Our guide for the evening was Brendan Larvor from the University of Hertfordshire where he specialises in the history and philosophy of mathematics and science.
Over a couple of hours we had an entertaining and thoughtful discussion on mathematical proof (discussion is the whole point of Big Ideas) during which I tried to take some notes while also trying to work out why I did not quite agree with the hypothesis presented. That is my excuse for why the following notes are unstructured and probably inaccurate.
Some pictorial proofs are easy and Brendan used the small panes in the windows behind him to prove the that the numbers in the sequence 1+3+5+7+.. are always square numbers (1+3 = 4 which is 2**2 and 1+3+5 = 9 which is 3**2). In pictures this can be seen by adding one more block to two sides of a square to make a slightly bigger square.
Similarly, it's is visibly obvious that 1/2 + 1/4 + 1/8 + ... = 1 as each additional step, 1/2**n, finishes half way between the previous step and 1. Drawing this on a number line makes it obvious.
The invention of mathematical notation made other proofs possible. It is easy to prove that 1/2 + 1/3 + 1/4 + 1/5 + ... is infinite as the sequence can be grouped into an infinite number of segments each of which adds up to more that 1/2, i.e. 1/3 + 1/4 and 1/5 + 1/6 + 1/7 + 1/8.
Mathematical notation is still not always capable of finding proofs. It is fairly easy to see physically that one knot on a line can pass by another (simply by loosening one of them) but there is no formal proof for this. Of course it may be that a proof is not possible because the question is badly formed, and that in itself would be interesting if we could prove it.
There are various standard ways of proving things. Mathematicians like to find other ways to prove things that have already been proved (we may have a proof for Fermat's Last Theorem but it is not the proof that Fermat had, if he had one) and to find other generic ways of proving things. Proof is very important in Mathematics.
The realm of Logic introduced words into Mathematics and, therefore into mathematical proofs. For example, if A implies B and B implies C then A implies C.
This logic is not generally well understood and people still use arguments like you are a man, men play golf therefore you play golf.
Using words to describe mathematics lets linguists into the game and that was the part of the discussion that I had problems with. Just because linguists have rules it does not mean that it makes sense to extend them to the use of language in mathematics. I think that Mathematics is supreme here and if there are any conflicts then it is the upstart linguists that have got a problem to solve.
Some proofs come without insight or meaning. For example computers have solved the four colour map problem by number crunching all possibilities that does not help us to understand why it is true.
It is often said, and I would tend to agree, that the best solutions are simple and elegant but it this just pretension (quite possibly!) or is there something more right about a simple proof than a complex proof. If Fermat did have a proof for his Last Theorem then it was a lot simpler than the one we have now.
We also touched on the world of proof in other areas where proving something absolutely is much harder. For example, Climate Change is accepted by most scientifically minded people, myself included, but without a parallel world that does not have Man on it it is impossible to say absolutely what the impact of Man is. Similarly economists do not have another global economic system to compare ours with.
Big Ideas on Mathematical Logic was two hours of brain stimulating talk with a group of intelligent and interested people in a room above a pub. It was a happy birthday!