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This paper was written as an assignment for Ian Walton's Math G - Math for liberal Arts Students - at Mission College. If you use material please acknowledge it.

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Ryan Olein

Final Paper - Math G S01

4/03/01

Hilbert’s Problems:

 

David Hilbert was born in what was then East Prussia in 1862. He received his doctorate from the University of Konigsberg in 1884. He continued on and taught at Konigsberg University from 1886 – 1895. Hilbert’s contributions to mathematics specifically algebra, number theory, and geometry were quite significant. In 1900 he was invited to the International Congress of Mathematicians in Paris to give an address. In this address he presented what have come to be called "Hilbert’s hit-list". This list is a series of twenty-three unsolved problems in mathematics. Some of these problems have remained unsolved even today. This list has become a chart by which mathematicians have noted their progress. Solving one of these problems is like attaining the holy grail of mathematics to some; "it would certainly include you in the honors class of the mathematical community." (1) What Hilbert accomplished with this address was as significant as any work in mathematics that he did.

He began his address by capturing the imagination of all the mathematicians at the conference, by pondering the joy of knowing what the future of mathematics held. He then taught a brief history lesson by stating a pattern that he noticed. He said that every generation of scientists has its problems; he then pointed out that these problems are either solved or discarded as worthless by the next generation. Next, he inspired the mathematicians that were in Paris this day by showing that if we found these problems and solved them now that we could look at what the immediate future of mathematics holds in store for us. Finally he made the challenge by listing exactly what he thought these problems were. After this day was done it is my belief that David Hilbert emmerged as one of the great mathematical inspirations of his time. It is my intention with this paper to look into this monumental list that Hilbert has compiled and explore some of the ideas he presented. Further more we have the luxury of looking in on this situation many years after the fact so we will be able to glimpse that future Hilbert spoke of; by looking at what mathematicians have discovered in the pursuit of the solutions to these problems.

It has been reported that Hilbert cited two problems specifically in his address in Paris, so these problems seem like a logical place to start. The two problems were Fermat’s last theorem and the three bodies problem.(2)

Fermat said with his last theorem that except for certain cases that the diophantine equation is unsolvable.(3)

Xn + Yn = Zn

A mathematician by the name of Kummer was inspired by the challenge to solve this mathematical impossibility. Kummer’s work lead to the discovery of the unique law of decomposition of the numbers of a circular field into the prime factors.(4) A discovery that has a profound significance in the mathematical arenas of algebra and the theory of functions. It is quite amazing to think of how Hilbert inspired the mathematicians of his time. This story reminds me a wonderful quote that I came across by Hilbert. Hilbert said, " This will not do. Physics is obviously far too difficult to be left to physicists!" This quote refers to some advice he gave to a physicist by the name of Heisenberg. Apparently when Heisenberg was trying to figure out his matrix equations, Hilbert suggested that he try and find the differential equation that would correspond to his matrix equations. Heisenberg obviously didn’t take this advise because it was Schrodinger that eventually solved the matrix problem with what now is the apparently obvious correlation between these two branches of mathematics.(5) This story further illustrates Hilbert’s grasp of the bigger mathematical picture, and his ability open the eyes of other mathematicians to his way of looking at mathematics.

The second problem that Hilbert mentioned in his Paris address was the three bodies problem. Hilbert used this problem in his presentation more as an example of how solving this generation’s problems can lead to future discoveries. Hilbert cites Poincare’s principles of celestial mechanics as the fruits of the labor of simply trying to solve the three bodies problem. He also spoke on how these principles are invaluable to today’s practical astronomy.

The first problem that Hilbert viewed as essential to modern mathematics as it happens is the "hotel infinity" problem. Better known as the continuum hypothesis in Cantor’s set theory. Hilbert believed the verification of this problem to be very important. He was he heard saying that no one will expel us from the paradise that Cantor created. Cantor developed a number aleph-zero, which is the cardinality of the natural numbers. He then looked at the cardinality of the real numbers and concluded that one infinity is larger than another. Cantor received much criticism for this hypothesis and it eventually lead to a nervous breakdown. Hilbert once explained this hypothesis as a hotel with an infinite number of rooms numbered 1,2,3 and so on. If a bus with an infinite amount of passengers were to arrive at the already full hotel, the manager of the hotel could just ask each guest in the hotel to move to the room number that was double the one they currently occupied. This would free up every other room for the new guests. This also holds true if an infinite number of buses with an infinite number of passengers on each arrives. It was for this reason that Hilbert believed that Cantor’s continuum hypothesis was correct.(6)

Hilbert’s next problem off of his hit list was to prove the compatibility of the axioms of arithmetic. This problem is a great example of what Hilbert’s hit list accomplished, for it has been surrounded by controversy. Hilbert’s theory was "that a definite number of logical steps based upon them can never lead to contradictory results".(7) Many believe that this problem remains unsolved. G. Kreisel believes that Hilbert was correct with his program for most branches of mathematics, but he believes that it fails for arithmetic and metamathematics. I mentioned earlier that I thought this was a great example. I stated this because despite the controversy over whether or not this problem is solved, many changes in the theories of logic have come from it. So whether or not we can ever say that Hilbert’s second problem has been solved, it has certainly been successful.

Hilbert’s third problem asks whether or not any polyhedron can be taken apart and put back together to form a cube of the same volume. Please bear with me on this one it is a little complicated. Hilbert thought the best way to phrase this question was in the terms of the volume of a pyramid. To do this you need to use a method call the devil’s staircase, which I understand is very complicated. Hilbert questioned whether this process was necessary and finally concluded that any polyhedron could not be dissected to form a cube. It was very soon after Hilbert proposed this problem in Paris that it was solved. An individual by the name of Max Dehn confirmed that a regular tetrahedron could not be decomposed into a cube of equal volume.(8) There were many interesting offshoots to this problem. First in 1924 Stefan Banach and Alfred Tarski proved that it is possible to cut a sphere into six pieces and then reassemble them into two spheres each of the same volume as the original sphere. ?? The Hodge conjecture has also been viewed by some as an offshoot of Hilbert’s third problem. The Hodge conjecture states, "that for particularly nice types of spaces called projective algebraic varieties, the pieces called Hodge cycles are actually (rational linear) combinations of geometric pieces called algebraic cycles."(9)

Hilbert’s fourth problem refers to my speech on Euclidean geometry. Hilbert’s fourth problem concludes by citing a new geometry that stands next Euclidean geometry. He derived this conclusion as follows. First if we take normal Euclidean geometry and exclude the axiom of parallels we arrive at the geometry of Lobachevsky. Second Hilbert states that if you take Euclidean geometry and exclude the axiom that says for three points of a straight line one and only one point lies between the other two you arrive at Riemann’s geometry. Finally Hilbert assumes that all the axioms of Euclidean geometry hold true, and the proposition that in every triangle the sum of two sides is greater than the third side is assumed as an axiom. We find a new geometry that stands next to Euclidean geometry. Hilbert concludes that a person named Minkowski as the basis for his arithmetical investigations developed this geometry.(10)

For the sake of my own sanity I will only briefly mention Hilbert’s third problem, because it looks to me to still be written in German despite the fact that it is not. In Hilberts own word’s " How far lie concept of continuous groups of transformations of manifolds is approachable in our investigation without the assumption of differentiability" or " is every locally Euclidean group a lie group" and according to Hilbert the answer is yes.(11)

I would like to jump now to Hilbert’s 8th problem because I find it rather interesting. Hilbert's 8th problem deals with Euler’s zeta function. I find this interesting because my video project was on Euler circuits, so it’s fun to see what else this mathematician has done. The interpretation that Hilbert uses of this problem is that of Riemann. Riemann states that the zeros of the zeta function all have real part of one half. Riemann believed the frequency of prime numbers among the natural numbers didn’t follow a pattern. Riemann did believe however that this frequency related to the zeta function described above.11 I also find this problem very interesting because this interpretation is not a proof. Further this problem has never been proven despite some false alarms. Further, the Clay mathematical institute has placed a million-dollar reward for solving this problem. Maybe we should be working on solving this one in class??

I think Hilbert used a very interesting definition for the word "solved". He believed that proving a formula or problem impossible was equivalent to solving it. I think what Hilbert realized above all, is that it doesn’t really matter if we solve the twenty-three problems that he proposed or not. But in the effort to solve them we will make discoveries that are invaluable to the sciences. To my knowledge there is only one of Hilbert’s original problems that still truly puzzles mathematicians. That problem is the Riemann zeta problem which I mentioned above. Many people believe this to be the most important open problem in pure mathematics.

In my research I also came across people who have tried to create new lists for the modern generation. None of them were as profound or all-encompassing as Hilbert’s, perhaps this has to do with the incredible evolution of modern mathematics. Regardless, these new lists are a testament to the accomplishments and the importance of David Hilbert’s original Twenty-three problems. Hilbert’s work in mathematical theory and the foundations of math have elevated him, as one of the greatest mathematicians of the twentieth century. But I still believe the most important thing he did for math was to focus an entire generation of mathematicians on cutting edge mathematics that changed the world.

 

References

1) author unknown http://www.math.ukans.edu/~engheta/bio/hilbert.html

quote from Hermann Weyl

2) Hilbert, David Mathematical Problems

http://deptinfo.unice.fr/fedou/ENSEIGNEMENT/OFI/GODEL/problems.html

3) http://deptinfo.unice.fr/fedou/ENSEIGNEMENT/OFI/GODEL/problems.html

4) http://deptinfo.unice.fr/fedou/ENSEIGNEMENT/OFI/GODEL/problems.html

5) O'Hear., Anthony. German Philosophy since Kant, Cambridge University Press, 1999

6) V.G. Boltianskii. Hilbert's Third Problem, Halsted Press, New York, 1978

7) Hilbert, David, http://deptinfo.unice.fr/fedou/ENSEIGNEMENT/OFI/GODEL/problems.html#prob2

8) V.G. Boltianskii. Hilbert's Third Problem, Halsted Press, New York, 1978

9) Deligne, Pierre. http://www.claymath.org/prizeproblems/hodge.htm

10) http://deptinfo.unice.fr/fedou/ENSEIGNEMENT/OFI/GODEL/problems.html

11) http://deptinfo.unice.fr/fedou/ENSEIGNEMENT/OFI/GODEL/problems.html

 

This paper was written as an assignment for Ian Walton's Math G - Math for liberal Arts Students - at Mission College. If you use material please acknowledge it.