Last week I was a participant on a panel organized around our campus theme for the year – On Being Human. The overall question we addressed, as I saw it, was “Is mathematics created or discovered?” Two of the panelists argued for the Platonist position that the objects of mathematics reside as ideal entities in the universe, one showed that very young children have some sense of fundamental mathematical concepts, and I argued that mathematics is a fundamentally human endeavor. I basically presented the views, as I understood them, of George Lakoff and Rafael E. Núñez (L&N) in their book, Where Mathematics Comes From, How the Embodied Mind Brings Mathematics into Being. My presentation was somewhat impassioned and imprecise. I want to summarize here the position I took, hopefully with less passion but probably with the same imprecision.
The main idea of the book is that “We understand mathematics using the conceptual system that the mind and body afford”, and that since human conceptual systems are mostly automatic and unconscious, we need to use the tools of cognitive science to study mathematical thought. The mathematical structures that we have are those that we can understand through metaphor using the basic conceptual forms that we as human beings with these particular bodies USE in this particular environment (the Earth). These concepts are used to form more complex concepts by building with metaphors using “conceptual blends.” We only understand anything really through the concepts necessary for living on earth and using only the particular neural structure of our brains. To me this means that a newborn brain without a body can only do the most fundamental mathematics, counting small numbers of objects for example and that alien creatures with different bodies in different environments could be using a different mathematics.
As a math instructor this thesis resonates. When I prepare a lecture I first try to discover how I understand a topic. I have to probe my unconscious for this. The process has the real feel of rationalizing after the fact as when, after long pondering, we make a major decision and then start listing the reasons. The results are symbols on a page that generally has little to do with my visceral understanding of a topic. I next have to devise a way of imparting this understanding to my students. Over the years I have learned to use bodily examples and to speak of motion and position for many algebraic algorithms. I line my students up in a number line and use the classroom seats as points on a Cartesian Coordinate system for instance. I am always trying to connect to the underlying metaphors that students have. I generally believe that if a student has precise control of their native language, they can learn basic algebra. Lastly the ideas in L&N’s book help me understand one of the problems in the history of mathematics, namely the late adoption of negative numbers. New metaphors needed to be developed so our mathematical forefathers could overcome the basic concept of number as distance.
In the last part of my presentation, I presented the straw-man, “The Romance of Mathematics” that L&N construct. In summary the Romance says that mathematics is disembodied and objectively real, independent of human beings – π is π everywhere in the universe, that math is the language of nature, and that reason (logic) is mathematical and can be done by computers. I argued along with L&N that this belief besides being false is pernicious and can do social harm (page 341). This “romantic” view tends to encourage the formation of an elite – people who get math and only discuss their ideas with other mathophiles. It encourages a “stratification of society” into the techies and the non-techies and can lead to anti-science reactions and less optimal societal decisions.
My presentation was, I am sure, even less coherent than this post, but I think it provide a useful counterpoint to the Platonic view. I also think that reading L&N and considering its arguments have made me a better teacher.