Linear algebra is often seen as the study of vectors and matrices, but its geometric aspect is often ignored in most treatments of the subject in books and university courses. In truth, linear algebra, or more accurately linear algebraic geometry, is the study of n-dimensional space and linear transformations on that space; here, vectors are the manifestation of a "difference" between two points in affine space (the most fundamental space in linear algebra) and matrices are in essence "representations" of the linear transformations between spaces. The usual concepts of linear algebra, such as the determinant, basis, Gaussian elimination and subspaces, are consequences of the geometric outlook of the subject, as opposed to the usual algebraic emphasis that the subject contains. Linear algebra naturally extends to the study of Lie algebras, whose roots in algebra and geometry are often ignored in favour of its connections to analysis and topology.
Kolmogorov (1933) formed probability theory using only elementary ideas from point-set topology and measure theory. In his framework, one relies on building a closed topology of subsets of a certain set, called the sample space, and defining a real measure which maps each element of this closed topology, more commonly called a sigma-algebra, to a number between 0 and 1. Something that may be of interest is whether one could define a measure onto an arbitrary field; it seems that relying on the ideas of Kolmogorov and generalising as much as possible is the ideal way to understand probability theory in an abstract sense, which makes it equally appealing to the pure/theoretical mathematician as it is to statisticians and applied mathematicians. Anothing thing that may be appealing to the interested is to see whether ideas from rational trigonometry can be used to give more power to Kolmogorov's approach to abstracting the ideas of probability and ultimately apply it to current statistical approaches.
Last updated: 19 August 2021