Guided tour to theorem country

What's a theorem? An assertion you can prove is true, at least that's what I'm asserting following yesterday's seminar at an Oxford University Continuing Education (OUCE) day.

If you've seen the web site Theorem of the Day, then you'll get a flavour of the day, created by the presenter, Robin Whitty. That web site is challenging though and so was the day. However, if you like thinking, and thinking in good like minded company, then you'd have enjoyed it too. The company included physicists, sixth formers, an eminent professor of biology, chemists, a member of the society of mathematics and her husband who had a PhD in maths, though not in the same area she hastened to assure me, and retired maths teachers.

Starting with some maths language and shorthand helped me remember some of the maths I used to know, and others in the audience were equally forthcoming with questions, which made for a relaxed atmosphere. Theorems covered included:

• Euclid's Infinity of Primes
• Ramsey's Theorem (I can see how to relate that to how many people you know)
• Contraction Mapping Theorem
  
• Hardy-Ramanujan Asymptotic Partition Formula - partitioning is so simple that you teach it to infant school children - it's how many different ways can you add up a number, like 5 is 2+3, and also 4+1, and 1+1+3 and so on
• Erdos-K-Rado Theorem on intersecting permutations - we got a bit confused on this one because Robin initially told us to think of Rubik's cube, but it isn't quite the same.
• The Robbins problem - I like Boolean algebra
• Morley's Miracle - that was an amusing story of a school teacher who discovered this by accident. If you tri-sect the angles of a triangle then at the intersections of the trisection lines, you can make the vertices's of a triangle. The miracle is that no-one ever noticed before Morley did, but then drawing a trisection isn't very easy.

We looked at the philosophy of maths - Are the truths of mathematics invented or discovered? A Canadian high school philosophy competition asked this question. The context, rules and winning answer can be downloaded from here.

Finally we had a discussion about whether computers 'do' maths. There's a Faustian battle between mathematicians: you have a choice between geometry and algebra. If we use computers, then we turn to algebra to compute rather than geometry that allows our intuition.

For example, someone had been presenting partitions from using the Pascal's triangle. At the end of the lecture, one person, Corteel, saw a bijection along the rth diagonal within the binomial expansion that matched the rth row of the partitioning - a computer couldn't have seen that!

I might go to another OUCE maths day - like the one next year on the history and consequences of calculus.