Editing David Hilbert (section)

===Axiomatization of geometry===
{{Main|Hilbert's axioms}}

The text ''[[Grundlagen der Geometrie]]'' (tr.: ''Foundations of Geometry'') published by Hilbert in 1899 proposes a formal set, called Hilbert's axioms, substituting for the traditional [[Euclid's elements|axioms of Euclid]]. They avoid weaknesses identified in those of [[Euclid]], whose works at the time were still used textbook-fashion. It is difficult to specify the axioms used by Hilbert without referring to the publication history of the ''Grundlagen'' since Hilbert changed and modified them several times. The original monograph was quickly followed by a French translation, in which Hilbert added V.2, the Completeness Axiom. An English translation, authorized by Hilbert, was made by E.J. Townsend and copyrighted in 1902.<ref>{{harvnb|Hilbert|1950}}</ref><ref>[[G. B. Mathews]](1909) [http://www.nature.com/nature/journal/v80/n2066/pdf/080394a0.pdf The Foundations of Geometry] from [[Nature (journal)|Nature]] 80:394,5 (#2066)</ref> This translation incorporated the changes made in the French translation and so is considered to be a translation of the 2nd edition. Hilbert continued to make changes in the text and several editions appeared in German. The 7th edition was the last to appear in Hilbert's lifetime. New editions followed the 7th, but the main text was essentially not revised.{{efn|Independently and contemporaneously, a 19&nbsp;year-old American student named [[Robert Lee Moore]] published an equivalent set of axioms. Some of the axioms coincide, while some of the axioms in Moore's system are theorems in Hilbert's and vice versa. {{citation needed|date=December 2020}}}}

Hilbert's approach signaled the shift to the modern [[axiomatic method]]. In this, Hilbert was anticipated by [[Moritz Pasch]]'s work from 1882.  Axioms are not taken as self-evident truths. Geometry may treat ''things'', about which we have powerful intuitions, but it is not necessary to assign any explicit meaning to the undefined concepts. The elements, such as [[point (geometry)|point]], [[Line (geometry)|line]], [[plane (geometry)|plane]], and others, could be substituted, as Hilbert is reported to have said to [[Schoenflies]] and [[Ernst Kötter|Kötter]], by tables, chairs, glasses of beer and other such objects.<ref>{{cite book |author=Otto Blumenthal |title=Lebensgeschichte |year=1935 |volume=3 |pages=388–429 |publisher=Julius Springer |editor=David Hilbert | series=Gesammelte Abhandlungen |url=http://gdz-lucene.tc.sub.uni-goettingen.de/gcs/gcs?action=pdf&metsFile=PPN237834022&divID=LOG_0001&pagesize=original&pdfTitlePage=http://gdz.sub.uni-goettingen.de/dms/load/pdftitle/?metsFile=PPN237834022%7C&targetFileName=PPN237834022_LOG_0001.pdf& |access-date=6 September 2018 |archive-url=https://web.archive.org/web/20160304122623/http://gdz-lucene.tc.sub.uni-goettingen.de/gcs/gcs?action=pdf&metsFile=PPN237834022&divID=LOG_0001&pagesize=original&pdfTitlePage=http%3A%2F%2Fgdz.sub.uni-goettingen.de%2Fdms%2Fload%2Fpdftitle%2F%3FmetsFile%3DPPN237834022%7C&targetFileName=PPN237834022_LOG_0001.pdf& |archive-date=4 March 2016 |url-status=dead |author-link=Otto Blumenthal}} Here: p.402-403</ref> It is their defined relationships that are discussed.

Hilbert first enumerates the undefined concepts: point, line, plane, lying on (a relation between points and lines, points and planes, and lines and planes), betweenness, congruence of pairs of points ([[line segment]]s), and [[criteria of congruence of angles|congruence]] of [[angle]]s. The axioms unify both the [[Euclidean geometry|plane geometry]] and [[solid geometry]] of Euclid in a single system.