the Frege, Russell-Whitehead, Hilbert program.
#2: Computer-assisted proofs have "solved" some important problems, but no human being can individually say why.
#3: There is sometimes no assurance of global consistency.
#2 examples include Appel & Haken on four-color theorem (1976), Tom Hales on
Kepler problem (1998), the 1970s Finite Simple Group collaboration culminating in the 26 sporadic groups led by the Monster, but with Michael Aschbacher (Caltech) sewing up loose ends through 2004 and still admitting the possibility that there might be another finite simple group out there in Platonic possibility which is different from all others; that skepticism amplified by Jean-Perre Serre.
"We have thus arrived at the following situation. A problem that can be formulated in a few sentences has a solution more than ten thousand pages long. The proof has never been written down in its entirety, may never be written down, and as presently envisaged would not be comprehensible to any single individual. The result is important, and has been used in a variety of other problems.... but it might not be correct."
See also: "Science in the Looking Glass: What Do Scientists Really Know?", E. Brian Davies, Oxford U Press, 2003.
It's abundantly clear to me that mathematicians are the modern voodoo masters, they not alone.
If faced with a simple problem of arithmatic the Statistician will provide a range of answers largely irrelevant to the original query.
The mathematician will provide an absolute answer accompanied by an vague and nebulous proof.
The accountant will simply ask you what you want the answer to be.
Jason wrote: What is fun, on the other hand, is going to a conference, meeting the author of the paper, and asking him to explain what, exactly, he did. It's quite rare for the explanation to take more than a few minutes.
Sounds like a good reason for electronic math journals to contain videos of each of the various papers' authors giving exactly the sort of brief, clear explanation you describe. In fact, such expositions (perhaps with more detailed bits and citations linked) might eventually substitute for the papers themselves.
"The sinister reason is that crystal clear writing would immediately make apparent the triviality of most research papers." amen
As for computer checking of proofs, I know some people who are working on that, but with current technology, it'd require that mathematics papers become completely unreadable and essentially formal logic (at least, that's what they tell me, really not my area) and I think I'd rather risk a wrong result than incomprehensible proofs of that sort.
Jud, the Journal of Number Theory is attempting to start up video abstracts. See the following link:
The community needs to get used to this sort of thing, but I think it'll be more useful for the up-and-coming generation, who are growing up on the internet and youtube.
Computers can absolutely verify most (if not all) possible proofs. The problem is that most mathematicians (at least for now) cannot (or at least don't) do the formal logic to state their proofs rigorously.
If it doesn't take intuition or high level pattern recognition, a mechanistic system can do it. If verifying a proof requires these, it isn't a proof.
BTW: At least the proof of the 4 color theorem does actually make sense. Saying "no human being can individually say why" is a cop-out and fundamentally incorrect. Reducing a problem to a finite set of cases and then demonstrating the theorem holds for all those cases isn't exactly hard to grok. An algorithm can be proved (in some cases), and more usefully, a proof (a valid one at least) can be transformed into an algorithm.
Another thing might be mathematicians' habit of acting as if a proof were true long enough to see if it goes somewhere interesting, and only then going back and doing the hard work of making sure it actually is true.
An image of a primate and a large area of herbaceous plants with long, curved yellow fruit is floating across my imagination.
Wouldn't an algorithm capable of verifying any proof be equivalent to the one deciding whether some other algorithm will always stop?
I don't think so. If you write your proof in formal logic, all it has to do is check a database of other theorems and statements, as well as the types of logical inferences allowed to make sure that each step is ok individually. However, you'd first need to get mathematicians to write in formal logic, and that's not going to happen. Look at the proofs of things in "Typographical Number Theory" in Godel, Escher, Bach to see just how bad basic statements get.
Automated proof verification is, in principle, trivial: there are a finite number of axioms and a finite number of rules of inference, so you just have to check that each line follows via a rule of inference from one of the axioms or one of the previously-proved lines. Writing proofs that are so checkable is inconceivable, and such proofs (even of trivial results) would all be incomprehensible to human readers.
I'm surprised that no one has mentioned a far simpler reason that math papers are terse and sometimes difficult to decipher: when I write down a proof of a new theorem, I've been working with the objects involved and thinking about their properties for weeks or months or years. My familiarity level with the ideas in my papers at moment of publication is extremely high -- probably higher than that of anyone else. Given such a high level of familiarity, I've gone over certain logical chains of argument sufficiently many times that I don't worry about providing a lot of supporting commentary. It's not surprising that someone who hasn't spent so long thinking about these ideas can't follow the logical chain immediately -- the important question is whether they can do it with a reasonable amount of effort. My experience (as a grad student in combinatorics) includes very few instances of impenetrable proofs on topics I claim to understand.
I recently read a book on Riemann's Hypothesis which mentioned that a lot of 'proofs' in moderm maths have an implicit "If R's H is true" at the start of them.
That's until one was pwned with a similar 'proof' that started with "If R's H is not true..." and led to the same conclusion.
There was no agreement that that 'proof' was universally true, just that it was no longer answering an important question as it's value was only in it's shedding light on the Hypothesis itself.
Eddie: that doesn't sound like a pwning to me, that sounds like complementary work. A proof that RH => X and another proof that !RH => X together amount to a proof of X. Unless you're an Intuitionist, of course.
"The sinister reason is that crystal clear writing would immediately make apparent the triviality of most research papers." - that makes me feel really great about my own rejected papers, thanks! :-(
I think the real reason isn't so sinister, and it isn't because paper is expensive. It's because writing out a proof in full takes time and care. Faced with the choice between carefully writing a proof, debugging it, typesetting it, debugging the typesetting, etc, and writing "it is easily verified that...", many mathematicians will go for the latter. The same goes for the lack of clarity: it's hard enough writing something that's correct, it's at least twice as hard writing something that's correct and intelligible.
The current trend in automated proof systems, as I understand it, is for systems that allow one to construct and re-use general lemmas. Once you've built up the beginnings of a library of such lemmas, you don't need to do so much low-level stuff (just as computer programmers don't spend most of their time writing assembly language!). I don't know if they've yet reached the point of being useful to working mathematicians, though.
I try to make my papers as clear and easy to read as possible, if only because
I'd like to be able to understand them myself 5 years later. I have run into a bit
of trouble. Once I had to cut out 1/3 of a 30 page paper because the editor
felt that it was not "scientifically necessary". One or two people have criticized
me because some of my papers read more like colloquium talks, despite the
fact that "only the experts" will look at them. On the other hand, the American
Mathematical Society has a major prize for exposition.