Earlier this year, I took a week to sing the praises of Leonhard Euler, whose 300th birthday we celebrate this year. There used to be a famous list of the three greatest mathematicians, Archimedes, Newton and Gauss, which most people now extend to four, adding in Euler. This list takes us to the 1850s, and there are many more recent mathematicians who can be fairly added to this list now. David Hilbert, born in 1862, is an easy consensus pick. (Also, let's give him some credit for style. That's one big pimpin' hat.)
The University of St. Andrews maintains the best website of mathematical biographies on the 'Net. Their summary of Hilbert reads as follows.
Hilbert's work in geometry had the greatest influence in that area after Euclid. A systematic study of the axioms of Euclidean geometry led Hilbert to propose 21 such axioms and he analysed their significance. He made contributions in many areas of mathematics and physics.
I will use as little actual math as possible as I sing Hilbert's praises. In math, when a proof is done without going into the details, it is called a hand waving proof. This will be a hand waving proof of Hilbert's greatness.
The first sign of mathematical greatness is by doing important work in several different fields, and here Hilbert clearly qualifies. There are many different branches of math, and the first major split is between algebra and geometry, which anyone who took high school math knows are different. If you took analytical geometry, you'll know that equations in x and y can be used to define geometric shapes like lines, parabolas and circles.
Let me explain one of Hilbert's theorems in general terms. Take any silhouette, like the one shown here. The edge between the two colors is defined as a closed curve in math. This particular silhouette is two closed curves, as there are two separate black areas. Pick any closed curve on a flat surface and any tolerance, say in this case, 1/1000 of an inch. Hilbert proved there must be a polynomial equation in x and y that will a draw a closed curve that will stay within the requested tolerance of the requested shape.
Hilbert's first great work, his Basis Theorem, was not without detractors. The great basis theorem before Hilbert belongs to Gordan, and uses a constructive approach. Hilbert worked hard trying to extend Gordan's work using Gordan's methods, but this proved very difficult, so he switched to a more abstract approach. The problem was, the most important place to publish this work in German was in the Annalen, at that time edited by Gordan and his good friend Felix Klein. Gordan, not wanting to see his way of doing things discarded, opposed publication, and it was assumed Klein would agree out of loyalty. But Klein read Hilbert's paper, which Hilbert said he would not change to spare Gordan's feelings, and Klein liked what he saw and asked for more of the same. The paper was published in 1888, and Klein prefaced the work with the statement,
I do not doubt that this is the most important work on general algebra that the Annalen has ever published.
Hilbert's reputation grew. In 1900, the international mathematical meeting was held in Paris, and Hilbert gave a talk outlining what he thought were the 23 most important unsolved problems in math at that time. Many were solved within months of Hilbert's talk. One was famously proved impossible to solve by the Incompleteness Theorem of Kurt Gödel. Some are unsolved to this day. To this day, mathematicians talking in shorthand will say "That's Hilbert's 7th." or "In Hilbert's 10th Problem...". When the international meeting rolled around in 2000, there wasn't a great universalist like Hilbert to come forward and make a new list everyone could agree upon.
For all his greatness as a mathematician, I would probably not have David Hilbert as the leadoff man in my week of heroes but for what he did to advance the career of Emmy Noether. Emmy, daughter of mathematician Max Noether, had done her doctoral work on the basis theorem problem, working very diligently to extend the work of Gordan. Frustrated by the lack of progress, not unlike Hilbert some twenty years before, in 1908 she took a more abstract approach and made some important breakthroughs. Hilbert himself considered her work the future of algebra, and in 1915 invited her to teach at Göttingen, one of the pre-eminent universities in the world in the field of math, where Hilbert was the head of the department. At the time, there were no women working at any European universities in mathematics, and the heads of the university were against the hire, even with her backing by the great Hilbert. It took four years for her to finally be hired full-time, working in the interim as a Privatdozent, which meant she wasn't paid by the university, but instead gave lectures and the students would pay her after class, like a street musician is paid. (And I thought working at community colleges is a tough way to make a living.)
The arguments against her hiring were against her having a vote in the academic senate, and to spare the feelings of the poor soldiers who would come home from the war to learn at the feet of a woman. Hilbert's statement on the topic was simple. "I do not see that the sex of the candidate is an argument against her admission as a Privatdozent. After all, the university senate is not a bathhouse."
The great man retired in 1930. In 1933, Emmy Noether was removed from the faculty at Göttingen, not for being a woman but for being a Jew. She emigrated to the United States and taught at Bryn Mawr. Another of her intellectual admirers, Albert Einstein, arranged for her to lecture at the Institute of Advanced Studies at Princeton. Whether Hilbert, who wasn't Jewish, would have done any better for Emmy Noether against the Nazis that his successor did, we will never know. A few years later, a Nazi newspaper came to interview the great German hero in retirement. Asked what great things he saw in the future for German mathematics at his old university, he simply answered, "There is no important mathematics being done in Göttingen now."
I will end this tribute to Hilbert with the quote that exemplifies the optimism of the scientific spirit of the early 20th Century, a spirit we can only hope to recapture. In 1905, he ended a talk with these six words:
Wir müssen wissen, wir werden wissen.
Translated into English, Hilbert said "We must know, we shall know."