Letters to a Young Scientist Page 2
What are we to make of this? For one thing, it is a humbling picture of the origin and the fate of one person’s genes. The fact is that sexual reproduction chops apart the combinations that prescribe each person’s traits and recombines half of them with somebody else’s genes to make the next generation. Over a very few generations, each parent’s combination will be dissolved in the gene pool of the population as a whole. Suppose you have a distinguished forebear who fought in the American Revolution, during which another roughly 250 of your other direct ancestors lived, including possibly a horse thief or two or three. (One of my 8 great-great-grandfathers, a confederate veteran of the Civil War, was a notoriously tricky horse trader, if not quite a thief.)
Mathematicians like to take the measurement of exponential growth from just counting jumps from one generation to the next, to the much more general state to fit a large population over a particular moment in time (to the hour, minute, or shorter interval as they choose). This is done with calculus, which expresses the growth of population in the form dN/dt = rN, which says in any very short interval of time, dt, the population is growing a certain amount, dN, and the rate is the differential dN/dt. In the case of exponential growth, N, the number of individuals in the population at the instant is multiplied by r, a constant that depends on the nature of the population and the circumstances in which it lives.
You can pick any N and r that interests you, and run with these two parameters for as long as you choose. If the differential dN/dt is larger than zero and the population (say, of bacteria or mice or humans) is allowed in theory to increase at the same rate indefinitely, in a surprising few years the population would weigh more than Earth, than the solar system, and finally than the entire known universe.
It is easy to produce fantastical results with mathematically correct theory. There are a lot of models that fit reality and produce factual implications that can jolt us into a new way of thinking. A famous one learned from exponential growth of the kind I’ve just described is the following. Suppose there is a pond, and a lily pad is put in the pond. This first pad doubles into two pads, each of which also doubles. The pond will fill and no more pads can double at the end of thirty days. When is the pond half full? On the twenty-ninth day. This elementary bit of mathematics, obvious upon commonsense reflection, is one of many ways to emphasize the risks of excessive population growth. For two centuries the global human population has been doubling every several generations. Most demographers and economists agree that a global population of more than ten billion would make it very difficult to sustain the planet. We recently shot past seven billion. When was the Earth half full? Decades ago, say the experts. Humanity is racing toward the wall.
The longer you wait to become at least semiliterate in math, the harder the language of mathematics will be to master—again the same as in verbal languages. But it can be done, and at any age. I speak as an authority on this subject, because I am an extreme case. Having spent my pre-college years in relatively poor southern schools, I didn’t take algebra until my freshman year at the University of Alabama. My student days being at the end of the Depression, algebra just wasn’t offered. I finally got around to calculus as a thirty-two-year-old tenured professor at Harvard, where I sat uncomfortably in classes with undergraduate students only a bit more than half my age. A couple of them were students in a course on evolutionary biology I was teaching. I swallowed my pride and learned calculus.
Admittedly, I was never more than a C student while catching up, but I was reassured somewhat by the discovery that superior mathematical ability is similar to fluency in foreign languages. I might have become fluent with more effort and sessions talking with the natives, but, being swept up with field and laboratory research, I advanced only by a small amount.
A true gift in mathematics is probably hereditary in part. What this means is that variation within a group in ability is due in some measurable degree to differences in genes among the group members rather than entirely just to the environment in which they grew up. There is nothing that you and I can do about hereditary differences, but it is possible to greatly reduce the part of the variation due to the environment simply by raising our ability through education and practice. Mathematics is convenient in that it can be achieved by self-instruction.
Having gone this far, I believe I should go on a bit further, and explain how fluency is achieved by those who wish to attain it. Practice allows elementary operations (such as, “If y = x + 2, then x = y - 2”) to be effortlessly retrieved in memory, much like words and phrases (such as “effortlessly retrieved in memory”). Then, in the way verbal phrases are almost unconsciously put together in sentences and sentences are built into paragraphs, mathematical operations can be put together with ease in ever more complex sequences and structures. There is, of course, much more to mathematical reasoning. There are, for example, the positioning and proving of theorems, the exploration of series, and the invention of new modes of geometry. But short of these adventures of advanced pure mathematics, the language of mathematicians can be learned well enough to understand the majority of mathematical statements made in scientific publications.
Exceptional mathematical fluency is required in only a few disciplines. Particle physics, astrophysics, and information theory come to mind. Far more important throughout the rest of science and its applications, however, is the ability to form concepts, during which the researcher conjures images and processes in visual images by intuition. It’s something everyone already does to some degree.
In your imagination, be the great eighteenth century physicist Isaac Newton. Think of an object falling through space. (In the legend, he was attracted to an apple falling from the tree to the ground.) Make it from high up, like a package dropped from an airplane. The object accelerates to about 120 miles an hour, then holds that velocity until it hits the ground. How can you account for this acceleration up to but not beyond terminal velocity? By Newton’s laws of motion, plus the existence of air pressure, the kind used to propel a sailboat.
Stay as Newton a moment longer. Notice as he did how light passing through curved glass sometimes comes out as a rainbow of colors, always ranging from red to yellow to green to blue to violet. Newton thought that white light is just a mix of the colored lights. He proved it by passing the same array of colors back through a prism, turning the mix back into white light. Scientists were later to understand, from other experiments and mathematics, that the colors are radiations differing in wavelength. The longest we are able to see creates the sensation of red, and the shortest the sensation of blue.
You likely knew all that already. Whether you did or not, let’s go on to Darwin. As a young man in the 1830s, he made a five-year voyage on a British government vessel, the HMS Beagle, around the coast of South America. He took that long period to explore and think broadly and deeply about the natural world. He found, for example, a lot of fossils, some of extinct large animals similar to modern species like horses, tigers, and rhinoceroses—yet different in many important ways than these modern equivalents. Were they just victims of the biblical flood that Noah failed to save? But that couldn’t be, Darwin must have realized; Noah saved all the kinds of animals. The South American species were obviously not among them.
As the young naturalist went from one part of the continent to another, he noticed something else: some kinds of living birds and other animals found in one locality were replaced by closely similar yet distinctly different kinds in another. What, he must have thought, is going on here? Today we know it was evolution, but that answer was not open to the young man. Anything that so openly contradicted holy scripture was considered heresy back home in England, and Darwin had trained for the ministry at the University of Cambridge.
When he finally accepted evolution, during the voyage back home, he soon began puzzling over the cause of evolution. Was it divine guidance? Not likely. The inheritance of changes caused directly by the environment, as suggested earlier by the French z
oologist Jean-Baptiste Lamarck? Others had already rejected that theory. How about progressive change built into the heredity of organisms that unfolds from one generation to the next? That was hard to imagine, and in any case Darwin was soon figuring out another process, natural selection, in which varieties within a species—varieties that survive longer, reproduce more, or both—replace other, less successful varieties in the same species.
The idea and its supporting logic came in pieces to Darwin while walking around his rural home, riding in a carriage, or, in one important case, sitting in his garden staring at an anthill. Darwin admitted later that if he couldn’t explain how sterile ant workers passed on their worker anatomy and behavior to later generations of sterile ant workers, he might have to abandon his theory of evolution. He conceived the following solution: the worker traits are passed on through the mother queen; workers have the same heredity as the queen, but are reared in a different, stultifying environment. One day, during this lucubration, when a housemaid saw him staring at an anthill in the garden, she made reference to a famous prolific novelist living nearby when she said (it is reported), “What a pity Mr. Darwin doesn’t have a way to pass his time, like Mr. Thackeray.”
Everyone sometimes daydreams like a scientist at one level or another. Ramped up and disciplined, fantasies are the fountainhead of all creative thinking. Newton dreamed, Darwin dreamed, you dream. The images evoked are at first vague. They may shift in form and fade in and out. They grow a bit firmer when sketched as diagrams on pads of paper, and they take on life as real examples are sought and found.
Pioneers in science only rarely make discoveries by extracting ideas from pure mathematics. Most of the stereotypical photographs of scientists studying rows of equations written on blackboards are instructors explaining discoveries already made. Real progress comes in the field writing notes, at the office amid a litter of doodled paper, in the corridor struggling to explain something to a friend, at lunchtime, eating alone, or in a garden while walking. To have a eureka moment requires hard work. And focus. A distinguished researcher once commented to me that a real scientist is someone who can think about a subject while talking to his or her spouse about something else.
Ideas in science emerge most readily when some part of the world is studied for its own sake. They follow from thorough, well-organized knowledge of all that is known or can be imagined of real entities and processes within that fragment of existence. When something new is encountered, the follow-up steps will usually require the use of mathematical and statistical methods in order to move its analysis forward. If that step proves technically too difficult for the person who made the discovery, a mathematician or statistician can be added as a collaborator. As a researcher who has coauthored many papers with mathematicians and statisticians, I offer the following principle with confidence. Let’s call it Principle Number One:
It is far easier for scientists to acquire needed collaboration from mathematicians and statisticians than it is for mathematicians and statisticians to find scientists able to make use of their equations.
For example, when I sat down in the late 1970s with the mathematical theorist George Oster to work out the principles of caste and division of labor in the social insects, I supplied the details of what had been discovered in nature and in the laboratory. Oster then drew methods from his diverse toolkit to create theorems and hypotheses concerning this real world laid before him. Without such information Oster might have developed a general theory in abstract terms that covers all possible permutations of castes and division of labor in the universe, but there would have been no way of deducing which ones of these multitude options exist on Earth.
This imbalance in the role of observation and mathematics is especially the case in biology, where factors in a real-life phenomenon are often either misunderstood or never noticed in the first place. The annals of theoretical biology are clogged with mathematical models that either can be safely ignored or, that when tested, fail. Possibly no more than 10 percent have any lasting value. Only those linked solidly to knowledge of real living systems have much chance of being used.
If your level of mathematical competence is low, plan on raising it, but meanwhile know that you can do outstanding work with what you have. Such is markedly true in fields built largely upon the amassing of data, including, for example, taxonomy, ecology, biogeography, geology, and archaeology. At the same time, think twice about specializing in fields that require a close alternation of experiment and quantitative analysis. These include the greater part of physics and chemistry, as well as a few specialties within molecular biology. Learn the basics of improving your mathematical literacy as you go along, but if you remain weak in mathematics, seek happiness elsewhere among the vast array of scientific specialties. Conversely, if tinkering and mathematical analysis give you joy, but not the amassing of data for their own sake, stay away from taxonomy and the other more descriptive disciplines just listed.
Newton, for example, invented calculus in order to give substance to his imagination. Darwin by his own admission had little or no mathematical ability, but was able with masses of information he had accumulated to conceive a process to which mathematics was later applied. An important step for you to take is to find a subject congenial to your level of mathematical competence that also interests you deeply, and focus on it. In so doing, keep in mind Principle Number Two:
For every scientist, whether researcher, technologist, or teacher, of whatever competence in mathematics, there exists a discipline in science for which that level of mathematical competence is enough to achieve excellence.
A relativistic jet formed as gas and stars fall into a black hole; artist’s conception. Modified from painting by Dana Berry of the Space Telescope Science Institute (STScI). http://hubblesite.org/newscenter/archive/releases/1990/29/image/a/warn/.
Three
THE PATH TO FOLLOW
THE PURPOSE OF THIS LETTER is to help orient you among your colleagues.
When I was a sixteen-year-old senior in high school, I decided the time had come to choose a group of animals on which to specialize when I entered college the coming fall. I thought about spear-winged flies of the taxonomic family Dolichopodidae, whose tiny bodies sparkle like animated gemstones in the sun. But I couldn’t get the right equipment or literature to study them. So I turned to ants. By sheer luck, it was the right choice.
Arriving at the University of Alabama at Tuscaloosa, with my well-prepared and identified beginner’s collection of ants, I reported to the biology faculty to begin my freshman year of research. Perhaps charmed by my naïveté, or perhaps recognizing an embryonic academic when they saw one, or both, I was welcomed by the faculty and given a stage microscope and personal laboratory space. This support, on top of my earlier success as nature counselor at Camp Pushmataha, buoyed my confidence that I had the right subject and the right university.
My good fortune came from an entirely different source, however. It was choosing ants in the first place. These little six-legged warriors are the most abundant of all insects. As such, they play major roles in land environments around the world. Of equal importance for science, ants, along with termites and honeybees, have the most advanced social systems of all animals. Yet, surprisingly, at the time I entered college only about a dozen scientists around the world were engaged full-time in the study of ants. I had struck gold before the rush began. Almost every research project I began thereafter, no matter how unsophisticated (and all were unsophisticated), yielded discoveries publishable in scientific journals.
What does my story mean to you? A great deal. I believe that other experienced scientists would agree with me that when you are selecting a domain of knowledge in which to conduct original research, it is wise to look for one that is sparsely inhabited. Judge opportunity by how few there are of other students and researchers in one field versus another. This is not to deny the essential requirement of broad training, or the value of apprenticing yourself to resea
rchers and programs of high quality. Or that it also helps to make a lot of friends and colleagues of your age in science for mutual support.
Nonetheless, through it all, I advise you to look for a chance to break away, to find a subject you can make your own. That is where the quickest advances are likely to occur, as measured by discoveries per investigator per year. Therein you have the best chance to become a leader and, as time passes, to gain growing freedom to set your own course.
If a subject is already receiving a great deal of attention, if it has a glamorous aura, if its practitioners are prizewinners who receive large grants, stay away from that subject. Listen to the news coming from the current hubbub, learn how and why the subject became prominent, but in making your own long-term plans be aware it is already crowded with talented people. You would be a newcomer, a private amid bemedaled first sergeants and generals. Take a subject instead that interests you and looks promising, and where established experts are not yet conspicuously competing with one another, where few if any prizes and academy memberships have been given, and where the annals of research are not yet layered with superfluous data and mathematical models. You may feel lonely and insecure in your first endeavors, but, all other things being equal, your best chance to make your mark and to experience the thrill of discovery will be there.