On one side of the divide we have mathematicians playing a parlor game called Bourbaki with other mathematicians. The game goes like this. Think up definitions for a handful of cute nouns, verbs, adjectives and adverbs: oblate corkscrewed doubly-banded sub-farkleoid. Now write a stream of papers classifying the nouns with their adjectives and composing the verbs with their adverbs. Tenure is only another 10 turns of the crank away. This mathematics 100% content-free; synthetic problems worked in synthetic settings."Playing Bourbaki" is exactly the kind of math I like.
On the other side of the Great Divide we have mathematicians talking to students. Here we have politically correct (or at least not politically incorrect) problems cast in severely dumbed-down non-Bourbaki mathematics. Have the student push the numbers around for an hour or two and mix in a couple x's and y's until they start to feel really good about saving the world from something.
Except for Part VII, The Influence of Mathematics, the Companion is all on the Bourbaki side of the Great Divide. This is not to say that there aren't some execellent sections. When an author really knows the subject you become convinced as you read that you understand it too. Barry Mazur on Algebraic Numbers is a wonderful example as is Computational Complexity by Oded Goldreich and Avi Wigderson. And the biographies in Part VI are by and large written with a light yet informative and insightful touch.
When I was researching Laguerre planes for my paper, what I found is that they are this pretty little math object, and that was enough for me. I had a wonderful time through all of the hard work I put in. It was only later, when I started to think about my presentation, or about my friends reading [under duress] the paper, that I started to feel bad about not knowing what the point of Laguerre planes is.
Do they have a point? Possibly. They are related to other things in geometry, and some things in geometry do have a point. (No pun intended.) But damned if I can figure out enough of the literature to know what the point is. And I don't really care.
I may begin my presentation by openly acknowledging this - that as far as I am concerned, Laguerre planes are just a pretty little math object, and look, isn't it shiny?