
Mathematicians
You will find a very comprehensive list of mathematicians at History
of Mathematics, which is a site you should bookmark. There are
so many worthy mathematicians that I could include but I have
restricted myself to a few whose lives have interested me.
 Sophie Germain was inspired by reading about the death of
Archimedes. He was so absorbed in a geometry problem that he didn't
do as a Roman soldier demanded and was killed. Sophie decided that
mathematics must be an amazing subject for anyone to be so immersed
in it that they couldn't sense danger. She taught herself Latin and
Greek and because her parents opposed her interest in mathematics (in
their eyes it was not a subject suitable for a thirteenyearold
girl), she read Newton
and Euler
secretly at night.
 As a woman she was denied prizes for solving mathematical
problems, so she often had to pretend to be a man in order to carry
on her research. She corresponded with the distinguished
mathematician Gauss and helped save his life when Napoleon invaded
his home town.
 She made significant advances on Fermat's
Last Theorem and is one of the few female mathematicians
wellknown in the mathematical community (the most wellknown being Emmy
Noether).
 Known as the father of computing his work has recently been
acknowledged in a series on British television about the cracking of
the Enigma code during the Second World War. Without his genius,
British and American intelligence would not have been able to gain
the German secrets they did, and, as Churchill admitted, the war
could have dragged on for much longer.
 Despite all he did for Britain, including inventing the first
computer, he was hounded after the war by the Intelligent Services
because of his homosexuality, and died in mysterious circumstances in
Wilmslow, Cheshire.
 His legacy includes much of the theory of logic, Turing
Machines, which tell us what computers are or aren't capable of
and the Turing
Test for whether machines are intelligent.
 There's a very detailed site devoted to him at The
Alan Turing Home Page.
 Like many geniuses Kurt Gödel sometimes appeared rather
strange to the average person. He starved himself to death because he
thought he was being poisoned.
 His greatest work was in logic. Until 1931 all mathematicians
thought that the whole of mathematics could be constructed by
starting with axioms and proving theorems. Indeed in 1900 the great
mathematician David
Hilbert talked about proving the consistency of mathematics 
that is that there is no possibility of a contradiction (any
contradiction would in fact be a disaster bringing down the whole of
mathematics). At the time mathematicians had made so many advances
that, like physics, they thought, in some sense, they were about to
codify the whole of mathematics.
 Like physics, they were mistaken and it was Gödel that
showed that whatever axioms you used there were always theorems you
couldn't prove. Furthermore, it was impossible to prove a set of
axioms was consistent using only those axioms. This is very much
bound up with the idea that you can't program a computer to solve all
mathematical problems, which means that there is a need for human intuition.
 This caused a revolution in mathematics as sensational as
relativity and quantum theory did in physics.
 See also The Kurt
Gödel Society.
 Ramanujan (as he is universally known) was that rarest of
mathematicians, selftaught. His field was number theory and his work
so impressed the Cambridge mathematician G.H.Hardy
that he was invited to Cambridge from India. Sadly, the British
weather made him ill.
 He produced profound work, much of which has never been
surpassed. Hardy's book tells
the story that when Hardy visited Ramanujan as he lay dying, he
started the conversation with the remark that his taxi was number
1729, which seemed to be a very dull number. Ramanujan replied that
it was very interesting number since it was the smallest number that
could be expressed as the sum of two cubes in two different ways.
(Exercise: what are those cubes?).
 Fermat is of course wellknown for his Last
Theorem. This is a bit of a misnomer. It wasn't the last theorem
he thought he'd proved but the last of his theorems that anyone else
had managed to proved. He was an amateur mathematician, though his
love of mathematics distracted him from his duties as a lawyer.
 Like many of his day, he didn't write out proofs but communicated
his results to others. He was able to prove most of them, though
sometimes he was mistaken and it is likely that his 'proof' of the
Last Theorem was flawed.
 Apart from number theory (see Fermat's
Little Theorem) he made contributions to differential calculus
well before Newton,
who is usually credited with its discovery. Thus Fermat could find
the gradient of the tangent to a curve and, in other words, he knew
how to differentiate some 40 years before Newton.
 The story goes that the night before a duel at the age of twenty,
he wrote down much of his results in a letter as he expected to die.
He complained that he had so much to write yet so little time to do so.
 The story may well have been embellished somewhat, The
Galois Story, but there is no doubt that Galois at twenty had
made some very impressive discoveries in the area of group theory.
Indeed the fact that it is impossible to give a formula for solving
polynomials of degree more than four is a result that is part of what
is now known as Galois Theory.
 Cantor is best known for his theory of infinite sets. The idea of
infinity was poorly understood and its use sometimes led to
contradictions. Cantor was able to put the idea onto a firm basis.
His theories are now mainstream but at the time they were opposed by Kronecker,
who only believed that the numbers 1,2,3 ... exist and 'all else
is the work of man'.
 Certainly Cantor's work seemed strange. He showed that there were
the same number of integers as there are rationals, but that there
were more real numbers. When you consider that between every pair of
irrational numbers there is a rational number and between every pair
of rational numbers there is an irrational number, yet the
irrationals are more numerous, you begin to see how strange it all
looks and why it was difficult to accept at the time. You'll find
proofs of these statements at What
is a number?
 In modern terminology, the set of rationals is dense in [0, 1]
but form a null set. That is, any number between 0 and 1 can be
approximated arbitrarily closely by by a rational number, yet the
rationals in [0,1] are negligible and can be discarded for many
mathematical purposes (see An
Introduction to the Theory of Numbers for more details).
Mathematics is a lot stranger and, so more interesting, than at first sight.
 Bertrand Russell was a philosopher as well as a mathematician and
he studied logic and set theory which border on both disciplines. His
name has been given to Russell's Paradox which is about sets which
are members of themselves. This was stated in many forms, one of the
clearest being Russell's
Barber Paradox:
 A barber in a village only shaves those villagers who do not
shave themselves. Who shaves the barber?
 Think about this paradox because it became very important in the
theory of sets and a version of the paradox is the key to Gödel's
results on the axiomisation of maths. Russell's ideas ushered in
a new era of logic, though an unfortunate sideeffect was that it
destroyed a book that Frege
had just published.
 Russell later became a prominent campaigner for world peace.
 Everyone has heard of Pythagoras, if only because of the theorem
that bears his name. He was the leader of a group of disciples called
the Pythagoreans who had some interesting beliefs. Very little is
known about Pythagoras himself, and the mathematical results may well
belong to his followers as a group rather than the man himself.
 One thing puzzled the Pythagoreans. They knew that rational
numbers could be measured in the sense that ½ was measured by
dividing a line of unit length in two. They also knew thatwas
not rational (see Root
Two, Ask
Dr. Math, A
Geometric Proof) and yet could be measured by the hypotenuse of
a rightangled triangle whose other sides have length 1. They never
solved this 'paradox'.
 Escher wasn't a mathematician but an artist. However, his graphic
drawings appeal to mathematicians as they illustrate mathematical
ideas such as tilings of a plane or impossible figures.
 Have a look at his drawings at
 Escher's
World
 M.C.
Escher Optical Illusions and Tessellations
 Neal's Escher Page
 Brendan's
M.C.Escher Page
 By the way, Escher's drawings are copyright so you probably
shouldn't copy them. The sites claim to have permission to show the
pictures, but if that isn't true they may shut down like many others
have been.
 You can't fail to be impressed by his drawings and you may well
recognise some of them. There are lots of books of Escher's work.
Unfortunately one of the best books The Graphic Work of M.C. Escher
is now out of print, but if you can find it then it's well worth buying.

I have listed mathematicians from the past. However, you might like
to search out two famous mathematicians who are very much alive, John
Conway and Ian Stewart. John Conway is best known as the
inventor of the Game of Life
and Ian Stewart brings mathematics alive for the general public Books,
Royal Institution lectures.

