Will the next Erdös be someone who hangs around at home, reads a lot of math blogs, and posts solutions to open problems in the comments?

## Erdös 2.0

**Tagged**erdos

Will the next Erdös be someone who hangs around at home, reads a lot of math blogs, and posts solutions to open problems in the comments?

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First he has to actually *be* the next Erdős (note double acute accent). Then he can worry about how he gets published, or not published.

First he has to actually *be* the next ErdősThat’s true, but it’s not as much of a restriction as one might think. There are, at the very least, dozens of living mathematicians at the level of Erdős. What made Erdős special and attracted so much attention was his extraordinary personal story. If he had settled down and accepted a permanent job and taken on the ordinary responsibilities of life, he would have been a famous chaired professor, but no popular books would have been written about him. By contrast, even a somewhat less talented mathematician who was as eccentric as Erdős could become as famous.

Paul Erdos was very unusual, not just a high-performing mathematician who happened to like international travel. Erdos got attention for writing more mathematical papers (and maybe more scientific papers) than anyone in history. That makes him a very extreme outlier, not one among dozens. It didn’t hurt that this output made him the symbol and leading prophet of a field that went from marginal to gigantic in his lifetime (combinatorics) or that many of his results and conjectures could be presented in media-friendly sound bites. It’s not so different from the case of Edward Witten in physics, for example, or the public interest in Grothendieck or Ramanujan long after their academic careers. The achievements are so singular that they attract attention to whatever interesting personal story might also be there.

Paul Erdos was very unusual, not just a high-performing mathematician who happened to like international travel. Erdos got attention for writing more mathematical papers (and maybe more scientific papers) than anyone in history.Yes, but most of those papers weren’t very good – he published a paper about every thought that ever entered his head and every conversation he ever had within another mathematician. He did write some excellent papers, maybe one or two hundred of them (which is itself an extremely impressive total). However, most of Erdos’s 1500+ papers were mediocre at best. That’s how he wrote so many of them.

In his later years, for most of his papers, he didn’t pick the problem, do most of the work, or write up the results. The general pattern was that that he visited some university, the combinatorialists there consulted with him on their problems, he offered whatever help he could while he was there (which might be quite substantial), and then they wrote up the results and added him as an author. When he worked with leading mathematicians working on important problems, this could lead to great work. However, that was true for only a small minority of his papers.

That makes him a very extreme outlier, not one among dozens.There are a lot of brilliant mathematicians in the world. For example, Serre, Yau, Witten, Thurston, Grothendieck, Gromov, and Gelfand are absolutely on the same level as Erdos (if not higher), both in terms of mathematical accomplishments and in terms of influence on the research community. And this is far from a complete list. I’ve restricted it to people 50 and over, to give enough time and perspective to judge their impact. If we were to include all living mathematicians, the list would presumably double in length. Plus I chose only names where I felt there should be no debate, which is an awfully strict standard. A fair comparison with Erdos would probably include several times as many names.

So I’m confident that there are several dozen living mathematicians at about the same level as Erdos, but I guess it depends on your standards for what constitutes a great mathematician. If you go by publication counts, then Erdos certainly wins.

One could make a stronger case that Erdos was the 20th century’s great combinatorialist. This is subject to debate: Szemeredi’s best work impresses me much more than Erdos’s best work, Noga Alon is as strong a problem solver as Erdos ever was, and this is not even getting into algebraic vs. probabilistic combinatorics (Rota played the same prophet role as Erdos in a different community).

It’s not so different from the case of Edward Witten in physics, for example, or the public interest in Grothendieck or Ramanujan long after their academic careers. The achievements are so singular that they attract attention to whatever interesting personal story might also be there.Grothendieck and Ramanujan would have attracted practically no interest from the general public if it hadn’t been for their extraordinary personal stories. For example, Ramanujan was a great mathematician, but Hilbert and Poincare were substantially greater, and neither one is remotely as well known. Witten is a little different: my impression is that the interest in him is not primarily due to his personal story or his amazing accomplishments, but rather because of string theory. (Theories of everything always attract lots of attention, string theory is a particularly exotic and interesting-sounding theory of everything, and the controversy attracts even more attention nowadays.)

I think the above underrates the sheer weirdness of Ramanujan’s mathematics (for its day), which comes out clearly from Hardy’s comments on the early letters, when he knew little or nothing of Ramanujan’s story and none of the “Ramanujan anecdotes” had even happened yet: the results Ramanujan had sent him “must be true because, if they were not true, no one would have the imagination to invent them.” In that sentence the whole of the Ramanujan legend is implicit.

“Erdos got attention for writing more mathematical papers (and maybe more scientific papers) than anyone in history.”

I think Euler takes that accolade, Erdos was the collaborator par excellence

“As for his place in the world of Mathematics, we quote Bruce C. Berndt: ‘Paul Erdős has passed on to us Hardy’s personal ratings of mathematicians. Suppose that we rate mathematicians on the basis of pure talent on a scale from 0 to 100, Hardy gave himself a score of 25, J.E. Littlewood 30, David Hilbert 80 and Ramanujan 100.’”