Thought is not a monolith
Sunday July 23, 2023
Any particular argument can in principle be formalized in a particular system of assumptions and inference rules, but flexibility in level of formality and use of different systems for different arguments is both practical and inevitable.
The plurality of systems of thought is particularly transparent in mathematics, where for example to know whether a statement of geometry is correct one must first ask which system of geometry is determining correctness. In one system the interior angles of a triangle always add to two right angles, but in another this is not the case. Conflicting results do not imply cognitive dissonance when each is correct in its respective system. The ability to choose among systems is a great power.
Mathematical formal systems are in a sense complete articulations, but they are also generally quite verbose and not necessarily easily understood. In another sense, by their very self-completeness formal systems close themselves off from meaning beyond their symbols, requiring something beyond themselves and in that sense being incomplete.
In practice, it is rare to completely formalize an argument. Ideally, the nearest acceptable assumptions and clearest steps of reasoning are described clearly and expediently. Often, assumptions and reasoning are just implied, or even elided. There may be more or less uncertainty in excavating the underlying argument.
Natural language is an imperfect tool. Spoken, written, or purely in thought, a statement of natural language is a phenomon not purely tied to any particular semantics. As in the Shannon–Weaver model of communication, language is a signal, and it is the only part of the system that we have meaningfully direct access to. Everyone speaks their own constantly-evolving language. Much of philosophy can be seen as postcards from Sisyphus visiting the ruins of Babel.
Meaning is built based on language, but the meaning cannot be directly inspected. Reading my own notes, I may find that now they mean nothing to me, or I may otherwise understand them differently from what I was understanding as I wrote them. It isn't clear that it is possible to understand a statement twice in exactly the same way. In the case of communication with others, the best we can hope for is roughly compatible understandings.
In stimulating meanings, even if not always perfectly identical, natural language is a powerful tool. By relying on shared understandings, which can be built, natural language can be very efficient and effective. Very often, it is the best option. Care should be taken to avoid the extremes of its weaknesses.