### The Mathematical Vocation

#### Posted by David Corfield

After a visit in 1939 to the Monastery of the Prophet Elijah on the Greek island of Santorini, the Oxford philosopher R. G. Collingwood entered into discussion with his students about the value of monastic life. It appears that the students were a little perplexed to find their prejudice that monks were “at worst idle, self-indulgent, and corrupt; at best selfishly wrapped up in a wrongheaded endeavour to save their souls by forsaking the world and cultivating a fugitive and cloistered virtue” clash with their admiration for

…the atmosphere of earnest and cheerful devotion to a sacred calling, the dignity of the services and beauty of their music, the eager welcome and the loving hospitality, and above all the graces of character and mind which the life either generated in those who had adopted it or at least demanded of aspirants to it and thus focused, as it were, in the place where the life went on. (‘Monks and Morals’,

Essays in Political Philosophy, Oxford, 1989)

Collingwood then draws the students’ attention to a vocation they do value.

Suppose a man devotes his life to the study of pure mathematics. Is he to be condemned for living on a selfish principle? Not, as my friends readily admitted, on the ground that pure mathematics cannot feed the hungry. Pure mathematics, apart from any consequences which may ultimately come of it, is pursued because it is thought worth pursing for its own sake. In order to judge its social utility, then, you must judge it not by these consequences but as an end in itself.

What is more, you cannot judge the social utility of a mathematician by asking whether he publishes his results. Unless there is value in being a pure mathematician, there is no value in publishing works of pure mathematics; for the only positive result these works could have is to make more people into pure mathematicians; and a society which does not think it a good thing to have one pure mathematician among its members will hardly think it a good thing to have many.

The social justification of pure mathematics as a career in any given society, then, is the fact that the society in question thinks pure mathematics worth studying: decides that the work of studying pure mathematics is one of the things which it wants to go on, and delegates this function, as somehow necessary for its own intellectual welfare, to a man or group of men who will undertake it. A test for this opinion is that the society in question should be grateful to the pure mathematician for doing his job, and proud of him for being so clever as to be able to do it; not that every one else should rush in to share his life, but that even if his neighbours feel no call to share it they should honour him for living as he does. The fact that they do so honour him is a proof that they want a life of that kind to be lived among them, and feel its achievements as a benefit to themselves. (pp. 145-146)

## Re: The Mathematical Vocation

I think the debate would have seemed much clearer in 1939 than it does now. In particular, the quote “Pure mathematics, apart form any consequences which may ultimately come of it, is pursued because it is thought worth pursing for its own sake.” presupposes that for things that were called pure mathematics as they were developed (ie, where there is a strong consensus, not borderline issues) to ever have “practical” consequences is rare.

In a modern view, modern physics, computer science and mathematical modelling of all sorts, tends to bring in

someresults from various branches of mathematics that are called pure mathematics (eg, number theory, construction of computable reals, etc). So the argument that publishing pure mathematical research cannot be any factor in deciding the “utility” of a mathematician becomes very difficult to agree with.Of course the question still exists in a different from: how does one define social utility for working on something that is very unlikely to consequences that have practical consequences, and vanishingly likely to have direct practical consequences?