Mathematics appears in two basic forms:
- Formal as axiomatic mathematics as games of words.
- Constructive as sets of instructions telling what to do.
The basic example of formal mathematics is Euclidean geometry based on five axioms expressed in terms of undefined concepts such as points and lines, with the first axiom being
- There is a unique line through any two distinct points.
The fifth axiom is the so called parallel axiom stating that given a line L and a point P not on the line, there is a line through P which does not meet L, a line parallel to L.
The idea of Euclidean geometry is that all of geometry comes out as consequences of the axioms by logical reasoning, with then the following basic questions
- Are the axioms without contradiction?
- Can some axiom be removed without changing anything?
Constructive mathematics is not defined in terms of axioms, but instead in terms of procedures telling what to do, like a computer program.
Starting at the turn to the 20th century and culminating in 1930s, a fierce debate raged between formalists (Hilbert, Russell) and constructivists (Brouwer) concerning the foundations of mathematics. The formalists won that battle and cleansed mathematics departments from constructivists, who spread out to form the new field of computer science outside math departments as essentially constructive mathematics.
Today constructivists equipped with the computer have taken the lead. This asks for a reform of mathematics education traditionally dominated by formalists and this is DigiMat.
Constructive math is expressed as