Paradoxes of Causal Infinitism
Let us workshop this essay about some logical problems faced by causal infinitism: the view that there can be an infinite chain of causes for an effect
Paradoxes of Causal Infinitism
Contradiction seems to result in any case of causal infinitism—forward, backward, simultaneous. An infinite regress of causes for an effect, that is to say, appears to be logically absurd no matter the direction of flow—indeed, even if cyclical. The following two paradoxes aim to highlight this fact.
Angel Paradox
Imagine that at each prior cause in a chain of causes receding infinitely back there is a unique angel that will blow its horn if and only if no angel at any prior cause already blew its horn. Now, pick any one of these causes, Cn. A horn is blown at a cause prior to Cn and so not at Cn. How so? Well, a horn would have been blown at a cause prior to Cn if a horn had not been blown at a cause even prior to that. In other words, an angel at Cn-1—a cause prior to Cn—would have blown its horn if no angel at a cause prior to Cn-1 (Cn-2) already blew its horn. (To spell this out for those struggling to connect the dots, realize that at Cn-2 a horn is either blown or not. If a horn is blown at Cn-2, then that means a horn is blown at a cause before Cn. If a horn is not blown at Cn-2, then that means a horn is blown at Cn-1—in effect, at a cause before Cn.)
But now we land in contradiction. Since a horn is blown, a horn is blown at a specific cause. That cause, of course, can be described as Cn—Cn being, after all, any arbitrary moment. For reasons just explained, though, no horn is blown at Cn. So a horn is blown at Cn and yet no horn is blown at Cn. We land in such contradiction by supposing that the chain of causes recedes to infinity. The contradiction goes away if the chain has a first member. For in this case the angel at the first cause blows its horn and all the rest of them do not blow their horns.
Captain Howdy Paradox
Imagine that during the witching hour—from 3am to 4am—a demon (Pazuzu) does a palm dance to see if he is going to possess a little girl (Regan MacNeil): he is inside Regan if and only if his palms are forward at the end of the series; he is not inside Regan if and only if his palms are backward at the end of the series. At 3am his right hand is level with his chin and his left hand is down at his waist, both hands palm-forward. At 3:30am he switches to a palm-back position. At 3:45am he switches to palm-forward. He keeps switching at every halfway mark, flipping his hands instantaneously (as a deity could) such that the end of the dance cannot culminate in his palms being somewhere in-between being palm-forward or palm-backward.
Now, since there is a later moment where palms are backward for each moment where palms are forward, it is absurd to say that he is inside Regan at the end (4am). And since there is a later moment where palms are forward for each moment where palms are backward, it is also absurd to say that he is not inside Regan at the end. In the worst case, then, Pazuzu is somehow both inside Regan and not inside Regan at the same time and in the same respect: a contradiction. In the best case, the series simply does not decide the final state: whether Pazuzu is or is not inside Regan is not settled by progressing through the series.
Even though the best case avoids contradiction, it still admits that there cannot be a regress of causes for an effect. Think about it. Either a final state never comes about or a final state does come about. Now, if a final state never comes about, then that amounts to admitting that there cannot be an infinite regress of causes for an effect. If a final state does come about, then that amounts to admitting that there cannot be an infinite regress of causes for an effect. For assuming the best-case scenario (namely, that a final state is not settled by the series), a final state comes about only if that final state is uncaused and so not caused in anyway by the series.
Responses?
I have heard several responses to these paradoxes that attempt to vindicate the possibility of an actually infinite regress of causes for an effect. As a great admirer of Spinoza, someone who endorses the reality of infinite chains of causes receding back ad infinitum (never bottoming out), I myself search for ways out of these paradoxes.
Some approaches are quite radical, such as denying the principle of contradiction on which these paradoxes depend—perhaps adding, in what amounts to a sort of aesthetic reinforcement, that dialethias are an essential part of the beauty of the world. Less radical is what we might call the “Multiverse Defense,” versions of which I am more prone to hear.
According to the Multiverse Defense, which I will put in the language of the Angel Paradox, at every causal step there is a bifurcation of realities: in one universe, the angel at a certain cause blows its horn; in another universe, it refrains. There is no contradiction because every potential scenario where an angel might blow its horn is realized in some universe or another. In Universe A, an angel at Cn blows its horn. In Universe B, an angel at C n−1 blows its horn, and so on. This continues across an infinite number of universes.
The problem with the Multiverse Defense should be clear. We can just limit our focus merely to a specific universe: universe A or B or so on. Consider universe A, for example. An angel at each cause in the infinite chain of causes in Universe A will blow its horn if and only if no prior angel has blown its horn. But then for any given cause Cn, a horn must have been blown at some prior cause—which, of course, is itself describable as Cn.
The Angel Paradox, given these boundaries, presents a stark challenge to causal infinitism within any given universe. The multiverse defense, as appealing as it might be to some, is effectively sidelined.
