According to Lee Smolin, the eight stages by which mathematics develop from the study of relations among natural objects are as follows:
1. Exploration of the natural case (e.g., countable objects, 3D spatial geometry, etc.)
2. Formalization of natural knowledge (e.g., arithmetic, trigonometry)
3. Exploration of the formalized natural case.
4. Evocation and study of variations on the natural case (e.g., non-euclidean geometry)
5. Invention of new modes of reasoning (e.g., axioms)
6. Unification of cases within more general frameworks (e.g., Riemannian geometry)
7. Discovery of relationships between constructions generated autonomously within mathematics
8. Discovery of the applicability of nature of knowledge developed internally.
Smolin rejects the idea that the study of mathematics constitutes the exploration of a Platonic realm separate from physical reality. Rather, he thinks that it represents an evocation from reasoning about items in the natural world. Thus, its applicability to physics occurs precisely because it is, historically, rooted in reasoning about the physical universe.
I need to learn more about this. So far it makes perfect sense!
