What Is an Infinite Regress? Explanations That Never Reach the Bottom

A regress arises when a proposed account of something invokes another instance of the same thing, which then requires its own account on the same terms, and so on without end. If every belief must be justified by another belief, then justifying any belief requires an endless chain. If every event must have a cause, the chain of causes has no first member. If a property must be explained by something possessing that property, the explanation never lands. The objection in each case is not that infinity is impossible but that the explanation has not done its work, since at no point does anything get explained, only deferred. That is what distinguishes a vicious regress from a merely infinite series, and the distinction is what most of the argument about particular cases turns on.

Faced with a regress, there are a limited number of moves and each has a name and a cost:

• •Foundationalism, stopping the chain at items that need no further support, which requires explaining what makes those special
• •Coherentism, replacing the chain with a web in which items support each other, which faces the objection that a coherent set can be entirely false
• •Infinitism, accepting the endless chain and denying it is vicious, which is a minority position taken seriously
• •Denying the principle that generated the regress, which is frequently the right answer and requires identifying which assumption to give up
• •Distinguishing kinds, so that the terminating item is of a different type and does not require the same treatment
• •Accepting a circle at some level, which is usually regarded as worse than the regress

The pattern appears in most areas of philosophy and in several outside it. The regress of justification is the foundation of epistemology, and the choice between stopping, weaving and continuing organises the subject. Cosmological arguments for a first cause proceed by claiming that a causal regress is vicious and must terminate, and the standard objections question whether it is vicious and whether the terminating item is exempt from the principle. Arguments about rule following ask how a rule determines its own application, since any further rule specifying that faces the same question. Regress objections appear in ethics, in philosophy of mind and in accounts of meaning. Outside philosophy the pattern turns up whenever an explanation smuggles in what it was supposed to explain, which is the structure of the objection to accounts of intelligence that posit a small intelligent controller inside.

One version of the argument deserves separate attention because of how far its consequences reach. If understanding a rule requires an interpretation of it, then the interpretation is itself something to be understood and requires a further interpretation, which never terminates and leaves nothing determining how the rule applies to any new case. Wittgenstein pressed this and concluded that the picture of understanding as grasping an interpretation must be wrong, since following a rule is something people demonstrably do, and that what grounds it is training and shared practice rather than a further mental item. That conclusion has been read in several ways and remains contested, and the argument itself is widely accepted as identifying a real problem. It matters beyond philosophy, since any account of meaning, of law or of computation that locates correctness in an interpretation faces the same difficulty.

Not every infinite series is an objection, and treating all of them as vicious produces bad arguments. A sequence of numbers continuing without end is unproblematic. A statement being true, and it being true that it is true, and so on, generates an infinite series of truths and no difficulty, since each follows from the first rather than being required to support it. The distinction that does most of the work is between a series where each member depends on the next for its status and a series that merely continues. The first fails to deliver what was promised, since nothing is ever grounded, and the second is just long. Applying that test to a particular case is not always easy, which is why arguments of this kind remain live rather than being settled once and applied thereafter.

An account invoking another instance of what it was explaining defers rather than explains, however long the chain. Foundationalism stops it, coherentism replaces it with a web, infinitism accepts it, and denying the generating principle is frequently the right move. A series where each member depends on the next for its status is vicious, and one that merely continues is not, which is the test worth applying.