Dec 14

Spiritual Principle #3

Spiritual Principles 5 Comments »

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, a \in \mathbb{R}, is the interval (a-\varepsilon,a+\varepsilon), for some \varepsilon. In other words, it is the set of all points within \varepsilon of a. (an open interval)

Given a set, A \subset \mathbb{R} and an infinite sequence, \{x_n \}_{n=1}^{\infty}, the sequence is said to be frequently in the set A if for everyN \in \mathbb{Z}^+, there exists some n \geq N such that x_n \in A. On the other hand, the sequence is said to be eventually in A, if there exists some N \in \mathbb{Z}^+ such that for every n \geq N, we have x_n \in A. 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?

written by SplineGuy

Dec 13

Spiritual Principle #2

Spiritual Principles No Comments »

Discretize Spaces

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 [a,b], let’s say we want to approximate it with some quadratic function, p(x) = ax^2+bx+c. A quadratic function can be thought of as an element of a 3-dimensional function space with basis functions, p_1(x) = 1, p_2(x)=x, p_3(x)=x^2. 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 p_4(x)=x^3. 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?

written by SplineGuy

Nov 30

Spiritual Principle #1

Spiritual Principles 1 Comment »

A Well Ordering Principle

In set theory, we describe a set X as being totally ordered if there exists a binary relation, \leq, that is antisymmetric, transitive and total.

  • A binary relation, \leq is antisymmetric if a\leq b and b \leq a \ \Rightarrow \ a=b
  • A binary relation, \leq is transitive if a \leq b and b \leq c \ \Rightarrow \ a \leq c
  • A binary relation, \leq is total if either a \leq b or b \leq a for every a, b

Now, a totally ordered set is said to be well-ordered if every subset has a least element, that is, \forall \ A \subset X, \ \exists \ m \in A \ \ni \ m \leq a \ \forall a \in A. 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, \{1, 2, 3, 4, \ldots \} 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, \{ \ldots, -3, -2, -1, 0, 1, 2, 3 \ldots \} 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?

written by SplineGuy