Open main menu
Home
Random
Recent changes
Special pages
Community portal
Preferences
About Wikipedia
Disclaimers
Incubator escapee wiki
Search
User menu
Talk
Dark mode
Contributions
Create account
Log in
Editing
Hilbert's fourth problem
(section)
Warning:
You are not logged in. Your IP address will be publicly visible if you make any edits. If you
log in
or
create an account
, your edits will be attributed to your username, along with other benefits.
Anti-spam check. Do
not
fill this in!
==Original statement== Hilbert discusses the existence of [[non-Euclidean geometry]] and [[non-Archimedean geometry]] <blockquote> ...a geometry in which all the axioms of ordinary euclidean geometry hold, and in particular all the congruence axioms except the one of the congruence of triangles (or all except the theorem of the equality of the base angles in the isosceles triangle), and in which, besides, the proposition that in every triangle the sum of two sides is greater than the third is assumed as a particular axiom.<ref name="Hilbert1900">Hilbert, David, "Mathematische Probleme" [[Göttinger Nachrichten]], (1900), pp. 253–297, and in [[Archiv der Mathematik und Physik]], (3) '''1''' (1901), 44–63 and 213–237. Published in English translation by Dr. Maby Winton Newson, {{cite journal | last1=Hilbert | first1=David | authorlink1=David Hilbert | journal=[[Bulletin of the American Mathematical Society]] | title=Mathematical Problems | volume=8 | date=1902 | issue=10 | pages=437–479 | doi=10.1090/S0002-9904-1902-00923-3 | doi-access=free}}. [A fuller title of the journal Göttinger Nachrichten is Nachrichten von der Königl. Gesellschaft der Wiss. zu Göttingen.]</ref> </blockquote> Due to the idea that a 'straight line' is defined as the shortest path between two points, he mentions how congruence of triangles is necessary for Euclid's proof that a straight line in the plane is the shortest distance between two points. He summarizes as follows: <blockquote> The theorem of the straight line as the shortest distance between two points and the essentially equivalent theorem of Euclid about the sides of a triangle, play an important part not only in number theory but also in the theory of surfaces and in the calculus of variations. For this reason, and because I believe that the thorough investigation of the conditions for the validity of this theorem will throw a new light upon the idea of distance, as well as upon other elementary ideas, e. g., upon the idea of the plane, and the possibility of its definition by means of the idea of the straight line, ''the construction and systematic treatment of the geometries here possible seem to me desirable.''<ref name="Hilbert1900" /> </blockquote>
Edit summary
(Briefly describe your changes)
By publishing changes, you agree to the
Terms of Use
, and you irrevocably agree to release your contribution under the
CC BY-SA 4.0 License
and the
GFDL
. You agree that a hyperlink or URL is sufficient attribution under the Creative Commons license.
Cancel
Editing help
(opens in new window)