William Bricken Ph.D, Iconic Arithmetic Volume I: The Design of Mathematics for Human Understanding (Unary press, 2019).
p. 141 The most important characteristic of these axioms is that two of them specify how to delete Structure. Both implicate only one form (labeled A), so that they both require only simple pattern-matching. Remarkably, this leaves all of the complexity of numeric algebra isolated in one Pattern Transformation.
p. 211: A final difference between sets and bounded tallies is that set theory is thoroughly dependent upon first-order logic for its definitions and its axioms, whereas boundary forms instead depend upon pattern matching.
# Proof by Pattern-Matching and Substitution
Axioms *statically* define patterns that are equivalent, and *dynamically* permit transformation between patterns. Structural axioms are implemented by matching a given structure to a permitted pattern and then replacing it by a certified equivalent structure. Two axioms identify void-equivalent forms, permitting deletion of structure. The third permits rearrangement. This makes computation and Verification short and elegant. (p. 152)
p. 361 Models of Multiplication
Functional Programming is when functions, not objects or procedures, are used as the fundamental building blocks of a program. Functions in this sense, not to be confused with Cee Language functions which are just procedures, are analogous to mathematical equations: they declare a Relationship between two or more entities.