3. Words differently arranged have a different meanings and
meanings differently arranged have a different effect.
- Blaise Pascal (1623-1662)
The study of any mathematics
requires an understanding of the nature of
deductive reasoning; frequently, geometry
has been singled out for introducing this
methodology to secondary school students.
This topic introduces the terminology
essential for a discussion of deductive
reasoning so that the extraordinary
influence of the history of geometry on the
modern understanding of deductive
reasoning will become evident.
Introduction
Components
Axiomatic
Systems
Example
Finite
Projective
Planes
Properties
Enrichment
4. Deductive reasoning takes place in
the context of an organized logical structure
called an axiomatic ( or deductive) system.
One of the pitfalls of working with a
deductive system is too great a familiarity
with the subject matter of the system. We
need to be careful with what we are
assuming to be true and with saying
something is obvious while writing a proof.
Introduction
Components
Axiomatic
Systems
Example
Finite
Projective
Planes
Properties
Enrichment
5. Remember:
it is crucially important in a proof to use
only the axioms and the theorems which
have been derived from them and not
depend on any preconceived idea or
picture.
diagrams should be used as an aid, since
they are useful in developing conceptual
understanding.
Introduction
Components
Axiomatic
Systems
Example
Finite
Projective
Planes
Properties
Enrichment
6. Components of an Axiomatic
System
Undefined Terms
Certain terms are left undefined to
prevent circular definitions. Examples of
undefined terms (primitive terms) in
geometry are point, line , plane, on and
between.
Undefined terms are of two types:
terms that imply objects, called elements,
and terms that imply relationships between
objects, called relations.
Introduction
Components
Axiomatic
Systems
Example
Finite
Projective
Planes
Properties
Enrichment
7. Components of an Axiomatic
System
Defined Terms
They are not necessary, but in nearly
every axiomatic system certain phrases
involving undefined terms are used
repeatedly. Thus, it is more efficient to
substitute a new term, that is, a defined
term, for each of these phrases whenever
they occur.
Example:
lines that do not intersect = parallel lines
Introduction
Components
Axiomatic
Systems
Example
Finite
Projective
Planes
Properties
Enrichment
8. Components of an Axiomatic
System
Axioms
Statements that are accepted without
proof.
Early GREEKS:
Axiom = an assumption common to all
sciences
Postulate = an assumption peculiar to
the particular science
being studied
Introduction
Components
Axiomatic
Systems
Example
Finite
Projective
Planes
Properties
Enrichment
9. Components of an Axiomatic
System
Introduction
A system of logic
From
the
axioms,
other
statements can be deduced or proved
using the rules of inference of a
system of logic (usually Aristotelian).
Components
Axiomatic
Systems
Example
Finite
Projective
Planes
Properties
Enrichment
10. Components of an Axiomatic
System
Introduction
Theorems
New statements which are
deduced or proved using the axioms,
system of logic and previous
theorems.
Components
Axiomatic
Systems
Example
Finite
Projective
Planes
Properties
Enrichment
11. Axiomatic Systems
Introduction
Four Point Geometry
AXIOMS 4P:
1. There exist exactly four points.
2. Each two distinct points are on
exactly one line.
3. Each line is on exactly two
points.
Components
Axiomatic
Systems
Example
Finite
Projective
Planes
Properties
Enrichment
12. Four Point Geometry
Consistency
Introduction
Components
Axiom 4P.1 explicitly guarantees the
existence of exactly four points. However,
even though lines are mentioned in
Axioms 4P.2 and 4P.3,new cannot
ascertain whether or not lines exist until
theorems verifying this are proved, since
there is no axiom that explicitly insures
their existence.
Axiomatic
Systems
Example
Finite
Projective
Planes
Properties
Enrichment
13. Four Point Geometry
Consistency
Introduction
Components
Axiom 4P.2 and 4P.3. like many
mathematical statements, are disguised
“if… then” statements. Axiom 4P.2 should
be interpreted as follows: If two distinct
points exist, then
these two points are
on exactly one line. Axiom 4p.3 should be:
If there is a line, then it is on exactly two
points.
Axiomatic
Systems
Example
Finite
Projective
Planes
Properties
Enrichment
14. Four Point Geometry
Introduction
Consistency
Components
MODEL A
3
Undefined
Term
Points
Lines
1
4
On
2
Interpretation
Dots denoted 1, 2, 3,
4
Segments illustrated
by the figure.
A dot is an endpoint
of a segment or viceversa
Axiomatic
Systems
Example
Finite
Projective
Planes
Properties
Enrichment
15. Four Point Geometry
Introduction
Consistency
Components
MODEL B
Undefined Term
Points
Lines
On
LINES
Axiomatic
Systems
Interpretation
Letters A, B, C, D
Columns of letters below
Contains, or is contained in
A
B
A
C
A
D
B
C
B
D
C
D
Example
Finite
Projective
Planes
Properties
Enrichment
16. Four Point Geometry
Introduction
Independence
The independence of this axiomatic
system is demonstrated by the
following three model, all of which
interpret points as letters of the
alphabet and lines as the columns of
letters indicated.
Components
Axiomatic
Systems
Example
Finite
Projective
Planes
Properties
Enrichment
17. Four Point Geometry
Introduction
Models for Independence
Components
Model 4P. 1
Axiom 4P.1:
Axiom 4P.1’:
Points
A, B
There exist exactly four
points.
There do not exist four
points.
Lines
A
B
Axiomatic
Systems
Example
Finite
Projective
Planes
Properties
Enrichment
18. Four Point Geometry
Introduction
Models for Independence
Components
Model 4P. 2
Axiom 4P.2:
Axiom 4P.2’:
Each two distinct points are on
exactly one line.
There are two distinct points
not on one line..
Points
A, B, C, D
Lines
A C
B D
Axiomatic
Systems
Example
Finite
Projective
Planes
Properties
Enrichment
19. Four Point Geometry
Introduction
Models for Independence
Components
Model 4P. 3
Axiom 4P.3:
Axiom 4P.3’:
Each line is on exactly two
points.
There are lines not on exactly
two points.
Points
A, B, C, D
Lines
A A B C
B D D D
C
Axiomatic
Systems
Example
Finite
Projective
Planes
Properties
Enrichment
20. Four Point Geometry
Introduction
Completeness
If all models of a system are pairwise
isomorphic, it is clear that each model has
the same number of points and lines.
Hence, if all models of the system are
necessarily isomorphic, it follows that the
system is complete.
Components
Axiomatic
Systems
Example
Finite
Projective
Planes
Properties
Enrichment
21. Four Point Geometry
Introduction
Completeness
In Four Point geometry, it is
clear that Model A and Model B are
isomorphic.
Components
Axiomatic
Systems
Example
Finite
Projective
Planes
Properties
Enrichment
22. Axiomatic Systems
Introduction
Four Point Geometry
Components
THEOREMS:
1. The four point geometry has
exactly six (6) lines.
2. Each point of the geometry has
exactly three (3) lines
Axiomatic
Systems
Example
Finite
Projective
Planes
Properties
Enrichment
23. Examples
Introduction
Three Point Geometry
Components
AXIOMS:
1. There exist exactly three points.
2. Each two distinct points are on
exactly one line.
3. Not all the points are on the same
line.
4. Each two distinct line are on at least
one point.
Axiomatic
System
Examples
Finite
Projective
Planes
Properties
Enrichment
26. Examples
Introduction
Five Point Geometry
Components
AXIOMS:
1. There exist exactly five points.
2. Each two distinct points have exactly
one line on both of them.
3. Each line has exactly two points.
Axiomatic
System
Examples
Finite
Projective
Planes
Properties
Enrichment
29. What is a MODEL?
Introduction
A model of an axiomatic system is
obtained if we can assign meaning to the
undefined terms of the axiomatic system
which converts the axioms in to true
statements about the assigned concepts.
It is
consistency.
also
used
to
Models are in two types.
establish
Components
Axiomatic
Systems
Example
Finite
Projective
Planes
Properties
Enrichment
30. What is a MODEL?
TWO TYPES :
Concrete Models
Abstract Models
A model is concrete if the meanings
assigned to the undefined terms are objects
and relation adapted from the real world. A
model is abstract if the meanings assigned
to the undefined terms are objects and
relations adapted from another axiomatic
development.
Introduction
Components
Axiomatic
Systems
Example
Finite
Projective
Planes
Properties
Enrichment
31. Properties
Introduction
Definition 1
An axiomatic system is said to
be consistent if there do not exist in
the system any two axiom, any axiom
and theorem, or any two theorems
that contradict each other. This can
be proved using a model.
Components
Axiomatic
Systems
Example
Finite
Projective
Planes
Properties
Enrichment
32. Properties
Consistency
Introduction
Components
If the model is obtained by using
interpretations that are objects and
relation adapted from the real world, we
have established absolute consistency. If a
model is obtained using the interpretations
from another axiomatic system, we have
demonstrated relative consistency.
Axiomatic
Systems
Example
Finite
Projective
Planes
Properties
Enrichment
33. Properties
Introduction
Definition 2
An axiom in an axiomatic
system is independent if it cannot be
proved from the other axioms. If
each axiom of a system is
independent, the system is said to
be independent.
Components
Axiomatic
Systems
Example
Finite
Projective
Planes
Properties
Enrichment
34. Properties
Independence
Independence is not necessary
requirement for an axiomatic system;
whereas, consistency is necessary. The
verification that an axiomatic system is
independent is also done via models. The
independence of Axiom A in an axiomatic
system S is established by finding a model of
S’ where S’ is obtained by replacing Axiom A
with a negation of A.
Introduction
Components
Axiomatic
Systems
Example
Finite
Projective
Planes
Properties
Enrichment
35. Properties
Introduction
Independence
Thus, to demonstrate that a system
consisting of n axioms is independent, n
models must be exhibited – one for each
axiom.
Components
Axiomatic
Systems
Example
Finite
Projective
Planes
Properties
Enrichment
36. Properties
Introduction
Definition 3
An axiomatic system is complete
if every statement containing undefined
terms of the system can be proved valid
or invalid, or in other words, if it is
possible to add new independent axiom
to the system.
Components
Axiomatic
Systems
Example
Finite
Projective
Planes
Properties
Enrichment
37. Properties
Introduction
Completeness
It is impossible to demonstrate
directly that a system is complete.
However, if a system is complete, there
cannot exist two essentially different
models. This means all models of the
system must be pair wise isomorphic and
the axiomatic system is categorical.
Components
Axiomatic
Systems
Example
Finite
Projective
Planes
Properties
Enrichment
38. Properties
Introduction
Definition 4
Two models of an axiomatic system
are said to be isomorphic if there exists a
one-to-one correspondence Φ from the set
of points and lines α onto the set of points
and lines of β that preserves all relations.
Components
Axiomatic
Systems
Example
Finite
Projective
Planes
Properties
Enrichment
39. Properties
Introduction
Definition 4
In particular if the undefined terms of
the system consist of the terms “point”,
“line”, and “on”, then Φ must satisfy the ff.
conditions:
1. For each point P and line l in α, Φ(P)
and Φ(l) are a point and line in β.
2. If P is on l, then Φ(P) is on Φ(l)
Components
Axiomatic
Systems
Example
Finite
Projective
Planes
Properties
Enrichment
40. Properties
Introduction
Definition 5
An axiomatic system is categorical if
every two models of the system are
isomorphic.
thus for a categorical axiom system
one my speak of the model; the one and
only interpretation in which its theorems
are all true.
Components
Axiomatic
Systems
Example
Finite
Projective
Planes
Properties
Enrichment
41. Properties
Introduction
Definition 6
In a geometry with two undefined
terms, the dual of an axiom or theorem is
the statement with the two terms being
interchanged.
EXAMPLE:
The dual of “ A line contains at least
two points”, is “A point contains at least two
lines”.
Components
Axiomatic
Systems
Example
Finite
Projective
Planes
Properties
Enrichment
42. Properties
Introduction
Definition 6
The dual of a statement is
obtained
by
replacing
each
occurrence of the word “point” by the
word
“line”
and
vice-versa,
(consequently,
the
words
“concurrent” and “collinear” must
also be interchanged.
Components
Axiomatic
Systems
Example
Finite
Projective
Planes
Properties
Enrichment
43. Properties
Introduction
Definition 7
An axiomatic system in which
the dual of any axiom or theorem is
also an axiom or theorem is said to
satisfy the principle of duality.
Components
Axiomatic
Systems
Example
Finite
Projective
Planes
Properties
Enrichment
44. Finite Projective Planes
An axiomatic system for an
important
collection
of
finite
geometries known as finite projective
planes. In a finite projective plane,
each pair of lines intersect; that is,
there is no parallel lines. This pairwise
intersection of lines leads to several
other differences between projective
planes and Eucledian planes.
Introduction
Components
Axiomatic
System
Example
Finite
Projective
Planes
Properties
Enrichment
45. Finite Projective Planes
AXIOMATIC SYSTEM
Introduction
Components
Undefined Terms: Point, Line, Incident
Defined Terms:
Points incident with
the same line are said to
be collinear. Lines
incident with the same
point are said to be
concurrent.
Axiomatic
System
Example
Finite
Projective
Planes
Properties
Enrichment
46. Finite Projective Planes
AXIOMATIC SYSTEM
Axiom FPP.1: There exist at least four distinct
points, no three of which are collinear.
Axiom FPP.2:
There exists at least one line
with exactly n + 1 (n > 1) distinct points
incident with it.
Axiom FPP.3:
Given two distinct points,
there is exactly one line incident with both of
them.
Axiom FPP.4:
Given two distinct lines,
there is at least one point incident with both of
them.
Introduction
Components
Axiomatic
System
Example
Finite
Projective
Planes
Properties
Enrichment
47. Finite Projective Planes
Introduction
Components
Any set of points and lines
satisfying these axioms is called a
projective plane of order n. Note
that the word “incident” has been
used in place of the undefined term
“on” in this axiom system.
Axiomatic
System
Example
Finite
Projective
Planes
Properties
Enrichment
49. Finite Projective Planes
CONSISTENCY
Introduction
MODEL B
Points
Segments
illustrated in the
figure.
Axiomatic
System
Lines
Dots
denoted
1, 2, 3, 4,
5, 6, 7
Components
1
4
Example
5
6
2
Finite
Projective
Planes
7
3
Properties
Enrichment
50. Finite Projective Planes
Introduction
Models A and B have three
points on each line, three lines on
each point, and a total of seven
points and seven line. To determine
if finite projective planes exist with
more points and lines, it is clearly
impractical to employ trial-and-error
procedures.
Components
Axiomatic
System
Example
Finite
Projective
Planes
Properties
Enrichment
51. Finite Projective Planes
Introduction
Components
Geometers developed a series
of theorems that lead to a general
result regarding the number of
points and lines in a finite projective
plane of order n.
Axiomatic
System
Example
Finite
Projective
Planes
Properties
Enrichment
52. Finite Projective Planes
Introduction
The proofs of these theorems
are simplified by noting that this
axiom system satisfies the principle
of duality, which Coxeter has
described as “ one of the most
elegant properties of projective
geometry. ( Coxeter, 1969, p. 231)
Components
Axiomatic
System
Example
Finite
Projective
Planes
Properties
Enrichment
53. Finite Projective Planes
Introduction
Components
In an axiomatic system that satisfies
the principle of duality, the proof of any
theorem can be “turned into” a proof of a
dual theorem merely by dualizing the
original proof. To show that an axiom
system has the property of duality, it is
necessary to prove that the duals each
axiom are the theorems of the system.
Axiomatic
System
Example
Finite
Projective
Planes
Properties
Enrichment
54. Finite Projective Planes
Introduction
Theorem FPP.1 (Dual of Axiom FPP.1)
There exist at least four distinct lines,
no three of which are concurrent.
Components
Axiomatic
System
Example
Theorem FPP.2 ( Dual of Axiom FPP.3)
Given two distinct lines, there is exactly
one point incident with both of them.
Finite
Projective
Planes
Properties
Enrichment
55. Finite Projective Planes
Introduction
Theorem FPP.3 (Dual of Axiom FPP.4)
Given two distinct points, there is at
least one line incident with both of them.
Components
Axiomatic
System
Example
Theorem FPP.4 ( Dual of Axiom FPP.2)
There exists at least one point with
exactly n+1 (n>1) distinct lines incident with
it.
Finite
Projective
Planes
Properties
Enrichment
56. Finite Projective Planes
Theorem FPP.5
There is exactly
n +1 lines through a
point P.
Theorem FPP.6
l1 l2
Each of these
lines contains exactly n
+ 1 points, that is, n
P1 P2
points addition to P.
Introduction
P
Components
Axiomatic
System
ln+1
ln+2
Pn+1
Example
Finite
Projective
Planes
Properties
Enrichment
57. Finite Projective Planes
Theorem FPP.7
Introduction
Components
A projective plane of order n
contains exactly n 2 n 1 points and
2
n
n 1 lines.
Proof:
The total number of points is (n+1)
n+1 = n 2 n 1. A dual argument verifies
2
n
n 1.
that the total number is also
Axiomatic
System
Example
Finite
Projective
Planes
Properties
Enrichment
