Goedel’s Theorem is an application to mathematics of Aristotle’s thesis that thinking relies on first principles and that first principles are unprovable assumptions. This means that faith and hope are ineradicable features of human existence even in the exact sciences. The briefest summary of the implications of Goedel’s Theorem and the necessity for first principles is the notion that not everything that is true can be proven to be true.
Goedel’s Theorem states that an axiomatic system can be consistent and incomplete, inconsistent and complete, but never consistent and complete. Eternal verities can only be proven in relation to other eternal verities. Axiomatic systems exist on the rational plane of thought. Their rationally approximate and unprovable nature is due to their ultimate reliance on transcendent truths described in Plato’s realm of Forms. For instance, people contrast earthly justice with perfect justice, though the latter has never been instantiated in the physical realm. This implies some intuition of perfect justice, though no one has ever experienced such a thing.
Positivists and post-modern relativists are likely to regard each other as opposites. More than likely both will be liberals and in most cases share a contempt for religion and any notion of transcendence. As rationalists, they will also most likely reject emotional attachment to and especial preference for family, tradition, community, culture and the local physical landscape. The modern liberal instead is committed to being a citizen of the world and welcoming to all comers, no matter their basic hostility to the ethos of the host culture.
Einstein said he would walk to his office at The Institute of Advanced Study at Princeton just for the pleasure of walking home with Goedel.
The second option in Goedel’s Theorem is “inconsistent and complete.” The word “inconsistent” here means self-contradictory. An inconsistent axiomatic system necessarily contains falsehoods because there is no such thing as a true contradiction. Once a contradiction has been accepted as true, it can be used to prove anything. Inconsistent axiomatic systems can be used to give the appearance of “proving” falsehoods. To put it mildly, this is intensely undesirable. And this is why such systems are to be avoided.
The fact that no axiomatic system can be consistent and complete points to such systems’ ultimate reliance on the transcendent; on truths that escape syllogistic proof.
First principles are foundational assumptions embodying transcendent truths such as “human life is valuable.” If the notion of the transcendent is rejected and any Platonic realm of Forms, then there are no truths to philosophize about. Without the transcendent, there is only the rational and rationality is not creative; it is analytic. Rationality takes what is provided to it and subjects it to analysis; but rationality itself provides nothing of substance. Rationality is a tool and at most it can discover logical truths but it cannot prove the intrinsic value of anything. It can show that something has extrinsic value, but without intrinsic value, a thinker is stuck in an infinite regression.
In acknowledging that an axiomatic system is incomplete, a thinker is admitting that axioms are being relied upon that are not provable within the system. Thus, these axioms transcend the system. Rationalists do not want to admit that the transcendent exists, so they would much prefer a “complete” system. However, since “consistent and complete” is not an option, the rationalist must either abandon his project or embrace “inconsistent and complete.”
This explains why rationalists are so happy to contradict themselves. They really have no alternative if they want to avoid nihilism. Of course, having rejected any transcendent truth or value, they are nihilists by default. Logically, it makes no sense for a nihilist to argue about anything, so a consistent nihilist would fall silent. If a nihilist tries to “save” the rest of humanity from its illusions, this is to act in contradiction of the notion that all is futile. Thus, the nihilist is engaged in a performative contradiction.
Consistent rationalists are nihilists. In fact, some of them remain committed to the unprovable transcendental value, goodness and beauty of truth. This cannot be acknowledged without giving the game away. If the rationalist is to continue writing, a way must be found to have his cake and eat it too.
Rationalists have two, perhaps more, ways of generating contradictions. One is to surreptitiously appeal to a transcendental truth whose existence they have just denied. The other is to espouse a theory that is self-negating. If the theory is true, then it applies to itself, and is thus false according to the theory’s own precepts.
Evolutionary biologists interested in explaining morality will reduce morality to things like survival value or the pleasure center of brains and then declare that morality is thus “good,” a thoroughly morally realist assessment. Determinists write as though it is still rationally meaningful to try to persuade someone that he too should be a determinist. English teachers will say that all theories are the result of gender, class and race, but ignore the fact that if true, that truth applies to the theory that all theories are the result of gender, class and race and hence is not true. Liberals will embrace “tolerance” but pounce with vitriol on anyone who deviates from the liberal party line. Liberals will claim sympathy for all victims of scapegoating, but scapegoat white men and Christians with abandon. Liberals will pretend to have the utmost concern with bathrooms for transsexuals, but downplay the anti-homosexual motives of the Orlando shooter. Liberals will castigate conservatives for microaggressions but stay silent about Muslims pushing gays off tall buildings. Positivists will deny the existence of consciousness while using their consciousness to do the denying. Post-modernists at international conferences will claim that cultures are incommensurable and their differences preclude real communication, while in fact all agreeing about this in mutual comprehension. Moral relativists are not moral relativists about “tolerance.” Cultural relativists deny the existence of universal moral principles while promoting the universal moral principle that thou shalt not criticize another culture.
Last semester I had a student who admitted that cultural relativism was self-contradictory but suggested “let’s believe it anyway, because its consequences are so good.” Accepting contradictions is to abandon any claim to rationality; making meaningful argument impossible. If the student’s assertion is to carry any weight, it would be because the premises support the conclusion; a logical relation. Once logic is abandoned, genuine argument ceases. So, the student wants his interlocutors to accept the logical implications of what he says while jettisoning logic.
When someone lies for a good cause it raises a giant question mark regarding the goodness of that cause, especially when elementary rules of logic are repudiated, intentionally or not. Contradictions are lies. A lie contradicts reality. The chances are that the imagined good a person thinks he is promoting is in fact bad. It is not possible to be too categorical or emphatic that the imagined good is bad because there could logically be a different set of legitimate premises for the same conclusion.