"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)