allow me to represent 'non-math' people in some way, and argue that the communication gap is caused by the inadequacy of everyday language and its underlying concepts. Take "orderless" for example: if a collection is in itself orderless, then it is best represented after nicely sorting it. So, when its ordering contains no info, you can freely sort it (if you can). And that's different from 'has no property by which it can be ordered'. More such underlying concepts (in any language) make for troubled communication. Try ro read any publication, and you're swamped by nouns with a particular meaning, known to all who know about such things, but pointing (but often not) the uninitiated to other places where they are 'explained'. Ha! In concreto, I've tried to reconstruct http://www.math.umn.edu/~tlawson/old/18.704/symmetric1.pdf pg. 2, where the author tries to explain how to gently generate the character table of group S4 : "Let’s write the characters of these permutation representations in a table for S4. To do this, we have to do some work and figure out how many tabloids are fixed by each cycle type; " I've not succeeded. Been trying for days. The concepts are not clear enough to me. I follow C. E. Shannon in this: you need a common reference basis to exchange info. And such 'common concepts' are best acquired young, and by face-to-face interaction. It takes a darn good Knuthly writer to achieve this by text only. no? Wouter.