1. How does the Syllogism relate to categories?

My interpretation/understanding of Aristotle's definition of syllogism relates to the ideas of categories by its correlation to the idea of definable characteristics that in turn make it understood as specific functions. Similar to how Allen Turing created computation as a way to rationalize a specific set of commands to perform a range of functions, Aristotle's syllogism uses categorizes as a way to identify certain characteristics which can then form a more complex range of meanings.

2. Where in Barr's text do you find something different than geometry?

Throughout Barr's text it seems that the subject of geometry is a constant however on pg. 4 in which he explains the transformation of topology creating different functional uses of a given object, it becomes apparent that topological manipulations have the ability to alter what we recognize as a certain object(in this case a taurus to a coffee mug) while maintaining its fundamental organization.

3. Where do we not use geometry in architecture?

Perhaps when we focus on the use of the algorithm to perform a specific sequence of tasks, we are relying on the intelligence and looped system output of the equation to generate form rather than relying on particular geometric systems for their building potential.