"Nature's Numbers" is the second book of the "Science Masters" series that I have read. This book proved to be as interesting as "The Periodic Kingdom". This book concentrates on mathematics, not chemistry. In its own right, mathematics is the basic, underlying science behind all others. Chemistry and physics obviously, but also biology, anthropology, archaeology and all of the other "-ologies" owe most, or at least some, of their being to mathematics.
But, what, exactly, is Mathematics?
Is it numbers? Is it the 2+2 we learned in school? What is it? Stewart addresses this issue by forcing us to look at patterns in nature. For example, did you know that "in nearly all flowers, the number of petals is one of the numbers that occur in the strange sequence 3, 5, 8, 13, 21, 34, 55, 89"? (p. 4) What is the pattern? Simply add the two previous numbers (i.e. 3 + 5 = 8, 5 + 8 = 13) to obtain the number of interest. Many patterns exist in nature and only mathematics allows us to discover these patterns...but, is that what mathematics is? Partially, but mathematics contains more...it contains symmetry, beauty and artistic value...but only if you can see beyond the numbers. Stewart leads us on a tour of discovery of numbers, patterns and more...much more.
Wait! Mathematics is all about numbers..right? Not exactly. Numbers provide the raw materials for describing mathematics, but they are not mathematics by themselves. Numbers are the musical notes of mathematics...they show the pitch and tenor at any point of the music of math, along with some of the beauty of it. Numbers by themselves are static, we add operators to provide dynamics to the system. For the mathematical music to flow, we must also have functions and operations and transformations...ways to modify numbers to move the "music of math" along.
Mathematicians then string together series of functions and operations and transformations to form symphonies called "proofs." A proof is the ultimate mathematical construction. Interesting enough, but proofs are used to prove that the functions and operations and transformations actually work. Wait a minute...we said that you use the operators to form proofs and proofs to prove the operators, right? That is confusing! Well, even mathematicians must allow for a starting point or assumption in their proofs. Thus, some things are just..."agreed axioms" (p. 39). These agreed axioms are the starting points for these strings of operators, for these symphonies.
So, what is mathematics? The music of science and nature.
Well...if mathematics allows us to uncover patterns, what other uses does it have?
Hundreds, thousands, millions, billions... The story of the discovery of The Calculus (that wonderful body of mathematical reasoning that allows us to study rates of change) contains an excellent example of the use of mathematics. Before Newton and Leibniz discovered and developed the Calculus, humankind could not understand any type of rate of change...espiecially a 2nd order rate of change like acceleration (acceleration is the rate of change of velocity, which itself is a rate of change of distance). Most people that drive a car intuitively understand acceleration, but without calculus, we could not predict how acceleration affects things. Thus, calculus allows us to fully understand and control a rate of change. But, what good does that do us? It allowed us to build a car that had brakes that would stop the car in a quick and predictable (or at least somewhat predictable) fashion. Brakes are just a negative acceleration...and they are very important :-) !
Without my going into detail, Stewart also talks about Newton's discovery of gravity and Kepler's discovery of the motion of planets as examples of the use of mathematics. Notice that most of these examples are natural phenomena...remember that math is the music of science and nature.
OK...what if there is no pattern?
Many natural occurrences show no pattern...or do they? This area of mathematics has just started to come to the forefront. Chaos theory is the popular name, nonlinear dynamics is the official title. "Chaos is not just complicated, patternless behavior; it is far more subtle. Chaos is apparently complicated, apparently patternless behavior that actually has a simple, deterministic explanation" (p. 113)...some of which we just have not yet discovered.
An interesting phenomena detailed in Nature's Numbers is the "butterfly effect." This effect controls our weather. The more official title is "sensitivity to initial conditions." Systems, like weather, that ascribe to this effect change when the initial conditions change. Sensitivity to initial conditions arises due to the growth of measurement error as we proceed into the future from any given point. Since we are limited in our abilities to measure a phenomena, as we go further and further into the future, we cannot predict with good accuracy...thus we can predict tomorrow's weather, but not next week's.
Frankly, Stewart does a good job of explaining this phenomena. I have read several books on Chaos Theory and his does the best job of explaining it clearly. I am unable to reproduce his clear explanation without copying it here. Just buy the book and read this section if you have any interest in chaos theory. After reading this book, I plan to go back and read James Gleick's Chaos: Making a New Science to further understand Chaos Theory.
After spending a chapter on Chaos, Stewart spends one on "complexity theory." Another new idea in mathematics, it equals chaos for challenge and interest. Basically, "complexity theory" states the nature will simplify even the most complex of actions. Stewart uses an example of "foxes chase rabbits" (p. 127). This statement appears simple...to our general knowledge, it is a true statement. But, when you look at the details of how a fox knows to chase a rabbit, the locomotion with which he actually chases and how he even sees the rabbit, we quickly get lost in the details. But, does that make the statement untrue? No...foxes really do chase rabbits. Nature generated something simple out of a whole series of complexities.
My Take on Nature's Numbers
All of the examples and premises in Nature's Numbers take their beginning and their meaning from our natural world. Mathematics is not some symbolic, number oriented thing...it is the revelation of the underbelly of nature...things in nature happen for a reason, a logical reason...mathematics attempt to show us these reasons.
Stewart's basic premise of this book reveals to us, the readers, that mathematics encompasses more than numbers or patterns...it shows us that math originates in nature and supports the natural world. By understanding mathematics, we can view nature on a whole new level. A level that not only entails the surface beauty, but also the deeper, underlying beauty of the patterns and shapes of nature.
This book helps you begin your journey to developing this deeper sense of the beauty of nature. It aids us in understanding that all of nature's creations occur for a reason...not just some random occurrence...even the weather patterns!
Although I have studied my fair share of mathematics, I did not understand this simplicity of the science. This simplistic beauty of our world is beyond most people...they do not take the time or expend the effort to see it...they simply exist on a superficial plane. Anyone spending the time to read this review should spend the time to read the book. It, like all of the "Science Masters" series simplifies and exposes a difficult subject for all to see. Now, a final thought from Ian Stewart....
"...nature's patterns are 'emergent phenomena.' They emerge from an ocean of complexity like Botticelli's Venus from her half shell - unheralded, transcending their origins. They are not direct consequences of the deep simplicities of natural laws; those laws operate on the wrong level for that. They are without doubt indirect consequences of deep simplicities of nature, but the route from cause to effect becomes so complicated that no one can follow every step of it" (p.146).
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