Letters to a Young Mathematician (14 page)

Dear Meg,

Yes, it is a bit of a dilemma. Tenure and promotion depend on your own personal record of teaching and research, but there are attractions in working with others as part of a team. Fortunately, the advantages of collaborative research are becoming widely recognized, and any contribution that you make to a team effort will be recognized too. So I think you should focus on doing the best research you can, and if that leads you to join a team, so be it. If the research is good, and your teaching record is up to scratch, then promotion will follow, and it won’t matter whether you did the work on your own or as part of a joint effort. In fact, collaboration has definite advantages; for instance, it’s a very effective way to get experience in writing grant proposals and managing grants. You can start out as a junior member of someone else’s team, and before long you will be a principal investigator in your own right.

Attitudes are changing, fast. In the past, mathematics was mostly a solo activity. The great theorems were discovered and proved by one person, working alone. To be sure, their work was carried out alongside that of other (equally solitary) mathematicians, but collaborations were rare and papers by three or more people were virtually nonexistent. Today, it is entirely normal to find papers written by three or four mathematicians. Something close to ninety-eight percent of my research in the last twenty years has been collaborative; my record is a paper with nine authors.

This is small potatoes compared to other branches of science. Some physics papers have well over a hundred authors, as do some papers in biology. The growth of collaboration has sometimes been derided as an adaptation to the “publish or perish” mentality, whereby tenure and promotion are determined by how many papers someone has published in a given period of time. An easy way to increase your list of publications is to get yourself added to somebody else’s paper, and the payback is simple: you add them to yours.

But I really don’t think this tit-for-tat behavior is responsible, to any significant degree, for the growth of coauthorship.

The reason for the huge cast of authors on some physics papers is straightforward. In fundamental particle physics, the joint work of a huge team, carried out over several years, is typically condensed into a paper of
perhaps four journal pages. The team will include theorists, programmers, experts in the construction of particle detectors, experts in pattern recognition algorithms that interpret the extremely complex data obtained by the detectors, engineers who know how to construct low-temperature electromagnets, and many others. All of them are vital to the enterprise; all fully deserve recognition as authors of the resulting report. But the report is usually short and to the point. “We have detected the omega-minus particle predicted by theory: here is the evidence.” Someone may get a Nobel Prize for those four pages. Probably whoever is listed first.

Big science involves big numbers of people. The same goes for gigantic biology projects such as genome sequencing.

Something similar has been going on throughout science. The root cause is that science and mathematics have become increasingly interdisciplinary. For example, you’ll recall that one area of interest to me is the application of dynamics to animal movement, and my early papers were all written in collaboration with Jim Collins, an expert in biomechanics. They had to be: I didn’t know enough about animal locomotion and Jim wasn’t familiar with the relevant mathematics.

My nine-author paper summed up two three-year projects to apply new methods of data analysis to the spring and wire industries. More than thirty people were involved in this work; for publication we reduced the list
of authors to those who had been directly responsible for a significant aspect of the results. Some of us worked on theoretical aspects of the mathematics; some worked out new ways to extract what we needed from the data we could record; others carried out the analysis of those data. Our engineers designed and built test equipment; our programmers wrote code so that a computer could perform the necessary analysis in real time. Interdisciplinary projects are like that.

For the last twenty years, funding bodies worldwide have advocated the growth of interdisciplinary research, and rightly so, because this is where many of the big advances are being made, and will continue to be made. At first they didn’t get it quite right. The idea of interdisciplinary research was praised, but whenever anyone put in a proposal for such work, it went to the existing single-discipline committees, who of course did not understand substantial parts of the grant application. For example, a proposal to apply nonlinear dynamics to evolutionary biology would be turned down by the mathematics committee because they and their referees had no expertise in evolution, and then it would be rejected by the biology committee because they didn’t understand the math. The result was that the funding agencies encouraged interdisciplinary research in every way except kicking in funds.

No one, mind you, was doing anything
wrong
. It was virtually impossible to justify a decision to spend money
on the dynamics of evolution when that took money away from top-class projects about algebraic topology or protein folding, say. To the credit of those involved, the system has changed, for the better, and one can now obtain funding for science and math that straddles several disciplines. An important consequence is the creation of entirely new disciplines, such as biomathematics and computational cosmology. Another is the blurring of traditional subject boundaries.

Setting these political factors aside, there is another reason why collaborative publication has increased dramatically. A social reason. Working in groups, with colleagues, is a lot more fun than sitting in your office with your computer. To be sure, sometimes you need solitude, to sort out a conceptual problem, formulate a definition, carry out a calculation. But you also need the stimulus of discussions with others in your own area, or in areas where you wish to apply your ideas. Other people know things that you don’t. More interestingly, when two people put their heads together, they sometimes come up with ideas that neither of them could have had on their own. There is a synergy, a new synthesis, something that Jack Cohen and I like to call
complicity
. When two points of view complement each other, they don’t just fit together like lock and key or strawberries and cream; they spawn completely new ideas. As your career progresses, you may well come to appreciate the delights of collaboration, and value the help, interest,
and support of colleagues whose minds complement your own.

Unfortunately, there can sometimes be a downside to collaboration. Choosing the wrong collaborator is a surefire recipe for disaster. Unfortunately, this can happen with entirely reasonable and competent people; it’s a matter of personal “chemistry,” and not always easy to predict. The main thing is to be aware of the possibility, and to leave yourself an exit strategy.

Some years ago two mathematicians that I know were coauthoring a book. They had no trouble agreeing on the mathematical content, or the order in which the material was presented. They just couldn’t agree on the punctuation. It got to a stage where one of them would go through the entire manuscript putting in commas, and then the other would take them all out again. And round and round it went. The book did get written, but they have never collaborated on another one. They remain the best of friends, though.

Everyone involved in a collaboration must bring something useful to it. They don’t have to do the same amount of work; one person may be the only one who knows how to do a big calculation, or write a complicated program for the computer, while another may contribute a crucial idea that ends up as two lines in the middle of a proof. As long as everyone contributes something essential, that’s fair. No one objects to all coauthors getting some of the credit for the final result.
But if one of the participants is just along for the ride, which occasionally happens, then it makes everyone else happier if their name does not appear on the final paper, book, or report. And it often makes the person concerned happier, too. This need not be a sign of laziness; sometimes the project changes direction in a way that could not have been anticipated, and a contribution that originally looked essential may turn out not to be needed.

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.

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 twelve 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.