Hungarian mathematician Alfréd Rényi (1921–1970) is the author of several important discoveries in discrete mathematics topics, such as graph theory, combinatorics and number theory. He once wrote, “When I’m unhappy, I do math to be happy. And when I’m happy, I do math to stay happy.” I think many of my colleagues identify with these principles.
Rényi is also indebted to the important discovery that “a mathematician is an apparatus for transforming coffee into theorems”—although he is often credited to his colleague and compatriot Paul Erdös (1913–1996), another great coffee consumer and prolific producer of coffee. theorems. The veracity of Rényi’s law is validated, moreover, by the dramatic testimony of none other than Henri Poincaré (1854–1912).
In the book “Science and Method”, published in 1908, Poincaré describes his process of mathematical discovery. “For 15 days I struggled to prove that there could be no such function as what I have since called Fuchsian functions. I was very ignorant. Every day, I would sit at my desk and stay there for an hour or two. I would try various combinations. , and didn’t get any results.”
But one night everything changed. “In the evening, against my habit, I drank black coffee and couldn’t sleep. The ideas flowed incessantly. I felt them clashing with each other, until pairs intertwined, so to speak, to form stable combinations. The next morning, I had proved the existence of a class of Fuchsian functions. All that remained was to write the results, which took only a few hours.”
It was the starting point of one of Poincaré’s greatest contributions, the theory of automorphic functions. This name was proposed by the German mathematician Felix Klein (1849–1925). Poincaré called Fuchsian functions, after Lazarus Fuchs (1833–1902), when the domain is the disk, and Kleinian functions, after Klein, in all other cases. But Klein felt that this did not do justice to the importance of his own work on the subject. Poincaré ironically dismissed his colleague’s objection, citing Goethe’s great poem “Faust”: “Name ist Schall und Rauch” (“names are nothing but noise and smoke” in German).
Reflecting on this and other stages of his work, Poincaré was led to distinguish the three phases of mathematical discovery: preparation, incubation and illumination.
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