Why and how to write history

My head's been filled the last few days with just how large a part of machine learning can be couched in terms of convex optimization - Fenchel duality, and all that - so it will be a relief just to spend a few minutes thinking about the history of mathematics. I have to do this as I 'm going to be speaking about it at a workshop in Berlin this September. I've promised to speak about:

Why and how to write a history of higher-dimensional algebra

In a recent paper 'How Mathematicians May Fail to be Fully Rational' [1], I advocated the adoption in the philosophy of mathematics of Alasdair MacIntyre's general notion of tradition-constituted enquiry. A central component of this notion requires of a rational tradition that it know the history of its successes and failures. This raises the question as to whether, were such a history to be written, it would fall foul of the criticism contemporary historians of mathematics have levelled at mathematicians' histories that they are largely 'Royal-road-to-me' accounts. I shall address this question in the context of a research programme known as 'higher-dimensional algebra', and consider the charge mathematicians may make in return that historians are unable to treat research programmes which run for decades, supported by tens or hundreds of mathematicians from many countries and institutions.

A few weeks ago I read a book by David Carr called Time, Narrative, and History 1986 Indiana University Press Bloomington, which I thoroughly recommend. The central thesis of the first part of the book can be summed up in a couple of quotations:

To be an agent or subject of experience is to make the constant attempt to surmount time in exactly the way the story-teller does. It is the attempt to dominate the flow of events by gathering them together in the forward-backward grasp of the narrative act. Mink and the other theorists are right to believe that narration constitutes something, creates meaning rather than just reflecting or imitating something that exists independently of it. But narration, intertwined as it is with action, does this in the course of life itself, not merely after the fact, at the hands of authors, in the pages of books. (61-62)

So, there's no pristine unnarrated experience which later undergoes narrative reformulation. Now, for the interpersonal dimension:

Thus concretely, when we recount to others what we are living through and what we are doing, such recounting, rather than the adventitious communication of an already prepared and clearly formulated message, is actually constitutive of the content of what is said, and through it constitutive of the temporal organization itself. Most people have had the experience that they do not quite know what they mean or intend until they try to communicate it to others. The content is all the more affected, of course, when the speaker is met by rejoinders, questions, and criticisms. Thus telling the story of my action or experience to others can reorganize it for me; telling the story of my life can serve to make a sense of it I have not been aware of before.

The social connection among persons, conceived in this way, is one of reciprocal communicative roles in the constitution of experiences, actions, and lives. Others are encountered by me, not only as audiences or sounding-boards for the sense-constitution of my own ongoing experiences and projects, but as engaged in their own narratives as agents and stroy-tellers, narratives to whose construction I may contribute in my role as audience and possibly critic. This at least one way to conceive of the social horizon of the individual's existence.

Now the concept of historicity, as put forward by Husserl and Heidegger, adds a crucial element to this picture. It affirms that my connection with the actions and experiences of others can take a special form, apart from the relation of reciprocal narration, a form we can describe as the relation of predecessors and successors. What is indicated is a priority, not only of time but also of accomplishemnt. In the case of the ongoing scientific project I take up the work already accomplished by others. The end of another's work becomes the beginning of my own. The other need not have finished his or her work, of course, but some conclusion has been reached which serves as my starting point. This is true whether the other's results can be used as a basis for my own, and built upon, or whether I must begin by undoing his or her work and starting over. In either case the work of others, rather than simply existing alongside my own, becomes its background and prior condition. (112-113)

And this leads us to the issue of a community's narrative. In a previous post, I suggested that a communal group of mathematicians can write a narrative of their research programme which will fall foul of historians' strictures, and yet which is nonetheless truthful. In fact, I rather think that to remain strictly at the level of historians' history would be to overlook some part of the truth. Let's continue with Carr:

...the narrative structure and narrational activity within communal existence is, as we have insisted, primarily practical in character; historical narrative, by contrast is cognitive and seeks an objective representation. The former is engaged in action and has an interest in its outcome; the latter is detached and disinterested, and aims only at truth. The second difference concerns the temporal standpoints of the narrators in each case. Our "practical" narrator is situated in medias res, whereas the historical narrator looks back at actions and events already completed. That gives the latter the well-known (and already discussed) advantage of hindsight over his subjects: he knows how things turned out, knows the difference between the intended consequences and the real consequences of their action, etc.

These differences between narrative agent or participant and narrative historian are operative and important: there is no denying the importance of temporal standpoint and of the difference in attitude (engaged or detached) in relation to a lived or performed sequence of human events. At the same time we should like to emphasize several respects in which these differences are mitigated. And we shall do this not by denying objectivity and hindsight to historical inquiry, but by attributing them to narrative-historical existence.

We have already pointed out ..., with respect to individual action and experience, that the narrativization that goes on there cannot be indifferent to truth where the past is concerned. Indeed, where the issue is not merely the shaping of an open future but the coherence of future, present, and past, it is important to be clear on what really happened; the past may be variously interpreted but it cannot be wished away or forcibly altered by an inventive narrative magination. So much of one's present capacities are in continuity with, and sometimes result from, past choices and experiences that getting straight one's past can be seem as a desideratum and even a necessary condition for a coherent life. This is, of course, one of the insights on which much psychotherapy is based, as we pointed out.

A concern for the truth of the past plays the same role in the case of the community. Members often debate the facts of the past, precisely because they are so important in the constitution of the present and the future. This is not to deny that the past is often manipulated, especially where social story-telling is political and persuasive in character. The personal past is often distorted too, deliberately or not. My point is merely that a genuine interest in the truth of the past is compatible with and indeed important for the practical narrative constitution of communal existence. Equally, objectively-oriented historical enquiry and research are not disqualified from playing a role in the ongoing political and social debnated of a community; on the contrary, they can and do contribute to them.

We are not commenting here, it should be noted, on the success with which truthfulness about the past is actually attained. Our point concerns the interest in or commitment to truth, and we are only saying that this is not restricted to history as a discipline. It is true that the discipline has among other things developed techniques for discovering and evaluating evidence in order to implement its commitment to truth. A justified suspicion that partisanship in the events of the day can distort our view of the past has led to the emphasis on detachment and objectivity. But these in turn, once achieved, can be put in the service of engagement in the present and the shaping of the future.

As for the hindsight which is characteristic of historical enquiry, this too is not exclusive to the latter, at least not formally. Socially constitutive narrative, like the narrative structure of individual life and action, has a prospective-retrospective form. In anticiapating the future, it aims at, and largely achieves, that quasi-hindsight that we characterized earlier, borrowing Schultz's term, as the future perfect. Far from waiting passively for things to happen, communities negotiate with the future and understand the present in the light of that future. 171-2

Carr's position would require historians and mathematicians to be brought into a much closer relationship, and rightly so:

Far from dealing with past events which are fixed and whose consequences are clear, historians here deal with events whose consequences are still being felt and are operative in the present. 173