The human condition craves uniformity and certainty. Unpredictability is a gross insult and offence to our sensibilities. That said, we are surrounded by a morass of chaos. Some folk seek rational refuge, but we are a small island surrounded by the havoc of absurdity.
The 17th century was a wondrous time for the relentless march of science and rational thought. After a millennium of intellectual darkness, religion in the West was receding from secular dominance at long last. It was a time for intellectual giants to shine. And of all the bright stars in the rational firmament, none was so bright as Sir Isaac Newton. Newton's tome, Mathematical Principles of Natural Philosophy (1687), represented a paradigm shift in Science and Mathematics. Newton's scholarship was so profound that the foremost scholars of the day could only gape in awe. Within this seminal work, Newton outlined his new mathematics, Calculus, and utilised this powerful and versatile mathematical discovery to complement and add new light to his laws of motion and gravity.
Newton's unravelling of nature's laws pointed to a deterministic, clockwork universe controlled by equations that predicted everything. Perhaps the mind of God had become clear, and God was a mathematician.
Newton used his new-fangled mathematics to calculate the motion of two celestial bodies, the Earth and the Sun. All well and good. However, when he tried to apply his equations to three celestial bodies, he confessed that the problem was beyond solution. His exact words: "...to define these motions by exact laws admitting of easy calculation exceeds, if not mistaken, the force of any human mind." I've discussed the intractable Three-Body Problem elsewhere on this esteemed blog- go seek and find.
It is time to introduce King Oscar of Sweden and Norway. The gracious king decided to mark his 60th birthday (1889) by offering a prize of 2,500 Crowns and a gold medallion to anyone who could solve a very difficult mathematical problem. Of the problems submitted, it was determined that the following question should be put forth: 'Is the solar system stable? Would the system remain, as if by clockwork, forevermore, or at some predetermined date, would collapse ensue? A very difficult question indeed. Solving this conundrum meant reevaluating the very equations that Newton considered unsolvable.
Enter our hero, the French mathematician Henri Poincare (b 1854), stage left. Poincare considered himself the man of the day and confidently set to work on the problem. First off, he had to solve the Three-Body Problem. He quickly realised the problem was too difficult and devised a shortcut. Firstly, he imagined two bodies interacting and introduced a speck of dust for his third body. He reasoned that the two bodies would be unaffected by the speck of dust, and thus, they would continue with ellipses around each other. The gravitational force of the two planets would attract the dust speck, and Poincare then attempted to work out the pathway the dust speck would take. The upshot of his analysis showed that the bodies under investigation would exhibit periodic paths, and periodic paths, by their very nature, are highly stable and repeatable. Obviously, the solution fell short of the rubric by a large margin. Newton's insight was vindicated, but Poincare's work set the stage for future contributions. Regardless of failing to come up with a definitive answer to the problem, the submitted paper was deemed impressive enough for Poincare to win the prize.
A Little Oopsie With Big Consequences
Poincares' paper was about to be published in the Swedish journal Acta Mathematica when one of the editorial staff posed a point of clarification concerning the rounding out of data. Each step in a mathematical proof requires rigorous attention. Poincare had made an assumption that seemed justified in his chain of reasoning. He assumed that rounding out some of his data points by small amounts would not significantly change the overall predicted orbits. He reassured his detractor that his data and reasoning were sound, but just in case, he would double-check. To his horror, he found that even the slightest corrections in his data could result in huge changes in the outcome of orbits (O Bugger!). Unfortunately for Poincare, by then, copies of his paper were already cascading from the press. In a desperate bid to save Poincare's reputation, the next few weeks were spent scooping up copies of his paper to prevent wide readership and limit the damage. Poincare offered to pay for the print run, which ironically exceeded the money gained for the prize.
In 1890, Poincare penned a second paper discussing the stark realisation that small deviations in the parameters defining stable systems could result in large deviations, culminating in unpredictable instability. His mathematical blunder led to the concept of Chaos Theory—a new and exciting branch of mathematical endeavour. However, it wasn't until meteorologist Edward Lorenz's rediscovery of chaotic dynamics in the 1960s that the concept became widely known. This story and others will have to await the second part of this saga.
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