In what way is mathematics dangerous? This past year, the Williams College Gaudino program has been looking at the idea of danger. Hence my question, in what way is mathematics dangerous? Maybe there is emotional danger, as the media mocks the typical math professor as nerdy and dorky. (These allegations have never been officially proven.) Maybe there is physical danger: There could be a mathematician somewhere whose life is threatened, being held hostage, forced to solve some extremely difficult calculus problem. (This seems a bit silly and far-fetched.)
After thinking about this for weeks, I’ve come to the conclusion that under the usual definition, math is not dangerous at all. I am not in physical danger, not involved in experiments involving rats, chemicals, or lasers. I’m also not invested in my field personally, unlike how Charles Taylor, in his monumental work “A Secular Age”, is invested in viewpoints of religion, economics, sociology, politics, and much more.
So let me offer another definition of danger:
Danger is to be placed in a situation where pain is highly likely.
In this way, mathematics can be extremely dangerous. I want to share with you instances in which mathematics has caused me pain, along with ways in which I am fearful of pursuing certain ideas.
My own field of research and expertise is in the field of topology, the study of shapes. About 10 years ago, a geoscience professor sat in on a graduate class I was teaching at Ohio State. Conversations with him made me think of connecting my work on how shapes deform to the field of cartography. My students and I wrote a nice paper on this, relating homotopy theory and cartographic generalization. When we submitted it to an appropriate cartographic journal, we got referee comments like this:
To be honest, I can’t make heads or tails of the theory, which seems a lot more complicated than it needs to be.
The complete absence of anything to do with geography reveals their complete and absolute ignorance of both the discipline of cartography and the discipline of geography.
There has been much pain as this paper was rejected over and over. However, at the end, there was good news: it was finally published by a journal on geoinformatics.
Now let me talk about a topic where I have not had the courage to put myself in danger; I am trying to avoid pain, pain due to humiliation. I work in the area of topology mostly because I have always loved to draw. In my research, within my mathematical community, I am known as one who is obsessed with visual presentation of information. This has led to a desire for me to bridge a gap, not between math and cartography, but between math and art.
From my perspective, most of the current buzz regarding math and visual art is unimaginative: Either the mathematics is elementary or the art is superficial. My dream is to create a visual work which not only pushes the boundaries of art, but leads to new mathematical insights. Though I have been encouraged by “art friends” and “math friends” to modify and officially submit my mathematical images, I am too embarrassed, too afraid to go down this dangerous road of humiliation: to claim that I have produced a work of art, rather than just an illustration.
So I did what any academic would do: I used my students as guinea pigs. I have offered two courses on math and art (2007: geometric designs; 2009: mural), along with a new one for this coming year (2012: visualization). Maybe someday I will be able to take that plunge in the face of danger.
[skeletal 4D associahedron from brushed metal 2′ x 2′ x 2′]