If you orient yourself temporally you may remember that back in August there was a online fracas involving mathematics. A teenage girl, doing her makeup before work, decided to take the opportunity to lay down for her TikTok followers her skepticism about the idea of math generally:
Who came up with this concept? “Pythagoras!” But how? How did he come up with this? He was living in the … well I don’t know when he was living, but it was not now, where you can have technology and stuff, you know?Grace Cunningham, TikTok user.
As was keenly observed by the many keen observers out there, the initial response was a pile-on that combined general misogyny with gen-Z hatred; it was the latest installment in the long running complaint about kids these days. This reactionary abuse was soon countered by a more positive wave of responses that acknowledged that her questions were not only legitimate, but exactly the kind of questions our curriculum does little to answer.
I don’t think many mathematicians are particularly satisfied about the way our subject is generally taught. At the university level I find it hard to love force marching students through rote material, stripping centuries worth of mathematics of all its scientific and historical context along the way. So obviously I am happy to see any student kicking back at what we inflict on them. But for those who have made mathematical communication their vocation it was a solid gold opportunity to evangelize. Euginia Cheng wrote a pdf answering Grace Cunningham’s formalized list of questions, and Francis Su wrote a twitter thread.
Grace Cunningham was calling the bluff on the pretenses of her education. In particular, the pretense that you should obviously be learning whatever we are telling you. “Why are we even doing this?” is a legitimate question in a mathematics course, and “why on earth did anyone prove these theorems in the first place” is an even better one. “How did people know that they were right,” presents the awkward truth that people most often are certainly incorrect about many things. What makes these questions awkward is that the people teaching you mathematics will frequently know little to nothing about the history and context within which the theory was developed. Mathematicians are terrible, as a rule, at scholarship, and the history of ideas within mathematics is an essentially distinct field. Most of the context that I have for the mathematics I do is essentially gossip, urban myth, and pablum. Fortunately, while we might be terrible historians we remain excellent gossips, so at least we have plenty of stories to tell.
(I should also concede that it is impossible to generalize in any way about most of my peers. Many of them are tremendously knowledgeable about all kinds of things and wonderful educators. I am, at least to some extent, either projecting or talking about our very worst failings.)
I was dissatisfied by the responses I found to Cunningham’s questions. Not least of all because I don’t think they really answer the questions. No actual historical context was given. The answers more resemble the kind of general motivation and propaganda we give students to encourage them to listen in class and do their homework. I think a good answer would address the fact that the people who developed much of classroom mathematics had some pretty wild ideas about what they were doing. Their motivations would be pretty alien to us, and is a far cry from their homework, exams, or getting a well paid job.
Just to make this explicit: How many of us who have ever taught or taken calculus a calculus course have even done any astronomy? Just from doing a little reading, an obvious observation seems to be that when people sat down to first learn calculus from Newton’s Principia, the big incentive for them was the promise of a serious set of answers about the Sun, the Earth, the Moon, the stars, and even comets. A modern mathematician explaining their motivation for calculus today is a little like a 21st century Western evangelical Christian explaining what the “Old Testament” is all about to an orthodox rabbi.
My modest reading has focused on the life of Isaac Newton. I read Jame’s Gleick’s biography of Newton (highly recommended) and I have a few more on the shelf. I already had some understanding that aside from developing calculus Newton was a heretic, alchemist, and later in life warden of the royal mint. I knew he lived through times of plague, apocalypse, dictatorship, conspiracies, and his work was a major part of the scientific revolution. Particularly pertinent to Cunningham’s question is the fact that for centuries after Newton’s death there was a suppression of the full range of Newton’s intellectual activities. It was only when John Maynard Keynes acquired a substantial portion of Newton’s surviving papers at auction that the truth came out. For a long time Newton’s preoccupations would be considered intellectually inconvenient for all those trying to boost his posthumous reputation, and that of British science with it.
The idea of knowledge as cumulative — as a ladder, or a tower of stones, rising higher and higher — existed only as one possibility among many. For several hundred years, scholars of scholarship had considered that they might be like dwarfs seeing further on the shoulders of giants, but they tended to believe more in rediscovery than progress. Even now, when for the first time Western mathematics surpassed what had been known in Greece, many philosophers presumed they were merely uncovering ancient secrets, found in sunnier times and the lost or hidden.Isaac Newton – James Gleick (pg 34-35)
Here is a not entirely fanciful reading of Newton’s life: starting his university career dissatisfied with the existing knowledge, and curious about the latest developments in astronomy, Newton develops his theory of calculus. But he is not yet really a scientist. He is still very much a wizard. A young man who has uncovered some profound secrets and is keen to discover more. He invests huge amounts of time and energy in alchemy and theology. The alchemy involved tracking down obscure texts that he hoped would contain the secret knowledge of transforming base metals into precious metals, and his notebooks from this period often amount to his copying out these texts. It also involved working with mercury, a poisonous metal known to drive the alchemists who used it to madness.
His theological interests were no less hazardous since they would have been viewed as clearly heretical to both the Protestant and Catholic religious authorities at the time. By studying the earliest Greek manuscripts he discovered that the concept of the Trinity — that the Godhead is three and one; Father, Son, and Holy Spirit — emerged late in the early church, and certainly couldn’t be considered part of the original Christian tradition. Newton concluded Jesus was not at the same level as God and had never claimed to be. At a time in England when having Catholic sympathies could land you in trouble, this was a dangerous view to have.
I would argue that Newton transformed from a wizard into a scientist the moment the German mathematician Leibniz independently derived his own theory of calculus. No longer had Newton uncovered a forgotten knowledge, but he had derived a theory that someone else could also derive. He was now entered into a race to establish the precedence for his own results — and this meant writing up.
For decades his tools of calculus had languished in notebooks and in his mind. Now he had to write them down, and he chose to present them in the style of Euclid’s Elements, with axioms, definitions, lemmas, theorems. And most intriguingly, in order to prove the correctness of his theory, he drew upon experimental data: astronomical observations from the newly establish Greenwich observatory and tidal charts. He was able to explain and predict natural phenomena that perplexed his contemporaries such as the sudden appearance of comets, and their unusual paths across the night sky. We can recognize this now as a prototype of the modern scientific method, but back then it was controversial, becoming part Newton’s dispute with Leibniz.
Newton wrote many private drafts about Leibniz, often the same ruthless polemic again and again, varying only by a few words. The priority dispute spilled over into the philosophical disputes, the Europeans sharpening their accusations that his theories resorted to miracles and occult qualities. What reasoning, what causes, should be permitted? In defending his claim to first invention of the calculus, Newton stated his rules for belief, proposing a framework by which his science — any science — out to be judged. Leibniz observed different rules. In arguing against the miraculous, the German argued theologically. By pure reason, for example, he argued from the perfection of God and the excellence of his workmanship to the impossibility of the vacuum and of atoms. He accused Newton — and this stung — of implying an imperfect God.
Newton had tied knowledge to experiments. Where experiments could not reach, he had left mysteries explicitly unsolved. This was only proper, yet the German threw it back in his face: ‘as if it were a Crime to content himself with Certainties and let Uncertainties alone.’Isaac Newton – James Gleick (pg 176-177)
Data is now the recognized currency of modern science, and theology is, well, theology. The mathematical analysis that makes calculus rigorous didn’t come until much later. Newton had started using infinite series in his calculus, but it was understood that you had to be careful because sometimes you could get some bad results.
When Cunningham asks her TikTok followers how early mathematicians knew they were right, in Newton’s case at least, it seems that there are three answers. Newton first convinced himself with arguments we would not consider mathematically rigorous along with his his own empirical observations. Decades later he convinced his peers by publishing a full written account of his theory (in Latin) that provided supporting data. Then a century or so later the full theory of mathematical analysis was developed.
These questions have complicated answers for Newton, but they are really no less complicated for us today, even if they are quite different answers. We live in the age of the arxiv, computer assisted proofs, machine learning, and bodies of work that amount to many hundreds of pages. I’m not going to lie; I love the drama of it all. Some would like to present mathematical proof and progress as being an enterprise free from being sullied with the humanity of its practitioners. For my part I am of the belief that the reasons people commit themselves to mathematics are more complicated than just the aesthetic appreciation of equations.