(Also part of The Lives of the mathematicians)

The following story is not only apocryphal, it is also patently false:

Picture, if you will, Alfred Nobel on his deathbed. He's setting up a trust fund which will give out prizes for those whose inventions or discoveries are of the greatest practical benefit to mankind. One by one, Nobel and his advisors go over the categories in which prizes will be given. The Curies should be given something, so prizes are set up for Chemistry and Physics. Doctors are always of benefit to mankind, so Medicine goes without saying. The list goes on. Finally, Nobel props himself up in his bed and asks: "And if I give a prize in Mathematics, who will be the first laureate?"

An aide leans forward and whispers a name into Nobel's ear.

"What? The bastard with whom my wife cheats on me?!" Nobel roars, "NEVER! There will be no Nobel prize in Mathematics!"

And that is (not) why there is no Nobel prize in Mathematics.

There is also no Nobel Prize in Economics, despite extensive media coverage. Determining the difference between a "Nobel Prize" "in" Economics and a real Nobel Prize is left as an exercise to the interested reader.

One of the reasons that there is no nobel prize in Mathematics might be the fact that the leading Swedish mathematician at the time, Gösta Mittag-Lefler, was a personal enemy of Nobel's.
Mittag-Lefler, a brilliant mathematician and the founder of the Mittag-Lefler Institute, was also a controversial and sometimes feared person. He would often participate in inflamed public debates, and he was a conservative in political matters. If a Nobel Prize in Mathematics had been created, Gösta Mittag-Lefler would have had a very good chance of receiving one of the first ones. Nobel would not have wanted this to happen.

It seems that the Nobel Archive at almaz.com/nobel has a long page on this subject, on http://almaz.com/nobel/why_no_math.html. They also focus heavily on Gösta Mittag-Lefler, but they have some evidence that Nobel was not so averse to him. They also mention ariels patently false story, noting in passing that Nobel had no wife.

Actually, the reason I find most convincing (and perhaps the most obvious) is the issue of practicality, and not some urban legend about personal rivalry. Mathematics is not a science; it is much more abstract than science, and as a (general) rule, the more abstract a concept is, the less practical it is as well (when was the last time quantum physics was put to something useful?

Entire branches of mathematics have been discovered and researched that have no relavence whatsoever to the real world (solving Fermat's last theorem comes to mind). Alfred Nobel created the prize with the intention of rewarding those who, through their actions, had a positive influence on the world. In other words, if one wished to win a prize, he would have to discover something that was more than just intellectually tittilating.

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