In Godel's Grip

"How deep to me are your thoughts, O God! How vast is the sum of them. If I try to count them, they are more than the sand." - Psalms, 139:17-18.

"Concepts without factual content are empty; sense data without concepts are blind ... The understanding cannot see. The senses cannot think. By their union only can knowledge be produced." - Immanuel Kant, Critique of Practical Reason.

Mathematics is full of simple, easily stated theorems that thus far have eluded proof. Goldbach's Theorem states that every even number can be expressed as the sum of two primes. The theorem has never found to be false, but it has not been proven in general. Perhaps there is a proof, even a simple one, just awaiting discovery. Perhaps we just need to furrow our brows and concentrate and human cleverness and ingenuity will uncover the proof. However, it may be one of those truths that Gödel said would forever elude our grasp.

Kurt Gödel was a small, bespectacled, owlish, introverted, and professorial mathematician and logician who turned the mathematics world upside down as the implications of his work became apparent. In 1931 pre-war Europe, the 24-year old Gödel published "Uber formal unentscheidbare Sätz der Principia Mathematica und verwandter Systeme" or "On Formally Undecided Propositions of Principia Mathematica and Related Systems." Although the work was short and dense, in one fell swoop the hubris of the purest of the pure intellectual disciplines, mathematics, was decimated.

Gödel proved that for any set of finite and consistent axioms sufficiently rich to encompass arithmetic, there are true statements that will be forever unprovable. In order to prove all truths either the set of axioms must be infinite in number or some of the axioms must be contradictory. With Gödel, what we could really know at the strictest level was not constrained by intelligence, diligence or cleverness but is inherently limited by logic itself. There are, therefore, true propositions that the severest skeptic could not affirm because of an impossible burden of proof.

With the discovery in physics of Heisenberg's Uncertainty Principle at about the same time, the tops of heap in intellectual enterprises, physics and mathematics began to bump head on into the limits of "knowability." Recently, chaos theory has suggested that since the unfolding of the universe may be so critically dependent upon initial conditions that there are practical limits of predictability for complex systems.

Of course the limits so carefully outlined by Gödel and Heisenberg are not a repudiation of rationality and logic but giant monuments to human intellectual capacities. Nonetheless, the boundaries encountered by the purest and most mature of the sciences and intellectual disciplines should provide an antidote to hubris. Just as blind faith is an abdication of responsibility, unhealthy skepticism is an abandonment of the entire man. Gödel's and Heisenberg's work requires the most resolute and honest skeptics to acknowledge that the strict adherence to perpetual and unrelenting doubt means certain truth is forever forbidden. To assert otherwise is an act of faith inconsistent with skepticism.

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