1. Religious Context
The mutakallimun held that the belief that God created the universe ex-nihilo was a tenet of faith that is so religiously fundamental that it is noninferentially known as part of Islam. The creation of the universe by God is mentioned in dozens of Quranic verses such as “All praise belongs to Allah, the Originator of the heavens and the earth” [35:1], and “O men, fear your Lord who created you from a single soul, and from it created its match, and spread many men and women from the two” [4:1] as well as many hadiths (e.g., “God existed without there being anything else besides Him” (Bukhari, 3191).
The falasifa, in the tradition of Aristotle, held that the universe was beginninglessly eternal, which entailed an infinite regress of contingent causes. The mutakallimun developed four major philosophical arguments for the impossibility of such an infinite series. One of these was an argument for the impossibility of an infinite series by correlation.
The crux of this argument is that cause and effect are correlatives, meaning that neither can exist without the other. But an infinite series of correlatives in one direction implies that the number of effects is greater than the number of causes. But it is impossible for a series of relatives to differ in quantity with their correlatives. Thus, an infinite series of correlatives is impossible (reductio ad absurdum).
2. Linguistic Definition
The root of tadayuf, dayafa, refers to one thing inclining towards another. It is said adaftu x to y: I made x incline to or be close to y, (Mu‘jam al-maqayis, 3:381).
3. Theological Context
3.1. Technical Definition
Tadayuf means two things, a and b, which are such that a’s relation to b is the cause of b’s relation a (as in sonhood and fatherhood). The conception of each of the two things depends on the conception of the other, (al-Ta‘rifat, 55).
3.2. The Full Argument
If there were an infinite series of correlatives such as causes and effects, an entailment would ensue whereby the number of effects would be one more than the number of causes, which is impossible. This may be explained as follows: all instances in a hypothetical (mafrud) infinite series are a paired cause and effect except the last effect, which is an effect but not a cause (ma‘lul wa-laysa bi-‘illa). Thus, the number of effects is greater than the number of causes by one. This entails that one effect exists without a cause, which is impossible, (Risala fi ithbat al-Wajib (al-Dawwani), p. 157).
If, on the other hand, the number of effects is the same (mukafi’an li) as the number of causes, then the last instance is a cause and not an effect, which entails that the series ends and has a beginning. It is, therefore, false that the series is infinite.
3.3. Non-Causal Correlatives
This argument applies to all correlatives, whether cause and effect or fatherhood and sonhood, (Risala fi ithbat al-Wajib (al-Dawwani), p. 158).
It also applies to the relation of before-ness and after-ness, such as an infinite series of created things, creature after creature without beginning. The correlativity in this case is that a) each instance is related to the one before it through the relation of after-ness and b) each instance is related to the one after it through the relation of before-ness, except for the (last) creature (makhluq), which does not yet have anything after it.
By the same line of reasoning described above, the supposition of this infinite series entails the existence of an instance that comes after another instance without there being anything before it, which is a contradiction.
3.4. Two-Way Infinite Series
One might object that this argument only works against an infinite series in one direction but not against a series that extends infinitely in both directions, having neither a beginning nor an end, in which case the number of relations and correlations will be equal. This was the position of many mutakallimun. Some mutakallimun, however, held that it is possible in that case to start with a specific effect and then complete the argument by looking back to those before it.
4. Contemporary Connections
Georg Cantor, the founder of set theory, developed a formal understanding of transfinite (“beyond finite;” for our purposes, “infinite”) sets. Cantor argued that some infinite sets that appear, at first glance to have different sizes (such as the infinite set of natural numbers, which appears to be smaller than the infinite set of integers) actually are the same size because the members of one set can be put into a one-to-one correspondence with the members of the other. Since Cantor, most mathematicians would think of the infinite collections of correlatives as sets of the same size because they can be put into one-to-one correspondence with each other (thus, the contradiction mentioned above would not arise).
William Lane Craig argues for the impossibility of an infinite series of causes by distinguishing between actual and potential infinities. Because of the counterintuitive consequences of potential infinities in the real world, he argues that an actual infinity is impossible (reinstating the possibility of the argument by correlation, which only requires the impossibility of an actual infinite).
