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I was wondering how mathematicians of today would treat, for example, Euler's proof of zeta(2).

In William Dunham's book 'Journey through Genius' ( http://www.amazon.com/Journey-through-Genius-Theorems-Mathematics/dp/014014739X ), the writer states (more or less) that most mathematicians of today wouldn't aprove of Euler's methods, as his treatment of the 'infinite' doesn't uphold to today's modern standards of rigour.

His evaluation of Zeta(2) and all other even zeta-arguments op to Zeta(26) was correct, however. Would Euler's proofs get published in a well-respected math journal?

(Of course, his papers would now be written in english and the would-be published results aren't known to the mathematical community, yet).

Thanks in advance,

Max Muller

PS: O.K. everyone, I think many of you have stressed some important points regarding this question. I can't choose one, which is why I have upvoted some of your answers and left the question as it is. Thank you for your thoughts.

PPS: I'm sorry for the confusing title of the question in its previous form. I hope you all think it is stated better now.

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Sufficient unto the day is the rigor thereof. -- E. H. Moore . –  Gerald Edgar May 12 '10 at 14:31
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I haven't read your question yet, but I'm pretty sure the answer to your title question is yes. –  Cam McLeman May 12 '10 at 14:46
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Ha Carn McLeman, your comment made me laugh and made me a bit proud as well (I've got a Swiss and Dutch passport). Please reconsider reading the question, though, I think it's well worth it to think about the answer! –  Max Muller May 12 '10 at 15:08
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I don't think the title and the body are asking the same question. –  Qiaochu Yuan May 12 '10 at 15:21
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Two different questions: (1) Would a paper written in 18th-century style using 18th-century rigor be published in a 21st-century well-respected math journal? (2) If Euler were alive today, would he use 18th-century style and rigor in his manuscripts, or 21st-century style and rigor? I think the answers are obvious... –  Gerald Edgar May 12 '10 at 15:49

9 Answers 9

I think the answer to your question is pretty well-addressed by Gerald in the comments. Let me just throw in a couple of points that wouldn't fit in a comment.

1) Euler knew what he was doing. He had a tremendous ability for mental calculation, and verified to his satisfaction that his sums converged. He was well aware of the divergence of the harmonic series, so it's not like he was unaware of the fundamental issues surrounding "treating the infinite." If a sequence converged in accuracy by one digit every couple of terms for 30 or 40 consecutive terms, this was good enough for him to be convinced.

2) Even if Euler's proofs were not the most rigorous, that is not to say that modern mathematicians don't appreciate the methods behind them. I'd say more often than not, the hard part of mathematics is figuring out what should be true. Any student armed with Fourier analysis (and probably less) could re-derive many of Euler's formulas ($\zeta(2)$ in particular) -- few, however, would be able to play with the series even heuristically to figure out what the nontrivial contribution to the sum was, and fewer still would arrive at $\frac{\pi^2}{6}$ without prior exposure. If Euler were to rediscover his results in today's academic atmosphere, I suspect he would be hailed for his great insight into what "should be happening," and have a very successful career providing graduate students amazing problems which needed details filled in.

3) Even today, heuristics form an important part of mathematical research, so the legacy (if you will) of Euler's approach is still alive and well. The Cohen-Lensta heuristics, as a more modern example (or maybe even analytic conjectures in general...maybe even something like BSD) might be considered as fundamental pieces of insight gleaned from heuristic reasoning and experimental data.

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Cam McLeman, thank you for this answer. I have no doubt Euler knew what he was doing, nor dor do I think anybody isn't amazed at the results he obtained and how. Everyone would 'hail' him for his insights nowadays, if he'd arrive at $ zeta(2)= pi^2/6$ now. I'm very impressed with his methods as well. It would be a pity, however, if his result wouldn't make it to a paper because his methods are simply 'not rigorous enough'. Another question comes to mind know: could mathematicians provide a firm(er) logical standing for Euler's methods, so his hypothetical 18th century paper would get accepted? –  Max Muller May 12 '10 at 17:00
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Well, just to adopt the opposite point of view, we can all marvel at Euler's greatness now in part because modern mathematics came along and was able to fill in the details. For every Euler, there's a thousand cranks whose "brilliant but non-rigorous ideas" actually do collapse after an application of modern rigor. I'm not sure how to respond to your last question -- there are many many ways of filling in the holes, to various extents modelling Euler's approaches. Isn't that partly what Dunham's book is all about? –  Cam McLeman May 12 '10 at 17:11
    
I strongly agree with you, Cam. As many others have stated, modern mathematicians can look at his methods in awe because (<-- I want that word in italics) he helped to establish the standards of rigour we're so proud of today. Euler's greatness lied in 'feeling' a bit what was right in mathematics, I think he developed an intuition to feel how to solve a problem. Dunham's book is mainly about the fact that the're a lot of ways to solve different problems. Of course, we can model Euler's approaches to fill in the holes, but the original idea he exhibited is the most important. –  Max Muller May 15 '10 at 23:40

Not to be overly negative, but I think even with the qualifiers you write on the bottom of your question the answer is no, for a completely trivial reason. This has nothing to do with rigor. It's just that Euler's tools are not advanced enough for a modern publication.

To give you an example close to my interests, Euler himself pioneered the theory of partitions. He realized that one should take generating functions for the number of partitions from various classes and prove identities for the resulting $q$-series to establish connections between them. I read his work (in translation, I am afraid) and it is lucid, beautiful and mostly rigorous. But (to answer your question) you can't take his obscure result and submit to a serious journal. This might have worked pre-1930 or so, but in modern times the journals are no longer satisfied with "new and correct" papers; the papers are also expected to have interesting technical innovations which might prove helpful. Euler's techniques by now are too well known and "standard" to be of interest...

Of course, this is not to say that Euler does not continue to publish books or even articles (see e.g. MR0818419).

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Perhaps a fairer interpretation of the question is about how a "modern-day Euler" would be received - someone whose education was up to date but whose sense of rigor was analogous to Euler's. One modern-day analogue I can think of is Feynman and the path integral. –  Qiaochu Yuan May 12 '10 at 19:41
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There also some famous mathematicians in modern times, whose standards of rigour and complete proofs do not please everyone. One example, which comes to my mind, is Thurston, where it took many years to fill his proofs in 'Geometry and Topology of 3-manifolds' with details and rigour (I don't know if it is done for everything). I've read that on these grounds, Serre was against to arward Thurston the Fields medal. Two other examples of mathematicians, who are more well-known for new ideas than complete and rigorous proofs are probably Gromov and Sullivan. –  Lennart Meier May 13 '10 at 12:30
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Thurston writes (in section 6) that his approach to rigor in the Geometrization program was intentional. He also argues that is was beneficial, for it mimics the way that people learn and live mathematics, which is actually quite different from what a stark formal proof or written paper might suggest. –  Greg Graviton Sep 28 '10 at 19:45
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I'm very late to this party, but I think it is important to point out that Thurston had complete and rigorous proofs of all the results he claimed. He discussed them with many people, and whenever pressed was able to produce as many details as people needed. He just chose not to write papers containing all the details of his proofs. The paper that Greg Graviton refers to contains his justification for this decision. –  Andy Putman Apr 10 '13 at 14:32
    
@Igor Pak: I don't think it is accurate to assert that "Euler's techniques by now are too well known and "standard" to be of interest..." Most of Euler's results are of course well understood, but his techniques, such as proof of the infinite product decomposition for the sine function, were properly interpreted only recently, through the work of Luxemburg, Kanovei, and others. The results are old, but the techniques are only beginning to be understood properly. –  katz Apr 10 '13 at 15:01

Even when he was wrong, Euler was frequently right.

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I guess we can say the same about Ramanujan too. :) –  Koundinya Vajjha May 13 '11 at 13:06

My own belief is that contemporary discounting of the validity of Euler's arguments is misplaced. Euler wrote arguments to the standards of his day. If he were writing now, he would write arguments to the standards of our day. As far as I know, and given his brilliance and insight, there's no reason to think that he wouldn't have been able to fill in any analytic details required to justify his arguments (which in the case under discussion are fairly straightforward).

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Just to stress a few points already addressed in comments and answers:

Euler in his time discovered many important facts and solutions to classical questions, advanced rigor and gave examples of the power of the recently created methods (infinitesimal calculus), popularized the science of his day (notably books dedicated to a German Princess), wrote some of the first textbooks in analysis (still pleasant reading today), gave strength to the prussian and russian academy of science, courtized by two of the most powerful powers of the day (the King of Prussia and the Czar of Russia), filled international academic journals, some of them he edited himself, with quality articles (in fact up to several decades after his death because of the sheer size of his output), fostered international cooperation, wrote in the most important languages of his day (latin, french, german, I think he also learned russian), published in applied science, was part of state scientific advisory commission, etc.

In fact Euler's work has been instrumental in progressively establishing the "rigor" some of us are so proud of.

So a better equivalent of his investigation of what we call now Zeta(2 n) and the Gamma function would be the solution of outstanding problems by one of the most recognized mathematician of his day building on recent work by one of his even more famous and established mathematician, Bernoulli, who was his PhD advisor and whose several family members have established positions in the scientific community.

I think he would have no difficulty publishing it. And his work would be quickly read and commented upon by many other mathematicians.

Even if we imagine a Leonard Euler finding himself straight-jacketed by the mathematical discourse and style of the XXIst century, he would pair up with another good mathematician to write scholarly articles, as Ramanujan and Hardy used to do at the beginning of the XXth in a mutually benefical couple.

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The important thing about Euler is he saw an approximately correct path to the correct solution. These kinds of people have always been the most regaled in mathematics. Whether or not he could write proofs to the degree we expect today is probably immaterial: he improved the human understanding of mathematics more than anyone in his generation and probably more than all but at most a handful of mathematicians in all of time. If he could continue to generate almost correct proofs/heuristics to produce correct answers to problems, he would no doubt find himself with employment as well as plenty of coauthors eager to check his pencil marks.

Mathematical rigor is important because intuition is too frequently wrong. But I think seeing the big ideas is still considered more valuable than getting the proof completely right.

Riemann's proof of the Riemann mapping theorem was flawed because it made use of the Dirichlet principle to a greater extent than is actually possible (as shown later by Weierstrass). Even if Riemann turned out to be somewhat wrong and the theorem did not hold to the generality he believed because the intuition of the day was that the Dirichlet principle was universally sound, would we not call it the Riemann mapping theorem (and instead name it after whoever gave the first complete proof)? In a similar-but-somewhat-different light, should Thurston's geometrization conjecture-now-theorem have a different name?

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I learned in my science history lessons, that the standards for mathematics varied over time.

In the ancient Greek time (let say, the period of Archimedes), it was preferred that a theorem had more than one proof. At least, it was not uncommon to give more than one proof.

So, one can argue, that current math articles would not be accepted by the well respected Archimedes.

I don't know the details, but don't assume that the standards of rigour were monotonic increasing over time.

Lucas

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That is an interesting perspective! Of course, you are aware that Archimedes himself has not "published" anything - his extant work was compiled over 700 years later using manuscripts and letters to Alexandrian scholars. –  Victor Protsak May 13 '10 at 21:11

Let me answer by rephrasing your question. How will Andrew Wiles' proof of Fermat's Last Theorem be seen in the year 2260? I think it will be acknowledged then as much as it is acknowledged today that the proof was a major step in the history of mathematics. However, I seriously doubt that his proof will be considered as 'rigorous' by the standards of the year 2260. A well-respected math journal will then require a proof formally verified by a symbolic engine. (See the Notices of AMS 2008, vol. 55, issue 11).

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I'm not so sure -- despite a multiple-millenium-long trend, it's not clear to me that the level of rigor required by the mathematical community will continue to be monotonically increasing. –  Cam McLeman Sep 28 '10 at 20:26

In answering this question, it is helpful to make a distinction between, on the one hand, what Reeder calls the "inferential moves" that Euler makes (see related thread Euler's mathematics in terms of modern theories?), and on the other, the mathematical objects he manipulates (infinitesimals, infinite integers, etc). This allows Reeder to observe that modern infinitesimal theories are far more successful in formalizing Euler's procedures ("inferential moves") than are $\epsilon,\delta$ techniques.

Traditional scholars like Ferraro (see thread linked above) were trained on the basis of conceptual frameworks that are inadequate to the task of making such an evaluation, and tend to receive the work of scholars like Laugwitz with hostility.

Laugwitz argued for an essential coherence of infinitesimal reasoning in both Cauchy and Euler, modulo certain "hidden lemmas" that need to be made explicit to meet a modern standard. I would adopt an optimistic position that many of Euler's greatest contributions are immediately publishable in contemporary journals, provided minimal changes are made so as to clarify the nature of the objects as well as the "hidden lemmas".

The verdict is still out on whether the MATHEMATICAL community (as opposed to that of the HISTORIANS of mathematics) will in the end side with Reeder's analysis or Ferraro's analysis.

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