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Friday, August 24, 2018

David Hilbert and mathematical problems


Photo source: Wikimedia Commons


David Hilbert (1862-1943) was a German Mathematician best known for his contributions to the axiomatization of geometry and Hilbert spaces. Hilbert is also known for compiling a list of 23 unsolved mathematical problems in 1900 for the International Congress of Mathematicians which became influential for the development of mathematics in the 20th century. Science historian A. D'Abro said,


"A more thorough study of Euclid's axioms and postulates proved them to be inadequate for the deduction of Euclid's geometry... Hilbert and others succeeded in filling the gap by stating explicitly a complete system of postulates for Euclidean and non-Euclidean geometries alike." (The Evolution of Scientific Thought from Newton to Einstein, 1927)


This post is a list of quotes from Hilbert.


Mathematics is indivisible


"Mathematical science is in my opinion an indivisible whole, an organism whose vitality is conditioned upon the connection of its parts." (


"The further a mathematical theory is developed, the more harmoniously and uniformly does its construction proceed, and unsuspected relations are disclosed between hitherto separated branches of the science." (


Mathematical problems


"A mathematical problem should be difficult in order to entice us, yet not completely inaccessible, lest it mock at our efforts. It should be a guide post on the mazy paths to hidden truths..." (Mathematical Problems, 1900)


"As long as a branch of science offers an abundance of problems, so long it is alive; a lack of problems foreshadows extinction or the cessation of independent development." (


"If I were to awaken after having slept for a thousand years, my first question would be: has the the Riemann hypothesis been proven? (Quoted in Mathematical Mysteries by Calvin Clawson)


"We can measure the importance of a scientific work by the number of earlier publications rendered superfluous by it." (Quoted in Mathematical Circles Revisited by Howard Eves, 1971)




"Besides it is an error to believe that rigour is the enemy of simplicity. On the contrary we find it confirmed by numerous examples that the rigorous method is at the same time the simpler and more easily comprehend. The very effort for rigour forces us to find out simpler methods of proof." (


" is ingrained in mathematical science that every real advance goes hand in hand with the invention of sharper and simpler methods..." (Mathematical Problems, 1900)