Non-Euclidean geometry and consistency

Euclidean Geometry

Remember we said that a mathematical system depends on its basic assumptions – its axioms.

These should be self-evident.

a + b = b + a

Euclidean Geometry

Axioms of Euclidean Geometry

Euclidean Geometry

1. It shall be possible to draw a straight line joining any two points

Euclidean Geometry

2. A finite straight line may be extended without limit in either direction.

Euclidean Geometry

3. It shall be possible to draw a circle with a given centre and through a given point.

Euclidean Geometry

4. All right angles are equal to one another.

Euclidean Geometry

5. There is just one straight line through a given point which is parallel to a given line

Non-Euclidean geometry

The last axiom of Euclid is not quite as self evident as the others.

In the 19th century, Georg Friedrich Bernard Riemann came up with the idea of replacing Euclid’s axioms with their opposites

Non-Euclidean geometry

• Two points may determine more than one line (instead of axiom 1)

• All lines are finite in length but endless (i.e. circles!) (instead of axiom 2)

• There are no parallel lines (instead of axiom 5)

Non-Euclidean geometry

People expected these new axioms to throw up inconsistencies….. But they didn’t!

Non-Euclidean geometry

Among the theorems that can be deduced from these new axioms are

1. All perpendiculars to a straight line meet at one point.

2. Two straight lines enclose an area

3. The sum of the angles of a triangle are grater than 180°

Do these make sense?!

1. All perpendiculars to a straight line meet at one point.

2. Two straight lines enclose an area

3. The sum of the angles of a triangle are grater than 180°

Do these make sense?!

They do if we imagine space is like the surface of a sphere!

1. All perpendiculars to a straight line meet at one point.

2. Two straight lines enclose an area

3. The sum of the angles of a triangle are grater than 180°

Non-Euclidean geometry

On the surface of a sphere, it can be shown that the shortest distance between two points is always the arc of a circle. This means in Riemannian geometry, a straight line will appear as a curve when represented in two dimensions.

Although these look curved, you can be sure the airlines are following the

shortest route to save money!

Straight lines in Riemannian geometry

Once we have clarified the meaning of a straight line in Riemannian geometry, we can give a meaning to the three theorems given earlier.

All perpendiculars to a straight line meet at one point.

Lines of longitude are perpendicular to the equator but meet at the North pole

Two straight lines enclose an area

Any two lines of longitude (straight lines) meet at both the North and South poles so define an area.

The sum of the angles of a triangle are greater than 180°

General relativity

According to Einstein’s general theory of relativity, the Universe obeys the rules of Riemannian geometry not that of Euclid. According to Einstein, space is curved!

Consistency

It would seem that it is easy to have a system of mathematics that is consistent. Not so!

Set theory

At the heart of set theory is a contradiction

Set theory

A feeling for the contradiction can be found in the following story;

Set theory

A barber had an affair with a princess. The king was very angry and wanted the barber executed. The princess begged for his life and the king agreed, provided that………

Set theory

……the barber went back to his village and only shaved all the inhabitants that did not shave themselves.

Set theory

“That’s easy” said the barber.

Is it?

Set theory

Another example is to imagine catalogues in a library. Some catalogues are for novels, some for reference, poetry etc.

The librarian notices that some catalogues list themselves inside, some don’t.

Set theory

The librarian decides to make two more catalogues, one which lists all te catalogues which do list themselves, and more interestingly, a catalogue which lists all the catalogues which do not list themselves.

Set theory

Catalogues which list themselves

Catalogues which do not list themselves

Set theory

Should the catalogue which lists all the catalogues which do not list themselves be listed in itself?

If it is listed, then by definition it should not be listed, and if it is not listed, it should be listed!

Gödel’s incompleteness theory

Kurt Gödel (1906-1978) was able to prove that it is impossible to prove that any formal system of mathematics is without contradictions.

Mathematicians’ certainty is an illusion!