Clippings

“The future historian of science concerned with the development of mathematics in the late nineteenth and the first half of the twentieth century will undoubtedly state that several branches of mathematics are highly indebted to Hilbert’s achievements for their vigorous advancement in that period,” Alfred Tarski has written. “On the other hand, he will have to note, perhaps with some wonder, that the influence of this man appears equally strong and powerful in some other domains which do not owe any exceptionally important results to Hilbert’s own research. An example of this kind is furnished by the foundations of geometry. I am far from underestimating the value of Hilbert’s contributions … in his [Foundations of Geometry], but I think that his most essential merit was the impulse he gave to organized research in this domain. A still more striking example is presented by metamathematics. Occasional considerations in this field preceded Hilbert’s Paris address; the first positive and really profound results appeared before Hilbert started his continuous work in this domain … [and] one does not immediately associate with Hilbert’s name any definite and important metamathematical result. Nevertheless, Hilbert will deservedly be called the father of metamathematics. For he is the one who created metamathematics as an independent being; he fought for its right to existence, backing it with his whole authority as a great mathematician. And he was the one who mapped out its future course and entrusted it with ambitions and important tasks.”

“The importance of scientific achievement is often not alone in the new material which is added to material already on hand,” Richard Courant has written. “Not less important for the progress of science can be an insight which brings order, simplicity and clarity into an existing but hard to reach area and thus facilitates or first makes possible the survey, comprehension and mastery of the science as a unified whole. …

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“For the analysis of a great mathematical talent,” [writes Otto Blumenthal], “One has to differentiate between the ability to create new concepts and the gift for sensing the depth of connections and simplifying fundamentals. Hilbert’s greatness consists in his overpowering, deep-penetrating insight. All of his works contain examples from far-flung fields, the inner relatedness of which and the connection with the problem at hand only he had been able to discern; from all these the synthesis – and his work of art – was ultimately created. As far as the creation of new things is concerned, I would place Minkowski higher, and from the classical great ones, for instance, Gauss, Galois, Riemann. But in his sense for discovering the synthesis only a very few of the great have equaled Hilbert.”

To Weyl, who himself made important contributions to mathematical physics, it seemed that “the maze of experimental facts which the physicist has to take into account is too manifold, their expansion too fast, and their aspect and relative weight too changeable for the axiomatic method to find a firm enough foothold, except in the thoroughly consolidated parts of our physical knowledge. Men like Einstein or Niels Bohr grope their way in the dark toward their conceptions of general relativity or atomic structure by another type of experience and imagination than those of the mathematician, although no doubt mathematics is an essential ingredient.”

The bright young newcomers who saw the famous Hilbert in action for the first time at [the Mathematics Club meetings] were struck by his slowness in comprehending ideas which they themselves “got” immediately. Often he did not understand the speaker’s meaning. The speaker would try to explain. Others would join in. Finally it would seem that everyone present was involved in trying to help Hilbert to understand.

“That I have been able to accomplish anything in mathematics,” Hilbert once said to Harald Bohr, “is really due to the fact that I have always found it so difficult. When I read, or when I am told about something, it nearly always seems so difficult, and practically impossible to understand, and then I cannot help wondering if it might not be simpler. And,” he added, with his still childlike smile, “on several occasions it has turned out that it really was more simple!”

Zermelo was … a nervous, solitary man who preferred whisky to company. He liked to prove at this time, which was before Peary’s expedition, the impossibility of reaching the North Pole. The amount of whisky needed to reach a latitude, he maintained, is proportional to the tangent of the latitude, i.e., approaches infinity at the Pole itself.

…Klein’s life had not been without its inner tragedy. The power of synthesis had been granted to him to an extraordinary degree. The other great mathematical power of analysis had been to a certain extent withheld. His ability to bring together the most distant, abstract parts of mathematics had been remarkable, but the sense for the formulation of an individual problem and the absorption in it had been lacking. “He was like a flier who, soaring high over the world, discovers and looks over new fields … but cannot land his plane in order to take actual possession, to plow and to harvest.” Perhaps Klein had himself been unaware of this deep schism… Certainly he had perceived “that his most splendid scientific creations were fundamentally gigantic sketches, the completion of which he had to leave to other hands.”