My Arg₃ in Context: Other Attacks on Church's Thesis

Over the past six decades, the possibility of CT's falsity has not only been raised,²⁵ but CT has been subjected to a number of attacks. While I obviously don't have the book-long space it would take to treat each and every attack, I think it's possible to provide a provisional analysis that is somewhat informative, and serves to situate the attack on CT I have articulated above. What this analysis shows, I think, is that Arg₃ is the best attack going.

Following R.J. Nelson, I partition attacks on CT into three categories:

CAT1

Arguments against the arguments for CT;

CAT2

Arguments against CT itself; and

CAT3

Arguments against doctrines (e.g., the computational conception of mind) which are said to presuppose CT.

Consider category CAT3 first. Perhaps the most promising argument in this category runs as follows. Assume for the sake of argument that all human cognition consists in the execution of effective processes (in brains, perhaps). It would then follow by CT that such processes are Turing-computable, i.e., that computationalism is true. However, if computationalism is false, while there remains incontrovertible evidence that human cognition consists in the execution of effective processes, CT is overthrown.

Attacks of this sort strike me as decidedly unpromising. For starters, many people aren't persuaded that computationalism is false (despite the many careful arguments I myself have given; cf. [2], [5]). Secondly, this argument silently presupposes some sort of physicalism, because the evidence for the effectiveness of cognition no doubt derives from observation and study of processes in the central nervous system. Thirdly, it is certainly an open question as to whether the processes involved are effective. At present, we just don't know enough about the brain, physics, and computability theory (and the interrelationships among these things) to conclude that cognition rests completely on effective processes. (To be a bit more specific, perhaps cognition arises from physical processes which are irreducibly analog and chaotic, a type of process mathematized in the final section of this very paper.)

What about CAT1? Well, my refutation of Mendelson falls within it -- and yet who would claim that what I have revealed about Mendelson's reasoning constitutes, by itself, a serious attack on CT? The same fundamental question derails even the work of those who intend to attack CT by attacking the time-honored rationales for it. For example, William Thomas [37] seeks to capitalize on the fact (and it is a fact, that much is uncontroversial) that the main rationale behind CT involves empirical induction -- a form of reasoning that has little standing in mathematics. Unfortunately, Thomas' observations don't threaten CT in the least, as is easy to see. Most of us believe, unshakably believe, that the universe is more than 3 seconds old -- but what mathematical rationale have we for this belief? As Russell pointed out, mathematics is quite consistent with the proposition that the universe popped into existence 3 seconds ago, replete not only with stars, but with light here on Earth from stars, and also with minds whose memories include those we have. More generally, of course, from the fact that p doesn't follow deductively from a set of propositions $\Gamma$, it hardly follows that p is false; it doesn't even follow that p is the slightest bit implausible.

We are left, then, with CAT2 -- the category into which my own attack on CT falls. How does Arg₃ compare with other attacks in this category? To support the view that my own attack is superior, let us consider a notorious argument from four decades back, one due to László Kalmár [19] (and rejected by none other than Elliott Mendelson [28]), and the only other modern attack on CT that I know of, one given by Carol Cleland [10], [9].²⁶