Scanned, proofed and corrected from the original magazine edition for enjoyable reading. (Worth every penny spent!)
Hilbert put forth a most influential list of 23 unsolved problems at the International Congress of Mathematicians in Paris in 1900. This is generally reckoned the most successful and deeply considered compilation of open problems ever to be produced by an individual mathematician.
— Excerpted from David Hilbert on Wikipedia.
In 1902 David Hilbert lectured on mathematical problems which were at that time awaiting solutions by the hands of competent mathematicians. This piece has been translated into English by Dr. Mary Newson
for the Bulletin of the American Mathematical Society.
Some of the problems are as follows:
Cantor's problem of the cardinal number of the continuum;
The compatibility of the axioms of arithmetic;
The equality of the volumes of two tetrahedra of equal bases and equal altitudes;
The problem of the straight line as the shortest distance between two points
Lie's concept of a continuous group of transformations without assuming that the functions defining the group are capable of differentiation;
The treatment of the axioms of physics as we treat the axioms of mathematics, placing in the first rank probabilities and mechanics;
The irrationality and transcendence of certain numbers;
Riemann's prime number formula;
Goldback's theorem, that every integer is expressible as the sum of two primes;
Is there an infinite number of pairs of primes differing by 2?
Is ax+by+c=o soluble in prime numbers x and y, where a, b are integral and a prime to b?
to apply the results obtained for the distribution of rational prime numbers to the theory of the distribution of ideal primes in a given number field k;
...and so on.
Poincaré is discoursed on the role of intuition and logic in mathematics, showing how, while intuition is often the source of discovery, it is logic which harmonizes and consolidates the creations of intuition.