A social mathematical machine is a networked system integrating humans and software to cooperatively solve mathematical problems. The software system MathChat impelments an instance of a social mathematical machine for collaboratively carrying out mathematical computations by entering instructions in a command shell. The software design of the MathChat system tightly integrates chat with command entry to support user communication and coordination. A theoretical model for the MathChat system is proposed. The theoretical model is based on the psychological theory of Distributed Cognition and the formalism of Message Passing in distributed computation, which allows formalizing the knowledge acquired by the system through repeated computations. In addition to elaborating on the software design and theoretical model, an application of the MathChat system to computational network analysis at Seton Hall University will be described.