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.
One 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
Using variables the equation becomes:
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.