What is Mathematics?  It sounds like a simple enough question.  Wouldn’t you expect that a high school or college student that has been formally engaged in mathematics education for more than 10 years would have an immediate (and relatively complete) response?

They don’t.

The truth is that mathematics is difficult to define, and even worse, there are many misconceptions regarding the true nature of math.

Fordham math professor, Dr. Robert H. Lewis, did his best to address the topic in an excellent article that first appeared online in 1999 (and has been revised several times since).

Many believe that mathematics is calculations and symbolic manipulations. This is both incorrect and problematic. When a calculus student believes that the primary objective of differential calculus is to be able to take derivatives, he focuses on the “tricks” such as the chain rule, product rule and power rule. The ability to take derivatives algebraically is a good thing, but if it becomes the focus, this same student might not be able to answer, “What is a derivative?” He might also lose sight of the mathematical process that led to the creation of these differentiation “tricks” as means to be more efficient.  This student might be frustrated by a lecture on limits or a proof of the chain rule.

The role of limits in calculus or the notion of a proof in mathematics are far more important for developing one’s sense of mathematics and developing one’s ability to reason mathematically.  If a proper understanding of math and a capability to reason mathematically are of value, then they must be taught as well.

Dr. Lewis’s article, through analogies (or “parables”), captures the misconceptions about math and the consequences of these misconceptions.

Unfortunately, the article does not provide anything better than a definition by analogy for mathematics.

In my courses, I regularly point to examples that I believe demonstrate mathematical reasoning. However, I can’t formulate a complete definition of what this is.  Like Dr. Lewis’s article, I tend to resort to analogies.

Why is it so difficult to answer, “What is mathematics?”