Euclidean Geometry as a Formalist System

·
The words __point__, __line__ and __plane__ are undefined
terms with no meaning. Other words, like triangle, are defined in
terms of them.

· One proves that certain statements about points, lines and circles follow from Euclid’s axioms. These statements have no meaning.

· Diagrams and visual intuition are useful but technically irrelevant. A computer could check the validity of any proof.

All of Mathematics as a Formalist System

·
The word __set__ is an undefined term. All mathematical objects
in all sub-fields can be defined in terms of sets.

· The axioms of set theory allow us to prove statements about sets, and hence about all mathematical objects.

· Any proof in any field could (in theory) be restated precisely enough for a computer to check whether it’s valid, meaning whether the conclusion follows from the axioms of set theory.

Difficulties with Formalism

· Real mathematicians don’t produce formal proofs. Most of us don’t know the axioms of set theory.

·
__Gödel’s Theorem__: For any axioms sufficiently
complex to study number theory, (1) the consistency of the axioms
could never be proven, (2) there are statements about numbers which
can neither be proved nor disproved.