Proof In Mathematics

Published in the Donmar Warehouse programme for Proof (2002).

What is your definition of a mathematical proof?

Scientists often talk about proving a theory, which means gathering enough evidence and data to support a hypothesis until it seems overwhelmingly likely that it is true. It is a bit like trying to prove that a suspect is guilty beyond all reasonable doubt. But there can be miscarriages of justice and there can be errors of science, because measurements and observation can never be absolutely perfect and accurate. In mathematics, however, there is the notion of a perfect proof, resulting in a theorem that is undeniable. Mathematical proofs depend on logic and argument. Mathematical proofs do not rely on imperfect measurements of the real world, so they can be perfect. For example, we laugh at some of the science theories of Ancient Greece, but we still teach the theorems of Euclid and Pythagoras. These are immortal truths. Are someproofs better than others? Well, all proofs are true, but some are more beautiful than others. Some proofs are short, simple and elegant. Others are beautiful because of their intricacy. Others are considered important because of their implications for other areas of mathematics. Perhaps the quality of surprise is also desirable. A proof that demonstrates something shocking or demonstrates something in a shocking way is a top class piece of mathematics.

Can you explain the level of emotional investment mathematicians put into their work?

The maths we learn in school is established. We are following a well-trodden path across a fairly mundane landscape. In contrast, mathematicians are exploring the unknown. They are searching for a proof of a theorem, a proof that might not even exist. They are using their intuition and experience to work out which path they should follow, and inevitably they will encounter chasms that will force them to backtrack and search for a new path. I think that the mystery of the unknown and a childish curiosity dominates the mind of the working mathematician. And as the proof emerges, then perhaps other emotions take over. A mathematical proof is a creation, just like a poem or a symphony, so there must be an overwhelming sense of passion and pride in creating something beautiful and profound. I sometimes make the comparison with music. Music is written in a notation that I cannot understand, but at least I can hear a symphony being played and appreciate its beauty. Mathematics is written in an equally impenetrable notation, and worse still there is no music associated with mathematics. The only way to truly appreciate mathematical ideas is to understand the notation. If there was a sort of music associated with mathematics, then many more people would see what drives mathematicians.

Can we prove the existence of mathematics?

Mathematicians argue about the existence of mathematics. Is it ‘out there', waiting to be discovered, or is it in our minds, waiting to be 'invented'? Some feel that mathematics is a bit like a game. Mathematicians select a few rules, then we play some games, and playing games is the equivalent of doing mathematics. The rules need to be consistent and should lead to a rich variety of games. In geometry, there are different sets of rules, which lead to different games and different results. Draw a triangle on a flat piece of paper and the angles always add up to 180 degrees, but play with a different set of rules and the result is different - a triangle drawn on a sphere could have angles that add up to 270 degrees.

Why can't computers prove every theorem?

Some mathematical questions involve infinity Is there an infinite number of primes? How many numbers up to infinity are sandwiched between a square and a cube number? Are some infinities bigger than others? Mathematicians can answer these questions using brilliant arguments. Computers cannot, because they cannot check every number up to infinity, because that would take eternity Mathematics is about inspiration, intuition, determination and experience, and these are qualities that computers lack. By the way, the answers are; 'yes 'there is an infinite number of primes, only 26 and zero are sandwiched between a square and a cube, and some infinities are indeed bigger than others.