Laurence phoned on Friday. His procurement training project at the MoD has been winning awards left right and centre � we decided it will be a good example of UK best practice for the supply chain management project I am involved with together with the two Mikes. I told him about the introduction of Kantian aesthetics that I read in Lycia and subsequently sent him some details. He suggested that I but the Dylan bootleg 7 from the Scorcese film which I have done.

Stephen B is back from walking the Lycian Way � the eastern half � including climbing Mount Olympos. We met up at the cluster-group meeting at the West Midlands RDA on Friday. I presented a paper on the numerical dimension to the four level strategy on a page which Viv and I have evolved for the automotive sector. Stephen reminded me about a ONS publication which goes by the code PA1003. It is based on various official registers like the VAT register and it has a report which looks at the size distribution of business units within regions analysed by Standard Industrial Classification. So it is possible to compare the way that the size distribution (by employment numbers) in the West Midlands has changed say over a six or seven year period.

I did a quick sketch on Friday evening and it looks as if at the high end � say units employing 500 or more there has been no net change. There may have been changes within the population � some units may have closed down and others been established or expanded from a smaller size. Similarly within the band there may have been a reduction in the average size. Nonetheless this stability is surprising.

At the lowest end of the distribution � say units employing less than twenty people � the numbers have declined but the rate of decline has been about the same within the region as they have been nationally. The main area of weakness seems to be the units which employ over 20 and less than 100 where the rate of decline has been much faster than the national average. If these results are right then they have potentially quite important implications.

Mrs C went to see the Puccini opera about the Wild West at the ROH on Saturday night. Not one of his best she says.

I have been looking at some of the pencil drawings Vita has been doing for a current project which compares Matisse and Cartier-Bresson. There are drawings and an acrylic copy of some Matisse portraits and then there is a pencil drawing of a Cartier Bresson photographic portrait which I think is pretty impressive. There is also the angle that the graphic style is nuanced in the direction of Matisse and then applied to the photograph. She seems to appreciate the AN material which I got from Brussells.

I have been reading more Yourgrau�s book on Godel and Einstein at Princeton which is extremely good. He looks at the way that G and E fitted into Central European intellectual life before they cut out for the US. G came from Brno � a town I have visited a couple of times and which I rather like � I have heard Moravian folk music and post-McCoy Tyner jazz there. It also has some stupendous mid century exhibition halls and is also the home of G Mendel who helped shape modern genetics.

Yourgrau explains the intellectual warfare that went on in Vienna between Mach who was a radical positivist and Boltzmann who was a mathematical modeller. He locates both G and E as artfully placed between these traditions. For example the approach that relativity adopts to measurement of the speed of light looks very positivist in as much as it posits that the measurement experiment will always give the same answer as an axiomatic fact. From there it dissolves categories such as space and time which had been metaphysical absolutes. But in fact Einstein is both pragmatic and opposed to progressing too far in that direction. At first he was opposed to Minowski�s mathematicisation of special relativity but then he used that approach to derive general relativity.

Godel mathematises axiomatic proof by using an arithmetic coding and certain properties of prime numbers and recursive functions. I think I have got the prime numbers bit more or less � any natural number is the product of a unique set of prime numbers. If you code the prime numbers into the axiomatic structure of arithmetic then each number stands for a statement in axiomatic arithmetic and you can use arithmetic to assess proof ie whether within the axiomatic theory one statement flows from another.

At the first level this is cunning � and at the second level it models the design of computers where numbers stand for other things and you manipulate the numbers through a few simple rules � eg the rules of axiomatic arithmetic.

Yourgrau is convinced that the Godel proof is the greatest bit of mathematical logic in the 20th century. The most interesting thing is why this should be the case and has to do with the context. On the one hand Frege and then Russell had thought it was possible to axiomatise logic and within one system to derive arithmetic � this looked like the big game and indeed Russell�s Principia Mathematic � his masterwork � is part of this. Frege�s work is considered so important that before I started Moral Science the faculty wrote to students suggesting they read his foundations of artithmetic which indeed I did. So the idea was that to do philosophy at all at the level they wanted to teach it you had to build on Frege.

Yourgrau explains that the project failed � first of all when Russell discovered his paradox � deep in the heart of set theory � but it was absolutely kyboshed by Godel who showed that mechanised proof-models would always be incomplete. That there would always be a statement that the model couldn�t decide. The thing is � why does this matter? Not everyone could see that it did matter � in particular Russell just couldn�t get it. His excuse was that he had given up mathematical logic for 18 years by the time that Godel derived his proof.

The answer is to do with recursive functions and in particular the work that Church did at Princeton. I have to say that I don�t have my head round this but I feel motivated for a variety of reasons.

Guildford�s greatest logician is undoubtedly Turing � not only did he go to Kings but he got Kings to pay for him to go to Princeton and study with both Godel and Church � then he came back and sat in on Wittgenstein�s seminars on the Foundations of Mathematics in Trinity � and we have transcripts of the debate. Then he went off and worked on Enigma. It would be fun to be more on top of the work of our local boy made good.

Yourgrau has a medium level position on the relevance of Godel. He thinks it shows that we will always need mathematical intuition and that people will always be able to do more than computers. Some people think that Godel proved the existence of God. Interestingly Godel did actually try to prove the existence of God � but not through his Incompleteness Theorem. He formalised Anselm�s Ontological Argument using modal logic � but he didn�t do this until after the invention and dissemination of the computer. There have to be some ironies there.

Anyway another reason for interest in all of this is the function of DNA. People have tended to think of DNA as a simple Turing machine eg one which copies and counts � some sort set of simple recursive operations. But just recently it has been decided that it operates at a higher level � for example rather than just executing the programming DNA might also do a bit of programming itself. So a chunk of DNA might have spent millions of years with one function and being passed on from cell � and then suddenly it adds another higher level function to itself � almost as if DNA have just described something simply suddenly decides its going to do metaphor as well � or move from rhyme to free verse.

I think that Yourgau is heading in the direction of black holes � and some odd properties of time. We�ll have to wait and see.