Hilbert Effect


Human beings tend to measure the influence of a person(s) on a particular field of study by associating their name to cornerstones. For example: Urysohn lemma, Tychonoff theorem, Gauss Lemma, Eisenstein Criterion, Chinese Remainder Theorem, Hensel Lemma, Langlands program, Diophantine Analysis, Clifford algebra, Lie Algebra, Riemann Surface, Zariski Topology, Banach–Tarski paradox, Russell paradox, Bernstein polynomial, Bernoulli Number ……

In mathematics their have been some fights about naming the cornerstones, which ended up creating a compund-name. For example, Lobachevsky-Bolyai-Gauss geometry (in textbooks it is generally referred as hyperbolic geometry), Bolzano–Weierstrass theorem (Bolzano prove it in 1817, later Wierstrass proved it again rigorously and popularized it), Schönemann–Eisenstein theorem (in textbooks it is generally referred as Eisenstein Criterion), ……

But, David Hilbert influenced mathematics at a whole new level. Apart from terms like Hilbert Cube (and many more..) named after him, he introduced exotic words in mathematics which are very popular in (research-level) mathematics. Following are some of the terms:

  • Eigen: This word troubled me a lot when I came across the term “eigen-vector” and “eigen-values” a couple of years ago. Hilbert used the German word “eigen”, which means “own”, to denote eigenvalues and eigenvectors of integral operators by viewing the operators as infinite matrices. You can find more information about the history of introduction of this term in mathematics in this web-page by Jeff Miller.
  • Entscheidungsproblem: It is german word for “decision problem”, but still mathematicians tend to use this particular term. For example, the famous paper by Alan Turing titled “On computable numbers, with an application to the Entscheidungsproblem“.
  • Syzygy: Interestingly, “syzygy” is greek word used in astronomy to refer to the nearly straight-line configuration of three celestial bodies in a gravitational system. In Hilbert’s terminology,  “syzygies” are the relations between the generators of an ideal, or, more generally, a module. For more details refer to this article by Roger Wiegand titled “WHAT IS…a Syzygy?“.
  • Nullstellensatz: It is german for “Set of zeros” (according to google translate). But today, just like syzygy, it has whole new meaning in mathematics. For more details, refer to this MathOverflow discussion: What makes a theorem *a* “nullstellensatz.”

Apart from the terms used in mathematics, Hilbert popularized the term “ignorabimus” in philosophy during his famous radio address. For more details read this short Wikipedia article.

It appears that mathematicians (sometimes) tend to use their creativity in naming theorems like Snake Lemma



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