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I had one of those very rewarding teaching moments yesterday in my General Physics lab. Some students and I got into a discussion of just what is physics, what is the connection between mathematics and physics, and whether the world is itself mathematical.
The nature of the world as mathematical is a common theme in the Chaos and Fractals course. I have developed a seminar module that has one main reading, and I keep finding supplemental readings every time I teach the course. I list a few of these here in order to collect them in one place - for future renditions of the course, and as a post that will hopefully generate some debate from interested readers.
The main reading is John Barrow's The Mathematical Universe in which he poses the question "The orderliness of nature can be expressed mathematically. Why?" This article is an excellent summary of the main schools of mathematical philosophy - realism, inventionism, formalism, and constructivism.
Anyone interested in deeper views of mathematical realism must read The Unreasonable Effectiveness of Mathematics in the Natural Sciences. This 1960 article by Eugene Wigner is definitely the "mother-of-all mathematics and the world" arguments for realism. In it he describes the uncanny connection between mathematics developed as part of pure, formal, abstract systems and physical observation, which naturally leads to ontological questions on the mathematical nature of the world:
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