Dissatisfied with the march of its symbols , I nudge the sphere onto a different streamline , fine-tuning its position until I see the unmistakable traces of the singularity I am seeking. I snap my fingers , and the sphere extrapolates itself into its own future and reports back what it finds. Suddenly there is a whole cloud of red spheres , all being car ried along by the fluid, like a shoal of fish that quickly spreads, swirling , putting out tendrils , flattening into sheets.
Then more shoals of spheres join the game-gold, purple, brown, silver, pink. I am in danger of running out of col ors. Multicolored sheets intersect in a complex geometric form. I freeze it, smooth it, paint it in stripes. I banish the spheres with a gesture. I call up markers , inspect their unfolded petals , pull some off and attach them to a translu cent grid that has materialized like a landscape from thinning mist.
I issue a new command. Title: A new chaotic phe nomenon in the three-body problem. Date: today. Then , the morn ing's research completed, I disengage from my Virtual Unreal ity Machine and head off in search of lunch. This particular dream is very nearly fact. We already have Virtual Reality systems that simulate events in "normal " space. Most of the bits and pieces of the Virtual Unreal ity Machine exist already.
There is computer-graphics soft ware that can " fly" you through any chosen geometrical object , dynamical-systems software that can track the evolv ing state of any chosen equation , symbolic-algebra software that can take the pain out of the most horrendous calcula tions-and get them right. It is only a matter of time before mathematicians will be able to get inside their own creations. But, wonderful though such technology may be, we do not need it to bring my dream to life. The dream is a reality now, present inside every mathematician 's head.
This is what mathematical creation feels like when you 're doing it. I 've resorted to a little poetic license: the objects that are found in the mathematician's world are generally distinguished by symbolic labels or names rather than colors. But those labels are as vivid as colors to those who inhabit that world. In fact, despite its colorful images , my dream is a pale shadow of the world of imagination that every mathematican inhabits-a world in which curved space, or space with more than three dimensions , is not only commonplace but inevitable.
You probably find the images alien and strange, far removed from the algebraic symbolism that the word "mathematics " con jures up.
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Mathematicians are forced to resort to written sym bols and pictures to describe their world-even to each other. But the symbols are no more that world than musical notation is music. Over the centuries , the collective minds of mathematicians have created their own universe.
Nature's Numbers: The Unreal Reality Of Mathematics
I don't know where it is situ ated-I don't think that there is a "where" in any normal sense of the word-but I assure you that this mathematical universe seems real enough when you 're in it. And , not X P R O L O G U E despite its peculiarities but because o f them, the mental uni verse of mathematics has provided human beings with many of their deepest insights into the world around them.
I am going to take you sightseeing in that mathematical universe.
I am going to try to equip you with a mathemati cian's eyes. And by so doing , I shall do my best to change the way you view your own world. Every night the stars move in circles across the sky. The seasons cycle at yearly intervals. No two snowflakes are ever exactly the same , but they all have sixfold symmetry. Tigers and zebras are covered in patterns of stripes, leopards and hyenas are covered in patterns of spots. Intricate trains of waves march across the oceans ; very similar trains of sand dunes march across the desert.
Colored arcs of light adorn the sky in the form of rainbows , and a bright circular halo some times surrounds the moon on winter nights. Spherical drops of water fall from clouds. Human mind and culture have developed a formal system of thought for recognizing , classifying, and exploiting pat terns. We call it mathematics. By using mathematics to orga nize and systematize our ideas about patterns , we have dis covered a great secret: nature 's patterns are not just there to be admired, they are vital clues to the rules that govern natural processes.
This was long before the theory that matter is made of atoms had become generally accepted. Kepler performed no experiments ; he just thought very hard about various bits and pieces of common knowledge. His main evidence was the sixfold symmetry of snowflakes , which is a natural conse quence of regular packing. If you place a large number of identical coins on a table and try to pack them as closely as possible , then you get a honeycomb arrangement, in which every coin-except those at the edges-is surrounded by six others , arranged in a perfect hexagon.
The regular nightly motion of the stars is also a clue, this time to the fact that the Earth rotates. Waves and dunes are clues to the rules that govern the flow of water, sand, and air. The tiger 's stripes and the hyena's spots attest to mathemati cal regularities in biological growth and form. Rainbows tell us about the scattering of light , and indirectly confirm that raindrops are spheres. Lunar haloes are clues to the shape of ice crystals.
There is much beauty in nature's clues , and we can a l l rec ognize it without any mathematical training. There is beauty, too, in the mathematical stories that start from the clues and deduce the underlying rules and regularities , but it is a differ ent kind of beauty, applying to ideas rather than things. Math ematics is to nature as Sherlock Holmes is to evidence. When presented with a cigar butt, the great fictional detective could deduce the age , profession, and financial state of its owner.
His partner, Dr.
Watson, who was not as sensitive to such matters , could only look on in baffled admiration, until the master revealed his chain of impeccable logic. If you are a Watson, it is just as baffling a trick, but I want to show you what it is like if you are a Sherlock Holmes. Patterns possess utility as well as beauty. Once we have learned to recognize a background pattern , exceptions sud denly stand out. The desert stands still , but the lion moves.
Against the circling background of stars , a small number of stars that move quite differently beg to be singled out for spe cial attention. The Greeks called them planetes, meaning "wanderer, " a term retained in our word "planet. One difficulty is that we are inside the Solar System, moving along with it, and things that look simple from outside often look much more complicated from inside.
The planets were clues to the rules behind gravity and motion. We are still learning to recognize new kinds of pattern.
Nature's Numbers: The Unreal Reality Of Mathematics - Ian Stewart - Google Livres
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