Hilbert's fourth problem
Encyclopedia
In mathematics
, Hilbert's fourth problem in the 1900 Hilbert problems was a foundational question in geometry
. In one statement derived from the original, it was to find geometries
whose axioms are closest to those of Euclidean geometry
if the ordering and incidence
axioms are retained, the congruence
axioms weakened, and the equivalent of the parallel postulate
omitted. A solution was given by Georg Hamel
.
The original statement of Hilbert, however, has also been judged too vague to admit a definitive answer.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Hilbert's fourth problem in the 1900 Hilbert problems was a foundational question in geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
. In one statement derived from the original, it was to find geometries
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
whose axioms are closest to those of Euclidean geometry
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
if the ordering and incidence
Incidence (geometry)
In geometry, the relations of incidence are those such as 'lies on' between points and lines , and 'intersects' . That is, they are the binary relations describing how subsets meet...
axioms are retained, the congruence
Congruence (geometry)
In geometry, two figures are congruent if they have the same shape and size. This means that either object can be repositioned so as to coincide precisely with the other object...
axioms weakened, and the equivalent of the parallel postulate
Parallel postulate
In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in Euclidean geometry...
omitted. A solution was given by Georg Hamel
Georg Hamel
Georg Karl Wilhelm Hamel was a German mathematician with interests in mechanics, the foundations of mathematics and function theory....
.
The original statement of Hilbert, however, has also been judged too vague to admit a definitive answer.