Musing upon Hilbert

Hilbert”, by Constance Reid, 290 pages (1972), ISBN 3-540-04999-1

Fame is a curious thing. Everyone knows of Einstein. Most people know of Newton. Many people recognize the names of Fermi and Feynman. Only the hard core have ever heard of Claude Shannon. Which brings us to David Hilbert.

Back in San Francisco during the Summer of Love, Constance Reid decided that one of the most significant mathematicians of the 20th Century deserved a biography, and set herself the challenge of writing that book.

In a mathematical sense, the biography is a disappointment. Most people who have heard of Hilbert are aware he was the developer of Hilbert space – but the reader of this book will learn almost nothing about what Hilbert space is, or why it was such a breakthrough in mathematics. Ms. Reid quotes an old French mathematician: “A mathematical theory is not to be considered complete until you have made is so clear that you can explain it to the first man whom you meet on the street”. As if to prove the difficulty of accomplishing that, Ms. Reid includes as an appendix “David Hilbert and His Mathematical Work”, an appreciation of Hilbert written in 1944 by Hermann Weyl. Mathematics has become a language all to itself.

Ms. Reid decided instead in this volume to focus on Hibbert’s life, tracing it from his birth in 1862 in what was then East Prussia (now Poland), not far from what was then Konigsberg (now Kaliningrad in Russia). Connections abound in this tale. Konigsberg’s most famous son was Imanuel Kant. The city had seven bridges where 18th Century locals used to while away Sunday afternoons trying to find a route to stroll across each of those bridges only once. The great Leonhard Euler had founded the mathematics of topology by proving the task was impossible. Young Hilbert pursued mathematics at the university there.

The book develops into a history of the mathematical institute at the University of Gottingen in central Germany, the former stomping ground of Carl Gauss. Hilbert joined the faculty in Gottingen in 1895 at the age of 33, and spent most of the rest of his life there. He later arranged to be joined by his fellow mathematician from Konigsberg, Hermann Minkowski, who had lectured Albert Einstein (apparently an indifferent student) in Zurich. Mathematics at Gottingen began to attract a galaxy of rising names in mathematics, in part due to Hilbert’s growing reputation.

The institution suffered during World War I and the years of privation that followed in Germany, but gradually regained its position in the 1920s. After the war, those victorious Europeans banned Germans from international mathematical conferences until 1928, when Hilbert led a group of 67 German mathematicians to a Congress in Italy.

Hilbert reached the university’s mandatory retirement age of 68 in 1930, but he remained involved in the active Gottingen mathematical community. And then came the rise of the Nazis, with their demand in 1934 for universities to expel Jewish professors. Most of the prominent mathematicians left for the United States, and Gottingen’s day in the sun was over. Hilbert declined physically, and died during World War II in 1943 at the age of 81.

Occasionally, low probability events occur. Key individuals happen to come together in the same place & time and push human development forward: Ancient Athens, Renaissance Italy, the Scottish Enlightenment, Gottingen, the San Francisco Bay Area in which Ms. Reid wrote this account of Hilbert’s life. All have a bright shining moment, and then it is over. Now instead of Nazis driving Jewish professors out of the universities, we have Woke mediocrities driving out of public life anyone who does not bow down to the shibboleths of the moment. Sadly, there is no longer a liberal United States to which such wanderers could flee.


As we stand, there isn’t much of a United States of any kind, much less classical liberal.


Applications of Hilbert Space to computation (and I don’t mean just quantum computation) strike me as under explored by mathematicians. Reversible computation, in particular, may have important implications for approximating the Algorithmic Information of a set of data since lossless compression can be viewed as reversible computation.

Imagine ML exploration of undirected graphs where the nodes are reversible gates. Feed in the dataset as a bit string on some arc. On all gate ports not on an arc (ie: “outputs” from the graph) count the number of 1s that appear. Add that to the size of the graph. That’s your ML “loss function”. This dramatically shrinks the search space since every graph is a lossless compressor of the dataset.

After the Launch Service Purchase Act of 1990 was basically ignored by NASA and my work with E’Prime Aerospace on commercializing the MX missile was nuked by the DoD (coincidentally) announcing the DC-X the very day of our meeting with VCs in LA, I decided the ocean was probably the best redoubt. That’s why I ended up working with Charlie Smith on a macroengineering plan to cultivate algae. However, given the intransigence by the Big Boys toward capitalizing the manufacturing of Algasol’s photobioreactor (the lynch pin of that plan), and the dramatic drop in solar PV I’m starting to think hydrogenotrophs may be a better base of the redoubt’s food chain.