Mathematics of Life [Rilegato]

There is a famous joke, an oldie, about the farmer who hired mathematicians to help him increase his milk yield. He got their report back, and read its initial sentence: "Consider a spherical cow..." Ian Stewart quotes the joke in _The Mathematics of Life_ (Basic Books) because it illustrates how mathematicians in their ivory towers can be far removed from the world of practical and messy living stuff. Mathematics was great for chemistry, and engineering, and physics, but if you studied biology, you could get by without being adept in math. Stewart, an emeritus professor of mathematics who has written many popular books about his field of expertise, says that not only has the divide between the two fields begun to shrink, but also that the driving force for the mathematics of the next century will be biology. His book is an agreeable introduction to this new arena for mathematics, and explains, without too many scary equations or too many spherical cows, the recent biological applications of topology, knot theory, game theory, multi-dimensional geometries, and more. Of course, biologists have always used mathematics to tally up population sizes or average heights, but that is arithmetic, and as important as arithmetic is, mathematics is more than numbers - it applies to shapes, processes, structures, and patterns, the very sorts of stuff that make up biology. Stewart is a clear writer, and there is plenty of biology he has explained here, along with examples, none too deep or daunting, of how the math promotes understanding of living stuff.

Stewart begins with a wonderful example of this sort of the importance of context with an examination of the Fibonacci sequence found in plants. The sequence is easily derived; start with a 1 and then another 1; every succeeding number is just the sum of the two immediately preceding numbers, as in 1, 1, 2, 3, 5, 8, 13, 21, 34, 55... Since we have a notion that "genes do everything," there isn't any reason that genes could not code for marigolds that had something other than 13 petals, or asters something other than 21 petals, or pineapples or pine cones with spirals that come in something other than 5 or 8 or 13. Those numbers, and the others in the series, show up repeatedly nonetheless. The answer is that the genes don't code for such patterns at all; they code for growth at the tip of a stem, and the buds of growth there push themselves around in arrangements that reflect an angular version of phi, the Golden Ratio, a ratio reflected in successive numbers of the Fibonacci sequence. Botanists knew in the 1850s that these numbers showed up all over in the plant world, but doing the counting and showing the numbers was all that could be done at the time; the mathematics of the growing stem tip could only be seen once its processes had been microscopically observed. Then, too, different mathematical models showed that if the packing numbers were slightly different, if the angles were a little more or less than phi, then the buds (or spirals or seeds) don't pack nearly so close together.

The chapters, always clear and crammed with biology as well as mathematics, take in biological problems ranging from the tiny (knot theory as applied to strands of DNA) to calculations of the possibilities of life elsewhere in the cosmos. Game theory is used to help explain how lizards compete for mates and also the possible mechanics of one species splitting into two new ones. Stewart mentions the input of game designers for the mathematical program Foldit, in which players hunt for the right way to fold a given protein, something that our cells do every microsecond but which is fiendishly difficult for computers to model. The construction of viruses is best understood by looking at them in the fourth or higher dimension, and Stewart does a fine job of explaining to us stuck in the third what those higher dimensions mean. The "reaction-diffusion" equations invented by the computer pioneer Alan Turing tell us lots about animal stripes and spots. Animal gaits can be easily modeled and reflect a simple neuronal circuit called a central pattern generator (this is an area of Stewart's own research). There's an old puzzle, which it seems to me is far from ever being solved: just why is it that math so beautifully models and promotes understanding of so many aspects of our world? The examples here immeasurably deepen the question.

Prof. Ian Stewart FRS is clever and well-regarded. For a long time, his book on Galois theory was on my to-read list. This book was a major disappointment. It started off, in prospect, as a possible five stars, but it rapidly slid down to two.

What are the problems? Too many to list, but here are some.

First, there is actually precious little mathematics here, esp. in the first hundred pages or so. Then the text is littered with statements that were almost literally painful to read. At one point, he observes that the number of bits required to encode the human genome is approximately the same as the capacity of a CD - thus 'we are roughly as complex as Seargent Peppers Lonely Hearts Club Band'. This is a _completely_ content free remark, for reasons that I am sure Prof. Stewart is aware of, when he is making any effort at all. He implies that we didn't 'really' know that a reef-knot cannot be untied, until topologists managed to prove it in this century. This is a serious confusion of models and reality. It is more accurate to say that we have known, _with absolute certainty_ that you cannot untie a reef-knot with fixed ends, we juat haven't bothered to shoe-horn that knowledge into the language of algebraic topology. Presumably we didn't know until this century either (because mathematics tells us that you can) that you could take a sphere the size of a football apart, and put it together as a sphere the size of the sun? This chapter ends up in a discussion of protein structure that I expected to build to some interesting mathematical theory for solving the protein folding problem (more a statistial physics problem than a mathematics problem, per se, I would have thought), but that ends up by saying nothing more than that there is surely some metric under which there is a continuously descending path in the potential space for a protein, because otherwise it wouldn't reliably fold (you don't say) but we don't know what that metric looks like - in the meantime some people have made the problem into a video game, and it turns out that there are people who are good at solving the problem. Cool! (I don't really think so, more sunday supplement cute, actually. What has that got to do with mathematics, or science, really?

He has a discussion of the structure of viruses that tells us (and not a lot of people know this) that some viruses have a structure that is found in 3d cross-sections of 4d lattices - I expected him at this point to provide a causal explanation of why this might be so, but the chapter just stops at this point.

There are various remarks about data analytics (e.g. clustering, classification tree construction) that avoid all concrete detail and look to be just plain misleading about computational complexity.

I could go on(really? you ask), but this review is already too long.

In 1968, J. Maynard Smith published a perfect little classic of a book, 'Mathematical Ideas in Biology' (CUP, 130pp in my copy). It is long out of print, but surely easy to track down. Stewart isn't in competition.

*****
"Though a complete understanding of how mathematics pries secrets out of nature requires long and rigorous study, Stewart conveys to general readers the fundamental axioms with lucidly accessible writing, supplemented with helpful charts and illustrations.... A rewarding adventure for the armchair scientist." -- Booklist

The versatility of mathematical approach has proven ideal, as a vital tool, to find an intuitive solution just about every problem. Mathematics quantitatively describe everything from the shape of viruses to the structure and function of DNA, and helps to explain the evolutionary games that led to the diversity of life on Earth. Mathematics is one of the fastest propellers for advancing science, and is considered "one of the greatest creations of mankind."
Ian Stewart, Britain's most prolific popularizer of mathematics could be introducing us to a revolutionary approach to an array of bioscience subjects that may have been traditionally considered descriptive, qualitative, and dull. Through a fascinating account on the historical exploration of biology, he portrays mathematics as the 'essential tension' promising new revolutionary perspective that will advance our understanding of the mysteries of life. Such mathematical approach determine all, from the shape of a flower to symmetrical viruses. Stewart leads us to believe that nature is a lot more interesting than most people ever imagined, telling us how biology is fun, and that Japanese researchers claimed an Ig Nobel Prize for demonstrating that slime molds can solve puzzles!

Stewart, like MIT Thomas Kuhn, perceives the advances of life science as leaps caused by revolutions in approach, and proposes its five tension points were the invention of the microscope, the systematic classification of the living creatures, evolution, detection of genes, and discovery of the DNA structure. But he strongly believes that truly fundamental changes to the way we thought about biology will be advanced by looking through the lens of Mathematics. The recent celebration of the human genome project's tenth anniversary, disappointingly ended. Scientists and the press are both blamed for creating false hopes for genomic research in human health. As the DNA era is running out of heat, biology is in desperate need of a fresh mathematical approach. While the work of biological scientists is basic to the future leap forward of biological and medical sciences, any breakthrough that has been expected, could not possibly deliver the awaited personalized drugs, and mass cure miracles, without the help of mathematical tools.

After reading "The Mathematics of Life," you can look at the world through a mathematical lens and see the beauty and meaning that is revealed. Julie Rehmeyer, a math columnist for Science News, summarizes Stewart findings, "A surprising number of plants have spiral patterns in which each leaf, seed, or other structure follows the next at a particular angle called the golden angle. The golden angle is closely related to the celebrated golden ratio, which the ancient Greeks and others believed to have divine and mystical properties. Leonardo da Vinci believed that the human form displays the golden ratio. Scientists were puzzled over this pattern of plant growth for hundreds of years. Even though these numbers were introduced in 1202, Fibonacci numbers and the Fibonacci sequence are prime examples of "how mathematics is connected to seemingly unrelated things."

Why would plants 'prefer' the golden angle to any other?
How can plants possibly 'know' anything about Fibonacci numbers?
I was eager to know about the golden angle, and find out how math could give a hint, just to offer the reader an advance appetizer. Initially, researchers thought these patterns might provide an evolutionary advantage by somehow promoting plants' survival. The golden angle is about 137.5º. Two radii of a circle C form the golden angle, if they divide the circle into two areas A and B, so that A/B = B/C. But recently, they have come to believe that the answer lies in the biochemistry of plants as they develop new leaves, flowers, or other structures. Scientists have not entirely solved the mystery, but a basic understanding of the process seems to be emerging. And the answers are sending botanists back to their electron microscopes to re-examine plants they thought they had already understood.

"The seeds of a sunflower, the spines of a cactus, and the bracts of a pine cone, all grow in whirling spiral patterns. Remarkable for their complexity and beauty, they also show consistent mathematical patterns that scientists have been striving to understand." -- Julie Rehmeyer

The Golden Ratio and Fibonacci Numbers