#### Non-Standard Analysis

My first paid work (as opposed to study) at the ISVR was on boundary-layer suction. I needed to brush up my knowledge of boudary-layer theory - the fluid dynamics lectures I'd had from the formidable Professor P O A L Davies as a BEng Engineering Acoustics and Vibration student in the late 80s, while fascinating and challenging, hadn't given me as solid a grounding as our current MEng Acoustical Engineering get nowadays.Around the same time I read

*The Problems of Mathematics*by Ian Stewart and was intrigued by the chapter about Abraham Robinson's Non-Standard Analysis. During my PhD Joe Hammond, my supervisor, had encouraged me to make contact with David Chillingworth in the Maths department and take his course on Advanced Calculus with Applications. Another fascinating course, this one from a pure mathematician (the 'application' turned out to be that if we were to cut out two particular cardboard shapes, glue them together with cork spacers, stand the result on its side and persuade a heavy enough beetle to walk along one of the perpendiculars, the structure would topple over when the beetle crossed a particular curve) and it introduced me to rigourous methods while showing me how little I, an engineer, knew about that whole area. Non-Standard Analysis seemed to offer a way to formalise the way engineers thought about infinitesimals and, Stewart suggested, allowed results to be obtained that would be much harder to derive by standard methods - 'canards' for instance.

The chapter's last section was called

*Logic for engineers*(no offence, eh?)

*and mentioned some areas of perturbation theory where it had been applied, one of which was boundary layer flow! This was exciting - perhaps I'd stumbled across a skeleton key that would enable me to unlock wonderful new results in boundary layer theory that couldn't be found any other way? I had to find out more, but all the references for the chapter seemed to be mathematical expositions of the method rather than applications, none more so than Robinson's original book on the subject which I flicked through but found very dense after Stewart's gentle introduction. I asked David Chillingworth if he knew who had applied Non-Standard Analysis to boundary layers. He didn't know but asked Ian Stewart. He couldn't remember where he'd got the boundary layer story from. I searched for references to Non-Standard Analysis in the engineering literature and found that Feri Farassat at NASA had been using them for infinitesimal shock thicknesses, which looked promising but the next time I met Feri I asked him about not only did he not know the boundary layer reference he warned me against using it for perturbation problems at all.*

I'd run out of leads when I met Geoff in the staff dining room. (In those less crowded days every table had paper napkins arranged alternately white and coloured - it was understood that the coloured ones were absorbent and were for mopping up spills, while the white ones were for sketching graphs and equations.) Geoff asked what I was up to and I told my tale, somehat surprised by the delight he seemed to be taking in it as I really wouldn't have expected him to have much interest in that sort of thing. His broad grin made it clear he knew something I didn't. The story soon unfolded: when he was at Cranfield College of Aeronautics Geoff had been friends with the inventor of Non-Standard Analysis Abraham Robinson who in the 1940s, despite his main field being logic and analysis, had thrown himself into aerodynamic theory as a contribution to the war effort and become a senior lecturer there. Geoff explained that they used have endless arguments about the importance of rigour and that Geoff had teased him that his ideas were all very well but irrelevant to anything he was interested in. So when 'Abie' published his comprehensive book

*Non-Standard Analysis*he took great delight in giving Geoff a copy and telling him "I've even put in a boundary layer example, just for you!". I went back to the Library and there it was hidden away at the back, a derivation of the basic equations in one paragraph. And if it hadn't been for Geoff it wouldn't even have been there at all.