Although I am currently writing a chapter on biographical portrayals of Newton “as a mathematician”, I am, by no stretch of the imagination, an historian of mathematics. The reason is, in large part, because I am not a mathematician. Now, I am also not a physicist, or a geographer, or a chemist or an astronomer, or a scientist (or woman of science) of any other description, having finished my formal scientific education with Higher Chemistry (a Scottish qualification, for those unsure, taken around the age of 16/17), yet I have written about aspects of the history of all of those disciplines.
I get away with it because history of science has long since moved away from focusing entirely on the content of the science and has embraced approaches that consider science in its widest social and cultural contexts. The battles between ‘internalists’ and ‘externalists’ in history of science are (mostly) behind us, and the subject is the richer for being able to take lessons from both spheres, and for including individuals with a range of backgrounds. History of mathematics, though, was one of the last remaining bastions of internalism.
Things are changing but the situation, as yet, remains distinctly different, despite there having been a workshop on the social history of mathematics way back in 1980. There is still a sense that, although the social, pedagogical and practical processes of doing science have been explored in other disciplines, mathematics is somehow different. As knowledge it is viewed as distinct, esoteric and disembodied, and its history has too often been told without situating it fully within the contexts in which it has been produced.
One of the problems is that mathematics as a whole tends to be treated as pure mathematics. Historically-speaking this is a problem as pure mathematics, as now understood, did not exist before the 19th century. Mathematics was, rather, a tool or series of tools that allowed natural philosophers to probe real-world problems further and to analyse data with greater levels of certainty. Historians of mathematics have, nevertheless, often treated their subject as divorced from reality, focusing instead on the sequence of ideas, as if they were simply sitting there waiting to be discovered.
Of course, even pure mathematicians are products of their time, with careers, recognition, training and development of ideas all being reliant on interaction between individuals, groups, societies and audiences. My friend and former PhD colleague, John Heard, is the expert on this, having written his thesis on ‘The evolution of the pure mathematician in England, 1850-1920’ (happily available for free download on Ethos). John delves into the social history of mathematics, finding explanations for the rise of the pure mathematician – a move relating to pedagogy, professional idenity, new ideas about funding for ‘pure’ research and aesthetics. His thesis is much recommended: it’s written clearly, and without the technical language that makes so much history of mathematics inaccessible to other historians.
As an aside, John notes that well-known historian of mathematics Ivor Grattan-Guinness complains in The Fontana History of the Mathematical Sciences that too few authors pay attention to the history of mathematics, and indeed that “the history of mathematics is largely absent from the “culture” of the educated public, historians and mathematicians included. The extent to which it is dismissed, abhorred, even derided, has to be experienced to be believed”. Yet this book is written with so much focus on the technicalities that its target audience is “the modern mathematician or student”. Conversely, by putting mathematics into its social and cultural context, John goes a long way to demonstrating why other historians should pay attention: “In order for the history of mathematics to be taken seriously as history, it needs to show points of contact with wider issues of interest to other historians and the lay public.” (pp. 28-29)
John’s supervisor, Andrew Warwick, has also done his bit for the social history of mathematics with another eminently readable book, Masters of Theory: Cambridge and the Rise of Mathematical Physics. Here he notes that, despite recent historiography, “mathematically formulated theories have continued, largely by default, to be regarded as the province of the history of ideas and therefore to have little or no place in social histories of science concerned with labor and material culture”. His pedagogical history corrects this by showing that, just as scientific practice, the production of mathematical theory requires “embodied skills, the tools of the trade, and specific locations”, not simply the historiographically-limiting notions of “contemplative solitude and cultural transcendence” (p. 12). Andy looks at Cambridge mathematical education, where fierce competition in the Tripos examination led to increasingly abstracted specialisation via written examinations, detailed problems that could be graded against each other, the coaching of students, and hard, solitary, focused study. He shows that the way mathematics was taught, both at Cambridge and its feeder public schools, had a fundamental impact on the style and content of mathematics and mathematical physics.
As well as relating to my Newton-as-a-mathematician essay, these books have seemed relevant to me when reading Stephen Curry’s recent articles (at THE and on Occam’s Typwriter) on the role of mathematics in the the teaching of biological sciences. His concern is that many school students are put off by a mathematics that seems “detached from reality, a jumble of indecipherable symbols written in an alien language that is easy to dismiss as irrelevant”. This seems unnecessary when, as he’s learned, many scientists who disliked maths at school nevertheless found themselves enjoying, or at least using, complex mathematics in their research. When they appreciated the usefulness of a particular process, the effort needed to understand it became worthwhile.
This marks a division in the role of mathematics that would not have been recognised in the 18th century. In the following century, as ‘pure’ mathematics developed to include “mathematical gyrations” that took analysis into the realms of negative and irrational numbers, or worse, there was deep anxiety about its meaning, usefulness and even morality. Teaching and researching pure mathematics was by no means an obvious thing to do, but John and Andy have shown how they were successfully imbedded in the relevant institutions and created our modern notion of ‘the mathematician’. This, in turn, has had its profound influence on the history of mathematics, which has only recently begun to weaken.
Stephen suggests that the source of our “divergent attitudes to words and numbers … can probably be found somewhere at the bottom of the fissure between the sciences and humanities explored by C.P. Snow in his famous ‘two cultures’ lecture”. My rant about Snow will have to wait for another post, but I would here suggest that the question of how mathematics is taught and viewed today can usefully be explored and explained through the discipline’s own history.
I am in no position to offer answers, but perhaps there is something to be said for encouraging schools to present mathematics as tools rather than (or as well as) a distinct set of ideas. Instead of solving artificial or ideal problems in maths textbooks, how much more satisfying would it be for some pupils to learn their mathematics by seeing its power in application to physics, chemistry and biology – and why not also to history, social studies, music, design & technology and more? The problem, of course, would be training teachers to be comfortable with handling such material.
1. Pure mathematics existed as a phrase only to distinguish it from mixed mathematics, which referred to the mathematics used directly in astronomy, optics, surveying, music and navigation. It nevertheless dealt with measurements and quantities, just not in relation to particular, tangible objects.
2. A phrase borrowed from D.P. Miller’s ‘The Revival of the Physical Sciences in Britain, 1815-1840’, Osiris 2 (1986), p. 109.