Monday, November 2, 2009

The Mathematics of Reason: The First Principles According To Boole.

In 1853, George Boole published his treatise, ”An Investigation of the Laws of Thought”, and gave the world “Boolean Algebra”, the mathematics of logic that ultimately made digital electronics and digital computing possible. To me, the most remarkable aspect of Boole’s work is his ability to resolve rational processes into simple equation form, even into tables for determining propositional truths.

His algebra varies only slightly from classical (numeric) algebra. It is necessary to envision sets, their intersection or non- intersection, rather than multiplication or division. For example, xy is the set that contains both x’s and y’s; this might not include all x’s or all y’s... but it could.

While there are other considerations, that one principle leads off to a remarkable conclusion. Here’s how it works:

If two entities are equal sets (identical), where y = x, then,
xy = x. (The intersection of the sets x and y are identical to the set x.)

This is the Principle of Identity.

And if y = x, then,

xx = x,

or
x2 = x.

next,

0 = x – x2,
0 = x ( 1 – x).

The sole solutions are x = 1 and 0, where 1 is a full set, and 0 is an empty or null set. Also, (1 – x) is the contrary set to x, where x is not a universal set.

This equation demonstrates several important things.

First, it represents the conjunction of both x and “not x”, making it a universal description. For example it could mean “truth” and “not truth”, covering the entire universe of possible validities. So it is a “universal” equation.

Second, it has only two solutions, 0 and 1. So in the case of “truth” and “not truth”, there is no intermediate value, meaning that only “true” and “not true” exist. This is the Principle of Excluded Middle. (Also called the Law of Duality, the principle of dichotomy in analytical thought.)

Third, it can be seen that x cannot be both 0 and 1 at the same time. This is the Principle of Non-Contradiction.

From just one equation, Boole demonstrates mathematically the axioms that underlie all rational thought.

Further, in order to demonstrate that dichotomy is the limit of human comprehension, Boole writes a trichotomy:

x = y = z (identical sets);

xyz = x;

then,

x3 = x ;

This factors into

x ( 1 – x )(1 + x) = 0;

The solutions are 0, 1, and –1. To illustrate the cognitive disconnect: If x = “all men”, and (1 – x) = everything that is not “all men”, then what does (1 + x) represent? Boole points out that this is surely beyond the comprehension of human minds. So trichotomies are outside the realm of rational thought, at least in this universe, and for human faculties.

As beautiful and remarkable as this is, it occupies only the first three chapters of Boole’s work. He goes on to analyze propositions, including if/then, and then it’s off into probability theory.