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The Algebra of the Atonement

April 26, 2012

The Algebra of the Atonement

My son, Andrew, asked me once, “Why didn’t God just give Adam and Eve a second chance?” It was a good question and it deserved a good answer. After thinking about it, my response was that He did something much better. In Christ, He fixed the problem of sin in a permanent way so much better than a second chance. My aim in this discourse is to show that Christ’s substitutionary atonement is mathematically (specifically algebraically) logical. It glorifies God as the ultimate problem solver, the master of divine mathematics.

 ImageOne frequent path to understanding abstract concepts is through more concrete models. However, sometimes it works the other way as well. Physical scientists typically are also quite comfortable with using another abstract tool, namely mathematics, to understand, explain, and model physical phenomena. Theoreticians, likewise, use mathematics to postulate unobserved phenomena. To a scientist, if something can be described in the mathematical language of equations and logical syllogisms, then it is considered as “understood.” It is even considered “mostly understood” if the equations are not an exact description of all aspects of the phenomenon but, in spite of simplifying assumptions nevertheless, provide a generalized approximation of it. Because mathematics has the principles of logic underpinning it, if something is mathematically sound, it is also implicitly logical. Is it possible to use the same skills a scientist uses to describe mathematically (and logically) a true spiritual phenomenon to demonstrate the soundness of its logic?

My personal understanding and appreciation of Christ’s substitutionary atonement has benefitted from this thought process. Ultimately, the atoning sacrifice of Christ provides satisfaction to God because He says that it does and that is really all that matters. We can trust in that and there is no further need for discussion. However, God goes to great length in scripture to explain it in terms we can understand. When scripture uses terms like “substitute,” several concepts come to mind such as a substitute teacher or a substitute motion. A less common (except to physical scientists, engineers, and mathematicians perhaps) is the idea of an algebraic substitution. However, if we are going to talk about substitutionary atonement in algebraic terms we need an algebraic equation or set of equations with a variable for which we can make a substitution. It may seem a bit of an oversimplification, but what reformed theology calls  the covenant of works can be stated as a logical IF-THEN-ELSE statement like in a computer program.

IF  my righteousness = God’s standard
THEN eternal life
                  ELSE physical death, hell, and eternal separation from God
GoTo Start

Using variables the equation becomes:

GoTo Start

Our federal representatives Adam and Eve along with us by imputation of their sin have failed the “IF” test and are living in the “ELSE” statement. The path given by the ELSE statement is hopeless because it never ends, the payment for sin is infinite man-years. If your head is spinning at this point and you’re wondering what’s the point, just be patient for about two more paragraphs.

In an algebraic substitution operation, a substitution is only allowed if the substituted variable has the same characteristics of the variable it is being substituted for. Can the substitutionary atonement of Christ be understood in terms of an algebraic substitution in to the IF THEN statement of the covenant of works? The doctrine of the substitutionary atonement has two components. First, Christ takes the penalty for our sins (D) and gives us his righteousness. The amount we owe God for our sins is infinite man-hours. Infinite man-hours is the price as shown below.

(finite man) x (infinite hours)= infinite man-hours

When Christ suffered and died he took our place because an infinite being suffered and was dead for a finite time. The payment was exactly what was needed, namely infinite man-years as shown below.

(infinite man) x (finite hours) = infinite man-hours

In Christ we are able to reach the end of the ELSE statement and go back to the Start. However where do we go from there? We go back to the original IF statement because we are “born again”. If we are in Christ, we now meet the qualifications of the IF statement and enter into fellowship with God and eternal life which has no end. It is eternal. Moreover, the righteousness that is imputed to us is sure and not dependent on our own efforts. This ensures that we NEVER fail the IF test.

There are several implications of this understanding. First, it is necessary that in the incarnation that Jesus be fully man and fully God. That’s the only way the “math” works. For Christ to substitute for us in the “ELSE” statement, he has to be fully man. For the full consequence of the ELSE statement to be met, he must also be God (infinite). Note that in this proposed understanding, God’s justice is preserved. The original covenant of works is not cast aside but a solution (the covenant of grace) to the dilemma (the covenant of works) is solved without abrogating the original covenant (the covenant of works). One goal of the Old Testament ceremonial sacrificial system was to provide a hint about the type of solution, a substitutionary atonement, that would one day be provided. For the Old Testament Jews, it was an annual rehearsing of the promise to come in the incarnation, death and resurrection of Christ.

Once the atonement was paid in full, Christ was resurrected as a testimony that the ransom indeed had been paid in full. The purchasing our salvation was a real cost to God, a non-fatal wound that he incurred on our behalf. It is a finite x infinite dimensioned gash that God the Son bears to this day as the resurrected Christ showed his wounds to the disciples and as pictured in the wounded lamb in John’s vision recorded in Revelation.

I know this is kind of a geeky approach to the atonement. My goal is not to “check God’s math” but to understand more deeply the salvation he has wrought and the above discourse does that as I view it through the lenses of my logic and science-trained mind and eyes.


Entropy and the Glory of God

March 31, 2009
Department of Entropy

Department of Entropy

Last week my lecture on chemical thermodynamics touched on entropy and the second law of thermodynamics. Fortunately, I was free to add some thoughts of my own on implications of these subjects because the class is at Wheaton College. My remarks dealt with 1) whether or not the second law of thermodynamics can be used to support or refute evolution and 2) speculation on whether the second law pre-dated or resulted from the fall of man in the Garden of Eden.

1) The second law of thermodynamics states that for all spontaneous reactions the entropy of the physical universe increases.  Written in equation form it is:

ΔSuniverse = ΔSsystem + ΔSsurroundings > 0

The creationist looks at this and says that the development of more complex organisms is a decrease in entropy (i.e. increased order equals decreased entropy) and that the increase in the entropy of the surroundings is not sufficient to offset the decrease in the entropy of the system. The only way around the second law is the input of energy from outside the universe, (a creator God who exists outside the physical universe). The evolutionist looks at the effect man has had on the earth as an increase in entropy greater than the decrease in entropy resulting from evolution. Both these arguments are difficult to prove because quantifying the terms ΔSsystem and ΔSsurroundings for the biosphere is impossible.

2) Since entropy is related to the degree of “disorder” in a system, the argument can be made that the universal trend toward disorder is a result of the fall and was not the way creation was originally intended to operate. However, I do not think it is necessary to associate the second law with the introduction of sin into the world. Clearly God creates order out of chaos as is described in Genesis. I would like to argue that God’s creation is not diminished in glory if it has an inherent trend towards disorder. If the trend towards disorder was absent, creation would not be dependent upon God’s sustaining power and God would not be glorified to the same degree as if creation is inherently dependent upon Him.

Now, in the second law of thermodynamics . . .

"Now, in the second law of thermodynamics . . . "

A published article similar to my “Providence and Probability” posting

January 23, 2009

In 2000, an article was published by Jason Colwell from Cal Tech that made may of the same points that I made in the “Providence and Probability” posting. The reference for the article is  International Journal for Philosophy of Religion 48: 131–138, 2000. The following link can be used to view the first page of the article: Colwell has another article worth reading. The reference is International Journal for Philosophy of Religion 53: 147–161, 2003. The title of the article is “The historical argument for the Christian faith: A response to
Alvin Plantinga.” A link to the first page is

Providence and Probability

January 15, 2009

Let me begin with a disclaimer about this blog entry. This entry falls into the category of speculative as opposed to certainty. That is to say, I’m holding at least some aspects of this entry loosely in my hand as compared to the death grip I use for those things that are absolutely certain such as the deity of Christ and the authority of scripture.

For some time now I have struggled with the tension between the concept of God’s sovereignty in the universe and with what little I’ve learned over the years as a scientist. First of all I’ve come to see that the concept of God’s sovereignty is not that of the ultimate puppet master–consciously, directly, and specifically determining and controlling the location and energy of every proton, neutron, electron (and all those sub atomic particles with the cool names that I don’t understand). In establishing the universe, God clearly set up rules for much of it to function autonomously. By these “rules” I mean what we call the laws of physics. Chemistry and biology are built on the foundation laid by the fundamental laws of physics. For example, chemical reactions are governed by thermodynamics and kinetics. They are reproducible and predictable and because of that reproducibility chemical processes can be developed and biological systems can be sustained.

At the atomic/molecular level there is a degree of what I will call ordered randomness. For example, there is a distribution of the velocities of gas molecules in a container at a given temperature, pressure, and number of gas molecules. The “temperature” of the gas is a measure of the average velocity of the gas molecules in the container. The shape of the velocity distribution plot is predictable and governed by basic statistics.  Which molecule is traveling at what direction and at what velocity at any given time is randomly determined. Yet in that randomness there is order.  The dynamic equilibrium that occurs between a liquid and it’s vapor is also “ordered randomness”. The partial pressure of the vapor above the liquid can be predicted using theory and experiment. However, which molecules are in the liquid and the vapor phase at any given time is constantly changing because the equilibrium is dynamic.

On the macro-scale we also see randomness that ranges from what clothes I put on in the morning to what lane I drive in on the tollway on my way into work. Granted, these choices are limited by the clothes in my closet and drawers and the number of lanes that are open (i.e. not under construction) on the tollway. In experimental data, the underlying randomness is often referred to as noise. Sometimes it is an artifact of the measuring instrument and other times it is inherent in the system being studied.

Thus to varying degrees this stochastic background noise permeates the universe on all levels. It obeys the laws of probability but it is random nevertheless. The first thesis I propose in reconciling this stochastic background noise with the sovereignty of God is that God has ordained this stochastic background noise in creation for at least two reasons. The first reason is that in a system with many moving parts, it is necessary, and secondly it gives Him freedom to work his sovereign plan in a “stealth” manner that most of the time appears to be completely natural-like riding with no hands appears to a small child  like the bicycle is steering itself.

First of all, let me offer two arguments as to why this stochastic background noise is necessary. The first argument is based on analogy. I am fully aware that arguments based on analogy are never sufficient to prove a point but they do serve to make us feel more comfortable with a point that at first makes us feel uncomfortable. Consider an automobile engine. In order for that engine to work there has be some tolerance in the dimensions of the moving parts. Pistons cannot fit too tightly in the cylinder or they will not move and there has to be some allowance in the design of the engine for dimensional changes due to frictional wear, erosion, and corrosion. These tolerances result in a limited randomness to the location and dimensions of the engine components at any given time. It is a necessary component of the engine design. The number of “moving parts” in the universe is an extremely big (but finite) number and requires a degree of tolerance in the design for it to function.

The second argument is that designing a universe that runs semi-autonomously or semi-automatically with a degree of stochastic randomness from the subatomic to the cosmic scale is an incredible feat. There are no sloppy patches or kludge fixes, it is an elegant system that speaks volumes about the power and wisdom of its designer (more on this in a future post). In short, it glorifies God.

In discussing how God uses this stochastic background noise to stealthily work his sovereign will I will use two scripture passages, one from the old testament and one from the new testament.  In I Kings 22:29-37 King Jehoshaphat of Judah goes into battle alongside  King Ahab of Israel against the Syrians.  To make himself less of a target, Ahab disguises himself as a common soldier while Jehoshaphat goes to battle wearing royal battledress. The Syrians had orders specifically to kill Ahab and for a while pursued Jehoshaphat (recognizable as a king by his garments) until they realized that he was not King Ahab. While this was going on in verse 34 it states, “But a certain man, drew his bow at random and struck the king of Israel between the scale armor and the breastplate. The would proved fatal and as the words of the prophet had spoken when the washed the blood out of the bottom of his chariot, the dogs licked up his blood and the prostitutes bathed in it (verse 38). In this passage God clearly superintends a “random” event (the arrow shot) to work his divine will. Prophecy ahead of the event called attention to the seemingly random event as in fact the specific will of God.

A similar event occurs in the Matthew 17: 24-27. In this passage Jesus instructs Peter to go to the sea (of Galilee) cast a hook and take the first fish he catches. He then tells Peter that when he opens the mouth of the fish he will find the exact amount of money (one shekel) to pay the half-shekel tax for Jesus and himself. Jesus did not tell him where to stand or where to cast the hook. He demonstrated complete control over a random event. It was regarded as a miracle because no one but God could achieve such a feat.

It is my opinion that God directly intervenes regularly in the realm of the stochastic background noise in our lives to work His will. Doing so in this manner exhibits an amazing degree of artistry and finesse because although the hands are not on the handlebars, He is still able to direct the bicycle in the direction he chooses. We do have freedom in the realm of stochastic background noise but those that think that the stochastic background noise is all there is are like those who leave the concert hall right after the cacophony of the orchestra tuning concludes and the buzz in the audience subsides mistakenly assuming that the concert is over. They are in fact missing out on hearing and playing in a magnificent symphony composed and created by master of the universe.

future blog topics

January 7, 2009

vanityfair3– Intelligent Design in Legoland
– The Critics Real Problem With Intelligent Design
– Entropy and the Fall
– Laboring in the Lab East of Eden
– All Work is Honorable
– Advice to Young Researchers (probably a series of posts)
– Soteriology, sanctification, thermodynamics, and kinetics