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The first three chapters in this project celebrated marvelous achievements by individual mathematicians, and also by the mathematical community which has managed to sustain and advance the body of knowledge in spite of extreme historical setbacks. Clouds of doubt accompanied, and sometimes inspired these successes. This final chapter will do more celebrating, but this time some of the problems will do more than create doubt -- they will prove to be insurmountable.
We will see a loss of the confidence, so characteristic in Enlightenment man, in the ability to understand mathematics and the universe. As rigorists attempt to establish more basic and logical foundations for geometry and other branches, our concrete understandings become mired in abstraction. The concepts of area and measure become so purely symbolized and postulated as to be unrecognizable to the lay person. These efforts, of course, had high ambitions. We wanted to have complete understanding of absolute concepts such as infinity, dimension, and time. For our troubles, we proved that these quantities are not absolute. We wanted to put all of mathematics on an ironclad logical footing broad enough to encompass the problem areas such as discontinuities, irrational and transfinite numbers. For our troubles, we showed that any formal axiomatic system that we will ever devise will necessarily have statements which can neither be proved nor disproved; and we have found such undecidable statements about our own number system.
These surprising setbacks should not be regarded, however, as defeats. The situation is not one where we have shown that the mathematician has no clothes. Rather, we have discovered that we were trying to dress math in a very naive Easter Sunday suit. We are just beginning to see the beauty and complexities of math's true wardrobe. Each time one fabric develops holes, a new weave is devised. The blocking of one avenue to knowledge leads to new investigative paths. We prove a lot -- notwithstanding knowing that we can't prove all. The science of probability has become more than an amusement associated with gambling, as scientific problems grow more complex, and absolute certainty in solutions gets further out of reach. We can attain levels of confidence in results, and they serve us as well as exactitude.
There are implications for math education in these recent developments. Do we preserve the naive view of math for the grade school students, those who must solve ever more complex problems in their world? When do we begin to give them a picture of a fallible, human math -- a math that tackles real problems and offers "best possible" rather than "correct" solutions. How do we then avoid the nearly inevitable adolescent disillusionment with math? One answer espoused by many reform-minded educators is to keep math learning open-ended and experimental. Eves seems to lend his support .
In 1734, Bishop Berkeley questioned the calculus of Newton and Leibniz, demanding to know if the infinitesimals are equal to zero or not. If zero, the integral sums to nothing, and the derivative is undefined. If not zero, then what? Newton himself struggled with the problem of the Moon's motion, even after he established the Universal Law of Gravitation. He realized that the problem of three bodies in mutual orbit was incredibly complex, and his comment was that the problem of the Moon was the only one which made his head ache .
In 1889 Henri Poincare submitted a paper to a panel of eminent European mathematicians. It claimed to have solved Newton's "three-body problem." It used a system of eighteen differential equations and many variable substitutions. The panel was barely able to read it, let alone check it. He was declared the winner of the first Oscar (well, in truth, a nice purse which was offered by King Oscar of Sweden). It remained for Poincare himself to discover a gross error in his work and spend many tense weeks repairing the damage .
More importantly, Poincare realized that, there being eight planets in the solar system and numerous moons, the analytical capability of differential systems was reaching a practical limit. He and Riemann are credited with taking a new view to these systems problems. It is sometimes called "intuitional" analysis. Poincare realized that the outcomes of a system as a whole were just as important as the fate of one of its particles. He also realized that new analytical techniques would be required. An example from basic algebra may serve to illuminate the point. The solution to the linear system
which is (2, 45), may be of use to someone, but what is of more interest mathematically is that the solution changes to (51, 6) when we merely change the parameter 2516 to 2517 .
At roughly this same time, in order to establish a rigorous logical foundation for calculus, Weierstrass, Peano, and Hilbert were devising non-standard functions which tested the notions of limit, continuity, and differentiability like no polynomial ever could. Their graphs, the ones simple enough to depict, were referred to as "monster curves." One family of curves, called "space-filling" require only one coordinate to approximate any point inside a square. See the figures below for views of this graph in its early stages. Cantor, meanwhile, in trying to establish a role for infinite sets, discovered an entire infinite family of "transfinite" numbers .
Even the very intuitive notion of dimension, when given one of various formal definitions, proved to be untamed. Cantor proved the equivalence of the set of points in the unit square (our "standard" for the two-dimensional measure) and the set of points of the unit segment. His comment: "I see it, but I don't believe it." You can see it in Figure 3.
Poincare and Hausdorff showed that, even though space seems confined to three dimensions, the mathematical definitions of point and distance extend to higher dimensions . Einstein made almost immediate application of this idea when, in his papers introducing relativity in 1905 and 1915, he showed that the universe is four-dimensional at the very least. Some of the monster curves were shown to behave as if their dimensions were greater than one. In the dragon curve below, what begins as a line becomes convoluted sufficiently to occupy a region. It's dimension, as the number of fold becomes infinite, is calculated with Hausdorff's formula to reflecs this dimensional gain. It's two! . From this and other monsters, fractals are born, but at the turn of the 20th century, most are much too complex to render visually. For more on the Dragon and other monsters see Bannon, Mathematics Teacher, March, 1991, pp. 178-185.
Adding to all this uncertainty is Heisenberg's Uncertainty Principle of 1927, which established the physical impossibility of measuring exactly the position and motion of atomic particles. The behavior of those particles, in similar locations and under similar influences, seemed unpredictable. This is now referred to as sensitivity to initial conditions .
Godel, in 1936, threw a wet blanket over Hilbert's hope that an axiomatic system could be devised that would unify all of mathematics and make it logically consistent. Godel proved that any system of axioms which is complex enough to include basic arithmetic properties will be "incomplete," in that there will be statements which cannot be proved true or false. He gave examples and showed how to construct such statements. In 1963, Paul Cohen proved that the continuum hypothesis of Cantor, which states that there is no transfinite number between the cardinals for the natural numbers set and the real set, is one of those undecidable statements .
Relativity, uncertainty, and incompleteness presented themselves as obstacles on the surface, but they really are superb achievements. Our human resources have not been wasted in the search for impossibilities. They have been redirected toward more fruitful ventures. Einstein's Theory has withstood all of the tests that 20th century discoveries and developments in technology have thrown at it.
In the 1920s Gaston Julia began a study of function recursion in the complex plane. Recursion is simply the continual recycling of the result of a function back into the function. Take a complex number to start, called the seed, and use it as the input in a function. Then use the output of the function as the next input, and continue to do so. See what happens eventually. While you're at it, see what happens to seeds which are nearby. Do they have similar fates, indicating stability in the region, or do they yield unpredictable outcomes? The sequences of outputs from the recursion are called orbits, and they have behaviors analogous to those of heavenly bodies. They can escape to infinity, much like the Voyager satellites have escaped the solar system. Other orbits may crash in toward zero, or settle into a stable periodic behavior, while others can become completely chaotic. These analytical tools have been shown to have widespread applications in the modeling of many natural systems in addition to astronomical ones. Figure 5 is a graph which distinguishes stable from chaotic and escaping orbits when seeds near the origin are subject to recursion in the function F(x) = x^2 - 0.72 - 0.251i. The boundary is a fractal and is called a Julia set. This type of analysis is the germ for creating the models which have proven useful in the study of complicated dynamical systems. This study is popularly referred to as the science of Chaos.
Julia had an idea that the graphs of the areas of stability and instability would be extremely intricate. He lived until 1978 which is about when computers gained the power to show Mandelbrot and others just how extraordinary the graphs were. Figure M shows the Mandelbrot set which classifies the outcomes of various complex quadratic functions.
But how could the graphs of complex functions in a complex plane possibly apply to anything of practical use? It turns out that they have many applications. Peitgen and Richter had a "smoking gun" experience when they discovered the Mandelbrot set, every tendril and bud, nestled "among cauliflower shapes with progressively more tangled knobs and furrows" in a magnified graph of theoretical magnetization properties. Their comment: "Perhaps we should believe in magic" . The study of dynamical systems continues to be one of the fastest growing fields today, and the story of how this field caught fire in the academic world has been marvelously told by Gleick. Surely Riemann, Poincare, Cantor, Julia and others must receive the credit for pointing the way toward this new science.
continue to Chapter 4, Part 2
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