Necessity and Infinity

Necessity and Infinity

Certain concepts lead naturally to the concept of infinity; arguably they entail the concept of infinity. Thus God, space, time, and number: anyone who grasps these concepts is apt to entertain thoughts of the infinite. God is infinitely powerful, space is unending, time goes on forever, and numbers never exhaust themselves. Even if there were some sound argument showing that these infinities are unreal, we would still naturally form the concept of infinity from the concepts in question. But are there infinite concepts that don’t wear their infinity on their face—are there implicitly infinite concepts? The concept of causality might be an example: for every event there is a cause, and every cause is an event. The chain of causation never terminates: there is no first cause (we might also argue that there is no last effect, since any effect is an event that must bring something else about). At any rate, the concept of causality leads naturally to the idea of an infinite series of causes.

            The concept of necessity is less obviously connected to the concept of infinity, but it has some claim to belong to this select group. We say that a necessary truth is one that is true in all possible worlds or in all conceivable circumstances, as an eternal truth is one that is true at all times—and in both these cases “all” ranges over an infinite totality. It is not that “2+2=4” is true in only a limited finite number of possible worlds; it is true in absolutely all of the infinitely many possible worlds there are. Certainly it must be true at all times, and times are infinite in number; but it must also be true across all of logical space, which is unlimited in extent. When we contemplate a necessary truth we therefore (implicitly) contemplate such an infinite totality: our minds turn to matters infinite. It is not so with contingent truths: here we just need to know a single world in which the proposition is true—we needn’t think beyond the actual world in the case of what is contingently true. But the concept of necessity forces us to consider a vast range of possibilities—it resounds across infinite logical space. When we think that necessarily 2+2=4 we are having a Very Big Thought—as with thoughts of God, space, time, and number. We are contemplating the infinite.

            This might help explain the phenomenology of modal thought—the sense we have that knowledge of necessary truth is somehow grand and far-reaching. It isn’t local and parochial. For what could be more extensive than infinity? To have knowledge of the infinite always feels awe-inspiring, even transcendent. It is the human intellect at its most capacious. This might also explain some of the suspicion under which the concept of necessity falls—it seems to incorporate more than we can legitimately encompass. It defies empiricism (there can be no “impression” of the infinite).

            There are orders of infinity, as we know from Cantor. Are there orders of necessity? Well, there is the distinction between nomological necessity and metaphysical or logical necessity. If there are infinitely many nomologically possible worlds, then the set of metaphysically possible worlds will be a still larger infinite set: so the two strengths of necessity correlate with two orders of infinity. That sounds right, because metaphysical necessity is broader in scope than nomological necessity. The concept of necessity extends across infinite space, infinite time, and infinite possibility. Necessity knows no bounds.

Colin McGinn