### The Sacred and the Profane

#### Posted by David Corfield

The categories mailing list recently engaged in some soul-searching as to why category theory remains such a well-kept secret and why its wonders are not better known. Many people proposed theories to explain this, while others were quick to deny the premise, pointing out that it is no longer a secret, as the mainstream work of Voevodsky, Lurie, and many others bears witness. I wonder, though, if this was quite the point at stake.

When I met up with Minhyong Kim before Christmas he made the very interesting remark that there’s a difference between treating some system of mathematics in a ‘sacred’ way and in a ‘profane’ way. This distinction was introduced by Dirk van Dalen in the Preface to his *Logic and Structure*:

Logic appears in a ‘sacred’ and in a ‘profane’ form; the sacred form is dominant in proof theory, the profane form in model theory. The phenomenon is not unfamiliar, one observes this dichotomy also in other areas, e.g. set theory and recursion theory. Some early catastrophes such as the discovery of the set theoretical paradoxes or the definability paradoxes make us treat a subject for some time with the utmost awe and diffidence. Sooner or later, however, people start to treat the matter in a more free and easy way. Being raised in the ‘sacred’ tradition my first encounter with the profane tradition was something like a culture shock. Hartley Rogers introduced me to a more relaxed world of logic by his example of teaching recursion theory to mathematicians as if it were just an ordinary course in, say, linear algebra or algebraic topology. In the course of time I have come to accept this viewpoint as the didactically sound one: before going into esoteric niceties one should develop a certain feeling for the subject and obtain a reasonable amount of plain working knowledge. For this reason this introductory text sets out in the profane vein and tends towards the sacred only at the end. (p. V)

When you see that $X$ can be used to achieve $Y$, you may react in two ways. You may say that it reflects to the greater glory of $X$ that $Y$ is made possible, or you may just be grateful to $X$ for helping to achieve the desired $Y$. If I show you how my new knife cuts through some wood, I may do so to allow you to admire the sharpness of the blade, rather than because I want two pieces of wood. You may respond by asking whether the knife can do what you take to be useful. We will then feel we haven’t quite understood one another. Perhaps then something similar is going on in the difference between a sacred attitude towards (higher) category theory and a profane one aiming at other ends.

Of course, you would like to ask your profane user about the choice of those other ends. Why are you so keen to see whether my machinery can help you? At some point in our questioning we should expect to be able to proceed no further, and to hear the reply, “That is simply what I want to know.” Gian-Carlo Rota called this the ‘bottom line’.

How do mathematicians get to know each other? Professional psychologists do not seem to have studied this question; I will try out an amateur theory. When two mathematicians meet and feel out each other’s knowledge of mathematics, what they are really doing is finding out what each other’s bottom line is. It may be interesting to give a precise definition of a bottom line; in the absence of a definition, we will give some examples.

To the algebraic geometers of the sixties, the bottom line was the proof of the Weil conjectures. To generations of German algebraists, from Dirichlet to Hecke and Emil Artin, the bottom line was the theory of algebraic numbers. To the Princeton topologists of the fifties, sixties, an seventies, the bottom line was homotopy. To the functional analysts of Yale and Chicago, the bottom line was the spectrum. To combinatorialists, the bottom lines are the Yang-Baxter equation, the representation theory of algebraic groups, and the Schensted algorithm. To some algebraists and combinatorialists of the next ten or so years, the bottom line may be elimination theory.

I will shamelessly tell you what my bottom line is. It is placing balls into boxes, or, as Florence Nightingale David put it with exquisite tact in her book

Combinatorial Chance, it is the theory of distribution and occupancy. (Indiscrete Thoughts, 51-52)

We could say then the mathematicians hold their bottom line to be sacred, and as such display a sense of *duty* in their actions towards it.

On the subject of duty, the philosopher R. G. Collingwood had this to say about choice of action,

…of the three reasons for choice [because it is useful; because it is right; and, because it is my duty], the third alone is a complete reason. Choice is always choice to do an individual action. Why do I do it? The answer ‘because it is useful’ explains only why I do an action leading to a certain end: not why, among the various possible actions which might have led to that end, I choose this and not another. The answer ‘because it is right’ explains only why I do an action of a certain kind, specified by the rule which I obey; not why, among the various possible actions conforming to that specification, I choose this and not another. But the answer ‘because it is my duty’ is a complete answer. What I do is an individual action; what it is my duty to do is an individual action; if what I do is my duty these two individual actions are one and the same. I do this and no other action because this and no other is the action it is my duty to do. (‘Duty’ in

Essays in Political Philosophy, p. 151)

His idea is that there’s nothing further to be said, as when you reply to your beloved’s question “Why do you love me?”, with “Because I do.” In the case of duty, we might say “Here I stand. I can do no other.”

More radical than a shift of mathematical bottom line, would be a move out of mathematics altogether, as has cropped up regarding John’s career changing decision. One effect of watching someone wrestle with their duty is to question whether you are following your own.

## Re: The Sacred and the Profane

The van Dalen quote is fascinating, because I “grew up” seeing proof theory as the “profane”, practical end of logic, and model theory and categorical logic as the “sacred” form. Categorical logic is what explains wild and mysterious things like Lawvere-Tierney topologies on categories of sheaves, whereas proof theory is what we use to design languages to keep web browsers from crashing. :)

What’s interesting is that how naturally my profane viewpoint arises naturally out of the philosophical ideas of Martin-Lof, which are explicitly sacred in motivation (albeit a different one than the categorical logicians). He explains his judgmental methodology as an attempt to reject the Fregean view of logic as being about truth only, and to return to the older Aristotelian vew of logic as being about all of reasoning, broadly construed. (He wants to do this while retaining all the technical advances of the Fregean revolution, of course.)

But when you start designing logical judgements to talk about how computer programs move around networks, this can’t help but demystify the machinery of logic, even though it might be “sacredly” justified as an example of bringing life and reason into closer harmony.

(As an aside, Martin-Lof’s judgmental methodology is something that has proven resistant to explanation via the standard tricks of categorical proof theory. I know Peter Lumsdaine has talked to Bob Harper about this, but I don’t know if they ever found a good story.)