The limitations, semantic and epistemic, of recursive-axiomatic deductive systems, are often harped on. But are not such limitations, after all, almost a truism ? What is important is not the 'axiomatic' or 'deductive' therein: it is the 'recursive' . The general theory of computatibility is as neglected as it is fundamental. Theorems are $\Sigma^0$. But arguments like those used for the Halting problem show that there are sets of arithmetical truths which are not in $\Sigma^0$. Thus recursive axiomatic-deductive systems are epistemically and semantically limited, incomplete. This is obvious. Much more interesting is the philosophical elucidation of the concept of computability and of the whole arithmetical hierarchy (and hierarchies beyond). And more interesting is the elucidation how we can apparently know and talk about sets whose elements we a priori cannot know or distinguish individually such as the set of numbers which codify arithmetical truths. If this is not a conclusive argument against extensionalism, then what is ? The algorithm is the intension, extension is a computational question about the intension. Also there is a tendency to confuse a formal system and its philosophical agenda. It is a happy truth that we can interpret ZFC however we want as long as we stick to the axioms and rules.
If incompleteness is a 'negative' trait of formal systems, there is also a surprising, significant, powerful 'positive' trait as well. This is the ability of formal systems to reflect, mirror, interpret and be embedded in each other. For one system to mirror within itself the meta-theory and semantics of another. And that this mirroring and interpretation is itself mirrored in a certain encompassing system. Incompleteness itself was established by Gödel to be a consequence of certain formal systems mirroring aspects of their own meta-theory. Mathematical logic is the mathematical treatment of the formal logic used to represent mathematics itself. It is a reflection-into-self of mathematics.
It is not so much formal systems that are 'limited' and 'incomplete' as our own attitude and science if we do not start studying axiomatic-deductive systems with proof systems in all levels of the arithmetical hierarchy and also with possibly infinitely long expressions. Even physics suggests this.
Formal systems are supposed to talk about things. When those used in mathematics are set up it is implicit that the expressions are meant to relate to objects, not mental contents of certain people at certain times. What will a philosophical formal system talk about ? The objects of science such as physics and biology ? Mental states and consciousness ? The semantic, ontological and epistemic problems of logic and mathematics themselves ? Or abstractly all of the above ?
But notice how many philosophical questions are framed: what is X ? how is Y possible ? For instance what are mathematical objects ? How is knowledge possible ? Questions about finding the nature or definition of X are paradoxical because if we do not know what X is how can the question even be asked meaningfully ? For instance: what is a snark ? We start from a situation of social linguistic usage and desire to go to the realm of essence. We also ask; what are numbers ? We use numbers and think we know what numbers are and yet we cannot say clearly and explicitly what they are (cf. Augustine's si nemo a me quaerat scio, etc.). This duality, division, shadow, in our cognition and consciousness does not seem to be approachable by formal systems. When we understand language there does not seem to be any conscious computation going on.
Another angle is to consider a mathematical approach to natural language. We are given a large data set of linguistic data plus descriptions of their social context. Our task is to give formal structure (morphological, syntactic, semantic, pragmatic, etc) to linguistic units so as to largely predict or explain computationally or deductively the massive data set - and extrapolate new data. For originally we needed both to explain and deduce known mathematical truths as well as deduce new ones. But we mentioned 'description of social context' : this itself is carried out within natural language. Also, how can we talk about mental states or the mind ? How can we examine or compare mental states ?
We wrote 'predict or explain computationally or deductively'. But in itself the computational complexity may be too great for our resources. Imagine if to prove anything arithmetically significant in ZFC or the Peano Axioms we actually needed billions of lines of proof. What difference would this be from just having a handy list of arithmetic validities which we could query ? In some cases looking for counter-examples might be more efficient than looking for proofs (the proposition or its negation). So measures of computational complexity are important, the way the length of the proof depends on the length of the theorem to be proven.
If we wish to focus on a formal analysis of natural language (which itself is highly problematic) then we are forced to restrict ourselves to a limited domain of language and linguistic practice; for either we are too general and simplistic or else we must be able to account for the philosophical use of language too, even in dealing with language itself. For instance, what does the word 'language' mean ' ? We are looking at the painting which we are inside in.
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