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Seeing Math

by Phillip Williams, Ph.D.

 

Many people seem to believe that, for math to appeal to a general audience, it is essential to demonstrate to them why it is useful. In contrast, I believe that this approach gets things almost completely backwards. In fact, math will never appeal to the average person unless the average person is consistently shown that math, even when (and sometimes, especially when) it is not useful at all, is intrinsically interesting. Further, the true usefulness of math is seen most clearly when this is appreciated. This is because progress and achievement in mathematics, which is the source of potential applications of mathematics, occur most fruitfully when people are able to leave practical considerations aside, at least temporarily, and consider math for its own sake. There is an interplay between applied math and pure math. Often a good mathematical question comes from a “real world” question. And often “real world” questions get answered, unexpectedly, by advances in pure math. For any of this to happen, however, people must commit to thinking about the math on its own.

Why is this? The answer is because doing mathematics properly requires one to think in a certain way about a certain sort of thing (mathematics), and eventually one becomes familiar with it and how it tends to go. Without experience thinking about mathematics as mathematics, one will not be able to reliably recognize the purely mathematical content of a practical question and think about that content in any kind of a fruitful way. With such experience, however, the mind gravitates towards such thoughts naturally.

A good example is the bridges of Königsberg puzzle. In the city of Königsberg, there is a collection of seven bridges and four land masses connected by these bridges in a particular way. In the 18th century, mathematician Leonhard Euler considered whether it was possible to walk through the city in such a way as to traverse each bridge once and only once (see figure 1).

The heart of mathematics

Some careful thought led Euler to abstract away features of the problem that were not relevant to its solution: one may safely ignore most of the geometric particularities of the land masses and the bridges. Ignoring all shape and size considerations, he “collapsed” each land mass to a single point, and each bridge became a line segment. The only thing that matters is which land masses are connected to which bridges (see figure 2).

The heart of mathematics

This was an early example of topological reasoning. In abstracting all of the geometric information away, Euler’s question becomes a graph theory question; in fact, in considering this question, Euler basically invented the subject of graph theory. In more modern language, the question becomes one of traversing edges on a graph. The answer to Euler’s question, by the way, is “no.” I encourage the interested reader, armed with Euler’s formulation of the problem, to try to prove this, before looking up the solution. Euler further developed a whole theory about exactly which sorts of graphs could be traversed in the sense described: these are graphs that have a “Euler path.”

In summary, Euler abstracted away exactly the right features to get an approachably abstract and purely mathematical question. He then discovered an elegant solution.

Euler’s work on this problem had general features which led to an explosion of advances in pure mathematics, arguably setting some of the foundations for two whole new fields of study (Topology and Graph Theory). Many of these advances allowed for further application and insight into many other “real world” problems, including many that proved more practical than Euler’s original musings about bridges. Topology is one of the largest and most important fields of mathematics in existence today, and deeply relevant to theoretical physics. Graph theory helps us better understand things like transportation and computer networks. These applications would not be possible if Euler (or someone like him) did not first think abstractly about a charming little problem about bridges. As advocates of math, we should train people to be more like Euler: to know deeply what truly mathematical content is, and thus to be able to spot it in the world around us—and build upon this.


Dr. Phillip Williams is a professor and researcher of number theory, algebraic geometry, and arithmetic dynamics. His popular publications can be found in Academic Questions, and at The James Martin Center.

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