Elementary recursive algorithms

They run our lives, if you believe the hype in the news, but there is no precise definition of "algorithms" which is generally accepted by the mathematicians, logicians and computer scientists who create and study them. My main aims here are (first) to discuss briefly and point to the few publications that try to deal with this foundational question and (primarily) to outline in Sections 4 and 5 simple proofs of two basic mathematical results about the "elementary recursive algorithms" from specified primitives expressed by recursive (McCarthy) programs.