Formal Systems and Applications Thereof (continued)

    The ideas expressed in Hofstadter's first masterpiece, Gödel, Escher, Bach: An Eternal Golden Braid, are powerful, if confusing. However I think he makes them about as accessible as can be expected. He makes humorous use of dialogue in illustrating some of his more far-fetched points. He weaves concepts from mathematics, psychology, philosophy, computer science, and even biology together in a way that draws on one's understandings of some in explaining the others. The beautiful thing, is that the fact he can do that mirrors what the book seeks to explain.

Form and Content

    Central to how any formal system is utilized in a real world setting is the distinction between how it operates and how it is interpreted. How it operates has to do with the axioms upon which it is based and the rules governing the creation of theorems from axioms and other theorems. How it is interpreted has to do with another key word.

Isomorphisms

    What makes a formal system useful, is a powerful isomorphism between the axioms and theorems of the system and some feature of reality. Abstract Algebra is useful in one instance, because if in the real world you have one apple, and someone gives you another, you have two apples. Abstract Algebra does not rely on that fact itself to produce the true statement 2 + 2 = 2, yet in that case and in any similar cases it will produce a theorem that corresponds accurately to our observable reality, given that we interpret it reasonably. The system was specifically developed to so, and is as such an extremely powerful tool in executing all kinds of tasks.

Meta-Mathematics

    Kurt Gödel understood this very well, and so took the path of using a formal mathematical system (number theory) to study itself. By "stepping outside of the system" he used the system itself to prove that it (number theory) was either inconsistent or incomplete.  The more wide reaching implication of Gödel's Theorem is, by the nature of his argument, which can be imported into any sufficiently powerful formal system, that every formal system is by necessarily either inconsistent or incomplete. That is to say it either contradicts itself, or it is not capable of producing every theorem of the system that reality dictates should be true.