"The graph of a continuous function consists of one piece; if it has a point below a line y=c and a point above this line, there must be a point on the graph at which the graph crosses the line y=c. What can be simpler than this argument? Why complicate matters by appealing to the least upper bound principle and by giving a rather involved indirect proof?
The same question can be asked about many "geometrically obvious" theorems for which mathematicians nevertheless give analytic proofs.
The answer is: Our geometric intuition is an invaluable guide, but, unfortunately, not an infallible one. There are statements that seem geometrically obvious, but are still false. The analytic proofs do not replace intuition; they reinforce it. Only our intuition and imagination, applied to particular cases, can suggest general theorems. Only a rigorous proof can assure us that intuition did not lead us astray."

-Lipman Bers (from Calculus, Holt, Reinhart and Winston, 1969)


"Compare Euclid, with all his flaws, to the most eminent of the ancient exponents of the convincing argument – Aristotle. Much of Aristotle’s reasoning was brilliant, and he certainly convinced most thoughtful people for over a thousand years. In some cases his analyses were exactly right, but in others, such as heavy objects falling faster than light ones, they turned out to be totally wrong. In contrast, there is not to my knowledge a single theorem stated in Euclid’s Elements that in the course of two thousand years turned out to be false. That is quite an astonishing record, and an extraordinary validation of proof over convincing argument."

-Robert Osserman (from C. Pugh's Real Mathematical Analysis, 2nd ed, Springer, 2015)