People who only know me as a musician are sometimes surprised to find out that I am also a Mathematician. The connection between Music and Mathematics is as mysterious as it is certain, and both Jazz and Mathematics are widely misunderstood art forms. For one who shares the joy of Jazz daily it may be easy to forget that many people’s perception of Jazz has been “smoothed” over by market forces or blinded by a smoky-downstairs prejudice that keeps them with the “squares”, cut off from music that provides an endless source of inspiration for enlightened people all over the globe. Of course all it takes is hearing the right group at the right time or being turned on to a classic side at a key moment and Jazz can begin rounding-off even the sharpest corners, Swing has a way of sticking around. Mathematics, too, is misperceived by most. But to discover the hidden harmonies and elusive rhythms of Mathematics is much more difficult. Those who equate Mathematics with memorization of formulas and tedious calculations will find it hard to believe that what Mathematicians seek is Beauty.
Mathematics does not live in it’s objects (e.g. “numbers”) but rather the relationships between objects, just as music is more than the sum of it’s notes. Mathematics is about ideas not “cold facts”. What a mathematician really wants to know is why something is true or false, and the question is really only relevant if the idea is “interesting”. I am most interested in the branch of Mathematics called Topology. Topologists study “spaces”, all different kinds of spaces of all kinds of dimensions (not just the 3-dimensions of physical space that seems to surround us). An interesting fact is that in every dimension except dimension 4 there is only one “way to do calculus” but in dimension 4 there are infinitely many different ways to do calculus. (Why this true is not easy to understand.) This notion of “being interesting” is sometimes motivated by a connection to the “physical world” but is equally (more?) likely to be a purely esthetic judgement. Oftentimes, any connections to the physical world come after the fact.
For instance, the great 19th century mathematician Bernhard Riemann pioneered the development of geometry in more than 3 dimensions and even described how these spaces could be “curved” in a complicated way. It wasn’t until the next century that Einstein used the 4-dimensional version of Riemann’s mathematics to harmonize space, time and gravity in his Theory of Relativity.
Mathematics is not only a solo gig. When two or three mathematicians get together to share their ideas on how to attack a problem, sessions can easily stretch out for hours and even days. The dynamic is not unlike jazz musicians improvising with each spurring the other to go farther than they would go alone.
Of course the ending can never be rehearsed and ultimately more questions may be raised than answered The fundamental element of Music is rhythm: the tension and resolution of harmony and phrasing only exists in time. Yet we gain something timeless. From Mathematics we also get something timeless but it’s harder to pat your foot to it. Research does have rhythms of it’s own but the phrasing can be tricky. Short lived tension and resolution is common: “Perhaps this is true…No, of course that can’t be!…Maybe if we…Ahh, yes…”. However, the long-lasting brilliant flash of insight occurs much less often. In the early 1900’s the father of modern topology, Henri Poincare, conjectured (but did not prove) a very simple characterization of the 3-dimensional sphere. In the 60’s groundbreaking work of Steven Smale delivered a proof of a generalized version’s of the “Poincare conjecture” in all dimensions greater than 4. The proof of the 4-dimensional case arrived in the 80’s as a result of Michael Freedman’s innovations. The originally stated 3-dimensional case, which was still unknown at the start of the 21st century, was finally proved by Grigori Perelman (following an approach of Richard Hamilton) — and the proof used delicate techniques from Analysis, completely different from the previous topological proofs in higher dimensions. This is not danceable but if you can hear what’s going on it is very listenable. Unfortunately, trying to explain what is really going on to a non-mathematician is like only showing the score of a symphony to someone who doesn’t read music. If you do catch a mathematician on a good day with a pencil and paper they may be able to give you a glimpse of what they’re up to, but for the most part Mathematics will remain “Music that only musicians can hear”.
For more of my musings on Music and Mathematics see the extended liner notes for Tone Twister on my “Activities” page, and also my published essay “Can one hear the sound of a Theorem?” (written for mathematicians and available in the book https://www.maa.org/press/maa-reviews/the-best-writing-on-mathematics-2012, or downloadable from http://www.ams.org/notices/201107/index.html).