Thursday, October 18, 2012

Enchantress of Numbers

It was recently Ada Lovelace Day. Ada Lovelace, who was the daughter of Lord Byron, is most famous for her collaboration with Charles Babbage and Luigi Menabrea, which in a sense makes her the author of the first English textbook on computers -- she translated Menebrea's account of Babbage's Analytical Engine and added in cooperation extensive notes of her own, including what are clearly things like computer programs. She thus sometimes gets the label 'the first computer programmer', which, like almost all such designations, is a bit questionable. However, there has recently arisen a sort of reaction to this which goes to an equally extreme length in the other direction, claiming that she was a mathematical incompetent. The arguments for this are somewhat varied. For instance, on one argument, Menebrea's text had a typo in which, instead of 'le cas' it had 'le cos', and the Countess Lovelace simply carried over the 'cos' despite the fact that it didn't make any mathematical sense in context. This is not a very strong argument, since the work seems to have been reviewed by Babbage, whom nobody thinks was a mathematical incompetent, and he didn't catch the error, either. And we get statements like this recent one from Julian Sanchez:

Dorothy Stein, the first of Lovelace’s biographers with sufficient training to seriously assess Ada’s frequent proclamations of her own extraordinary mathematical genius, concludes that Lovelace was scarcely the prodigy she imagined herself to be, and struggled to grasp concepts that would be standard fare in a modern high school course in AP calculus.

To which one is inclined to say, Well, obviously; a modern high school course in AP calculus presupposes lots and lots of ideas that were only just being developed by and diffused among the mathematical geniuses of Lovelace's day. Let's look at the dates of two extraordinarily important people in the development of the modern calculus and compare them with Ada Lovelace's dates:

Lovelace (1815-1852)
Cauchy (1789-1857)
Weierstrass (1815-1897)

Notice that there's just a bit of an overlap there; and it should be noted that Weierstrass only became known relatively late: the works that the mathematical community first started taking serious notice of were published after Lovelace's death. But it is Weierstrass who is usually credited with putting the calculus (including the limit concept, that bane of the high school AP student) on a rigorous basis. In Lovelace's day, people were still using a lot of guesswork; even well after her day you could still find competent British mathematicians (who, while often managing to do their own good work, were also often playing catch-up in using methods that had developed on the continent) trying to solve problems by methods that would be considered shockingly informal and wrongheaded today.

But behind Sanchez's hyperbole is Stein's argument that Lovelace, in her correspondence course with De Morgan, occasionally has difficulties with problems that would be fairly easy today. A serious examination of the matter, however, requires not looking at what we can do today, when notation has stabilized and there are lots of people are given extensive teaching in a formal setting, but at what could be done in Lovelace's day, in which notation for the calculus had not yet stabilized, many British mathematicians of undoubted competence were still uncomfortable with the continental ways of doing analysis, and someone like Lovelace was for all practical purposes studying the subject on her own with occasional help from people like DeMorgan (they met every few weeks and exchanged correspondence here and there between). We don't actually know why Lovelace had the difficulties she did; it could be that she was just not comfortable with the notation yet, or that she was making some silly mistake which she later corrected, or that it just hadn't clicked yet, or that De Morgan had explained something badly at some point, or that she wasn't really all that great at mathematics, or any number of other things. Even getting simple things wrong is not a sign of incompetence, especially when starting out; there isn't a mathematician in creation that would survive to remain in the competence box if the criterion were never making a mistake that someone else could easily avoid -- and this, it should be pointed out, is especially if the someone else lives a century and a half later. It is of a piece with people who claim that Lovelace's 'programs', as they are often called, are merely 'student exercises' without also pointing out that what we have of Babbage's 'programs' are often not any more sophisticated. I don't think the kind of study that is required for this sort of assessment has actually been done, and it's possible we simply lack the evidence to do so. It's entirely plausible to think that Babbage, who called her "the Enchantress of Numbers" was just excited to find someone so enthusiastic about his Analytical Engine and so willing to flatter him over it; and it is true that Lovelace liked to talk of her own intellectual abilities in overinflated rhetoric; but Babbage was not the only mathematician of note to comment on her high level of mathematical ability.

The debate, in short, is not yet played out, and it is premature to be dogmatic about the outcome.

But, of course, this is in some sense moot: people don't recognize Ada Lovelace day in order to celebrate the Countess of Lovelace; they recognize Ada Lovelace day in order to celebrate women in mathematics and science. Lovelace is the occasion.

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