Friday, November 08, 2002

Experience Mathematics #19 -- Euclid's axioms

Just like elements and sets, Points and Lines are undefined notions.

We can think of a line as a set of points. These satisfy certain axioms, such as: Given a line $l$ and a point $P$ not on the line, there is only one line that is parallel to $l$ containing the point $P$. Axioms are considered to be self-evident truths.

However, several gaps were found in Euclid’s axioms. For example, consider Euclid’s proof that the base angles of an isosceles triangle are equal. Suppose we have an isosceles triangle $ABC$, where the side $AB$ is equal to the side $AC$. Drop a perpendicular $AD$ from a vertex to the side $BC$. There is nothing in Euclid’s axioms that says that the point $D$ is between the points $B$ and $C$. Nevertheless, Euclid proves that the triangles $ABD$ and $ACD$ are congruent. From this it is easy to see that the base angles of an isosceles triangle are equal.

The great mathematician Hilbert completed Euclid’s work by listing a few more axioms. These included the betweenness axioms. For example, given three points $A, B and C$, one of the axioms said either $B$ is between $A$ and $C$, or $C$ is between $A$ and $B$ or $A$ is between $C$ and $B$.

To return to Euclid’s proof, some steps need to be added to show that $D$ is between $B$ and $C$. 

But that is not all. We could consider a geometry where given a line $l$ and a point $P$ not on the line, there are no lines parallel to $l$ containing the point $P$. Such a non-euclidean geometry exists on the surface of the Earth. So one of Euclid’s axioms cannot be considered to be a self-evident truth after all.

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