Languages allow us to express various aspects of our experiences using a structure. This structure can include categories, rules, symbols, sounds, and images. Language is deeply integrated into our perception, helping us navigate and simplify our complex experiences. It serves as a shared center of awareness for ourselves and our surroundings. Language represents the infinite use of finite resources.
Mathematics is arguably the most powerful language we have developed (or discovered?) in terms of its ability to generalize and delve into the layers of reality. Mathematics transcends the physical reality we encounter, and one of its key strengths is its capacity to help us think about and work with abstract entities, such as infinities. Russell and Whitehead began Principia Mathematica with the objective of building mathematics from the ground up and proving its infinite consistency.
Gödel was the first to prove, in his two-part Incompleteness theorem (later supported by Church and Turing through a more intuitive Recursive theory of calculus and Halting problem, respectively), that there exist statements in any formal system that cannot be proven true or false using the rules of that system. A formal language can never prove its own consistency, and any attempt to create a system to prove its own ability cannot exist due to the paradoxical result that arises from the proof.
How do informal languages differ from such languages? What aspect of consciousness cannot be accommodated within a formal language? Languages are our only means of preserving and creating constancy in an ever-changing world. Language is what a living being is left with as it moves through time, a condensed representation of the useful dynamics of the past. This interpretation is shaped by the language used by the entity that creates it.