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.
1. Sophie Germain (1776 - 1831)
- 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 thirteen-year-old
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 well-known in the mathematical community (the most well-known being Emmy Noether).
- 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.
2. Alan Turing (1912 - 1954)
- 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.
- 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.
3. Kurt Gödel (1906 - 1978)
- 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.
- 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.
4. Srinivasa Ramanujan (1887 - 1920)
- Ramanujan (as he is universally known) was that rarest of
mathematicians, self-taught. 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?).
5. Pierre de Fermat (1601 - 1665)
- Fermat is of course well-known 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.
- 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.
6. Evariste Galois (1811 - 1832)
- 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.
7. Georg Cantor (1845 - 1918)
- 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.
- 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?
8. Bertrand Russell (1872 - 1970)
- 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 side-effect was that it destroyed a book that Frege had just published.
- Russell later became a prominent campaigner for world peace.
- A barber in a village only shaves those villagers who do not shave themselves. Who shaves the barber?
9. Pythagoras (569 - 475 BC)
- 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 that
was
not rational (see Root
Two, Ask
Dr. Math, A
Geometric Proof) and yet could be measured by the hypotenuse of
a right-angled triangle whose other sides have length 1. They never
solved this 'paradox'.
- 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 that
10. Maurits Escher (1898 - 1972)
- 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.
- Have a look at his drawings at
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.