My Journey through the Mathematical Maze:

Most summers I travel in the literal landscape, exploring outdoor vistas and hiking the trails of the wide open spaces.  This summer, I chose another sort of journey.  

I enrolled in twelve credits of college calculus.  Starting in late May, I spent two hours of every weekday in a classroom and more than six hours a day in study.  Each course--they were simply called Calculus 1, 2, and 3--was a five week session that was itself a compressed version of the fifteen week course students would have encountered in the regular semester.  I was the only person among the students who took all three sessions in one summer.  As you may imagine, this gave me a unique perspective on the sweep of the subject.  At the end of the last of the three five week sessions, the professor assigned an essay in which each student was asked to present his or her own reflections on the learning process, starting from his or first introduction to calculus and culminating with the course we had just completed.

I have included the entire essay.  I hope you may share a bit of the joy of discovery I felt and be inspired to look back at your own mathematical explorations with some of the wonder you may have forgotten.

On the first day of Calculus 1, the professor draws a series of concentric circles on the board.  They are labeled—in order from innermost to outermost—limits, continuity, derivation, and integration.  These circles, the professor explains, represent the basic concepts of the calculus.  It appears to be a straightforward target, without context or meaning, and it is from here I begin my journey.  I dive directly into the heart of the target, and discover a different world.  Rather than the simple pattern seen from above, I find myself wandering in a grand garden maze whose complex and elegant design may be magnificently clear from above, but presents joyful and occasionally frustrating explorations and discoveries for the wanderers within.

Michelangelo once said that he did not create statues.  Rather he freed them from the stone in which they were trapped.  To some extent, the journey through the mathematical maze is something like this.  Each new discovery is both the culmination of a form of engrossing effort and sometimes deep frustration, and yet each, once discovered, is as clear and simple and beautiful as the statue emerging from the rough stone.  

Starting at the heart of the maze, I find nothing but high walls, alien terms, and strange forms—all alluringly interesting and mysteriously unknown.  There is as yet no hint of where the paths will lead.  The amazing symmetry, elegance and intricacy of the maze is completely invisible from within and is only gradually discovered, mostly by practicing and repeating certain patterns, by heeding to the messages left by those authorities who have gone before, and occasionally by joyful leaps of intuition.

I traverse the innermost circle along Zeno’s path, taking smaller and smaller steps towards a wall that is not a wall but a doorway into infinity.  By approaching the infinitely small, I enter into the world described by calculus, leading as it does to the notion of the instantaneous and the possibility of quantifying the continuous change of form.  It is wonderful to see the connection between the changing form of the graph and the derivative and to uncover the pattern—to be able to see the shape of the graph in the changes in the derivatives and the second derivatives.  A high point is that moment when after drawing graph after graph, I make the leap to predict and then prove that the slope of the sine will be the cosine—a discovery that is as remarkable to me as if I had been the first person to make it, although, like all such discoveries, it settles into the tool bag with the multiplication tables and the Pythagorean theorem, and I only recover the sense of wonder, as I look back and remember.

The transition to integration begins on a circuitous road whose direction and purpose is unclear.  All the hard won shining clarity of the earlier patterns seems to evaporate as once again, I enter into tortuous and thorny paths whose meanings do not immediately become clear.  Integration has its own beauty, and it is in this phase of exploration that I learn a number of tools by rote and only gradually begin to comprehend their place in the overall design as I start to be able to derive and discover some of the tools myself, long after I have been using them.  The exhilarating breakthrough for me in this part of the journey is the realization that there are more branches in the maze than even the experts have explored.  The sense of mysterious new trails waiting to be explored beckons.

As in any maze, the ways are not straight but turn back on themselves in intricate patterns that loop back to the beginning and enhance the experience.  Thus, I find myself once again addressing infinity directly and amazed at the power of the patterns emerging from the study of the sums of infinite series.  Yet again, I am entranced by the way patterns can be discovered and rediscovered in new contexts—to be able to transform intractable functions into sums of infinite series and by this strange method, uncover yet another route to understanding.

Even as I progress through the maze and became more and more sure that the map I am building in my mind is accurate, I am introduced to new details, new complexities, that at first, seem to turn clarity into ragged edges, but ultimately consolidate into a more complete and beautiful pattern than before.  I learn that the calculus I was beginning to comprehend is only the beginning of a larger design—that there are ways and means to change to different coordinate systems—polar and spherical and cylindrical—that I can envision not only the two dimensional graphs that first delighted me, but three dimensional and even higher dimensions can be described if not envisioned, that it is possible to re-envision the very functions themselves using different parameters, that I can even translate these patterns and concepts to fields of forces—vector fields, that describe aspects of motion and work that I can barely conceive.

Here, once again, the maze turns in upon itself and I am able to re-imagine the first insights that began the journey.  I revisit the so-called fundamental theorem of calculus—which had not been obvious or clear to me earlier—and see it anew.  I grasped in a way that I had only accepted before, the mysterious and amazing connection between the anti-derivate and the infinite sum which is the area under the curve.  Like the Pythagorean Theorem, like so much that one uncovers in mathematical explorations, it becomes at once obvious and miraculous that there is such a straightforward connection between such disparate things.  In the end, the pattern is completed by four more theorems that each connect more complex integrals to lesser ones—the fundamental theorem for line integrals of vector functions, the Stoke’s Theorem (which is a more complete form of the Green’s theorem) relating surface integrals to line integrals, and finally the Divergence theorem that relates volume integrals to surface integrals.

Even as I am reaching the end of the odyssey that has been my journey through this mathematical maze, I am realizing that my mental map is still woefully incomplete, or rather that my experience of the maze has only been at the surface level, that although the pattern I am now seeing is true, it is like the original target I saw on the first day—only the beginning of something more complex.  Like the patterns within patterns of fractal art, I suspect, that every aspect of what I have seen and explored this summer could be more deeply examined and depths and new patterns discovered.  The satisfaction of encountering what had already been described by so many before me is only a shadow of the anticipation I feel at the notion of exploring what more may come and possibly glimpsing occasionally those branches of the maze not yet fully mapped.