Men of Mathematics (E.T. Bell)

நான் இந்தப் பதிவை தமிழில் எழுத முயற்சி கூட செய்யப் போவதில்லை.

In my teens, I came across a book called Men of Mathematics by Eric Temple Bell, a minor mathematician. I spent countless hours reading and re-reading that book.

I was a tolerably good student at that time, especially good at high school level math. But while I could easily and frequently score 100 marks in calculus and algebra tests, I just could not understand how one can cancel some terms involving delta-x because it is close to zero but not others; or how one can divide by delta-x that is almost close to zero.

I really had trouble during my +1 year. My math teacher was one K.V. Srinivasan or KVS, an old school teacher (He had a B.A. in math, not B.Sc.) at St. Joseph’s, Chengalpattu. I repeatedly asked him about the delta-x issue and he couldn’t give me a satisfactory explanation. His explanations looked very weak to me – I am sure he was as tired of my questions as I was tired of his answers.

Similarly, just wtf is an imaginary number? I mean, what number, when squared, yields a negative number? I was able to use them well, without really understanding it. Or even the negative numbers – why in the world would multiplying two negative numbers yield a positive number? I was used to thinking about negative numbers as debt – but why would multiplying one debt by another give me a profit? I didn’t have any teacher, any book, any resource that could answer such fundamental questions. I also couldn’t see any use for trigonometry. Algebra – which was my favorite branch of math at that time – felt like just symbol manipulation. Geometry was one branch I never truly liked – primarily because my drawing skills were very poor – I couldn’t draw a circle even with a compass, my circles would never close. But also, I just couldn’t see the point of opposite angles of intersecting lines being equal or the artificial distinction between but equilateral and isosceles triangles and so on. I remember asking in 6th or 7th grade where all the theorems about isosceles angles are all used in real life and getting a huge scolding from my teacher.

I can keep going on. Basically, higher level math – anything beyond simple arithmetic – felt like just a load of useless, impractical, theoretical garbage.

Aside: Algebra was my favorite branch because I had discovered formulae like a^2-b^2 = (a-b) * (a+b) by myself independently. That was my first intellectual achievement ever. My only clue was the observation that the difference between successive squares differ by 2. I was writing up squares of all numbers upto 32 as part of my classwork and noticed that 24^2=576, 25^2=625, 26^2=676; i.e. 25^2-24^2 = 49 and 26^2-25^2=51 and noticed that the pattern can be extended forever i.e. 27^2-26^2 = 53, 28^2-27^2 = 55 and so on. I was 11 or 12 around that time. And it took me around 4-5 months of thinking and thinking about this pattern before I came up with the formula. I didn’t know algebra at that time, I actually expressed it in terms of words, Tamil words at that 🙂 Something like:

ஒரு வர்க்கத்திலிருந்து இன்னொரு வர்க்கத்தை கழித்தால் கிடைக்கும் எண் அவற்றின் வர்க்கமூலங்களின் வித்தியாசத்தை வர்க்கமூலங்களின் கூட்டலால பெருக்கினால் கிடைக்கும் எண்ணுக்கு சமம்

When I came across algebra in 9th standard (I think), it felt great! I suddenly had a language to think about these kind of things. But 2-3 years later, I had come to think of algebra as rote manipulation of symbols according to certain predefined rules, just useless theory.

Then I came across Bell’s book. Bell made math come alive for me. Suddenly, I understood limits after reading the chapters on Fermat and Newton. In fact, it made perfect sense. Though I didn’t expect to use it myself, I suddenly saw why calculus is useful; not just useful, but necessary to understand the laws of physics. I suddenly saw imaginary numbers in terms of a co-ordinate plane and things just clicked. Multiplication of one negative number by another? Just a directional change along the number line.

When I read about Descartes, suddenly geometry became a discipline that I could master, by coming from a different angle. I got pretty decent at co-ordinate geometry, and I attribute that to entirely Bell’s effect. In fact, I had a voice at the back of my mind going that I could have done this, I could have invented co-ordinate geometry myself, Decartes’ achievements aren’t that great. That feeling that I could have created analytical geometry myself was enough to hook me onto math forever.

I read about Poncelet and the romantic account of his discovery of Projective geometry from prison, it made great sense. I desperately wanted a textbook on projective geometry and couldn’t get one. When I joined engineering, that one chapter gave me a fundamental understanding of engineering drawing class. I was still pretty bad at drawing, but I could clearly understand and in fact even go beyond what was taught in class. One chapter – just 10 pages or so – made a discipline real for me.

Of course, there were things I didn’t quite understand as well e.g. Fourier series, Analysis in general, Ellipitic integrals, Group theory, Riemann Hypothesis, chapters on Hermite and Poincare, but it just didn’t matter. I still don’t understand many of these topics. But I found enough to fascinate me. Cantor’s infinities, the Dedekind cut, Zeno’s paradoxes, Weirstrauss’s continuous but non-differentiable curve, Non-Euclidean geometries (The instant I was told to imagine a sphere, I could see how the fifth axiom is not valid any more), Fermat’s Last Theorem, the list goes on.

I didn’t see the point of of several of these topics as well e.g. I didn’t think Hamilton’s quarternions would ever find any use. But now I was confident that somewhere there must be a use for it. I read that number theory didn’t have much practical use, but I found it the most fascinating of all branches of math. Now of course, primes are crucial in cryptography…

Bell had a habit of combining the biography with tantalizing bits of math. For instance, he just mentioned Wilson’s Theorem in passing and would make a remark like a high school student can prove it. I think I spent 3-4 months thinking about it before I came up with a proof of my own. I can still vividly remember the thrill when I realized that every number has an inverse in modulo arithmetic. Wilson’s theorem was one of my rare successes, though. I probably failed in trying to prove 90% of what I attempted. No matter. I don’t think I had that much fun learning, just trying to figure out things before and after those 12-15 months in my teens.

I also spent hours ranking the mathematicians in that book in my mind. Gauss was always the #1. Descartes never got a high ranking from me because I felt that if a high school student like me can derive analytical geometry, then Descartes doesn’t deserve lot of credit. But I remember ranking Archimedes, Fermat, Newton, Leibniz, Euler, Abel, Galois, Riemann and Cantor as the top mathematicians.

One of my great regrets is that I couldn’t convey my enthusiasm for the book to my daughters – or for that matter, anybody else. I even bought a copy for Jeyamohan, but I doubt whether he actually opened the book any time. 🙂

Later I came to know that Bell would now and then exaggerate to make the biography more interesting. I couldn’t care a rat’s ass. He had done his job – kindled a life long fascination with math in a govt school kid brought up mostly in villages. I just wish I had gone beyond the dilettante stage and actually learned math. I would have enjoyed working on number theory!

Aside: When my daughter went for a familiarization tour to her college (UC Santa Barbara), Yitang Zhang – one of the romantic stories of math today – was giving an introductory lecture on the prime gap. I think I was the only one who was listening; I was actually thrilled with that lecture! My wife and daughters still make fun of my asking questions at the end of the lecture. I could see how his proof that the prime gap is less than 70 million ties in with Goldbach conjecture, and I was remembering Bell gratefully…

Category: Math

12 thoughts on “Men of Mathematics (E.T. Bell)

  1. Quote One of my great regrets is that I couldn’t convey my enthusiasm for the book to my daughters – or for that matter, anybody else. Unquote
    May I know the name of the book in reference and you have echoed my feeling of how others are not showing interest when we talk about the topic of mathematics or mathematicians.

    I want to share my experience, when I went to theatre to see the film the man who knew infinity, I am the only one whe reacted when a dialogue delivered with names of Euler and others (my son and my nephew) could not understand the enthusiasm of mine.

    I have copy of the tamil film Ramanujan and I would have watched plenty number of times and when I met a lady of 70 years old who is also mathematics graduate and we enjoyed the film watching (with subtitles in English as she is not tamil) and felt mathematics has no language barrier.

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    1. One more thing I wanted to share, I came across a small clip of video which describes Ramanujan square based on his DOB and from that I have arrived with my own calculation which can be applied to any DOB and afterwards I prepare a small video clip based on that and sent to others on their birthday but they were not interested in that and I stopped that practice.

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      1. If you google Ramanujan’s magic square you can get it and the first row is his DOB and in that format I worked on my own formula to arrive for any DOB.

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