Thursday, 3 May 2012

Pearsall Smith and the goat at Portsmouth

Pearsall Smith and the Goat at Portsmouth

"Someone mentioned a goat, so I told them the story of the Goat at Portsmouth"

'In conversation, someone mentioned a goat, so I told them the Story of the Goat at Portsmouth, whereupon an awful thought crossed my mind: "Had I told them this story before?". But then a deeper chasm loomed before me: "Was it the case that whenever anyone mentioned a goat I always told the story of the Goat at Portsmouth?".'

This quaint metaphor is well know in my family, but seems largely unknown elsewhere. It was a favourite with my father, RGR West, psychiatrist and philosopher. My brother Michael points out that Father clearly read it in Logan Pearsall Smith's 'More Trivia' which came out in 1921. What a delight to discover that cerebral goldmine of quirky fragments (which you can read online here)! One lingers unintentionally, drawn from each disjointed gobbet to the next by its sheer inconsequentiality; the only common elements are the very 'Bloomsbury' figure of Pearsall Smith himself and the chintzy clarity of his Edwardian England; umbrellas, railway carriages, teacups and sofas. (I quote: "Though I sometimes lay down the law myself on public questions, I don't very much care to hear other people do it." )  So delightfully challenging, so carefully phrased! I therefore append Pearsall Smith's original.

THE GOAT

In the midst of my anecdote a sudden misgiving chilled me—had I told them about this Goat before? And then as I talked there gaped upon me—abyss opening beneath abyss—a darker speculation: when goats are mentioned, do I automatically and always tell this story about the Goat at Portsmouth?

From 'More Trivia' by Logan Pearsall Smith (1921) (Available online)

 

Tuesday, 1 May 2012

Goldbach's Conjecture

Goldbach's Conjecture

Every even number can be expressed as the sum of two primes.

One of the oldest unproven conjectures stems from a letter of 1742 from Goldbach to Euler. I was intrigued, for in seventy years I had not come across this type of concept, one which (as Euler declared) is certainly true even though it cannot be proven to be true.

I very soon convinced myself that the conjecture is sound, by an argument that many others have discovered and explored. The gap (g) between two adjacent prime numbers (p1, p2) grows as p1 increases, but grows far more slowly than p1 grows. For example the gap between the adjacent primes 977, 983, 991, 997 are (respectively) only 6, 7 and 6. As all primes (other that 2) are odd, any two added together will produce an even sum. It is easy enough to show for small even numbers (2n, 2(n+1), etc) that the conjecture holds: 4=3+1; 6=5+1, or 3+3; 8=7+1, or 5+3; 18=17+1, or 13+5, or 11+7.   Thus we see that as 2n increases the number of ways in which it can be expressed as the sum of 2 primes also increases. So, even though it is a laborious task for computer enthusiasts to find the necessary primes, it becomes progressively harder to imagine the falsification of the conjecture.

But that is not a proof that the conjecture must, for all values of n, be true a priori. It is at least in part an induction a posteriori.