Thoughts on the Formal Systems
- This particular concept stems largely from the little I had understood in reading Hofstadter's book, Bach, Godel, Escher: An Eternal Golden Braid (BGE)
Formal system is like a game, a puzzle. In a formal system, complex or simple, a rule, or a set of rules, is defined, and from those rules emerge a new world, a contrived one, insightful and equally beautiful as the most intuitive one.
I found the idea of formal system pleasing, as I often struggled with the purpose of the myriad volume of theorems and proof I was taught in my education, most notably in math courses. It is not the cute "when will I ever use --- in my life?" question that I have gained answered to - engineering has proven that you cannot really solve engineering problems without mathematical representation, and techniques of linear algebra and differential equations are incredibly useful in acquiring a solution for a problem. The question is - is it possible to study something with no regards to application? To clarify, is every scientific endeavor, a mathematical notion a product of attempting to understand, and ultimately manipulate the physical world of our perception? Is there a merit in investigating something for the sake of investigating?
I do sometime get the notion that everything must have some sort of relevance to the world lest you be called an impractical romantic full of quixotic ideas. (And it was with guilt and diffidence I hoped for that sense of almost religious exaltation one could feel in the appreciation of absolute beauty of an single idea. I didn't want to say that I care less about the application and more about the theory because it is more beautiful because I am, to my disappointment, quite self-conscious.) And formal system now provides a better excuse to be a impractical romantic, and masculine one too.
In example, I bring in linear algebra. I had consoled myself with matrix operations, finding determinants, adjoin, inverse was all interesting techniques of solving system of linear equations, a practical and engineering-ish endeavor indeed. But I suspect that development of these incredibly physically unrelatble ideas were done for the sole purpose of solving a linear equation to solve a practical problem. Rather, matrix itself is a formal system of that represents the set of linear equation we are more accustomed to seeing (another formal system reflecting of the world). It appears different, and functions under a different set of rules, but the information in one form in not lost in another form (isomorphism). What is more powerful- that by playing with the formal system of one kind, revelations are made that may have been nearly impossible to have made in another.
