大数据不仅在科学、工程与计算智能中有着广泛的应用，而且在人类感知、估计、量化、记忆和推理的认知机制中发挥着基础性作用。通过对大数据科学理论的基础研究，提出一组大数据系统的一般原理和分析方法。为了从形式上解释大数据的起源和本质，探讨大数据的认知基础及其数学模型，严格地引出了根植于科学、工程和社会各个领域的大数据的一般模式。研究发现大数据不再是传统实域上的纯数，而是一个前所未有的新型数学结构，称为递归类型化超结构（RTHS）。这一大数据系统的基本拓扑特性揭示了大数据工程的复杂性及其操作与处理的全新认知、理论挑战，以及可选解决方案。
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.
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