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.

set theory in and of itself is embodied, in that it is a synaptic structure and that is it. It is through this synaptic structure we see we are not independent of that synaptic structure, nor is it independent of us or the environment. Our brains are molded by the environment because there is no separation between our brains and the environment. at deeper levels we are our brains and we are the environment at the same time.

I’m with you. I found your blog because I am going to a local presentation on Embodied Mathematics and I was trying to read more about it… little did I know that it ends up being very related to other things I’ve been reading (Reuben Hersh in particular).

I absolutely love his viewpoint though the books I’ve started are a little long winded. I am kicking myself for dropping his “History of Math” class that I was enrolled in when I was in college… I didn’t realize what I might have gotten from that at the time.

But I agree, it is not a popular viewpoint. I currently teach high school math, and though we haven’t had a downright discussion, I can tell from small comments here and there that my colleagues don’t agree at all. One in particular, who happens to run our department, is very vehement about the “right way” to do everything in math… I mean, I’m with him that some answers are right and some are wrong… but who said there is just one way to write it down, talk about it, or interpret it?

I’ve already started trying to have discussions with my students about these ideas as well. I teach at a very humanities-based school, so I think relating math to philosophy and history really engages them… I even had the geometry students read Hersh’s “The ‘Origin’ of Geometry”, which was over some of their heads, but I think some of them really got the idea.

Thanks for the comment! I’m really happy to hear you are exposing your students to this. I wish I had heard about this in high school. It might have gotten me going on the math path earlier.

I didn’t decide to do math until college, but I’ve always felt that if I had heard what math is “really” like, then I would have been interested earlier. Good luck!

My thoughts exactly (about how I didn’t like math in high school but started to in college – it was actually a history of math-ish seminar that got me into it!) So my hope is to show more of the context now, especially to get kids interested at a school that’s not known for being mathy.