Math

October 3, 2014

A Brief Note on Birds

The essence of Mathematics resides in its freedom
— George Cantor

There seems to be a tighter bond between math and poetry than between man and the mind in his head. Well known poets who have long since dead such as John Keats when he wrote Lamia, or Edgar Allan Poe when he worked on Sonnet — To Science, or Walt Whitman when crafting When I heard the learn’d astronomer, do not seem to have recognized this tight bond. The same goes for those who are living and think that math and poetry are two things that have developed at opposite poles, and because of this cancel each other out. A slightly better view, even though it is still shallow, will say that math and poetry, or literature, are not enemies, but these two things are not on the same level, and it is literature that is superior to, that is more amazing than, math. This second view can also be felt in the fat thick novel of the most famous Japanese contemporary writer, Haruki Murakami.

 1Q84 1

In the first part of his novel IQ84 whose launch was greeted with much fanfare in Europe and America, Murakami wrote the following on mathematics and literature:

Where mathematics was a magnificent imaginary building, the world of story as represented by Dickens was like a deep, magical forest for Tengo. When mathematics stretched infinitely upward toward the heavens, the forest spread out beneath his gaze in silence, its dark, sturdy roots stretching deep into the earth. In the forest there were no maps, no numbered doorways…. Tengo began deliberately to put some distance between himself and the world of mathematics, and instead the forest of story began to exert a stronger pull on his heart… Someday he might be able to decipher the spell. That possibility would gently warm his heart from within.

In the previous paragraph in the same section, Tengo, the main male protagonist, had already affirmed that:

Math is like water. It has a lot of difficult theories, of course, but its basic logic is very simple. Just as water flows from high to low over the shortest possible distance, figures can only flow in one direction. You just have to keep your eye on them for the route to reveal itself. That’s all it takes. You don’t have to do a thing. Just concentrate your attention and keep your eyes open, and the figures make everything clear to you. In this whole, wide world, the only thing that treats me so kindly is math.

Those who are already quite familiar with the world of math may possibly knit their brows when reading Tengo’s confession. Although there is some truth to Tengo’s statement on mathematics being similar to water, mathematics is far more seductive, more diverse as well as more unsettling than what was understood by Tengo.

If the subject being discussed is the world of classical arithmetic, we can indeed agree with Tengo’s innocent sentence. Or this sentence: What do I like about math? When I’ve got figures in front of me, it relaxes me. Kind of like, everything fits where it belongs. In an area where mathematics has developed from Ancient Egyptian and Mesopotamian times up to 14th century Arabia, mathematics would indeed resemble a clear lake at dusk after planting season had ended and the blades of the paddy leaves and wheat were starting to grow and extend. A mathematical equation perhaps resembles water in a communicating vessel in a classroom or a network of small fountains in a temple where movement of cool translucent objects can easily be followed calmly, and no longer leave a residue of unresolved dark matters.

But in one’s experience, playing with mathematics is not just like playing in calm water, which soothes and acts as a balm for boredom. The effect of playing with mathematical numbers and symbols is far more intoxicating than that. Playing with mathematics is more like dancing in a trance-like state, in a transparent world, or, to be more accurate, in a universe where the dimensions of time and space evaporate. A mathematical problem that is known and clearly formulated, will lead us also to knowing the solution even before that solution eventually comes up. And there is only one solution, there can’t be others. On the other hand, if we know a solution, we can guess accurately its origin, even when its origin was lost in time. Mathematics, which transcends space and time, also frees itself from the background of the intelligence that worked on it.

The universality of mathematics that can make us cognitively go back and forth in the stream of time, free to traverse anywhere in space, and feel like we can converse with a variety of forms of intelligence that are more or less equal, naturally has a great influence. We become the center that manipulates and controls all similarities to our liking, and that gives an extraordinary sensation. At the peak of mathematical ecstasy that could continue for months, the Earth can feel as if it is shrinking, gravity thinning, and we step forward with weightless legs such as on the surface of the Moon. With a slight smile on our lips and a bright glow in our eyes, we step lightly into a universe where space sublimes and the past and the future seem to be in our grip. It is absolutely natural and sensible for a mathematician who was also the greatest scientist of Classical Greece, Archimedes of Syracuse, to say: give me a resting place, and I shall move the world.

Here mathematics does resemble poetry that soars like thunder and birds: they fly to blow up the light while ignoring the boundaries. If poetry hacks ─ and with that ─ expands the horizons of language, mathematics overcomes the dimension of space and time, and with it encapsulates the universe. The physical universe always presupposes space and time, while it is not always the case with the mathematical universe. In the game that ignores boundaries, the mathematical geniuses can seem to be crazy. If the smart and talented, to borrow Arthur Schopenhauer, hits a target no one else can hit, the genius hits a target no one else can see. And frequently, these geniuses pursue a target that even they themselves had never seen before, and that leads them further and further astray into the future. And when they return down to earth in their lifetime, they seem to return bringing experiences and a world view that originated 100 years too early. And that could mean that they could insert a premature disaster into the world that comes 100 years earlier.

That is what happened around the dawn of the 19th century, when a number of mathematicians started rebuilding mathematics on a basis that was far stronger than just common sense — on the basis of a really tight and rigor logic. Without having one manuscript of a provoking manifesto, they launched a movement that freed mathematics from the calculation to directly serve the needs of science and technology, especially war and construction needs, and work to build mathematics immersing itself in the freedom to create new structures, new language, a new view of the universe. Mathematics that is indifferent to the world and does not wish to have a reference to physical reality is only interested in answering the formal challenges and needs that arise from mathematics itself.

Before the advent of the 19th century, mathematicians could be described as relying on their trained common sense and intuition, and on visualizing their thoughts in mechanical methods and realistic geometry. Geometry that for thousands of years developed only in flat surfaces, started to be applied in curved surfaces, with more dimensions than just the 4 dimensions that we are familiar with. Algebra also soared in growth. Classical algebra became only one among several higher level types of algebra, where paragraphs of traditional symbols were replaced by single character which was then played according to an odd and non commutative algebraic law, where a x b was not itself the same as b x a.

Mathematics that search for these challenging possibilities indeed become increasingly abstract and farther and farther away from everyday experiences, and increasingly unclear as to what it is all for. Unconcerned about this, a number of mathematicians, such as G.E. Hardy, actually praised this as a virtue. These people believed that mathematics would indeed be its most splendid the less it could be used for mankind. The less useful mathematics was, the higher its value was. Mathematics that was beautiful and useless was of more value than mathematics that was useful and ugly. However, as long as logical rigorous and formal beauty were respected, no matter how high mathematics could fly beyond boundaries, it was only a matter of time before the thunderbird would be transformed into a stork flying low, pulled by gravity, and eventually come to perch on the rice field of practical application. On the other hand, this dislike of a practical usefulness can encourage the mathematicians to be more passionate about creating mathematics that as much as possible is useless, not caring if this type of mathematics cannot be measured for its scientific value; not even considered wrong. (In Edward Frenkel and Reine Graves’ short movie, Rites of Love and Math, the mathematician even tries to hide his math formula from the world.)

Not all mathematicians in the second half of the 19th century and in the first half of the 20th century could immediately accept the strictness and exploration of mathematics. There were even those who thought that this movement was dangerous, and its biggest victim was the world of mathematics itself that had been built by mankind for 4000 years. Everything that had been quietly sitting in its own place suddenly went through an upheaval. Certainties that for centuries had guided brilliant thinkers toppled from their places. Uncertainties that up till then had been tamed, restrained and pushed back from the continent of mathematical truth, instead seemed to fly up and sweep through the entire horizon. The farther and freer the mathematics explorers ventured, the more it was felt that the walls of space and time that had been transparent, became once again walls that got darker and darker, and even mathematics seemed to be besieged by darkness. The only thing that could be clearly seen was only the sprinkling of some mathematics problems that refused to be solved and kept on biting at the peace of the mathematicians.

In an exploration at the edge of the horizon where some mathematicians could only grope their way in the dark, a number of people appeared who could see things that most of mankind could not perceive – people whose work was considered to invite disaster. On the frontlines of mathematics, the atmosphere was indeed not as peaceful as a lake over which dusk has fallen, and where everything was sitting comfortably in its own place, such as understood by Tengo. On the frontlines, as is the case on the battlefield, a storm was blowing, even though there might not have been any bloodshed. In a realm whose gates were uncovered by George Cantor, David Hilbert, Bernhard Riemann, Kurt Gödel, Alan Turing, and several mathematicians of gigantic stature from all over the world, a cyclone tore through, taking some casualties.

Georg Cantor

George Cantor, the inventor of the set theory was one of those victims. His attempt to cultivate and subdue infinity which then gave birth to transfinite numbers aroused the anger and hostility of a number of mathematicians. Henri Poincare, the greatest French mathematician of that era who was called the Last Universalist because of his mastery of all mathematical branches known at that time, once called Cantor’s efforts a “grave disease” infecting the discipline of mathematics. Meanwhile Leopold Kronecker, a German mathematician who dedicated his life to algebra and numbers theory, spent some time describing Cantor as a “scientific charlatan”, a “renegade” and a “corrupter of youth.”

Mathematics is indeed not as simple as described by Murakami, or as understood by Tengo. Regarding mathematics, Tengo was only partly right. But indeed, Tengo was described as a math teacher for only students of junior high school. And this mathematical image itself was borrowed by Murakami to become a literary device to create a parallel reality: an arena for the flow of a story considered to be adequate to accommodate two moons in the Tokyo sky. In brief, Murakami used the spirit of mathematics that was always moving to make sense of the impossible.

Poetry is of course also able to illuminate and bring forth the impossible, and this it did, among others, by liberating the use of language from its ordinary practice so that language could realize and go beyond its limits. Mathematics illuminates and brings forth the impossible by seemingly doing the opposite: controlling the use of its language and making it purely a representation of logical operation, and not a representation of the physical world. Even though their treatment of language seems different, even total opposites, mathematics and modern poetry have both developed by upholding and defending the autonomy of “text.” This worship of text causes the efforts to measure the value of a mathematical structure by merely examining its correspondence to the world facts, to be misplaced. Although these efforts can be acknowledged in term of scientific interest, they are clearly irrelevant to mathematical needs. Whatever, modern mathematics is certainly not meant to correspond to reality: mathematics only struggles to answer the needs of its own narrative.

One hundred years have passed since the beginning of major exploration and tightening of mathematics known as mathematical formalization. What came out of this as seen during the last few decades was modern mathematics that, as written by David Bergamini, developed in two directions. The first direction was to be on the outward, decorated with a variety of successes and conquests — the ability to explain and solve problems. The other direction that was inward in nature was contemplation and soul searching — the nature and purpose of the highest mathematical abstraction.

Among its glories of conquest, 20th Century mathematics can refer to the two whole new kingdoms: the game theory and topology. If game theory is the analysis of strategy, whether in the hot games or the cold games of war, topology is the study of the properties of geometric shapes which do not change when the shape themselves are stretched, twisted, scrunched up or turned inside out. Of all the practical triumph of mathematics, the most amazing, of course, is Relativity which has irrevocably proved the awesome power that mathematics can wield over everyday life. Albert Einstein’s theory owed much to the geometry of curved surfaces developed by Carl Friedrich Gauss and Bernhard Riemann, which at the beginning did not promise any practical benefits at all. This geometry only logically and coherently explores a very challenging possibility but for thousands of years had never been conceived as it had not been needed at all.

Evariste Galois

On the contemplative side, two of the most notable developments are set theory and symbolic logic. If the set theory resulted in, among others, a new type of arithmetic to deal with infinity, symbolic logic is an attempt to reduce all forms of human reasoning into a mathematical notation. Both set theory and symbolic logic are abetted by a third form of mathematics, group theory, which plays a unifying role in analysis and reveals many unexpected similarities between the various realms of mathematics. Pioneered by Evariste Galois and developed by a number of mathematicians such as Robert Langlands and Edward Frenkel, this theory suggested the possibility of growing into a great theory unifying all branches of mathematics, even unifying mathematics and physics.

This great integration of mathematics and physics was also concluded by Max Tegmark. This mathematician and cosmologist had even asserted in Mathematical Universe Hypothesis (MUH) that the universe basically contains a mathematical structure, and that anything that is constructed mathematically can certainly be constructed physically. This mathematical universe hypothesis, of course, unfurled a number of interesting challenges to work a narratological criticism on it and prove its strength. This type of criticism of course presupposes and at the same time asserts that mathematics is essentially a vocabulary of language, a thesaurus of both words and grammar. This treasure has grown to produce the most abstract logical construction that has never existed in reality, as well as being the most practical construction that can eventually be utilized by the world. Mathematical structure, with all its logical rigor, is a kind of sentence that can develop into a piece of the story, with all the interwoven plots and subplots, with many sharp turns and potential solutions.

These sentences and story were used by natural scientists to set up a dialog with nature. The world of natural sciences, especially physics, is essentially an autobiography of the universe and what it holds. It is the living history of physical reality written as closely as possible to the facts as they lie. Scientists transcribe the story of this universe in a process somewhat similar to what is seen in the film “The Diving Bell and The Butterfly” (2007), based on the memoirs of Jean-Dominique Bauby. In this film, Jean-Do, who suffers from locked-in syndrome, relates his story to his transcriber just by blinking his left eye to indicate “wrong” and “right” to each question or statement made. In this universe we find the story be told through indirect answers in the form of empirical facts whose content is a response meaning “maybe” or “wrong” to all human conjecture.

The difference between Jean-Do and the universe is that the latter is not the source of a spirited and generous story that presents its narrative. This great and mighty cosmos may perhaps be even worse tempered than someone who was felled by a stroke and suffers from locked-in syndrome. In the case of Jean-Do, though he suffered total paralysis from head to toe, he still had the passion to communicate, even passion to be known. Even though it has never lied, this huge universe does not have the urge to speak specifically and passionately to mankind. If Jean-Do very much cares and in so doing, he really makes an effort to communicate to the transcriber so that the transcriber can understand what he is trying to say, the universe does not care at all and of course does not try to make it easy for mankind to understand. Mankind needs to struggle to understand the facts lay out by the universe, look for the pieces and put the story of the universe together properly. (Among those scientists very passionate at guessing the meaning of the universe whose indifference is even worse than a patient with locked-in syndrome, is Stephen Hawking who is also paralyzed resulting from a condition called amyotrophic lateral sclerosis).

In their efforts to obtain information and put together the story of the universe, the scientists have built up a hypothesis that gains in value in the form of a mathematical construction that gets increasingly beyond human intuition, and tester instruments that increasingly surpass human senses. Scientists who actively pursue this story of the universe are indeed greatly assisted in drafting and proposing conjectures by borrowing sentences and stories that develop in the conferences of thunderbirds. And, like crazy lovers, these scientists continue trying to build ever greater tester instruments. The cyclotrons which extend across three countries have for a long time now been considered not to be adequate for the needs of current theory testing. Looking at the trend, it is not impossible that mankind might be building a device as large as the Solar System, or even as huge as the Milky Way. These objects may be very large for humans at this time, but it really means nothing compared to the greatness of the universe.

What is interesting is that out of the busy-ness of transcontinental transcribing of cues and stitching together pieces of the story of the universe, mankind sees itself changing from just being a transcriber to gradually working as a co-author. At first they are amazed at seeing the awesomeness and mystery of the universe and all it contains, but with the increasing number of story pieces that they can put together, the more they understand the nature of the universe narrative. The giant stories of this universe themselves are stimulating and surprising, and constantly remove the presence of this super great writer’s figure who can plan everything down to the most subtle details. This great story seems to grow out of nothing, then forms itself through time unfolding for so long, through random events and multiplying an infinite amount. Through these events called opportunities, replication, and selection, a wonderfully rich story is created that allows for the appearance of a character that is self-aware.

The universe and all it holds including life developing within it is indeed an enormous richness and miracle. As a miracle, it is amazingly sensitive and fragile, and as such is extraordinarily valuable. If only this entire physical universe could start from the beginning again, it is hugely possible that this character called an intelligent being would not exist. The apex of all of the miracles, the most amazing thing found in all the readings and writing of the story of the universe, is that the character formed in that story does not only understand the story that gave birth to it. This character can even gradually see how that story, including the character itself, stores some weaknesses, a number of defects, that could, even challenge one, to be improved on.

Albert Einstein used to say that the most incomprehensible thing about the universe was that the universe could actually be understood. Now we can say that mankind can understand the universe because they are both one: both are driven by the spirit that also gives life to poetry, i.e. to grab what is infinite with finite material, and respect the rule that came out of this and ignore the rule that is not part of it. This “poetic force” also explains the phenomenon referred to by Eugene P. Wagner as The unreasonable effectiveness of Mathematics in Natural Sciences. The urge to reach infinity with finite material, shows itself in a variety of forms, and they can mutually and effectively illuminate each other.

An understanding of the poetic nature of mathematics and the universe is what makes the character move from fascinatum et tremendum to the existential pleasure of engineering. At the end of untold pleasures, he is again mesmerized by receiving a priceless blessing which he actually does not deserve to receive, which is the opportunity to continue the grand narrative of the universe and all it holds, so that it can develop better, be more perfect from what came before. Therefore, it seems inevitable that mankind will face the universe will be similar with Tengo’s view of the forest of story, and in particular the manuscript of Air Chrysalis which to him seemed so promising but written in a fairly poor manner. Reading and understanding this grand text of the universe, mankind will also feel a new sensation which encourages an unstoppable urge to ignore all risks and disasters to edit and continue the great text which is still far from being finished.

The editing and rewriting work naturally begins with editing and rewriting the character’s own biological constitution to fit the needs of a narrative on the scale of the universe. Immortality was pursued and never obtained by Gilgamesh thousands of years ago, but which now may be achieved through editing our biological constitution, is a natural response to the needs of a narrative on a scale of universe. Biological immortality itself, a centuries-long life span, is a very important element in the great game of universe which is extraordinarily amazing, whose arena was built on the basis that the mathematical universe, which is full of possibilities, is much broader, richer and wilder than the physical universe; that our physical universe which seems so majestic and haughty, is actually finite, and is plagued by a terrible burning desire to emulate the mathematical universe which is really infinite but can be reached by the imaginations of intelligent beings ever evolving. In short, as life attempts to imitate art, the universe also attempts to imitate mathematics, and the interplay between mathematics and the physical universe is an invitation for intelligence to play a role that is wonderfully fulfilling: the grandest gift in the universe.

Pustaka:

• David Bergamini, Mathematics. Life Science Library (Canada: Time Inc, 1972)
• Haruki Murakami, 1Q84 (New York: Vintage International, 2011)
• Edward Frenkel, Love and Math: The Heart of Hidden Reality (New York: Basic Books, 2013)
• Max Tegmark, Our Mathematical Universe: My Quest for the Ultimate Nature of Reality (New York: Alfred A. Knof, 2014).

Jakarta, Juni 2014

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One Response to “Math”

  1. Gps Tracking Says:

    Great people with a great mind will have a positive influences to many people around the world.


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