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.
Best Answer
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.