Mathematicians' histories and historians' histories

Returning to Leo Corry's thoughts on the history of mathematics, I think it is important to recognise that a corrective against simplistic, triumphalist story-telling was very necessary. Historians of science were many years ahead of their counterparts in mathematics, and have undertaken their own corrective work with zeal. A good example of this is one mentioned by Corry. How many times have you read about how the 1919 expeditions to measure the bending of light by the sun confirmed Einstein's theory of general relativity? It all sounds so simple: Einstein predicts a phenomenon which runs against Newtonian theory; scientists observe the phenomenon; and, anyone with a shred of rationality gives up on Newton's theory. However, historical research into this episode paints a very different picture, or perhaps I should say it paints very different pictures. The best known of these questions the way the data was selected (see John Waller (2003). Einstein's Luck: The Truth behind Some of the Greatest Scientific Discoveries. Oxford, England: Oxford University Press). However, in Matthew Stanley's “An Expedition to Heal the Wounds of War”: The 1919 Eclipse and Eddington as Quaker Adventurer, Isis, 94 (2003), 57-89, we read that:

The 1919 eclipse expedition's confirmation of general relativity is often celebrated as a triumph of scientific internationalism. However, British scientific opinion during World War I leaned toward the permanent severance of intellectual ties with Germany. That the expedition came to be remembered as a progressive moment of internationalism was largely the result of the efforts of A. S. Eddington. A devout Quaker, Eddington imported into the scientific community the strategies being used by his coreligionists in the national dialogue: humanize the enemy through personal contact and dramatic projects that highlight the value of peace and cooperation. The essay also addresses the common misconception that Eddington's sympathy for Einstein led him intentionally to misinterpret the expedition's results. The evidence gives no reason to think that Eddington or his coworkers were anything but rigorous. Eddington's pacifism is reflected not in manipulated data but in the meaning of the expedition and the way it entered the collective memory as a celebration of international cooperation in the wake of war.

Either way the old-style history was thoroughly misleading. For Corry there are extremely important lessons to learn from the new historians' history:

A complex mixture of social, institutional, political and cultural circumstances (all of them fully legitimate and human) stand at the background of this interesting chapter in the history of twentieth-century science. They must all be taken in consideration, together with the purely scientific issues involved here, if we want to make full sense of the impressively quick and sweeping process of acceptance of Einstein's new theory on the basis of the astronomical observations of 1919.

Being this the case, one may for a moment conjecture about possible scenarios that might have ensued, had the results of the expedition not been as readily accepted as they were (under the active influence and authority of Eddington and Dyson, and for the many reasons that guided their efforts in this direction) or if the results had showed a preference for Newton's theory over Einstein's. These are by no means imaginary scenarios and they could have easily materialized had the circumstances been different. It is important to remember, in this context, that once the measurements reported by the expedition (and by implication, the confirmation of Einstein's theory with the concomitant refutation of Newton's one) were published in British, (and later in German) newspapers, Einstein was immediately catapulted into world fame. He thus turned into a cultural icon that embodied for decades to come the ideal of the scientist as a secular saint working in isolation from the rest of the world. The events of 1919 played an important role in shaping much of the course of physical science in the twentieth century as well as of its public perception.

Corry describes the difference between the old and new forms of history of science in literary terms:

Science as Drama/Greek Tragedy
a) We know what will happen: drama arises because we know that it will happen
b) Human emotions, ideas, and behavior as products of, or responses to the unfolding of the human essence
c) Universal elements of the human situation and fate

Science as Epic Theater (Brecht)
a) "Things can happen this way, but they can also happen in a quite different way" (Walter Benjamin)
b) Human emotions, ideas, and behavior as products of, or responses to, specific social situations
c) Behavior people adopted in specific historical situations

Plurality, difference, contingency are what mark modern history of science. Things could by now be so very different. But have historians gone too far? Is it not to accord too small a role to the world to claim, for instance, that "The events of 1919 played an important role in shaping much of the course of physical science in the twentieth century as well as of its public perception."?

Now when we turn to mathematics, you might think the contingentist historian would have an easier job. Even if the world as mediated through physics is taken to play an important role in determining the mathematics we study, and even if we take there to be a single correct logic for mathematical reasoning, there's still plenty of scope to believe that mathematics could have gone in very different ways. But this brings us back to the issue of bumping into mathematical reality. Sometimes it feels as though whichever way a mathematician tries to move they can't help but knock into something. The issue for me is how to write history of mathematics which is historically sophisticated, and yet alive to this experience. A good test case would be to write the history of n-categories. Baez and Lauda have begun one here which treats the physicists' input into higher-dimensional algebra. I should imagine that historians would find this a little too much of a 'Royal Road to Me' (see Corry again). But let's turn the onus around. Are they able to treat the decades long development of a body of ideas, which involves scores of mathematicians from many institutions and countries? Or would this have to represent too great a concession to old-style history in what to them would be an arbitrary delineation of a portion of mathematical activity?