*But wait a minute: what could be more amazing than a young man discovering a fundamental force of nature while sitting under a tree? For starters, we have to recognize how foreign Newton’s ultimate idea about gravity was to philosophers, astronomers and mathematicians in the era of the Scientific Revolution. Newton provided an answer to a question that hadn’t even been asked yet. The problem with understanding the distant past is that we take our twenty-first century ideas and attitudes for granted. We think, for example, that the following is obvious: if the planets, like the Earth and Jupiter, regularly orbit the Sun, there must be something that causes them to follow their orbits. After all, if nothing caused them to orbit the Sun, they would fly off into deep space. *[my emphasis]

So what is wrong with this piece of #histSTM prose? Let us start with the second of my bold emphasised segments:

*Instead, for many generations, leading philosophers and mathematicians thought this: the circle is a perfect mathematical form, and the planetary orbits are circular, so they are ever-lasting aspects of the natural world. To them, the orbits were so perfect that nothing caused them to occur. They simply were.*

Whilst it is true that, following Empedocles, Western culture adopted the so-called Platonic axioms, which stated that celestial motion was uniform and circular, it is not true that they claimed this motion to be without cause. Aristotle, whose system became dominant for a time in the Middle Ages, hypothesised a system of nested crystalline spheres, which working from the outside to the centre drove each other through direct contact; a system that probably would not have worked due to friction. His outer-most sphere was moved by the unmoved mover, who remained unnamed, making the theory very attractive for Christian theologians in the High Middle Ages, who simple called the unmoved mover God. Interestingly the expression love makes the world go round originates in the Aristotelian belief that that driving force was love. In the Middle Ages we also find the beliefs that each of the heavenly bodies has a soul, which propels it through space or alternatively an angel pushing it around its orbit.

All of this is all well and good but of course doesn’t have any real relevance for Newton because by the time he came on the scene the Platonic axioms were well and truly dead, killed off by one Johannes Kepler. You might have heard of him? Kepler published the first two of his planetary laws, number one: that the planetary orbits are ellipses and that the sun is at one focus of the ellipse and number two: that a line connecting the sun to the planet sweeps out equal areas in equal time periods in 1609, that’s thirty-three years before Newton was born. Somewhat later Cassini proved with the support of his teachers, Riccioli and Grimaldi, using a heliometer they had constructed in the San Petronio Basilica in Bologna, that the earth’s orbit around the sun or the sun’s around the earth, (the method couldn’t decide which) was definitely elliptical.

By the time Newton became interested in astronomy it was accepted by all that the planetary orbits were Keplerian ellipses and not circles. Kepler’s first and third laws were accepted almost immediately being based on observation and solid mathematics but law two remained contentious until about 1670, when it was newly derived by Nicholas Mercator. The dispute over alternatives to Kepler’s second law between Ismaël Boulliau and Seth Ward was almost certainly Newton’s introduction to Kepler’s theories.

Turning to the other two bold emphasised claims we have:

* **Newton provided an answer to a question that hadn’t even been asked yet. The problem with understanding the distant past is that we take our twenty-first century ideas and attitudes for granted. We think, for example, that the following is obvious: if the planets, like the Earth and Jupiter, regularly orbit the Sun, there must be something that causes them to follow their orbits. After all, if nothing caused them to orbit the Sun, they would fly off into deep space.*

And:

*The question of what caused the planetary orbits was not even on the table for astronomers in those days**.*

I’m afraid that Herr Kepler would disagree rather strongly with these claims. Not only had he asked this question he had also supplied a fairly ingenious and complex answer to it. Also quite famously his teacher Michael Maestlin rebuked him quite strongly for having done so. Kepler is usually credited with being the first to reject vitalist explanations of planetary motion by souls, spirits or angels (anima) and suggest instead a non-vitalist force (vir). His theory, based on the magnetic theories of Gilbert, was some sort of magnetic attraction emanating from the sun that weakened the further out it got. Kepler’s work started a debate that wound its way through the seventeenth century.

Ismaël Boulliau, a Keplerian, in his *Astronomia philolaica* from 1645 discussed Kepler’s theory of planetary force, which he rejected but added that if it did exist it would be an inverse-square law in analogy to Kepler’s law of the propagation of light. Newton was well aware of Boulliau’s suggestion of an inverse-square law. In 1666 Giovanni Alfonso Borelli, a disciple of Galileo, published his *Theoricae Mediceorum planetarum ex causis physicis deductae* in which he suggested that planetary motion was the result of three forces.

Famously in 1684 in a London coffee house Christopher Wren posed the question to Robert Hooke and Edmond Halley, if the force driving the planets was an inverse-square force would the orbits be Keplerian ellipses, offering a book token as prize to the first one to solve the problem. This, as is well known, led to Halley asking Newton who answered in the positive and wrote his *Principia* to prove it; in the *Principia* Newton shows that he is fully aware of both Kepler’s and Borelli’s work on the subject. What Newton deliberately left out of the *Principia* is that in an earlier exchange it had in fact been Hooke who first posited a universal force of gravity.

As this all too brief survey of the history shows, far from Newton providing an answer to a question that hadn’t been asked yet, he was, so to speak, a Johnny-come-lately to a debate that when he added his contribution was already eighty years old.

The Institute of Arts and Ideas advertises itself as follows:

*So the IAI seeks to challenge the notion that our present accepted wisdom is the truth. It aims to uncover the flaws and limitations in our current thinking in search of alternative and better ways to hold the world.*

Personally I don’t see how having a leading philosopher of science propagating the lone genius myth by spouting crap about the history of science fulfils that aim.

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Yesterday was World Maths Day and the Royal Society tweeted portraits of mathematicians with links to articles all day, one of those tweets was about Emmy Noether. The tweet included a paraphrase of a well known quote from Albert Einstein, after all what could be better than a quote from old Albert the greatest of the great? Well almost anything actually, as the Einstein quote is highly demeaning. As given informally by the Royal Society it read as follows:

*Emmy Noether was described by Einstein as the most important woman in the history of mathematics*.

What Einstein actually wrote in a letter to the New York Times on the occasion of her death in 1935 was the following:

*In the judgment of the most competent living mathematicians, Fräulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began. In the realm of algebra, in which the most gifted mathematicians have been busy for centuries, she discovered, methods which have proved of enormous importance in the development of the present-day younger generation of mathematicians.*

In the same year, but before she died, Norbert Wiener wrote:

*Miss Noether is… the greatest woman mathematician who has ever lived; and the greatest woman scientist of any sort now living, and a scholar at least on the plane of Madame Curie.*

Now I’m sure that the Royal Society, Albert Einstein and Norbert Wiener all meant well, but take a step back and consider what all of them said in their different ways, *Emmy Noether was pretty good for a woman* [my emphasis].

Emmy Noether was one of the greatest mathematicians of the twentieth century, male or female, man or woman, about that there is absolutely no doubt, to qualify that praise with the term woman is quite simple demeaning.

In my mind it triggers the text of Melanie Safka’s mega pop hit from 1971, *Brand New Key*:

*I ride my bike, I roller skate, don’t drive no car*

*Don’t go too fast, but I go pretty far*

*For somebody who don’t drive*

*I been all around the world*

** Some people say, I done all right for a girl** [my emphasis]

On twitter, space archaeologist, Alice Gorman (@drspacejunk) took it one stage further, in my opinion correctly, and asked, “Dare I cite Samuel Johnson’s aphorism about the talking dog?” For those who are not up to speed on the good doctor’s witticisms:

*I told him I had been that morning at a meeting of the people called Quakers, where I had heard a woman preach. Johnson: “Sir, a woman’s preaching is like a dog’s walking on his hind legs. It is not done well; but you are surprised to find it done at all.”* – Boswell: Life

Can we please in future when talking about Emmy Noether resist the temptation to quote those who affix their praise of her mathematical talents with the term woman and just acknowledge her as a great mathematician?

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His algebra lies at the base of computer circuit design and of most computer programming languages and Booleans power the algorithms of the ubiquitous search engines. As a result two years ago the bicentenary of his birth was celebrated extensively and very publically. All of this would have been very hard to predict when his work on the algebra of logic first saw the light of day in the nineteenth century. His first publication *Mathematical Analysis of Logic* (1847) was largely ignored by the wider world of mathematics and his definitive presentation of his logic *An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities* fared little better, initially attracting very little attention. It was only some time after his death that Boole’s logical works began to attract deeper interest, most notably in Germany by Ernst Schröder and in America by Charles Sanders Peirce.

In 1883 Peirce published *Studies in Logic*: *by Members of the Johns Hopkins University*, edited by himself it contained seven papers written largely by his students. Of central interest is the fact that it contains a doctoral thesis, *On the Algebra of Logic*, written by a women, Christine Ladd.

Christine Ladd’s life story is a casebook study of the prejudices that women, who wished to enter academia suffered in the nineteenth and early twentieth centuries. Born 1 December 1847 (the year Boole published his first logic book) in Windsor, Connecticut the daughter of Eliphalet and Augusta Ladd, she grew up in New York and Windsor. Her mother and her aunt Julie Niles brought her up to believe in education for women and women’s rights. Her mother died in 1860 but her father initially supported her wish for advanced education and enrolled her at Welshing academy in a two year course for preparing students for college; she graduated as valedictorian in 1865 but now her father opposed her wish to go on to college. Only by arguing that she was too ugly to get a husband was she able to persuade her father and grandmother to allow her to study at the women’s college Vassar. She entered Vassar in 1866 but was forced by financial difficulties to leave before completing her first year. She now became a schoolteacher until her aunt helped her to finance her studies and she returned to Vassar.

At Vassar the pioneering female astronomer Maria Mitchell took her under her wing and fostered her developing interest in physics and mathematics.

Due to the fact that women could not do experiment work in laboratories she was forced to choose mathematics[1] over physics, a decision that she regretted all of her life. She graduated from Vassar in 1869 and became a secondary school teacher of mathematics and science in Washington, Pennsylvania. Over the next nine years she published six items in *The Analyst*: *A Journal of Pure and Applied Mathematics* and three in the *American Journal of Mathematics*. More importantly she took a very active part in the mathematical questions column of the *Educational Times*, the journal of the College of Preceptors in London, a profession body for schoolteachers. This mathematical questions column was a very popular forum for nineteenth century mathematicians and logicians with many leading practitioners contribution both question and solutions. For example the nineteenth-century Scottish logician Hugh McColl published his first logical essays here and Bertrand Russell’s first mathematical publication can also be found here[2]. Ladd contributed a total of seventy-seven problem and solution to the *Education Times*, which would prove highly significant for her future career.

In 1878 she applied for and won a fellowship to study mathematics at the Johns Hopkins University. Her fellowship application was simply signed C. Ladd and the university had assumed that she was male. When they realised that she was in fact a woman, they withdrew their offer of a fellowship. However the English professor of mathematics at Johns Hopkins, James J. Sylvester, who knew of Ladd’s abilities from those *Educational Times* contribution insisted on the university honouring the fellowship offer.

At the time Johns Hopkins did not have a very good reputation but Sylvester did, in fact he was a mathematical star, not wishing to lose him the university conceded and allowed Ladd to take up her three-year scholarship. However her name was not allowed to be printed in circulars and basically the university denied her existence. At the beginning she was only allowed to attend Sylvester’s classes but as it became clear that she was an exceptional student she was allowed to attend classes by other professors.

In the year 1879 to 1880 she studied mathematics, logic and psychology under Charles Sanders Peirce becoming the first American women to be involved in psychology. Under Peirce’s supervision she wrote her doctoral thesis *On the Algebra of Logic*, which was then, as mentioned above, published in 1883. Although she had completed all the requirements of a doctoral degree Johns Hopkins University refused to award her a doctorate because she was a woman. They only finally did so forty-four years later in 1927, when she was already seventy-eight years old.

In 1882 she married fellow Johns Hopkins mathematician Fabian Franklin and became Christine Ladd-Franklin, the name by which she is universally known today. As a married woman she was barred from holding a paid position at an American university but she would lecture unpaid for five years on logic and psychology at Johns Hopkins and later at Columbia University for thirty years.

In the 1880s she developed an interest in vision and theories of colour perception publishing her first paper on the subject in 1887. She accompanied her husband on a research trip to Germany 1891-92 and used the opportunity to study with the psychologist Georg Elias Müller (1850–1934) in Göttingen

and with the physiologist and physicist Hermann von Helmholtz (1821-1894) in Berlin.

In 1894 she returned alone to Germany to work with physicist Arthur König (1856–1901), with whom she did not get on and whom she accused of having stolen her ideas, and again in 1901 to work with Müller.

As a result of her researches she developed and published her own theories of colour vision and the causes of colour blindness that were highly influential.

Ladd-Franklin was a tireless campaigner for women’s rights and even persuaded the inventor of the record player, Emile Berliner, to establish a fellowship for female professors, the Sarah Berliner postdoctoral endowment, in 1909, which she administered for the first ten years and which is still awarded annually.

She herself continued to suffer rejection and humiliation as a female academic. In 1904 the British psychologist Edward Titchener (1867–1927) founded a society for experimental psychologists, “The Experimentalists”, and although he knew Ladd-Franklin well her barred her, as a woman, from membership. A decision, which she fought against in vain for many years. Women were only permitted to attend following Titchener’s death.

Despite the discrimination that she suffered Christine Ladd-Franklin published many papers in the leading journals and her work was held in high regard. She died of pneumonia, aged 82, in 1930. Today the American Association for women in Psychology have an annual Christine-Ladd Franklin Award, awarded for significant and substantial contributions to the Association.

Although she struggled against prejudice and discrimination all of her life and never received the formal recognition that should have been her due, Christine Ladd-Franklin made significant contributions to the fields of Boolean algebra and colour vision for which she is highly regarded today. Through her fighting spirit and unbending will she helped open the doors of scientific research and academia for later generations of women.

* *

[1] It is interesting to note that barred from access to academia and its institutions a small but significant number of women managed to some extent to break through the glass ceiling in logic and the mathematics in the nineteenth century, because these are subjects in which one can make an impression with nothing more than a pencil and a piece of paper.

[2] In my days as a logic historian I spent a not very pleasant two weeks in the British Newspaper Library in Colindale (the tenth circle of hell), amongst other things, going through the *Educational Times* looking for contributions on the algebra of logic. During this search I came across the Bertrand Russell contribution, which I showed, some time later, to a leading Russell scholar of my acquaintance, who shall remain here nameless. Imagine my surprise when shortly afterwards an article was published by said Russell expert explaining how *he had discovered* Russell’s first ever mathematical publication in the *Mathematical Questions* column of *The Educational Times*. He made no mention of the fact that it was actually I who had made the discovery.

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The first major problem with this piece is the title; in reality it should read four contributions that Brahmagupta made to the history of mathematics with his *Brāhmasphuṭasiddhānta* and one development in Indian mathematics, which failed to transfer outside of India.

The first four elements of the list are the number system, zero, solutions of quadratic equations and rules for negative numbers, which are all, as I said above, taken from Brahmagupta’s *Brāhmasphuṭasiddhānta*, which was written in the seventh century CE. Both zero and negative numbers are parts of the number system so we really only have one item not three but I will return in detail to this and the quadratic formula later. First I want to deal with the fifth item on the list, basis for calculus.

This is something I blogged about several years ago in a brief outline of the history of calculus. What we have here is the so-called Kerala School of mathematics, which flourished in the 14^{th} to 16^{th} centuries and did some quite remarkable work on infinite series, anticipating work that was first done in Europe in the 17^{th} century. This work is indeed the basis on which calculus stand, however there are various caveats that need to be made here about any potential influence on the world. First the extent to which the Kerala School anticipated calculus is debatable. George Gheverghese Joseph from whose book *The Crest of the Peacock*: *Non-European Roots of Mathematics* (Penguin) I first learnt of the Kerala School is convinced that what they had is a full blown calculus, whereas Kim Plofker in her excellent *Mathematics in India* (Princeton UP) is far less convinced. However the real problem is that although Joseph sets up a plausible route of cultural transfer from Kerala to Europe, all investigations have drawn a blank and there is absolutely no evidence for such a transfer. As far as we know the Kerala School flourished and died without influencing the history of mathematics outside of their own circle. This is not an uncommon phenomenon in the history of science.

Let us return to Brahmagupta. His text is indeed the text that introduced the so-called Hindu-Arabic decimal place value number system to the world outside of India, first to the Islamic Empire and then through them to medieval Europe. However this wasn’t the only place value number system from antiquity and not even the only decimal one. The Chinese also had a decimal place value number system and historians of mathematics still don’t know if the Chinese influenced the Indians or the Indians the Chinese or whether the two systems developed totally independently of each other. Of course the Babylonians also had, much earlier than the Indians, a place value number system but a base sixty (sexagesimal) one not a base ten (decimal) one. There was certainly knowledge transfer between Babylon and India did the Indians get the idea of a place value number system from the Babylonians? We do know that the Indians took over a lot of their astronomy from the Greeks and Greek astronomers used the Babylonian sexagesimal place value numbers system in their astronomical texts, did a knowledge transfer take place here? A lot of unanswered questions but although we do have the decimal place value numbers system from Brahmagupta there are still a lot of open questions as to where he got it from.

With zero as a number we are on safer ground, although the Babylonians did develop and use a place holder zero, as did the Greeks in their astronomical texts, it really does appear that zero as a number, and not just a place holder, is a genuine unique India invention. There is however even here an important caveat; Brahmagupta thought one could divide by zero, which as every modern school kid knows is not on.

Turning to negative numbers, whilst Brahmagupta does indeed correctly describe their use in his *Brāhmasphuṭasiddhānta* he wasn’t the first to do so. In this case the Chinese beat him to it in *The Nine Chapters on the Mathematical Art, *which dates from 202-186 BCE, so some eight hundred years before Brahmagupta. The author of the article write that “European mathematicians were reluctant to accept negative numbers as meaningful” but so were Islamic mathematicians and also some prominent later Indian mathematicians.

In his piece the author write:

*In his seminal text, the astronomer Brahmagupta introduced rules for solving quadratic equations (so beloved of secondary school mathematics students)*…

Whilst it is true that Brahmagupta presents what is now know as the quadratic formula the Babylonians knew how to solve them at least two thousand years earlier. They however used two formulas for the two solutions based on the so-called reduced quadratic (where the parameter for x^{2} is reduced to 1 by division). The Babylonians of course rejected negative and imaginary solutions. Euclid solves quadratic equations geometrically, which is why we call them quadratic, meaning square). So there were methods for solving quadratic equations long before Brahmagupta.

Whilst by no means whishing to diminish the undoubted Indian contributions to the history of mathematics, what I am trying to make clear here is that any aspect of the history of mathematics or science has a context, a pre-history and a post-history and to ignore those aspect when presenting any given aspect automatically produces a distorted and misleading picture.

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*The format of the show focuses on Davies and three other guest panelists answering questions that are extremely obscure, making it unlikely that the correct answer will be given. To compensate, the panelists are awarded points not only for the right answer, but also for interesting ones, regardless of whether they are right or even relate to the original question, while points are deducted for “answers which are not only wrong, but pathetically obvious”– typically answers that are generally believed to be true but in fact are misconceptions. These answers, referred to as “forfeits”, are usually indicated by a loud klaxon and alarm bell, flashing lights, and the incorrect answer being flashed on the video screens behind the panelists.* [my emphasis]

Given the section that I have highlighted above the Twitter account should have points deducted to the sounds of a loud klaxon and an alarm bell accompanied by flashing lights for having tweeted the following on 12 September

*It wasn’t until 1992 that the Catholic Church finally admitted that Galileo’s views on the solar system were correct* – @qikipedia

This is of course complete rubbish. In what follows I will give a brief summary of the Catholic Church’s ban on heliocentrism, as propagated by Galileo amongst others.

The initial ban on propagating heliocentrism as a proven theory, one could still present it as a hypothetical one, was issued by the Inquisition in 1616. Interestingly whilst the books of Kepler and Maestlin, for example, were placed on the Index of Forbidden Books, Copernicus’ *De revolutionibus* was not but merely banned temporarily until corrected, which took place surprisingly rapidly; correction meaning the removal of the very few passages where heliocentricity is presented as a fact. By 1621 *De revolutionibus* was back in circulation for Catholic astronomers. Galileo’s *Dialogo* was placed on the Index following his trial in 1632.

Books openly espousing heliocentricity as a true fact, which was more that the science of the time could deliver, were placed on the Index by the Catholic Church, so all good Catholics immediately dropped the subject? Well no actually. The ban had surprising little effect outside of Italy. Within Italy, astronomers kept their heads below the parapet for a couple of decades but outside of Italy things were very different. Protestant countries, naturally, totally ignored the ban and even astronomers in Catholic countries on the whole took very little notice of it. The one notable exception was René Descartes who dropped plans to publish his book *Le Monde, ou Traite de la lumiere* in 1633, which contained his views supporting heliocentricity, the full text only appearing posthumously in 1677. Quite why he did so was not very clear but it is thought that he did it out of respect to his Jesuit teachers. However, Descartes remained the exception. Galileo’s offending *Dialogo* quickly appeared in a ‘pirate’ edition, translated into Latin in the Netherlands, where later his *Discorsi*, would also be published. I say pirate but Galileo was well aware of the publication, which had his blessing, but officially knew nothing about it.

Within Italy once the dust had settled Catholic astronomers began to publish books on heliocentricity that opened with some sort of nod in the direction of the Church along the lines of, “The Holy Mother Church has in its wisdom condemned heliocentricity as contrary to Holy Scripture…” but then continued something like this “…however it is an interesting hypothetical mathematical model, which we will now discuss.” This face saving trick was accepted by the Church and everybody was happy. By the early eighteenth century almost all astronomers in Italy, with the exception of some Jesuits, were following this course.

In 1758 the ball game changed again as the then Pope basically dropped the ban on heliocentricity, although this was done informally and the formal prohibition stayed in place. The publication of a complete works of Galileo was even permitted with a suitable preface to the *Dialogo* pointing out its faults. From this time on Catholic astronomers were quite free to propagate a factual heliocentricity in their publications.

This was the situation up till 1820 when an over zealous Master of the Sacred Palace (the Church’s chief censor), Fillipo Anfossi, refused to licence a book containing a factual account of heliocentricity by Giuseppe Settele. Settele appealed directly to the Pope and after deliberations the ban on heliocentricity was formally lifted by the Church in 1821. The next edition of the Index, which didn’t appear until 1835, no longer contained books on heliocentricity. Anfossi and Settele only feature in the history of science because of this incidence.

So to summarise, the Church only banned factual claims for the heliocentric system but not hypothetical statements about it, so this is how Catholic astronomer got around the ban. In 1758 the Pope informally lifted the ban clearing the way for Catholic astronomers to write freely about it. In 1821 the ban was formally lifted and in 1835 books on heliocentricity were removed from the Index, so where did QI get their date of 1992 from?

In 1981 the Church constituted the Pontifical Interdisciplinary Study Commission to re-examine the Galileo trial, which came to rather wishy-washy conclusions. In 1992 the Pope held a speech formally closing the commission and saying that the whole affair had been rather unfortunate and that the Church had been probably wrong to prosecute Galileo.

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This was followed by an equally excellent and equally deflating essay by Eduardo A Escobar an expert on cuneiform tablets. And so another hyped sensation is brought crashing down into the real world. Both put downs were endorsed by Eleanor Robson author of *Mathematics in Ancient Iraq*: *A Social History* and a leading expert on Babylonian mathematics.

Last week saw the next history of mathematics press feeding frenzy with the announcement by the Bodleian Library in Oxford that an Indian manuscript containing a symbol for zero had been re-dated using radio carbon dating and was now considered to be from the third to fourth centuries CE rather than the eight century CE, making it the earliest known Indian symbol for zero. This is of course an interesting and significant discovery in the history of mathematics but it doesn’t actually change our knowledge of that history in any really significant way. I will explain later, but first the hype in the various Internet reports.

We start off with Richard Ovenden from Bodleian Libraries who announced,** “The finding is of “vital importance” to the history of mathematics.”**

The Guardian leads off with an article by Marcus Du Sautoy: *Much ado about nothing: ancient Indian text contains earliest zero symbol*. Who in a video film and in the text of his article tells us, “This becomes the birth of the concept of zero in it’s own right and this is a total revolution that happens out of India.”

The Science Museum’s article *Illuminating India: starring the oldest recorded origins of ‘zero’, the Bakhshali manuscript*, basically repeats the Du Sautoy doctrine,

Medievalists.net makes the fundamental mistake of entitling their contribution, *The First Zero*, although in the text they return to the wording, “the world’s oldest recorded origin of the zero that we use today.”

The BBC joins the party with another clone of the basic article, *Carbon dating reveals earliest origins of zero symbol*.

Entrepreneur Cecile G Tamura summed up the implicit and sometimes explicit message of all these reports with the following tweet, * One of the greatest conceptual breakthroughs in mathematics has been traced to the Bakhshali manuscript dating from the 3rd or 4th century at a period even earlier than we thought.* To which I can only reply, has it?

All of the articles, which are all basically clones of the original announcement state quite clearly that this is a placeholder zero and not the number concept zero[1] and that there are earlier recorded symbols for placeholder zeros in both Babylonian and Mayan mathematics. Of course it was only in Indian mathematics that the place-holder zero developed into the number concept zero of which the earliest evidence can be found in Brahmagupta’s *Brahmasphuṭasiddhanta* from the seven century CE. However, this re-dating of the Bakhshali manuscript doesn’t actually bring us any closer to knowing when, why or how that conceptual shift, so important in the history of mathematics, took place. Does it in anyway actually change the history of the zero concept within the history of mathematics? No not really.

Historians of mathematics have known for a long time that the history of the zero concept within Indian culture doesn’t begin with Brahmagupta and that it was certainly preceded by a long complex prehistory. They are well aware of zero concepts in Sanskrit linguistics and in Hindu philosophy that stretch back well before the turn of the millennium. In fact it is exactly this linguistic and philosophical acceptance of ‘nothing’ that the historian assume enabled the Indian mathematicians to make the leap to the concept of a number signifying nothing, whereas the Greeks with their philosophical rejection of the void were unable to spring the gap. Having a new earliest symbol in Indian mathematics for zero as a placeholder, as opposed to the earlier recorded words for the concept of nothingness doesn’t actually change anything fundamental in our historical knowledge of the number concept of zero.

There is a small technical problem that should be mentioned in this context. Due to the fact that early Indian culture tended to write on perishable organic material, such as the bark used here, means that the chances of our ever discovering manuscripts documenting that oh so important conceptual leap are relatively low.

I’m afraid I must also take umbrage with another of Richard Ovenden’s claims in the original Bodleian report:

* **Richard Ovenden, head of the Bodleian Library, said the results highlight a Western bias that has often seen the contributions of South Asian scholars being overlooked. “These surprising research results testify to the subcontinent’s rich and longstanding scientific tradition,” he said.*

Whilst this claim might be true in other areas of #histSTM, as far as the history of the so-called Hindu-Arabic numbers system and the number concept zero are concerned it is totally bosh. Pierre-Simon, marquis de Laplace (1749-1827) wrote the following:

*“It is India that gave us the ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit. But its very simplicity and the great ease which it has lent to computations put our arithmetic in the first rank of useful inventions; and we shall appreciate the grandeur of the achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity.”*

I started buying general books on the history of mathematics more than 45 years ago and now have nine such volumes all of which deal explicitly with the Indian development of the decimal place value number system and the invention of the number concept zero. I own two monographs dedicated solely to the history of the number concept zero. I have four volumes dedicated to the history of number systems all of which deal extensively with the immensely important Indian contributions. I also own two books that are entirely devoted to the history of Indian mathematics. Somehow I can’t see in the case of the massive Indian contribution to the development of number systems that a Western bias has here overseen the contributions of South Asian scholars.

This of course opens the question as to why this discovery was made public at this time and in this overblown manner? Maybe I’m being cynical but could it have something to do with the fact that this manuscript is going on display in a major Science Museum exhibition starting in October?

The hype that I have outlined here in the recent history of mathematics has unfortunately become the norm in all genres of history and in the historical sciences such as archaeology or palaeontology. New discoveries are not presented in a reasonable manner putting them correctly into the context of the state of the art research in the given field but are trumpeted out at a metaphorical 140 decibel claiming that this is a sensation, a discipline re-defining, an unbelievable, a unique, a choose your own hyperbolic superlative discovery. The context is, as above, very often misrepresented to make the new discovery seem more important, more significant, whatever. Everybody is struggling to make themselves heard above the clamour of all the other discovery announcements being made by the competition thereby creating a totally false impression of how academia works and how it progresses. Can we please turn down the volume, cut out the hype and present the results of academic research in history in a manner appropriate to it and not to the marketing of the latest Hollywood mega-bucks, blockbuster?

[1] For those who are not to sure about these terms, a placeholder zero just indicates an empty space in a place value number system, so you can distinguish between 11 and 101, where here the zero is a placeholder. A number concept zero also fulfils the same function but beyond this is a number in its own right. You can perform the arithmetical operations of addition, subtraction and multiplication with it. However, as we all learnt at school (didn’t we!) you can’t divide by zero; division by zero is not defined.

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It is clear to me that biographies, particular popular ones, play a very central roll in the creation of the great men and lone genius myths. Now don’t misunderstand me I am not condemning #histSTM biographies in general; I have one and a half metres of such biographies on my bookshelves and have consumed many, many more that I don’t own. What I am criticising is the way that many such biographies are written and presented and I am going to make some suggestions, with examples, how, in my opinion such biographies should be written in order to avoid falling into the great man and lone genius traps.

The problem as I see it is produced by short, single volume, popular biographies of #histSTM figures or the even shorter portraits printed in newspapers and magazines. Here the title figure is presented with as much emphasis as possible on the uniqueness, epoch defining, and world-moving importance of their contribution to the history of science, technology or medicine. Given the brevity and desired readability of such works the context in which the subject worked is reduced to a minimum and any imperfections in their efforts are often conveniently left out. In order to achieve maximum return on their investment publishers then hype the book in their advertising, in the choice of title and/or subtitle and in the cover blurbs. A good fairly recent example of this was the subtitle of David Loves Kepler biography, *How One Man Revolutionised Astronomy*, about which I wrote a scathing blog post.

The authors of such works, rarely themselves historian of science, also tend to ignore the painfully won knowledge of historians and prefer to repeat ad nauseam the well worn myths handed down by the generations – Newton and the apple, Galileo and the Tower of Pisa and so on and so forth.

#histSTM biography does not have to be like this. Individual biographies can be historically accurate, can include the necessary context, and can illuminate the failings and errors of their subjects. Good examples of this are Westfall’s Newton biography *Never at Rest* and Abraham Pais’ Einstein biography *Subtle is the Lord*. Unfortunately these are doorstep size, scholarly works that tend to scare off the non-professional reader. Are there popular #histSTM works that surmount this problem? I think there are and I think the solution lies in the multi-biography and the theme-orientated books with biographies.

A good example of the first is Laura J Snyder’s *The Philosophical Breakfast Club*: *Four Remarkable Friends Who Transformed Science and Changed the World*. Despite the hype in the subtitle this book embeds its four principal biographies in a deep sea of context and because all four of them were polymaths, manages to give a very wide picture of Victorian science in the first half of the nineteenth century.

Another good example is Jenny Uglow’s *The Lunar Men*: *The Friends Who made the Future*, once again a terrible subtitle, but with its even larger cast of central characters and even wider spectrum of science and technology delivered by them we get a true panorama of science and technology in the eighteenth and nineteenth centuries. Neither book has any lone geniuses and far too many scrambling for attention for any of them to fit the great man schema.

Two good examples of the second type are both by the same author, Renaissance Mathematicus friend and Twitter sparring partner, Matthew the Mancunian Maggot Man, aka Matthew Cobb. Both his books, *The Egg and Sperm Race*: *The Seventeenth Century Scientists Who Unravelled the Secrets of Sex, Life and Growth*

and *Life’s Greatest Secret*: *The Race to Crack the Genetic Code*

deal with the evolution of scientific concepts over a relatively long time span. Both books contain accurate portraits of the scientists involved complete with all of their failings but the emphasis is on the development of the science not on the developers. Here, once again, with both books having a ‘cast of millions’ there is no place for lone geniuses or great men.

These, in my opinion, are the types of books that we should be recommending, quoting and even buying for friends and relatives not the single volume, one central figure biographies. If more such books formed the basis of peoples knowledge of #histSTM then the myths of the lone genius and the great man might actually begin to fade out and with luck over time disappear but sadly I don’t think it is going to happen any day soon.

Having mentioned it at the beginning I should say something about Paul Engle’s *Conciatore*.

This is a single volume, one central figure biography of the seventeenth-century glassmaker Antonio Neri, who was the first man to write and publish a book revealing the secrets of glassmaking. His revealing of the trade secrets of a craft marks a major turning point in the history of technology. Up till the seventeenth century trade secrets were just that, secret with severe punishment for those who dared to reveal them, including death. Later in the century Joseph Moxon would follow Neri’s example publishing a whole series of books revealing the secrets of a whole range of trades including the first ever textbook on book printing his *Mechanical Exercises or the Doctrine of Handy-Works*. Paul’s book is a biography of Neri but because of why he is writing about Neri it is more a history of glassmaking and so sits amongst my history of technology books and not with my collection of #histSTM biographies. Here the context takes precedence over the individual, another example of how to write a productive biography and a highly recommended one at that.

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MC: *What would have happened if Einstein fell under a tram in 1900? What difference would it have made, for how long?*

Me: *Not a lot, Poincaré was almost there and others were working on the various problems. I’d guess at most a ten-year delay*

MC: *So are there any true examples of ‘great men’ or is science all over-determined?*

My instantaneous response to Mathew’s last comment was yes there are great men in the history of science and Einstein was certainly one of them but not in the sense that people usually mean when they use the term. It is this response that I will try to unpack and elucidate here.

When people describe Einstein as a great man of science what they usually mean is that if he hadn’t lived, see Matthew’s original question, we ‘wouldn’t have the theories of relativity’ or ‘physics would have been held back for decades or even longer’. Both of the expression in scare quote are ones that occur regularly following statements along the lines of if X hadn’t existed we wouldn’t have Y and both are expressions that I think should be banned from #histSTM. They should be banned because they are simply not true.

Let’s take a brief look at the three papers Einstein published in 1905 that made his initial reputation. The paper on quantum theory, for which he would eventually get his Nobel Prize, was, of course, in response to Planck’s work in this field and was a topic on which many would work in the first half of the twentieth century. The so-called black body problem, which sparked off the whole thing, was regarded as one of the most important unsolved problems in physics at the turn of the century. Brownian motion, the subject of the second paper, was another hot topic with various people producing mathematically formulations of it in the nineteenth century. In fact Marian Smoluchowski produced a solution very similar to Einstein’s independently, which was published in 1906. This just leaves Special Relativity. The problem solved here had been debated ever since it had been known that the Clerk Maxwell equations did not agree with Newtonian physics. We have both Lorentz and FitzGerald producing the alternative to the Newtonian Galilean transformations that lie at the heart of Einstein’s Special Relativity theory. The Michelson-Morley experiment also demanded a solution. Poincaré had almost reached that solution when Einstein pipped him at the post. The four dimensional space-time continuum now considered so central to the whole concept was delivered, not by Einstein, but by his one time teacher Minkowski. Minkowski’s formulation was, of course, also central for the General Theory of Relativity; the solution for the field equations of which were found independently by Einstein and Hilbert, although Hilbert clearly acknowledged Einstein’s priority.

Without going into a lot of detail it should be clear that Einstein is solving problems on which a number of other people are working and making important contributions. He is not pulling new physics out of a hat but solving problems over-determined by the field of physics itself.What about other ‘great men’? The two most obvious examples are also physicists, Galileo and Newton. I’ve already done a major demolition job on Galileo several years ago, in which I show that everything he worked on was being worked on parallel by other highly competent scholars that you can read here. And a more recent version here.

So what about Newton?As should be well known Leibnitz and Newton both developed calculus roughly contemporaneously, even more important, as I explained here, they were both building on foundations laid down by other leading seventeenth-century mathematicians. Newton was anticipated in his colour theory of white light by the Bohemian scholar Jan Marek Marci. As I’ve explained here and here Newton was only one of three people who developed a reflecting telescope in the 1660s. Robert Hooke anticipated and probably motivated Newton on the theory of universal gravity and Newton’s work on dynamics built on the work of many others beginning with Tartaglia and Benedetti in the sixteenth century. His first law of motion was from Isaac Beeckman via Descartes and the second from Christiaan Huygens from whose work he also derived the law of gravity. Once again we have a physicist working on problem of his time that were being worked actively on by other competent scholars.

I think this brief analysis that the work of these ‘great men’, Einstein, Galileo and Newton, was to a large extent over-determined that is dictated by the scientific evolution of their respective times and their finding solutions to those problems, solutions that others also found contemporaneously, does not qualify them as special, as ‘great men’.

Having said all of that I would be insane to deny that all three of these physicists are, with right, regarded as special, as great men, so what is the solution to this seeming paradox?

I think the answer lies not in the fact that they solved the problems that they solved but in the breadth and quality of their work. Each of them did not just solve one major problem but a whole series of them and their solutions were of a quality and depth unequalled by others also offering solutions. This can be illustrated by looking at Hooke and Newton on gravity. Hooke got there first and there are good grounds for believing that his work laid the foundations for Newton’s. However whereas Hooke’s contribution consist of a brief series of well founded speculations, Newton built with his *Principia* a vast mathematical edifice that went on to dominate physics for two hundred years. Put simply it is not the originality or uniqueness of their work but the quality and depth of it that makes these researchers great men.

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**Hello JSTOR my old friend**[1]

**I’ve come to search in you again**[2]

**Because a reference softly creeping**

**Left its seeds while I was reading**[3]

**And the paper that was gnawing at my brain**

**Still remains**[4]

**Within the stress of research**[5]

**Through restless links I searched alone**

**Papers from journals I do not own**

**Neath the halo from my desk-lamp**

**I turn my collar to the research lab**[6]

**When my eyes were stabbed by the pain**

**Of a sleepless night**

**As I tried to write**

**Through the stress of research**[7]

**And in the flickering light I saw**

**Ten thousand deadlines maybe more**[8]

**Within the stress of research **

**Post-doc said, a****h you do not know**

**Research like a cancer grows**

**Hear my words that I might teach you**

**Read my diss’ and it might reach you**

**But my sources like undergrads they failed**

**Adding to the stress of research**[9]

**Then the faculty bowed and prayed**

**To bureaucratic gods they made**

**And the REF flashed out its warnings**

**Low impact scores were alarming**[10]

**And the graphs and words from students**

**Were projected on the classroom walls and lecture halls**

**Folks breaking under the stress from research**[11]

Composed 31 August 2017

Extended 5 September 2017

[1] Clare @mcclare95

[2] JSTOR @JSTOR

[3] Thony Christie @rmathematicus

[4] Vivek Santayana @viveksantayana

[5] Thony Christie @rmathematicus

[6] Eric Keeton @w0wkeeton

[7] Vivek Santayana @viveksantayana

[8] Vivek Santayana @viveksantayana

[9] Eric Keeton @w0wkeeton

[10] Vivek Santayana @viveksantayana

[11] Eric Keeton @w0wkeeton

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The major scientific theme of the day has been, of course, Monday’s total eclipse over America. In the lead up we have seen a lot on the Internet about eclipse maps. Eclipse maps are maps that show/predict the shadow path of the eclipse usually differentiating between those areas experiencing a full eclipse and those only experiencing various degrees of partial eclipse. On 17 August the website Atlas Obscura had an article on eclipse maps with the title *How Edmond Halley Kicked Off the Golden Age of Eclipse Mapping *that featured Halley’s 1715 eclipse map. Now this title contains a serious history of astronomy error and Atlas Obscura were unfortunately not the only ones to make it in the lead up to Monday’s great solar event.

Edmond Halley did not kick off the Golden Age of Eclipse Mapping, the seventeenth-century mathematician and astronomer Erhard Weigel (1625–1699) (who you can read about here) did with an eclipse map published in 1654 sixty-one years before Halley’s effort.

Halley wasn’t even second in the eclipse map stakes as Weigel’s student Johann Christopher Sturm (1635–1703) (who you can read about at the Weigel link) published one 1676, thirty-nine years before Halley.

Both Weigel and Sturm were known to the Royal Society, of which Halley was both a member and for a time an employee, and Sturm was also a member, so there is a strong possibility that Halley knew of the efforts of his German colleagues and cannot even be regarded as an independent inventor.

If you want all of the dope on eclipse maps then I highly recommend the excellent Eclipse-Maps website, which can fill all of your eclipse map desires whatever they might be. It is the source of the three eclipse maps shown here.

Another eclipse related false claim is the one presented below:

Now Ibn al-Haytham (c.965–c.1040) is one of the most important figures in the history of optics and he put the pinhole camera effect to very good use in his optical researches but he can’t be said to have invented it. You don’t actually have to build a ‘camera’ to display the pinhole camera effect and there are plenty of images on the web of people projecting images of the eclipse onto some sort of background through a hole in a hat, through a colander, through the holes in a salt cracker etc., etc.

The earliest known description of the pinhole camera effect can be found in the so-called Chinese Mozi writings, which date from the fifth century BCE, so about one and a half thousand years before Ibn al-Haytham lived. A description of the pinhole camera effect can also be found in the writings of Aristotle (384–322). In his *Problems* Aristotle wrote:

*Why is it that an eclipse of the sun, if one looks at it through a sieve or through leaves, such as a plane-tree or other broadleaved tree, or if one joins the fingers of one hand over the fingers of the other, the rays are crescent-shaped where they reach the earth? Is it for the same reason as that when light shines through a rectangular peep-hole, it appears circular in the form of a cone?*

* *As you can see he’s even describing using the effect to view a solar eclipse. As a small bonus, the name camera obscura for the pinhole camera (the origin of the term camera) was coined by Johannes Kepler.

My third rant of the day leaves the direct field of history of science and moves into the sphere of science communication and philosophy of science. Also provoked by the eclipse several different versions of the following meme have been circulating in the Internet over the last few days. I don’t know who originated it but Neil deGrasse Tyson has been aggressively tweeting a shorter version.

Now I’m a one hundred per cent supporter of science and the scientific method (whatever that might be) and the results that they produce in their attempts to explain our world but I find the analogy drawn here simplistic, naive and anything but helpful. I will endeavour to explain my thoughts on the matter.

Put very simply people are making the mistake here of comparing apples with oranges. A solar eclipse and its scientific explanation are of a very different type to the science of evolution or vaccines and all the other things that denialists reject.

First of all there is a time dimension. Already in the second half of the first century BCE Babylonian astronomers were pretty good at explaining solar and lunar eclipses and could predict lunar ones accurately and at least predict when a solar eclipse could theoretically take place. This knowledge was acquired through many centuries of astronomical observation. So, we are talking about more than two thousand years for people to digest and accept the science behind solar eclipses. In contrast to this, the theory of evolution and the scientific explanation for vaccines are both products of the nineteenth century and less than two hundred years old, far less time for people to digest and accept.

The second factor and the more serious one is complexity. Once you accept that the sun, the moon and the earth are just three balls rotating through the heavens, something accepted in Europe around five hundred BCE, – whether your model is geocentric or heliocentric doesn’t make any real difference to the explanation – then the scientific explanation of an eclipse is, to put it mildly, trivial. In fact it can be easily demonstrated in any classroom using a powerful torch (that’s a flashlight for Americans), a basketball and a large inflatable terrestrial globe. I’ve even seen it demonstrated using a torch and three children as the sun, the moon and the earth. There is not an awful lot you have to understand.

If we now turn to evolution or vaccines we are in a wholly different ball game. The theory of evolution is a highly complex scientific theory based on a vast amount of scientific material. The same can be said of the science behind the theories of disease and the use of vaccines to combat some of them. These are not scientific results that can be lucidly explained by a simple classroom demonstration in a couple of minute.

A third factor is personal involvement. There is a certain distance between a human being and the object of astronomy. It is true that we are dependent on the sun for our existence but on the whole we don’t connect to celestial objects on a very personal level. Things are very, very different with both the theory of evolution and vaccines. The theory of evolution says very directly where we as a species come from and that people have difficulty getting their heads around the fact that we are, over a long period of time, descended from some sort of proto-ape-like creature, in fact from the very same proto-ape-like creature as chimpanzees and gorillas shouldn’t come as a surprise. Remember that infamous Victorian quip, “You might think that your grandfather was an ape, sir but your grandmother!” It’s very easy to mock but it’s a hell of a long stretch to convince people to believe the theory of descent. Add to this the complexity of the actually mechanisms of evolution and that you are going to have problems convincing people to accept them shouldn’t surprise anybody.

All the above can be repeated for the theory of disease and the explanations for the function of vaccines; it’s all very, very complex and difficult to swallow for many people. Add to this the fact that vaccine damage is a reality. Before anybody tries to teach me how to suck eggs, I am well aware of the fact that the risk of any given child suffering vaccine damage is by several factors lower than the risk of death or serious brain damage, from say measles, for a child in a non-vaccinated population. But this statement has two problems, firstly ‘my child’ could be damaged by the vaccine, people are emotional, and secondly people don’t understand statistics. Any scientific explanation that involves statistics is likely to set the recipient in a state of panic.

As science communication, or spreading the science gospel, I find the meme above underwhelming to say the least. To say in an arrogant, sneering tone that if you accept the scientific explanation for, trivial phenomenon, A then you have to accept the scientific explanation for, anything but trivial, B is in my opinion anything but helpful and is more likely to antagonise than convince.

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