Hilbert discovered and developed a range of fundamental ideas in many areas. He worked on invariant theory, the axiomization of geometry, and the notion of Hilbert space. This is one of the foundations of functional analysis. Hilbert and his students supplied much of the mathematics needed for quantum mechanics and general relativity. He was one of the founders of proof theory and mathematical logic. He was also one of the first people to make the distinction between mathematics and metamathematics, and warmly defended Georg Cantor's set theory and transfinite numbers.
In 1895 Hilbert became Chairman of Mathematics at the University of Göttingen, at that time the best research center for mathematics in the world. He remained for the rest of his life. Among his students were: Hermann Weyl, the champion of chess Emanuel Lasker, Ernst Zermelo, and Carl Gustav Hempel. John von Neumann was his assistant. At the University of Göttingen, Hilbert was surrounded by a social circle of some of the most important mathematicians of the 20th century, such as Emmy Noether and Alonzo Church.
The text "Grundlagen der Geometrie" ("Foundations of Geometry") was published by Hilbert in 1899. It proposed a formal set, Hilbert's axioms, instead of the traditional axioms of Euclid. They avoid weaknesses in those of Euclid, whose works at the time were still used textbmathematics is his 1900 presentation of a set of problems that set the course for much of the mathematical research of the 20th century.
He put forward a number of unsolved problems at the International Congress of Mathematicians in Paris in 1900. This is reckoned the most successful and deeply considered compilation of open problems ever to be produced by an individual mathematician. Later he expanded his list to 23 problems.
In 1920 he proposed explicitly a research project in "metamathematics", which became known as Hilbert's program. He wanted mathematics to be formulated on a solid and complete logical foundation. He believed that in principle this could be done, by showing that:
He seems to have had both technical and philosophical reasons for formulating this proposal.
After 1912, Hilbert turned his focus to physics. At that time, he worked in general relativity and mathematical physics. His work in these fields is also important.
Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell (18 May 1872 – 2 February 1970) was a British philosopher, logician, and mathematician. He was born in Wales, but spent most of his life in England. He worked mostly in the 20th century. He won the Nobel Prize in Literature in 1950.
Bertrand Russell wrote many books and articles. He also tried to make philosophy popular. He gave his opinion on many topics. He wrote the essay, "On Denoting", which has been described as one of the most influential essays in philosophy in the 20th Century. He wrote on very serious issues as well as everyday things.
He was a well known liberal as well as a socialist and anti-war activist for most of his long life. Millions looked up to Russell as a prophet of the creative and rational life. At the same time, his stances on many topics were extremely controversial. From 1931 until his death, he was a member of the House of Lords.
Born at the height of Britain's economic and political ascendancy, he died of influenza nearly a century later when the British Empire had all but vanished, its power dissipated in two victorious, but debilitating world wars. Russell's voice carried enormous moral authority, even into his early 90s. Russell supported nuclear disarmament a lot, but did not support the American war in Vietnam even when it was popular.
In 1950, Russell was made a Nobel Laureate in Literature "in recognition of his varied and significant writings in which he champions humanitarian ideals and freedom of thought". He died of influenza.
In his 1949 speech, "Am I an Atheist or an Agnostic?", Russell expressed his difficulty over whether to call himself an atheist or an agnostic:
In Russel's book "The Impact of Science on Society" he wrote:
This is a selected bibliography of Russell's books in English sorted by year of first publication.
Note: This is a mere sampling, for Russell also authored many pamphlets, introductions, articles and letters to the editor. His works also can be found in any number of anthologies and collections, perhaps most notably, "The Collected Papers of Bertrand Russell", which McMaster University began publishing in 1983. This collection of his shorter and previously unpublished works is now up to 16 volumes, and many more are forthcoming. An additional 3 volumes catalogue just his bibliography. The Russell Archives at McMaster also have more than 30,000 letters that he wrote.
Additional references:
A. Russell
B. Secondary references:
Alfred North Whitehead
Alfred North Whitehead, OM (15 February 1861 – 30 December 1947) was an English mathematician who became a philosopher. He was born in Ramsgate, Kent, and died in Cambridge, Massachusetts, USA. He wrote on algebra, logic, foundations of mathematics, philosophy of science, physics, metaphysics, and education. He is the co-author, with Bertrand Russell, of the important "Principia Mathematica".
Mask
A mask is a type of clothing which covers the face. It is similar to a veil, but more close-fitting.
There are different reasons why people use masks. A person can use a mask so others do not know who they are (as when committing a crime). Or, a mask may keep someones face safe from harm (like a gas mask so harmful gases do not get into the lungs).
There are many practical uses for masks in everyday life. Mostly, the masks used are "semi-masks" which cover only the bottom half of the face, the mouth and nose. An example is the use of masks in operating theatres, and
by wood-workers. In surgery, the purpose is to keep bacteria off the patient; in wood-working the purpose is to keep wood fibres out of the lungs.
In Venetian masks only the top half of the face is covered, to prevent recognition. Ice hockey masks, used in the Hannibal Lecter movies, cover the whole face to protect it from the hockey puck moving at high speed.
The word 'mask' might have come from other languages. The French "masque", the Italian "maschera", the Spanish "máscara", the Latin (not classical) "mascus/masca (meaning "ghost")" or the Arabic "maskharah (meaning "jester" or "man in masquerade")" all use the same word.
One of the oldest masks found is the Death Mask. This stone mask is 9000 years old and is probably the oldest mask in the world. It is kept in Musée de la bible et Terre Sainte, Paris.
There are also three masks from the same period, kept in The Israel Museum, Jerusalem.
Masks are used in many magical or shamanistic rituals in non-western societies. The celebration or appeasement of animistic spirits takes place with dance and song in ancient performances in many parts of the world. In these events performers use costumes and masks to mimic the supposed spirits.
From this kind of beginning, the use of masks in ancient Greek theatre may have started. The actors and chorus in this early theatre all used masks.
Masks have been used to protect people against air-borne diseases. They were worn during the plague in the Middle Ages; during the influenza epidemic of 1918, and the COVID-19 epidemic. Gas masks were issued in World War II even though phosgene gas was never used in that war (it had been in WWI).
During the 2019-20 coronavirus pandemic, a new variation of trikini, with two pieces for the body and a face mask was designed. Indian film actress Kajol popularized the idea of this trikini.