Dusko Pavlovic raises some interesting points - I just want to comment quickly on a couple of them. Firstly, it may be that (like those annoying "semiconvergent" series in numerical analysis), attempts to clarify something pass through a point of maximum simplicity after which they get worse again. Given all the indications that we can't have the kind of foundations that were once sought after (Goedel and All That), perhaps we are just looking for that point? The other comment addresses the idea that children use complex grammatical structures without completely understanding them. Sure they do - but they don't always use them *correctly*. Neither, of course, do adults. The trouble with mathematics is that unlike most of what we use natural language for, its groundrules don't (despite certain recent articles) permit statements on the 'well, it's mostly right and everybody knows what I mean' level. Most natural language usage jumps out of the system all the time... which gets into the AI debate. In mathematics, the medium is much closer to being the message. We try not to get into situations similar to the old chestnut "Time flies like an arrow, while fruit flies like a banana." , supposed by some to show that computers can't parse English. (I'm not sure that it does, any more than if I say 'Professor Smith is visiting next week' and you don't know whether to ask 'Is he?' or 'Is she?' , that shows that *you* don't know how to parse English...) But natural languages do, because in "natural" situations there is something outside the linguistic universe that can be referred to usefully. -Robert Dawson