Few words elicit as much varied reactions in randomly chosen mature human beings capable of independent thoughts and opinions, such as Mathematics. In the eyes of many of them it is simply a bare necessity, something that must be learnt in order to progress smoothly in life and to build a decent career, including passing the high school examination. Among these people, there are some who dislike it wholeheartedly and find great comfort in forgetting all the formulae taught by the quintessential “maths teacher”, as soon as they can. Another group is perhaps a bit more tolerant towards it, although they do not have any particular affinity for Mathematics, neither do they find it repulsive nor do they shun it completely out of apparent compulsion.
On the other hand, there are some, albeit a minority in comparison to the sheer number of the earlier groups, who love Mathematics and vouch for its beauty and elegance with an unmistakable passion. It is certainly not the objective of this article to justify the attitude towards Mathematics of one group over the others. Indeed, the author is a firm believer in the age old saying that variety is the spice of life. However, in view of the widely varying viewpoints of mankind towards a particular discipline, it certainly merits discussing the various facets of Mathematics with an open mind.
What is mathematics After All?
To begin with, let us embrace the most fundamental query in our context. “What is Mathematics”, after all? Unsurprisingly, this has been asked many times before, by some of the greatest minds in the history of humanity. A simple Google search will reveal that there exists a book, and a fantastic one without a shadow of doubt, written by two masters of the game, that goes by the precisely same name. While I do not expect the majority of the readers to feel particularly enthusiastic about studying the somewhat voluminous treatise, I have no doubt that anyone doing so will benefit greatly, should she choose to do so. It seems impossible to arrive at a universally acceptable answer to our query, simply because of the incredibly large scope of the concerned discipline.
Nevertheless, it should not (and will not) stop us from framing our individual opinions on the subject matter. As guided by my personal working experience, Mathematics, in my view, is primarily the study of numbers and shapes, and of abstractions arising out of it, strictly within the logical framework of certain predefined “rules” that seem natural in some sense. It is quite amazing to yours truly that such an apparently simple “definition” essentially covers a huge part of all the scientific endeavors of mankind, starting from the very beginning of human civilization. The reader is heartily encouraged to pause here and critically analyze the validity of the above claim from her own point of view. I have absolutely no qualms even if she disagrees with me. It is not the agreement in thought that I seek here, rather the thought itself!
How Did it All Begin?
Let us now come to the inevitable questions of how it all began and who were the pioneers of Mathematics in its earliest days. The answers to both the questions remain obscure, although that seems hardly surprising in light of the fact that a considerable part of human history remains unwritten and uncharted. This of course does not stop us from making a guess, especially regarding the first query. Let us try to imagine the unwritten lives of a group of earliest cavemen who have just started domesticating certain animals for their own good, including but not necessarily limited to a steady supply of milk and wools. It is absolutely essential to keep count of the animals belonging to the group, except the fact that numbers have not been introduced yet!
To address the collective need, one of the members suddenly comes up with a brilliant plan. When the flock goes out for grazing in the morning, why not pile up stones, precisely “one” for each animal! When they return in the evening, once again consider the pile in the same way, “one” for each animal. If some stones remain unassociated, some animal(s) must be missing. A very practical problem solved in a most convenient way, based on a brilliant idea of one-to-one correspondence between stones (which are abundant) and animals (which are precious). So that’s more or less how numbers were born! The common link between a stone and an animal is not at all related to the physical properties of either of them, rather it is the number “1” itself! Of course, the symbol “1” is immaterial here, all that matters is the abstract idea that allowed this correspondence. We will never know the originator of this marvelous idea, but perhaps that does not matter. The creator is best found not in the name, but the creation itself!
Falling in Love with Mathematics
Scattered throughout human history are the stories of extraordinary gentlemen (and women) falling in love with Mathematics. Since time immemorial, these individuals have shaped the progress of Mathematics, sometimes at the cost of their lives (have you heard the story of the ancient Greek scholar and philosopher Hippasus who was allegedly drowned at sea for refuting the sacred Pythagorean belief that “natural numbers” are the fundamental building blocks of our world?), and this trend will definitely continue in one form or other. The rationale behind my conviction is quite simple.
The interplay of logic and imagination, that can rightfully be called “a song of ice and fire” and is very often found interwoven in the very fabric of Mathematics, is too elegant to be overlooked by the brightest of minds. While it is almost impossible for a more trivial mind (such as that of yours truly) to properly appreciate the beauty of the bigger picture of Mathematics, we can all enjoy the stories of the great masters, in order to taste the true flavors of the subject. Although many stalwarts have enriched Mathematics by their brilliant contributions, including our very own Indian prodigy Srinivasa Ramanujan (do watch the Hollywood movie “The man who knew infinity” if you haven’t already), one particular story involving the “Prince of Mathematics” Carl Friedrich Gauss stands out. Not in the least for its simplicity and pure elegance. Young Gauss, still in the primary classes, was once asked by his teacher to add all the whole numbers from 1 to 100, after he mischievously created a lot of disturbance in the class.
Presumably, the teacher firmly expected Gauss to spend quite some time over the problem, giving him the much-needed opportunity to continue teaching the other students without any further botheration. However, to his utter surprise, Gauss solved the problem in the blink of an eye by applying the following ingenious methodology. The genius of Gauss took no time in observing that one can pair up the given numbers in such a way that the sum of the numbers in each pair is always 101: 1 + 100 = 2 + 99 = 3 + 98 = … = 101. Since there are 50 pairs, each summing to 101, the required sum must be 101*50=5050!! Admittedly, the method sounds quite easy, once the detailed solution is spelt out. However, contrary to the perceived elementary nature of the solution, the brilliance of Gauss shines through the fact that his method essentially presents a neat formula for the sum of first “n” natural numbers, usually taught today in the theory of arithmetic progressions. Indeed, follow the grouping suggested by Gauss to obtain:
1 + 2 + 3 + … + n = n(n+1)/2.
The reader should ask herself about the case of “n” being odd, as opposed to the case of “n” being an even number, such as 100. It is widely claimed, and for good reasons, that reading about the lives of great mathematicians and the associated anecdotes allows one to look at the beauty hidden in Mathematics from a higher plane and an altogether different perspective.
The Beauty of Mathematics and the Beholder
Before we finish this article, let us once again come back to the very beginning of it for one last time. What could be the possible reasons behind so many people having a serious antipathy towards Mathematics? Mathematicians do claim emphatically that the beauty hidden in the many layers of their beloved subject is indeed unparalleled. However, let us also not forget the prophetic words coming from none other than the revered Greek philosopher Plato: Beauty lies in the eyes of the beholder. In my (undoubtedly limited) experience of interacting with the people from various cross-sections of the society, the “abstract nonsense” of Mathematics is seldom associated with beauty of any kind, especially by the category of people more interested in addressing real world problems, leading to a definite improvement in the quality of lives of average human beings.
Aren’t Physics, or, Engineering, more suited to this specific purpose? In the humble opinion of the author, of course, they are, but only with the active help of certain parts of Mathematics. Indeed, the applications of Mathematics in our daily lives are so abundant that we sometimes tend to forget it altogether quite conveniently. To paraphrase in the words of the great detective (albeit fictitious) Sherlock Holmes, “It is so overt that it is covert”. Mathematics is all around us, silently shaping almost every aspect of our lives, from balancing the cheque books to decorating the homes. When we comfortably withdraw money from the ATM, little do we realize that how our very own Indian prodigy Srinivasa Ramanujan and his prolific studies on number theory (one of the most important branches of pure Mathematics, once strongly considered to be completely devoid of any real-world application!) made it possible. Interestingly enough, Ramanujan never thought of any such applications while working on his seminal research, purely for the joy of creation.
A Little Puzzle to Storm Your Brain
Therefore, it might not be entirely pointless to suggest that there are many other results in the theoretical realm of Mathematics which are waiting patiently for some potential applications having the power to fundamentally alter our life as we know it. Having said all these, I certainly do not claim Mathematics (at least in its current form) to be free from all sorts of limitations. While it has achieved extraordinary success in describing the outer world and controlling the relevant parameters to our immense benefit, much remains to be done towards understanding the inner world, which is at least as important as the former. I, for one, have never heard of the Mathematics of Mother’s love, have you? But perhaps we will, one day! After all, it has always been the priority of Mathematics “to explore new horizons, and to boldly go to places where no one has gone before!”. To celebrate the spirit of Mathematics in my own way, and also to illustrate the power of logical deduction associated intricately with it, allow me to take the liberty of ending the present article with the following simple puzzle:
Imagine that you have arrived at a crossroads having two distinct routes, while travelling alone in a dense forest and you find two unknown persons sitting there. You also find a placard announcing that only one of them always speaks the truth, and the other one always gives a false reply. It also tells you that only one of the routes will take you to your desired destination, while grave danger awaits you on the other. You have no way of identifying the individual who speaks the truth, or the route leading you to your goal. How can you know the correct route, by asking only one question to only one of the two persons?
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