Confessions of an ex-mathematician

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My mathematical autobiography 

(Curriculum Vitae)

Doctoral education

As a doctoral student under Alexander Varchenko, I realized that the highly abstract “pure” mathematics I had grown up with in India had gone out of fashion. Varchenko was himself a student of the great mathematician Vladimir Arnold, who famously said, “Mathematics is part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap”.

Although my Ph.D. thesis was considered decent and was published in a well-known journal called Annals of Combinatorics, I realized that nobody would ever read it. Nobody understood or cared that I had found some "duality" using symbolic manipulations (called "proofs"). I further realized that I didn't enjoy the unproductive labor of writing papers that nobody would read, on problems nobody cared about.

Moments of reflection

Unfortunately, it was the case back in the day (and probably still is today) that nearly all papers in mathematics had little or no impact. In a typical mathematics department, no two individuals understood each other's research. The mathematical community had become the butt of jokes for solving toy problems. As one saying goes: “A mathematician is someone who can tell you about a problem you didn’t know you had, and solve it in ways you couldn’t understand”.

I learned the hard way that for a mathematical paper to be worthwhile, it has to solve someone else's problem, not my own. I was also convinced that important ideas are most likely to arise through intuition about the real world, not by digging into mathematics itself.

Unfortunately, due to my third-world education, my knowledge of physics was nil, along with my understanding of mathematical thinking. I had learned a small area of mathematics by mentally wrestling with symbols, rules, and logic — which is akin to learning a language by reading a grammar book. Varchenko had to work very hard to clear the weeds and sow the seeds of experimental thinking by constantly repeating the four words, "Give me an example!"

The mathematics of musical sounds

Thankfully, despite my limited scientific education, my Indian heritage came to the rescue. I had a passion for music and, like many children from my generation, grew up as a fan of Bollywood songs. My interests later expanded to Ghazals and Indian classical music. I independently studied some Indian music theory of ragas and was intrigued by its mysterious scales.

A simple and intriguing question arises: Why are there 7 notes in a raga scale? (For example, there could have been 9 or 10 notes. The presence of the number "7" also suggests a mathematical problem).

Grace & awe

By the grace of God, I stumbled upon some scientific literature on music. I was utterly surprised to find that while we Indians were busy confusing each other with old theories, the West had conducted an enormous amount of experiments on the nature of musical sounds. One of the gurus in this field is William Sethares, famous for his experiments on "consonance". I wanted to apply his pioneering ideas to vocal music and incorporate my insights from Indian music theory.

Thankfully, Sethares became very interested in my ideas, leading to a lengthy collaboration that resulted in my paper titled "Local Minima of Dissonance Functions", published in the Journal of Mathematics and Music.

The paper was an instant hit, attracting enthusiasm and interest from mathematicians, sound engineers, and musicologists. I had the privilege of collaborating with experts in the field, like Pantelis Vassilakis, Thomas Fiore, and Emmanuel Amiot, in the process of writing the paper.

Reviews of paper on music

The following two reviews suggest that Indian vocal music provides key insights into the understanding of consonance. The paper could have a significant impact on the theoretical framework and experimentation of sound perception.

Anonymous review, probably by William Sethares:

Perhaps the easiest way to view this paper is as a generalization of Sethares' 1993 paper with a similar name. The paper begins by "lifting" the partials of the sound to a log space, where many of the calculations (especially the derivatives) become easier. Sethares' dissonance function is then translated into the lifted space, and a variety of mathematical results are shown at a deeper, more fundamental level. The paper defines families of timbres related by "spectral envelopes" (formants) rather than by simple transposition. This adds another layer of complexity but also makes the results closer to what might normally be meant by "the same sound" at a different pitch. The major results (which go quite a bit beyond those in Sethares' paper) involve an investigation of the conditions under which timbres lie on the boundaries of the timbre space. There is a lot of interesting work here, and I think the paper will make a nice contribution to JMM […]

Anonymous review, probably by Pantelis Vassilakis:

Key research objective of the study: to identify the conditions describing the boundary of a spectral distribution’s dissonance minima. This is a very useful objective. Accomplishing it would conceivably permit algorithmic/automatic recognition of dissonance minima and, more interestingly, dissonance minima time-profiles of time-variant spectral distributions of sound signals, even in real-time. Generation of real-time dissonance time profiles is already achievable via Vassilakis, P.N. and Fitz, K. (2007), which includes a necessary modification to the dissonance model used in the study under review (see further below). Automation of dissonance minima identification would provide a very welcomed addition/contribution to the area. It would support experimental testing of a large range of hypotheses exploring the relationship between a) sensory consonance/dissonance and b) a variety of intellectual or affective responses to sound in a variety of contexts. The theorems explored are also well thought out and appear to validly support accomplishing the study’s objective. I have not checked the theorems’ procedural validity and full mathematical soundness, leaving this task to reviewers whose expertise best serves this purpose […]

Those familiar with reviews of mathematical papers know that they are usually quite drab and disinterested, usually meaning, "Looks okay, some interesting results, don't know what it really means." However, the ones above are overly generous and kind (as if I have discovered quantum-relativity). I personally don't think it is a big deal. It's a simple application of basic principles of mathematical thinking that I was fortunate to learn under Varchenko. I am also especially glad that my interests in Indian music eventually led to a mathematical paper.

A spiritual journey

I mentioned God only once above, but the entire journey was accompanied by spontaneous spiritual changes in my life. Most of my life, my ego was consumed in trying to prove myself by learning complicated mathematical language and solving problems from the back of a book. In fact, I was quite disappointed that my doctoral thesis was rather simple and did not contain enough fancy language. Varchenko understood my psychological problem and reassured me, saying: "Yes. It is simple. But, that’s how good mathematics should be." The paper on music is even simpler, understandable to an undergraduate, and unlike my thesis, thankfully, was read by a lot of people.

Ultimately, I am grateful to God for the opportunity to dedicate a small piece of work to my great Guru, "Sri Sri" Alexander Varchenko, in gratitude for his immense generosity and relentless efforts to liberate me from the matrix of the ego. And what were his thoughts on my efforts? His response was one of delight, a note that rang with the warmth of his words, "Dear Debu, Thank you for sending me the papers. I looked through all three. I am impressed and I am very glad you are working actively in the theory of sound. Please keep me informed". Upon reading the reviews, he later added, "I am glad your work has very positive reviews.”

These were not ordinary words from Varchenko, although he was being overly generous towards his student. His favorite activity was destroying people for speaking complicated nonsense and for daring to utter the word "proof". So, for once, I felt I had done something worthwhile in my mathematical career.

Retirement from mathematics

That being said, it was also a good time to bid farewell to professional mathematics. Although I should have written a couple more papers and always wanted to work on a keyboard capable of playing ragas more melodiously, I had to stop somewhere (for personal reasons). For ordinary mathematicians like me, one good piece of work is enough — you need "one original idea".

In professional mathematics, one has to be prolific and write papers regularly, whether they are worth reading or not. For personal limitations of attributes and temperament, and lack of motivation, I am not suited for such work. I would rather take my own time and write something interesting when I can.

Today, teaching and learning have exploded over the Internet. Many top universities have made their lectures available for free, potentially leveling the playing field for students globally. In my generation, most places were completely isolated from mainstream trends in mathematics. Given my humble background, by God’s grace, I have tried my best in research. Also, conventional teaching is no longer relevant today, as the best of people have already published their videos with beautiful high-tech animations and visualizations.

So, overall, an early retirement from mathematics seemed like a good choice for me. Luckily, it also came at a time in life when I had enough time left to pursue another career in the field of Law and perhaps contribute to education in other ways in the changing environment.