The Abridged Ajani

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Mathematics as a language sounds strange, when the only thing we can recall whenever someone asks of us to “do Maths” is to take out a piece of paper and start filling it up with numbers and symbols unable to be understood to the untrained mind. This should be merely a technicality of speech. The symbols and numbers we write form mathematical language, and it is to the field of Mathematics how the English language should be to English literature.

We use English not simply to express ourselves as Shakespeare does, obviously, but perhaps to describe an activity, converting the scene from pictures to words. This is just like how mathematical language is to physics. We also use English to categorize people and things for easier understanding, which reminds me of the equations of Mathematics.

How did Mathematics, this parallel language, evolve alongside Man, even as we were changing the way we speak to each other with our natural languages evolving? How did this language unify from the many ways of writing into the standard Arabic numeric labels, and how has it from there been able to give birth to the many other extensions of Mathematics even we humans can sometimes not understand, like differentiation, integration, complex numbers, chaos theory etc.

Mathematics today, because of its extensions on arithmetic into integration and the lot, has become almost like a pseudo-dead language. It exhibits virtually all of the characteristics a “dead language” like Latin, with only the best-specialized academics able to understand and speak it “like a native”.

At this point of time, I am slightly reminded of Bertrand Russell’s Principia Mathematica, Russell seems to work backwards in his evolution however, logic must have been a formal afterthought of the first quantifying systems. Not to say, however, that logic is a rational product of Mathematics, but perhaps, we should start from the first empirical count.

One, two, three… I believe that the first counting system simply employed, or coined, new words for the concept of the numbers, that would eventually become 1, 2, 3… Out of convenience, and perhaps practice, the earliest ways of writing numbers was actually in tally marks or similar, I would suppose. For the sake of writing, I shall refer to numbers and symbols, of all cultures and time, with the conventional Arabic numerals “1, 2, 3″… and the conventional infix “+, -, /, *” for simplicity and understanding, but it is to be noted that these notations actually vary from culture to culture, era to era.

One also cannot forget how many different fundamental tenets of Mathematics there was. The base-10 system we use today could have been base-60 for the ancient Babylonians.Despite all the differences, Mathematics remained a language contiguous, with all the differences forming the basis for “mathematical dialects”. These dialects were, at the core, unified by a common system of logic, which began with counting. Placing two leaves together made one leaf and one leaf add up to two. The observable hence became our basis for what we today call Logic - though Logic, like its cousin Mathematics, has went through its fair share of evolution through time, but has successfully broken into different languages of itself, not necessarily unified. (Dual-valued logic may be “logical”, but “fuzzy logic” and multi-valued systems show as much promise as the former in answering problems.)

As early as the Babylonians, however, humans would already have had the mathematical fields of algebra, geometry and even trigonometry as part of the standard mathematics canon. At this point of time, one can still accept that mathematics was an empirical subject. Why the change from counting to equations? Maybe the Babylonians were counting the average yield of crops per square metre of land for harvest purposes - there was still pretty much an application of Mathematics to the observable world.

When the concept of ‘perfect numbers’ and ‘prime numbers’ crept into Mathematics, it became clear that Mathematics started to become somebody’s playground, much like English is to word pun crafters. Such numbers were conceptualized in both Egypt and Greece, but popularly it is attributed to the work of Greeks. Advanced civilizations such as Egypt and Greece in their days must have had much time to ponder upon such intricacies of Mathematics - “knowledge for knowledge’s sake”. In fact, Philosophy and Mathematics for the Greeks like Pythagoras were virtually complementary if not the same. The search for the Absolute led many men into the realm of abstract Mathematics, while some others preferred material Physics while there was still a great deal unknown about the observable world. I believe this was when Mathematics started to accelerate towards the abstract. In fact, the Greeks had by then even thought of ‘integration’ in the form of the ‘method of exhaustion’, something modern mathematicians call the ‘precursor of integration’.

Not forgetting contributions from the East, Indian mathematicians experimented with permutations and combinations, logarithms and the like. The ‘inventors’ of mathematical fields, I daresay, must have had the ultimate creativity that is no longer associated with mathematicians today. Completely new radical methods of solution are extinct, while modern mathematicians merely expand the knowledge of a given field of Mathematics. There are little “new” fields of Mathematics to be pioneered anymore - would anyone creative enough dare do it?

The Chinese synthesized, or otherwise must have independently worked out, shorter and more-powerful methods from existing knowledge to really beautify Mathematics a lot. Algebra was expanded to include binomial theorems and matrices. Ways to shorten long calculations made Mathematics elegant - this would have a lot of influence on how Mathematics is viewed today despite its rationalist nature now.

The Arabs also dealt with non-Euclidean geometry. This brought Mathematics to its esoteric level of today, in a sense, as non-Euclidean shapes had not even real-world foundations to begin with. As mathematicians hurried to apply existing concepts to non-Euclidean shapes, many concepts were either re-written or expanded upon to encompass such virtual ideas. In fact, strangely, Mathematics, with its open-ness to the virtual and abstract henceforth, was to become more and more suited to be the official language of contemporary Physics, where truth actually can defy common sense. By now, the language of Mathematics was now huge with an extensive vocabulary and as Arabic numbering spread over the world, Mathematics broke down language barriers. With its acceptance to discuss abstract, non-existent shapes, Mathematics was no longer a field of knowledge that was held by its own constraints and scope, but was already a fully-fledged language, to be employed to describe anything it could. Mathematics therefore became a language of itself.

Little revolution in the history of Mathematics was to occur till the times of the Renaissance and the subsequent Enlightenment. With an intellectual keenness, people like Galileo and Kepler looked to the skies and began to seriously use mathematical language to describe physical phenomena - hence, Mathematics functioned as the language of Physics. The power of Mathematics over English, most importantly, was in its precision and extendability. In computing terms, one could say that Mathematics was a more ‘powerful’ language than English, specific to the field of Physics. Equations became as much a description as a prediction - the language of Mathematics proved to be infinitely more powerful than the natural languages of Man. It was neater, more precise and it mapped the Universe excellently to the eyes of Man. Without the conveniences of Mathematics, one could never have seen how mathematical the Universe could seem - a later starting point for the anthropic principle, which was to condemn science as an inaccurate measure of objective truth.

I like to think that Mathematics and mathematical language influence one another. English literature often coined and/or re-defined several words, and the language was expanded, to be later used by others. This is akin to Mathematics. The birth of Newtonian and Leibniz calculus was very-much a synthesis of mathematical knowledge. Building upon geometry, algebra and many other mathematical fields, calculus if personified would be a bard of mathematics.

Why do I wonder about it anyway?

A year ago, I had decided to continue with an education in Physics for my GCE ‘A’ Levels. That was after I was greatly inspired by the book “Parallel Worlds” by Michio Kaku. I must say that I was very much ‘misled’ by the book as I misunderstood a formal Physics education. Instead of the wormholes and relativity theories I had been reading about in plain English, I was immediately overwhelmed by a multitude of mathematics and rational proofs as I started to ponder about the role of Mathematics in the Sciences.

A certain Mr Donovan Lee was invited to my college recently to give a talk entitled ‘Exploring the Mathematical Wormhole’, which is well, an introductory, mathematical, rather than plain English, exploration of wormholes. In it he declared that ‘mathematicians discovered relativity before physicists‘. After the talk I was intrigued about the meaning behind the single sentence, much more than I was with everything else he had said for the hour-long lecture. I failed to completely understand the mathematics behind wormholes, even as I could explain it, with limited success, using some English and some imagination. Discussing it briefly with my friends, one suggested that if I wanted to understand the bizarre mathematics behind it, I should use Physics to reason.

Now, I recall vaguely, that String Theory, the best answer so far to the much fabled Theory of Everything for Physics, actually had it roots in pure, abstract Mathematics. The building blocks of the Universe, too small and impossible to observe, were actually the shape of strings, fundamentally. Now, if it was too ‘impossible’ to empirically observe, how did they arrive at this conclusion then? There I realized - it was mathematics. How should I use Physics to reason, if it was Mathematics that had reasoned Physics so well beforehand?

That, is my main concern. Is it valid to call Mathematics a language of Physics? Even so, would it be okay to use the language to figure out missing pieces of the puzzle of Physics? With virtually nothing outside of Physics with such case I know of, it is a question I cannot properly answer. Is String Theory science? In the most accurate wording, String Theory is correct for science because it was based upon Science, though built using Mathematics. Perhaps it is this kind of strange “back-copying” that really proves the power of mathematical language.

Then, if so, is it possible to invent a language with the power and abstraction of Mathematics? Mathematics started out from counting, could we invent a separate language from the simplest of life? A necessity drives the invention of a language. I cannot name any examples for which we could even think of wanting a new language for - I do not think that the first “counters” could have imagined that eventually what they did would evolve into a system to solve the mysteries of the Universe.

There is a reductionist concept which states that psychology can be reduced to biology; biology to chemistry; chemistry to physics. If Mathematics can not just describe Physics, but in fact form a subset of it in answers, could physics be reduced too to mathematics? It is scary to think that the ancient Greeks can be right, after all. Yet we are left with a paradox - what is Mathematics? Are the numbers the Truth? Is the later calculus Truth? Mathematics is actually not as constant a field of knowledge as it is absolute - what is Mathematics? The mathematical language can still evolve, can it not? How can we ever apply abstract Mathematics to our existence? Maybe we could, if we expand it reverse of the reductionist argument, till we finally meet back psychology. Lost is what we initially came to prove, in the end.

Mathematics, an abstraction that is so beautiful, but with it so esoteric, sadly.

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