1. Mathematics
Non-euclidian Maths
Powerpoint Templates

2. Once upon a time Euclid had 5 axioms
It shall be possible to draw a straight line joining any 2 points.
A finite straight line may be extended without limit in either direction.
It shall be possible to draw a circle with a given centre and through a given point.
All right angles are equal to one another.
There is just one straight line through a given point which is parrallel to a given line.

3. But then along came Riemann and his Contraries
Two points may determine more than one line (axiom 1)
All lines are finite in length but endless
(axiom 2)
There are no parallel lines
(axiom 5)
a. All perpendiculars to a straight line meet at one point.
b. Two straight lines enclose an area.
c. The sum of the angles of any triangle is greater than 180 degrees.

4. Georg Friedrich Bernard Riemann (1822-66)

5. But then along came Riemann and his Contraries

6. Albert agrees…space is curved

7. Riemann
Two points may determine more than one line (axiom 1)

8. Riemann
All lines are finite in length but enless (axiom 2)

9. Riemann
There are no parallel lines
(axiom 5)

10. M.C. Escher

11. M.C. Escher

13. Systems work within their own axiom sets
Riemann vs Euclid
Neither have even been proven wrong, and yet they contradict each other???
Systems work within their own axiom sets
Godel ( ), Austrian

14. Kurt Godel (1906-78) – Incompleteness Theorm
It is impossible to prove that any mathematical system is free from contradiction
He didn’t prove that Maths has contraditions, just that it’s impossible to prove it doesn’t

15. Kurt Godel – Incompleteness Theorm
However,
Maths hasn’t had any contraditions since it’s formalisation 2500 yrs ago – so most mathematicians mostly ignore Godel’s theorm

16. So if Riemann, Godel & Einstien have contradicted Euclidian maths, why do we still use it?
Because it works!
Why does it work?