Back of the envelope reasoning for robust quantitative problem solving

Humans routinely answer questions, make decisions, and provide explanations in the face of incomplete knowledge and time constraints. From everyday questions like "What will it cost to take that vacation?" to policy questions like "How can a carbon taxing scheme affect climate change?" we often do not have all the knowledge, time and computational resources to come up with a precise, accurate answer. This thesis describes and formalizes Back of the Envelope (BotE) reasoning - the process of generating rough quantitative estimates. We claim that a core collection of seven heuristics: mereology, analogy, ontology, density, domain laws, balances and scale-up achieves broad coverage in BotE reasoning. We provide twofold support for this claim: (1) by evaluation of BotE-Solver, an implementation of our theory, on thirty five problems from the Science Olympics, and (2) by a corpus analysis of all the problems on Force and Pressure, Rotation and Mechanics, Heat, and Astronomy from Clifford Swartz's book (2003), "Back-of-the-envelope Physics." An aspect of estimation is learning about quantities: what is reasonable, high and low, what are important points on the scale. We call this facility for quantities as quantity sense. We present the Symbolization By Comparison (SBC) theory of quantity sense. This theory claims that quantity sense consists of qualitative representations of continuous quantity, or symbolizations, which are built by process of comparison. The computational implementation of the SBC theory, CARVE, is evaluated in a functional manner. The representations generated by CARVE help generate more accurate estimates.