by Ian Stewart
In big science, where huge teams are involved, the project plan usually stays fairly rigid and people drop out only if they leave and are replaced. But mathematical collaborations tend to be loose and spontaneous, and if there is a project plan, the first item on the list is to be ready to change the plan.
It helps a lot to be relaxed and tolerant. That doesn’t prevent arguments; quite the contrary. The best of friends can have long, loud, and emotionally heated disputes during a research project. Psychologists now think that the rational part of our brain rests on the emotional part: you have to be emotionally committed to rational thinking before you can think rationally. With some of my collaborators, neither of us feels the project is getting anywhere unless we have a shouting match every so often. But the shouting stops as soon as we both sort out who is right, and there is no residual resentment. We are relaxed about having an argument; we are not so relaxed that arguments do not happen.
Never enter into a collaboration merely because you have been convinced that you should. Unless you are genuinely interested in working with someone, don’t. It doesn’t matter how big an expert they are, or how much grant money the project would bring in. Stay away from things that do not interest you.
On the other hand, I do find that it pays to have broad interests. That way, the list of things you should stay away from is much smaller. I once had a fascinating lunch with a medievalist who was an expert on the use of commas in the Middle Ages. Nothing came of that interaction, but it does occur to me that my two friends might have benefited from his presence on their book-writing team.
21
Is God a Mathematician ?
Dear Meg,
It was very good to see you in San Diego last month. I’m ashamed to say I’d rather lost touch with your parents since they moved to the country. I wrote to them and was glad to hear your dad is on the mend.
People react to getting tenure in interesting ways. Most continue their teaching and research exactly as before, but with reduced stress. (This does not apply in the UK, by the way, because tenure was abolished there twenty years ago.) But I do remember one colleague who earnestly declared his intention to publish no more than one paper every five years for the rest of his career. That, he said, was the frequency with which good ideas came to him. It was an honest attitude, but possibly not a wise one. Another devoted himself almost exclusively to consulting work; within two years he’d left the university to start his own company. He now has a vacation home on one of the Caribbean islands. Apparently he got tired of “cheap and cheerful.”
You, I see, have reacted by growing philosophical.
The physicist Ernest Rutherford used to say that when a young researcher in his lab started talking about “the universe,” he put a stop to it immediately. I’m more relaxed about such talk than Rutherford was. My main reservation is that the territory should not be reserved solely for philosophers.
Two and a half thousand years ago, Plato declared that God is a geometer. In 1939 Paul Dirac echoed this, saying, “God is a mathematician.” Arthur Eddington went a step further and declared God to be a pure mathematician. It is certainly curious that so many philosophers and scientists have been convinced of a fundamental link between God and mathematics. (Erd~s, who thought God had other fish to fry, still believed He kept a Book of Proofs close at hand.)
God and mathematics both strike terror into the heart of common humanity, but the connection must surely run deeper. This is not a question of religion. You needn’t subscribe to a personal deity to be awestruck by the astonishing patterns in the universe or to observe that they seem to be mathematical. Every spiral snail shell or circular ripple on a pond shouts that message at us.
From here it’s a short step to seeing mathematics as the fabric of natural law and dramatizing that view by attributing mathematical abilities to a metaphorical or actual deity. But what are laws of nature? Are they deep truths about the world, or simplifications imposed on nature’s unutterable complexity by humanity’s limited brainpower? Is God really a geometer? Are mathematical patterns really present in nature, or do we invent them? Or, if real, are they merely a superficial aspect of nature that we fixate on because it’s what we can comprehend?
The reason we cannot answer these questions definitively is that we human beings cannot step outside ourselves to obtain an objective view of the universe. Everything we experience is mediated by our brains. Even our vivid impression that the world is “out there” is a wonderful trick. The nerve cells in our brains create a simplified copy of reality inside our heads and then persuade us that we live inside it, rather than the other way around. After hundreds of millions of years of evolution, the human brain’s abilities have been selected not for “objectivity” but to improve its owner’s chances of survival in a complex environment. As a result, the brain is not at all a passive observer of nature. Our visual system, for example, creates the illusion of a seamless world that envelops us completely, yet at any instant our brains are detecting only a tiny part of the visual field.
Because we cannot experience the universe objectively, we sometimes see patterns that do not exist. About two thousand years ago, one of the strongest pieces of evidence for the existence of a geometer God was the Ptolemaic theory of epicycles. The motion of every planet in the solar system was held to be built up from an intricate system of revolving spheres. How much more mathematical can you get? But appearances are deceptive, and today this system strikes us as nonsensical and overly complex. It can be adjusted to model any kind of orbit, even a square one. Ultimately it fails, because it cannot lead us to an explanation of why the world should be this way.
Compare Ptolemy’s wheels within wheels to Isaac Newton’s clockwork universe, set in motion at the moment of creation and thereafter obeying fixed and immutable mathematical rules. For example, the acceleration of a body is the force acting on it divided by its mass. This one law explains all kinds of motion, from cannonballs to the cosmos. It has been refined to take relativistic and quantum effects into account in the realms of the very small or the enormously fast, but it unifies an enormous body of observational evidence. The tiny ripples discovered recently in the cosmic microwave background show that when the big bang went off, the universe did not explode equally in all directions. This asymmetry is responsible for the clumping of matter without which you and I wouldn’t have a leg—or a planet—to stand on. It’s an impressive verification of the modern extensions of Newton’s laws, and it shows that patterns need not be perfect to be important.
It is no coincidence that Newton’s laws deal with forms of matter and energy that are accessible to our senses, such as force. If we ride on a fairground roller-coaster, we feel ourselves pulled off our seats as the vehicle careers over a bump. But again our brains are playing tricks. Our senses do not react directly to forces. In our ears are devices, the semicircular canals, that detect not force but acceleration. Our brains then run Newton’s law in reverse to provide a sensation of force. Newton was “deconstructing” his sensory apparatus back into the laws that made it work to begin with. If Newton’s laws hadn’t worked, then his ears wouldn’t have worked either.
We have gotten much better at spotting the artificiality of putative patterns like Ptolemy’s, systematic delusions created by a mathematics that is so adaptable that it can explain anything. One way to eliminate these delusions is to favor simplicity and elegance: Dirac’s provocative point, and the true message of Occam’s razor.
One of the simplest and most elegant sources of mathematical pattern in nature is symmetry.
Symmetry is all around us. We ourselves are bilaterally symmetric: we still look like people when viewed in a mirror. The symmetry is not perfect—normally hearts are on the left—but an almost-symmetry is just as striking as an exact one, and equally in need of explanation. There are precisely 230 symmetry types of crystals. Snowflakes are hexagonally symmetric. Many viruses have the symmetry of a dodecahedron, a regular solid made from twel
ve pentagons. A frog begins life as a spherically symmetric egg and ends it as a bilaterally symmetric adult. There are symmetries in the structure of the atom and the swirl of galaxies.
Where do nature’s symmetric patterns come from? Symmetry is the repetition of identical units. The main source of identical units is matter. Matter is composed of tiny subatomic particles, and all particles of a given type are identical. All electrons are exactly the same. The famous physicist Richard Feynman once suggested that perhaps there is only one electron, batting backward and forward in time, and we observe it multiple times. Be that as it may, the interchangeability of electrons implies that potentially the universe has an enormous amount of symmetry. There are many ways to move the universe and leave it looking the same. The symmetries of a spiral snail shell or the drops of dew spaced along a spider’s web at dawn can be traced back to this pattern-forming potential of fundamental particles. The patterns that we experience on a human scale are traces of deeper patterns in the structure of space-time.
Unless, of course, those deeper symmetries are only imaginary, the modern version of epicycles.
That the universe we experience is a contrivance of our imaginations, however, does not imply that the universe itself has no independent existence. Imagination is an activity of brains, which are made from the same kind of materials as the rest of the cosmos. Philosophers may debate whether the pattern that we detect in a tiger’s stripes is really present in an actual tiger; but the pattern of neural activity evoked in our brains by the tiger’s stripes is definitely present in an actual brain. Mathematics is an activity of brains, so they at least can on occasion function according to mathematical laws. And if brains really can do that, why not tigers too?
Our minds may indeed be just swirls of electrons in nerve cells; but those cells are part of the universe, they evolved within it, and they have been molded by Nature’s deep love affair with symmetry. The swirls of electrons in our heads are not random, not arbitrary, and not—even in a godless universe, if that is what it is—an accident. They are patterns that have survived millions of years of Darwinian selection for congruence with reality. What better way to build simplified models of the world than to exploit simplicities that are actually there? Imaginary systems that get too far removed from reality are not useful for survival.
Intellectual constructs like epicycles or laws of motion may be either deep truths or clever delusions. The task of science is to provide a selection process for ideas that is just as stringent as that employed by evolution to weed out the unfit. Mathematics is one of its chief tools, because mathematics mimics the pictures in our heads that let us simplify the universe. But unlike those pictures, mathematical models can be transferred from one brain to another. Mathematics has thus become a crucial point of contact between different human minds; and with its aid, science has come down in favor of Newton and against Ptolemy. Even though Newton’s laws—or, more to the point, their modern successors, relativity and quantum theory—may eventually turn out to be delusions, they are much more productive delusions than Ptolemy’s.
Symmetry is a better delusion still. It is deep, elegant, and general. It is also a geometric concept. So the geometer God is really a God of symmetry.
Perhaps we have created a geometer God in our own image, but we have done it by exploiting the basic simplicities that nature supplied when our brains were evolving. Only a mathematical universe can develop brains that do mathematics. Only a geometer God can create a mind that has the capacity to delude itself that a geometer God exists.
In that sense, God is a mathematician; and She’s a lot better at it than we are. Every so often, She lets us peek over her shoulder.
Notes and References
Page and baffled by open questions like the Riemann hypothesis. The Riemann hypothesis concerns Riemann’s zeta function ζ (z), which makes it possible to convert questions about prime numbers into questions in complex analysis. It states that if ζ (z) is zero then either z is twice a negative integer or the real part of z is ½. See Karl Sabbagh, Dr. Rie-mann’s Zeros, Atlantic Books, London 2002.
Page the prevalent belief that the human sperm count is falling. P. Bromwich, J. Cohen, I. Stewart, and A. Walker, Decline in sperm counts: An artefact of changed reference range of “normal”? British Medical Journal 309 (2 July 1994) 19–22.
Page Leonardo of Pisa, also known as Fibonacci. “Fibonacci” means “son of Bonaccio.” This nickname was probably invented by Guillaume Libri in the nineteenth century, and certainly does not go back much earlier.
Page the only plausible symmetric network that could explain all of the standard gaits of four-legged animals. M. Golubitsky, I. Stewart, J. J. Collins, and P.-L. Buono, Symmetry in locomotor central pattern generators and animal gaits, Nature 401 (1999) 693–695.
Page as some say the Bible does, but only if you take an obscure passage extremely literally. 1 Kings 7:23 reads, “And he [Hiram on behalf of King Solomon] made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it round about.” If we assume the geometry is a circle, and if we assume the measurements are exact, then the circumference is three times the diameter: π = 3. But the passage is clearly not intended as a precise mathematical statement.
Page Mission to Abisko. Mission to Abisko (eds. J. Casti and A. Karlqvist), Perseus, New York 1999, 157–185.
Page Occasionally someone invents such a piece of machinery out of the blue, and proves all the experts wrong. A classic case is Louis De Branges’s proof of the Bieberbach conjecture. See Ian Stewart, From Here to Infinity, Oxford University Press, Oxford 1996, 206.
Page Sir Peter Swinnerton-Dyer, has offered a simpler explanation of Fermat’s claim. P. Swinnerton-Dyer, The justification of mathematical statements, Philosophical Transactions of the Royal Society of London Series A 363 (2005), 2437–2447.
Page Archimedes knew how to trisect an angle using a marked ruler and compass. Given angle AOB, draw BE parallel to OA, and the circle center B through O whose radius equals CD, where C and D are the marks on the ruler (thick line). Place the ruler so that it passes through O, while C lies on the circle and D lies on BE. Then angle AOC is one-third of angle AOB. See Underwood Dudley, A Budget of Trisections, Springer, New York 1987.
Page given a chessboard with two diagonally opposite corners missing, can you cover it with thirty-one dominoes? The left figure shows the chessboard with its missing corners. The right figure shows a typical attempt to cover it: two squares are left uncovered.
In contrast, if the two missing corners are adjacent to each other, then the puzzle is easily solved:
Page it is impossible to trisect the angle using an unmarked straightedge and compass. The first proof was given by Wantzel. See Ian Stewart, Galois Theory, Chapman and Hall /CRC, Boca Raton 2004.
Page a short calculation shows that with rare exceptions, the cubic equation associated with angle trisection is not like that. Ian Stewart, Galois Theory, Chapman and Hall /CRC, Boca Raton 2004.
Page Mathematicians are proud to trace their academic lineage through thesis advisers. There is a website dedicated to doing just that: http://www.genealogy.ams.org/
Page All of my Portuguese daughters have remained in mathematics. The story of the first, Isabel Labouriau, is one of the many fascinating biographical histories in a wonderful book about women in mathematics:Complexities (eds. Betty Anne Case and Anne M. Leggett), Princeton University Press, Princeton 2005.
Page a penetrating article in 1981 in Mathematics Tomorrow. Timothy Poston, Purity in applications, in Mathematics Tomorrow (ed. L. A. Steen), Springer, New York 1981, 49–54.
Page such gems as the law of quadratic reciprocity. This theorem, first proved by Gauss, states that if p and q are odd primes, then the equation x2 = mp + q has a solution in integers if and only if the related equation y2 = nq + p has a solution, except that if both p and q are of the form 4k + 3, then one equation
has a solution and the other does not. See G. A. Jones and J. M. Jones, Elementary Number Theory, Springer, London 1998.
Page the Titius–Bode law. This empirical pattern in the spacing of the planets was discovered by Johann Titius in 1766 and published by Johann Bode in 1772. Take the series 0, 3, 6, 12, 24, 48, 96, in which each number except the first is twice the preceding number. Add 4 to each term and divide by 10, to get 0.4, 0.7, 1.0, 1.6, 2.8, 5.2, 10.0. Omitting 2.8, these are very close to the distances from the sun to Mercury, Venus, Earth, Mars, Jupiter, and Saturn, respectively, measured in astronomical units. (By definition, the distance from Earth to the sun is one astronomical unit.) The asteroid Ceres neatly filled the gap at 2.8.
Page Karl Weierstrass found a simple continuous function that is differentiable nowhere. K. Falconer, Fractal Geometry, Wiley, New York 1990.
Page On the Enfeeblement of Mathematical Skills. J. Hammersley. On the enfeeblement of mathematical skills by “Modern Mathematics” and by similar soft intellectual trash in schools and universities, Bulletin of the Institute of Mathematics and its Applications 4 (1960) 66.
Page his “shape of a drum” paper is a gem. M. Kac. Can one hear the shape of a drum? American Mathematical Monthly 73 (1966) 1–23. Given the spectrum of tones that can be produced by a vibrating membrane in the plane, can you deduce its shape? Kac proved that its area and perimeter can be deduced. The general question was answered in the negative by C. Gordon, D. Webb, and S. Wolpert, One can’t hear the shape of a drum, Bulletin of the American Mathematical Society 27 (1992) 134–138.
Page Hammersley’s 2004 obituary in the Independent on Friday. Independent on Friday, 14 May 2004.
Page John Barrow argues the case like this. Mission to Abisko (eds. J. Casti and A. Karlqvist), Perseus, New York 1999, 3–12.
Page The Man Who Knew Infinity. Robert Kanigel, The Man Who Knew Infinity, Scribner’s, New York 1991.