I study maths, and I have found that a useful way of thinking about two propositions A and B being equivalent is to regard them as being two different ways of saying the same thing, or equivalently, interchangeable.
However, what I find puzzling from a philosophical perspective is why it may seem that one of the equivalent statements is much more apparent than the other. For example, it has been proven that Euclid's parallel postulate is equivalent to all of the following:
- two lines that are parallel to the same line are also parallel to each other
- there exists a triangle with internal angles summing to 180°
- every triangle in the plane has internal angles summing to 180°
- the square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the legs of the triangle (Pythagoras's theorem)
I have listed them in the (subjective) order "most obvious" to "least obvious". In particular, the idea that (2) and (3) are equivalent is really quite puzzling. Intuitively, (2) seems to be a much less stringent condition than (3) - it is much more apparent, even though both provably say exactly the same thing!
My initial thought was that the fact that we perceive a difference indicates only that we understand some topics better than others i.e. that all equivalent statements ought to have the same apparency, but that is not true for us because we do not have enough knowledge. But I am not convinced that this fully explains (2)-(3).