Gregory Chaitin has devoted his life to the attempt to understand what mathematics can and cannot achieve, and is a member of the digital philosophy/digital physics movement. Its members believe that the world is built out of digital information, out of 0 and 1 bits, and they view the universe as a giant information-processing machine, a giant digital computer. In this book on the history of ideas, Chaitin traces digital philosophy back to the nearly-forgotten 17th century genius Leibniz. He also tells us how he discovered the celebrated Omega number, which marks the current boundary of what mathematics can achieve. This book is an opportunity to get inside the head of a creative mathematician and see what makes him tick, and opens a window for its readers onto a glittering world of high-altitude thought that few intellectual mountain climbers can ever glimpse.

The Goldbach conjecture states that all positive even integers greater than or equal to four can be expressed as the sum of two odd prime numbers.

Cristian Goldbach (1690-1764) submitted his conjecture to Euler who dismissed it as trivial, however as of today (2003) it has not yet been proved.

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Though you might not guess it from trying to read some of the research papers, the whole point of mathematics is to make things simpler. No one has taken this more seriously than the topologists, a rarefied breed of thinkers who insist that the world, however messy and diverse it may appear, is really made of just two basic shapes, the doughnut and the sphere.

Actually it's a bit more complicated than that — the doughnuts can have more than one hole, for example, and the topologists don't limit themselves to the usual three dimensions. Lately, they have been preoccupied with claims that a Russian mathematician has solved a famous century-old problem involving what might be called hyperdoughnuts and hyperspheres existing in an imaginary four-dimensional space.

143-Year-Old Problem Still Has Mathematicians Guessing
The Riemann hypothesis, first tossed off by Bernhard Riemann in 1859 in a paper about the distribution of prime numbers, is still widely considered to be one of the greatest unsolved problems in mathematics.