Tag: Mathematics

A beautiful introduction to the Holy Grail of mathematics

Two years back, exactly to this day, I had visited Alliance Francaise in Delhi with a friend. They were going to add new titles to their library, so some part of the older collection had been put up for sale. As expected, they had books and editions not usually found in bookshops or online bookstores. Amongst the gems I picked up that day was Marcus du Sautoy’s The Music of the Primes. In all honesty, I purchased the book because it was a hardcover with an intact dust jacket, it was on mathematics, the book was in perfect condition and didn’t have a single pen or pencil mark.

The book would turn out to have two interesting connections with my earlier forays into the world of maths, and I was oblivious to both at that time.

The first one was the author of the book. A few years earlier I had watched a BBC Four series titled The Story of Maths. It was presented by Sautoy and although I hadn’t gathered his name, his face had stuck.

The second was what the book was about – the Riemann Hypothesis, the most important unsolved problem in mathematics.

The importance of the hypothesis can be gauged from the fact that it is the only problem that occurs on both Hilbert’s list of twenty three problems and the seven Millennium Prize problems – the two most important lists of unsolved problems that have come up over the last century and a quarter and which have provided an impetus and given a general direction to mathematical research. The former were presented by David Hilbert at the International Congress of Mathematicians at Paris in 1900, and the latter were put forth by the Clay Mathematics Institute in the year 2000.

The Riemann Hypothesis was put forward by Bernhard Riemann, one of the most important mathematicians of all time, in the year 1859. Riemann brought about a shift in perspective in the philosophy of mathematical research. He believed it was more important to understand the hidden structure of maths than to try to solve specific questions. In that sense, he heralded a revolution in the psychology of approaching mathematical problems, a culture that continues to this day. In fact, half a century later, Einstein would discover that Riemann’s new mathematical language, of which the Hypothesis was merely an incidental observation, was perfectly suited to express his transformative ideas of special and general relativity.

Over time, hundreds of other results have come up which proceed by assuming either the truth or falsity of the Riemann Hypothesis. Thus the resolution of this hypothesis, either way, will have huge implications for the mathematical edifice.

At its heart, the Riemann Hypothesis asks a simple question – is there some hidden pattern in the distribution of prime numbers?

Despite being the building blocks of arithmetic, prime numbers are really not that well understood. Why does their distribution seem so random? Are they following some pattern? Is there some logical structure that permeates them, and which could be used to catch a glimpse about their mysterious world? Given a prime number, how long do we have to count upwards till we encounter another one?

Mankind’s quest to understand prime numbers actually has a pretty rich history that goes back over two thousand years to Euclid who, in his book Elements published around 300 BCE, provided a very simple argument to prove that there are an infinite number of primes. Over time a number of mathematicians have worked on it. In fact, the list of mathematicians whose work has either directly or indirectly helped in our understanding of primes sounds like a who’s who of the history of mathematics – Euler, Fermat, Gauss, Dirichlet, Fourier, Hilbert, Riemann, Ramanujan, Hardy, Gödel and Turing.

The modern history of the story, as well as this book, starts with Gauss who brought about the first fundamental shift in how we think about prime numbers. Instead of asking when the next prime number will occur, he asked, instead, how many primes occur up to any given number ‘n’. John Napier had come up with his logarithm tables just a few decades before, and Gauss realised he could use them to convert multiplications of huge numbers into simple addition. He developed his ideas and came up with his path-breaking approximation of n/log(n).

It was Riemann, however, who shifted gears and brought about the second fundamental shift by opening up a new landscape to understand the problem. He transformed the puzzle of the distribution of primes into the properties of a certain curve in three dimensions. The Riemann Hypothesis essentially states that the set of coordinates where the height of the curve is 0, follow a particular pattern.

The Music of the Primes provides an extremely enriching and exhilarating vision of the developments in the study of primes over the last two hundred odd years when, really, most of the progress has been made. Among the many interesting things I came to know, two have really stood out.

The first one was realising that in 1976 a group of mathematicians, working on a theorem put forth by the Russian mathematician Yuri Matiyasevich, came up with a formula in 26 variables using which it is possible to generate all the prime numbers. The second was realising that even though we have such a formula, it is not as valued today because the focus has now shifted following Riemann’s gear change, and it was precisely this striving to make sense of the hidden structure and behaviour of mathematics that has led to connections between, lo and behold, prime numbers, quantum theory and chaos theory. That’s right. Please do yourself the favour of reading the previous sentence again, and then kindly proceed to pick up your jaw that may have fallen to the floor.

Sautoy is the Simonyi Professor for the Public Understanding of Science at the University of Oxford, a chair that was created in 1995 and was occupied by Richard Dawkins till 2008, when Sautoy took over. His choice for the title of the book reflects his ability to realise that referencing the Hypothesis would end up restricting his audience to a very niche subset of mathematical enthusiasts, whereas a title such as The Music of the Primes is at once both mysterious and evocative, and would be able to vibe even with the general public.

A watershed proof in the history of mathematics

While growing up, we sometimes come across certain special mathematical problems which have the following three properties – they are easily stated, are simple enough to understand and are either unsolved or require the knowledge of advanced mathematics, certainly for that age, to be tackled. Such problems perform an especially important role of stimulating our young minds even as they provide us a fleeting glimpse of the beautiful world of mathematics that lies beyond our school texbooks. Prime examples of such problems are Fermat’s Last Theorem, the Goldbach Conjecture, the Seven Bridges of Königsberg and so on.

The Four Colour Theorem (4CT) falls in this unique category of problems.

The 4CT says that in any continuous map containing an arbitrary number of countries, the maximum number of colours needed to colour them such that no two adjoining countries are of the same colour, is four. If two countries meet at a point, then they are not considered to be adjoining.

The story starts in the middle of the nineteenth century in England with two Guthrie brothers – Francis and Frederick, the latter of whom was studying under the famed mathematician Augustus De Morgan, the formulator of the well-known De Morgan’s Laws of set theory.

Francis came up with the Four Colour Conjecture (now theorem) in late 1852 when he noticed that he only needed four colours to colour all the counties of England such that adjoining counties were of different colours. He discussed this with Frederick who then consulted De Morgan. Therefrom, there was no turning back.

The problem was not as popular among mathematicians in the initial decades, and it took another half a century before it started to be taken seriously by mathematicians on the other side of the Atlantic. Nevertheless, all this while, it continued to invoke the occasional public interest and draw in amateurs and non-mathematicians to dabble in it.

The 4CT was proven by contradiction and at its heart lay the idea of a “minimal criminal” – if the theorem is false, then there must exist a particular smallest possible map that necessarily requires at least five colours to be coloured. Two core ideas formed the bedrock of this method of attack – “unavoidable set”, which is a set of configurations of shapes such that at least one member of such a set must be present in every map, and “reducible configuration”, which is any arrangement of countries that cannot exist in a minimal criminal.

So, if we are able to find a non-empty “unavoidable set of reducible configurations”, which is a set of reducible configurations such that at least one member of the set must be present in every map, then that would prove that a minimal criminal could not exist, thereby proving the 4CT.

As the twentieth century progressed, much of the effort towards proving the theorem was done along these lines, and it was finally in the year 1976 that Kenneth Appel, an Assistant Professor, and Wolfgang Haken, a visiting professor at the University of Illinois at Urbana-Champaign, finally proved the 4CT, having extensively used the computing resources available at the university. They had come up with a candidate set of 1,482 possible configurations and it took over twelve hundred hours of computing time on the university’s supercomputer to verify them all, one by one.

The proof of the 4CT was a watershed moment in the history of mathematics as it was the first major theorem that was proved using a computer. This raised serious philosophical questions at the time regarding what is a proof, and whether something that couldn’t be verified by hand could be trusted.

In addition, the proof’s brute-force method was also criticised for lacking beauty and being inelegant, something even Appel accepted.

Robin Wilson does a pretty good job of preparing the mathematical foundation for the important ideas linked with the proof that come up in the second half and there are ample diagrams to help the reader visualise the arguments that are being presented. In fact, I cannot remember reading any other science book with as high a ratio of diagrams to text.

The first one third of the book is a breezy read that the reader will sail through, and it is only when the ideas start to come together from this point on that the book demands attention and careful reading to connect the dots. The brute force nature of the proof means that the reader does lose context a few times when the details are being discussed but, thankfully, Wilson refers to earlier parts of the text at a few important stages of the book to help the reader align his thoughts vis-à-vis the path they have taken to reach the final proof. In fact, barring some brief sections that may be difficult to understand (which is natural, considering the historical significance of the problem), I believe most of the book can be understood, with some effort, even by high school students who are even slightly mathematically inclined.

All in all, Four Colours Suffice is a welcome addition to one’s library.

An involved introduction to a modern mathematical beauty

The Poincaré Conjecture is one of the seven Millennium problems that were announced by the Clay Mathematics Institute in the year 2000. These were problems deemed to be the most important open problems in mathematics, and whose solutions, or even attempts towards the same, were expected to lead to pivotal advancements in our mathematical knowledge. Although not explicitly stated, the kind of problems chosen and the general acceptance of the list in the mathematical community means it could, potentially, significantly affect the general direction of mathematical research in this century.

The conjecture goes back over a century when Henri Poincaré, a leading French mathematician of the time, and who is also considered to be “The Last Universalist” i.e. someone who excelled in all the fields of mathematics that existed during his lifetime, came up with the following gem in 1904 –

“Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.”

Do not stress if you cannot understand a word of it for, and I admit this with shame, even after completing the book I was not clear what all the terms used above meant, and I could only afford some basic level of clarity once I consulted a friend of mine who has done post-graduate studies in mathematics.

This is a fairly technical book, and although it is possible to understand certain sections and even chapters in their entirety, prior exposure to topological concepts and terms are an essential requirement in order to fully appreciate the beauty of the ideas presented. Of course, I could easily blame the author for writing the book that way, but the truth is that it is immensely difficult to break down such involved mathematics for a genuinely layman audience. This conjecture took over a century to prove; surely I couldn’t have hoped to understand it by reading a few hundred pages?

O’Shea has actually done a decent job in conveying the significance of the problem – the conditions under which it arose; how it affected its field in the years to come; the psychological effect of having a problem that an entire generation of mathematicians knew about, even if they were not directly linked with that discipline; its social and historical context and how its solution, and efforts towards the same, affected all of the above.

To understand the problem, however, in all its technical beauty, would be a bit too much to expect from someone without decent mathematical exposure. Significant knowledge of the discipline is required to understand why the conjecture couldn’t have been solved back then, what tools and techniques arose in the quest for its solution, and how it affected the growth of its own and related fields in the years to come, both in terms of the kind of problems that it solved, and the kind it created for the next generation of mathematicians to ponder over.

While I did understand the historical significance of the conjecture, the technical beauty all but eluded me and although I do remember and understand some parts of the book, and some vestiges of the arguments do remain in my intellect, they are in a form that betrays my ability to pass them on to anyone else in an intelligible manner. In that sense, can I even claim to have read the book?

Having said that, will I suggest reading this book?

My gold standard for writing popular maths is Fermat’s Last Theorem by Simon Singh, against which I have come to compare all the popular mathematics books I have read since. If you have read that one, know that The Poincaré Conjecture demands a bit more effort on the reader’s part.

If you haven’t, and if you have some background in mathematics, or if you loved the subject growing up, you could take up this book, but maybe skip through the parts that get into the intricacies of the mathematical concepts. I have personally felt the gradual arrival of the moment, in some books, when my mind says it is all going over my head. In most of the cases, especially if one doesn’t have the requisite background, heeding that request is beneficial.

However, if you have studied mathematics after high school, then I would request you to wade through the work and put in slightly more efforts. You will still not get everything, but maybe enough to appreciate the sheer beauty of the ideas.

And, if you have been exposed to a bit more advanced mathematics, and if you have some prior knowledge of the topological vocabulary used in the work, it may be worthwhile to hang in tight and read with full concentration. You may just get it! Reading is not always a leisurely way to pass time. It can also be challenging, and that is why we read in the first place, to peer into a field far removed from our daily lives, waiting with mysteries to delight and leave us in awe.