Morality and Mathematics

Posted on 18 August 2018


I have a vivid memory from my first or second year of college when discussing mathematics with an older student. I don’t remember the specific topic, but at one point he pointed out to me that mathematical truths are only true relative to particular axioms and definitions that we have assumed, and that they are not true independently, in and of themselves. Moreover, the axioms are merely assumed to be true, and do not rest on any deeper foundation. This may not have been the first time I considered this, but in this particular instance the insight was especially profound for me. None of our mathematical theorems is true in any context-independent way. And yet, mathematics is the paradigm of certain knowledge. On the one hand, once we clearly specify a particular set of axioms and definitions, there is no ambiguity or doubt or uncertainty about what statements are and are not true relative to that context. (For example, no postmodern critique or skeptic can ever show that there is a largest natural number, or that the Pythagorean theorem is not true in Euclidean geometry.) On the other hand, in mathematics we are free to choose any axioms and definitions we like, and true statements will change accordingly. (Finite fields can have a largest number, and the Pythagorean theorem is false in non-Euclidean geometry.) So there is no absolute, context-independent truth.

For me personally, this insight had implications far beyond mathematics. For example, in situations where I would find my own ideas in conflict with the ideas of someone else, instead of following the impulse to prove myself right and them wrong, or figure out who is “really” right, I would instead seek to understand in what sense each is true in its own context. How do our assumptions or definitions differ? Of course, there is always the possibility that we do share common assumptions and definitions but one of us is simply being inconsistent. But the effect of the insight was to give others the benefit of the doubt, to look at apparently conflicting positions as alternatives that can be true in their own context instead of as opposed to each other in a dogmatic battle for truth. Many years later, I wrote a playful parable about this insight here.

I think it is fair to say that this insight from mathematics has had for me a moral dimension, insofar as it has helped to support personally an attitude of openness and interest in superficially conflicting ideas and opposing viewpoints. This kind of openness is essentially a form of love. The golden rule would have us consider the perspectives of others not as opposed to our own but as another possibility to be understood on its own terms, on equal footing with our own.

The insight has also helped clarify my thinking about certain general issues of morality, such as the problem of moral relativity. On the one hand, the insight implies that moral principles are not absolute, context-independent truths. They are either assumptions, or based upon assumptions. And such moral relativity implies that, ultimately, there is no absolute moral foundation. This can raise the concern that everyone can then have their own personal morality and assert that there is no basis for saying their morality is any less valid than anyone else’s. But this is no more a concern than it is a concern that mathematics allows each person to choose their own definitions and axioms and develop their own mathematical theorems. They are free, as a matter of principle, to do so. In practice, however, if a mathematician wants to be a member of a community, they are obliged to use conventional definitions and focus their research on areas of mathematics that are considered by the community to be relevant. For example, I’m free to make up my own idiosyncratic definitions for common mathematical concepts like “associative” and “commutative”, use non-standard notation instead of “+” to represent addition, or adopt different axioms for well-established mathematical objects like groups, rings, and fields. But I can’t do all of that and expect to have my work considered relevant by others. To be part of a community means to share common conventions, assumptions, terminology, notation, and so on. The same, I would submit, is true for morality. In order to live with others harmoniously, we need to share at least some basic moral principles. They need not be absolute to serve this function. Individuals whose morality deviates in significant ways from the society in which they live will have problems living within that society. An analogy that I find helpful to illustrate this is the US-Mexican border. On the one hand, its existence and location is not an objective truth. It is a relative truth, based upon an agreement between the governments of the US and Mexico. But its status as a relative truth does not make it subject to arbitrary whims of each individual. Quite to the contrary, if individuals ignore the established conventions (i.e., laws and regulations) relating to the border, they will suffer very real consequences. So, the point here is that the relativity of truth does not imply that “anything goes” or cause us to degenerate into anarchy. Mathematics does just fine with the relativity of truth. In fact, it sets a fine example of how to look at relative truth, including our morals in society: we should strive to make our assumptions, definitions, conventions, etc. as clear as possible so as to avoid confusion and conflict. And we should develop ways to arrive at consensus regarding standards that are adopted by each community or society, so that members of that community can work harmoniously together.

(The above is an excerpt from personal correspondence with Moral Math pioneer Sarah Voss, August 2018, written in response to her solicitation of my perspective on the topic. -TJM)