The 13th Book
This is a continuation of the last article where I have shared six books for maths and logic under the following two categories. (Read the last article)
(i) Evolution and History of Mathematics
(ii) Mathematical and Logical Puzzles
In this article, we will explore a few more books on maths and logic from the following categories
(iii) Some Cool Topics in Mathematics
(iv) Great Pupils of Mathematics
I know the original plan was to introduce three books in each topic to make it 12 books in total, as also portrayed in the title for this series. However, at the last moment, I have decided to include one more book in this series. This will make the cumulative count go up to 13 books in total. Thirteen, some would call it unlucky, has a particular significance for the additional book I have added to the series. The book is called Prime Obsession. Let me start with this thirteenth book to begin with and I will share the rest of the books after that.
(iii) Some Cool Topics in Mathematics
Now let me reveal the reason why number 13 has a connection with this book. Thirteen, apart from being an unlucky number for some, is also a prime number. Prime numbers are defined as integers that could be divided by itself or 1 to produce integer solutions. 2 is prime since it can be divided by just 2 and 1 to produce integer solutions (1 & 2). However, 4 is not a prime since it can be divided by 1, 4 and also 2 to produce integer solutions.
Analysis of Distribution of Primes
Wear your analyst cap and let us do a quick analysis of prime numbers to understand the message in this incredible book.
Question: How many prime numbers are there less than 10?
Answer: 4 (i.e. 2, 3, 5, and 7)
Similarly, I have displayed a table (taken from the book) to display a number of primes below one thousand, one million, one billion and so on. As you could see in this table, there are 168 primes less than 1000 (i.e the second column (π(N)). The last column is the first column divided by the second column i.e. 1000/168=5.9524)
|N||Primes less than N (π(N))||N/π(N)|
As an analyst, ask yourself is there a pattern in the last column. Ok, here is a hint: subtract the 3rd row in the last column from the 2nd row. The result is 6.7868 (i.e. 12.7392-5.9524) . Repeat this for the successive rows (as shown below in the table below). Yes, sir there is a pattern, they are all hovering very close to seven. Therefore, we could create a simple model taking natural log of N. The error percentage between our model (ln(N)) and actual value is also displayed in the last column.
|N/π(N)||Row Difference||ln(N)||Error %|
This is a phenomenally simple model and highly accurate as most analysts would agree. Error term is shrinking, as the N is getting larger. Carl Fredrik Gauss (of famous normal distribution) first discovered this property of prime also known as the Prime Number Theorem (PNT). Further PNT and zeta function were linked through the Riemann hypothesis proposed Bernhard Riemann. The Riemann hypothesis is expressed in one sentence as “All non-trivial zeros of the zeta function have real part one-half”. I do not know about you but I had no idea what this sentence meant before reading this fascinating book by John Derbyshire. I would recommend that you read this book to learn more.
We have discussed prime numbers in some detail while discussing John Derbyshire’s book above. Some people do maths for maths’ sake (a modification of phrase ‘art for art’s sake’). However, others would ask, why this whole fuss about prime numbers? Why does one need them for practical purposes? One of the practical uses of prime numbers is in cryptography. Modern day cryptography that involves hiding your email passwords, bank passwords, etc. from the rest of the world is through the usage of prime numbers. The Code Book tells the interesting history of cryptography and science of secrets. It starts with the reason for the beheading of Mary Queen of Scots in 1587. English deciphered Queen Mary’s encrypted message and revealed her plan for treachery against Queen Elizabeth. This book has one of the best descriptions of the Enigma Machine from the Second World War. Finally, Simon Singh’s book discusses the application of quantum mechanics in the field of ciphering.
I know some of you have noticed it. I agree this is a book about evolution and biology and not a book on maths. However, I must quickly add Richard Dawkins’ book in my opinion gives one of the best descriptions of game theory in popular science literature. The book conveys the message that the primary purpose of life is to propagate, reproduce and procreate. We, including animals and plants, are biological machines trying to fulfill this primary purpose. The idea behind a selfish gene is to maximize the chances of survival for the species and not for the individual. Altruism and benevolence, according to the book, is a by-product of this primary purpose. In the twelfth chapter titled Nice Guys Finish First, you will find a great description of the game theory. The interrelationship between prisoner’s dilemma (game theory) and biological optimization of survival machine is wonderfully captured in this book.
I was in the last semester of my Master’s course in Physics. That was when I came across Non-Linear-Dynamics (NLD) and chaos theory in one of the courses. I immediately fell in love with the field. I still remember coding a Mandelbrot set on FORTRAN to produce an aesthetically pleasing image on GNU plot. The incredible aspect of the Mandelbrot set is that it is a simple iterative equation Zn+1=Zn+C. A simple do-loop can generate the wonderfully intricate image shown on the right (taken from Wikipedia). The two axes of this two-dimensional plot are the real and complex axis. I think I will have a separate discussion on chaos theory sometime soon on YOU CANalytics. However, for now, let us come back to the book by James Gleick. The book starts with the Butterfly Effect, the phenomenon discovered by Edward Lorenz in 1961. He was working on a very simple weather model with three interlinked differential equations using his computer. What he noticed became one of the greatest accidental scientific discoveries and the reason why we cannot predict the weather for a distant future (say 6 months from now). A slight difference in the initial condition can bring about two completely different results for prediction. In Lorenz’s case, the accidental difference in initial condition was at the fourth decimal place i.e. 0.000X. This phenomenon is known as the Butterfly Effect. Gleick’s book then delves into several topics in NLD such as bifurcation, strange attractors, universality and fractals (Mandelbrot and Julia set). This book is a good introduction to chaos theory.
I could easily add 4-5 more books to this category. However, I feel I will introduce those books in some later discussions on YOU CANalytics. To tell you quickly, other books I had in mind are
The Signal and the Noise – (Cool Topic: Bayes Statistics) – by Nate Silver
Thinking fast and slow – (Cool Topic: Behavioral Economics) – by Daniel Kahneman
Reckoning with Risk – (Cool Topic: Medical Risk and Bayes Theorem) – by Gerd Gigerenzer
Drunkard’s Walk – (Cool Topic: Randomness) – by Leonard Mlodinow
(iv) Books on Great Pupils of Mathematics
The book was written in 1937 and is still considered as one of the most definitive works to capture the lives and works of the great mathematicians. As you could guess, and several have complained, that this is a book on men of mathematics, without any reference to women of mathematics. Despite this criticism, the impact of this book can be measured through the numerous references it has garnered. The book starts with the lives of Zeno (5th Century BC), Eudoxus (4th Century BC), and Archimedes (2th Century BC), the greats of Greek mathematics. Then it covers others great mathematicians such as:
Fermat described as the prince of amateurs
Gauss described as the prince of mathematicians
And my favorite, Euler, described as analysis incarnate
Finally, the book ends with Cantor (1845 – 1918). Like several authors, I use this as a reference book as well. For instance, once I was confused about the number of famous mathematicians in the Bernoulli family. The book has a neat family chart of mathematicians in Bernoulli family and there were ten of them! This is a good book to have on your bookshelf.
When I purchased E.T.Bell’s book I was desperately searching for two great mathematicians in his book. They were Srinivasa Ramanujan (1887–1920) and Kurt Gödel (1906–1978). I did not know then that they were contemporary to E.T.Bell and hence could not make it into his book.
To get an understanding of the life and mathematics of Ramanujan, Robert Kanigel has produced a perfect book. It is reasonably rigorous to explain his mathematics to an unaware audience. Additionally, after reading the book you will understand Ramanujan’s life much better.
However, I could not find a similar book for Kurt Gödel. Gödel’s Proof by Nagel and Newmann is a thin book with less than 100 pages. It is a good book to grasp the concepts and logic of the revolutionary 1931 paper by Gödel. This paper shook the foundation of mathematics. The incompleteness theorem proposed by Gödel demolished the dreams of mathematicians like Bertrand Russell and David Hilbert. The dream was to find a complete and consistent set of axioms for the entire field of mathematics. I would still be keen on reading a complete book on Gödel. Please let me know if you have some suggestions.
This is my list of books on maths and logic. However, I am more than keen on receiving suggestion or learning about your list of favorite books on the topic. Please leave a message in the Leave a Reply section right below. Look forward to hearing back from you.