Risto

This past weekend while reading Descartes' Meditations on First Philosophy I was struck by the similarities between Descartes' descriptions of human finitude and natural infinitude, and the maxims of Kurt Gödel's Incompleteness Theorem. I'm no expert (or amateur) on Gödel's Incompleteness Theorem, however it seems as though Gödel's Theorem is a formal expression and proof of aspects of Descartes' philosophy.

For example, Descartes states that "[f]or although the idea of substance is in me by virtue of the fact that I am a substance, that fact is not sufficient to explain my having the idea of an infinite substance, since I am finite, unless this idea proceeded from some substance which really was infinite." Unfortunately for humans, "the nature of the infinite is such that it is not comprehended by a being such as I, who am finite," and "it is the essence of a finite intellect not to understand many things." Tying this back to Gödel; in the book Incompleteness: The Proof and Paradox of Kurt Gödel by Rebecca Goldstein, the author describes some of the conclusions we can draw from the Incompleteness Theorem, and notes that "[t]he limits of formalization, of our attempt to reduce all ... knowledge to the specified rules of a system, are not congruent with the limits of our knowledge. Our ... knowledge exceeds our systems." In other words, "aspects of ... reality ... escape our formal systematizing (although not our knowledge)"; "[w]hereof we cannot formalize, thereof we can still know." Hence the human ability to possess knowledge about nature exceeds the human ability to explain and understand the legitimacy of that knowledge, but nonetheless our (finite) knowledge proceeds by legitimate processes from (the infinity that is) nature.

What's important to note here is that while incompleteness validates the logic of inductive inferences, it also invalidates the idea that humans can be objectively certain about their empirical beliefs. As Descartes notes in his reply to the second set of objections, "there is more objective reality in the idea of an infinite substance than there is in the idea of a finite substance." This is what Bronowski was getting at in The Origins of Knowledge and Imagination when he noted that "[n]ature is not a gigantic formalizable system. In order to formalize it, we have to make some assumptions which cut out some parts. We then lose the total connectivity. And what we get is a superb metaphor, but it is not a system which can embrace the whole of nature ... [because] no formal system embraces all the questions that can be asked." Therefore, because "the reach of all formal systems is limited," "[w]hen you axiomatize an arithmetical or mathematical system, you automatically impose a limit on it," and conceptually speaking, "you cut the universe in half." Axiomatization is of course inescapable, and as a result of this human communications will always be ambivalent to some degree. Quite simply; "[t]here is no way of exchanging information that does not demand an act of judgment." Thus humans may legitimately claim to understand aspects of nature, but only insofar as their finite physiology (and it seems to me that in particular their limited capacity for humility) permits. Humans exist as formal systems, and formalization necessitates natural incompleteness. Human incompleteness means the species has an innate inability to achieve total connectivity, and although humans possess the ability to contemplate the Cartesian infinity that is nature, they are forever blocked from fully apprehending it, regardless of scientific advancements.

The implications of this are manifold. Consider for example the most recent disagreement or misunderstanding you had when dealing with a friend or family member. Now, consider the mathematical certainty of the fact that incompleteness is your permanent lot in life. Finally, reflect on the ramifications of incompleteness for our most cherished mathematical models, as applied for example in economics and politics. Externalities and collateral damage anyone?