The text below is a section of the book "Where mathematics comes from: how the embodied mind brings mathematics to being" by George Lakoff and Rafael Núñez.
As we have just noted, a significant part of mathematics itself is a product of historical moments, peculiarities of history, culture, and economics. This is simply a fact. In recognizing the facts for what they are, we are not adopting a postmodernist philosophy that says that mathematics is merely a cultural artifact. We have gone to great lengths to argue against such a view.
The theory of embodied mathematics recognizes alternative forms of mathematics (like well-founded and non-well-founded set theories) as equally valid but about different subject matters. Although it recognizes the profound effects of history and culture upon the content of mathematics, it strongly rejects radical cultural relativism on empirical grounds.
In recognizing all the ways that mathematics makes use of cognitive universal and universal aspects of experience, the theory of embodied mathematics explicitly rejects any possible claim that mathematics is arbitrarily shaped by history and culture alone.
Indeed, the embodiment of mathematics accounts for real properties of mathematics that a radical cultural relativism would deny or ignore: conceptual stability stability of inference, precision, consistency, generalizability, discoverability, calculability, and real utility in describing the world.
This distinguishes an embodied view of mathematics from a radical relativist perspective. The broad forms of postmodernism recognize the effects of culture and history. But they do not recognize those effects of embodiment that are not arbitrary. It is the nonarbitrariness arising from embodiment that takes mathematics out of the purview of postmodernism.
Moreover, the embodiment of mind in general has been scientifically established by means of convergent evidence within cognitive science. Here, too, em- bodied mathematics diverges from a radically relativistic view of science as just historically and culturally contingent. We believe that a science based on convergent evidence can make real progress in understanding the world.
I think that just the fact that the authors felt the need to clarify why their ideas do not align to the postmodern view is quite interesting. But also, I enjoyed their explanation.