Tag: Newton

The wrangler’s insecurity.

[This is my third post on Newton. Previous posts: one and two.]

If you were to take a look around you during a math department seminar or colloquium, you would witness the audience’s attention begin to drift as the talk sunk further into detail and became increasingly difficult to follow. Losing interest in a talk is more or less expected, and the professional mathematicians in the audience come prepared. Maybe they bring a paper to read, or possibly exam scripts to grade. Sometimes they will turn to a fresh page of their notebook and begin doing some actual mathematics of their own.

As a graduate student at McGill, I remember watching a postdoc fill up a page with long exact sequences and all kinds of diagrams, the notation veering into doodles as he got stuck at what must have been a familiar dead-end. It was a rare, voyeuristic glimpse into someone else’s solitary mathematical practice. I later asked this postdoc — whose notebook I presumed was full of such pages — if he ever went back and reread what he had written. No, he admitted, with a guilty smile.

Which was a relief. Not only because my own notebooks were full of repetitious dead-ends, but also because I too almost never went back to review anything I’d written.

Much has been made of Imposter Syndrome among academics — doubting whether we have truly earned whatever position we have reached given how paltry our contributions can sometimes feel. There is a related sense of insecurity to be found in wondering if you are doing mathematics correctly. To be clear, I don’t mean whether a proof we have written up is sound, but whether or not our process of formulating and devising them is the the “proper” way. As if there might be a correct way (or even a professional way) of doing mathematics.

These are not new concerns to have.

Isaac Newton was incredibly secretive in his work and did not have anything approaching students as we might describe them. But after his death, the calculus he developed would form the foundation of modern mathematical and scientific education at Cambridge.

Those who scribbled hastily on those exam papers were students, above all, of Newton’s mathematical physics. Though Newton had not cultivated a following during his own tenure at Cambridge, by the end of the eighteenth century the principals laid down in the Principia — and in particular the mathematical contents of that book — formed the basis for an intensely competitive system of testing at the university by which students were ranked in descending order based on their results on terminal examinations. known as the “Mathematical Tripos.” (The origin of the term Tripos is uncertain, but it may refer to the three legged stool on which students originally sat to take the oral examinations.)

The Newton Papers – Sarah Dry, pg 85

The material, and especially the notation, would be modernized as European influences arrived, but Newton did not lose his centrality. The manner in which he actually arrived at his great insights became a matter of interest. There is a great distinction between how discoveries are made, and how they finally appear on the page. Everyone knew how they personally went about doing mathematics, and even how their tutors told them to do mathematics, but was that the same as how Newton went about making his original discoveries? For all they knew, it might have all been provided to him by divine revelation.

On the Quadrature of Curves. See the Cambridge website to view more scans of Newton’s papers.

In 1872 a means of settling the question presented itself. Newton’s papers — or at least a large portion of them, covering far more than mathematical physics — had resided for nearly 150 years in the library of one of England’s aristocratic houses: Hurstbourne park. But now the Earl of Portsmouth was donating the scientific portion of papers back to the University of Cambridge.

Newton had been famously coy about his own methods, suggesting that he had kept his true means of discovering the Principia private and had only cast them publicly in the language of geometry. The question was therefore whether he adhered to the rigorous, manly, and above all morally upright techniques of thinking that Cambridge undergraduates were coached to acquire. To answer this Stokes and Adams were forced to consider whether Newton himself should — or could– be held accountable to the techniques that were mastered in his name. The Newton papers had the potential to probe more deeply the shadowy divide between patient work and divine inspiration, offering the promise of settling not simply what Newton had done but how he had done it. […] the question had a special urgency at Cambridge where the moral value of study was paramount. In that respect the Newton papers mattered for every undergraduate preparing for the Tripos and for what the Tripos itself stood for. Would the man who served as a model for what should be learned also reveal himself through his private papers, as a model for how to learn?

The Newton Papers – Sarah Dry, pg 88

Sarah Dry, author of The Newton Papers, a chronicle of the journey Newton’s writings took after his death, presents an interesting comparison of the two mathematicians, John Couch Adams and George Gabriel Stokes, who were tasked with making sense of his old notebook papers.

John Couch Adams (1819-1882),

On the one hand was John Couch Adams, whose ability to compute mathematically in his head was the stuff of Cambridge legend. This savant-like ability came with a tenacious reluctance to write anything down. This reluctance cost English astronomers the first opportunity to observe Neptune. Having deduced, from Uranus’ orbital irregularities, where a mystery planet should be found in the night sky, he failed to explain himself clearly to the astronomical bigwigs, who had little patience for the recent graduate. Roughly a year later, in 1846, the Frenchman Urbain Le Verrier managed to solve the problem and pointed his country’s own telescopes in the right direction. Adams was left with nothing but his incomplete written accounts and undated papers declaring his discovery, making establishing precedence impossible. Not that he seemed much bothered by losing out on the glory. He was personally very satisfied simply to have managed the computation.

Sir George Gabriel Stokes, 1st Baronet (1819-1903)

George Gabriel Stokes (of Navier-Stokes and Stokes’ Theorem) on the other hand, wrote compulsively, both mathematically and in personal correspondence (often to ease his own insecurities). Later in life he became editor of the Philosophical Transactions of the Royal Society, then the foremost journal in science, and this involved dealing with a huge amount of correspondence. Unfortunately he was a hoarder of papers of all and every kind, filling the rooms at his disposal with tables on which to pile up his papers. This was all compounded by his inclination towards procrastination

They might have made a formidable team, had their temperaments combined to negate the other’s weaknesses. Instead the project to deliver a verdict on the value of Newton’s papers and reveal his way of thinking was subject to great delay. Of the two however, it seems that Adams was the one who was most readily able to probe the documents deeply. Sarah Dry quotes Glaisher (Adam’s obituarist) as saying:

[…it was a] difficult and laborious task, extending over years, but once which intensely interested him, and upon which he spared no pains. In several instances he succeeded in tracing the methods that Newton must have used in order to obtain the numerical results which occurred in the papers. The solution of the enigmas presented by these numbers written on stray papers, without any clue to the source from which they were derived, was the kind of work in which all Adam’s skill, patience, and industry found full scope, and his enthusiasm for Newton was so great that he had no thought of time when so employed. His mind bore naturally a great resemblance to Newton’s in many marked respects, and he was so penetrated with Newton’s style of thought that he was peculiarly fitted to be his interpreter. Only a few intimate friends were aware of the immense amount of time he devoted to these manuscripts of the pleasure he derived from them.

John Glaisher — Memoir of the life of John Couch Adams

What Adams was doing, in his own manner, was nerding out. As with the discovery of Neptune, it seems that his motivations were overwhelmingly personal, and less in service to the scientific community. Imagine a referee today reading the paper under review very carefully, but forgetting to take notes and neglecting to get back to the editor. Nevertheless, conclusions were eventually drawn out of their little committee and a report of their findings was presented.

Here was confirmation that Newton had indeed worked by process of refinement that inevitably included false starts and error. In this sense, Newton revealed himself to be less an otherworldly genius and more a figure with whom the Cambridge wranglers could identify, a tireless worker in the mathematical trenches, where progress was made by increments rather than leaps. Adams knew the feeling well. In 1853 he had published an important paper pointing out errors made by Laplace in determining lunar motion and promising to provide the correct calculations soon; it had taken him six long years to get the final numbers. Here, in the papers, was evidence that Newton had worked just as hard to come up with his results.

The Newton Papers – Sarah Dry, pg 105

By the time I had finished my PhD I had produced a sizable pile of used dollar-store notebooks. Browsing through them I could recognize the contours of what I’d spent the past four years trying (and occasionally succeeding) to do. I might even have reconstructed from the pictures and computations I had written out what I might have actually been thinking at the time. And aside from myself, there are a few people in the world who could possibly make sense of their contents. It all went in the recycling. If somehow one of my notebooks did manage to survive, and made its way into the hands of future scholars, I would be alarmed to consider them giving the content more than cursory attention.

The hundreds (?) of pdfs that I have produced, now sitting out there in the cloud stand a far better chance at outliving me. And not just my published work and arxiv pre-prints (which number in the tens). But everything I ever committed to a latex document in my own personal space up there in the cyber heavens. Among all the discarded drafts that might find evidence of something interesting. Not only what I managed to do, but also what I failed to do. What I thought I had succeeded in doing, but had in fact betrayed my own good sense. When I have found mistakes, I am occasionally mindful enough to leave a short note in all-caps to make it clear where the point of failure lies. There is a great deal we can learn from knowing what other mathematicians have tried and failed to do.

Unlike physical notebooks, our cloud storage is password protected. Digital inheritance is already “a thing”, but it seems unclear to me how it will work out in practice. Kafka left his manuscripts in the possession of Max Brod with the instructions that they be destroyed in the event of his death. Brod told Kafka himself that he certainly wouldn’t, and indeed when Kafka died at the age of 40 as a consequence of tuberculosis, Brod set about getting Kafka’s work published. I haven’t taken a survey, but I would imagine that most young writers have made no attempt to ensure their passwords and unpublished estates are in suitable hands. In principal it is possible to submit a request to Google for access to the accounts of the deceased, but I can’t imagine there are any guarantees. I certainly have no idea what the terms and conditions that I have accepted have to say about such eventualities.

Newton died a man of wealth and importance. With neither wife nor children he had no direct descendants, but he did have a slew of half-nephews, half-nieces, and children of his half-sisters. The assets of obvious value were split between them. Those assets of less obvious value — the leftover pile of notebooks and “reams of loose and foul papers” fell into the possession of Catherine Conduitt. She was one of Newton’s half-nieces, and wife to John Conduitt, who had actively assisted Newton in his duties as master of the Royal Mint. This was the consequence of some rather wild tying-up of loose ends:

Newton had died while holding the post of master of the Mint, which in those days required that its holder assume personal responsibility for the probity of each new coinage of money. That meant that at Newton’s death he had nominal debts amounting to the entire sum of Great Britain’s national coinage. John Conduitt agreed to take on this debt until the coinage had been certified, accepting liability for any imperfections in the coins. In exchange for assuming this risk, he asked for, and was granted Newton’s manuscripts.

The Newton Papers – Sarah Dry, pg 15

The Conduitts took ownership of these papers with the view of producing a biography, and begin the work of securing Newton’s posthumous reputation. They became the first in a long line of people who had access to the papers, but lacked the tools really required to properly make sense of them. Their daughter, Kitty Conduitt married John Wallop who would become the Earl of Portsmouth, and papers would enter the library of Hurstbourne Park, seat of the Portsmouth family. (That is to say they fell into the possession of the aristocracy.) And it was there that they would remain, save occasional minor forays, and the recovery of the substantial portion of scientific papers by Stokes and Adams. What finally shifted the remaining papers out into the open was the fall of the English Aristocracy. In 1936, under the financial pressure of death duties and a recent divorce, Gerald Wallop, the ninth Earl of Portsmouth had the papers put up for auction at Sothebys.

John Maynard Keynes, 1st Baron Keynes (1883-1946)

If there is a hero in Dry’s account of the Newton Papers, it must be John Maynard Keynes. His heroic virtue being exceptional taste and judgement. Having begun collecting books as a child (possibly his first foray into speculation) he developed a rather prescient sense for what should be considered valuable. Unlike the majority of collectors he shared the marketplace with, Keynes was actually interested in reading the books themselves. He was less interested in the superficial qualities: illuminations, illustrations, binding, or an illustrious list of prior owners left him unmoved.

Keynes’s new style of collection was self-consciously intellectual, as opposed to aesthetic or literary. It asserted that a particular history of ideas or chain of thought linked certain men through the ages. And it projected the implicit assumption that its creator was an inheritor of both the material and the intellectual masterpieces of a previous age. Keynes was a thoroughgoing Bloomsburyite in his respect. The paintings on the wall, the rugs on the floor, the furnishing in the room, and the books on the shelves were never just things: they were the physical embodiment of ideas and values whose display was a source of both aesthetic pleasure and moral reinforcement. A book in the hand, like the good life in Bloomsbury of the Sussex countryside, linked the life of the mind with that of the physical world.

The Newton Papers – Sarah Dry, pg 147

You might already get the sense that Dry sees Keynes as simply bringing a new set of beliefs to the table, complete with their own set of limitations. Indeed, Keynes considerable contribution to our modern impression and understanding of Newton as half magician and half scientist, was really a very hot take based on an initial reading. He was the one who announced that the papers reveal Newton devoted great time and energy to the disreputable pursuits of alchemy and heretical theology. Yet the fact that so many of Newton’s papers have remained together and in the possession of the University of Cambridge can be attributed to his prescience sense of the papers’ importance.

Abraham Shalom Yahuda (on the right) (1877–1951)

Kaynes was only one of two major buyers at Sotheby’s. Abraham Yahuda, a scholar of ancient languages, bought most of Newton’s theological writings. Yahuda had found himself alienated from his own field of scholarship, due to recent developments in Higher Criticism applied to biblical scholarship. The Documentary Hypothesis was a shocking new line of textual analysis that argued the origins of the Torah were of combination and synthesis with earlier texts. As a consequence, these texts cease to resemble one coherent whole revealed to man, and begin to look more like artifacts of history and culture.

For Yahuda this was a vision of criticism taken to extremes, the text reduced to nothing but error, the possibility of meaning dissolving amid a multiplicity of authors, leaving only commentary, a Talmud with no Torah left in it. He thought in particular that too many sources were being attributed to the Pentateuch and that too many “experts” were exerting themselves “in the art of text alterations and source-hunting.” Thus “the original text was distorted and disfigured and in its place was offered a quite new text of pure invention.” In Newton, who himself sought to return a blemished Christianity to its purer origins, Yahuda found a kindred soul. Interpreting ancient texts didn’t require robbing them of fixed meaning. Both Newton and Yahuda sought instead to find a singular truth amid the variations.

The Newton papers – Sarah Dry, pg170

As a consequence of Yahuda’s desire to find an ally in Newton, those theological papers now reside in The National Library of Israel.

The final portion of Dry’s book concerns the subsequent attempts at synthesis of the material. The fact of the matter is that the task was simply impossible. There is too much material, covering too many subjects for any grand unifying conclusions to be drawn. It was hard to even put together a definitive edition of the Principia that covered all the different editions as well as Newton’s own marginalia. When finally published it was controversial due to the inevitable editorial decisions to include or not include certain material.

It is worth making one final point clear. I have never read Newtons’ Principia. I don’t believe you could find a research mathematician alive who has — unless their research happens to be the history of mathematics. It is a book whose significance is measured in its influence. Many decisions in its composition — in particular the modelling Newton’s Laws on the axioms in Euclid’s Elements — were very important. But you should not read it. When we discuss “great” books, there is usually the tacit understanding that we are missing out if we have not actually read the book. I am quite certain that we have not missed out.

On English Magic

Jonathan Strange and Mr Norrell, Susanna Clarke’s alternate history fairy-tale, opens with one of its characters asking a question that carries through the rest of the novel’s thousand pages.

“Mr Segundus wished to know,” he said, “why modern magicians were unable to work the magic they wrote about. In short, he wished to know why there was no more magic done in England.”

This is an England that had once been a very magical place, yet is no longer so. Over the decade “history” the novel covers (1806-1817) we see two new magicians arrive to provide their spells in service to their county in the Napoleonic Wars. The magic of previous generations had been lost, or forgotten, or become dysfunctional in some way. This was apparently despite the many books about/of magic that had been written by the very real magicians of the past, making their secrets and practices clear. Indeed, as the novel opens, England has many leaned societies of magicians, but these members are exclusively of a “theoretical” type — quite unable to cast a single simple spell.

The title characters are our heroes, of a sort. Mr Norrell, an uncharitable and unsociable Yorkshire gentleman who had devoted his youth to carefully studying the remaining books of magic, while also hoarding them away from others. Jonathan Strange, the more sympathetic of the two, is of a more obsessive and intuitive character, sociable and likable, ready to befriend Norrell, and complement his own innate ability by becoming Norrell’s apprentice.

Reviewers have noted the imbalance between the two protagonists, with Strange being the more compelling of the two, yet only actually arriving in the narrative proper a third of the way in. I personally found a great deal interesting in Norrell, however, when I recognized parallels between him and the Isaac Newton I had recently been reading about. Indeed, as I previously described, many of Newton’s pursuits could be described as attempts to recover magical techniques or knowledge from the past that had become lost or forgotten. Norrell’s inclination towards either preserving his recovered knowledge and even monopolize magic are reflected in many of Newton’s own inclinations. The ultimate difference being that science is not magic, and Newton himself was indebted to many of his contemporaries, (most controversially to John Flamsteed for astronomical data). But if you were trying to imagine the mentality of a man like Isaac Newton, I think you could do much worse than consider the character of Norrell.

It can be considered a kind of rule in story telling that you make a promise at the start of a story and you must deliver on it by the end. The question of why there is no magic done in the England certainly makes a clear promise that some kind of light will be shed on the matter. While neither the reader, nor the characters, get direct or complete answers to that question, we do however learn a great deal that is interesting on the subject. Plenty can be deduced a careful reader — enough to leave the book satisfied.

Jonathan Strange and Mr Norrell was a publishing sensation. Bloomsbury invested heavily in marketing it, imagining that it would allow them to expand beyond their foundation of Harry Potter sales. I think it is fair to say that almost twenty years on, it is regarded as a classic. In as far as such a thing could be said at this point. I certainly found the length no obstacle, and by the final third I was enraptured by the characters’ unfolding trajectories and their ultimate ends.

It is easy to say that a novel is just words on the page, but the word “just” is doing a great deal of heavy lifting. I find myself increasingly paying attention to what goes on in an individual paragraph the way that film buffs concern themselves with actors, cinematography, and special effects. Take the following paragraph, which demonstrates quite well the Austen-esque prose styling along with Clarke’s ability to capture the regional richness of England which I either never really encountered or appreciated before in English fiction.

At no. 9 Harley-street Lady Pole’s country servants were continually ill at ease, afraid of going wrong and never sure of what was right. Even their speech was found fault with and mocked. Their Northamptonshire accent was not always intelligible to the London servants (who, it must be said, made no very great efforts to understand them) and they used words like goosegogs, sparrow-grass, betty-cat and battle-twigs, when they should have said gooseberries, asparagus, she-cat and ear wigs

pg 173

The list of alternates given in that paragraph pass by quickly in the way that good set dressing, special effects, and cinematography do, but this is harder to pull of than you might imagine. The following paragraph similarly stood out to me, again for it’s command of its setting, but also for the kind of fantastical whimsy that the likes of Lord of the Rings, Harry Potter, and Neil Gaiman manage to tap into. If it could be meaningfully produced industrially you’d find it sold in the Harry Potter stores scattered all across England.

Whereupon Mr Strange told them how, to his certain knowledge, there had been four copies of The Language of Birds in England not more than five years ago: one in a Gloucester bookseller’s; one in the private library of gentleman-magician in Kendal; one the private property of a blacksmith near Penzance who had taken it in part payment for mending an iron-gate; and one stopping a gap in a window of the boy’s school in the close of Durham Cathedral.

pg 281

For those of you reading this who are put off by a thousand page door stop, there is also the BBC miniseries adaptation that I hear was pretty good. And if you would prefer a shorter novel, her second, Piranesi was released last year, is much shorter, and also very good. The New Yorker wrote a good profile of Clarke discussing it.

There has been a murder in Gathertown

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.