In truth, the 20th century produced a tremendous amount of new mathematics, especially with the growth of mathematical physics and the advent of computers. With the move towards specialization, however, much of modern mathematics is simply inaccessible to the average person. Imagine the whole edifice of mathematics as a giant tree with the ancients (such as the Pythagorean Theorem) forming the roots and the trunk of the tree, and modern mathematicians breaking off into specialities along different branches. An example of a twig sprouting off of a branch would be Topology (a sub-field of Geometry), however, it would only be understandable to someone who devoted months, maybe even years, to that one particular area of study. The number of sub-fields, twigs spawning twiglets, grows ever larger. The sheer breadth of topics has been growing exponentially and it continues to grow, making learning these topics prohibitive.
So what do mathematicians really do? How do they eke out new twiglets on their little branch of this enormous tree? Mathematicians play. They test new ideas, search for generalities, look for beautiful relationships. In professor Paul Lockhart’s brilliant polemic “A Mathematician’s Lament,” he claims that “To do mathematics is to engage in an act of discovery and conjecture, intuition and inspiration, to be in a state of confusion” (p 8). The process of mathematics begins with an observation or a question and then that notion is toyed with. The process culminates, if the mathematician is clever or lucky enough, with an airtight proof of the conjecture. Though his work is not exactly modern, an example from the life of famous mathematician Leonhard Euler (1707-1783) will help illustrate this.
Legend has it that in the mid-18th century, the locals of Königsberg (now Russia) made a game of walking the seven bridges spanning the Pregel River and that good fortune would come to anyone who could traverse the bridges without backtracking or walking over the same bridge twice. See for yourself: attempt to trace a path with your finger but don’t go backwards or over the same bridge twice. Fun, huh?
Leonhard Euler wasn’t even 30 years old at the time, but his reputation was well-known, having enrolled in university at 13 years old, completing his doctorate by his teenage years, and writing on diverse topics such as navigation, sound propagation, and mathematics. He would eventually be regarded as one of the most brilliant and prolific mathematicians in history, working until his death at 76.
Euler likely treated this problem like most mathematicians would, playing with it just as you might. This brief video brings it to life and shows you how Euler was able to get to the heart of the problem (tracing the seven lines), dissect it, and ultimately solve it.
Euler's revelation provided the first result in a major branch of the tree of mathematics — that of Topology, a sub-field of Geometry — which has blossomed over the past 200+ years. While new developments in modern mathematics rarely produce such fertile results as Euler and his creation of Topology, nonetheless, the fun is there and the inexorable moving forward of the subject continues.
Director of Studies Rick Tony is a member of Solebury School’s Math Department. This year, he teaches Advanced Placement Calculus AB.