Lecture Notes 1
- Brief Historical Sketch
- Modern Applications
The need for undefined terms
Vish (Vicious Circle) : Start with any word in a dictionary and continue to look up
words used in the definition until some word gets repeated for the first time.
" Vish illustrates the important principle that any definition of a word must inevitably
involve other words, which require further definitions. The only way to avoid a vicious circle
is to regard certain primitive concepts as being so simple and obvious that we agree to leave
them undefined. Similarly, the proof of any statement uses other statements; and since we
must begin somewhere, we agree to leave a few simple statements unproved. These primitive
statements are called axioms." - Coxeter, Projective Geometry, pg. 6.
Example: Vish using The American Heritage Dictionary
- Point : = A dimensionless geometric object having no property but location.
- Location : = A place where something is or might be located.
- Place : = A portion of space.
- Space : = A set of points satisfying specified geometric postulates.
- Point : = ....
Can be traced back to Gino Fano (1892) with some ideas going back to von Staudt (1852).
There are two undefined terms : points, lines. There is also a relation between them called
on. This relationship is symmetric so we speak of points being on lines and lines being on
A Three-Point Geometry:
Theorem 1.1 : Two distinct lines are on exactly one point.
- There exist exactly 3 points in this geometry.
- Two distinct points are on exactly one line.
- Not all the points of the geometry are on the same line.
- Two distinct lines are on at least one point.
Theorem 1.2 : The three point geometry has exactly three lines.
A Four-Line Geometry:
Theorem 1.3 : The four line geometry has exactly six points.
- There exist exactly 4 lines.
- Any two distinct lines have exactly one point on both of them.
- Each point is on exactly two lines.
Theorem 1.4 : Each line of the four-line geometry has exactly 3 points on it.
The plane dual of a statement is the statement obtained by interchanging the terms point and
The plane duals of the axioms for the four-line geometry will give the axioms for the four-
point geometry. And the plane duals of Theorems 1.3 and 1.4 will give valid theorems in the
Theorem 1.7 : Each two lines have exactly one point in common.
- There exists at least one line.
- Every line of the geometry has exactly 3 points on it.
- Not all points of the geometry are on the same line.
- For two distinct points, there exists exactly one line on both of them.
- Each two lines have at least one point on both of them.
Theorem 1.8 : Fano's geometry consists of exactly seven points and seven lines.
5'. If a point does not lie on a given line, then there exists exactly one line on that
point that does not intersect the given line.