On the mathematical incompleteness of Ethics

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The German mathematician of the last century, Kurt Gödel was intensely preoccupied with the formalisms of abstract reasoning. He came to some unfortunate conclusions, in the form of his Incompleteness Theorems, bringing an end to some rather optimistic theories about provability: the quality of any truth to be traced back with a set of theorems to known axioms. This is much relevant to the topic of ethics, since a similar optimism guides our discussions as well — the hope that we can determine some arguably objective good or bad nature for the world surrounding us.

First, let us define a formal system as being complete if any statement can be determined to be true or false based only on statements also inside the system. (For example, googling why are apples healthy, prompts me with the following reasoning: “Apples are rich in fiber, vitamins, and minerals … These substances help neutralize free radicals. Free radicals are reactive molecules that can build up as a result of natural processes and environmental”). Moreover, a system is consistent if there exists no statement that is simultaneously true and false.

The first theorem of incompleteness states that in a consistent system of logic, there exists statements that are true but cannot be proven as such inside the system. Hence the system is incomplete, it “requires” some outside intervention in order to fully explain its workings. The second theorem states that a system cannot prove its own consistency — yet again, such a conclusion requires statements, “truths”, outside the system itself. (As an example, one should look no further than when a child indefinitely asks “Why?”, which quickly exhausts the parent’s patience)

Let us say we sit in a train, ready to leave the station, watching outside. Right next to us, another train prepares to depart. Suddenly, it starts moving and, for a moment, one cannot help but be confused — is it us standing still, and the other train moving, or the other way around? And it is precisely when the last cart of the other train passes completely that we can decide on the nature of our state. For a brief moment, we lived in Gödel’s first theorem — our system, limited to the mere sight through the window (and not some other tools or knowledge) was not able to prove the nature of a statement inside of itself, hence the momentary confusion. It was only by deploying the external information of the scenery after the other train has passed that we could reach our conclusion.

Thus, the conclusions of these formal discussions do have practical implications. The generic formal system Gödel describes can represent our discussion on ethics. For example, in this system, “One should help another” can be considered a true statement, while “One should harm another” can be considered a false statement. What can we derive from Gödel’s incompleteness theorems? Interestingly, this analogy, which leads to the conclusion that ethical systems are incomplete and require “external” intervention, seems to point to some deeply intuitive human beliefs. It seems to demonstrate one fundamental longing of man, that of a higher power. Indeed, it is only some transcendental phenomena that man feels can contain the keys to the completeness of his reasoning and existence.

Religion has been the historical generator of moral compasses, especially because it introduces a system with the prospect of external confirmation. That is, any statements in the system of the world of the mortal can be proven true or false by deploying not only statements in this system itself, but statements in the system of the gods, as well. Before anything, this reasoning must be viewed as a confirmation of Gödel’s theorem s— long before him, humanity accepted to irrevocably lack (at least) some of the answers. For the believer, the words of God are axioms and the nature of any statement can be traced back to the intention of God. Ethics is, in this case a much simpler discussion, at least a theoretically complete one.

For the modern man, religion is an unsatisfactory explanation. It relies on too much belief and too little scientific method. But the question remains: how is man to solve the uncomfortable dilemma of incompleteness? Where the religious deploy the external, transcendental powers of God, the modern man deploys the similarly external and transcendental properties of reason. Fundamental understanding is just as much an external system that promises to solve the incompleteness of our own. It promises deterministic explanations of our phenomena, which in turn, promise the pre-requisites of moral judgement. One may decide on the nature of an action given its outcome. But again, this only goes to show the mirage of completeness, built on top of the deployment of an un-accessible external — this external being a just-as-worshiped determinism.

Gödel’s theorems seem to be a sentence to our hopes of discussing ethics, leaving us in the purgatory of indefinitely incomplete arguments. But this discouragement for answers shall only encourage us in putting forward questions! At the end of the day, our very existence requires a course of action, and one shall look not only outwards but also inwards for the hints of these decisions. In the finite span of time our everyday decisions allow us, there is a certain feeling for what is good and what is bad. And even if such classification is abusively limited, in the end, one still has to make the choice: either you do it, or you don’t.

Shout-out to this random discussion I found online, and to my friend, Bogdan, who has always threw Gödel’s works in our discussions. For far more out-reaching topics of discussion check his “Is mathematics constructed or discovered?” or “On Deustche’s Unproven Theorem Paradox”.