This chapter introduces the two main themes that structure this book: architectural thinking and properties. It identifies philosophical parallels and precedents for examining these two and their hypothesised relationships. This is followed by an overview of the relationship between architectural thinking and theory on the one hand and architectural properties and making on the other. The chapter explores the history of mathematical thinking about architectural space and form, demonstrating how various concepts have historically been represented in architecture. It identifies key symbolic (representational), semiotic (communicative), phenomenal (relating to perceptions), creative (relating to inspiration) and generative (algorithmic) applications. An example of a simple plan is used to explain how symbolic and representational thinking could be embedded in architecture. Then, potential mathematical properties of space and form are considered, the most important of which are those that can be measured (mensuration), simulated and predicted (statistics, modelling), or correlated (to empirical data). Some simple properties are explored by discussing three fragments of architectural plans by Palladio, Le Corbusier and Eisenman. The chapter concludes with an overview of the structure and content of the book.