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Mathematics

Non-euclidian Maths

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- 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.

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.

Georg Friedrich Bernard Riemann (1822-66)

But then along came Riemann and his Contraries

Albert agrees…space is curved

Riemann

- Two points may determine more than one line (axiom 1)

Riemann

- All lines are finite in length but enless (axiom 2)

Riemann

- There are no parallel lines
- (axiom 5)

M.C. Escher

M.C. Escher

Riemann vs Euclid

Neither have even been proven wrong, and yet they contradict each other???

Systems work within their own axiom sets

Godel (1906-78), Austrian

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

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

So if Riemann, Godel & Einstien have contradicted Euclidian maths, why do we still use it?

Because it works!

Why does it work?