"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.
Eventually or Frequently?
The two terms, "eventually" and "frequently," stood out to me as having fairly interesting definitions when the came up some introductory courses in analysis and toplogy while I was in graduate school. Specifically, I am referring their use in the context of infinite sequences. Now to lay the foundation for the illustration I'd like to make, recall that the definition of an epsilon-neighborhood of a point, , is the interval , for some . In other words, it is the set of all points within of . (an open interval)
Given a set, and an infinite sequence, , the sequence is said to be frequently in the set if for every, there exists some such that . On the other hand, the sequence is said to be eventually in , if there exists some such that for every , we have . We can define convergence of a sequence as saying that a sequence converges to a point if it is eventually in every epsilon neighborhood of some real number.
TRANSLATION: To the non-mathematician, think of an infinite sequence simply as an infinite list of numbers. The sequence is frequently in a set, if no matter how far you go out in the list, there is another item in the list after that, which is in the set. On the other hand, the sequence is eventually in a set if you can go far enough out in the list so that all the elements after that point are in the set. The convergence of a sequence means that we can get arbitrarily close to some point by going far enough out in the list.
So, what is the spiritual principle?
I like to think of my life, more specifically, my faith development as a sequence of discrete moments where I demonstrate my faith in Christ. Throughout each day, I have opportunities to obey God's call on my life or choose my own selfish desires. I involve myself in activities that can lead me closer to my Savior or further away. I think we'd all agree that there is NO guarantee in the Christian life that each day that goes by, NECESSARILY, draws us closer to God and to a godly character. That is, just because I am a Christian, doesn't mean that I am better today than I was yesterday.
However, (and this is a big however), I believe that the work of the Holy Spirit is to shape us into the character of Christ, eventually. Think of the character and life that God has designed you for. That is the point toward which our life is converging, once we have been saved. Now, we may oscillate near and far, but we will eventually be closer. If you could measure the closeness to that point, we could say that given a level of character close to that of Christ, there is some point in time past which we will be that close. If you read Paul, in his letters, he often speaks of all the aspects of character to which we should apply ourselves, and he always does so with the expectation that we CAN obtain it. We are converging to the character of Christ.
Now, without grace, we may be frequently in the neighborhood of that point but we cannot converge. It is only through grace, that we may truly converge to the mind of Christ.
What do you think?
My last entry mentioned that the next post would focus on why I find approximation theory so interesting. I've been chewing on an idea / illustration for some time now that I want to run by my readers. Much of my research in has been spent in using finite dimensional function spaces to approximate much more general spaces of functions. In most cases, the goal is to find a linear combination of basic functions that will be "close" to some function from a larger, infinite dimensional space.
For example, a simple infinite dimensional space is the class of continuous functions over a closed interval. These are generally nice functions to work with but it is still a very "large" space. Given a continuous function over an interval , let's say we want to approximate it with some quadratic function, . A quadratic function can be thought of as an element of a 3-dimensional function space with basis functions, . In approximating a function, we are trying to find the coefficients of these basic functions so that we have a "close" function to the original function. There are many ways to measure this closeness, but the idea is that each basic function contributes additional information that helps the approximating function get closer to the original function. If we chose to use cubic polynomials instead of quadratics, we have one more basis function . There is a theorem, which I will probably post later as a favorite theorem, called the Weierstrass approximation theorem that implies that for any continuous function over a closed interval, there exists some polynomial that is arbitrarily close to that function. But, the caveat is that the more accurate you want the approximating function to be, the more basis functions you'll need to contribute information. For polynomials, this translates to higher and higher degrees. As a side note, it turns out that there are much better functions for approximating than general polynomials, such as piecewise polynomials or splines. And don't get me started on the beauty of splines.
Now, for the spiritual application. I see this as a perfect explanation for the necessity of community in our faith. Let God, himself, be the infinite space that we wish to understand (e.g., obtain an approximate knowledge of). Our minds, being finite, are limited and unable to fully comprehend the character and nature of God. Our experiences throughout our life help to inform our faith but these experiences are also finite and limited to a single individual. I see each person as a basis element in a finite dimensional space. Meaning that to better understand God, we can use other's experiences and knowledge to contribute to our understanding of who God is. We listen and learn from each other and thus our approximate understanding of God becomes closer and closer to an accurate picture of God. Obviously there exists the possibility that individuals can take away from our understanding thus we must listen and learn with discernment but, in general, we are better able to know God as part of his body, the community of faith.
It was Henry Blackaby, in Experiencing God, who described the four fold way to know the will of God, through the Bible, the Holy Spirit, Circumstances and other believers. Each, in a limited way, serves to better inform us and help us gain and more accurate, but still approximate knowledge of God.
The nice thing about numerical analysis, my chosen field, is that we are able to specifically quantify the error. We can know in some sense a bound on our error, and know how far off we could possibly be in our approximation. Perhaps, the analogy fails here but I'd like to believe that necessarily, the more elements we use to develop our knowledge of the Holy, the more accurate our image of our Father in Heaven.
What do you think?
A Well Ordering Principle
In set theory, we describe a set as being totally ordered if there exists a binary relation, , that is antisymmetric, transitive and total.
- A binary relation, is antisymmetric if and
- A binary relation, is transitive if and
- A binary relation, is total if either or for every
Now, a totally ordered set is said to be well-ordered if every subset has a least element, that is, . In other words, if a set has the property that every subset, or part, of it will have a least element. For example, the set of positive integers, is well-ordered since no matter which group of those numbers you pick, there will be a smallest one. On the other hand, the set of all integers, is not well-ordered since you could take a subset that contains all the numbers down to negative infinity and thus, has no least element.
By now, I have probably lost all of my current readers so let me get to my point. As I was pondering this concepted of well-ordered sets, I recognized a something applicable outside of the world of mathematics. In today's world, we are told to leave our work at the office, to leave our family life at home and to keep our faith to ourselves, all so as to not offend anyone. So, we compartmentalize. We put each part of our life separate from the other. Now there is something to be said for this but clearly the faith that we have cannot be kept separate from the rest of our life.
So I propose a new definition. Let us say that a well-ordered LIFE is one in which every part of our life has a "least element." And by "least element", I am thinking in terms of a foundational element or a most important element or maybe even a guiding element.
This definition brings new light to the often quoted passage, "Seek first the kingdom of God." When Jesus said this, he could have meant that we are to put Him first, chronologically. The first thing we are to do each morning is seek the kingdom. When Jesus said this, he could have meant that we are to put Him first, by importance. For example, any time there is a scheduling conflict between church and family, work or play, we must put the church first. Now, certainly our life is better off through prioritizing our faith, and yet, doesn't Jesus mean more than either of these. The principle of seeking first the kingdom is summarized well by the well-ordering principle. Every area of our life should be built on our faith, informed by our faith, permeated by our faith and inseparable from our faith. That is putting the kingdom first and that is a principle I need to work on.
Do you have a well-ordered life?