http://www.percona.com/ppc2009/PPC2009_Boring_MySQL.pdf:

Avoid operate in the database if it can be done elsewhere

I have to admit I do not quite appreciate this statement. Can someone elaborate onto it?

Will it mean the we ought to enforce integrity around the application level rather than in the database level?

Or will it mean another thing completely?

We've not a way of knowing. Alone you never know what he meant by that's the writer, Baron Schwartz. A fast scan of his 35mm slides indicates he is not a fool, so I am confident he does not mean enforce integrity within the application not the database. He most likely means don't overload the database along with other processing that you can do more effectively elsewhere.

Dont add business logic towards the db if it can be done in the application level.

Database should be utilized for its title suggests - to keep data.

Typically of thumb, try keeping all logic inside low, otherwise the application becomes harder to keep and much more bug prone.