A few days ago, I gave a lecture at Haberdashers Askes Crayford Academy in United Kingdom about mathematics. The audience was a hall full of young students aged between 11 and 13 years. The purpose of the lecture was to show the students how relevant mathematics is in our daily lives, whether at the work place or at home. Below is the message I wanted to get across to them. At the start of the lecture, I asked them what mathematics meant to them. As expected, one student answered that mathematics is about numbers. He has summarised what the majority of people think of mathematics. Mathematics is this and much more. And so, with that premise clearly set, I began my lecture. Time being a strict constraint, I was not able to elaborate on the topic but here I can expand more on this.
Mathematics, first and foremost, is a language. We might not see it that way, just as we understand that English or French or Urdu is a language. When we speak of languages then we immediately think of verbal languages like the ones I have mentioned. There are different sorts of languages: programming languages such as JAVA or Python; there are non-verbal ones such as Body Language or Sign Language; and there are more abstract ones such as mathematics. Language is a tool for communication. It is about getting a message across. So how does mathematics fit into this definition? And what message or messages do we deliver using mathematics?
Physics, and science in general, is the study of natural phenomena. We observe the world around us, we make measurements, we analyse how things are and how they behave, we make predictions about how things will evolve and we find out why things are the way they are. In all of this, we need to be able to quantify what we are doing. We need to be able to assign some value to what we are observing and measuring. More importantly, we need to be able to make sense of what it is we are studying. That which allows us to make sense of the world, at a very fundamental level, is mathematics. If the alphabets are the basis of a spoken language then numbers are the basis of that language we call mathematics. This is why, it seems, that the first response to the question ‘What is maths?’ is inevitably ‘numbers’. This is how we perceive it, this is how it is first taught.
So what is it that we are trying to communicate using this language called mathematics? If the ‘alphabets’ of this language are what we call numbers then this gives us a clue. There is a whole branch of mathematics that is dedicated to the study of numbers and it’s not just simple arithmetics that I’m talking about here. Before we can even compute what 1 plus 1 is, we have to come up with a way to represent those numbers. Hence where the numerical symbols come into play. We are familiar with these symbols: 1, 2 and 3. We know what they represent. They each represent a certain quantity. A quantity of what? Just a quantity of stuff, of things, of anything we can assign a value to. I look at my left hand and what do I see? I’ve got a number of appendages. If I can somehow link these appendages to something else such as a bunch of bananas on my kitchen table then I can find a way to represent those bananas using these appendages. Therefore, by raising only shortest of these appendages (in this case my little finger) then I can represent the same number of bananas as there are raised fingers. This very basic way of representing things or, more precisely, a number of things, using some symbol (here, my fingers) is at the very root of mathematics. It is about having this insight, this abstraction of thought that one kind of thing (bananas) can be represented by something else (fingers). And, even more importantly, the number of things represented by that something else is intended to be exactly the same as the number of thing in question. When we say that mathematics deals with the abstract, we don’t necessarily mean that it is about stuff which has nothing to do with real things. Abstract, here, doesn’t imply something that exists only as a concept. Abstract, here, means to abridge or to shorten. It shortens the connection between two seemingly different things. It links two things, be they real stuff or some metaphysical concept. In this case, we are linking together the fact that the one finger I’ve raised is exactly the number of banana I’m asking for. Finger is being linked to banana. More appropriately, the number of fingers is being linked to the number of bananas. There is a good deal of mathematics that has to do with the abstract in the sense of the conceptual and non-real stuff. But this kind of mathematics comes much later in its development. Yet, ultimately, it comes full circle. But we have yet to realise that. Right now, it is about understanding what is the root of mathematics and what it means.
So, numbers. We can represent them using our fingers to enumerate how many bananas we are asking for or we can use some other way to represent that number. Hence, we come up with those symbols such as 1, 2, 3 and so on. These symbols have been filtered down through the millennia from the Eastern to the Western world. The Roman symbols for numbers, I, II, III and so on, are not used for counting anymore. What we use are descendants of Arabic and Indian symbols for numbers: 1, 2, 3 et cetera. By no means where these two collection of symbols the only ways to represent numbers. We have to go way back to some ten thousand years ago in the Mesopotamia. I will not elaborate on this as to recount the history of numbers requires in itself another series of lectures. It suffices to say, however, that mathematics began with the need to count things and therefore with a way to represent that which needs counting. Numbers are at the base of mathematics. As I have mentioned before, the story does not end here. There are other foundations that we need to explore such as geometry and algebra. In short, mathematics is this language we use to represent things that have to do with counting or quantity.
Like any other language, the more one is exposed to it, the more one uses it, the more fluent one becomes in that language. Mathematics is not hard. I do not agree with that lame excuse people use to justify why they do not like or understand mathematics. At some point in their schooling they lost touch with what mathematics is about and thus lost interest in learning that language. Imagine if French was taught to a native Hindi speaker. They have different alphabets, different grammar, different syntax and so forth. It will require some dedication, that is true, to learn the basics of the new language. It will require practice and discipline. But that is not to say that this is all the language is about. Mere arrangement of letters and words and sentences. Mere recitation and memorisation of paragraphs and chapters. Mere rote learning from books and regurgitation from tomes. This is not how one learns French or any other language for that matter. There is more depth to French that agglomeration of alphabets. Reducing mathematics to numbers is equivalent to saying that a poem is just a collection of words. One has to have a sense of aesthetics to appreciate a poem. Mathematics is no different. One has to have a sense of aesthetics to appreciate the depth and breadth of mathematics. This sense of aesthetics is in all of us and it has to be nurtured and cultivated and pruned. One simply does not quote Voltaire or Shakespeare without appreciating what is being said, even at the most elemental level. It is not writing an essay or discourse on the meaning of ‘To be or not to be” that is required. But, at the very minimum, appreciate the beauty and elegance of that statement. When one quotes a beautiful equation from mathematical theorem then what is expected is an appreciation of the gravitas of the statement being made by the equation.
This appreciation of beauty does not come to everyone. We might not immediately like a piece of art but this sense of aesthetics can be cultivated in anyone. We simply require the will and means to do so. Without this will to appreciate beauty, we cannot appreciate art. We cannot be forced to accept that the Mona Lisa painting is the most mesmerising painting in the history of humankind. We have a sense of judgement, of subjectivity, we have an opinion. Some might say that the Mona Lisa is not as great as it is deemed to be. Others might argue that Mozart is a better composer than Beethoven. Again, we have it in us to judge a piece of art. The ability to discern good music from bad comes naturally to all of us regardless of what type of music we like. We might not be an actor but we appreciate the good acting skills of a performer when we see one. The point I am trying to make here is that mathematics is like any other art form. We have it in us to judge whether one piece of mathematics is beautiful or not. We have it in us to appreciate the elegance of an equation. We have the capacity to judge the aesthetics of a mathematical theorem. Mathematics is an art. It is not as subjective as, say, photography but it is an art nevertheless. Why, therefore, does this art form seem so remote and difficult to connect with for most people? It should not be like this. We have to teach students how to appreciate this language, this art just as we teach them to appreciate the genius writing of Shakespeare or the beautiful compositions of Bach.
There is only so much beauty one can extract from numbers and simple arithmetics. Numbers, as I have mentioned earlier, were invented for the purposes of counting and keeping records – not unlike what accountants do today. Symbols had to be devised to represent those numbers. These symbols might not carry any beauty in themselves but they helped simplify what would otherwise be a difficult task of keeping records of transactions and measurements. Their elegance lies in the fact that they simplified complicated systems. Let’s take an example which, I think, would fit in the early years of when humankind came up with agriculture. The nomadic lifestyle was no longer required when people were able to settle in one place and work the land and grow their food. The need to roam about in search of food gradually faded into rearing cattle and cultivating crops. The basics of commerce also take roots in this process of agriculture. Exchanging one source of food for another, for example, is the early bartering system which would evolve in the more familiar transactions of circulating bank notes. So, to keep track of one’s cattle and how much to exchange for a bale of wheat was essential in ensuring one is not short of supplies. If one sheep would be traded for three jars of oil, for instance, then one has to be able to keep track of such transactions in an efficient manner. This is where using numbers and symbols representing those numbers were useful. The basics of arithmetics, that is, adding and subtracting numbers, most likely came about with the need to keep track of simple bartering transactions. Is there beauty in such mathematics? To see how this simple system of counting can be beautiful, we have to think of any alternative system to arithmetics. The fact that any other means of keeping a record of transaction that does not involve basic arithmetic does not (or no longer) exist means that the most efficient and, hence, elegant way of solving accountancy problems survived over the millennia. It’s something that just works! As simple as that. Arithmetics is, excuse the pun, as easy as 1, 2, 3. For the very fact that it is the most efficient and simple solution, I deem it to be elegant. If you do not believe me on this then I challenge you to come up with a better system than arithmetics and numbers where counting is involved.
Just as a hammer is a tool that simplifies nailing things to other things, mathematics is a tool that simplifies a given problem. Now, not all problems can be solved using mathematics but that does not mean that mathematics is useless or unnecessary. It has its applications, albeit limited. Mathematics is not a physical tool, unlike a hammer. But it helps fix things. Also like the hammer, it helps break things down. And by that I mean it helps us to analyse things. Analysis is simply the process of breaking down something complicated into the constituent bits and pieces which are manageable and comprehensible. Mathematical analysis is the process of breaking down a system using mathematical logic and reasoning so that we can better understand what problems we are faced with. Would it cost less to travel further to buy cheaper petrol? Will there be enough tiles to cover a bathroom wall? How much flour would you need if you’re using half as much butter in baking your cake? All these simple problems are easily solved using basic arithmetics. Yet because they are so easily solved using the simplest of mathematics, we often take for granted how useful and efficient a tool mathematics is. And it is for this very reason that I find mathematics to be a most elegant language and tool.
The second point I raised was about an emergent property of mathematics. You see, the more you familiarise yourself with that language and the more you use the tool to solve various problems, the more you observe certain patterns emerging. Encountering the same type of problem over and over again makes you realise that they must come from something more fundamental, that they must be related in some fundamental way. Hence, rather than trying to find a different solution for each and every similar problem, a mathematician seeks the most fundamental of solution that solves every problem. Again, this economic use of resources and means to resolve something shows how efficient and elegant mathematics is as a tool. On top of being a language, mathematics is the study of patterns. This takes us back to the origins of geometry. Geometry is the study of shapes and their relationship. From the simplest of lines to triangles and squares to the most elaborate multi-dimensional objects, geometry allows us to make sense of the patterns and structure around us. Geometry began with the Greeks; that is not to say that before them shapes did not exist. Ever since humankind has taken the metaphorical pen to the metaphorical paper, ever since the human species began recording their thoughts and dreams on a permanent surface, we drew the most basic of shapes: the lines and curves. It is likely that the origins of writing and language evolved side by side with the origins of drawing and painting. Mathematics is not just about concepts but also about symbols, that is, about how you represent those concepts. Whether we use numbers, letters or shapes or any other kind of symbol, it is about representing the abstract into something more visual and tangible.
Once patterns are seen forming and emerging, they are put under scrutiny. Sometimes, you have patterns about patterns. Altogether, thus, it is about discovering the underlying structure to how things are and why they are they are the way they are. Why should a flower have 13 petals arranged in a particular way? Why should the formation of clouds or crystals adopt a given shape? How do such and such shapes fit together to produce a given pattern? What is the shape of the universe? All of these questions have their answers anchored in geometry. It is only after digging deeper into the fundamental structure and pattern of the universe that we appreciate just how elegant things are. We might have an eye for spotting a nice pattern or we might see heaven in a flower but to truly appreciate how beautiful things are we have to appreciate the mathematics behind them. The spots on a leopard or the stripes on a tiger or the shape of a shell is all due to fundamental laws of mathematics. The laws of mathematics that govern the formation of patterns play an important role in determining whether spots or stripes will be formed on a zebra, for example. One can certainly get by appreciating the arts and aesthetics that abound our world but, the point being made here is, one will have an even deeper appreciation of what it is that underlies such beauty and elegance in the universe if one is aware of the mathematical foundations of our world.
The link between geometry, the study of shapes, and that of arithmetics, the study of numbers, were initially thought to be disjointed. As mathematicians ploughed further into the different fields of mathematics, the seemingly disconnected topics turned out to be related. Soon, what used to be a purely a problem in geometry (such as determining where two lines intersect each other), algebra proved to be an even more intuitive and reliable tool to solve such a problem. The mathematics behind solving simultaneous equations is also relevant to finding the point where two lines cross each other. Algebra is another branch of mathematics that is as fundamental and important as arithmetics. But unlike arithmetics, it is more versatile. And it versatility arises because it has the capacity to deal with the abstract rather than just with what is concrete. Numbers refer to things we can deem to be countable and tangible. Algebra relates to things which might not be limited to numbers and fixed quantities. For example, if one has to determine what volume V of liquid a cylindrical container can hold, one has to calculate the following: A x H, where A is the area of the base of the cylinder and H is its height. V = A x H is a simple mathematical formula that can be used to determine the volume or capacity of the container. Regardless of whether the height is 10 cm or 30 cm, using this formula will give you the volume. Alternatively, if you know the volume and the area then you can determine the height. In this case, H = V/A. Algebra is about finding general rules to how things work. And those rules are written as formulae. A formula, therefore, is a rule that one follows to determine a certain quantity. It assembles related quantities in one neat group. Of course, algebra has many more flavours than just solving formulae. At its peak, it ties in with the rest of mathematics in that it shows the underlying symmetry that binds different topics of mathematics together. It brings numbers and patterns and shapes together into a more powerful tool while reinforcing their individual importance. The mathematics used to solve quadratic equations is the mathematics used for drawing parabolas and is also the mathematics used to plotting flight paths to the Moon. If this alone is not a sign of how powerful mathematics can be then I don’t know how else to convince you of its magnificence.
Mathematics is not limited to what is found in textbooks. It is not just an academic transit, during our sojourn at school, that will gradually fade like an old holiday photograph as we sail through life’s journey. It is, in fact, the vehicle that takes us through the peripeties of our life. It is the foundation upon which a modern society is built. None of the major advancements in science, engineering, technology, medicine, architecture, design or even art would have been possible had it not been for progress in mathematics.
It is easy to take for granted how useful mathematics is in our everyday life. Whether it is about statistics or arithmetics or calculus, we depend on mathematics on so many different levels. A lot of the mathematics happens behind the scenes and this is probably why we do not realise how important it is to society. Even if we do come across some basic mathematics, whether in the form of calculating the bill at a restaurant or interpreting the statistics in the sports section of the newspaper, we still take it for granted by assuming that this is the most that mathematics will ever be relevant to us. Or, wrongly, we think that any other form of mathematics other than basic addition and subtraction is too complicated for us and better left to professors and experts. Before one can appreciate mathematics for its elegance and aesthetics, one has first to be able to recognise where and how the different forms of mathematics are being used. That way, we will be made aware of how useful it is and how relevant it is to us. Only then would we be able to appreciate it as an art.
So where is mathematics used? Let’s start with us. The clothes we wear come from factories which are operated, in parts, by machines and computers. These machines and computers had to be designed and engineered. As soon as you touch on anything related to engineering and technology, you instantly rely on their mathematical foundations. Remove that foundation and everything that sits comfortably above it collapses, taking us as a society down with it all. Same goes for the food we eat. There is a good deal of chemistry and biology and technology involved in the engineering, cultivation and distribution of food products. The vaccines we’ve been inoculated with to protect us from so many diseases and defects, have all been developed and tested thanks to progress in biology. Once again, these rely on the mathematics used in biology to ensure the results obtained from years of medical research are sensible and reliable. Computers are undeniably the epitome of how mathematics runs the modern world. The spread of mathematics is worldwide but also across so many fields of science and technology. To enumerate all of them would take a lot time. It is enough to say, however, that we have a lifetime to find out for ourselves how mathematics is a useful tool for us. Only then would we find out how important it is for all of humankind.
The way I see it is as follows: mathematics is like poetry; to really appreciate its beauty, we have to be familiar with that language and, at the same time, nurture that sense of aesthetics about this art form. Most people who speak and use English on a regular basis must surely have heard of Shakespeare and perhaps even read at least one of his works. If the same could be done for mathematics it would be a step towards improving awareness about this great subject. There are numerous mathematicians that have made tremendous contributions to our world over the millennia. Names such as Pythagoras, Euclid and Newton should not remain bounded in old textbooks and libraries. Their works should become common knowledge and their legacy should be made more relevant to all of us. This kind of endeavour starts right from the early years at school. Students should not be taught mathematics simply as a means to obtaining a good certificate. Students should be taught mathematics as an essential tool for appreciating the value of nature and of the world around us. We should aim to make every student become a mathematician. By this I mean that we should cultivate their sense of aesthetics. For mathematics is none other than the most elegant of all arts, the most fundamental of all sciences and the most trustworthy of all truths.