To analyze the task of mental arithmetic with external representations in different number systems we model algorithms for addition and multiplication with Arabic and Roman numerals.Expand
Three different ways in which systems of axioms can contribute to the discovery of new notions are presented and they are illustrated by the various lattices have been introduced in mathematics.Expand
The domain of numbers provides a paradigmatic case for investigating interactions of culture, language, and cognition: Numerical competencies are considered a core domain of knowledge, and yet the development of specifically human abilities presupposes cultural and linguistic input by way of counting sequences, the cross-linguistic variability of which has implications for number representation and processing.Expand
Using four examples of models and computer simulations from the history of psychology, I discuss some of the methodological aspects involved in their construction and use, and illustrate how the existence of a model can demonstrate the viability of a hypothesis that had previously been deemed impossible on a priori grounds.Expand
This paper provides a general framework for assessing the efficacy of these typologies relative to certain desiderata, and it uses this framework to discuss the two influential typologies of Zhang & Norman and Chrisomalis.Expand
This paper focuses mainly on the propositional fragment of the Begriffsschrift, because it embodies the characteristic features that distinguish it from other expressively equivalent notations.Expand
Moritz Pasch (1843–1930) gave the first rigorous axiomatization of projective geometry in Vorlesungen uber neuere Geometrie (1882), in which he also clearly formulated the view that deductions must be independent from the meanings of the nonlogical terms involved.Expand