The big data play an indispensable role not only in a wide range of science fields and engineering applications, but also in the cognitive mechanisms of the sensation, the quantification, the qualification, the estimation, the measurement, the memory, and the reasoning of human beings. This paper reviews the basic studies of the theoretical foundations of the big data science, as well as a coherent set of general principles and analytic methodologies for the big data systems. The cognitive foundations of big data are explored in order to formally explain the origin and the nature of the big data. A set of mathematical models of the big data are created to rigorously elicit the general essences and patterns of the big data across pervasive domains in science, engineering, and society. A significant finding about the big data science is that the big data systems in nature are a recursively typed hyperstructure (RTHS) rather than pure numbers. The fundamental topological properties of the big data reveal a set of denotational mathematical solutions for dealing with the inherited complexities and unprecedented challenges in big data engineering.

big data science  /  big data engineering  /  mathematical models  /  recursively typed hyperstructures (RTHS)  /  cognitive computing  /  computational intelligence