6 - The Three Worlds of Mathematics

Summary

We now have the constructs to formulate the development of mathematical thinking in terms of ‘three worlds of mathematics’ arising from a sensori-motor-linguistic foundation building on the set-befores of recognition, repetition and language and the met-befores from previous experience that may be supportive or problematic in new situations.

Mathematical thinking involves the compression of mathematical structures into thinkable concepts connected into knowledge structures that are blended together, leading to sophisticated crystalline concepts that have an inevitable mathematical structure.

The Three Worlds

The three set-befores common to us all are foundational in the development of three mental worlds of mathematics:

A world of (conceptual) embodiment building on human perceptions and actions developing mental images verbalized in increasingly sophisticated ways to become perfect mental entities in our imagination;

A world of (operational) symbolism developing from embodied human actions into symbolic procedures of calculation and manipulation that may be compressed into procepts to enable flexible operational thinking;

A world of (axiomatic) formalism building formal knowledge in axiomatic systems specified by set-theoretic definition, whose properties are deduced by mathematical proof.