"Do not worry about your difficulties in Mathematics. I can assure you mine are still greater." ~Albert Einstein
Mr. A: (closing the book in his hands) “I wish they would write books without equations. There’s so much I’d like to learn, but every time I try, I run into equations that I simply cannot comprehend. I’ve never really liked Mathematics and equations.”
Mr. G: “Oh, I think I can relate to that. I’ve always been fascinated by the mysteries of the universe; just never could understand the equations. And there are so many in Physics. At some point I just gave up, and moved on to more accessible pursuits.”
Mr. A: “Well, I’m glad to hear I’m not the only one facing this problem. I guess it’s true what they say, misery loves company. Professor T, you certainly do not have this problem. You must think of us as a couple of illiterates?”
Prof. T: (putting down his coffee mug) “Au contraire. It is always a pleasure to talk to you Gentlemen over coffee. You tend to bring up thought provoking issues.”
Mr. A: “Professor T, I understand that your work involves a lot of Mathematics. Perhaps you could shed some light on the issue at hand? I’m not quite ready to give up yet. What must I do?”
Prof. T: (with a smile of admiration) “I admire your attitude, and I’d be more than happy to share my thoughts on the matter. Let me start by asking you this, have you had the experience of learning a foreign language?”
Mr. A: “Yes. As a matter of fact I am currently in the process of learning a language.”
Prof. T: “Well then, you might want to consider Mathematics as just another foreign language.”
Mr. A: “Hmmm … I would that I could, I just don’t see how.”
Prof. T: “Let me clarify that by taking a brief look at the process of learning any foreign language. Consider a language with a script that you are not familiar with. When you look at a sentence for the first time, it may be the case that you cannot even pick out individual characters, and you certainly cannot understand the meaning. The whole sentence appears no more than a pattern of lines on the paper. Are you with me so far?”
Mr. A: “I think I follow. There are many languages that appear to me as some incomprehensible pieces of art”
Prof. T: “Right. So how does one get from there to reading the most profound literary work in that language?”
Mr. A: “Well, lots of studying and lots of practice I guess. It takes a lot of time and hard work to reach the level at which advanced literature can be studied.”
Prof. T: “Precisely. Practice is a must. It also requires repetition. The more you use the language, the better you get at it. However, there is a step by step process involved here that we must not overlook. It all starts with learning the alphabet. We then learn how to join the alphabet to create words. Grammar dictates the rules for formulating valid phrases and sentences. Vocabulary provides us with valid words, and so on and so forth. At a certain point we reach the stage where we can understand the everyday use of language. But we continue learning, until we reach the skill level required to understand more complex literature; both poetry and prose. To summarize, we start by working on the basics i.e. by building a strong foundation. And with enough time and effort, master the language. Agreed?”
Mr. A: “Agreed.”
Prof. T: “So if learning any language entails a tedious and lengthy process, why should learning Mathematics be any different?”
Mr. A: “But Professor T, I still cannot quite see Mathematics as a language.”
Prof. T: “Mathematics is the language of abstractions. It is precise and unambiguous. It helps us formulate abstract structures and their properties. Numbers, operators, function, theorems, proofs etc.; these are all what constitutes the elements and style of the language that is Mathematics. You know, the most fascinating aspect of Mathematics is the portability of the abstract structures. It is due to this portability that the same structures can be used across disparate scientific disciplines.”
Mr. A: “But all I see in Mathematics is numbers and computations. What is this abstraction that you speak of?”
Prof. T: “You raise an important point. In my opinion, it is a common mistake to consider the whole of Mathematics as Arithmetic. The computations and calculations are essentially Arithmetic. As you’ve pointed out, this is what we come across most of the time. However, this is not all. Pure Mathematics is about abstract structures and their properties. A Mathematician creates (or discovers) abstract structures that have no physical reality. She proves theorems about the properties associated with these structures. Then, at a later point, someone finds applications of these structures. The numbers and calculations only enter the picture when computations are performed for a specific (concrete) problem.”
Mr. A: “I see. I’ve heard people talk about mathematical beauty. What is that all about? It cannot be about number crunching, right?”
Prof. T: “There’s a certain beauty in every language. However, what we often refer to as beauty in literature is that of the piece; a poem, a novel etc. In Mathematics, a proof can be like a beautiful poem. The step by step process of a proof may be mundane, or elegant. It depends on a specific proof, just like one poem may be considered ordinary, while the other a work of genius. In order to appreciate the beauty of an elegant proof, one must be well versed in the language of Mathematics. It is hard to appreciate (or like) something that one does not understand.”
Mr. A: “Wow! this is all very interesting. I’ve never thought of Mathematics in this manner.”
(Prof. C arrives at the coffee house and joins the others. Mr. A mentions what the conversation is about.)
Mr. A: (looking at Mr. G, who appears to be lost in contemplation) “You’re awfully quite there Mr. G. Bored by our discussion perhaps?”
Mr. G: “No, not at all. My apologies if it appears that way. I have been listening intently, yet I fail to see how viewing Mathematics as a language helps us with the apparently incomprehensible equations that we encounter in books. Would someone be so kind as to enlighten me on the matter?”
Mr. A: “Good point. Professor T?”
Prof. T: “The point I wanted to make is this: Mathematics is just like any other language; one must start with the basics. It takes time and effort to familiarize one with the syntax and semantics, the grammar and vocabulary. It may be a slow and tedious process, but eventually one reaches the level where seemingly complex equations do not pose any significant challenge. Also, the more one works with pure Mathematics, the more one appreciates the elegance and beauty of abstractions. A lot of us never get to experience what pure Mathematics is about. And perhaps that is the reason why it all seems so inaccessible.”
Mr. G: “I think I get your point. It is indeed encouraging to see Mathematics as just another language. Nevertheless, I feel that there is something about Mathematics that makes it a lot more challenging than just another foreign language. Sometimes I find it hard to comprehend even the basic concepts.”
Prof. T: “Ah, you’ve stumbled upon yet another characteristic of language that we often overlook. In certain languages, there are phrases, idioms, proverbs etc. that cannot be translated into other languages. These often have contextual relations to the culture associated with the language. Unless one understands the culture, such language constructs hardly make any sense at all. A major difficulty in Mathematics stems from the way of thinking it requires. It makes us step out of a certain comfort zone, and think differently. Consider the concept of visualization. We often find it easy to understand structures and objects that we can visualize. In Mathematics, this isn’t always possible. E.g. due to our limited sensory perception, we can only visualize structures in three dimensions. Abstract mathematical structures are often explored in infinite dimensional space. To be able to understand the concept of infinite dimensional space requires a new way of thinking. One must develop it over time.”
Mr. A: “So what should I do in order to pursue my passion to understand and unravel the mysteries of the universe?”
Prof. C: “Professor T, if you don’t mind, may I respond to Mr. A’s question?”
Prof. T: “Oh please. Go ahead.”
Prof. C: “First and foremost, take a deep breath. Do not let your passion consume you. Do not let it transform into frustration. You must find a way to channel it towards a productive outlet. Mathematics is a very broad field of study. It might be wise to start by identifying the branch of Mathematics that is the most important to you at this point. You must then build the vocabulary and learn the rules that govern your target field. Once you are fluent in one branch of Mathematics, you’ll find it much easier to learn the others. And so will begin your journey into the world of the abstract. It is a long road, I know. But trust me, it is definitely worth the time and effort. Persistence and perseverance are the key ingredients. It will open new doors for you, and make it possible for you to understand the workings of the universe; from fundamental particles to life, and from this world to the cosmos. It will be a liberating experience.”
Mr. A: “Oh, this is wonderful. I can already feel the power of Mathematics. I am so glad we had this conversation today. Professor T, Professor C, I must thank you for providing me with the insight and encouragement I needed at this point. I’ll start right away.”
Mr. G: “The spirit of Mathematics has been rekindled in me. Mr. A, I think I’ll join you. We share the interest in understanding the cosmos. Perhaps it would be a good idea to collaborate?”
Mr. A: “Certainly. Let’s head to the library and see where we must begin.”
"Yes, we have to divide up our time like that, between our politics and our equations. But to me our equations are far more important, for politics are only a matter of present concern. A mathematical equation stands forever." ~Albert Einstein
