Tuesday, October 21, 2008

Metaphors and Mathematics 1

When asked to describe mathematics we often resort to metaphor rather than attempt to provide strict definitions. These pictures from high school math textbooks from the 1930s are an example of this tendancy.

The simple hierarchies of these images resolve the complicated relationship between mathematics and science by appealing to our desire for an organic unity among disciplines, giving mathematics a foundational role within the general concept of science. These images are appealing, but do not stand up to scrutiny.

The simple relationship between mathematics and science becomes complicated when mathematics is described, as it sometimes is, as a science itself. It's definition as "the science of space and quantity" is further complicated by the caveat that it is an exact deductive science, unlike the usual inductive kind. Following this line of thinking further, mathematics is then described as a kind of meta-science, or a limit point to which science might aspire - science emptied of all of its empirical content, a science of pure thought. While some view mathematics as a foundation for science, others as a supra-science, the emerging field of experimental mathematics brings mathematics back into the empirical fold, reducing it (or elevating it) to a science like any other. So, mathematics can be seen as root, branch, or even the form of the tree itself.

Thinking about these things for even a short while evokes some sympathy with Bertrand Russell's remark that "mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true."

The "series" of posts on this is:
Metaphors and Mathematics 1
Metaphors and Mathematics 2
Metaphors and Mathematics 3


  1. Hello,

    I was very interested by your post, and surprised to realize that mathematics are not that easily defined. I work on taxonomy, a branch of natural sciences which is defined too either as the root, a branch or the tree of natural sciences. Could you precise the sources of your pictures, and give some examples of papers debating on the definition of mathematics? That would be very helpful.

    thank you very much !

  2. Sorry for the delay in answering these questions.

    The image at the bottom of the post was originally from a display at 1933 World's Fair in Chicago - "A Century of Progress". It was a tableau called The Tree of Knowledge by Mr. John Norton - it has been reproduced in a few textbooks. The folks at the special collections branch of the library of the University of Chicago tell me that appears in the "Official Catalog of Exhibits in the Division of the Basic Sciences" for the 1933 fair. I suspect that the image at the top was inspired by Norton's, it appeared in a Canadian high-school math textbook from the 1950s.

    On the topic of what is mathematics, I would recommend Google - much ink has been spilled on this topic. One title that comes to mind is Richard Courant's "What is Mathematics?" Also, just the other day I saw that Paul Lockhart is coming out with a book called "Measurement" that sounds like it will provide a nice overview of what mathematics is from the perspective of a contemporary working mathematician and educator.