The authors present in this paper a closely and carefully reasoned development of the design of a non-von Neumann computer architecture. Directly executing a Very High Level Language (VHLL) is the concept driving the design. A VHLL is described as one where the user states a problem in problem domain terms without needing any knowledge of the underlying hardware, software, or execution control to achieve solution. The VHLL used in this paper is one tailored for mathematicians doing algorithm development. The authors argue that this language and other VHLLs would most successfully work on a demand-driven direct execution architecture. Their examination of the underlying control structures and process communication led to a design that has many specialized processors with distributed execution control. The details of how and why processes are distributed over processors is also described. All critical design issues for a specific computer architecture are addressed.