Nature and Influence of The Problems
Hilbert's problems ranged greatly in topic and precision. Some of them are propounded precisely enough to enable a clear affirmative/negative answer, like the 3rd problem (probably the easiest for a nonspecialist to understand and also the first to be solved) or the notorious 8th problem (the Riemann hypothesis). There are other problems (notably the 5th) for which experts have traditionally agreed on a single interpretation and a solution to the accepted interpretation has been given, but for which there remain unsolved problems which are so closely related as to be, perhaps, part of what Hilbert intended. Sometimes Hilbert's statements were not precise enough to specify a particular problem but were suggestive enough so that certain problems of more contemporary origin seem to apply, e.g. most modern number theorists would probably see the 9th problem as referring to the (conjectural) Langlands correspondence on representations of the absolute Galois group of a number field. Still other problems (e.g. the 11th and the 16th) concern what are now flourishing mathematical subdisciplines, like the theories of quadratic forms and real algebraic curves.
There are two problems which are not only unresolved but may in fact be unresolvable by modern standards. The 6th problem concerns the axiomatization of physics, a goal that twentieth century developments of physics (including its recognition as a discipline independent from mathematics) seem to render both more remote and less important than in Hilbert's time. Also, the 4th problem concerns the foundations of geometry, in a manner which is now generally judged to be too vague to enable a definitive answer.
Remarkably, the other twenty-one problems have all received significant attention, and late into the twentieth century work on these problems was still considered to be of the greatest importance. Notably, Paul Cohen received the Fields Medal during 1966 for his work on the first problem, and the negative solution of the tenth problem during 1970 by Matiyasevich (completing work of Davis, Putnam and Robinson) generated similar acclaim. Aspects of these problems are still of great interest today.
Read more about this topic: Hilbert's Problems
Famous quotes containing the words nature and, nature, influence and/or problems:
“When, in the course of human events, it becomes necessary for one people to dissolve the political bands which have connected them with another, and to assume the powers of the earth, the separate and equal station to which the laws of nature and of nature’s God entitle them, a decent respect to the opinions of mankind requires that they should declare the causes which impel them to the separation.”
—Thomas Jefferson (1743–1826)
“What art can paint or gild any object in afterlife with the glow which Nature gives to the first baubles of childhood. St. Peter’s cannot have the magical power over us that the red and gold covers of our first picture-book possessed. How the imagination cleaves to the warm glories of that tinsel even now! What entertainments make every day bright and short for the fine freshman!”
—Ralph Waldo Emerson (1803–1882)
“A bestial and violent man will go so far as to kill because he is under the influence of drink, exasperated, or driven by rage and alcohol. He is paltry. He does not know the pleasure of killing, the charity of bestowing death like a caress, of linking it with the play of the noble wild beasts: every cat, every tiger, embraces its prey and licks it even while it destroys it.”
—Colette [Sidonie Gabrielle Colette] (1873–1954)
“Hats have never at all been one of the vexing problems of my life, but, indifferent as I am, these render me speechless. I should think a well-taught and tasteful American milliner would go mad in England, and eventually hang herself with bolts of green and scarlet ribbon—the favorite colour combination in Liverpool.”
—Willa Cather (1876–1947)