Jeremy Gray’s “The Hilbert Challenge”, using diagrams, and using simple special cases

David Hilbert’s famous speech to the International Congress of Mathematicians in 1900 is included as an appendix to this book.

The speech was a listing of what Hilbert saw as the main open problems of maths at the time, and how they might be approached.

Interestingly, in this speech to professional mathematical researchers, Hilbert cites two of the main ideas on how to approach problems that we teach to sixth-form maths students today.

One: do a diagram. Or many different diagrams, visualising the problem in different ways

“Arithmetical symbols are written figures and geometrical figures are drawn formulas; and no mathematician could do without these drawn formulas, any more than in calculation he could dispense with the insertion and removal of brackets…”

Two: start by trying to solve a simplified version or special case of the problem, and then build up to the full problem

“Perhaps in most cases when we unsuccessfully seek an answer to a question, the cause of the failure lies in the fact that problems simpler and easier than the one in hand have either been incompletely solved, or not solved at all. Everything depends, then, on finding those easier problems and solving by means of methods as perfect as possible and devices capable of generalisation”.

Between the mid-19th century and the 1930s, mathematics changed in a big way. It became the study of patterns, structures, and systems, rather than of formulas and calculations.

Read here about Lejeune Dirichlet’s contribution to that shift, and here about Emmy Noether’s contribution.

Maybe the central figure in the shift, and the central figure in early 20th century maths, was the German mathematician David Hilbert.

In 1900, at an International Congress of Mathematicians, Hilbert made a speech in which he outlined 23 problems to map out the main directions in mathematical research for the new century.

Some of those problems were solved quickly. One proved to have been stated too unclearly by Hilbert to be solved. Some remain unsolved. Many have been solved only relatively recently, and often the solution has turned out to be very different from what anyone in Hilbert’s day expected. A whole heap of new problems, for example those associated with the Langlands program, have gained attention. The rise of electronic computers has made combinatorics, discrete mathematics, and algorithmics much more central than they were back in 1900.

Jeremy Gray’s book telling the story of the problems and the attempts to solve them gives very valuable elements of a map of 20th century mathematics.

The book is described as suitable for “the general reader with an interest in mathematics”. That is, I think, a “general reader” with a maths degree. Nevertheless, the book is written clearly enough that a keen A level student can read it and, while finding some bits too difficult to follow in detail, follow the general drift of the book and learn a lot from it.