This document summarizes key concepts in mathematics including: 1. Euclidean geometry is the first mathematical theory described using axioms developed by Greek mathematician Euclid, with five fundamental postulates including one about parallel lines. 2. In the 19th century, it was shown that the parallel postulate is independent of the others, allowing for non-Euclidean geometries like spherical geometry where lines are great circles and triangle angles sum to more than 180 degrees. 3. Kurt Godel's incompleteness theorems proved that any sufficiently powerful mathematical system cannot prove its own consistency, showing the limits of what can be proved from axioms alone.

1. WikiStage ENP 2014
Am I Right ?
Presented by :
Tarik Zakaria BENMERAR

2. Mathematics to understand Truth
A simple and a abstract view of the world
Deep understanding of the subject
Generalized facts about the subject

3. The Origin of Mathematical Truth
Axioms
Mathematical
Knowledge (Truth)
Mathematical
Logic
Unprovable
Facts
(Supposed
True)

4. Euclidian Geometry
1. The first mathematical theory described using
axioms.
2. Employed by the Greek mathematician Euclid.
3. The basis of the plane and solid geometry.

5. Euclidian Geometry
Euclide Postulates
1. A straight line segment can be drawn joining any two
points.
2. Any straight line segment can be extended indefinitely in a
straight line.
3. Given any straight line segment, a circle can be drawn
having the segment as radius and one endpoint as center.
4. All right angles are congruent.
5. If two lines are drawn which intersect a third in such a way
that the sum of the inner angles on one side is less than
two right angles, then the two lines inevitably must
intersect each other on that side if extended far enough
( Parallel Postulate).

6. Euclidian Geometry
Euclide Postulates
1. Euclid used only the first four postulates
(absolute geometry) to prove the first 28
propositions of his book : Elements.
2.But, was forced to use the fifth postulate to
prove the 29th proposition.

7. Euclidian Geometry
Some mathematicians asked, if the fifth
postulate is :
A theorem proveable from the first four
postulates
1
2
3
4
5

8. Euclidian Geometry
Or :
An independant axiom unproveable from
the first four postulates
1
2
3
4
5

9. Euclidian Geometry
In 1823, Janos Bolyai and Nicolai
Lobachevsky indepedently found that the
fifth postulate is an independant axiom.
So, we can create a new consistent
geometry with the four first postulates being
true, but the fifth postulate being false.
1 2 3 4 5’

10. Non Euclidian Geometry
(Spherical Geometry)
The lines are the
great circles of the
sphere
Spherical Triangles

11. Non Euclidian Geometry
(Spherical Geometry)
In Spherical Geometry, non-parallel lines
meet in two points, and the sum of triangle
angles are greater than 180 degrees.

12. Moral of the story
We can be both right even if we disagree

13. Godel Incompletness Theorems
Axioms
Mathematical
Knowledge
The whole
knowledge
Can the mathematical knowledge reach the
whole knowledge ?

14. Godel Incompletness Theorems
• In 1931, Kurt Godel proved that any arithmetic
theory is necessary incomplete.
• Some theorems require an infinite number
of steps to prove.
Some knowledge cannot be reached from the
axiom-based mathematical knowledge

15. Godel Incompletness Theorems
Can we prove that a theory is consistent (Doesn’t
contain contradictions)?
1
2
3
4
5
Not 5
Contradiction

16. Godel Incompletness Theorems
• Godel also proved that such demonstration is
impossible (Arithmetic).
• Any « complex » mathematical theory
cannot ensure its consistency.
• The mathematical world can break at any
moment.

17. Moral of the story
I am not sure if I am right
I am not sure if I am consistent

18. Some advices
• Learn to hear and accept others opinions.
• There is always something to learn from
others.
• Break the rules and explore new knowledge.
• Some truths are supposed to be true, but
can be false in certain context.

19. Some keywords
Decidability
Computability
Halt Problem
Choice Axiom

20. Thank you very much