People who are seen as mathematicians are famous for uttering this line all the time, “mathematics is beautiful.” Whenever someone says this, some people nod in agreement, while still others nearby are probably wishing they saw the same beauty in math.
I would like to take a moment to analyze this statement and, ultimately, refute it. Not only is math not beautiful, it is not easy, hard, frustrating, fun, enlightening, or empowering. Nor does math have any other quality anyone would like to attach to it.
Here’s why I say this: math does not exist, at least not in the way we have traditionally thought about it.
In January we were visited by Dr. Leslie Steffe. I will respectfully label him as the “guru” whose work largely informs my team’s work here at the AIMS Center. One evening he led the AIMS Colloquium – the video can be viewed here.
While addressing our team, Dr. Steffe began with this question: “Is mathematics discovered, or is it constructed?” I would encourage you to ponder this question a moment before reading further.
Dr. Steffe then led the group through a series of activities and discussions supporting the idea that mathematics is a construction of our minds. I highly recommend a viewing of the colloquium video for a better understanding of Dr. Steffe’s ideas and the philosophy of constructivism, but the ideas might be summed up in the following way. There is no math inherent in nature. There is no “eight” or “pi” or “addition” or “algebra” out there waiting to be discovered and applied to our lives. These things are all constructed by us. Furthermore, they are constructed by us individually in such a way that none of us truly understands them in exactly the same way. Each of us has unique experiences and perspectives. Likewise, each of us has a unique understanding of our world, including the mathematical ideas we have developed.
Math is not a set of rules or algorithms. We know this because different cultures use different algorithms. For example, here is a nice article outlining other cultural approaches to the addition algorithm. Algorithms are simply someone else’s construction of math, or a culture’s collective understanding which we might attempt make sense of.
I say math is not beautiful because math, as an object, does not exist. What does exist is our understanding of our world. This understanding can grow, adapt, reform, and help us make sense of our experiences. This understanding of the world, including our construction of mathematical ideas, can be a beautiful thing for each of us.
So instead of saying that math is beautiful, we can say, “Your construction of math is beautiful to you, and my math is beautiful to me, so let’s see how we can find what is similar in our understandings.”