The Structure of Mathematical Knowledge

This report develops a conceptual framework in which to talk about mathematical knowledge. There are several broad categories of mathematical knowledge: results which contain the traditional logical aspects of mathematics; examples which contain illustrative material; and concepts which include formal and informal ideas, that is, definitions and heuristics.

Symbolic Mathematical Laboratory

A large computer program has been developed to aid applied mathematicians in the solution of problems in non-numerical analysis which involve tedious manipulations of mathematical expressions. The mathematician uses typed commands and a light pen to direct the computer in the application of mathematical transformations; the intermediate results are displayed in standard text-book format so that the system user can decide the next step in the problem solution. Three problems selected from the literature have been solved to illustrate the use of the system. A detailed analysis of the problems of input, transformation, and display of mathematical expressions is also presented.

Boundaries and Topological Algorithms

This thesis develops a model for the topological structure of situations. In this model, the topological structure of space is altered by the presence or absence of boundaries, such as those at the edges of objects. This allows the intuitive meaning of topological concepts such as region connectivity, function continuity, and preservation of topological structure to be modeled using the standard mathematical definitions. The thesis shows that these concepts are important in a wide range of artificial intelligence problems, including low-level vision, high-level vision, natural language semantics, and high-level reasoning.

Experimental Markets for Product Concepts

Market prices are well known to efficiently collect and aggregate diverse information regarding the value of commodities and assets. The role of markets has been particularly suitable to pricing financial securities. This article provides an alternative application of the pricing mechanism to marketing research - using pseudo-securities markets to measure preferences over new product concepts. Surveys, focus groups, concept tests and conjoint studies are methods traditionally used to measure individual and aggregate preferences. Unfortunately, these methods can be biased, costly and time-consuming to conduct. The present research is motivated by the desire to efficiently measure preferences and more accurately predict new product success, based on the efficiency and incentive-compatibility of security trading markets. The article describes a novel market research method, pro-vides insight into why the method should work, and compares the results of several trading experiments against other methodologies such as concept testing and conjoint analysis.