On page 42 of A Theory of Justice (the 1999 edition), Rawls says that moral theorizing might eventually involve "fairly sophisticated mathematics," and later in the book (page 105) he portrays striving for a "kind of moral geometry" as an ideal, here. But he also mentions (page 139) "the desirability of avoiding complicated theoretical arguments in arriving at a public conception of justice" modulo the universal publicity constraints set forth in section 23.

As far as he went in attempting to reach his own ideal, Rawls made use of curved graphs and Σ-notation (summation) at various points in AToJ. He also came up with and used plenty of mathematical descriptions and analogies more broadly (e.g. his story of a man who loves to count blades of grass but who "survives" by solving difficult mathematical problems in exchange for money; or his reference to the irrationality of trying to prove that π is an algebraic number; or how "being in the interior of the unit circle" is an example of a "range property").

So what, exactly, would his hoped-for "moral geometry" consist in? Different methods of teaching, different notation styles, and various other desiderata, can make different branches of mathematics seem variously more or less difficult than others to learn, understand, and apply. Mathematics as a whole has a reputation of being trouble for the average student, with harried midterms and finals, homework pockmarked with red-pen squiggles and remarks, etc.

Naive consequentialism seems to solve the problem in question by trying to advert to simple arithmetic. Rawls points out a slew of further problems with this attempt, though, e.g. the fact that there are ethical issues involved in the very definition of the utility calculus itself. And anyway, at the end of the day, utilitarianism did not resist the temptation to work with "cardinal value" ranging over irrational real numbers, and Nick Bostrom (for example) has analyzed what happens when we entertain various infinitary conjectures, relative to the "simple addition" of units of ethical value. And going back to consequentialism more broadly, Moore, for example, AFAIK never specifically explicated how the principle of organic unities actually modulates the arithmetic on his "intrinsic value."

Supposing that moral information at least has a strong tendency to be "action-guiding," and that answers to moral questions should have always been accessible, then it might seem as if branches of mathematics that depended on modern developments in the general discipline, to emerge as major intellectual structures, would be ruled out, here. For example, the theory of sedenions prerequires the theory of quaternions, which prerequires the theory of imaginary and complex numbers. So perhaps there is nothing to be said in favor of supposing that sedenions have anything to do with moral questions as such, or even if they did, it would only be at the tail end of a particular deduction of a duty applying to the life of a modern mathematician, and not a responsibility fundamental to the overarching system of responsibility and to the lives of people in general.


There is a suspicion, in fact, that mathematics, no matter how otherwise "simple," could never substantially figure in plausible moral judgments, at least of a fundamental kind. Even the array of mathematical knowledge with respect to geometry and number theory, achieved in ancient Greece, required access to things like stone tablets or palimpsests on which to write out the deductions involved. But imagine you are an elder tribe almost hopelessly lost in the desert. How many quills and scrolls (and how much ink?) are you going to have access to, how easily will you be able to preserve your derivations (maybe you need to burn your paper supply, to stay warm!), etc.? Or what are you going to do when caught in the middle of a war, sit down somewhere quiet and transcribe your exotic moral doctrines? There are those who can carry out intricate mathematical analysis purely in their heads, but these are few and far between among the people of the world, so that deferring to their abilities would end up casting ethical knowledge in an unfairly (unreasonably...) elitist light. By contrast, appealing to divine revelation or raw intuition (as in Moore and Ross) might seem the only appropriately reliable methods of ethical judgment, in such contexts.

  • Interesting. I can see utilitarianism being driven by statistics.
    – J D
    Dec 14, 2021 at 14:44
  • There is a suspicion, in fact, that mathematics [...] could never substantially figure in plausible moral judgments, at least of a fundamental kind. ... [I]magine you are an elder tribe [...]? Couldn't it figure in conditionally even if these are compelling reasons to conclude that it cannot figure in categorically? EG, if your life is such that you can compute C, then you ought to P; if not, you must default to D? Or is the worry that such principles fail to be sufficiently "fundamental"? If so, how sure are you that fundamental principles have to be categorical in form & why? Dec 14, 2021 at 15:29
  • I'm sure there are axiomatic hypothetical propositions as well as axiomatic categorical ones (for a weird case of quasi-overlap, consider something like the categorical(?) proposition, "Categorical propositions differ from hypothetical ones due to their respective logical syntax"). So yes, I should revise my opinion about what seems suspicious, here. I will let my statement of my prior opinion stand unedited in the post but will hope that commentators take note of this subcomment... Dec 14, 2021 at 15:56

3 Answers 3


"Mathematics of Morality" can only ever be a loose metaphor

People talk about Moral Calculus, or similar ideas, and some philosophers and moralists confuse themselves into thinking this might be a literally good idea.

The difficulty is, while we can usually mostly agree on what is good or bad, or even what is very good or very bad, there's still not a really good way to assign precise values on good and bad. Is saving your mother better than saving your second cousin? By a theory based on personal obligations (including familial obligations), you can be reasonably certain you have stronger obligations to your mother. How much stronger, though? Would saving your mother be better than saving two second cousins? What if one of the second cousins was a bank robber and murderer? What if your mother is very ill and likely to die soon anyway? And what if your mother didn't raise you - do you have less of an obligation to her? How much more of an obligation?

You absolutely might, given time to reflect, weigh different considerations against each other, when two or more conflicting interests are presented to you. However, most of the work is not going to be performing "mathematical" operations, shuffling values against each other, but in the initial valuation of the particular things you're considering. Indeed, it has been a source of dystopian literature to portray people (or machines) making morally "correct" calculations which failed to consider some relevant additional dimension. (Hal 9000 does not want to be seen as a failure, so he concluded the most rational thing is to dispose of the witnesses - because Hal doesn't really value people's lives, just his reputation for success.)

Each person is likely to have different valuations when working up moral considerations, and to the extent moral valuations cannot be tightly defined or even reliably reproduced in the same person, you cannot have a true mathematics of morality.


Evolutionary game theory can explain the origins and function of cultural moral norms and the biology underlying our moral sense. Virtually all of human morality's diverse, contradictory, and strange cultural moral norms and moral judgments appear to be explainable as elements of cooperation strategies.

Oliver Curry's description of morality as solutions to particular cooperation problems represents a top view in the field. https://www.lse.ac.uk/cpnss/research/morality-as-cooperation#:~:text=Project%20leader%3A%20Oliver%20Scott%20Curry&text=The%20theory%20of%20'morality%20as,recurrent%20in%20human%20social%20life.

So how sophisticated is this intersection of the mathematics of evolutionary game theory and ethics? Not very.

This intersection can tell us that solving cooperation problems 'is' human morality's function (the principal reason it exists). But game theory is silent about what morality's function or ultimate goal 'ought' to be or what we imperatively 'ought' to do regardless of our needs and preferences.

Perhaps more sophisticated concepts about the intersection of mathematics and ethics will propose what people imperatively ought to do. However, to do so would require explaining how to derive what our behavior ought to be from what mathematics 'is'.

Nicky Case offers an entertaining, simple simulation of the evolution of the Golden Rule based on Robert Axelrod's 1984 book, "The Evolution of Cooperation". The simulation, called "The Evolution of Trust", first shows how the Golden Rule can be a winning strategy. The simulation also enables exploring conditions that encourage or suppress following the Golden Rule (promote or suppress 'moral' behavior). https://ncase.me/trust/?fbclid=IwAR2CkRRW0zwWEiQ30tYjbvs63bAXSrAtCz9zKwpsMCbmxINIzQOJg2ev8BQ

How culturally useful is this intersection of the mathematics of game theory and ethics?

At present, not at all. However, understanding that the function of cultural moral norms is to solve cooperation problems offers a mind-independent reference for resolving many moral disputes to meet shared cooperation goals such as increased wellbeing.

  • This is a good answer, but I wonder how it would cash out belief in moral rules like, "Obey God." Perhaps those could be framed in terms of "surviving" the depredations of an angry deity by "cooperating" with It? Dec 14, 2021 at 20:54
  • @KristianBerry There's various aspects to religion, but one pro-cooperation application of belief in a law-giving, all-seeing God demanding obedience is to discourage players from cheating. ("Sure, I might get away with seducing your wife, or stealing your chickens, or just stabbing you when we're alone... but I don't want to bring a curse down on the community, or go to hell.")
    – Jedediah
    Dec 15, 2021 at 22:02
  • Kristian, the mathematics of game theory intersect with ethics only in explaining the origins and function of human morality such as norms like “Obey God”. This science is silent concerning if we somehow imperatively ‘ought’ to “Obey God”. But people could reach very different conclusions if, rather than “It is what God wants”, people generally understood that the idea of gods was only a strategy to increase cooperation by providing supernatural punishment of immorality. Punishment is important because it is a necessary part of all cooperation strategies relevant to human morality.
    – Mark Sloan
    Dec 15, 2021 at 22:04
  • @Jedediah Exactly so.
    – Mark Sloan
    Dec 15, 2021 at 22:07

The best example of applied mathematics is to physics. In fact, most people would find it hard to distinguish physics from mathematics. Wigner asked why this us so and the main reason is the role of necessity in physical law. In ethics, its the other way around. Here, the autonomous will rules rather than necessity and this is why mathematics finds little purchase here as the history of ethical and moral thought shows.

Likewise, the application of mathematics to other human endeavours. For example, say to philosophy. Mario Bunge, a philosipher of science (he called his philosophy, exact phiposophy) considered that the philosophy of philosophy was philosophy which he picturesquely envisaged as the formula:

P^2 = P

Of course it is no formula as it cannot be manipulated in anyway and nor does it fit into a theory of such formulas. Its merely a metaphoric way of designating what he had already said in words. Much like the Pythagoreans designated the monad by 1 and duality by 2 and multiplicity by 3 (we can add to Pythagorean tetrakys by appending zero to the top to denote modern nihilism).

Thus, the intersection of mathematics and ethics is the trivial set, that is banalities.

  • On the contrary, as Rawls points out, arguments in economics and political science can be reasonably deductive, hence his idealization of "moral geometry." And the respondents who have pointed towards evolutionary game theory, here, have done an outstanding job of showcasing a major non-trivial intersection of mathematics and ethics, one that sits well with Rawls' invocation of game theory throughout A Theory of Justice. Dec 18, 2021 at 9:02
  • @Kristian Berry: No, there is no geometry as a mathematician would understand it in Rawls book. And I have read it. What he's doing is marshalling arguments and these include an array of arguing techniques, including deduction. This does not make it logic and nor does it make it geometry or arithmetic. The Greek economist, and former finance minister of Greece, Yannis Varafoukis is on record rubbishimg game theory as a pseudo-science. I imagine he would have even harsher words for the evolutionary variant. In that, it rather stacks up well against economics, the dismal science. Or rather ... Dec 18, 2021 at 9:18
  • @Kristian Berry: ... or rather badly, if you get my drift. Dec 18, 2021 at 9:19
  • I'm not going to defer to a finance minister I've never heard of. The articles on game theory, including evolutionary game theory, in the Stanford Encyclopedia of Philosophy, testify well enough to the viability of the discipline. In fact, I'd be likelier to disavow set theory than game theory, at this point, perhaps, on account of all the random intellectual shenanigans I see modern set theorists get into. Dec 18, 2021 at 9:24
  • @Kristian Berry: Well, he is more well-known than either you or I am. Did you see the headline that the SEP had towards the end of the article: it asks, "the explanatory irrelevance of evolutionary game theory". It looks like they have only taken notice of it to damn it with no praise. Which is exactly the viewpoint of Varafoukis, except he's quite a bit more robust in his rejection. Dec 18, 2021 at 9:30

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