David Peat

130 From Certainty to UncertaintyThis brings us to yet another paradox of chaos, for, although theequations of a system are totally deterministic, the final results cannever be calculated exactly. Even the fastest and largest computers arefinite. They may carry out a calculation to ten decimal places, which isgood enough for most purposes. But this means that there is alwaysuncertainty in the final decimal place—one part in ten billion. Thisseems unimportant until one realizes that this tiny uncertainty is beingiterated around and around in the calculation. Under critical conditionsthe cycling of even an almost vanishingly small uncertainty beginsto grow until it can dominate the entire result.The meteorologist Edward Lorenz discovered this in 1960 whenhe was iterating some simple equations used in weather prediction. Tospeed up the calculation he dropped some decimal places and, whenthe calculation was finished, to his great surprise he discovered that theresulting weather prediction was vastly different from his initial, moreaccurate calculation. A small uncertainty in the initial data of theweather system had swamped the final calculation.By analogy to the computer calculation, when a real weather systemis in an unstable condition a small perturbation can produce aradically different change of weather. With a system balanced on aknife-edge or at a bifurcation point, even the flapping of a butterfly’swing can send it in a totally different direction. The ancient Chinesedrew attention to the interconnectedness of all things by saying thatthe flapping of a butterfly’s wings can change events on the other sideof the world. In chaos theory this “butterfly effect” highlights the extremesensitivity of nonlinear systems at their bifurcation points. Therethe slightest perturbation can push them into chaos, or into some quitedifferent form of ordered behavior. Because we can never have totalinformation or work to an infinite number of decimal places, therewill always be a tiny level of uncertainty that can magnify to the pointwhere it begins to dominate the system. It is for this reason that chaostheory reminds us that uncertainty can always subvert our attempts toencompass the cosmos with our schemes and mathematical reasoning.There is yet another reason why a system, deterministic in principle,can be unpredictable in practice. Leaving the limitations of computersaside, it is impossible to collect all the data needed to character-