"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. Among the postulates missing in Euclid's list was the celebrated postulate of Archimedes, according to which, by placing an indefinite number of equal lengths end to end along a line, we should eventually pass any point arbitrarily selected on the line. Hilbert, by denying this postulate, just as Lobatchewski and Riemann had denied Euclid's parallel postulate, succeeded in constructing a new geometry known as non-Archimedean. It was perfectly consistent but much stranger than the classical non-Euclidean varieties. Likewise, it was proved possible to posit a system of postulates which would yield Euclidean or non-Euclidean geometries of any number of dimensions; hence, so far as rational requirements of the mind were concerned, there was no reason to limit geometry to three dimensions."
Quote Details
Added by wikiquote-import-bot
Unverified quote
0 likes
Academics from GermanyAgnosticsMathematicians from GermanyPhysicists from GermanyLogicians from Germany
Original Language: English
Available Languages (1)
Sources
A. D'Abro, The Evolution of Scientific Thought from Newton to Einstein (1927) pp. 37-38
https://en.wikiquote.org/wiki/David_Hilbert
Revision History
No revisions have been submitted for this quote.
Categories
David Hilbert
1862 – 1943
deutscher Mathematiker
46 quotes on TrueQuotesView all quotes by David Hilbert →
Related Quotes
"If I were to awaken after having slept for a thousand years, my first question would be: Has the Riemann hypothesis b…"
"One of the supreme achievements of purely intellectual human activity."
"If one were to bring ten of the wisest men in the world together and ask them what was the most stupid thing in exist…"
""Mathematics is a presuppositionless science. To found it I do not need God, as does Kronecker, or the assumption of …"
"One can measure the importance of a scientific work by the number of earlier publications rendered superfluous by it."
"The art of doing mathematics consists in finding that special case which contains all the germs of generality."
"Mathematics knows no races or geographic boundaries; for mathematics, the cultural world is one country."
"Wir dürfen nicht denen glauben, die heute mit philosophischer Miene und überlegenem Tone den Kulturuntergang propheze…"
"Every kind of science, if it has only reached a certain degree of maturity, automatically becomes a part of mathematics."
"But he (Galileo) was not an idiot,... Only an idiot could believe that scientific truth needs martyrdom — that may be…"