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.