Infinite Regresses: Vicious and Benign

The peripatetic (not Peripatetic) Kevin Kim once asked me:

Are all infinite regresses (regressions?) vicious? Why the pejorative label? Of the many things I don't understand, this must be near the top of my list, and it's an ignorance that dates back to my undergrad Intro to Philosophy days. When I first read the Thomistic cosmological proofs, I found myself wondering why Aquinas had such trouble countenancing the possibility that, as the lady says, "it's turtles all the way down."

Without a first, there can't be a second... so what? It doesn't follow that there must be a first element to a series. What makes a temporally infinite series (of moments, causes/effects, etc.) impossible?

Here is the answer I gave him, considerably expanded and updated:

1. No, not all infinite regresses are vicious. Some are, if not 'virtuous,' at least benign. The term 'benign' is standardly used. The truth regress is an example of a benign infinite regress. Let p be any proposition. And let 'T' stand for the operator 'It is true that ( ).' Clearly, p entails T(p). The operation is iterable. So T(p) entails T(T(p)). And so on, ad infinitum or ad indefinitum if you prefer. The resulting infinite series is wholly unproblematic. Whether you call this a progression or a regression, it doesn't cause any conceptual trouble.

Example. Let Snow is white be our base proposition. This proposition cannot be true unless it is also true that It is true that snow is white is true. And so on infinitely. It doesn't matter whether we think of infinity as potential or as actual. If potential, one can always add another member to the series. If actual, the entire infinite series is 'out there' in full actuality.

2. The truth regress is not vicious because it does not arise in the course of giving an explanation. The notion of explanation in play here is philosophical rather than empirical-causal. It would require a separate post properly to exfoliate the notion of philosophical explanation. But anyone with philosophical aptitude will have an intuitive sense of it. (And if you don't have an intuitive sense of it? Then I say you lack philosophical aptitude!) Consider the proposition Snow is white. The proposition is true, but what makes it true? A philosopher with realist inclinations will say that there is need of something extrapropositional to make it true. These philosophers, and I am one of them, think that some contingent truths need an ontological ground of their being-true: they need truth-makers. These realist philosophers proffer a type of explanation, a peculiarly philosophical explanation. They say something along these lines: Contingent atomic truths (whether true propositions or sentences or beliefs, or whatever) are true BECAUSE there are truth-makers in reality external to these propositions or sentences or beliefs. The 'because' in the preceding sentence is not being used empirically-causally. A truth-maker is not the empirical cause of the truth of Snow is white, say.

Or consider the question whether a red object, say a red tomato, is red because in English we apply the predicate 'red' to it, or whether instead we apply the English predicate 'red' to it because it is red. The question -- a 'Euthyphro question' if you will -- has nothing to do with the empirical causes of the redness of red tomatoes. We are asking a different sort of question, one the answer to which is a different sort of explanation. My realist answer would be that it is the antecedent redness of the tomato that grounds and justifies our application of 'red' to it, or rot to it if we are speaking German, or rouge to it if we are speaking French. 'Red' is true of the tomato BECAUSE it is red, where 'because' obviously does not have an empirical-causal sense. And not the other way around. The redness explains the correct applicability of 'red'; it is not the correct applicability of 'red' that explains the redness. Of course, 'red' is correctly applicable to red tomatoes. But it is not the correct applicability of the predicate that makes the items red. The following is uncontroversially true:

(*) Necessarily, for any x, x is red if and only if English 'red' is correctly applicable to x.

But (*) involves no explanation, philosophical or empirical-causal. The following do:

(**) Necessarily, for any x, x is red BECAUSE English 'red' is correctly applicable to x.

(***) Necessarily, for any x, the English 'red' is correctly applicable to x BECAUSE x is red.

As a realist, I say that (***) is true and (**) false. But all accept the uncontroversial (*).

Although much more needs to be said, the above conveys part of the meaning of 'philosophical explanation.' If you can understand the three bolded propositions above, and how they differ from each other, then you have an understanding of philosophical explanation.

Now my point about the truth regress is that, since it is not involved in any explanatory project, it cannot be vicious. A vicious infinite regress, then, is one that arises as part of a philosophically explanatory project. This is not a definition. I am merely drawing your attention to one necessary feature of vicious infinite regresses.

3. Now let us consider how the charge of vicious infinite regress might arise. Suppose someone asks why the universe exists. He is given the explanation: Because God created it. But since God cannot do anything, let alone create, unless he exists, one can legitimately ask about the existence of God: Why does God exist? The idea here is that if U requires an explanation of its existence, then any G one introduces to explain U also requires an explanation of its existence. The idea is that it would be arbitrary to demand an explanation of U but not of G. For if G exists without explanation, as a matter of brute fact, then that might also be the case for U. Now if G needs an explanation, and G cannot explain itself, then G needs an explanation in terms of a numerically distinct entity G*. But G*, to do its job, must also exist and will be liable to the same question, namely, Why does G* exist? The reasoning is plainly iterable and generates an infinite regress of explanantia: G, G*, G**, . . . .

But is the regress vicious? Intuitively, yes. We want to know why U exists. We want an ultimate explanation. After all, it is a philosophical question and we seek a philosophical explanation. We are not seeking an empirical-causal explanation of some item or items within U. Philosophy seeks ultimate explanations. But there is no ultimate answer if the entity G invoked to explain U is subject to a question about its existence.

So an atheist might mount the following infinite regress argument against the theistic explanation of the universe in terms of a creator god. "The theist posits God, something 'outside' the universe, to explain the universe. But if the universe needs explanation, then so does God. Contrapositively, if God does not need explanation, then neither does the universe. The theist, however, refuses to accept the existence of the universe as a brute fact. So he posits God as ground of the universe. But the theist's explanation is worthless since it involves him in a vicious infinite regress: any G he posits to explain U will itself need an explanation in terms of G*, and so on infinitely. Since the regress does not terminate, the theist does not succeed in rendering intelligible the existence of the universe."

4. How might one rebut an infinite regress argument (IRA) such as the one I have just sketched on behalf of the atheist? There are at least three possible strategies.

a) One might try to show that the regress does not arise, that there is no regress. Thus the theist of our example might argue that he is not saddled with a regress. There are two ways the theist might proceed to show this.

a1) For the regress as sketched to get off the ground, it had to be assumed that both U and G exist, and in the same way. But the theist can easily deny this. He can say: U exists contingently while G exists necessarily. The theist thereby denies the atheist's assumption that if U requires an explanation of its existence, then any G one introduces to explain U also requires an explanation of its existence. For if G is a necessary being, then it does not require an explanation of its existence. G explains U, which needs an explanation because it is contingent, but G does not need an explanation because it is necessary. In traditional terms, G is causa sui. So there is no infinite regress. There is a regress all right, but it terminates in one step, the step from U to G. One regresses in one step from the contingently existing universe to its necessarily existent ground.

a2) One might try to show that the theist is not forced into an explanatory regress since an explanation of Z in terms of Y can be successful even if one can give no explanation of Y. For example, if a crop failure is explained in terms of drought, that can count as a true explanation of the crop failure even if no account can be given of the drought. Why must a satisfactory explanation of some phenomenon Z require an explanation of all of Z's causal antecedents? Thus it is worth the theist's while to explore the following possibility: the existence of U is a brute fact, and the existence of G is a brute fact; but U exists because God created it. Therefore, U has an explanation in terms of the existence and creative activity of God despite the fact that God's existence has no explanation.

b) One might try to show that there is an infinite regress, but that it is benign because non-explanatory. This strategy does not apply to the God argument, but it does apply to the truth regress.

c) One might try to show that there there is an infinite regress, but that it is benign because, although it is explanatory, it is successfully explanatory. Suppose U is explained by G, G by G*, G* by G** and so on infinitely. Suppose that each G is a contingent being. Suppose further that there is an actual infinity of Gs. Then every G has an explanation, and no G is unexplained. If the regress were potentially infinite, then there would be a failure of explanation. But if the regress is composed of an actual infinity of members, then every member of the regress from U on out has an explanation in terms of an 'earlier' (whether logically or temporally) member.

5. Strategy (c), though, could be used by the atheist to argue that the universe can explain itself, and thus has no need of a transcendent explanans or an infinite series of transcendent explanantia. Suppose, contrary to current cosmology, that the universe has an infinite past, and that each phase of the universe is caused by an earlier phase. Suppose further that there is nothing problematic in the notion of an actual (as opposed to potential) infinity, and that there is a good answer to the question of how, given the actual infinity of the past, we ever arrived at the present moment. Granting all that, the infinite regress of causes is benign. For if the universe has an infinite past, then every phase of it has a cause. But if every phase of the universe is caused, then there is a sense in which the universe is self-caused, in which case it is in no need of a transcendent cause.

6. I have very serious doubts about the (c) strategy, whether used by a theist or by an atheist. I assume as unproblematic the notion of actual infinities. But even if the universe has an actually infinite past and that each of its phases is caused by an earlier phase, so that no phase or part of the universe is uncaused, this is consistent with the universe's contingent brute-factual existence. That every phase of the universe has an explanation does not amount to an explanation of the universe itself. Note that one cannot explain why the universe exists by saying that it always existed. For even if there is no time at which it did not exist, there remains the question why it exists at all. The universe is contingent: it might not have existed. So even if it exists at every time, that does not explain why it exists. To say that the universe always existed is to say that it has no temporal beginning, no temporally first cause. But this gives no answer to the question why this temporally beginningless universe exists in the first place. Adding that each phase is caused by an earlier phase changes nothing.

The situation is not relevantly altered if we suppose the universe has an actually infinite series of transcendent causes, each of which is contingent. For then the whole system is contingent and we are left with the question why the whole system exists.

7. Kevin Kim asked, "What makes a temporally infinite series (of moments, causes/effects, etc.) impossible?" Well, nothing makes it impossible. It is thinkable that the universe have an actually infinite past, a beginningless past, and that each phase of the universe is caused by an earlier phase. But as I have already stated, this does not explain why the universe exists in the first place.

In sum, I find untenable the notion that there are explanatorily successful actually infinite regresses.

Comments

[Big Hominid]

Thanks for the shout-out. I hope all's well.

Kevin

[Bill Vallicella]

You're welcome, Kevin.