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
Foundations of mathematics
(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!
=== Synthetic vs. analytic geometry === Since the introduction of [[analytic geometry]] by [[René Descartes]] in the 17th century, there were two approaches to geometry, the old one called [[synthetic geometry]], and the new one, where everything is specified in terms of real numbers called [[coordinates]]. Mathematicians did not worry much about the contradiction between these two approaches before the mid-nineteenth century, where there was "an acrimonious controversy between the proponents of synthetic and analytic methods in [[projective geometry]], the two sides accusing each other of mixing projective and metric concepts".<ref>Laptev, B.L. & B.A. Rozenfel'd (1996) ''Mathematics of the 19th Century: Geometry'', page 40, [[Springer Science+Business Media|Birkhäuser]] {{ISBN|3-7643-5048-2}}</ref> Indeed, there is no concept of distance in a [[projective space]], and the [[cross-ratio]], which is a number, is a basic concept of synthetic projective geometry. [[Karl von Staudt]] developed a purely geometric approach to this problem by introducing "throws" that form what is presently called a [[field (mathematics)|field]], in which the cross ratio can be expressed. Apparently, the problem of the equivalence between analytic and synthetic approach was completely solved only with [[Emil Artin]]'s book ''[[Geometric Algebra (book)|Geometric Algebra]]'' published in 1957. It was well known that, given a [[field (mathematics)|field]] {{mvar|k}}, one may define [[affine space|affine]] and projective spaces over {{mvar|k}} in terms of {{mvar|k}}-[[vector space]]s. In these spaces, the [[Pappus hexagon theorem]] holds. Conversely, if the Pappus hexagon theorem is included in the axioms of a plane geometry, then one can define a field {{mvar|k}} such that the geometry is the same as the affine or projective geometry over {{mvar|k}}.
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)