Software engineering involves modelling very different domains (e.g., business processes, typesetting, natural language, etc.) as software systems. The main idea of this course is that this kind of modelling is also important when tackling classical mathematics. In particular, it is useful to introduce abstract datatypes to represent mathematical objects, to specify the mathematical operations performed on these objects, to pay attention to the ambiguities of mathematical notation and understand when they express overloading, overriding, or other forms of generic programming. We shall emphasise the dividing line between syntax (what mathematical expressions look like) and semantics (what they mean). This emphasis leads us to naturally organise the software abstractions we develop in the form of domain-specific languages, and we will see how each mathematical theory gives rise to one or more such languages, and appreciate that many important theorems establish "translations" between them.

Mathematical objects are immutable, and, as such, functional programming languages are a very good fit for describing them. We shall use Haskell as our main vehicle, but only at a basic level, and we shall introduce the elements of the language as they are needed. The mathematical topic treated have been chosen either because we expect all students to be familiar with them (for example, limits of sequences, continuous functions, derivatives) or because they can be useful in many applications (e.g., analytic functions, Laplace transforms).