I’m really going to try to get off this topic soon. It is probably getting kind of old. What is the “romance of mathematics?” Lots of people talk about this (I think Hersh calls it the “front end”). It basically boils down to the view that math is objective and external to humans. If other intelligent life formed in a distant galaxy we would be able to communicate with them through math since it is universal. Math is the ultimate science, since it is absolutely precise and infallible. As such, math itself is the only science worthy to characterize the nature of math. And so on…
Embodied mathematics clearly rejects this viewpoint. Whenever it seems that we are studying objects external to us, or that math is appearing in nature (physics, fractals, etc) we are forgetting that this is being interpreted through our embodied mind first. We have no direct access to the external world. It only seems as if math is inherent in these places, since we have no other way of seeing it.
It is absurd to think that other life forms would develop the same mathematics as us. Their embodied mind might be completely different. They may have no conception of the metaphors we use. Mathematics has varied across cultures and history from different concepts in humans. Now we shouldn’t get too carried away. It is not purely culturally contingent. The similarities of math across cultures is perfectly explained from the embodied standpoint. Humans, regardless of culture, have many cognitive similarities. We embody certain mathematical concepts in the same way (e.g. we use image schema, conceptual blends, conceptual metaphors, etc).
There are no “absolute foundations” for math.
Mathematical truth is based on whether or not our embodied understanding of the subject is in accordance with the situation in question. Simpler: Truth is dependent on embodied human cognition.
Not only is Platonism rejected in the above sense, but formal reductionism is also rejected (these two being about 99% of working mathematicians’ viewpoints). Saying that all mathematics can be reduced to set theory and logic is false since that is ignoring the fact that mathematics is ideas. Formal reduction is a great metaphor with lots of use, but it is not all of math.
This is really only a brief philosophical analysis of this huge new discipline. I haven’t decided if I’m going to continue with another post tomorrow. If I do, then it will be on how this relates to my main interest (and it seems like the authors were semi-aware of this) of developing an aesthetic theory for mathematics.