Convergence & Determinism

The sort of determinism advocated in Daniel Dennett’s Elbow Room¹ shatters on the plausible idea that an infinite causal series can converge to a limit in the finite past. Before we begin, however, this thesis has nothing to do with backward causality. Backward convergence need not imply backward causality.

Dennett writes:

“If determinism is true, then our every deed and decision is the inexorable outcome, it seems, of the sum of physical forces acting at the moment; which in turn is the inexorable outcome of the forces acting an instant before, and so on to the beginning of time…”

To add a little precision, let F_n denote the totality of forces acting at time t_n. On a deterministic view, there is some law-like map f such that each state is the necessary outcome of its predecessor: F_n = f(F_n-1).

What this overlooks is that an infinite series can converge to a finite value. In analysis, one classic example is ∑_k=1^∞ (1/2)^k = 1. There is no contradiction in supposing that a non-vitiating infinite regress of causal states might converge to a limit L at some point in the finite past.³ If so, an event E could be the result of an infinite causal series that converges upon L, and nothing prior to L would bear on E causally.

To deny this possibility is to commit a fallacy akin to those exposed by Zeno’s paradoxes.² A little Calculus goes a long way!

In my book Gödel’s God Theorem (GGT), I offer a Leibnizian perspective on contingency that stands in diametrical opposition to the necessitarianism of Spinoza. I present this as a way out of modal collapse. But free will is something more than mere contingency. A monadological view, I believe, is the key to developing a sound theory of freedom. Monads are sufficiently independent of external causal influence, allowing them to achieve a form of self-determinism. Indeed, Leibniz makes determinism the handmaid of freedom by defining freedom as monadological self-determination. GGT gestures toward some ways of developing such a theory further.

Notes

1. Daniel C. Dennett, Elbow Room: The Varieties of Free Will Worth Wanting (Cambridge, MA: MIT Press, 1984), 5. ↩︎
   
2. One might object: why assume an infinity of causes? The universe itself could be infinite with no definite starting point. Yet even if the universe is finite, Leibniz’s notion of contingent actuality suggests that elements of the contingent world exhibit a never-ending descriptive complexity, such that we are never quite done discovering new properties (once we learn how to look). On this view, Creation—though possibly temporally finite—remains infinitely complex, which may help explain why the dictum “reality is stranger than fiction” resonates so strongly. In Gödel’s God Theorem, I show in the chapter on my ♦–Cosmological Argument (♦CA) that the hypothesis of an infinite universe in no way diminishes the force of a Leibnizian cosmological argument. Hence one reason why ♦CA is superior to the so-called Kalam Argument. ↩︎
3. Philosophers distinguish between vitiating (or vicious) and non-vitiating infinite regresses. A vitiating regress is incoherent or self-defeating. A non-vitiating infinite regress, by contrast, can be logically consistent and even explanatory—much like a convergent series in mathematics. Aristotle, by contrast, typically held that all actual infinities (and thus infinite regresses) were impossible. However, the set of real numbers disrupts Aristotle's view. ↩︎