Sunday, 28 December 2014

The axiom of choice

I have an invisible OCD. I call it 'thinking'. I'm drawn to abstract structures: language, music, ethical discourse as logic systems, mathematical equations and algorithms, architecture—anything, really, that can be derived from a set of first principles.

One of these things that I have obsessed over is called 'set theory' where the 'axiom of choice' resides.

Set theory, among all mathematics, has got to have the ugliest symbolism. If we were to assiduously follow GH Hardy's edict that 'beauty' be the final arbiter of mathematics, the only saving grace of set theory would be the implications of its statements.

In not so many words, the axiom of choices says:

Axiom of Choice Let C be a collection of nonempty sets. Then we can choose a member from each set in that collection. In other words, there exists a function f defined on C with the property that, for each set S in the collection, f(S) is a member of S. (http://www.math.vanderbilt.edu/~schectex/ccc/choice.html)

Ok...

Eric Schechter of Vanderbilt University (just quoted above) says of "the last great controversy of mathematics" (ie, the axiom of choice) that:

The controversy was over how to interpret the words "choose" and "exists" in the axiom:

  • If we follow the constructivists, and "exist" means "find," then the axiom is false, since we cannot find a choice function for the nonempty subsets of the reals.
  • However, most mathematicians give "exists" a much weaker meaning, and they consider the Axiom to be true: To define f(S), just arbitrarily "pick any member" of S. (ibid)
I, personally, think that the words "choose" and "exist" are merely consequential grammatical elements of the broader vulgate statement of the axiom and adjunct to the main notion of "property"—"...a function f defined on C with the property that..."

This abstract notion of 'property' appears highly flexible and general, yet highly constrained—a given result is never mere whimsy but always derived from a set of rules/operations that do not quarter any exception (else it would not comprise of a set as such).

Mario Livio the author of The Equation That Couldn't Be Solved (2005) writes of this notion of 'property':

The properties that define a group are:

1. Closure. The offspring of any two numbers combined by the operation must itself be a member. In the group of integers, the sum of any two integers is also an integer (e.g., 3 + 5 = 8).
2. Associativity. The operation must be associative—when combining (by the operation) three ordered members, you may combine any two of them first, and the result is the same, unaffected by  way they are bracketed. Addition, for instance, is associative: (5 + 7) + 13 = 25 and 5 + (7 + 13) = 25, where the parentheses, the "punctuation marks" of mathematics, indicate which pair you add first.
3. Identity element. The group has to contain an identity element such that when combined with any member, it leaves the member unchanged. In the group of integers, the identity element is the number zero. For example, 0 + 3 = 3 + 0 = 3.
4. Inverse. For every member in a group there must exist an inverse. When a member is combined with its inverse, it gives the identity element. For the integers, the inverse of any number is the number of the same absolute value, but with the opposite sign: for e.g., the inverse of 4 is -4 and the inverse of -4 is 4; 4 + (-4) = 0 and (-4) + 4 = 0. (Mario Livio, 2005. p. 46)

On a clock calculator (modulo 6), Gauss found that any given prime number has congruence with either 5 or 1 for prime numbers greater than 3. How many rotations are required to obey this congruence is not predictable, but there is an interesting property in that for any twin primes the first must occur at 5 o'clock and the second must occur at the immediately following 1 o'clock.

Jay

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