This is a guide for students writing a substantial course essay or bachelors thesis in the history of mathematics.
The essence of a good essay is that it shows independent and critical thought. You do not want to write yet another account of some topic that has already been covered many times before. Your goal should not be to write an encyclopaedia-style article that strings together various facts that one can find in standard sources. Your goal should not be to simply retell in your own words a story that has already been told many times before in various books. Such essays do not demonstrate thought, and therefore it is impossible to earn a good grade this way.
So you want to look for ways of framing your essay that give you opportunity for thought. The following is a basic taxonomy of some typical ways in which this can be done.
Critique. A good rule of thumb is: if you want a good grade you should, in your essay, disagree with and argue against at least one statement in the secondary literature. This is probably easier than you might think; errors and inaccuracies are very common, especially in general and popular books on the history of mathematics. When doing research for your essay, it is a good idea to focus on a small question and try to find out what many different secondary sources say about it. Once you have understood the topic well, you will most likely find that some of the weaker secondary sources are very superficial and quite possibly downright wrong. You want to make note of such shortcomings in the literature and cite and explain what is wrong about them in your essay, and why their errors are significant in terms of a proper understanding of the matter.
The point, of course, is not that finding errors in other people’s work is an end in itself. The point, rather, is that if you want to get anywhere in history it is essential to read all texts with a critical eye. It is therefore a good exercise to train yourself to look for errors in the literature, not because collecting errors is interesting in itself but because if you believe everything you read you will never get anywhere in this world, especially as far as history is concerned.
Maybe what you really wanted to do was simply to learn some nice things about the topic and write them up in your essay as a way of organising what you learned when reading about it. That is a fine goal, and certainly history is largely about satisfying our curiosities in this way. However, when it comes to grading it is difficult to tell whether you have truly thought something through and understood it, or whether you are simply paraphrasing someone else who has done so. Therefore such essays cannot generally earn a very good grade. But if you do this kind of work it will not be difficult for you to use the understanding you develop to find flaws in the secondary literature, and this will give a much more concrete demonstration of your understanding. So while developing your understanding was the true goal, critiquing other works will often be the best way to make your understanding evident to the person grading your essay.
For many examples of how one might write a critique, see my book reviews categorised as “critical.”
Debate. A simple way of putting yourself in a critical mindset is to engage with an existing debate in the secondary literature. There are many instances where historians disagree and offer competing interpretations, often in quite heated debates. Picking such a topic will steer you away from the temptation to simply accumulate information and facts. Instead you will be forced to critically weigh the evidence and the arguments on both sides. Probably you will find yourself on one side or the other, and it will hopefully come quite naturally to you to contribute your own argument for your favoured side and your own replies to the arguments of the opposing side.
Some sample “debate” topics are: Did Euclid know “algebra”? Did Copernicus secretly borrow from Islamic predecessors? “Myths” in the historiography of Egyptian mathematics? Was Galileo a product of his social context? How did Leibniz view the foundations of infinitesimals?
Compare & contrast. The compare & contrast essay is a less confrontational sibling of the debate essay. It too deals with divergent interpretations in the secondary literature, but instead of trying to “pick the winner” it celebrates the diversity of approaches. By thoughtfully comparing different points of view, it raises new questions and illuminates new angles that were not evident when each standpoint was considered in isolation. In this way, it brings out more clearly the strengths and weaknesses, and the assumptions and implications, of each point of view.
When you are writing a compare & contrast essay you are wearing two (or more) “hats.” One moment you empathise with one viewpoint, the next moment with the other. You play out a dialog in your mind: How would one side reply to the arguments and evidence that are key from the other point of view, and vice versa? What can the two learn from each other? In what ways, if any, are they irreconcilable? Can their differences be accounted for in terms of the authors’ motivations and goals, their social context, or some other way?
Following the compare & contrast model is a relatively straightforward recipe for generating reflections of your own. It is almost always applicable: all you need is two alternate accounts of the same historical development. It could be for instance two different mathematical interpretations, two perspectives emphasising different contexts, or two biographies of the same person.
The compare & contrast approach is therefore a great choice if you want to spend most of your research time reading and learning fairly broadly about a particular topic. Unlike the critique or debate approaches, which requires you to survey the literature for weak spots and zero in for pinpoint attacks, it allows you to take in and engage with the latest and best works of scholarship in a big-picture way. The potential danger, on the other hand, is that it may come dangerously close to merely survey or summarise the works of others. They way to avoid this danger is to always emphasise the dialog between the different points of view, rather than the views themselves. Nevertheless, if you are very ambitious you may want to complement a compare & contrast essay with elements of critique or debate.
Verify or disprove. People often appeal to history to justify certain conclusions. They give arguments of the form: “History works like this, so therefore [important conclusions].” Often such accounts allude briefly to specific historical examples without discussing them in any detail. Do the historical facts of the matter bear out the author’s point, or did he distort and misrepresent history to serve his own ends? Such a question is a good starting point for an essay. It leads you to focus your essay on a specific question and to structure your essay as an analytical argument. It also affords you ample opportunity for independent thought without unreasonable demands on originality: your own contribution lies not in new discoveries but in comparing established scholarly works from a new point of view. Thus it is similar to a compare & contrast essay, with the two works being compared being on the one hand the theoretical work making general claims about history, and on the other hand detailed studies of the historical episodes in question.
Sample topics of this type are: Are there revolutions in mathematics in the sense of Kuhn? Or does mathematics work according to the model of Kitcher? Or that of Lakatos or Crowe? Does the historical development of mathematical concepts mirror the stages of the learning process of students learning the subject today, in the manner suggested by Sfard or Sierpinska? Was Kant’s account of the nature of geometrical knowledge discredited by the discovery of non-Euclidean geometry?
Cross-section. Another way of combining existing scholarship in such a way as to afford scope for independent thought is to ask “cross-sectional” questions, such as comparing different approaches to a particular mathematical idea in different cultures or different time periods. Again, a compare & contrast type of analysis gives you the opportunity to show that you have engaged with the material at a deeper and more reflective level than merely recounting existing scholarship.
Dig. There are still many sources and issues in the history of mathematics that have yet to be investigated thoroughly by anyone. In such cases you can make valuable and original contributions without any of the above bells and whistles by simply being the first to really study something in depth. It is of course splendid if you can do this, but there are a number of downsides: (1) you will be studying something small and obscure, while the above approaches allow you to tackle any big and fascinating question you are interested in; (2) it often requires foreign language skills; (3) finding a suitable topic is hard, since you must locate an obscure work and master all the related secondary literature so that you can make a case that it has been insufficiently studied.
In practice you may need someone to do (3) for you. I have some suggestions which go with the themes of 17th-century mathematics covered in my history of mathematics book. It would be interesting to study for instance 18th-century calculus textbooks (see e.g. the bibliography in this paper) in light of these issues, especially the conflict between geometric and analytic approaches. If you know Latin there are many more neglected works, such as the first book on integral calculus, Gabriele Manfredi’s De constructione aequationum differentialium primi gradus (1707), or Henry Savile’s Praelectiones tresdecim in principium Elementorum Euclidis, 1621, or many other works listed in a bibliography by Schüling.
Expose. A variant of the dig essay is to look into certain mathematical details and write a clear exposition of them. Since historical mathematics is often hard to read, being able to explain its essence in a clear and insightful way is often an accomplishment in itself that shows considerable independent thought. This shares some of the drawbacks of the dig essay, except it is much easier to find a topic, even an important one. History is full of important mathematics in need of clear exposition. But the reason for this points to another drawback of this essay type: it’s hard. You need to know your mathematics very well to pull this off, but the rewards are great if you do.
Whichever of the above approaches you take you want to make it very clear and explicit in your essay what parts of it reflect your own thinking and how your discussion goes beyond existing literature. If this is not completely clear from the essay itself, consider adding a note to the grader detailing these things. If you do not make it clear when something is your own contribution the grader will have to assume that it is not, which will not be good for your grade.
Here’s another way of looking at it. This table is a schematic overview of different ways in which your essay can add something to the literature:
The table shows the state of the literature before and after your research project has been carried out.
A Describe project starts from a chaos of isolated bits of information and analyses it so as to impose order and organisation on it. You are like an explorer going into unknown jungles. You find exotic, unknown things. You record the riches of this strange new world and start organise it into a systematic taxonomy.
You need an exotic “jungle” for this project to work. In the history of mathematics, this could mean obscure works or sources that have virtually never been studied, or mathematical arguments that have never been elucidated or explained in accessible form.
An Explain project is suitable when others have done the exploration and descriptions of fact, but left why-questions unanswered. First Darwin and other naturalists went to all the corners of the world and gathered and recorded all the exotic species they could find. That was the Describe phase. Darwin then used that mass of information to formulate and test his hypothesis of the origin of species. That was the Explain phase.
Many areas of the history of mathematics have been thoroughly Described but never Explained.
What if you find that someone has done the Explain already? If you think the Explain is incomplete, you can Critique it. If you think the Explain is great you can Extend it: do the same thing but to a different but similar body of data. That way you get to work with the stimulating work that appealed to you, but you also add something of your own.
Likewise if you find two or more Explains that are all above Critique in your opinion. Then you can do a Compare & Contrast, or a Synthesise. This way you get to work with the interesting works but also show your independent contribution by drawing out aspects and connections that were not prominent in the originals.
See also History of mathematics literature guide.