The authors describe systematic methods for uncovering scientific laws a priori, on the basis of intuition, or "Gedanken Experiments". Mathematical expressions of scientific laws are, by convention, constrained by the rule that their form must be invariant with changes of the units of their variables. This constraint makes it possible to narrow down the possible forms of the laws. It is closely related to, but different from, dimensional analysis. It is a mathematical book, largely based on solving functional equations. In fact, one chapter is an introduction to the theory of functional equations.

The book describes up-to-date applications and relevant theoretical results. These applications come from various places, but the most important one, numerically speaking, is the internet based educational system ALEKS. The ALEKS system is bilingual English-Spanish and covers all of mathematics, from third grade to the end of high school, and chemistry. It is also widely used in higher education because US students are often poorly prepared when they reach the university level. The chapter by Taagepera and Arasasingham deals with the application of knowledge spaces, independent of ALEKS, to the teaching of college chemistry. The four chapters by Albert and his collaborators strive to give cognitive interpretations to the combinatoric structures obtained and used by the ALEKS system. The contribution by Eppstein is technical and develops means of searching the knowledge structure efficiently.

The book describes up-to-date applications and relevant theoretical results. These applications come from various places, but the most important one, numerically speaking, is the internet based educational system ALEKS. The ALEKS system is bilingual English-Spanish and covers all of mathematics, from third grade to the end of high school, and chemistry. It is also widely used in higher education because US students are often poorly prepared when they reach the university level. The chapter by Taagepera and Arasasingham deals with the application of knowledge spaces, independent of ALEKS, to the teaching of college chemistry. The four chapters by Albert and his collaborators strive to give cognitive interpretations to the combinatoric structures obtained and used by the ALEKS system. The contribution by Eppstein is technical and develops means of searching the knowledge structure efficiently.

The authors describe systematic methods for uncovering scientific laws a priori, on the basis of intuition, or "Gedanken Experiments". Mathematical expressions of scientific laws are, by convention, constrained by the rule that their form must be invariant with changes of the units of their variables. This constraint makes it possible to narrow down the possible forms of the laws. It is closely related to, but different from, dimensional analysis. It is a mathematical book, largely based on solving functional equations. In fact, one chapter is an introduction to the theory of functional equations.