You can think of mathematics as a building. Every building has a foundation, a solid basis upon which we can construct more complex structures brick by brick. In mathematics, specifically geometry, these foundations were laid long ago by Euclid in ancient Greece.

Euclid developed the Euclidean axioms: simple statements accepted as true without proof.

  • Given any two points in a plane, you can draw a straight line connecting them.
  • Given a point and a radius in a plane, you can draw a circle.
  • Given a straight line and a point not on it, exactly one line through the point is parallel to the given line.

These axioms form the basis for proving more complex theorems. By tracing a theorem’s logical chain back to the axioms, we establish its truth, but, as mathematics evolved, new challenges emerged, like these famous ones:

  • Trisecting an arbitrary angle.
  • Squaring the circle (constructing a square with the same area as a given circle).
  • Doubling the cube (constructing a cube with twice the volume of a given cube).

These puzzled brilliant minds for centuries.

Today, we know they are impossible under those constraints, does this mean the axioms are flawed? Consider David Hilbert, who aimed to fully axiomatize mathematics. His goal: a complete system where every statement is either provable (true) or disprovable (false). Kurt Gödel shattered this dream with his Incompleteness Theorems, that prove that any consistent formal system powerful enough for basic arithmetic is incomplete (containing true statements it cannot prove) and cannot prove its own consistency.

Like the geometry problems (true but unconstructible with limited tools), mathematics has inherent limits. Euclidean geometry relies on broader mathematics to resolve such gaps, yet remains trustworthy. Gödel reveals mathematics as an open endeavor. How do we grasp truths beyond proof? Perhaps through our uniquely human gifts: creativity, intuition, and insight.