How is there math after calculus?

I served at a Graduate Teaching Assistant for 5 of my 6-year PhD program.  For nearly each of these semesters I was an Instructor of an undergraduate math course.  My students eventually realized that I spent most of my time outside of the classroom working towards my PhD.  With this epiphany came the question "How is there math after calculus?!"

The best analogy I have for explaining mathematics is music.  We all understand what music is and have been exposed to a variety of genres of music.  Musical genres include classical, jazz, folk, heavy metal, techno, gospel, country, and the list goes on.  What characteristics distinguish one genre from another?  Excluding lyrics, if any, an easy way to categorize music is according to which instruments are used.  Classical music does not include bongos and heavy metal music does not require oboes.  It is also quite difficult to "order" music.  One would not say that there is a necessary or obvious progression from learning country music to techno.

However, did you ever take a music class in school?  My elementary school sent us home with recorders in 4th grade and that was my first experience learning to play an instrument.  There are a limited number of notes one can play on a recorder, but it allows a student to learn the concept of reading music.  In 5th grade I began taking piano lessons.  This introduced significantly more notes and chords, but the foundation of reading music had already been laid.

This is how I will explain mathematics.  The areas of mathematics are many, just as the genres of music are many.  In some sense, mathematicians classify areas based on the objects being study.  For example, topology is the study of "properties that are preserved through deformations, twistings, and stretching of objects."  While harmonic analysis is the study of "decomposition of functions using Fourier series and the Fourier transform."  In a sense, the objects of study are the instruments which define the genre or area of mathematics.

To study these branches of mathematics, one must have an ability to read mathematics.  This is not as innocent as reading a novel, just as reading sheet music is not so trivial.  So how do we learn to read mathematics?  We start with the basics.  We start with mathematical symbols, notations, objects, and rules found in the standard algebra and geometry courses.  These can feel uninspiring and devoid of meaning just as learning the recorder felt to me!  However, there is much to gain from this foundation.  

Calculus is part of the foundation required for many areas of Analysis, including Harmonic Analysis, but it is not on its own a gateway to higher mathematics.  Other areas of study contribute significantly to the foundation needed before pursuing higher-level mathematics.  Such areas include logic or an introduction to proofs.  Though I am biased, I also see as much beauty in mathematics as I do in music.

For quick look at how modern mathematics is roughly organized, check out this video!