To Infinity and Beyond: A Cultural History of the Infinite
Mind Tools: The Five Levels of Mathematical Reality
“The world is colors and motion, feelings and thought…and what does math have to do with it? Not much, if “math” means being bored in high school, but in truth mathematics is the one universal science. Mathematics is the study of pure pattern, and everything in the cosmos is a kind of pattern.”
In the above quotation, the first paragraph of Rudy Rucker’s latest book, observe the word “pure.” Mathematical patterns are pure, timeless concepts, uncontaminated by reality. Yet the outside world is so structured that these patterns in the mind apply to it with eerie accuracy. Nothing has more radically altered human history than this uncanny, to some inexplicable, interplay of pure math and the structure of whatever is “out there.” The interplay is responsible for all science and technology.
Perhaps it is a dim awareness of the explosive role of mathematics in altering the world, together with the low quality of math teaching in this country, that accounts for the growing number of books intended to teach mathematics to those who hated it in school. The two books here under review are general surveys, in the tradition of such popular classics as Edward Kasner and James Newman’s Mathematics and the Imagination. Unlike most such surveys, each book is organized around a unifying concept.
For Eli Maor, an Israeli mathematician now at Oakland University in Rochester, Michigan, the unifying concept is infinity. “Finite mathematics,” a term that has come into recent use, is precalculus math in which infinity is avoided as much as possible, yet even in the most elementary math there is no way to escape completely from the concept. As Maor points out, counting numbers go on forever, and straight lines are endless in both directions. Textbooks on finite math have chapters on probability, but what is meant when you say the odds are equal that a flipped coin will fall heads or tails? “We tacitly assume,” writes Maor, “that an infinite number of tosses would produce an equal outcome.”
Maor begins his admirable survey with the concept of limit. In one of Zeno’s notorious paradoxes, a runner can’t get from A to B until he first goes half the distance. Now he must run half the remaining distance, then half the still remaining distance, and so on into an infinite regress. Because at any time the number of distances yet to be traversed is infinite, how can he reach B? Worse than that, how can he begin? If the distance is sixteen miles, he must first run eight miles. To go eight he must go four. Again, the halves form an infinite sequence. How does he get started? Of course mathematicians are no longer troubled by such paradoxes of motion, but it is impossible to resolve them without a clear notion of the limits of infinite sequences of magnitudes in both time and space.
Maor’s well-chosen examples are wide-ranging. Archimedes determined the value of pi (the ratio of a circle’s circumference to its diameter) by calculating the perimeters of inscribed and circumscribed regular polygons. By increasing the number of sides of these nested figures he was able to squeeze the value of pi between inside and outside polygons that came closer and closer to the limit of a circle. In this way he got pi correct for the first time to what today we call two decimal places. Last year a Japanese supercomputer calculated pi to more than 134 million digits.
At present no one knows if certain patterns, say a run of a hundred sevens, occur somewhere in the nonrepeating endless decimals of pi. Are we entitled to say the run is either there (wherever “there” is) or not there? Here the concept of infinity generates a curious split in the philosophy of mathematics. A Platonic realist would answer “of course,” the run of sevens is there or not, but there are mathematicians called constructivists who will have none of this. The “either/or” cannot be asserted, they insist, until such a run is actually found, or until someone proves in a finite number of steps that the run must or cannot “sleep” in pi, as William James once put it. It is, of course, legitimate for a constructivist to say that a run of a hundred sevens does or doesn’t exist in the first billion decimals of pi, because there are algorithms (procedures) for answering this question in a finite number of steps.
Proving whether certain types of numbers belong to finite or infinite sets is a major ongoing task of number theory. Maor gives Euclid’s elegant proof that the number of primes is infinite. (A prime is an integer greater than 1, divisible only by itself and 1.) Twin primes are primes that differ by two, such as 3,5 and 11,13. Are they infinite as well? Twins of monstrous size have been found by computers, but whether there is an infinity of them remains unanswered. Maor reports that in 1982 a computer software company offered $25,000 for the first proof of this old conjecture.
The harmonic series, which has so many applications in physics, is 1/1 + 1/2 + 1/3 + 1/4 + …, where the dots indicate an infinity of the reciprocals of the counting numbers. (The reciprocal of x is 1/x.) As the number of terms increases, the partial sums get larger. Do these sums converge (approach a limit) or diverge (increase without limit)? Because the terms are increasingly smaller, one suspects the sum converges. Amazingly, it doesn’t, though the divergence becomes increasingly sluggish. It takes 12,367 terms to reach a sum that exceeds 10; to exceed 100 the number of terms required has forty-four digits.
Maor provides many curiosities involving this remarkable sequence. If you remove all the terms that contain a specified digit in the denominator, the series converges. For example, if you remove all fractions that contain 9, the sequence converges on a sum slightly less than 23. Suppose you remove all fractions with denominators that are not prime. The series still diverges. On the other hand, the reciprocals of twin primes (assuming they are infinite) have been shown to converge.
Maor’s discussion of numbers comes to a climax with chapters on Georg Cantor’s revolutionary discovery that it is possible to define “transfinite” numbers that stand for an infinite heirarchy of infinities. The smallest—Cantor called it aleph-null—counts the integers, as well as any infinite subset of the integers. For instance, there are as many primes as there are integers. The proof is simply to put the two sets of numbers into one-to-one correspondence:
1 2 3 4 5 ….
2 3 5 7 11 ….
Any set of objects that can be put into correspondence with the integers is called countable. Cantor was able to show that the set of all integral fractions is countable, but the set of irrational numbers (numbers that can’t be expressed as integral fractions, such as pi and the square root of 2) is not countable. Cantor called the number that counts the real numbers (rational and irrational) aleph-one, or C (for continuum), because, as Maor shows, it counts the number of points on a line segment. Cantor believed that 2 raised to the power of aleph-null is the same as C, and he proved that an endless ladder of alephs can be generated simply by raising 2 to the power of higher and higher alephs.
Turning to geometry, Maor covers a variety of fascinating topics, such as “pathological curves” of infinite length that enclose a finite area, and surfaces of infinite area that surround a finite volume. A section on inversion explains how a circle or sphere can be turned inside out to put all its points into correspondence with all outside points on an infinite plane or in infinite space. There is an old mathematical joke about how to catch a tiger. You invert the space outside an empty cage. This puts the tiger (along with everything else) into the cage.
A chapter on the Dutch artist Maurits Escher reproduces many of his pictures (some in color) that involve infinity, such as his marvelous mosaics of birds and animals that tile the infinite plane. (The plane is said to be tiled if the shapes completely cover it, without gaps or overlaps, like the hexagonal tiles of a bathroom floor.) The jacket of Maor’s book has an Escher drawing of a globe covered with loxodromes. These are helical paths followed by ships and planes that travel at a constant angle (not a right angle) to the earth’s meridians. The paths spiral around the poles, making an infinity of revolutions until they strangle the poles.
Maor ends his survey by leaving pure math for the disheveled outside world. Discussions of modern cosmology and particle physics raise deep questions about the infinitely large and the infinitely small. Does spacetime extend forever, or is it finite but unbounded like the surface of a sphere would be for flatlanders living on it? Are there other universes out there in some sort of hyperspacetime? Does the infinitely small stop with a truly fundamental particle (the latest speculation is that the basic units are infinitesimal strings), or is matter an infinite regress of endlessly smaller entities, like an infinite nest of Oriental wooden dolls?
Rudy Rucker, who holds a doctorate in set theory, is a professor of computer science at San Jose State University, in California. He is well known to science-fiction readers for his far-out fantasies, including White Light, a novel based on Cantor’s alephs. Another novel assumes that as you shrink down into smaller and smaller levels of reality you eventually enter the same universe you started from. Rucker’s previous nonfiction books, including Infinity and the Mind and The Fourth Dimension, mix mathematics with occasional bizarre science-fiction themes. Mind Tools, a survey of math organized around the modern concept of information, is a similar blend.
Rucker divides mathematics into what he calls five archetypes or modes of thought: Number, Space, Logic, Infinity, and Information. A section of his book is devoted to each mode, the first four approached from an information perspective. To explain the modes, Rucker considers a human hand.
From the perspective of number, the fingers model the integer 5, but scores of other numbers count such quantities as hairs, wrinkles, cells, lengths of fingers, areas of nails, weight, temperature, blood-flow rate, electrical conductivity, and so on. Viewed as space, the hand is a three-dimensional solid. Because it lacks holes, it is topologically equivalent to a ball. (Topology studies properties that remain the same when an object is continuously deformed.) The hand’s blood vessels branch in a pattern that mathematicians call a tree. Parts of the hand’s surface are concave, parts convex. From a logic point of view the hand is a machine about which all sorts of “if…then” statements can be made: if it clenches, knuckles get white; if it touches fire, it jerks away; if it digs in dirt, fingernails get black.
Infinity enters when you consider the hand as an abstract solid with an alephone infinity of points. As an actual solid, if the nested-dolls conjecture holds, it may have an infinity of components. Viewed as information, the hand grew in accord with detailed instructions coded by the body’s DNA. Information about the hand’s past is embodied in such traces as scars and freckles. How many questions would someone have to ask about your hand to build a replica? What is the shortest computer program that would give this information?
Rucker likes to substitute familiar words for technical jargon. Instead of saying the world is a mixture of discreteness and continuity, he speaks of spottiness and smoothness. The usual references to the wave/particle duality of quantum mechanics are replaced by talk of lumps and bumps. Sometimes an electron acts like a discrete lump, sometimes like a bump in the shifting patterns of a wave field. Which is more fundamental, a particle or its field? This, says Rucker, is like asking which is more fundamental, a person or society? He invokes Niels Bohr’s famous aphorism: “A great truth is a statement whose opposite is also a great truth.” Bohr called this the principle of complementarity. He was so intrigued by the Oriental yin-yang symbol of complementarity that he put it on his coat of arms.
Following in the mental steps of his great-great-grandfather, the German philosopher Hegel, Rucker is a monist who believes that in some ultimate sense, like the circle that surrounds the yin and yang, all is One. There is no need, he writes, to distinguish a particle from its field, or a person from society. “Reality is one, and language introduces impossible distinctions that need not be made.” Need not be made? To a pluralist—William James for example—the distinctions have to be made, not just because language forces them but because that’s how the universe is fragmented. I once ran across a couplet by some unknown poet whose name I long ago forgot, though not the lines:
If all is One,
Who will win?
In some transcendent sense, monism may prevail, but the White Light of Hegel’s Absolute is stained by Shelley’s dome of many-colored glass, and without the distinctions we couldn’t think, talk, or live. Indeed, Rucker could not have written his book without thousands of distinctions in pure mathematics. As for the outside world, nothing is perfectly smooth. Everything has lumps.
Such metaphysical animadversions need not hinder a pluralist from enjoying Rucker’s lively explorations. His number section tells how to use your fingers as flip-flops for binary counting. This leads to a discussion of logorithms, figurate numbers (numbers modeled by spots in patterned arrays), giant numbers, and the numerology of interesting numbers from 1 to 100. Ninety-one is particularly interesting. It counts the spots in a triangular array of thirteen spots on the side, in a hexagonal array of six spots on the side, and the number of balls in a pyramid with six on the side of its square base. It is the sum of the cubes of 3 and 4, and when you write it in base-9 notation it is 111. Twenty-three is the smallest integer Rucker found relatively boring.
The section on space allows Rucker to introduce tiling theory, with special attention to an extraordinary discovery in 1974 by the British mathematical physicist Roger Penrose. Penrose found a pair of quadrilateral figures, usually called “kites and darts” because of their shapes, that tile the plane in only a nonperiodic way. A periodic tiling is one on which you can outline a region that tiles the plane by translation (shifting without rotating or reflecting), like the bricks that tile a brick wall. On a nonperiodic tiling, no such region can be outlined. It is of course possible to tile the plane nonperiodically with replications of a single shape as simple as a triangle or square, but such shapes also tile periodically. Whether there exists a single shape that will tile only nonperiodically is one of the major unsolved problems of tiling theory.
The amazing thing about Penrose’s kites and darts is that the only way they will cover the plane, without gaps or overlaps, is nonperiodically. Mathematicians—notably John Conway, now at Princeton University—at once began finding all sorts of astonishing properties of Penrose tiling, when a few years ago a wholly unexpected event took place. Crystals were constructed with atoms arranged in a nonperiodic pattern based on a three-dimensional analog of Penrose tiles! Hundreds of papers have since appeared about these strange “quasicrystals.” It is a superb instance of how a discovery in what can be called pure recreational mathematics suddenly found a totally unexpected application to the shaggy world “out there.”1
The same Conway invented the most profound of all computer recreations, the cellular-automaton game of Life. A cellular automaton is a structure of cells, each of which can assume a certain number of states. At each “tick” of time, the states simultaneously alter according to “transition rules” that govern the passage of information to a cell from a specified set of “neighbors.” Cellular-automata theory is now a hot topic on the fringes of math, with many applications to robot theory and artificial intelligence. Edward Fredkin, at MIT, has conjectured that the universe itself may be one vast cellular automaton. As Rucker points out, this vision is similar to Leibniz’s dream of a cosmos composed of isolated “monads” that “have no windows,” incessantly changing in obedience to transition rules decreed by God. Viewed this way, the universe is playing a computer game so awesomely complex that the fastest way anyone will ever be able to predict its future states is just to let the game go on and see what happens.2
Discussions of classical curves (including some with such splendid names as Pearls of Sluze and the Nephroid of Freeth) lead Rucker into the exciting new field of fractals, a remarkable kind of irregular pattern that Benoit Mandelbrot was the first to investigate in depth. A fractal is an infinitely long curve or infinitely complex pattern that always looks the same if you keep enlarging portions of it. Mandelbrot called them fractals because he found an ingenious way to assign them fractional space dimensions. During the last ten years, following Mandelbrot’s brilliant leads, fractals have found hundreds of applications in science and aesthetics.3 A coastline, the surfaces of mountains, the surface of the moon are familiar approximations of fractals. As a camera gets closer to the moon, photographing smaller and smaller craters, the surface still looks the same. Computer programs are now generating fractal music, and fantastic fractal landscapes for science-fiction films. The topic propels Rucker into one of his wild conjectures, but you’ll have to consult his chapter “Life is a Fractal in Hilbert Space” to get the details.
Rucker’s next section, on logic, begins with Aristotle’s syllogisms, followed by the propositional calculus and the predicate calculus, the two lowest levels of symbolic logic. Next comes a stimulating discussion of Kurt Gödel’s famous undecidability proof that in any formal system, complicated enough to include arithmetic, true theorems can be stated that can’t be proved within the system. Goldbach’s conjecture, for instance—that every even number is the sum of two primes—could, in the light of Gödel’s theorem, be undecidable. If so, mathematicians may be doomed never to find a counterexample, or to prove the conjecture true.
Rucker examines Gödel’s theorem from his five perspectives; he ties the discussion into the theory of Turing machines (idealized computers), and a theorem of Alonzo Church’s that says that no algorithm (step-by-step procedure) exists that will in a finite time tell whether an arbitrary statement in a complex formal system (one more complex than the propositional calculus) is true. The section ends with musings on how dull life would be if Gödel’s and Church’s theorems did not hold. “Our world is endlessly more complicated than any finite program or any finite set of rules. You’re free, and you’re really alive, and there’s no telling what you’ll think of next.”
The section on information carries Rucker into questions about infinity. Cantor’s alephs are explained; then, going the other way, the infinitesimally small numbers of a modern approach to calculus called nonstandard analysis. Bishop George Berkeley ridiculed the infinitesimal magnitudes in the calculus of Newton and Leibniz, but now, thanks to the labors of Abraham Robinson, infinitely small quantities are as respectable as Cantor’s alephs. The section leads into subtle information theorems recently established by Gregory Chaitin and his colleague Charles Bennett. Rucker paraphrases their theorems in a characteristically cryptic way:
Speaking more loosely, Chaitin showed that we can’t prove that the world has no simple explanation. Bennett showed that the world may indeed have a simple explanation, but that the world may be so logically deep that it takes an impossibly long time to turn the explanation into actual predictions about phenomena.
To make it even simpler: Chaitin shows that we can’t disprove the existence of a simple Secret of Life, but Bennett shows that, even if someone tells you the Secret of Life, turning it into usable knowledge may prove incredibly hard. The Secret of Life may not be worth knowing.
Hegel had a compulsion to group ideas into triads of thesis, antithesis, and synthesis. His great-great-great grandson’s book ends, not with a triad but a pentad:
My purpose in writing Mind Tools has been to see what follows if one believes that everything is information. I have reached the following (debatable) conclusions.
1) The world can be resolved into digital bits, with each bit made of smaller bits.
2) These bits form a fractal pattern in fact-space.
3) The pattern behaves like a cellular automaton.
4) The pattern is inconceivably large in size and in dimensions.
5) Although the world started very simply, its computation is irreducibly complex.
So what is reality, one more time? An incompressible computation by a fractal CA [cellular automaton] of inconceivable dimensions. And where is this huge computation taking place? Everywhere; it’s what we’re made of.
You can find out more about Penrose tiling in Chapter 10 of Tilings and Patterns, a beautiful book by Branko Grünbaum and G.C. Shephard (W.H. Freeman, 1986).↩
Even the ridiculously simple transition rules of Life, concerning cells with only two states and eight neighbors, create patterns impossible to predict. An entire book about Life, and its philosophical implications, is William Poundstone's The Recursive Universe (Morrow, 1984).↩
On fractals, see Benoit Mandelbrot's masterpiece, The Fractal Geometry of Nature (W.H. Freeman, 1982), and The Beauty of Fractals, by H.O. Peitgen and P.H. Richter (Springer-Verlag, 1986).↩
An Infinity of Points March 3, 1988
You can find out more about Penrose tiling in Chapter 10 of Tilings and Patterns, a beautiful book by Branko Grünbaum and G.C. Shephard (W.H. Freeman, 1986).↩
Even the ridiculously simple transition rules of Life, concerning cells with only two states and eight neighbors, create patterns impossible to predict. An entire book about Life, and its philosophical implications, is William Poundstone’s The Recursive Universe (Morrow, 1984).↩
On fractals, see Benoit Mandelbrot’s masterpiece, The Fractal Geometry of Nature (W.H. Freeman, 1982), and The Beauty of Fractals, by H.O. Peitgen and P.H. Richter (Springer-Verlag, 1986).↩