In conversation with Dmitry Vostokov, renowned diagnostician, scientist, author, inventor, and educator
My abiding interest in mathematics is at best peripheral, which is why it hardly ever reflected in my legendary academic Performance – whether in the form of scores, marks, points, or grades. Fortunately, the scholastic blemish could not disturb the furniture of my intuitive interest in the history and philosophy of mathematics and the evolution of mathematical metaphysics, which includes re-reading the biographies of and tributes to (not necessarily the works of) legends like Aryabhata, Brahmagupta, Bhaskara, Gauss, Pythagoras, Euclid, and Archimedes; versatile thinkers and mavericks like Albert Einstein, Lokmanya Tilak, Prof. D D Kosambi, and Bertrand Russell; as also delving deep into the literature devoted to deciphering the minds and methods of self-taught genius Srinivasa Ramanujan and his prolific mentor, G H Hardy, author of the seminal essay ‘A Mathematician’s Apology.’
Much later, in the course of my chequered career, when I took to teaching math to high school students as part of a social development project, I relived the fear that a subject like calculus invokes even among the most sincere of students, and I intuitively sensed the perplexment on their faces at having to commit many a formulaic monster to memory with no clue whatsoever about their genesis and , and graphing the apparently simple (the teacher says so!) parabolic happenings and shifting gradients of the queer-looking quadratic function y = x^2 on the other.
Now, with the help of Eli S Pine’s radical approach of demystifying differentiation and integration under the simpler names of ‘slope-finding’ and ‘area-finding’, I am able to minimize the panic gripping most young minds, if not completely weed it out, before moving on to the overwhelming intricacies. There was no Pine book to guide me during my student years which is my cherished excuse for the poor grades.
Thankfully, I need no excuse whatsoever to discuss a few issues surrounding the larger cause of mathematics with Dmitry, a towering thought leader-practitoner commanding invaluable insights into pure and applied Math.
More about Dimitry can be accessed here.
Excerpts from the Q & A…
What, in your opinion, needs to be done to teach conceptual mathematics in a way that inspires students to think beyond formulae and methods?
I think pictures are very important and as are connections to various common domains outside the traditional mathematical physics topics, such as literature or image manipulation. A typical example I observed in people is a fascination with fractal images or digital image manipulation and then a sudden understanding that this is mathematics – but not the one we do at school. If you teach math to software developers, teach ideas from functional programming and how they are connected to pure math. Somehow, the latter connection didn’t occur to me until very recently. So, all math books that start with Visual or end with Illustrated in their title are highly recommended. Great examples are Topology Illustrated and Visual Group Theory.
I personally find the ‘mathematical circles’ culture of Russia and Bulgaria an inspiring movement – does it explain why Russians are generally good at Math?
It is probably an illusion. When people good at math (whether they attended “circles” or not, I didn’t) arrive in the West due to imposed selection processes, this gives such an impression because we have a different statistical distribution. The same conclusion also applies to arrivals from other countries. Just compare the number of math textbooks and popular books in Russia and the English-speaking West. Some say that mathematical proofs are the feature of the so-called Russian math, but I always hated proofs at school, but not nowadays – I have the Proof Patterns book on my reading list. A nice historical read: A Russian Teacher in America http://toomandre.com/my-articles/engeduc/ARUSSIAN.PDF But it should be noted that it was published 30 years ago, and it was not Russia but still the Soviet Union conceptually – teaching was the same everywhere, and the author emigrated before the mass wave in the mid-1990s. Since mathematics was a relatively safe area to do research in the Soviet Union – it additionally attracted people – hence these circles (in a different meaning, no pun intended). What is described in Toom’s article about America, I also observed as a “tourist” in post-wild capitalist Russia in mid-2000 (since I already lived in Ireland) because, in addition to the top-level Moscow State University earlier (state meant Soviet Union state) where people came to become researchers, I also attended a lesser-level university where people came for a degree (and my observations let me believe this was the same in late Soviet Union years). So, in retrospect, there are the same patterns everywhere.
In our earlier conversation, you mentioned that you view and use mathematics as a conceptual tool for theoretical software diagnostics. Can you elaborate?
Abstraction unifies disparate areas of mathematics. And we have disparate areas in software diagnostics too. When abstracting problems in one platform, it is much easier to do problem diagnostics (and solving) in another. You don’t have to repeat the same learning path. But I mean conceptualizing via pattern languages, not formalizing.
We find so many claims (and counterclaims) of solving the Riemann hypothesis – what is your personal take on the conjecture that all of the nontrivial zeros are on the critical line?
On an impulse, I think of the book In Search of the Riemann Zeros: Strings, Fractal Membranes, and Noncommutative Spacetimes, which was on my reading list for years but thank you for reminding me. All three ingredients in a subtitle interest me greatly, hope I can apply some of them to theoretical software diagnostics, but I also don’t want to fall into the camp full of Sokal affair people (https://en.wikipedia.org/wiki/Sokal_affair). Although, I personally found wild metaphors interesting conceptually and a great source of inspiration. Windows crash dumps are full of prime numbers: the most famous is number 5, access denied. Please note that the Windows memory access violation code is not prime but a composite number – 3221225477. But I don’t want to go further into this Numerological connection. As a side note, Riemann’s manifold thought inspired the book The Riemann Programming Language (ISBN: 978-1906717605), which I’m still writing. It is rather Riemannian geometry that interests me. Another recollection, in 2003, my exposure to cryptography inspired my curiosity for prime numbers and number theory in general, especially to understand the proof of Fermat’s last theorem.
Why did you hate Euclidean Geometry at school?
I think that I didn’t see any connection to my interest in Chemistry, and all the books I was reading didn’t mention this geometry. Also, possibly, I just hated proofs. But I managed to get a good mark in the intermediate examination. The final two school years included stereometry, but there were no exams, so I skipped that subject completely. However, my parents gifted me a two-volume Geometry by Marcel Berger (translation by Mir publisher), and it looked much algebraic and analytic inside and, in some sense, shaped my view of how modern mathematics looks like. Unfortunately, I managed to read only a few pages then, so I plan to restart and see how I improved in 35 years.
Conversely, what explains your natural affinity for category theory, topology, and the variety of logics?
Earlier learning of Chemistry – algebraic, rule-based, and full of diagrams – great patterns. Graph theory and topology in Organic Chemistry and Biochemistry, group theory in Crystal Chemistry, and algebraic relations and reaction inference rules in Inorganic Chemistry. Additionally, calculus and statistics in Physical Chemistry, linear algebra, and mathematical analysis in Quantum Chemistry gave me a foundation for understanding machine learning.
You say you have a very algebraic style of thinking which helped you in clearing examinations? Does that mean you disregard geometric patterns at all, or do you use them intuitively?
Since I knew examination-level algebra well and there were 5 questions (4 algebra + 1 geometry), I decided to concentrate on algebra only to get a good mark for sure. Regarding geometric patterns, I think I used them intuitively; I just didn’t want to study all these formally. But nowadays, I’m pretty much interested in everything I didn’t have an interest in at school or University. Another example is philosophy – I didn’t have any clue about it except ideas from Democritus and Marx-Engels. But now I have my own monist philosophical ideas.
What in your reckoning could be done to make calculus interesting for those who fear it – either instinctively or courtesy of terrible teachers who take the charm away through military diktats rooted in mechanical memorizations of formulae?
I learned calculus through physics, the first volume of Feynman’s lectures. I also noticed some calculus textbooks that now use Excel as a calculation medium. There needs to be some motivation; why do we need to learn calculus? For me, it was to understand quantum chemistry. Nowadays, it is probably basic ideas of machine learning that inspire people to learn calculus if they didn’t study it at school or forgot. I think new calculus textbooks should add examples from machine learning in addition to physics. And definitely more pictures. I like the approach taken by Georg Glaeser in his book Math Tools: 500+ Applications in Science and Arts.
Do you feel that students need to be first made conversant with algebraic functions, exponents, inequalities, and trigonometry before they are brought face to face with calculus? In your personal tryst with calculus, which approach resonated naturally with you – the infinitesimal approach or the limit approach?
Some knowledge of what functions are is necessary, of course, as well as exponents, inequalities, and trigonometry – all this algebraic stuff with function graphs. It is truly foundational knowledge, even without calculus. It is also highly conceptual. But to learn calculus for the first time - I believe infinitesimals is the right approach – something along the lines of non-standard analysis (https://en.wikipedia.org/wiki/Nonstandard_analysis). It is how I grasped calculus. I learned about limits much later as a part of mathematical analysis, not calculus.
Who are your role models and idols among Math greats? Did any of the towering folks and also any of your teachers inspire you to delve deep into mathematics over the course of your career?
Cantor, Gödel, and the dynamic duo of Zermelo-Fraenkel– these names inspired my interest. And I read Gödel’s biography book much later. But I don’t recall inspirational teachers except perhaps one that taught mathematical analysis at Moscow University; I liked his jokes. But I learned all myself by reading before, and University didn’t add anything to it, perhaps because, in the background, I completely switched to programming – almost 15 years of not learning any math. My return to math was unexpected, when I started learning linguistics and other human languages inspired by my job of implementing rules of C++ language semantics in static code analysis tools, and this brought back my wide learning aspirations when I saw a popular book on string theory at that time in 2002.
Any thoughts that you wish to share with the community at large…
If one wishes to restart learning mathematics (and, in the process, learn things not taught at school or 3rd-level institution) after a long break like me – start with math history and philosophy books. Read the first chapters of many non-history math books, even advanced ones. It is ok to skip proofs entirely. Repetition is the key to success. Some popular wide expositions definitely help, such as Concepts of Modern Mathematics by Ian Stewart or more advanced All the Math You Missed, Second Edition, by Thomas Garrity (I used to read the first edition). These books should also help when reading basic category theory books to understand mathematical examples. If you are interested in physics, semi-popular mathematics can be found in the monumental The Road to Reality by Roger Penrose. I wish there were popular expositions of advanced topics such as non-commutative geometry. I recall that I started writing the book Software and Modern Mathematics (ISBN: 978-1906717582) mainly to teach myself related math but other projects swept it aside. Now, I would change the title to Software and Contemporary Mathematics and hope to resume sometime soon.