Going Gödel-ian - II

Since Gödel’s Incompleteness Theorem was first published, its significance with regards to the possibility of a complete base of knowledge has been debated. On the one hand, philosophers such as Patrick Grim have argued that Gödel’s Incompleteness Theorem proves that there is no possibility of a complete body of knowledge, since any body of knowledge can be modeled on a formal system, subject to Gödelian incompleteness. John Lucas argues that omniscience, or at least a transcendence of incompleteness, is possible, and that knowledge is generated beyond mere mechanical manipulations. Some, such as Gödel and me, have interpreted the theorem to mean that mathematics contains truths that are objective. In this section, I shall present these three interpretations of the theorem.

One of the easiest conclusions that can be drawn from Gödel’s Incompleteness Theorem is that, since any formal system must necessarily be incomplete, no body of knowledge or knowledgeable entity can be complete, and therefore omniscience is impossible. Essentially, if the entity attempting to be omniscient starts from a priori truths, the way Kant or Descartes would require, Gödel’s Incompleteness Theorem means that no set of a priori truths (mathematically, axioms) can lead to a comprehensive knowledge of the world, since there must always be statements that cannot be “known” by someone working entirely from a priori precepts. This may seem unusual, since the human mode of thinking is not restricted to working entirely within a system; our thought process is capable of taking a posteriori facts—observations—and incorporating them into our body of knowledge, as well as using logic on propositions that are not within a given system, as Gödel himself did to demonstrate that G must be true. However, our ability to think outside the box, as it were, is only granted when we artificially impose a box on ourselves in the first place. We are not capable of lifting ourselves beyond the hardware of our brains and thinking in ways that are not dependent on our initial configuration of neurons (the axioms) and their patterns of firing (the rules of inference)—at least, not yet. Humans are capable of thinking outside some boxes, but ultimately, we are still bound to something analogous to a rigid formal system, and therefore we will eventually reach a box outside of which we cannot think. This view specifically depends on humans or other knowledge-holding entities using a model of thinking and truth-acquisition that is analogous to a formal system, but so far no thinker has been discovered which does not learn according to the same mechanistic rules that imply incompleteness. Omniscience, in this view, is therefore impossible, since it is impossible for any being with a finite amount of knowledge and pre-defined methods of learning to achieve knowledge of everything around them.

This view takes for granted that human thought in particular and all thought in general is analogous to a formal system, and that it will therefore be subject to incompleteness. John Lucas, however, takes the Incompleteness Theorem as proof that the human mind is not analogous to a formal system, since it is capable of indefinitely thinking outside its box and using reasoning that is not strictly formal to derive conclusions that are, in terms of truth, just as legitimate as ones derived (that is, we can know that G is true, even though we cannot prove it, and therefore we can still know all the facts that are true). In particular, humans are capable not just of finding a posteriori pieces of knowledge that can be known, but are also capable of developing entirely new rules of inference based on these findings. Due to human pattern-recognition and extrapolation capabilities, Lucas says that human thought is not necessarily limited to the formal derivation of proofs from first principles, and therefore omniscience, or at least a capacity to think outside the box, is capable.

My personal views on the matter tend to be more related to the ultimate significance of mathematics and not the possibilities of omniscience as such. I believe that the Incompleteness Theorem demonstrates that truth in mathematics is something that surpasses artificial creations. Since a statement being true and a statement being derived from axioms and rules of inference are not synonymous, it means that it is meaningful to talk about statements having objective truth that is not dependent on the standards of human interpretation or the fallacies of artificial systems. In terms of omniscience, I do think that the human mind is highly analogous to a formal system, and is therefore subject to the Incompleteness Theorem’s restrictions, although a posteriori knowledge and direct experience do allow us to “know” things that aren’t “proved,” although not necessarily in a way that transcends logical derivations. However, this does not prohibit omniscience in general, since one of the assumptions of the Incompleteness Theorem is that the formal system has a finite number of axioms. If, say, a divine being began with an infinite number of axioms, or an infinitely extensible number of axioms, then a complete knowledge that formal system might be possible.

The common theme in these interpretations of the Incompleteness Theorem is that it implies that anything sufficiently similar to a formal system can never achieve a full cognizance of the possible knowledge. While I believe that this does restrict human knowledge and its ultimate expansion, I do not believe that it applies to omniscience in general, and I believe that the expressive nature of these formal system analogues means that knowledge can be conveyed from a successfully omniscient entity to beings, such as humans, that are restricted by the nature of their thought process.