Sci-fi website Io9 recently posted two thought-provoking posts on mathematics that related in unusual ways to God. Though the writers of this website, by and large, are probably not deeply religious (they certainly do not seem so), it is worthwhile to examine what the mathematics they discuss has to deal with genuine religious belief and practice, and in the relationship between God and mathematics.

A Modal Proof of the Existence of God

Some decades ago, the eccentric mathematician Kurt Godel wrote a modal proof of God, using the branch of philosophy known as ontology [1]. He apparently did not make the proof as a sign of his own belief in the Lord God of Abraham, Isaac, and Jacob, but rather as a mathematical exercise of a particularly eccentric (and controversial) sort.

Nonetheless, though his proof of God (which amounts to assumptions that cannot be disproven exhaustively, a typically inadequate and typically German sort of divine proof) is itself defective on some levels when considered by the standards of belief to which genuine theists are held, it does hold intellectual appeal as a philosophical demonstration of a profound truth nonetheless. St. Anselm had written a similar proof of the existence of God, and such a proof is very strikingly similar to the presuppositional apologetics of Van Til and his followers in Theonomy. It is not the mathematical proof itself, but rather its use as an idle intellectual exercise rather than as a statement of personal belief that makes it defective.

Reading this proof, and examining on its importance, one hears the echoes of Romans 1:18-23, which ought to stand as a warning to the scientists and philosophers of our decadent age, as much as it was to our forefathers in the time of classical Greece and Rome: “For the wrath of God is revealed from heaven against all ungodliness and unrighteousness of men, who suppress the truth in unrighteousness, because what may be known of God is manifest in them, for God has shown it to them. For since the creation of the world His invisible attributes are clearly seen, being understood by the things that are made, even His eternal power and Godhead, so that they are without excuse, because, although they knew God, they did not glorify Him as God, nor were thankful, but became futile in their thoughts, and their foolish hearts were darkened. Professing to be wise, they became fools, and changed the glory of the incorruptible God into an image made like corruptible man–and birds and four-footed animals and creeping things.”

We can’t say we weren’t warned–we share in the same arrogance and the same sins as the Hellenistic philosophers (especially the Greek ones) whose moral decadence and hostility to God’s ways led them to depravity, division, and conquest by the Romans, and whose civilization was ultimately destroyed because of ungodliness. If we wish to avoid that fate, we ought to take the sovereignty of God seriously. It is not sufficient to prove the existence of God (itself not a terribly difficult task–it is most easily done by contradiction, given the paradox of the many and the one that exists in the philosophies of man that seek to deny God and that can only be resolved by a being who combines both attributes in unity, namely God. Van Til, Banhsen, and others have provided those proofs about the incoherence and contradiction inherent in the humanistic project, should anyone care to examine it.

Perhaps that is why Godel’s proof is so controversial. Though it seems to have been an errant intellectual exercise on the part of Godel himself, others who are hostile to God’s ways understand that beliefs and proofs have consequences, and they do not wish to be held accountable to the standard of God’s law, and hence must deny any proof of God’s existence, lest they enter into condemnation. More than Godel, such people understand the danger of any kind of proof of God, but such people are precisely those whom Paul speaks about as suppressing the truth in unrighteousness. Let us neither follow in such sin ourselves nor be found to approve of those who practice it.

God And Infinity

Georg Cantor, a 19th century mathematician born in Russia but raised in Germany, achieved lasting fame for using an obscure and often neglected branch of mathematics known as set theory (a branch of mathematics, it should be noted, that I am particularly fond of) in order to rigorously prove the nature of infinity [2]. Clearly the implications of infinity are somewhat massive to finite beings such as ourselves.

Cantor proved that there were different orders of infinity based upon one-to-one correspondence. For example, even and odd numbers, cardinal numbers, integers, and rational numbers all have one-to-one correspondence with each other and all belong on what is called the aleph-null level of infinity. This level of infinity is probably the easiest to grasp, because it leads to the foolish paradoxes of “infinity plus one” (which equals infinity), or of the “Can God create a rock so big that He can’t lift it” that show just how inadequate the understanding of infinite is for human reasoning.

However, there was a second-level (which we call aleph-one) level of infinity on the real numbers because there is no one-to-one correspondence between the real and rational numbers. There is always a way to ensure that the real numbers have “more” items than can correspond on any level with rational numbers, and therefore they exist on a higher plane of infinity. Other mathematicians have tried to make infinite levels of infinity by using other means (like power sets), positing that there are infinite levels of infinity, ultimately going to the aleph-aleph-null level. Cantor speculated that such a level of infinity was God, an issue that was (of course) controversial.

The problem that an infinite universe creates for finite beings is how infinity can be harnessed and controlled. We know as finite and limited beings ourselves, that our ability to control the chaos of the universe around is limited and very partial. At least we should be aware of that, as even a cursory analysis of fields from medicine, civil engineering, military history, and mathematics ought to make plain. At best we are able to create islands of order within chaos, or at worst our own attempts to control the threats of nature make our lives more complicated and unstable.

Likewise, there is at least some kind of unease in the mind of people who believe that random and purposeless processes can create order in the level of the infinite and transcendent. This unease is present because whenever someone reminds such nonbelievers of the truth they spout nonsense about the biological theory of evolution and quantum creativity. Philosophy of science or the nonsense of deconstructionism is the last refuge of the infidel. There is, after all, at least an aleph-null infinity of human nonsense theories.

[1] http://io9.com/5805775/proof-of-the-existence-of-god-set-down-on-paper?utm_source=io9+Newsletter&utm_campaign=81335c793a-UA-142218-29&utm_medium=email

[2] http://io9.com/5809689/a-brief-introduction-to-infinity?utm_source=io9+Newsletter&utm_campaign=fcab1b94c8-UA-142218-29&utm_medium=email