Saturday, 19 March 2011

Symbolic logic and rules of grammar

As a true believer in Inuit Qaujimaningit as a discourse I am drawn in and rendered a true believer by the idea of "information-rich" first principles. What I mean here in saying "information-rich" principles is that quality of a system having the ability to generate and produce novel logical outcomes solely from a set of principles or logical arguments (the ability to deduce theorems, in maths speak) not immediately or even tertiarily obvious in the set of existing logical statements themselves that make up the system.

One such system is the axioms of mathematics; another is the systematic grammatical rules of human languages. But I must confess that I think the notational symbolism of formal logic (in maths) is as ugly as things can get, and, if we follow GH Hardy's dictum, have no permanent place in the world.

The symbols comprise of such things as upside down and backwards E's, upside down A's, clunky epsilons and crossed out clunky epsilons, etc. yech! In this symbolic system the prettiest theorems of set theory do not have a hope of redemption the same way that the prettiest girls in a laxative commercial have something just plain wrong with them.

As an Inuk born and raised, I have an almost irresistible need to problem-solve or at the very least to try and understand a problem worthy of attention. Over the course of my career as a thinker, reader and connoisseur of all things excellent that human beings can produce, I've come to appreciate the beauty of mathematics and mathematical symbolism.

I find concise and elegant maths statements truly beautiful, such as Euler's equation: eπi + 1 = 0 and the Lorentz Transformation equations of Einstein's theory of relativity: x’ = x – vt/√1 – v2/c2 that say that the speed of light cannot be exceeded because the bottom, right part links velocity and the speed of light with 1 - [1] as a square root (simply, cannot divide by zero).

As mathematicians started to examine the foundations of mathematics, questions of whether its set of axioms was complete and self-consistent enough to not generate contradictions were raised. Seemingly intractable problems such as Russell's paradox of the barber who shaves only the men who do not shave themselves (does he shave himself or not?) and the logical yes and no answers to consistency and completeness questions arose.

One such yes and no answer to the completeness and consistency questions came from Godel whose numbering system as applied to the axioms of set theory "produced" one of those yes and no answers, very troubling for maths which relies upon the exclusion of the middle possibility (or third answer: yes and no).

There are other paradoxes and questions of infinite regress that fly about in this "Platonic realm", such as the statement: "everything that exists has a location; the location has a location, has a location..." But I think the "yes and no" and infinitely regressible statements have deep similarities with problems in linguistics.

The "differences" between a noun in the subject slot and the predicate slot for eg seem to suggest that these are two mutually excludible stand-alone nouns (Socrates is a man -types of contructs) that would produce the same yes and no answers as Godel's numbering system. But I think suggestions for trying to resolve these issues lies in maths itself and linguistics.

In the infinite regress problem of location, one could say that the location itself is a function of that which exists, and nip the bud right there; with the Godel proofs, though I'm not sophisticated enough to judge his results, I've always had this nagging suspicion that with all of his impressive intellect and logic he might have only produced a logical accident as one is prone to come across in linguistics:

Little green ideas dream furiously.

The above construct seems to obey all logic and grammar of linguistics, but does it really?


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