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By Constance Reid

"It provides a delicate portrait of an excellent man or woman. It describes properly and intelligibly on a nontechnical point the realm of mathematical rules during which Hilbert created his masterpieces. And it illuminates the heritage of German social heritage opposed to which the drama of Hilberts existence used to be performed. past this, it's a poem in compliment of mathematics." -SCIENCE

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Nevertheless, Kronecker remarked petulantly to Minkowski that he was going to stop sending papers to Hilbert if Hilbert did not send papers to him. Hilbert promptly composed a letter which managed to be formal and respectful but firm: "I remember exactly, and my list of mailed papers also shows it clearly, that I have taken the liberty of sending you a copy of each. paper without exception immediately after its publication; and you have had the kindness to send your thanks on postcards for some of the last mailings.

By January 30, however, having received two explanatory letters from Hilbert in the intervening time, Cayley was congratulating the young German: "My difficulty was an a priori one, I thought that the like process should be applicable to semi-invariants, which it seems it is not; and now I quite see .... " Hilbert had solved Gordan's Problem very much as Alexander had untied the Gordian Knot. At Gordium [Plutarch tells us] he saw the famous chariot fastened with cords made of the rind of the cornel-tree, which whosoever should untie, the inhabitants had a tradition, that for him was reserved the empire of the world.

And it was here that Hilbert found at last the powerful new tools he had been seeking. In a key work, in 1892, he took up the question of exactly what was needed to produce in actuality a full system of invariants in terms of which all the other invariants could be represented. Using as a foundation the theorem which he had earlier proved, he was able to produce what was in essence a finite means of executing the long sought construction. Although Hilbert was not the first to make use of indirect, non-constructive proofs, he was the first to recognize their deep significance and value 36 and to utilize them in dramatic and extremely beautiful ways.

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