The word "mathematics" comes from Ancient Greek μάθημα, meaning "that which is learnt", "what one gets to know", hence also "study" and "science". The word for "mathematics" came to have the narrower and more technical meaning "mathematical study" even in Classical times.
In Latin, and in English until around 1700, the term "mathematics" more commonly meant "astrology" (or sometimes "astronomy") rather than "mathematics"; the meaning gradually changed to its present one from about 1500 to 1800. This has resulted in several mistranslations. For example, Saint Augustine's warning that Christians should beware of mathematici, meaning astrologers, is sometimes mistranslated as a condemnation of mathematicians.
Several fields of mathematics feature prominently in the work of the Thalesians:
- Mathematical logic, also called formal logic, is a subfield of mathematics exploring the applications of formal logic to mathematics. It bears close connections to metamathematics, the foundations of mathematics, philosophy, and theoretical computer science.
- Theoretical computer science (TCS) is a subset of general computer science and mathematics that focusses on mathematical aspects of computer science such as the theory of computation, lambda calculus, and type theory.
- Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.
- Linear algebra is the branch of mathematics concerning linear equations, linear maps, and their representations in vector spaces and through matrices.
- Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense.
- Probability theory — the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treates the concept in a rigorous mathematical manner by expressing it through a set of axioms.
- Mathematical optimization is the selection of a best element, with regard to some criterion, from some set of available alternatives.
- Mathematical finance — also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modelling of financial markets.