Wednesday, June 24, 2009

The Axioms of Arithmetic

During the 20th Century, mathematicians decided that they had to properly define the foundations of maths. Work by philosophers such as Bertrand Russell showed that there were inherent contradictions in the subject that might put many important assumptions in doubt. As several theorems proved to be false, there were concerns that many branches of maths were built on houses of cards that may be found to tumble.

To try to resolve this, a list was produced of axioms which can be taken as self-evident. This means they can be assumed as first principles upon which all other theories can be built. However, even these axioms have not been immune to questioning - and, if you need axioms upon which you build the axioms, doesn´t that also mean you need axioms to build the axioms of the axoms....(ad infinitum)? Anyway, these are the seven Axioms of Arithmetic:

  1. For any numbers mn: m + n = n + m
  2. For any numbers m, n, k: (m + n) + k = m + (n + k)
  3. For any numbers m, n, k: m(n + k) = mn + mk
  4. There is a number 0 which has the property that, for any number n: n + 0 = n
  5. There is a number 1 which has the property that, for any number n: n x 1 = n
  6. For every number n, there is another number k such that: n + k = 0
  7. For any numbers m, n, k, if k ≠ 0 and kn = km, then m = n

Sunday, June 21, 2009

A truel

From Fermat´s Last Theorem, by Simon Singh,

"A truel is similar to a duel, except there are three participants rather than two. One morning Mr Black, Mr Grey and Mr White decide to resolve a conflict by truelling with pistols until only one of them survives. Mr Black is the worst shot hitting his target on average only one time in three. Mr Grey is a better shot, hitting his target two times out of three. Mr White is the best shot, hitting his target every time. To make the truel fairer Mr Black is allowed to shoot first, followed by Mr Grey (if he is still alive), followed by Mr White (if he is still alive) and round again until only one of them is alive. The question is this: Where should Mr Black aim his first shot?"

If you've not seen this question before, please feel free to drop me a comment with your answer and a short description of why.

This kind of question is part of the branch of maths which has been called Game Theory, a term coined by John von Neumann in 1944. This became fundamental in the study of military strategy, particularly during the Cold War. People of my generation will be familiar with the 1983 film War Games, in which a young Matthew Broderick attempts to teach game theory to a computer which is about to start a global nuclear war, using his Commodore Vic 20.


Friday, June 12, 2009

Fermat's Last Theorem

Pierre de Fermat (1601-1665) was a French mathematician who, together with Isaac Newton and Gottfried Leibnitz, is credited with the invention of calculus (calculations which allow the mathematical description of two factors relative to each other). He also made notable contributions to analytic geometry, probability, and optics. His first love, however, was number theory - the philosophy of mathematics. He was more interested in solving and setting problems, rather than his own fame and fortune.

One of these, "Fermat's Last Theorem" (named as such because it became the last of his theorems requiring proof), states that no three positive integers x, y, and z can satisfy the equation x^n + y^n = z^n for any integer value of n greater than two. Fermat set the challenge for mathematicians to prove the theorem correct. In a scientific sense, it can't be proved, since this would involve testing an infinite series of numbers. However, maths provides the opportunity to prove a theorem without the limitations of perception, and therefore is considered the purest form of truth and knowledge. Despite the simplicity of the equation, for three hundred years, the challenge beat all of the best minds in the world. Tantalisingly, Fermat left some indications when he died that he had come up with the proof himself. In 1995, softly spoken British mathematician Andrew Wiles, a maths professor at Princeton University, gave a talk at a conference at Cambridge in which he claimed to have met Fermat's challenge. It was the culmination of thirty years of work. He'd been fascinated by the probem since he came across it in a library book at the age of ten. The documentary below shows what was involved in him producing his proof, and the nightmare that resulted from it:

Saturday, June 6, 2009

The brain

This documentary about the human brain may be a bit old and dated now, but it does contain some particularly interesting information about the way we think:

video

Monday, June 1, 2009

Uncertainty and Schrodinger's Cat

Werner Heisenberg (5 December 1901 – 1 February 1976) was a German theoretical physicist who made fundamental contributions to quantum mechanics and is best known for asserting the Uncertainty Principle of quantum theory. In very simple terms, this states that for a particle such as an electron, the more precisely you know one physical property such as momentum, the less precisely you are able to state another property such as position. This is not a statement regarding the limitations of measurement, but rather a philosophy on the nature of the Universe (at the quantum level) - that events are essentially probablistic. This means that a particle could exist in one position, another, or even both points at the same time. See my earlier post (The Bell Experiment) for a better explanation.

Other physicists such as Albert Einstein and Erwin Schrodinger refused to believe in a probabilistic Universe, and defended the classical view of determinism (all events are determined by what has gone before - that every event has a cause). In fact, in his later years, Einstein became embittered by what he percieved as the vandalism of classical physics by theoreticians such as Heisenberg, Niels Bohr and Wolfgang Pauli.

In order to defend determinism, Schrodinger attempted to show that Uncertainty was, at heart, ridiculous by coming up with his famous thought experiment, concerning a cat in a box (no real cats were ever experimented on). This is explained in the following clip:

video

However, despite Einstein's and Schrodinger's objections, quantum mechanics continued to go from strength to strength and has survived many attempts to discredit or falsify it. It even provides an explaination for radioactive decay and the Big Bang (which determinism is unable to do), in the sense that quantum mechanics provides a framework in which an event occurs spontaneously, without a deterministic cause.

And Schrodinger's Cat? Again, in simple terms, a cat does not itself exist at the quantum level, although the particles it is made up of do. Therefore we cannot resonably expect a cat (as a whole entity) to behave in the same way as a sub-atomic particle.