Mathematics as Formal Structures

The formalist concept of mathematics was built in steps. The first step was taken by Hilbert when he investigated the foundation of geometry. It had long been recognised that Euclid’s Elements had flaws, as some proofs were not only reached through logical deduction, but also based on intuitive readings of figures and diagrams. While Euclid presented five axioms, Hilbert presented 20 axioms as the foundations of geometry. A second step was taken by the metamathematical programme, which turned mathematical theories into objects for systematic study. They were concerned with the independence of mathematical axioms, the consistency and completeness of mathematical theories, and the possibility of solving the decision problem. A third step consisted of specifying what a formal system is in terms of: the alphabet the system; the sequences of symbols that count as formulas; the set of formulas that serve as axioms; the rules of inference to apply when making deductions; the notion of proof; and the definition of a theorem. With this clarification of a formal system, formalism declares that mathematics is made up of formalisms.