In this article, the author suggests a new topology of numerical notation systems. This new typology organizes each system by two axes. The first of these axes is cumulative versus ciphered versus multiplicative and the other is additive versus positional. The author goes on to argue that, considering the intelligibility of number systems even in the absence of knowledge about their associated language, human number systems are a reflection of cognitive processes. Finally, the author uses this new typology to build a theoretical model of numerical system evolution over time. The article concludes with suggestions for utilizing this typology in future cross-cultural research.