The history of mathematics used to belong to mathematicians. Until about half a century ago, almost all research in history of mathematics was done by people with a thorough background in mathematics itself. Even first-rate mathematicians such as Bartel van der Waerden and André Weil wrote extensively on history with penetrating insight. But this has changed in step with increasing research specialisation in modern academia. Nowadays almost all research in the history of mathematics is conducted in various humanities departments. These new historians have brought a shift in the focus and methodology of the field, often explicitly distancing themselves from the mathematically-oriented approach of previous generations.
I maintain that it is crucial to keep the mathematician’s point of view alive in historical scholarship. Technical acumen and an empathic sense of how a mathematician thinks are essential tools for understanding past mathematics. And this is best done by one who lives and breathes mathematics on a daily basis in the classroom, at the colloquium, and by the coffee machine — in short, from within a department of mathematics.
Fields-medalist David Mumford put it well when relating his “personal experience reading Archimedes for the first time”: “after getting past his specific words and the idiosyncrasies of the mathematical culture he worked in, I felt an amazing certainty that I could follow his thought process. I knew how my mathematical contemporaries reasoned and his whole way of doing math fit hand-in-glove with my own experience. I was reconstructing a rich picture of Archimedes based on my prior. Here he was working out a Riemann sum for an integral, here he was making the irritating estimates needed to establish convergence. I am aware that historians would say I am not reading him for what he says but am distorting his words using my modern understanding of math. I cannot disprove this but I disagree. I take math to be a fixed set of problems and results, independent of culture just as metallurgy is a fixed set of facts that can be used to analyze ancient swords. When, in the same situation, I read in his manuscript things that people would write today (adjusting for notation), I feel justified in believing I can hear him ‘speak’.”
This way of doing history has widespread resonance in the global mathematical community. All mathematicians know the feeling of struggling to understand a mathematical work until it “clicks” and one feels certain that one has experienced the same idea as the author, regardless of whether he be centuries or millennia removed from us. Those of us who approach mathematical texts in this way know not to pay too much attention to superficial aspects of the presentation: scribbles of various kinds are merely imperfect representations of the author’s thought, whereas the digested “aha” insights we reach when we understand it are its true content. Mumford is right that modern historians, by contrast, are trained to categorically reject such a “gut feeling” approach and stick slavishly to the exact written word as if it were a veritable alien communiqué for which no concordance with our own ways of thinking may be assumed.
The new historiography has greatly advanced the field by offering more specialised perspectives than mathematicians alone ever could, such as histories deeply informed by broader social context and meticulous work on sources and editions according to the highest standards of textual critical apparatus. But amidst the zeal to exploit these new frontiers the field has been left with a leadership vacuum in its traditional core dominion. The time is ripe for a resurgence of the mathematician’s perspective, whose cross-fertilisation with modern developments will bring great fruits.
Here in Utrecht we keep alive this mathematical tradition of historical scholarship. We have a legacy of generations of quality history of mathematics being done in a Mathematical Institute that commands the highest international respect. We are widely recognised as the natural heirs of this way of doing history. It is not for nothing that Jan Hogendijk was awarded the European Mathematical Society’s inaugural Otto Neugebauer Prize, epitomising our continuity with the Göttingen mathematical tradition, while Henk Bos was awarded the Kenneth O. May Prize, the highest honour of the International Commission for the History of Mathematics.
I have taken up the role of torch-bearer of this movement in both words and deeds. I offered a big-picture vision for its enduring relevance and importance in a programmatic paper on the historiography of mathematics, and my more specialised works instantiate these ideals. A notable example is my paper reviving and defending the geometrical algebra interpretation of the history of Greek geometry: an issue where the battle lines have traditionally run along departmental divisions, and the older interpretations being advanced by mathematicians like van der Waerden and Weil have been singled out for criticism as emblematic of the dangers of the mathematically-oriented approach to history. With no mathematicians forthcoming anymore to challenge them, the humanistic historians who dominate the field today had been lulled into a consensus, to the detriment of the vitality of our field. The same dynamic is at play in many other cases as well, which is why the mathematician’s point of view has much to offer, not only in terms of subject-matter insights, but also for stimulating diverse and critical thought in the field.
My first publication, on the isoperimetric problem, symbolises how my point of origin is mathematics itself. Although it is a work of history in that it gives an exhaustive survey of historical solutions of the problem, it is clearly driven by a mathematician’s delight at beautiful proofs drawn from wide-ranging fields of mathematics, such as complex analysis and integral geometry.
My conception of the history of mathematics as being first and foremost about the development of mathematical ideas gives my work educational and expository appeal for a broad mathematical audience. My work on the history of the calculus, for instance, is fertile soil in this regard, and I have made the most of this in my free calculus textbook and several papers making classroom-relevant aspects of my research accessible to a wide readership.
This grounding in the mathematician’s point of view serves me well in my more historical work, where an intuitive sense of what makes sense mathematically often leads me to different interpretations than those who are guided more by contextual considerations external to the mathematical argument itself. I have taken on historians and philosophers along such lines for instance in my Copernicus paper and my paper on Leibniz’s early work on the foundations of the calculus, to name but two prominent examples.
Thinking like a mathematician also gives me a perspective on broader issues missed by historians and philosophers who keep technical mathematics at arm’s length. For example, in my dissertation I studied Leibniz and argued that from the corpus of his technical mathematical works there emerges a clear and unmistakeable picture of his conception of the purpose and method of geometry. This has wide-ranging implications for understanding the scientific and philosophical thought of that era generally, yet this perspective has been neglected since it is expressed “only” implicitly in the mathematical works. But as Albert Einstein said: “If you wish to learn from the theoretical physicist anything about the methods he uses, I would give you the following advice: Don’t listen to his words, examine his achievements.”
Mathematicians conversant with the history of science are also much needed to analyse technical issues. My paper on Copernicus is a case in point. In this article I refute an argument due to Swerdlow that has been considered crucial for over forty years and has been widely cited as decisive by historians who had not themselves worked through its technical mathematical basis.
On the other hand, mathematical understanding is not all it takes to do history of mathematics. Mathematicians who turn to history without background and training in this field often make grave errors of their own. I expose and refute many such errors of anachronism in my dissertation and elsewhere. One example is my paper on what is often called Leibniz’s proof of the fundamental theorem of calculus but which is actually nothing of the sort. The notion that this is Leibniz’s proof of this theorem is widely repeated in numerous sources. It is a notion that seems very plausible to anachronistic eyes looking only at a short piece of Leibniz in isolation, but in reality it is simply false, as becomes clear when the work is studied in its proper context. Another example is my paper rehabilitating Jakob Steiner’s geometry from anachronistic misjudgements. To clear up these kinds of things the field desperately needs proper professional expertise in both history and mathematics.
A key theme emerging from my dissertation, which I intend to build on in future work, is the influence of classical mathematics on general scientific and philosophical thought in the early modern period, which was much more comprehensive than recognised today. It was a widespread conviction at the time that if you seek truth, you must do what the geometers did; you must replicate their method and extend it to other branches of learning and philosophy. Descartes’s Discours de la méthode (1637) is explicitly written for this very purpose; indeed this famous manifesto on the method of doubting everything clearly proclaims that “I did not doubt” that “only mathematicians” had struck upon the right way of reasoning. Likewise Hobbes writes in his Leviathan (1651) that “geometry is the only science that it hath pleased God hitherto to bestow on mankind,” and proceeds to expressly fashion his philosophy in its image. Spinoza’s Ethica (1677) declares in its very title that its is “ordine geometrico demonstrata.” Newton opens his Principia (1687) with a preface outlining what “the glory of Geometry” consists in, in order to use its example to justify his innovative scientific methodology.
But what exactly did these authors mean when they spoke of “the geometrical method”? The complexities of this question are poorly understood by scholars and historians today. The 17th-century thinkers who invoked geometry were not referring to some superficial idea of geometry as conveyed by Plato or Aristotle. They were referring to the rich picture of the geometrical method that emerges from a thorough study of technical corpus of Greek geometry, as conveyed by advanced technical writers such as Pappus. Indeed they frequently refer to this technical tradition even in works that go well beyond geometry itself: Descartes cites Pappus in his Discours; Hobbes does the same in his Elements of Philosophy; Newton cites Pappus even in the very first sentence of his Principia. These authors were thoroughly versed in the technical Greek tradition, as their mathematical works show. By citing Pappus and other technical Greek material they are signalling very clearly that when they call upon geometry they mean not some simplistic idea of axiomatic-deductive method but a rich methodology only conveyed implicitly in the finest ancient works of advanced geometry.
Unfortunately modern scholars do not share these 17th-century thinkers’ excellent technical mastery of advanced classical geometry. Consequently, current scholarship has failed to appreciate the extent to which conceptions of the geometrical method permeates 17th-century thought. One indication of this is that the crucial Book 4 of Pappus’s Collection was translated into English for the first time only in 2010. Even more deplorably, Leibniz’s published mathematical works have never been translated into English at all, even though their crucial importance is universally acknowledged. Meanwhile, any philosophical treatise of even a fraction of the importance of these works has invariably been translated multiple times, betraying the skewed and anti-mathematical emphasis of modern scholarship.
Much of the richness and impact of the mathematical perspective has therefore been missed by modern scholars since it is not spelled out in philosophical prose, neither in Greek nor in early modern times. But mathematics speak loud and clear to anyone who cares to listen, and anyone who was serious about philosophy in those eras was obviously expected to know their geometry — much in the spirit of the famous inscription above the entrance to Plato’s academy. To understand 17th-century thinkers it is time for us to start taking their appeals to geometry seriously and recognise the full scope of the rich methodological picture they drew from advanced Greek geometry.