The Spontaneous Order of Mathematics

I have argued that mathematics is its own spontaneous order, the most abstract spontaneous order of the "practical knowledge" set, which includes the physical sciences and the technologies that have been invented to do science, with the latter two being mixed (abstract and concrete) and concrete, respectively. But math has not always been that way. And math has not always been a spontaneous order.

In fact, as Randall Collins points out,

In the 1700s the field [of mathematics] consisted mostly of analysis, exploring the branches of Leibnizian calculus and their applications in physical science. By around 1780, the belief had become widespread among leading mathematicians that  mathematics had exhausted itself, that there was little left to discover. (697)

Note that this describes a mathematics that is firmly rooted in the physical sciences. It is not its own order. The only mathematics being investigated is math relevant to rapid-discovery science. But then,

Unexpectedly, the following century was the most flamboyant in the history of the field, proliferating new areas and opening the realms of abstract higher mathematics.

The sudden expansion of creativity arose from shifts in the social bases of mathematics. Competition for recognition increased with a large expansion in the numbers of mathematicians. The older bases for full-time professional mathematicians had consisted of the official academies of sciences, notably Paris, along with Berlin, St. Petersburg, and a few others. [...] The [German] university reform extended to mathematics the emphasis on innovative research, as well as giving a distinctive slant toward pure knowledge apart from practical application. The process of disciplinary differentiation split math from physics and astronomy, encouraging the tendency to abstraction. (697)

This is pretty much how I describe how literature and the arts emerged into their own spontaneous orders from religion. With Modernist art and literature, we end up with extremely reflexive, highly abstract works of art and literature. This is a natural consequence of spontaneous orders. We end up with science about science, technology about technology, art about art, business about business within the market economy -- and math about math. This is a necessary consequence of the emergence of differentiated spontaneous orders, even if those orders continue to influence each other in their overlapping ecotones.