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.[1] 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”[2] 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.

Notes

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.

“that took analysis into the realms of negative and numbers” – is there an adjective missing somewhere?

Oops, not sure how that slipped out – irrational, I meant. Thanks! Will correct.

Fascinating article and very pertinent at the moment as the way we look at the utility of maths changes.

I have a question that has always puzzled me. Why was the word “pure” used to describe a field of maths? Do other languages use a similar word? In what sense is it “pure”? Is it untainted with applications?

At the moment I think there is a real tension between mathematicians (i.e. people who identifiy primarily with maths) and scientists who use maths who would not consider themselves mathematicians but use it a lot in their everyday life. Teaching maths to those who don’t necessarily like it that much is very much a cinderella vocation without the status of the hard “Pure” maths. I’m thinking of students who come to university to study biology and typically didn’t see the point of maths at school even if they did ok at it) Teaching these students is often described as “remedial” when in fact it’s not: it’s about explaining the maths in context rather than as a thing in its own right.

John Heard’s thesis is very useful at looking into the evolution and various meanings of the word ‘pure’ in the context of mathematics, which has always been contested. It relates closely to the developing argument that ‘pure’ research in science – untainted by interests, identified applications, the requirement of being profitable – should be regularly funded. It could be painted as a high-minded activity, although the promise, of course, was that at some point it would be applied to the real world and be beneficial to mankind.

It’s worth noting, though, that it’s the term ‘applied’ (in mathematics or science) which is actually the more recent. In mathematics the distinction was between abstract or pure and mixed, in other sciences no such division existed before about the 1830s. I believe that it was borrowed from both Kant and later French usage, but I’m afraid I’m not so familar with the terms used elsewhere. The rise of ‘applied’ reflects the 19th-century focus on utilitarian science, and the argument that this stuff was essential for national and public benefit. Pure, and later fundamental, research reflected the need to argue that some funded research should be what we would now call ‘blue-skies’.

Great post, as usual, Rebekah. Quite insightful and I will certainly download John Heard’s thesis.

I have not much knowledge neither of the history of mathematics nor of its evolultion, but somehow I have the feelig that, in the particular case of math, the boundary question of if pure math can be considered as much a science as other more traditionally recognized scientific disciplines, instead of, say, a kind of language or logical game not bearing any necessary relationship to the real worl, or empirical facts, or whatever one may call it, may have played a role in the process of recognising it together with other branches in any history of science movement, in general. This seems a bit paradoxical, though, when one (as was the case) brings the approach towards a more “externalist” side – which should actually suit it best (if, say, one considers pure math as siimilar to a language, for example).

Yes – lots of boundary work going on. John points to the drawing of analogies with art, or even exploration of unknown lands. The idea of mathematics as a language, though common, seems to be contentious among pure mathematicians.

Cool! And thanks for pointing me towards UK e-bank of theses. Found already at least two interesting ones for side-commentaries to my own (still in progress – yeah, I am quite slow, and specially now with a fractured arm – luckily right the left one. The right one was left in good condition (two puns intended)).

Mathematics is not ‘a’ language it’s a family of languages.

Do other languages use a similar word?

In Portuguese it is called “Matemática Pura” (as opposed to applied math), pretty much in a literal translation, as far as I understand it.

Chemistry also distinguishes between “pure” and “applied” chemistry. The IUPAC (International Union of Pure and Applied Chemistry) comes to mind. Pure chemistry treats the structure of molecules and chemical reactions as an intellectual exercise, whereas applied chemistry is using chemistry to make useful things for everyday life. The similarity to mathematics is pretty strong.

In German, the distinction is also between “rein” and “angewandte” Chemie, which translates to pure and applied. German also uses “technische” Chemie, which can sometimes imply chemical engineering more than “angewandte.”

When chemists formally made the division, so far as I know, is an unanswered question among historians of chemistry, although chemistry has always had pure and applied components.

Thanks for this, Peter. I’m pretty sure that pure and applied came into chemistry at the same sort of time as other sciences, i.e, early 19th century. Chemistry also, though, has the more familiar divisions of organic and inorganic, analytical and synthetic.

It strikes me too that, as well as the arguments about how science should be funded and how much it should be directed, there is also the discussion about what science, on the one hand, gives mankind (beneficial applications) and, on the other, gives the individual (a mental training). Here, in pedagogy, we find some of the strongest arguments for the role of ‘pure’ science or mathematics – for example from William Whewell.

History of mathematics is indeed another area that is kind of off in its own world, but they are doing some good work on the Continent in particular, which integrates mathematical history into other history rather well. I’ve enjoyed the work of Henk Bos, and Tinne Kjeldsen’s stuff has been very useful to me, and this crew in Paris seems to be doing some good work as well. We had Catherine Goldstein from the Paris group come give a talk at Imperial a little while back, and supposedly David Aubin is dropping by some time, and I’m very much looking forward to that.

However, on the larger historiographical point, I think this whole “internalists” vs “externalists” battle is a bit of professional mythologizing about what we’ve moved beyond so we can feel more comfortable with where we are now. I’m sure there were some people who held to hard positions, but based on my own readings of older history, I don’t think it actually describes historiographical history very well. I think there may have only been a few people who really took hard-line positions on this.

I’m particularly uncomfortable when this gets into talk about “the last remaining bastions of internalism”. We can

saythat it’s OK, because we all very sensibly agree that internalism/externalism divides are nonsensical, and so getting rid of internalism is about integration, not elimination. But we have to take into account the state of the fine art of historiography maintenance. You imply that while you aren’t particularly into intellectual history, that’s OK because there are others out there who are, and so, really, it’s all a fine, happy situation. And you can — and should — cite Andy Warwick’s fine integration of intellectual and cultural work to show that this is so. But if you actually go and talk to Andy about it, it doesn’t take much prodding to get him to tell you (and I don’t think it’s a big secret) how pessimistic he is about the state of the intellectual history of science, which he’ll say is what he got into the field to do.So, really, if the so-called resolution to internalism vs. externalism is just an agreement that the divide is

philosophicallysilly, while thehistoriographicalreality of the situation is the decimation of the intellectual history of science, that’s a big issue. If cultural and intellectual history can only in principle be seamlessly integrated, but the reality is that the former erodes the latter, my inclination is not to press for integration, but to recognize internalist/intellectual history of science as a legitimate genre, encourage it to do its thing, ask about and celebrate its accomplishments, recognize the bridges that exist, and build new ones where possible. Ultimately, intellectual history of science is very difficult, and there’s not a lot of reward for it, which makes it fragile. We shouldn’t take it for granted that it can survive assimilation.Thanks for the recommendations for reading. You’re right, of couse, that there’s some excellent work out there, and that much of it’s done on the Continent.

Perhaps ‘last bastion of internalism’ was not the best description for the differences I’ve found with history of mathematics. Possibly what I really meant was that it is one of the areas that has – in this country – least successfully integrated with professional history of science and that a significant amount of work done is produced by retired mathematicians rather than historians of mathematics.

I was a bit flippant about the whole internalism/externalism thing, mainly assuming a wider readership (I wish!) of the post. I agree that the issues have not gone away, and that there’s a question of how the more technical work can find its audiences and maintain its professional position. (I think technical rather than intellectual might be the more appropriate word here, as I feel that I deal with plenty of ideas in my work, not all scientific.) I much admire those who can do this work well, and acknowledge my own inadequacy and very much appreciate those who are able to make use of insights from both ‘sides’ (and explain them to me!).

Becky, I think you’re right here about drawing a distinction between technical and intellectual history, since a lot of histories of practice do end up being histories of ideas or ideals that are supposed to be implicit to those practices.

The Paris group, I think, is going to be the place to look for our cues on a successful integration of mathematics into other histories. In fact, Goldstein’s presentation was about the oscillation of the French mathematical community between commitments to pure and applied work surrounding mathematicians’ participation in World War I (particularly applied ballistics). Of course, on close inspection, there are a lot of gradations that can be parsed out of the history here (clearly applied work, the theory underlying the applied work, abstract mathematics, “foundations”, etc.)

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A student on an undergraduate History of Mathematics course with the Open University once commented to me that she felt a lot of her fellow students from the more arty side struggled sometimes because they hadn’t the appreciation of why certain developments were exciting or important whereas she, with a good mathematical education and a feel for the subject, understood better how it fitted together into a narrative. The non-mathematical students had to take the course authors’ interpretations on trust.

Extrapolating from one story, I wonder if History of Mathematics is more generally shunned because it’s difficult for a potential researcher to understand and evaluate without good mathematical skills?

Weirdly, I just took a mental break from trying to redo a section of my book on the Traveling Salesman Problem (TSP) to cruise through my bookmarks. Much of the literature on linear programming (LP) emphasizes its practical utility, and thus its military funding. So, I am actually trying right now to explain why problems in LP like the TSP had an intellectual appeal to high-level mathematicians, who could often switch easily between practical problems and theoretical issues suggested by those problems. I find the absence of such explanations is common in both the cultural and the technical literature. The latter usually just assumes that a particular advance is of innate interest (here is how Tool Q, which we still use, was invented), rather than trying to explain it as part of research interest, style of work, or class of problems. For the former, “this work was funded by the military”, or something like that, usually passes for a sufficient discussion of context.

I think there are similar problems in trying to explain why someone was a “good physicist” or a “good mathematician” — often the literature just points to a discovery or a list of discoveries, or simply just testimonies to the fact that they were “good” without ever really trying to articulate the qualities of how they chose and solved problems.

I absolutely agree about this missing element, although it is not just the approach or knowledge of the historian that accounts for it, but also the available evidence. Unless your subject choses to reflect on what they’re doing, it can be very difficult to recapture motivations and processes that may have seemed obvious at the time.

Having that kind of material certainly helps, but it’s also often possible to put the big prize-winning work in the context of “lesser” papers, and draw a lot out that way. Also, I think it can be possible to actually

dothe work and get a sense of what made it interesting, particularly if you’ve also followed along with the work of the prior history. You can’t be sure this is what your actors felt, but it can lend insight.I’ll probably do a post before too long at Ether Wave Propaganda about historians who gain appreciation through a tactile encounter with the problem. Principe and Newman mention the importance of their lab work in their Focus articles on chymistry; also see Martin Rudwick’s fascinating “Places and Specimens” section of his bibliography in

Bursting the Limits of Timebook (I think he might say more inWorlds Before Adambut I don’t have that in front of me right now) where he talks about how useful it can be to actually visit the geological features his actors talked about. Of course, museum collections are important here too.I remember once doing a mathematical recreation I ran up against in one of my sources (if you have twelve identical coins, except one is not the same weight; find the odd coin using a balance scale in only three trials). Not being an especially good mathematician, it took me a couple of hours, but I always felt it was worth it for the insight into how the problem of economizing the act of calculation weighed on my actors, even on their lunch breaks.

Sometimes, it’s a simply matter of putting emphasis on things the actors or technical historians say in an off-handed way. So, let’s say they say “the TSP differed from the transportation problem in that it imposed a certain kind of constraint”, it’s helpful for the historian to point out that problems can be fundamentally altered by changing the constraints in ways that present a novel intellectual challenge. For the actors that sort of thing can go without saying.

Thanks for your comment, Sam. I think what you say is true, but that it largely reflects the fact that history of mathematics is often taught and researched differently from other parts of the history of science. ‘Internalist’ or technical history of natural philosophy and physics, optics, chemistry and so on is just as difficult for those without a scientific background as mathematics. The point is that both history of science and mathematics *can* be taught in ways that help all students undertand why developments were exciting, important or interesting, within the context of the period being studied. The more technical stuff in all fields may remain challenging, and comes back to Will’s point above about the challenge of nurturing technical, intellectual history.

However, I do think there is at least one bonus for those who lack a scientific background, which is an ability to look at past science with less baggage in terms of current knowledge. When the student you refer to spoke of getting a grip on a narrative that her arty peers missed, I suspect that means that she could trace a line towards what she was taught today. That kind of view can lead to linearity and hindsight in the historical story that misses the sense of what people *at the time* thought they were doing, as well as losing sight of areas of research that were once considered important but proved to be dead ends.

Thanks for an interesting article – it’s reminded me that I should spend some time reading up on “proper” history of mathematics! (Like many maths undergraduates my introduction to mathematical history was through E. T. Bell, a 1930s Californian maths professor who notoriously presented it as a triumphant progress towards the creation of 1930s Californian maths professors.) However, as a maths teacher I’d advise caution when trying to draw conclusions about maths pedagogy from the philosophy or history of maths. In particular, presenting mathematics through its applications to other disciplines is one of those ideas that we all feel ought to work much better than it does in practice. I think there are two problems: the extra load imposed on the students by the contextualisation, and the loss of mathematical coherence.

The trouble with setting a contextualised maths problem is that the student has to grasp the scientific or other context, the model (often implicit) which relates this to a mathematical problem,

andthe mathematical method required. Depending on how much guidance they’re given, it’s very easy for students to become completely overwhelmed or to formulate the model in a way that leaves them unable to solve it with the mathematical tools to which they have access. In practice, to steer students towards the techniques they can employ successfully, the problems are often made rather artificial and the maths ritualistic. There’s a memorable example of overload in chapter 4 of David Copperfield:“Even when the lessons are done, the worst is yet to happen, in the shape of an appalling sum. This is invented for me, and delivered to me orally by Mr. Murdstone, and begins, ‘If I go into a cheesemonger’s shop, and buy five thousand double-Gloucester cheeses at fourpence-halfpenny each, present payment’… I pore over these cheeses without any result or enlightenment until dinner-time, when, having made a Mulatto of myself by getting the dirt of the slate into the pores of my skin, I have a slice of bread to help me out with the cheeses, and am considered in disgrace for the rest of the evening.”

(George Orwell points out that those double-Gloucester cheeses are the characteristically over-the-top Dickensian touch, and I can’t help feeling that they’re also the distractingly vivid detail that makes the sum so impenetrable for young David.)

The other problem is that the sequence in which maths is required by most subjects (physics, as the most mathematicised science, might be an exception) doesn’t correspond well with the natural sequence in which these ideas are developed mathematically – at least once one has got beyond the level of elementary arithmetic. (I recommend trying to design the syllabus of a first-year Maths for Engineers module as a way of discovering how inflexible is the order in which material can be introduced.) Either one has then to teach the maths in advance of the applications, or to teach the applied methods before the students are in a position to understand how these methods work. The first option returns us to where we started, with “artificial or ideal problems”, while the second degenerates into teaching maths as a voodoo ritual which elicits the answers through a supernatural agency known as “the method for…”. There’s also the issue of the practice required to internalise mathematical ideas before they can successfully be applied. Reading some advocates of contextualised maths teaching I can’t help imagining football players whinging that they’ll never have to use gym equipment in a real match so it’s not fair making them work out during their training sessions…

None of this is to argue, of course, that teachers can’t or shouldn’t learn from the intellectual history of the discipline – it’s just that I think we have to distinguish between the way in which maths developed historically (partly through applications), the way in which maths can be developed logically (mostly post hoc) and the way in which maths can productively be taught – often a very complicated compromise between the other two!

Many thanks for your comment. I will absolutely defer to you and others on the issue of mathematics education, I was just interested at how many of the comments on Stephen Curry’s original blogpost raised these issues and reflected some of the arguments that appeared at the beginning of the 19th century. Their view of mathematics as a discipline also linked nicely to what John Heard has written, and to the differences between history of science and history of mathematics. These are all useful for helping us think about the image of mathematics generally speaking.

With my comment about teaching I did not, in fact, mean that mathematics classes ought to go all focusing on applications, but rather that some maths ought to be brought into other lessons at secondary level. It ought to be possible to use some statistics in history lessons, for example, or weights, measures, conversion and percentages in Home Economics etc, as well as not shying away from mathematical content in other science and technology lessons. Using it in these lessons, with data you’ve collected or objects you’re handling, would feel much more appropriate than working your way through problems set in textbooks.

This, on the one hand, might develop a bit more basic mathematical literacy among the non-scientific (seeing it as an aid to daily life: a useful tool rather than scary abstract concept), and, on the other, help budding scientists and mathematicians to see the role of mathematics in a wide range of areas. I would not suggest ending ‘pure’ mathematics in maths class – I enjoyed it at school much more once we got on to proper bits of algebra and away from problems based on Sally, Ann and Michael having so many sweets to share out (or cheese in the cheesemongers)…

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