A mathematical roadmap for quantum chemistry

This text accompanies the Mathematics canvas.

Each box contains one field, or topic, and they are connected to other boxes with the arrow indicating the direction from "more general" to "less general and more special". Click headings in this canvas to go to relevant sections in this note.

Of course, many more arrows could be drawn, and many more boxes could be added. In order to make the graph not too cluttered and confusing, we keep only main links.

NOTE: Topics marked with a ❤️ are considered of prime importance to any quantum chemist, regardless of their specialization.

Logic and Set Theory

Logic is a branch of mathematics and philosophy that deals with reasoning, the principles of valid inference, and the structure of propositions, including but not limited to mathematical statements. It is usually subdivided into propositional logic, dealing with statements that are true or false but not both, and predicate logic, which add the ability of logic to express properties of objects and relationships between them. For example the argument "If P then Q. Q is not true. Then P is not true" uses propositional logic. The statement "there exists an x such that Joke(x) and Funny(x)" is an existential statement from predicate logic.

Logic provides the formal framework used to analyze and construct mathematical proofs, ensuring that conclusions follow from premises in a valid and systematic way.

Set theory is the branch of mathematics that studies sets, which are collections of objects. It forms the foundation for much (all?) of modern mathematics, providing the language and basic concepts used to describe and analyze mathematical structures. Set theory forms the foundation in the sense that every other mathematical theory can be formalized in the language of set theory. Indeed, in set theory, every object is a set.

Most mathematicians are aware of formal set theory, but the "version" used in most contexts is "naive set theory", which in a more informal manner defines mathematical sets compared to the rigorous axiom based constructions. As Russel's Paradox shows us, naive set theory has some pitfalls. Thus, in mathematics, one must resort to formal set theory in situations that can be described as "borderline".

Although formal set theory is unlikely to be applied in the study of quantum chemistry, the basic notation is widely-used and an important language tool when specifying computational methods and their implementation.