Sets for foundational representations? A design case study with probability and distributions

Ideas about sets are foundational to our understanding of many knowledge domains. And cognitive science tells us that the representation (notation or visualization) we use to encode the knowledge of a domain substantially determines what we can think and how easily we can reason about that do-main. Therefore, how a representation encodes ideas about sets may sub-stantially determine how readily we can comprehend, solve problems and learn about its domain. So, how should we design representations for knowledge rich domains to ensure that concepts about sets are readily ac-cessible and also effectively integrated with the domain’s other concepts? A case study is presented in which a representation for sets (Set Space Dia-grams) is taken as a foundation for a representation for probability theory (Probability Space Diagrams) and then further extended as a representation for statistical distributions (Distribution Space Diagrams). Together the three representations constitute a unified framework that conceptually inte-grates knowledge across the three domains.