Claudia Pombo

In this paper we study the mathematical object matrix and compare its historical and psychological origins. The literature on history of mathematics tells that this science started with the study of numbers and shapes, giving rise to the fields of arithmetic and geometry. The historical approach has been leading to an interpretation of algebraic structures as secondary concepts, derived from systems of equations representing numerical or geometrical constrains. Our epistemological view supports the argument that matrices originate from a especial form of tabulation, which had and still have a widespread use, appearing in different sciences and forms of expression. Originally, these tables are not numerical and became matrices when numbers were introduced in them. This view leads to the assumption that these algebraic structures are just as fundamental as numbers and figures. Therefore we can assume that algebra is one of the most fundamental domains of mathematics, forming a basic trilogy with the other two fields. This review gives rise to new interpretations in the physical realm, where observational languages are built a priori on suitable mathematical domains. It can show contrasting views between the historical approach and the epistemological view. Observational realism is our guiding epistemology.

The art of memory started with Aristotle’s questions on memory. During its long evolution, it had important contributions from alchemists, was transformed by Ramon Llull and apparently ended with Giordano Bruno, who was considered the best known representative of this art. This tradition did not disappear, but lives in the formulations of our modern scientific theories. From its initial form as a method of keeping information via associations, it became a principle of classification and structuring of knowledge. This principle, which we here name differentiation with stratification, is a structural design behind classical mechanics. Integrating two different traditions of science in one structure, this physical theory became the modern paradigm of science. In this paper, we show that this principle can also be formulated as a set of questions. This is done via an analysis of theories, based on the epistemology of observational realism. A combination of Rudolph Carnap’s concept of theory as a system of observational and theoretical languages, with a criterion for separating observational languages, based on analytical psychology, shapes this epistemology. The ‘nuclear’ role of the observational laws and the differentiations from these nucleus, reproducing the general cases of phenomena, reveals the memory art’s heritage in the theories. Here in this paper we argue that this design is also present in special relativity and in quantum mechanics