The famous Christian apologist, William Lane Craig, is famous for, among other things, arguing that God cannot do logical impossibilities. One example that he mentions is the idea of whether it is possible for God to make "2 + 2 = 5" true. Because such a thing is logically impossible, it is mentioned that God cannot do such a thing.
At first, this seems to make a lot of sense. But, when I look more deeply into this, I find problems with this. I first ask, "What is a mathematical formalism?" Is such a thing a reality in-and-of-itself? Or, is it simply a tool that we create to help us with the engineering tasks of everyday life?
It seems to me that mathematics is simply a tool that we use. If this is the case, I wonder about what has brought about this particular mathematical formalism. I imagine perhaps somebody collecting a bunch of berries from bushes in a forest. Divvying out the berries to his family, he grabs 2 from the pile. He then grabs 2 more from the pile and puts it on top of the 2 that he previously grabbed. He then notices that this result is the same as if he had simply grabbed 4 from the pile in the first grab. This, then, could perhaps be an origin of addition. Putting back in the pile could be an origin of subtraction. Grabbing the same quantity from the pile several times in a row could be an origin of multiplication. And, perhaps cutting a large animal into pieces for the family to eat could involve an origin of division. From this, the pieces could be put together to create a mathematical formalism.
Now, is it possible that there could be a mathematical formalism in which "2 + 2 = 5" is true? It seems to me that it is possible to create such a mathematical formalism. If Euclidean geometry, which appears to correspond to common experience, can be transformed into Riemann geometry by a change of defining assumptions, then perhaps the same can be done to create a new mathematical formalism in which "2 + 2 = 5" is true.
But, what use is a non-standard formalism if it doesn't express truths in our experienced reality? Perhaps the more relevant question is whether there exists a possible world where the common experience of grabbing 2 things from a pile and subsequently grabbing 2 more things will have the same result as simply grabbing 5 things from a pile. This clearly defies our common experience. But, is such a world possible? Perhaps one such possible world could have some metaphysical property where an interaction always occurs when the second collection of 2 items is grabbed and added to a pile of 2 items, resulting in the creation of an additional item to the pile. Suppose that this occurred for every collection of any substance such that it would appear to be a general law of nature. In such a case, the observed empirical result would be "2 + 2 = 5". Thus, a mathematical formalism matching this experience could be constructed. In fact, when Quantum Physics was first discovered, the results were so different from Newtonian Physics that this same process was followed. Tests were run. Observations were made. And a new mathematical formalism was created to match the observed results.
But, I can imagine William Lane Craig objecting to me here, saying "The claim is that within the standard formalism where 2 + 2 is defined as 4, then 2 + 2 cannot equal 5, and God cannot make it so." The Russian philosopher, Shestov, argued that it was possible that God might ask us to believe something that deeply betrayed our common understanding, such as the idea that "2 + 2 = 5" is true. To believe God despite the observed reality was said to be an act of faith. Alvin Plantinga describes this view as "extreme fideism," where he defines fideism as a conflict between faith and observed reality.
Godel's Incompleteness Theorem states that, for any formal system, there exist true statements about that system which cannot be represented in that system. Working with a mathematical proof for Godel's Incompleteness Theorem, we can find that mathematics is inconsistent. And, in its inconsistency, we can form the statement "2 + 2 = 5". Logic and reason themselves are also formal systems which similarly can be shown to be incomplete in this manner. In fact, if reason is simply a function of our brains, is it not possible that some aspects of reality as it exists (beyond the phenomena we experience) can contain truths which our brain cannot understand? If so, is it possible that "2 + 2 = 5", for our standard experienced mathematical formalism, is a truth beyond the understanding of the human brain?
All of this theory comes to practical use when we discuss the Trinity. God is One, but God is Three. What does this mean? Traditionally, it was said that there is one God with three persons. Augustine simply used the word "person" as a technical term because it sounded better than saying "three I-know-not-what". In the same way, I prefer to use the word, "member", but I do not wish to imply that each member is only a part of the whole substance of God. Augustine argued that, in every action, all three members of the Trinity are acting. Augustine also argued that, at Jesus' baptism, only the Father spoke from the sky and only the Spirit appeared as a dove. How can these two be reconciled?
Thomas Aquinas argued that, being the same substance, all three members of the Trinity share all substance properties equally in totality. On Aquinas' account, the only distinction between the members of the Trinity are in relational properties, that is, the mere fact that the Father is the "Father of the Son" and the Son is the "Son of the Father". Karl Barth described this by saying that there is only one Divine-I, one personality among the members though every member exists together at the same time. Thomas Aquinas is trying to understand the oneness of "with God and was God" in John 1:1 and "in the very nature of God" in Philippians 2 and "I and the Father are One" in John 10. But then we have to address passages in the Bible where Jesus submits to
the Father in Philippians 2, Jesus prays to the Father in John 17, the
Spirit intercedes to the Father in Romans 8, and the Father speaks to
Jesus at His baptism. How can these two be reconciled?
If the initial premises that Augustine and Aquinas use are true, and if their logical form is valid, then the conflict that we see in their reasoning should be understood as an anomaly, but true nonetheless. Somehow, someway, both are true. This doesn't mean that we stop doing more detailed study to try to understand the matter more clearly. But, as it stands today, there is a sense in which one could conceive of the statement that "2 + 2 = 5" is true.
"For we walk by faith, not by sight." --2 Corinthians 5:7 (HCSB)
