Question

Why is Axiom $5$, in the list of Euclid's axioms, considered as a 'universal truth'? (Note that the question is not about the fifth postulate.)

Answer 1

Axiom $5$ states that the whole is greater than the part.

This axiom is known as a universal truth because it holds true in any field, and not just in the field of mathematics.

Let us take two cases: one in the field of mathematics and one other than that.

Case I

Let t represent a whole quantity and only $a,b,c$ are parts of it.

$t=a+b+c$

Clearly, $t$ will be greater than all of its parts $a,b$ and $c$.

Therefore, it is rightly said that the whole is greater than the part.

Case II

Let us consider continent Asia. Then, let us consider a country India which belongs to Asia. India is a part of Asia and it can also be observed that Asia is greater than India. That is why we can say that the whole is greater than the part. This is true for anything in any part of the world and is thus a universal truth.