"The miracle of the appropriateness of the language of mathematics to the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve".
from The Unreasonable Effectiveness of Mathematics in the Natural Sciences by Nobel Laureate Eugene Wigner (1960)
An article by the author Mario Livio, who recently wrote the book Is God a Mathematician?, appeared in Plug Magazine in December. After reading the article, I immediately added his book to my reading list.
Essentially, the question that both the article and book address is whether or not the effectiveness of mathematics to model and predict natural events is unusual relative to other historical attempts to do so. And if so, why does mathematics possess this unreasonable effectiveness in the natural sciences?
Livio categorizes the effectiveness of mathematics into two facets, labeling them as active and passive. By active, he is referring to those instances where scientists actively use mathematics to “light their path” through complexity of natural phenomena. They develop new models and use these models to answer questions about the world around them.
Even more interesting to me is the second facet. In the passive sense, the effectiveness of mathematics is demonstrated when mathematics in its purest sense is developed simply for the sake of mathematics but eventually becomes a very powerful model for an aspect of nature. Pure mathematics studies patterns and structures within the mathematical objects themselves, with no concern for the applicability of the mathematics to the real world. In fact, a true pure mathematician would see a physical, real-world application of their theorem as something that takes away from the purity and beauty of mathematics.
While I’m sure his book (that I’ve not yet read) contains a number of examples of these, the article mentioned above provides the example of knot theory. “Knots, and especially maritime knots, enjoy a long history of legends and fanciful names such as ‘Englishman's tie,’ ‘hangman's knot,’ and ‘cat's paw’.” As Livio tells it,
Knots became the subject of serious scientific investigation when in the 1860s the English physicist William Thomson (better known today as Lord Kelvin) proposed that atoms were in fact knotted tubes of ether (that mysterious substance that was supposed to permeate space). In order to be able to develop something like a periodic table of the elements, Thomson had to be able to classify knots—find out which different knots are possible.
As we all know, the existence ether was later disproven, but an entire theory of mathematics had been born out of this. The concept of a mathematical knot developed, theorems for classification of these knots were proven and knot theory continued as area of interest to many pure mathematicians.
The surprising part is when physicists in the late 20th century discovered a connection between aspects of knot theory and the burgeoning field of string theory.
In particular, string theorists Hirosi Ooguri and Cumrun Vafa discovered that the number of complex topological structures that are formed when many strings interact is related to the Jones polynomial. [The Jones polynomial is used in the classifications of knot variations]
This connection is completely unexpected and most surprising. In a particular twist, this application of a mathematical idea was once developed for a theory about the fundamental substance of all matter, namely ether, which was eventually discarded. It returns much later to explain a great deal about today’s modern theory about the fundamental substance of all matter.
Now for my input. GOD IS A MATHEMATICIAN! As I tell all of my students, one of the primary reasons I study mathematics and call myself a mathematician is that it has been established as the language of science and science is, at its very core, the study of God’s creation. Due to the “unreasonable effectiveness” of mathematics, its clear to me that God’s nature has a very strong mathematical component. That God follows patterns that are detectable to his creatures tells me at least two things: (1) God is personal and wants us to discover him, and (2) God is faithful to follow his established pattern. He is immutable and unchanging. The effectiveness of mathematics by no means proves these traits of God, but from my viewpoint as a man of faith and a mathematician, it serves as very strong evidence for them.