In the comments to my posting on the incoherence of Narnia, it has become apparent that some of the respondents are deeply confused about the relationship between mathematical and empirical truth. Symptoms of this confusion have included a superficially plausible but mistaken application of the Law of the Excluded Middle and an an attempt to invoke Gödel’s Incompleteness Theorem to suggest constraints on our ability to obtain empirical truth.
It’s time to bust some myths…
I think what these confusions mainly demonstrate is that even most of the bright college-educated geeks who comment on my blog have managed to avoid learning anything about philosophy or foundational mathematics. This is mostly not their fault; foundational mathematics is pretty recondite stuff, and the way philosophy is taught at most American universities (e.g. as a branch of literature, or worse as a kind of political indoctrination) is such that a lot of people walk out of philosophy courses less able to reason effectively than when they walked in.
I was, at one time, a mathematician with a strong interest in foundational mathematics and philosophy. Before that I had the good fortune to learn General Semantics from my grandfather at the age of twelve, an experience that gave me extremely effective filters against the kinds of map-vs.-territory confusion endemic in traditional philosophy. The effect of this background (and a lot of thinking) is that I happen to be an expert on the specific philosophical issue of the relationship of mathematical truth to empirical truth, and on some closely related areas in confirmation theory and epistemology.
All this laying-out of credentials is to motivate you to go read The Utility Of Mathematics, an essay on this topic I wrote back in the 1990s. Go read it now…
…and now that you have, I hope it will be clearer why the errors I called out above are in fact errors. In case it isn’t, a bit more education follows.
The “Law” of the Excluded Middle (“For any propositions x, either x is true or not-x is true”) is a specific property of some two-valued logics. Before you can use this “Law” to reason about the empirical world, you have to establish that the essential features of the empirical system you are describing are in fact captured sufficiently well for predictive purposes by a two-valued (or “Aristotelian”) logic.
Usually this will not be the case. We are commonly fooled into thinking that the empirical world obeys Aristotelian logic because the two-value assumption is wired into almost all natural languages. But in most cases Aristotelian logic is as wretched a guide as Aristotelian physics. (Most of General Semantics is techniques for getting rid of Aristotelian mental reflexes conditioned in by our birth
My essay should also make clear why trying to draw conclusions straight from Gödel’s Incompleteness Theorem about the limits of human empirical knowledge is doomed. The GIT is about marks on paper, formulae in a formal system. To make it suggest anything about empirical reality, you have to ring in a lot of complicated and fragile assumptions about how exactly pieces of reality behave as models of certain specific and complex formal systems.
In the words of that famous recipe book: first, catch your rabbit. To apply GIT to human empirical reasoning, you’d have to start by showing that Peano arithmetic and classical two-valued predicate logic are a sufficient formalism to capture the entirety of human empirical reasoning.
Three words: Not. Gonna. Happen.
Be careful out there. The relationship between logic and/or formal mathematics (on the one hand) and observable empirical reality (on the other) is a helluva lot thinner and more contingent than most people think. Please consult a qualified expert before putting it under load.