Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics

by John Derbyshire

From the point of view of analysis, however, a series is just a sequence in thin disguise. The statement “The series converges to 2” is mathematically equivalent to: “The sequence converges to 2.” The fourth term of the sequence is the sum of the first four terms of the series, and so on. (The term of art for this kind of sequence is the sequence of partial sums.)

From the point of view of analysis, however, a series is just a sequence in thin disguise. The statement “The series converges to 2” is mathematically equivalent to: “The sequence converges to 2.” The fourth term of the sequence is the sum of the first four terms of the series, and so on. (The term of art for this kind of sequence is the sequence of partial sums.)

A lay person might think that it would be satisfying enough to know the number to half a dozen places of decimals. No: mathematicians want to know it exactly, if they can. Not just because they are weird obsessives, but because they know from experience that getting that exact value often opens unexpected doors and throws light on the underlying math. The mathematical term of art for this exact representation of a number is “closed form.” A mere decimal approximation, however good, is an “open form.” The number 1.6449340668… is an open form.

Hilbert had never liked this gloomy philosophy. Now, with all the world (or at any rate the scientific-mathematical part of it) listening, he gave it a last resounding kick. We ought not believe those who today, with a philosophical air and a tone of superiority, prophesy the decline of culture, and are smug in their acceptance of the Ignorabimus principle. For us there is no Ignorabimus, and in my opinion there is none for the natural sciences either. In place of this foolish Ignorabimus, let our resolution be, to the contrary: “We must know, we shall know.”