1. Syllogism relate to categorism
Categories are based on distinct identities. I believe that catergories are aided by syllogism which reveals (intentionally and unintentionally) an array of finite characteristics and function.

2. Barr
What about the entire underlying notion that is based on viewing geometric shapes topologically according to their geometric definition. For example, on page 12 there is a pentagon and a topological pentagon. The only thing that makes these similar is the fact that they each have 5 verticies and 5 edges. Whether or not the lines are straight is irrelevant. That is not to be considered by the viewer. It's a differnt state of mind in a way.

3. No geometry in architecture
Other than the obvious negotiating and writing of contracts one could argue that the assembly of material schedules does not use geometry. Yes the physical materials relate to geometry but simply specifying a particular brand and assigning materials to their respective locations does not require geometry.