A Reply to Thony Christie

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I'm grateful to Thony Christie for devoting a post on his deservedly popular blog to my recent post on Scientific American's guest blog. Thony claims not to be interested in the argument of my piece, which is a shame (I'll get to that later), but raises some objections to my claims about pre-eighteenth-century mathematics. I'll take a little space here to respond to Thony's historical objections, and I want to use my response to try to convince him (and his readers) to care about my piece's argument as well.

Thony starts by catching me in a bit of a cliche, about ancient astronomers marking seasons. It's a figurative usage, but Thony is right to say it could be misleading about just what these elites did with their mathematics. I encourage those interested in this history to read Eleanor Robson's groundbreaking Mathematics in Ancient Iraq, which explains the long historical development and diversity of what has often been lumped together as "Babylonian mathematics." By comparing these astronomers to a modern civil service, Thony helps make my point: as described later in the piece, mathematical expertise from the late eighteenth-century onward became a crucial gateway into many privileged areas of modern civil service, with France a leading example.

Thony's next objection is fairly subtle. He claims that uses of mathematics in early societies tended to benefit the public, and I agree. That does not make them a "public good": something available to everyone, that all can make use of, not subject to rivalry or proprietary control. Merchants and tradesmen surely shared mathematical techniques amongst each other, but (as I originally claimed) they did so as tricks-of-the-trade, as protected means to get ahead and conduct business that were not expected to be shared by all. I think the same applies to the rhetoric and distribution of Fibonacci's Liber Abbaci that Thony brings up later. Thony also claims advanced math was not a part of European history for most of the period during which I claim it was a philosophical pastime of the privileged. This cuts, for instance, the rich history of Islamic court mathematics out of the European history in which it emphatically belongs. I had recent social histories of both Greek and Islamic mathematics most in mind when writing that passage.

The core of Thony's criticism is his objection to my use of Robert Recorde and John Dee to illustrate two different views of the place of mathematics. Thony is absolutely right to observe that both men had rich and fascinating lives and works. I've written and spoken about both in more academic venues, which doesn't make me an orthodox authority but also doesn't make me the dilettante Thony suggests I am. I'm willing to concede Thony's claim that it was a bit of a stretch to illustrate Recorde's approach to less practical mathematics with a quote from the Ground of Artes, which was billed as a practical book but went beyond conventional practical arithmetics in a few respects. Then again, Recorde billed all of his books as practical ones, even those like the Pathway to Knowledg (on Euclidean geometry) that were eminently impractical. That was why I identified him as a forerunner of elite mathematicians today claiming that mathematics was really everywhere.

I do, as Thony observes, neglect Dee's many efforts to expound mathematics at a variety of levels in England. (If you don't like Dee as your English standard bearer for keeping mathematics close to one's chest, try Thomas Harriot.) Both Dee and Recorde used their mathematical expertise to rise to privileged positions of authority, but while Recorde mostly insisted math was perfectly perspicuous, Dee tended to focus instead on math's power. It was a pitch aimed at certain kinds of patrons: political and economic elites who could help Dee advance his own interests in all aspects of mathematics. Just as Recorde is a forerunner of modern mathematicians claiming that mathematics is really everywhere, Dee is a forerunner of mathematicians (often the same ones) courting influence through the power of mathematics. To me, it is a suggestive comparison to set up the latter part of the argument that Thony ignores completely, not the final word on the careers of Recorde and Dee.

What Thony finds most shocking is my claim about the scientific revolution. Following Steven Shapin and many who have written since his classic 1988 article on Boyle's relationship to mathematics, I chose to emphasize the conflicts between the experimental program associated with the scientific revolution and competing views on the role of mathematics in natural philosophy. Thony pretends that naming some figures remembered today both for mathematics and for their contributions to the scientific revolution contradicts this well-established historical claim. But to take just his most famous example, Newton's prestige in the Royal Society is generally seen today to have had at least as much to do with his Opticks and his other non-mathematical pursuits as with his calculus, which contemporaries almost uniformly found impenetrable.

To be sure, Shapin's claim (which he was not alone in arguing) initially represented for some a shocking revision to the historiography of the scientific revolution. One of the things that made it most shocking is that the very developments from later periods that I mentioned in my piece also shaped historical accounts of the scientific revolution, placing an outsized emphasis on an often anachronistic picture of the mathematization of nature at the expense of other kinds of changes to the ways a range of actors sought to understand the world. Why did later scholars emphasize math so much in this history? Because from mid-eighteenth century onward, math was an indisputable key to their intellectual and social authority. They took their present circumstances and read them backward, placing math at the center of their forebears' worlds, in ways that further reinforced their own claims to mathematically-grounded privilege.

That is why I don't think Thony can so simply ignore the main argument of the piece in disputing some of the evidence behind it. How math is seen today affects not just our policies for the future of math but also our understanding of the mathematics and mathematicians of the past. It is important to know not just the career trajectories and publication histories of those like Recorde and Dee, but also how interpretations of figures like these have changed in different times and places. What aspects of primary sources we take at face value and what aspects we challenge or contextualize has a lot to do with what we think is important about scientists, institutions, and knowledge in the present. But, even more to the point, however you interpret Recorde's and Dee's rhetoric and career trajectories it remains true that mathematics (especially of the kind that was not considered "mixed mathematics") in Early Modern England was the privileged domain of a small elite. Moreover, the later social history of mathematics helps explain why some truisms about the past may not be as solid as one might believe.

I applaud Thony for his commitment to getting the history right, especially in pieces that can reach a relatively wider audience than a specialist conference or seminar. I share that commitment unequivocally. For me, part of getting the history right--the most important part--is looking for its implications and significance as something more than an ordered collection of facts and observations. My goal with the Scientific American post was to use history to help its readers see the current place of mathematics in the public differently. You have to understand that goal in order to evaluate the emphases I placed on different aspects of history--emphases which Thony found strange and off-putting, but which I hope this brief rebuttal has shown to be sound and defensible. The shape of the mathematical profession presents challenges to historians, policymakers, mathematicians, mathematics educators, and the public alike. I aim to help us all see that those challenges are more interconnected than they may first seem.

Update

On his blog, Thony has added a reply to my reply. My own reply to his reply to my reply was short enough that I left it to the comments section of his blog, but I reproduce it here for your reference:

Hi Thony,
Thanks for your re-reply! I think my re-response is brief enough to deposit here rather than a separate page.
You’ve made my point about abbacus schools for me: my entire argument turns on the importance of distinguishing access *in principle* from access *in practice*.
We’ll just have to disagree about the shape of Europe, but I hope you’ll acknowledge that, semantics aside, "European history" is much richer and more complete if one includes the parts of the Islamic world that were in close contact with those lands with the present political classification of "Europe". To rule out Baghdad, e.g., because it doesn’t line up the right way on a map, strikes me as narrow and anachronistic, especially since historical conceptions of the boundaries of Europe have always been in flux.
Perhaps I wasn’t being clear about Harriot: the man you describe is the one I had in mind. Surely you don’t think his mathematics was *everywhere*?
But we really *really* seem to disagree about the scientific revolution. That's ok; we wouldn't be the first pair to have fundamental disagreements about it! I do indeed rely a great deal on authorities who have studied the period in greater detail than I have, and as those go Shapin is no slouch. This is not the place, however, to rehearse his CV, nor to give a full bibliography of the many other authorities who have informed my own understanding of the period.
Sure, Newton was admired and respected for his Principia, but who actually read and understood it? According to the historiography I know, not many people. You know this, of course, and you also know how controversial many claims about mathematics were in and beyond the Royal Society.
You can also rest assured that I have no quarrel with your claim that mathematics was important in the 17th century. I know enough about the period to know that you’ve necessarily left out an enormous pile of evidence about the uses and roles of mathematics. But look again at the claim you imply this refutes, that "the new promoters of an experimental science that was (at least in principle) open to any observer were suspicious of mathematical arguments as inaccessible, tending to shut down diverse perspectives with a false sense of certainty." The fact that not everyone was dubious of mathematics does not contradict my own assertion that a good many promoters of the new experimental science were indeed dubious. In fact, your own post backs me up on the latter point. So we agree after all!
In that case, I'd encourage you to revisit the rest of my Scientific American essay, which describes how those (including the intellectual heirs of several you note here) who took a different approach to mathematics than the Boyleans and Baconians came to use mathematics to secure their positions as elites. Their legacy continues to shape mathematics' place in society today, and it's important to recognize how, as well as what that implies for society's engagement with this powerful field of knowledge.
– Michael

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