In late 1978 one of Hubbard's graduate students, Frederick Kochman, approached Mandelbrot at a conference and showed him Hubbard's pictures. Mandelbrot "didn't seem very interested," Kochman recalls. Yet shortly thereafter Mandelbrot wrote a letter to Hubbard inviting him to IBM to discuss his work. In the letter, which Hubbard kept, Mandelbrot wrote: "When sampling the works of Fatou and Julia, I had thought of doing these things myself, but had not mustered the courage. Nevertheless I can claim that I was awaiting your pictures for a long long time."
Hubbard says he went to IBM early in 1979 and, while there, told Mandelbrot how to program a computer to plot the output of iterative functions. Hubbard concedes that he did not appreciate the full significance of his own images then and that they showed only pieces of the Mandelbrot set. He also admits that Mandelbrot developed a superior method for generating images. Nevertheless, Hubbard says he was and continues to be "outraged" that Mandelbrot did not give him credit in the 1980 paper and later writings. "It was a breach of mathematical ethics," he asserts.
Mandelbrot recalls seeing "one impressively early drawing of a Julia set" by Hubbard but denies that it contributed to his own discovery. In response to Hubbard and Douady's charge that he is stingy in granting credit, Mandelbrot says he has also been accused of overcitation. He adds that his failure to cite the early finding of Brooks and Matelski might have spared them "derision" for "their failure to do anything with it."
What about the suggestion of Hubbard, Matelski and Brooks that the true discoverer of the Mandelbrot set is Fatou, who was the first to define the set and speculate about its properties? Brooks even proposes that "if Fatou had had access to modern computing facilities, he could have and would have drawn pretty much the same pictures that Matelski, Mandelbrot and I did." Mandelbrot calls such speculation pointless and insists that Fatou's definition of the Mandelbrot set does not constitute discovery. "Definition counts for nothing," he says. "You have to say why something is important."
Other mathematicians familiar with the case are somewhat bemused. "It seems strange to me that there should be such a fuss," remarks John Milnor of Princeton University. He maintains that neither Brooks and Matelski nor Mandelbrot did anything mathematically important. "Hubbard and Douady are the first ones to really obtain some sharp results," he says, "and let us know something about the set."
The dispute over precedence, Milnor suggests, may spring from a clash of different mathematical cultures. "In pure mathematics," he explains, "there is a tradition of letting others praise your work." Mandelbrot, he notes, is in applied mathematics.
"Mathematical developments don't take place single-handedly," William P. Thurston of Princeton points out, "and it's pretty common that things are not named after the first person to develop them. The Mandelbrot set follows that pattern." He suggests, however, that no one would begrudge recognizing Mandelbrot's achievements if he would reciprocate more himself. "He could be a bit more magnanimous," Thurston says.
Sullivan, who has also been acclaimed for his studies of the Mandelbrot set, calls himself "sort of a defender of Mandelbrot." Mandelbrot deserves to have the set named after him, Sullivan says, because his efforts brought the set to the attention of both the public and of the pure-mathematics community.
The fact that it was only "by coincidence" that the set proved later to be mathematically significant, Sullivan says, in no way diminishes Mandelbrot's achievement. "That's the wonderful thing about mathematics," he adds. "Even amateurs can make important contributions."
So who did discover the Mandelbrot set? Sullivan calls the question meaningless. Perhaps. Sheldon Axler, editor of the Intelligencer, plans to publish a letter pointing out that the Hungarian mathematician F. Riesz reported on work related to the set in 1952 .
The final answer, if pursued, seems likely to recede in a blur of ever finer detail.