Metaethics and Mathematical Constructivism

Co-written with Claude

People have different ideas about what it means for things to be right or wrong. Sometimes we disagree when we’re mistaken about facts. Other times we seem to disagree about values. But there are also cases to be made that some of our disagreements about values are also disagreements about facts.

Underlying these disagreements are questions of moral realism. Are there facts about morality that exist to be discovered? Are there infinite varieties of moralities? Does it make sense to say that someone is wrong about right and wrong at all?

There’s a hidden element in all of these questions, which is a question of the nature of facts and truth in the first place.

I realize that questioning the nature of facts and truth gets a pretty bad rap these days, and justifiably so, but I promise that I’m just really into epistemology, and I do believe that things can be true, and that mostly this works out to be boring.

But sometimes it isn’t boring! And understanding precisely what we say and hear sometimes really changes what conclusions we draw.

The classical approach to truth is that propositions are either true or false. It’s pretty clear that this can’t cover everything—for example, “This sentence is false.” is totally unresolvable in this way. An appropriate way to resolve this in mathematics is to instead say that a proposition can have a valid associated proof, or a valid associated disproof. This makes claims of truth much clearer—obviously we don’t have every proof laid out in front of us, so some things must be in the set of propositions that we, at minimum, do not yet have a proof of. For some examples, like the Liar Paradox above, we can in fact prove that neither a proof or disproof can be found. And it turns out the statement “it is possible to prove, disprove, or prove the non-existence of a proof about any statement” is itself unprovable.

Computer science takes this a step further with computational complexity—which basically measures how long a proof of a statement is, in these terms, which is more practical (it’s easier to read, verify, and trust shorter proofs), but doesn’t overcome the unprovability problem. Determining how long a program takes to run requires a program that takes so long to run it’s unmeasurable! And even some programs that do terminate would take longer than the lifespan of the universe.

But in practice software engineering has been a massive part of the world economy for decades, so a lot of programs that run in reasonable human amounts of time are very useful after all.

And analogously, most of the disagreements that people have about truth are, in fact, grounded by chains of justification that can be followed down in a conversation.

If this is the nature of truth in general, why should we expect moral truths to be different? Moral realism suggests that moral claims are true or false in an objective sense, and the process of understanding why something is good or bad can feel like a process of discovery! But taken naively, it also suggests that there’s a simple truth to whether things are right or wrong, in a way that feels incompatible with the morally gray nature of the world we live in. And especially in the context of divisive thought experiments that may or may not relate to reality.

For interpreting these moral experiences, undecidability has advantages! The idea that for an action to be moral or immoral requires a proof of its morality or immorality allows that in some cases no proof may exist! Or in other cases, additional assumptions may be required for the proof to exist.

At a very basic level, Haidt’s moral foundations serve as a very basic form of this. The foundations serve as the basis of a vector space, and the proof of morality of actions (given an evaluation along these foundation dimensions) is the observation of the partial ordering of the vector space according to some inner product. This certainly seems to point toward multiple moral theories being internally consistent, which is inconvenient for practical uses, but also gives a richer set of tools for analyzing moral disagreements. Outlining the full proof, or full argument, may reveal persuasive structure even under different axioms. And tools like length of proof, or reductions of assumptions, or different proof techniques provide plenty of directions to examine and improve discourse.