Analytical geometry has never existed. There are only people who do linear geometry badly, by taking coordinates, and they call this analytical geometry. Out with them!

For example, it is well known that Euclidean geometry is a special case of the theory of Hermitian operators in Hilbert spaces.

It is indubitable that a 50-year-old mathematician knows the mathematics he learned at 20 or 30, but has only notions, often rather vague, of the mathematics of his epoch, i.e. the period of time when he is 50.

Certain foreigners, invited as spectators to Bourbaki meetings, always come out with the impression that it is a gathering of madmen. They could not imagine how these people, shouting (sometimes three or four at the same time) about mathematics, could ever come up with anything intelligent.

Quoted in D MacHale, *Comic Sections * (Dublin 1993)

Now ... the basic principle of modern mathematics is to achieve a complete fusion [of] 'geometric' and 'analytic' ideas.

Should we teach modern mathematics, *American Scientist* **61** (1973), 16-19.

We have not begun to understand the relationship between combinatorics and conceptual mathematics.

*A Panorama of Pure Mathematics: As seen by N. Bourbaki*, (New York 1982)

JOC/EFR April 2011

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