What is the Difficulty of Proving Euclid's Fifth Parallel Postulate?

It is plainly obvious that the fifth parallel postulate is not as straightforward as other axioms proposed by Euclid. Postulating that two points determine a line is virtually intuitive—how else would a line connect two points but by being directly between them? However, the fifth postulate, concerning the behavior of parallel lines, is not as self-evident. The statement that two lines cutting a straight line in such a way that the inner angles on one side add up to less than two right angles will meet if extended far enough, involves a level of subtlety that challenges even the most rigorous thinkers.

The Difficulty in Proving the Fifth Postulate

One could conceivably question the truth of this postulate. For instance, one might wonder, "How can you know that the lines will meet somewhere beyond the solar system, or perhaps beyond the limits of our galaxy?" Despite these doubts, no one seriously questioned its truth until the advent of non-Euclidean geometry.

The difficulty lies not in its truth, but in its inherent complexity. It is impossible to prove within the framework of Euclidean geometry. There was never any doubt that this statement was true; the issue was whether it could be derived from simpler, more fundamental axioms.

Attempts to prove the fifth postulate as a theorem rather than an axiom were met with failure. Mathematicians wanted to break it down into simpler pieces, much like the other axioms. However, it soon became clear that this was not possible. The fifth postulate is the wholistic fact that defines flat space and separates it from all other geometries. It is a singular determining factor, making it inherently complex and complexifying any attempts to simplify it further.

Postulates and Theorems: A Distinction

Before the advent of non-Euclidean geometry, all postulates were considered plainly obvious and self-evident. Postulates are statements that are accepted as true without proof. This is why Euclid had it as a postulate in the first place. However, the fifth postulate was seen as strangely complex for an axiom—it involves definitions that are not easy to define succinctly and makes the statement inherently difficult to break down into simpler components.

Once the concept of other geometries was introduced, the uniqueness of Euclidean geometry became clearer. The other axioms establish what geometric means. The fifth postulate is the most powerful factor in choosing which geometry to study most intensively, adding another layer of complexity to the proof.

Non-Euclidean Geometries: A New Frontier

Denying or modifying the fifth postulate opens up the realm of non-Euclidean geometries, which have proven to be quite useful in various applications. These geometries challenge the traditional understanding of space and have expanded our knowledge of geometry in profound ways. The quest to prove the fifth postulate as a theorem instead of an axiom led to the discovery of these new and fascinating geometries.

Ultimately, the difficulty of proving Euclid's fifth postulate is not just a matter of proving its truth, but understanding the underlying principles of geometry and the elegance of a more complex, powerful, and beautiful system.