Augustin-Louis Cauchy
Weierstrass’ function f(x), which is continuous everywhere but only
differentiable on a ste of measure zero. The formula is at the top; the
graphs correspond to k=2 (red), 3 (green) and 4 (blue).
Georg Cantor
The Koch ‘snowflake curve’ (1904) is constructed by an infinite sequence of
iterations (convergent), as in the picture. The limit is everywhere continuous, but
has no tangent anywhere. It was revived in the 1960s by Mandelbrot as an example
of a fractal.
How the Cantor set is constructed ‘classically’; and an
example of one which arises ‘naturally’ in iterating the
function f(z)=z2+c. (The cantor set is the set of all points
which don’t tend to infinity under iteration – e.g. fixed or
periodic points.)
Gottlob Frege
• ‘The complete reliability of mathematics is an illusion, it does
not exist, at least not unconditionally.’
• ‘Indeed there is one branch of mathematics today over which
opinion is divided, and some consider right what others
reject. This is the so-called set theory, in which the certainty
of mathematical deduction seems to be becoming completely
lost.’
• (O. Perron ‘On Truth and Error in Mathematics’, 1911; cited
by Gray (see bibliography, 2004)