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By Root 723 0

a moment. Imagine what would have happened had the logicist endeavor been entirely successful. This would have implied that mathematics stems fully from logic—literally from the laws of thought. But how could such a deductive science so marvelously fit natural phenomena? What is the relation between formal logic (maybe we should even say human formal logic) and the cosmos? The answer did not become any clearer after Hilbert and Gödel. Now all that existed was an incomplete formal “game,” expressed in mathematical language. How could models based on such an “unreliable” system produce deep insights about the universe and its workings? Before I even attempt to address these questions, I want to sharpen them a bit further by examining a few case studies that demonstrate the subtleties of the effectiveness of mathematics.

CHAPTER 8

UNREASONABLE EFFECTIVENESS?

In chapter 1, I noted that the success of mathematics in physical theories has two aspects: one I called “active” and one “passive.” The “active” side reflects the fact that scientists formulate the laws of nature in manifestly applicable mathematical terms. That is, they use mathematical entities, relations, and equations that were developed with an application in mind, often for the very topic under discussion. In those cases the researchers tend to rely on the perceived similarity between the properties of the mathematical concepts and the observed phenomena or experimental results. The effectiveness of mathematics may not appear to be so surprising in these cases, since one could argue that the theories were tailored to fit the observations. There is still, however, an astonishing part of the “active” use related to accuracy, which I will discuss later in this chapter. The “passive” effectiveness refers to cases in which entirely abstract mathematical theories had been developed, with no intended application, only to metamorphose later into powerfully predictive physical models. Knot theory provides a spectacular example of the interplay between active and passive effectiveness.

Knots

Knots are the stuff that even legends are made of. You may recall the Greek legend of the Gordian knot. An oracle decreed to the citizens of Phrygia that their next king would be the first man to enter the capital in an oxcart. Gordius, an unsuspecting peasant who happened to ride an oxcart into town, thus became king. Overwhelmed with gratitude, Gordius dedicated his wagon to the gods, and he tied it to a pole with an intricate knot that defied all attempts to untie it. A later prophecy pronounced that the person to untie the knot would become king of Asia. As fate would have it, the man who eventually untied the knot (in the year 333 BC) was Alexander the Great, and he indeed subsequently became ruler of Asia. Alexander’s solution to the Gordian knot, however, was not exactly one we would call subtle or even fair—he apparently sliced through the knot with his sword!

But we don’t have to go all the way back to ancient Greece to encounter knots. A child tying his shoelaces, a girl braiding her hair, a grandma knitting a sweater, or a sailor mooring a boat are all using knots of some sort. Various knots were even given imaginative names, such as “fisherman’s bend,” “Englishman’s tie,” “cat’s paw,” “truelover’s knot,” “granny,” and “hangman’s knot.” Maritime knots in particular were historically considered sufficiently important to have inspired an entire collection of books about them in seventeenth century England. One of those books, incidentally, was written by none other than the English adventurer John Smith (1580–1631), better known for his romantic relationship with the native American princess Pocahontas.

The mathematical theory of knots was born in 1771 in a paper written by the French mathematician Alexandre-Théophile Vandermonde (1735–96). Vandermonde was the first to recognize that knots could be studied as part of the subject of geometry of position, which deals with relations depending on position alone, ignoring sizes and calculation with quantities. Next in