Alonzo Church was a very important figure in 20th Century philosophy, especially at its intersection with mathematics. For example, he was pivotal in articulating what has come to be known as the Church–Turing Thesis. (On one formulation, the thesis has it that every computation meeting certain conditions—in technical parlance, every mechanical or effective computation—can be carried out by a Turing Machine (a specific type of simple computer).)
Key to the articulation of that thesis was a certain approach to philosophy, involving a combination of reflection, formalisation, and further reflection. To a first approximation, the approach involves initial reflection on the target subject matter, followed by formalisation so that a determinate theory of part of the subject matter is produced, followed by further reflective testing of the theory and then its re-formalisation, its rejection, or (though this is rare) its provisional acceptance. The role of formalisation—provision of a determinate theory—is to facilitate testing of the theory by making as explicit as possible its connections with other theories and claims, and more generally by making as explicit as possible its pattern of commitments. Clarity about those commitments can then facilitate the generation of testable predictions. Although the outputs of reflection are accorded weight in this kind of procedure, both in producing and in testing theories, none is taken to be sacrosanct: the aim is optimal capture of the fruits of reflection, and that might involve the sacrifice of putatively self-evident darlings.
It’s not implausible to see the approach, at least from a certain height, as an attempt to pursue philosophy via something like the methods that have achieved success in pure mathematics. (At least, that is so if we are willing to allow that little or nothing in pure mathematics is transparently self-evident and, so, immune from potential rejction.) Furthermore, Church himself suggests that, from the same sort of height, the method bears comparison with approaches pursued in the empirical exact sciences. Here, for example, Church sketches a notion of observation that aims to be neutral between the respective outputs of perception and reflection:
“But the preference of (say) seeing over understanding as a method of observation seems to me capricious. For just as an opaque body may be seen, so a concept may be understood or grasped. And the parallel between the two is indeed rather close. In both cases the observation is not direct but through intermediaries—light, lens of eye or optical instrument, and retina in the case of visible body, linguistic expressions in the case of the concept. And in both cases there are or may be tenable theories according to which the entity in question, opaque body or concept, is not assumed, but only those things which would otherwise be called its effects.” (Church, ‘The need for abstract entities in semantic analysis’, American Academy of Arts and Sciences Proceedings, 80, 1951: 100–113.)
Church’s model appears to be approximately this. Opaque bodies present a plurality of appearances to sight, mediated by light, &c.. And theorising about such bodies is, in part, a matter of seeking the underlying principles that organise the appearances. In seeking such principles, we may begin by attempting to capture all the appearances. However, it is likely that the pursuit of optimal theories—theories that are maximally simple, general, elegant, &c.—will lead us to view some of them as mere appearances. Similarly, concepts present a plurality of appearances to the understanding, mediated in this case by language. And theorising about concepts is, in part, a matter of seeking the principles that organise those appearances.
Before we can get into a position properly to assess Church’s proposal, two natural questions of clarification are these. First, how exactly are we to understand the correlative notion of appearances in the case of the understanding? Are these particular judgments, or intuitions, or something else? Second, how should we think of the analogy between the role of media in sensory perception—that is, the role of light, eye, and instrument—and the role of language in the operations of the understanding?