No matter which class I am teaching, for quite some time the first reading assigned has been an article on Goedel’s Theorem. The reason is to emphasize that any attempt to make an axiomatic system of any moderate complexity consistent and complete (able to determine whether any statement within the system is either true or false) will fail. This is because, at least when it comes to mathematics, Goedelian propositions will be generated by the system that are true, can be seen to be true, but are not provable. Goedel’s stand-in for all such propositions is the statement “this statement is not provable within this axiomatic system.” If this statement is true, then it is not provable. If it were to be false, and was provable, then it would again be proved that it is not provable, since you would have just proved a statement that says it is unprovable! I add to this that the axioms upon which axiomatic systems are based are by definition, not provable, their truth being self-evident. So, axiomatic systems contain unprovable truths coming and going.
When teaching ethics, after covering Gödel, the first thing I do is to point out our intuitive understanding of the truth and validity of reciprocity/justice/fairness. If someone were to give you a cup of your favorite coffee at the appropriate time and you were to punch them in the face, barring some convoluted back story, this would be grossly unjust. The truth of reciprocity is captured by phrases like “one good turn deserves another” and “do unto others as you would have them do unto you.” In practice, no normal person doubts the truth of that notion.
Fairness and reciprocity are axiomatically true. Their truth is self-evident. If anyone claims to doubt their truth, this person is almost certainly a liar and a hypocrite, at least on this topic. When this hypothetical person approaches another in a spirit of friendliness and politeness, only to be greeted with unbridled rudeness and hostility, he is likely to feel offended or at least to question the mental stability of the other person.
Ludwig Wittgenstein had some sensible things to say on topics like these. He spends a few pages of his aphoristic writings wondering what it would mean to be mistaken that one was speaking or writing in English. There is not really a standard of certainty that goes beyond knowing such a thing. Part of his point is to throw a monkey wrench into the useless, theoretical musings of his analytic philosophy contemporaries. Continue reading