[Archimedes] concentrated his ambition exclusively upon those specu-
lations which are untainted by the claims of necessity. These studies, he
believed, are incomparably superior to any others, since here the grandeur
and beauty of the subject matter vie for our admiration with the cogency
and precision of the methods of proof. Certainly in the whole science of ge-
ometry it is impossible to find more difficult and intricate problems handled
in simpler and purer terms than in his works. Some writers attribute it to
his natural genius. Others maintain that phenomenal industry lay behind
the apparently effortless ease with which he obtained his results. The fact is
that no amount of mental effort of his own would enable a man to hit upon
the proof of one of Archimedes' theorems, and yet as soon as it is explained
to him, he feels he might have discovered it himself, so smooth and rapid is
the path by which he leads us to the required conclusion.
Plutarch Life of Marcellus [Scott-Kilvert's translation]
It may be observed of mathematicians that they only meddle with such
things as are certain, passing by those that are doubtful and unknown. They
profess not to know all things, neither do they affect to speak of all things.
What they know to be true, and can make good by invincible argument,
that they publish and insert among their theorems. Of other things they
are silent and pass no judgment at all, choosing rather to acknowledge their
ignorance, than affirm anything rashly.
Barrow Mathematical Lectures
For [A. N.] Kolmogorov mathematics always remained in part a sport.
But when ... I compared him with a mountain climber who made first
ascents, contrasting him with I. M. Geffand whose work I compared with
the building of highways, both men were offended. ' ... Why, you don't
think I am capable of creating general theories?' said Andrei Nikolaevich.
'Why, you think I can't solve difficult problems?' added I. M.
V. I. Arnol'd in Kolmogorov in Perspective