Mathematical versus moral truth

This piece was inspired by a discussion between Magnus Vinding and Brian Tomasik, as well as a private discussion between Magnus Vinding and myself.

Some philosophers say that moral claims like killing is bad can be proven true or false. This position is called moral realism and according to a recent PhilPapers survey on the views of philosophers, it is a very common view among philosophers. It’s important to understand that moral realists not only claim truths in terms of some underlying assumptions! They don’t just say “If our goal is to reduce suffering, then torturing squirrels is objectively bad.” (In Kant’s terminology this would be called a hypothetical imperative by the way.) They claim that the goal of reducing suffering itself or some other goal can be true or false by itself.

One particular comparison that pops up in discussions of moral realism is that between morality and mathematics. After all, there are absolute (“a priori”) truths without assumptions in mathematics, right? Well…

Truth in mathematics

Until the beginning of the 20th century, most mathematicians probably would have argued that mathematics finds absolute truth without making assumptions. (Interestingly, Greek mathematician Euclid had axiomatized his geometry some two thousand years ago. However, the axioms were seen as “undoubtably/obviously true” without much further thought. Nevertheless, the idea of axiomatization, i.e. basing mathematics on assumptions, was not completely unknown.) But then mathematics had a foundational crisis mainly about problems like Russel’s paradox, which showed that you can easily run into some serious problems, if the foundations of mathematics aren’t studied in a rigorous manner and judgment of correctness of inference is based solely on “intuitive obviousness”. (An extremely accessible and overall great introduction to the topic is the graphic novel Logicomix.) To get rid of contradictions in mathematics, people tried to systematize its foundations. It turned out that the most solid way to go was explicit axiomatization: assuming a small set of statements (called axioms) that all the rest of mathematics can be deduced from. Luckily, a very small set of pretty basic assumptions called Zermelo-Fraenkel set theory is enough for deducing all of mathematics. And axiomatization is still the standard way to talk about truth in mathematical logic. (There are, however, different philosophical interpretations, some of them saying that the axioms are just undoubtedly true.)

Note that when some axiom system implies 2+2=4, this very statement of axioms => 2+2=4 is still not true in an absolute sense as it requires definitions of what all the symbols in that statement mean. And these definitions can only be made when using other symbols, for example those of natural language. (And the higher level proof of axioms => 2+2=4 can only be made by assuming some properties about these meta-symbols.) At some meta-…-meta-level we have to stop with a set of statements/definitions that just have to be assumed true and understood by everyone, or we have an infinite regress. (Humans usually stop at the language level, which is somehow obvious to everyone.) Or we get cycles (“a set is a collection” and “a collection is a set”). That’s the Münchhausen trilemma.

Maybe more importantly, axioms => 2+2=4 is not really the statement that we want to be absolutely true. We want 2+2=4 without underlying assumptions.

So, how did people do mathematics before axioms, then? How did they manage to avoid finding underlying axioms? Well, they always assumed some things to be obvious and did not require further justification. And these things were not even on the level of axioms, yet, but more like commutativity (n+m=m+n, e.g. 1+2=2+1). So, when proving some mathematical hypothesis, they wrote things like n+m=K and therefore also m+n=K and people didn’t ask why, because commutativity of addition is obvious. This is still how it is today: People know that if you continue to ask why they will end up at something like the Zermelo-Fraenkl axioms. And even if they don’t view the Zermelo-Frankl axioms as the foundation of mathematics, almost everyone agrees about commutativity of addition and basic arithmetics in general, whether generated by the Zermelo-Fraenkl or the Peano axioms.

So, this sounds terrible. How do we know that 2+2=4 if all of this is based on axioms that can’t be proven? Well, for one, mathematics is mainly relevant as a model of reality, which can be used to make testable predictions: if we take two separable things and add another two separable things then we will have four things. (That does not work for clouds, you see.) If we had an axiom system in which the natural definition of addition makes 2+2=5 true, then this addition would not be useful for predicting the number of things you have in the end. So, that’s one way to justify the use of Zermelo-Fraenkl: as a gold mine of simple models that can be used to make correct predictions.

There’s another silver lining. The sets of axioms that are usually used are extremely simple and basic. For example, one of the Zermelo-Fraenkl axioms states that if you have a couple of sets, then there is another set which contains all the elements of the sets. While it is philosophically troubling that proofs can’t verify “absolute truths” without some assumptions, it’s obviously not very productive to ask: “But what if the union of two sets really does not exist?” Just like it is not very productive to put into question the rules of logic and probability theory. Without assuming such “rules of thought”, it is not even clear how to think about doubting them! (Thanks to Magnus Vinding for pointing that out to me.) So, some things must be assumed if you don’t want your brain to explode.

Also, some of the axioms can be seen as so basic that they merely define the notion of sets. So, you may view the axioms more as definitions of what we are talking about than as some assumptions about truth.

So, some set of assumptions (or at least a common understanding of symbols) is simply necessary for doing any useful mathematics, or any useful formal reasoning for that matter. (Similarly, you need probability theory and some prior assumptions to extract knowledge from experience. And according to the no free lunch theorems of induction, if you choose the “trivial prior” of assuming all possible models to have the same probability you won’t be able to learn from experience either. So, you need something like Occam’s razor, but that’s another story…)

There is not much wrong with calling mathematical theorems “true” instead of “true assuming the axioms of … with the underlying logical apparatus …”, because the meaning does not vary greatly among different people. Most people agree about assuming these axioms or some other set of axioms that is able to produce the same laws of arithmetic. And there is also a level on which everybody can understand the axioms in the same way, at least functionally.

(Interestingly, there are exceptions to universal agreement on axioms in mathematics. The continuum hypothesis and its negation are both consistent with the Zermelo-Fraenkl axioms (assuming they themselves are consistent). This has led to some debate about whether one should treat the continuum hypothesis as true or false.)

Truth in ethics

So, let’s move to ethics and how it can be justified. The first thing to notice, I think, is that making correct claims in ethics requires additional assumptions or at least terminology. Deducing claims about what “ought to be” or what is “good” or “bad” requires new assumptions or definitions about this “oughtness”. It’s not possible to just introduce a new symbol or term and prove statements about it without knowing something (like a definition) about it. (This is probably what people mostly talk about when they say that moral realism is false: Hume’s is-ought-gap which has never been argued against successfully.)

But one needs to be able to make normative statements, i.e. statements about goals. Acting without a goal is like trying to judge the validity of a string of symbols without knowing the meaning of the symbols (or like making predictions without prior assumptions): not very sensible. Without goals, there is no reason to prefer one action over another. So, it does make sense to introduce some assumptions about goals into our set of “not-really-doubtable assumptions”.

Here are a few options that people commonly choose:

  • using the moral intuition black box of your brain;
  • using a set of rules on a meta-level, like “prefer simple ethical systems”, “ethical systems should be in the interest of everyone”, “ethical imperatives should not refer in any way to me, to any ethnic group or species” etc.;
  • using a lot of different, non-coherent (“object-level”) rules, like the ten commandments, the sharia, the golden rule, egoism etc., and deciding on specific questions by some messy majority vote system;
  • using some specific ethical imperative like preference utilitarianism, the ten commandments + egoism, coherent extrapolated volition etc.

(Note that this classification is non-rigorous. For example, there is not necessarily a distinction between rules and meta-rules. Coherent extrapolated volition can be used in the imperative “fulfill the coherent extrapolated volition of humankind on September 26, 1983” or in the meta-level rule “use the moral system that is the result of applying coherent extrapolated volition to the version of humankind on September 26, 1983”.  Also, most people will probably keep at least some of their intuition black box and not override it entirely with something more transparent. Maybe, some religious people do. Eliezer Yudkowsky would argue that affective death spirals can also lead non-religious people to let an ethical system override their intuition.)

Diverging assumptions

What strikes me as the main difference between assuming things in the foundations of mathematics and assuming some foundations of (meta-)ethics, is that there are many quite different sets of assumptions about oughtness and all of them don’t make your brain explode. It’s extremely helpful to have mathematical theories of arithmetic (or at least geometry) to produce useful/powerful models. And as soon as you have this basic level of power, you can pretty much do anything, no matter whether you reached arithmetic directly or via Zermelo-Fraenkl. Without the power of arithmetic, geometry or something like that, an axiomatic system goes extinct, at least when it comes to “practical” use in modeling reality. (Similar things apply to choosing prior probability distributions. In learning from observations, you can, in principle, use arbitrary priors. For instance, you could assume all hypotheses about the data that start with a “fu” to be very unlikely a priori. But there is strong selection pressure against crazy priors. Therefore, evolved beings will tend to find them unattractive. It’s difficult to imagine how the laws of probability theory can be false…)

For ethical systems the situation is somewhat different. Most people base their moral judgments on their intuition. However, moral intuition varies from birth and continues to change through external influences. There does not seem to exist agreement on meta-level rules, either. For example, many people prefer simple ethical systems, while others put normative weight on complexity of human value. People disagree about object-level ethical systems: there are different consequentialist systems, deontology and virtue ethics. And there are plenty of variations of each of these systems, too. And, of course, people disagree a lot on specific problems: the moral (ir)relevance of non-human animals, the (il)legitimacy of abortion, whether punishment of wrongdoers has intrinsic value, etc. So, people don’t really disagree on the fundamentals of mathematics, but they fundamentally disagree on ethics.

And that should not surprise us. Humans have evolved to be able to work towards goals and to learn about abstract concepts. Arithmetics is an abstract concept that is useful for attaining the goals that humans tend to have. And thus, in a human-dominated society, the meme of arithmetics (and axiomatic systems that are sufficient for arithmetic) will spread.

I can’t identify similar reasons for one ethical system to be more popular than all others. Humans should have evolved to be “selfish” (in the sense of doing things that were helpful for spreading genes in our ancestral environment) and they are to some extent (though, in modern societies, humans aren’t very good at spreading genes anymore). But selfishness is not a meme that is likely to go viral: there are few reasons for a selfish person to tell someone else (other than maybe their close kin) that they should be selfish. (Some people do it anyway, but that may be about signaling…) So, one should expect that meme to not spread very much. Including relatives and friends into moral consideration is a meme much more virulent than pure egoism. People tend to raise their children that way to stop them from exploiting their parents and from competing too hard with their brothers and sisters. Generosity and mutual cooperation among friends is also an idea that makes sense to share with your friends. Life in larger society makes it important to cooperate with strangers especially if they pay taxes to the same government that you do.

But as opposed to these somewhat ethical notions that can be selfish to spread, most people tend to think of ethics as giving without expecting something in return, which is not favored by evolution directly. So, while we should expect the notions of egoism, and cooperation with kin, friends and even members of one’s society (potentially at the risk of being a “sucker” sometimes) to be universal, there is much less direct selection pressure towards, say, helping animals. Explaining “real altruism” (not cooperation, signaling etc.) with evolutionary psychology is beyond this post, but I suspect that without strong selection pressures in any specific direction, there is not much reason to assume that there is much homogeneity. (The continuum hypothesis can be seen as a precedent for that in the realm of mathematics.)

Diverging assumptions make moral “truth” relative in a practical sense. A moral claim can be true on the set of assumptions held by many people and wrong on another set of assumptions also held by many people. So, instead of saying “true” one should at least say “true assuming hedonistic utilitarianism” or “true assuming that simple ethical systems are to be preferred, ethics should be universalizable and so on”. (Unless, of course, you are talking within a group with the same assumptions…)

Lack of precision

Another difference between ethics and logic is that ethical statements are much less precise than mathematical ones. Even before the foundations of mathematics were laid in the form of axiom systems and logical rules of deduction, there was little disagreement on what mathematical statements meant. That’s not the case for ethics. Two people can agree on killing is wrong and disagree on meat consumption, the death penalty and abortion, anyway, because they mean “killing” in different ways. People can agree on ethics being about all sentient beings and have the same knowledge about pigs and still disagree on whether they are ethically relevant, because they don’t agree on what sentience is.

Lack of precision makes moral truth ill-defined for most people. If your goal is to reduce suffering and you don’t have a precise definition of what suffering is, then you can’t actually prove that one set of consequences contains less suffering than another even if it is intuitively clear. If you have a plain intuition box, things are even worse. And thus one problem in normative ethics seems to be that few people are able to say what kind of argument would convince them to change their mind. (In the empirical sciences, statements are frequently blurry, too, of course, but usually much less so – scientists usually are able to make falsifiable claims.)


With moral “truth” being relative and usually ill-defined, I don’t think the term “truth” is very helpful or appropriate anymore, unless you add assumptions (as a remedy to relativity) that are at least somewhat precise (to introduce well-definedness).

As a kind of summary, here are the main ways in which moral realists could disagree with this critique:

  1. “Truth in mathematics (and the principles of empirical science, e.g. Occam’s razor, Bayesian probability theory) is absolute. At some level there is a statement that requires neither further definitions nor other assumptions. This makes it possible to also prove things like 2+2=4 without underlying assumptions, where 2,+, = and 4 can be translated into other things that don’t require further definition.”
  2. “The trilemma arguments about mathematical truth don’t apply to moral truth. While mathematical truth requires some underlying assumptions, moral truth does not require underlying assumptions. Or at least no further assumptions about ‘oughtness’.”
  3. “There is some core of assumptions about ethics that everybody actually shares in one variation or another similar to the way people share arithmetics and these can also be turned into precise statements (to everyone’s agreement) that make moral truth well-defined. With derivations from these underlying assumptions, most people could be convinced of, for example, some specific view on animal consciousness.”
  4. “Even though people disagree about the assumptions and/or their conceptions of ethics are non-rigorous and therefore nontransparent, speaking about ‘truth’ (without stating underlying assumptions) can somehow be justified.”

As I have written elsewhere, I am not entirely pessimistic about reaching a moral consensus of the sort that could warrant calling moral claims true in the sense that objection 3 proposes. I think there are some moral rules and meta-rules that pretty much everybody agrees on: the golden rule, morality being about treating others, etc. And many of them seem to be not completely blurry. At least, they are precise enough to warrant the judgment that torturing humans is, other things being equal, bad. Works like that of Mayank and Daswani and myself show that concepts related to morality can be formulated rigorously. And Cox’s theorem, as discussed and proved in the first chapters of E.T. Janes’ book on probability theory, even derives the axioms of probability from qualitative desiderata. Maybe, though I am not sure about this, there is a set of axioms that many people could agree on and that is sufficient for deriving a moral system in a formal fashion.

There are some models of this in the realm of ethics. For example, Harsanyi proved utilitarianism (though without actually defining welfare) under some (pretty reasonable) assumptions and Morgenstern and von Neumann proved consequentialism under another set of very reasonable assumptions. Unfortunately, somebody has yet to come up with an axiomatization of moral intuitions that is convincing to more than just a few people…


Adrian Hutter made me aware of an error in an earlier version of this post.

3 thoughts on “Mathematical versus moral truth

  1. Joshua Greene makes a very similar point in chapter 7, section “Is morality like math?” of his (very nice) book “Is morality like math?”:

    “I’m a big fan of reason. This whole book—indeed, my whole career—is devoted to producing a more reasoned understanding of morality. But there is a rationalist vision of morality that, in my opinion, goes too far. According to the hard-line rationalists, morality is like math: Moral truths are abstract truths that we can work out simply through clear thinking, the way mathematicians work out mathematical truths. Kant, for example, famously claimed that substantive moral truths, such as the wrongness of lying and stealing, could be deduced from principles of “pure practical reasoning.” Today, few people explicitly endorse this view. Nevertheless, many of us appear to have something like hard-line Kantian rationalism in mind when we insist that our own moral views, unlike those of our opponents, are backed up by reason. Many people state, or imply, that their moral opponents hold views that can’t be rationally defended, the moral equivalents of 2 + 2 = 5.

    What would it take for morality to be like math? For it to be thoroughly reasoned? Mathematicians are in the business of proving theorems. All proofs begin with assumptions, and the assumptions of mathematical proofs come from two sources: previously proven theorems and axioms. Axioms are mathematical statements that are taken as self-evidently true. For example, one of Euclid’s axioms for plane geometry is that it’s possible to connect any two points with a straight line. Euclid doesn’t argue for this claim. He just assumes that it’s true and that you, too, can see that this must  be correct. Because all theorems derive from previous theorems and axioms, and because theorems do not go back indefinitely, all mathematical truths ultimately follow from axioms, from foundational mathematical truths taken as self-evident.

    If morality is like math, then the moral truths to which we appeal in our arguments must ultimately follow from moral axioms, from a manageable set of self-evident moral truths. The fundamental problem with modeling morality on math is that, after centuries of trying, no one has found a serviceable set of moral axioms, ones that (a) are self-evidently true and (b) can be used to derive substantive moral conclusions, conclusions that settle real-world moral disagreements.”


  2. Yes, some moral truths are just like math. It’s all clear when you can grasp the entire structure of explanatory knowledge as a coherent *whole*. The human brain wasn’t designed to do this, and that explains the lack of agreement in morality. My top-level domain model of reality clearly shows where ‘axiology’ fits in:

    See group of 3 knowledge domains top-right (Phenomenology, Decision&Game Theory, Axiology).

    Most of the confusion is (as you suspected) because consciousness is currently not well-understood. Ultimately Axiology is a high-level knowledge domain built up from the concepts in the domain ‘phenomenology’. But it’s clearly a fundamental property of reality, exactly analogous to knowledge domains ‘cosmology’ (in physics) or ‘set theory’ (in mathematics).

    The correct principles are a combination of 3 general principles(ultra-finite 3-level recursion):

    Virtue ethics > Consequentialism > Deontology

    The correct platontic ideals also a combination of 3 ideals (ultra-finite 3-level recursion):

    Beauty > Liberty > Perfection

    By the way, the principle of 3-level ultra-finite recursion means the continuum hypothesis has to be true. Math separates to 3 levels exactly analogous to the levels of axiology listed above:

    Computation > Number Theory/Algebra > Analysis

    Continuum (real numbers) are on level 3 (analysis). Integars sit on level 2 (algera). So there can’t be any other level of cardinality between the integars and the reals.

    There are no ‘absolutes’ ether in math or moralty, it’s just a coherent network of beliefs. But we don’t need to bridge ‘is-ought’ for moral realism and universality to work (options 3 and 4 in your list can save realism).

    Everything is going to be all right. There are worlds enough, and time…


  3. Pingback: Moral realism and AI alignment – The Universe from an Intentional Stance

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