The persistent rumor that Nobel did not establish a prize in mathematics because Mittag-Leffler had an affair with Nobel’s wife is certainly incorrect. Nobel never married. But the other version of this rumor, founded on hostility between Nobel and Mittag-Leffler, may be correct though there is no documentation to support it. Certainly there appear to have been ill feelings between the two men: according to a letter from J.L.Synge to H.S.Tropp [Tropp], Fields told Synge that this was the case, and Synge remarks that he later confirmed this himself in Sweden.

There is no doubt that Nobel and Mittag-Leffler knew each other. Mittag-Leffler was one of the most prominent Swedish scientists at the time. In 1890 Nobel turned down Mittag-Leffler’s proposal to fund a professorship for Sonya Kovalevsky at the Stockholms Högskola (later Stockholm University) where Mittag-Leffler was a professor and one of its most powerful members. (Kovalevsky was on the faculty there from 1884 until her death in 1891.) The Högskola was named as a beneficiary in Nobel’s first will (1883), but not in his final will (1896). According to [p. 53, Crawford], the rector of the Högskola, a chemist named Otto Pettersson, and Svante Arrhenius, a physicist, let it be known “that Nobel’s dislike for Mittag-Leffler had brought about what Pettersson called the `Nobel flop’ ” (the term he used to describe the dropping of the Högskola from Nobel’s will).

There seems little doubt too that Mittag-Leffler had many detractors – “Mittag-Leffler had a great ambition to succeed in the many endeavors to which he applied his organizational skills. The judgments of many of his contemporaries about his person were not positive” (p.334, [Lehto]). One wonders whether the hostility between Nobel and Mittag-Leffler, and the friendship between Fields and Mittag-Leffler, were factors in Fields’s establishment of his award. Ironically, Mittag-Leffler (as well as Arrhenius) was, in the first few years after Nobel’s death in 1896, of “decisive importance … in shaping the decisions and hence the international standing of the [Nobel] prizes” [p.8, Crawford].

[Crawford] Elisabeth Crawford, The Beginnings of the Nobel Institution, The Science Prizes, 1901-1915, Cambridge University Press (Cambridge) & Editions de la Maison des Sciences de l’Homme (Paris), 1984.
[Lehto] Olli Lehto, Mathematics Without Borders, A History of the International Mathematical Union, Springer-Verlag (New York), 1998.
[Tropp] Henry S. Tropp, The Origins and History of the Fields Medal, Historia Mathematica 3 (1976), 167-181.