Thursday, October 07, 2010

I is for Incompleteness

One of the most important results in mathematics and logic in the past century came from Kurt Gödel, an Austrian logician and mathematician. His two incompleteness theorems force a sort of white-knuckled humility upon mathematicians everywhere. However, for the most part, outside of mathematicians, logicians, and philosophers, Gödel’s results remain largely unknown and seem largely irrelevant but the theorems actually have far-reaching impacts for almost any discipline. The two theorems are as follows:

Theorem 1: No consistent algorithm/axiomatization of formal arithmetic is complete.

Theorem 2: For all systems of arithmetic (A), if A is consistent, then there is no proof in A of the consistency of A.

These might seem highly technical and irrelevant, but let’s look at what the theorems actually mean and what some of their consequences are.

To understand either theorem requires some knowledge of terminology:

Arithmetic is one of the mathematical sciences (actually, the mathematical science of the integers)

An algorithm is a system of steps or procedure for solving a problem

An axiomatization is a set of axioms (or self-evident truths) which forms the starting-point for a mathematician desiring to arrive at new conclusions in mathematics

A set of axioms is consistent if and only if none of the axioms are mutually exclusive. That might seem somewhat abstract, but it makes sense. For instance, let’s say one of my axioms was “a certain number exists” and then another of my axioms was “that same number does not exist.” This is an absurdity and this set of axioms would be considered inconsistent.

A set of axioms is complete if and only if that set of axioms can be used to arrive at every truth in its particular science. For instance, a set of arithmetic axioms (or an axiomatization for arithmetic) is complete if it can be used to show every arithmetical truth.

It is desirable that an axiomatization be consistent, because if it is inconsistent, then it will arrive at contradictory conclusions. In other words, an inconsistent axiomatization leads to false conclusions, which is obviously undesirable. It is desirable that an axiomatization be complete so that we can arrive at every true conclusion regarding our particular science (for instance, arithmetic).

Again, this may seem abstract, but I am reaching a point, so please bear with me. Noting all of this information, then, informally we may state Gödel’s incompleteness theorems as follows:

No set of axioms can be proven to be consistent using that set of axioms. Furthermore, even if you could prove that it was consistent, it couldn’t be complete. That is to say, there is no way to show that a set of axioms will never lead to falsehood. However, say you knew you had a consistent set of axioms. That set of axioms cannot possibly arrive at all mathematical truth.

Okay, now to the fun part. Why is all this crazy stuff about mathematical systems of axioms important? The implications of these theorems are huge. Since those theorems are true, it appears to be true that, at least through mathematical methods, not every mathematical truth can be shown to be true. In other words, there are true things that we cannot prove. And if that’s true in mathematics, it’s certainly true in other disciplines as well. This is a strong argument for the existence of objective truth, because it shows that there are truths that are true even though it is impossible for us to prove them – that are true independent of our own minds.

It is also an important conclusion for refuting people who deny the existence of immaterial reality. Why? People who deny immaterial reality deny it on the basis that it can’t be proven using the empirical sciences (with the exception of certain sophists who choose to deny the existence of any reality at all - material or immaterial). But we know there are truths that can’t be proven via deductive reasoning. It is a mathematical fact (which, ironically enough, has been proven). Obviously this, of itself, is not sufficient for showing the existence of immaterial reality, but it is enough to silence those who argue against its existence which might open them up to our own arguments for the existence of immaterial reality.

To conclude, then, Gödel proved his two incompleteness theorems which (together) show that a complete knowledge of mathematical truths is outside the capability of any mathematical system (that is to say there are mathematical propositions that are true that are also unprovable). This has repercussions for our debates with people who believe in the supremacy of mathematics or physics rather than theology and her handmaiden philosophy. As such, even though the theorems themselves might be a little too technical to bring into a discussion, its consequences are (if not essential) highly useful for the Catholic participating in the New Evangelization.

Mariae et Jesu Semper Servus Sum,


Source: Philosophy of Mathematics: A Contemporary Introduction to the World of Proofs and Pictures, 2nd Edition, written by James Robert Brown


Jeb said...

Joe said...

Okay, Gödel's Incompleteness Theorems do not prove the existence of God (which appears to be the assertion of the author to whom you linked) ... and I never claimed it did. I didn't even touch the issue of the existence of God. What I touched was the issue of the existence of immaterial reality and the existence of objective truth.

This is a quote from my post above that seems relevant here: "Obviously this, of itself, is not sufficient to show the existence of immaterial reality, but it is enough to silence those who argue against it..." In other words, the closest to denying the existence of immaterial reality anyone can get, given these theorems, is a reasoned uncertainty concerning the existence of immaterial reality rather than an explicit denial.

Hopefully that makes sense.

Thank you for your post,