03.08.08
Undefined Terms
I hate language. Language is imprecise. Language is malleable. When I want to communicate precisely, I’d rather not have the ability to combine reasoning and verbal play. I might enjoy using verbal tricks for fun, but sometimes it is best to be forced to say exactly what you mean. Yet even in symbolic logic, we have a problem! Amid all of the “definition” “theorem” and “proof” of even the most rigorous treatise on logic or math, there is still the issue that the definition of a word or symbol requires the usage of other words and symbols. Of course, as in any remotely philosophical issue, it was Aristotle who first noticed this. However, nobody really has a solution for this problem! If you read Euclid’s elements (well, maybe not the entire series) and you interpret his so called “definitions” in a way that he did not intend, you are really reading something totally different from what he wrote!
Even worse, I’m not even attempting to define the words I use in this post! For some reason, I have a strange faith that some part of what I am writing is communicated through a bunch of pixels on your screen. As my intellectual hero of the moment Wittgenstein said: “Propositions cannot represent logical form: it is mirrored in them. What finds its reflection in language, language cannot represent. What expresses itself in language, we cannot express by means of language. Propositions show the logical form of reality. They display it.”
Thus, as much as I hope this blog to be clear and precise, I obviously cannot confront the flaws in my language in a blog. Most of the time, I will begrudgingly use ordinary language. From time to time, I will math it up or even use symbolic logic. I hope to talk about math (including philosophy of math), logical positivism, the efficacy of language, economics, empiricism, politics, and anything else where I feel like a rational point can be made. Maybe next time I’ll be more explicit about what my terms mean.
The Reactionary Epicurean said,
March 21, 2008 at 12:35 am
Your worries about mathematical propositions being self-referential/ungrounded/reliant on unprovable axioms can be resolved in one of two ways:
1) Mathematics is a game we play. The work of mathematics is the construction of arbitrary definitions (axioms) of abstract structures, and we then prove theorems that tell us how these structures relate.
2) Some axioms are true (in a transcendent way outside of any logical system) and some are false.
Gregory Chaitin, among others, has pointed out that the real work of mathematics is not that of proving theorems from axioms, but of deciding which axioms to accept.
I also have quibbles with your objection to blending form and content, but that’s the subject for a much longer conversation.
musicheck said,
March 22, 2008 at 11:46 pm
I stand for choice 1a) Mathematics is a game we play with meaningless symbols according to meaningless rules, but the activity of mathematics makes them meaningful. Our axioms are are not arbitrary- they come from a combination of aesthetics and empirical intuition (e.g. what is useful for physics).
Furthermore, about Chaitin, its been said that the majority of mathematicians never cite an axiom once in their work. There is plenty of work left to be done in even the simplest axiomatic systems, just as Schoenberg once said there was plenty of music left to be written in C major. One can work in the analog of C major or 12 tone serialism, and one chooses based on which produces what one considers nicer results. I would also say that we pick our axioms based on what theorems we want them to prove (e.g. Zermelo’s use of the axiom of choice to prove the well ordering theorem.). We already have intuitions of how we want things to behave, and the axioms we choose reflect this. There is nothing arbitrary, and it is indeed the theorems which our axioms can prove that are the final goal.