A FRACTAL MODEL OF CREATIVITY IN SCIENCE
The same and not the same
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| Fractal (1) |
Computer models of complex biological structures like branching plants, circulatory systems, and lung airways suddenly got better when the fractal revolution came along. Around 1975, Benoit Mandelbrot coined the term fractal to describe objects that look similar at all scales. Fractal objects are also called self-similar.
As an example, look at the trunk of a big, well-formed oak or maple tree in winter, when the branches are bare. Notice the angles that the first large branchings make with the trunk, and notice the ratio of diameters of trunk to first branchings. Then look at large branches, and note the same things about how they form smaller branches, and so on to the smallest twigs. Each size level is just a smaller version of the larger ones, with the same branching angles and size ratios, and with the same fractional distances between branching levels. This pattern, which looks much the same at any level of magnification, is a fractal.
Many natural objects exhibit this kind of self-similarity. The snowflake is a dramatic example. The angles at which water molecules interact are stamped at every branch. In living organisms, self-similarity might reflect an underlying economy in growth and development, because it suggests that a single procedure or program is being employed repeatedly as a living organism develops, so that a relatively small set of instructions, carried out at each level, results in a complex structure.
Artists, musicians, and poets have played with fractal structures in their creations, and the introduction of fractal patterns sometimes seems to give a “naturalness” to human artifices. The discovery of fractal patterns in creative works that date from before the fractal revolution suggests that self-similarity arises naturally as artists strive for satisfying composition and natural textures.
A fractal flea
So nat'ralists observe, a flea
Hath smaller fleas that on him prey,
And these have smaller fleas that bite 'em,
And so proceed ad infinitum
And the great fleas themselves, in turn,
have greater fleas to go on,
While these again have greater still,
and greater still, and so on.
(2)
Philosophy—of science?
I first formally encountered the philosophy of science in an undergraduate course at North Carolina State University in the early 1960s. I was looking for an interesting way to satisfy one of my general-education requirements, during a time when educators—even science educators—were still confident that it was a good thing for undergraduates to be liberally educated, and to be informed particularly well about their own Western culture.
I see now that, because I was a rather naive, sheltered, small-southern-town kind of student, stupidly believing that I had already cast off many of the prejudices under which I grew up, I was very fortunate to be an undergraduate before the educational upheavals of the later 1960s. This unrest threw public, undergraduate general education into disarray, and it never really recovered. (But I saw firsthand, as a teacher, that the 60s did less damage in private, liberal-arts colleges, some of which remain bastions of broad education.) I am glad that I graduated while most students were still required to pay homage to the idea of Western liberal education, because I received much-needed help in understanding Western culture, which was yet (is yet?) to reach the textile mill town where I grew up. I might not have fared as well in the anything-goes buffet-style undergraduate programs that proliferated in the next few decades.
I readily admit—in fact, I insist—that I needed more guidance than a more worldly student from a larger town and a professional family, and I am glad I got it while the getting was good. In part because of this soon-to-be-disrupted style of education, I was required, even in the rather restrictive curricular demands of the engineering, agriculture, and applied-science culture of a school like North Carolina State University, to take courses from all major disciplines, including some, like the social sciences, that I am sure I would have avoided. One of those disciplines was philosophy.
I was a science major, and one of the options, philosophy of science, sounded to me like learning about how scientists think and work. The course was only somewhat like what I expected. The reading material and discussions were difficult for me. The instructor appeared to expect rigor and sound logic, but although I made a good grade, I must admit that I was never sure how or whether I achieved anything resembling rigor. In particular, it was not obvious that the content of the course applied to what I was doing and studying in the hopes of being a scientist, or at least a science teacher. My lasting impression of the course is that it was simply slippery, often leaving me with the feeling that surer footing was somehow eluding me, or that that’s just philosophy.
Nevertheless, the course fueled a lifelong interest in the subject, which is one of the best things that a college course can do. Often, particularly in science, a person’s college major courses have the most impact on their working career—their living—while those “other” courses, sometimes unobtrusively, change their lives. Almost dutifully, I continued to read philosophers that appeared to be respected, including Carl Popper, Stephen Toulmin, and Thomas Kuhn, but I continued to be frustrated that most of what I read did not give me as much insight into science as did the scientific literature itself, especially those issues of Scientific American and Science that regularly appeared in my mailbox.
Recently, I was forced to admit that at least thus far (and it’s getting pretty late), the academic subject called philosophy of science has been a disappointment. And now I realize that it is all because I had the seemingly reasonable expectation that the philosophy of science would be about science.
Big mistake
Certain academic disciplines sound like they are about something, but are really not. For example, too much of what is called literary criticism is not really about literature at all (more in an essay to come). In the same vein, a prominent historian of science once told me that he was disappointed that his field had become an ingrown, arcane, academic discipline that had little to do with science’s history. Shortly afterward, he turned disappointment into action, and left a tenured position in a highly ranked department at a large, prominent university. He moved from one coast to the other, to a small liberal-arts college, where he hoped to engage students in thinking about how the contributions of prominent scientists got us to where we are in science, and where we might be heading. His proposed activities certainly sounded like the history of science to me, so I still did not understand exactly what he was leaving behind. His words came back to me when I began to realize that the philosophy of science seemed to be letting me down in the same way.
My readings were always more satisfying when I tackled philosophical works by scientists themselves. Jacob Bronowski’s The Common Sense of Science, Richard Feynman’s The Character of Physical Law, and Jacques Monod’s Chance and Necessity made me feel that I was reading people who knew something about science, who were interested in how science relates and compares to other disciplines, and who thought about the broader meaning of a model of nature based on scientific findings. Later on, Steven Weinberg’s Dreams of a Final Theory, was another favorite. In that clear, thoughtful, and well-written book, Weinberg expressed the opinion—startling at first, but then comfortingly familiar—that the philosophy of science has had absolutely no impact upon science and its practice.
Stephen Hawking and Leonard Mlodinow, in their 2010 book The Grand Design, are blunt about the lack of pertinence of philosophy in helping us to understand the physical world in which we find ourselves. The following statement arrives early—in the book’s second paragraph:
Traditionally these are questions for philosophy, but philosophy is dead. Philosophy has not kept up with modern developments in science, particularly physics. Scientists have become the bearers of the torch of discovery in our quest for knowledge. (3)
Two books that I found especially disappointing, particularly in light of their prominence for a time, were Dancing Wu Li Masters by Gary Zukav and The Structure of Scientific Revolutions by Thomas S. Kuhn. I know of philosophers of science who still use them as texts, and I can’t imagine why. Zukav was looking for meaningful similarities between the world view of modern physics, with such surprises as indeterminacy and many-worlds theories, and the mystical philosophies of Asia. For my money, he simply did not find them, though he seemed to think he did -- many a QED when I could not see anything demonstrated. Kuhn focused on what he called paradigm shifts, which entail reinterpretation of data in the light of new overarching models. The grist of his mill were the great shifts in thinking that followed the insights of such thinkers as Newton, Darwin, and Einstein.
Neither author could produce in my ears a recognizable ring of truth when they spoke of what scientists do and how they think. They would say, in effect, “Science is like this,” and I would think, “Hmm—I don’t think of science that way.” They would say, “Scientists do this,” and I would think, “I am a scientist, and I know lots of scientists, and we don’t do that.” This failure to recognize science in the philosophers’ descriptions of science made me nervous about other things that they had to say.
Missing the broader view
In addition, both books had in common a very galling shortcoming that, to my knowledge, none of their critics pointed out. Both writers focussed on the really big upheavals in scientific thinking—Copernicus to Galileo, Galileo to Newton, Newton to Einstein, everybody to Darwin—but they completely missed the similarities between these great moments and less sweeping, year by year, month by month, developments in science. Zukav found these iconic scientists enormously creative, but he seemed to see creativity nowhere else in science. It seemed to me that he was utterly unaware of day-to-day creativity at all levels of science.
Kuhn found his pet dynamic, the paradigm shift, in the big transitions; however, between the revolutions, he found only the inescapably pejorative “normal science,” when scientists were little more than automatons, filling in boring details, and reinterpreting previous results to fit the new view.
By the time I struggled through these books, I had done some scientific research myself—to be honest, only enough to earn a Ph.D and complete a year as a postdoc, plus some modest research with college undergraduates—so I felt a little bit like a scientist myself. In addition, I was familiar with the work of many scientists, from leaders in important fields like structural biology, all the way to colleagues of mine who found interesting little research niches to explore while making a career in teaching. The science periodicals that came across my desk held countless stories of startling, exciting, and creative discoveries in many areas of the major oceans of scientific work, but also in its quieter backwaters.
In my view, the Darwin-Newton-Einstein insights, with their total revisions of a world view and the accompanying brilliant strokes of creativity, happen on all scales in science, from the earth-shaking changes that affect whole disciplines, as Darwin restructured life science, to the better method that a lab technician finds to do a common task.
Creativity as a fractal
Does this sound familiar, like a form of self-similarity? I see a fractal structure in the creativity of science, and in its interpretative shifts in response to new findings. At every level of magnification, science sort of looks the same, with the possibility and impact of creativity always looming. For example, Darwin’s rare insight shook the foundation of all the life sciences. But new insights shake life science more gently, and more frequently, at the level of sub-disciplines like genetics, as new paradigms arise from comparing whole genomes; at the level of more restricted fields like signaling, as new paradigms arise from revelations about control of gene expression; at the level of specific research problems, as scientists reveal how small molecules of RNA control gene expression; at the level of individual research groups, as gene-regulation specialists who once focused entirely on proteins realize that small RNAs must also be in their picture; and even at the level of solving everyday experimental problems, as a technician suddenly realizes that something that seemed inconsequential, like exposure of a reaction mixture to light, really does matter, and begins to ask why.
Writers like Zukav and Kuhn, lacking any direct experience in the practice of science, had no way to recognize that the kinds of thinking that they thought unusual and earthshaking in science—creativity and paradigm shifts—actually permeate science at every scale. They constitute a major reason why scientists find enduring satisfaction working on “little” problems as well as big ones. Like the branches of the oak tree, creativity in science shows fractal self-similarity. At any level of magnification, creativity is there, driving science and scientists toward reliable knowledge.
Footnotes
(1) Image from http://harryseldon.thinkosphere.com/2010/01/10/winter-is-the-enchanting-fractal-season-snow-and-naked-trees
(2) Jonathan Swift, On Poetry, a Rhapsody, 1733; obtained from http://en.wikipedia.org/wiki/Ad_infinitum#cite_note-0; Augustus de Morgan, A Budget of Paradoxes, 1872, p. 377.
(3) Stephen W. Hawking and Leonard Mlodinow, The Grand Design, New York: Bantam Books, 2010.
