Written: Nov. 21, 2004.
I just finished reading Allyn Jackson's fascinating two-part article about the great mathematical genius Alexandre Grothendieck (that appeared in the Notices of the Amer. Math. Soc.) , and Pierre Cartier's extremely moving and deep essay `Une pays dont on ne conaitrait que le nom: Le "motifs" de Grothendieck'. (that appeared in the very interesting collection "Le Reel en mathematiques", edited by P. Cartier and Nathalie Charraud, and that represents the proceedings of a conference about psychoanalysis and math).
In Pierre Cartier's article, in addition to an attempt at a penetrating "psychoanalysis" he also gives a very lucid non-technical summary of Grothendieck's mathematical contributions. From this it is clear that one of the greatest giants on whose shoulders Grothendieck stood was Israel Gelfand, whom I am very fortunate to know personally (I am one of the few (too few!) regulars that attend his weekly seminar at Rutgers). I couldn't help notice the great contrast between these two Giants, and their opposing styles of Genius.
Myself, I am not even an amateaur psychoanalyst, but motives and psi aside, I can easily explain why Grothendieck stopped doing math a long time ago (hence, died, according to Erdos's nomenclature), while Gelfand, at age 91, is as active and creative as ever.
First and foremost, Grothendieck is a dogmatic purist (like many of the Bourbakists). He dislikes any influences from outside mathematics, or even from other subareas of math. In particular, he always abhored mathematical physics. Ironically, as Cartier explains so well, many major applications of his ground-breaking work were achieved by interfacing it with mathematical physics, in the hands of the "Russian" school, all of whom were disciples of Gelfand. As for Combinatorics, forget it! And don't even mention the computer, it is du diable. As for Gelfand, he was always sympathetic to all science, even biology! In fact he is also considered a prominent theoretical biologist. Gelfand also realizes the importance of combinatorics and computers.
Also people. Grothendieck was a loner, and hardly collaborated. On the other hand, Gelfand always (at least in the last sixty years) works with other people. Gelfand is also very interested in pedagogy, and in establishing math as an adequate language.
Grothendieck spent a lot of energy in rebellious political causes, probably since in his youth he was an obedient bon eleve. On the other hand, Gelfand was already kicked out of high-school (for political reasons), so could focus all his rebellious energy on innovative math.
So even if you are not quite as smart or original as Gelfand and Grothendieck (and who is?), you will still be able to do math well into your nineties, if you follow Gelfand's, rather than Grothendieck's, example.
Added Jan. 28, 2022: Read interesting comments by Luc Haine
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