Am I Right ? Tarik Zakaria BENMERAR - WikiStage ENP
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Les diapos de mon talk lors du Wikistage ENP 2014.
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Am I Right ? Tarik Zakaria BENMERAR - WikiStage ENP
- 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.
- 20. Thank you very much