Neverness (68 page)

   For me it was terrifying to find myself once again journeying through that dark nebula that was the Entity. The dense, interstellar dust, the glowing hydrogen clouds, the cancerous black bodies, and always those goddamned mysterious moon-brains, as Bardo would say - whenever I fell out into realspace, I had difficulty imagining why I had once again, despite myself, returned to this strange hell. I was still full of the horror of war, and the afterimage of Bardo's
Blessed Harlot
as it disappeared haunted me. I wondered where he was, almost moment by moment, wondered how he would face his death? I wondered where my fellow pilots were. I could not track their lightships across the Entity because the manifold was like bubbling, black mud. Too bad. Often, I wondered at the Entity's purpose. Did She really want us to witness the death of a star? Or was it all just a cruel trick, Her way of exterminating the soul of an Order which had grown stale, obnoxious and bellicose?

   If She - this goddess whom the warrior-poet had once called Kalinda of the Flowers - if it was important to Her that we quickly reach Gehenna Luz, why didn't She give us more help? Specifically, I wondered why She didn't show us the solution to the Continuum Hypothesis. If we could prove the Hypothesis, we could have mapped from Perdido Luz to Gehenna Luz in a single fall, in almost no time. Why had She provided us laborious mappings through Her twisted interior if a much simpler solution existed? Ah, but what if there was no solution? Or what if a solution existed, and She did not know - or care - what it was? (As a historical note, I should mention that there is an ancient, unrelated theorem of the same name. The Old Continuum Hypothesis states that there is no infinite set with a cardinality between that of the set of natural numbers and the set of points in space. For a century, this remained impossible to prove or disprove, until one of the first - and last - self-programming computers discovered the axioms of Generalized Set Theory and decided the question once and for all.)

   Of course it was arrogant and foolish of me to suppose that I might prove what the Entity perhaps could not. But for all my pains and adventures, I was still an arrogant man. I wanted badly to prove the Hypothesis, I needed to prove it, and to prove it before another pilot such as Soli proved it. All my life I had dreamed of proving it, and now great secrets lay before me if only the pure light of inspiration would illumine this most famous of theorems. I floated naked within my ship's pit, all the while wondering where this inspiration might come from. From slowtime I passed into the white light of dreamtime, and the manifold opened to my mind. Strange are the pathways of a goddess's brain: I entered a rare Lavi torison space and began in-folding through what I prayed would be a finite set of folds. Time slowed. I seemed to have forever to think my thoughts. My thoughts were like the dull glow of an oilstone; my thoughts were as weak as the light of a coldflame globe through a drifting cloud of snow on a winter night. I did not know where to seek inspiration. The great brain of my ship lay before me; its neurologics surrounded me in a web of electric intelligence, but it had been designed to compute, to reason by symmetry and heuristics, to manipulate logic structures, to store information, to do a million things which complemented and added to the mental powers of a human brain without replacing it, I could face my ship, forever and be forever lost to the ecstasy of the size of a brain, I thought, did not necessarily determine its talent for creating mathematics. Perhaps even, the Entity - and here I was being utterly foolish - had little real interest or talent for pure mathematics. And then I had another thought as clear as the Timekeeper's glass: If I were to prove the Great Theorem, the inspiration would have to come from somewhere within myself

   I am a mathematical man. I am a curious man. I have always wondered at the nature of mathematics, and at my own nature as well. What
is
mathematics? Why should mathematics describe the laws of the universe so exactly? Why should our minds' seemingly arbitrary creations and discoveries fit so well this mad, swirling blizzard we call reality? For example, why should gravity (to use the model of newtonian mechanics) act between two objects according to the inverse of the
square
of the distance separating them? Why doesn't it act according to the second and a half power, or the point zero one five and so on power? Why is everything so tidy and neat? It may be, of course, that the human brain is so puny that it can discover only the simplest, the most obvious of universal laws. Perhaps there remains an infinity of laws so hopelessly complicated that they would be impossible to state. Had gravity acted more complexly, The Newton probably never would have found an equation to describe it. Who knows what wonders will forever remain hidden from the mathematical sight of man? This explanation, however, favored by the eschatologists, still does not explain why mathematics works as it does, or why it even works at all.

   What is mathematics? I have turned this question in my mind, turned and returned to this mystery all my life. We create mathematics as surely as we create a symphony. We manipulate our axioms with logic as a composer 'arranges musical notes, and so the holy music of our theorems is born. And in a different sense we also discover mathematics: The ratio of a circle's circumference to the diameter remains the same for human minds and for aliens of the Cetus cloud of galaxies. All minds discover the same mathematics for that is the way the universe is. Creation and discovery; discovery and Creation - in the end I believe they are the same. We create (or discover) undefined concepts such as point, line, set and betweenness. We do not seek to define these things because they are as basic as concepts can be. (And if we did try to define them, we would make the mistake of The Euclid and say something like: A line is breadthless length. And then, using other words we would have to define the concept "breadthless" and "length." And so on, and so on, until all the words in our finite language were eventually used up, and we returned to the simple concept: A line is a line. Even a child, after all, knows what a line is.) From our basic concepts we make simple definitions of mathematical objects we believe to be interesting. We define "circle"; we create "circle"; we do this because circles are beautiful and interesting. But still we know nothing
about
circles. Ah, but some things are obviously true (or it is fun to treat them as if they were true), and so we create the axioms of mathematics. All right angles are congruent, parallel lines never intersect, parallel lines
always
intersect, there exists at least one infinite set - these are all axioms. And so we have lines and circles and axioms, and we must have rules to manipulate them. These rules are logic. By logic we prove our theorems. We may choose the natural logic where a statement is either true or not, or one of the quantum logics where a statement has a degree or probability of trueness. With logic we transmute our simple, obvious axioms into golden theorems of stunning power and beauty. With a few steps of logic we may prove that in hyperbolic geometry rectangles do not exist, or that the number of primes is infinite, or that aleph null is the smallest infinity that exists, or that ... we may prove many wonderful things which are not obvious at all; we may do this if we are very clever and if we love the splendor of the number-storm as it rages and consumes us, and if we are filled with the holy fire of inspiration.

   What is inspiration? From where does it come? As I fenestered through the torison space, the Lavi Curve Theorem and the Second Transformation Theorem were as beautiful as diamonds, and I was full of wonder. Where does mathematics come from? How is it born? Yes, we have axioms and logic and concepts such as "line," but where do these abstractions come from? How is it that even a child knows what a line is? Why do the Darghinni, who are as alien as aliens can be, think according to the same logic as human beings?
Why should this be so?

   I segued through the last fold in the torison space; my ship dropped into realspace, like a flea popping out of the shaken robes of a harijan. I looked at the veiled stars of the Entity, and I thought of the age-old answer of the cantors. Mathematics is a special language, and language is born within the brain. Our brains have evolved for fifteen billion years from the brains of man-apes and back, from the simpler mammalian brains, from the ganglia and nerve clusters of creatures slithering or swimming through the warm salt water of our distant past. And back still further to the bacterial spores which carried life to Old Earth. But from where did these spores come? Did the Ieldra create them? Who created the Ieldra? What is life? Life is the information and intelligence bound within DNA, and life is the explosive replication of protein molecules, and life is the carbon, hydrogen, oxygen, and nitrogen which exist or are born with the cores of the stars. And the universe gives birth to stars; the universe is a vast, star-making engine; the universe brought forth Bellatrix and Sirius and the blue giant stars of the greater Ede Cluster; from stars such as Antares and the First Canopus the stuff of life is made. Every atom of ourselves was assembled in some faraway, heavenly fire. We are the children of the stars, the universe's creation. If our star-born brains conceive "line" and the other elements of language, should we be surprised that "line" is a natural and meaningful concept within that universe? Is it a wonder that the logic of the universe is our logic, too? The cantors are fond of saying that God is a mathematician. They believe that when we create the special language of mathematics we are learning to speak the language of the universe. We have all of us, we pilots and mathematicians, uttered the sounds of this language, in however an infantile and primitive form. Once or twice, while contemplating the wonderful
fit
of mathematics to the contours of spacetime and to the undulations of the manifold, I have felt that the universe was talking to me in its special vocabulary, if only I could listen. How could I learn to listen? How could I learn to speak more elegantly the pure tones of mathematics? What
is
inspiration?

   I journeyed on, and my ship seemed like a dark, stale tomb imprisoning me darker by far than the Timekeeper's stone cell. As a germinated seed seeks its way out of the ground into the light of day, I longed to break free of the old thoughtways that stifled me and restrained my inspiration. How I longed to prove the Great Theorem! But at the same time that I had longings, I had a certain dread, too. I wondered, again and again, at the nature of my own intelligence. From where did my powers of scrying and remembrancing spring? What other powers might I someday gain? If I did somehow prove my theorem, would the proof really be my own? Or would it be merely the creation of the Agathanian's information virus? Could I dare to call forth the seed of inspiration within me, to try to shape that seed as it grew, to taste the bittersweet fruit it might bear?

   I followed the Entity's mappings across a series of thickspaces. Once, I fell out into realspace as dark and empty as the intergalactic void. I nearly panicked, then. But I found that I was actually in the middle of a thickspace! The point-sources were stived as closely as the black eggs in the belly of a jewfish. How this could be so I did not know. Only stars or other matter (or intelligence) can deform space to create a thickspace. I quickly opened a window, and I segued into the manifold. I fell into dreamtime as I thought about this odd thickspace. If the brain of the Entity could contain such wonders as a starless thickspace, what wonders might lie within my brain? Suppose I really tried, tried so hard my eyes burned like coals and my brain's blood surged like an ocean - suppose I tried for the thousandth time to prove the Continuum Hypothesis?

   As soon as this thought hardened, the number-storm intensified. A tide of ideoplasts began to build and flow and rage before my inner sight. I was excited almost beyond control. For the thousandth time I contemplated the deceptively simple statement of the Hypothesis: that between any pair of discrete Lavi sets of point-sources there exists a one-to-one mapping. I broke the statement apart and examined the pieces. What, exactly was a Lavi set? What was a point-source? Was I sure I understood the difference between a Lavi set and a
discrete
Lavi set? How could I show the mapping was one-to-one, and more importantly, how might I construct the mapping to begin with? At first I fell into old thoughtways and rediscovered my old attempts to find a solution. Often I found myself reasoning in circles. I grew discouraged at the shallowness of my thinking. How could I prove this? How could I prove that? How could I break the rusted chain of my habitual, uninspired thoughts?

   I tried to restate the problem in a different form, hoping that a new way of looking at it might enable me to see the obvious. And though I did find an equivalent statement, it proved even more opaque than the original. I decomposed the Hypothesis, recombined it into a slightly different statement - all to no avail. In my mind, I represented the pieces of the Hypothesis with pictures in order to "see" relations that I might have overlooked. I generalized the Hypothesis to include all Lavi sets, and I played with mappings of specific Lavi sets which were quite well known; I tried proof by contradiction; I dissected related theorems (Bardo's Boomerang theorem, incidentally, is closely related, though simpler to prove); I followed long, dark corridors of reasoning down thousands of steps; I cursed and I rubbed my eyes and temples, and finally, when my beard and hair were rimed with crusts of dried sweat and I had nearly abandoned hope, I began to make wild guesses.

   I do not know how long I tried to prove the Hypothesis. Days, seconds, years - what did time matter? And yet it did matter. At any given time, Soli might be close to his moment of inspiration. The race went on, and measureless moments passed into endless days, and I began to think the Hypothesis was unprovable. For a long while I tried to show that it was unprovable, even though I did not really believe it could be so. My intuition - and a mathematical man should never ignore his intuition - something within me whispered that the Hypothesis was indeed provable, and more, that the proof would seem embarrassingly obvious once I had found it.
If
it could be found. If
I
could find it. If ... If one mapping between a pair of discrete Lavi sets of point-sources exists, then there are an infinite number of mappings; if one covers an n-dimensional cube with finitely many sufficiently small closed sets, then there are necessarily points which belong to at least n + 1 of these sets; if one stirs a bowl of blood tea for a thousand years, there will exist at least one point-one corpuscle of blood - which will remain fixed in its original position, undisturbed by the stirring; if then; if I examined the ideoplasts of the Tycho's Conjecture and the Tiling Theorem and the Fixed-Point Theorem, if I broke down the brilliant crystalline arrays into the simple shards of proof steps instead of clumping the arrays together, then I might better understand the inspirations leading to the proofs of these famous theorems. If I understood the proofs better, then I might better use the theorems to prove the Great Theorem.