Category Archives: History of Mathematics

Christmas Trilogy 2017 Part 3: Kepler’s big book

Johannes Kepler was incredibly prolific, he published over eighty books and booklets over a very wide range of scientific and mathematical topics during his life. As far as he was concerned his magnum opus was his Ioannis Keppleri Harmonices mundi libri V (The Five Books of Johannes Kepler’s The Harmony of the World) published in 1619 some twenty years after he first conceived it. Today in popular #histsci it is almost always only mentioned for the fact that it contains the third of his laws of planetary motion, the harmonic law. However it contains much, much more of interest and in what follows I will attempt to give a brief sketch of what is in fact an extraordinary book.

kepler001

A brief glace at the description of the ‘five books’ thoughtfully provided by the author on the title page (1) would seem to present a mixed bag of topics apparently in some way connected by the word or concept harmonic. In order to understand what we are being presented with we have to go back to 1596 and Kepler’s first book Mysterium Cosmographicum (The Cosmographic Mystery). In this slim volume Kepler presents his answer to the question, why are there only six planets? His, to our eyes, surprising answer is that the spaces between the planets are defined by the regular so-called Platonic solids and as the are, and can only be, five of these there can only be six planets.

Using the data from the greatest and least distances between the planets in the Copernican system, Kepler’s theory produces an unexpectedly accurate fit. However the fit is not actually accurate enough and in 1598 Kepler began working on a subsidiary hypothesis to explain the inaccuracies. Unfortunately, the book that he had planned to bring out in 1599 got somewhat delayed by his other activities and obligations and didn’t appear until 1619 in the form of the Harmonice mundi.

The hypothesis that Kepler presents us with is a complex mix of ideas taken from Pythagoras, Plato, Euclid, Proclus and Ptolemaeus centred round the Pythagorean concept of the harmony of the spheres. Put very simply the theory developed by the Pythagoreans was that the seven planets (we are talking geocentric cosmology here) in their orbits form a musical scale than can, in some versions of the theory, only be heard by the enlightened members of the Pythagorean cult. This theory was developed out of the discovery that consonances (harmonious sounds) in music can be expressed in the ratio of simple whole numbers to each other (the octave for example is 1:2) and the Pythagorean belief that the integers are the building block of the cosmos.

This Pythagorean concept winds its way through European intellectual history, Ptolemaeus wrote a book on the subject, his Harmonice and it is the reason why music was one of the four disciplines of the mathematical quadrivium along with arithmetic, geometry and astronomy. Tycho Brahe designed his Uraniburg so that all the architectonic dimensions from the main walls to the window frames were in Pythagorean harmonic proportion to one another.

Uraniborg_main_building

Tycho Brahe’s Uraniborg Blaeus Atlas Maior 1663 Source: Wikimedia Commons

It is also the reason why Isaac Newton decided that there should be seven colours in the rainbow, to match the seven notes of the musical scale. David Gregory tells us that Newton thought that gravity was the strings upon which the harmony of the spheres was played.

In his Harmony Kepler develops a whole new theory of harmony in order to rescue his geometrical vision of the cosmos. Unlike the Pythagoreans and Ptolemaeus who saw consonance as expressed by arithmetical ratios Kepler opted for a geometrical theory of consonance. He argued that consonances could only be constructed by ratios between the number of sides of regular polygons that can be constructed with a ruler and compass. The explication of this takes up the whole of the first book. I’m not going to go into details but interestingly, as part of his rejection of the number seven in his harmonic scheme Kepler goes to great lengths to show that the heptagon construction given by Dürer in his Underweysung der Messung mit dem Zirckel und Richtscheyt is only an approximation and not an exact construction. This shows that Dürer’s book was still being read nearly a hundred years after it was originally published.

kepler002

In book two Kepler takes up Proclus’ theory that Euclid’s Elements builds systematically towards the construction of the five regular or Platonic solids, which are, in Plato’s philosophy, the elemental building blocks of the cosmos. Along the way in his investigation of the regular and semi-regular polyhedra Kepler delivers the first systematic study of the thirteen semi-regular Archimedean solids as well as discovering the first two star polyhedra. These important mathematical advances don’t seem to have interested Kepler, who is too involved in his revolutionary harmonic theory to notice. In the first two books Kepler displays an encyclopaedic knowledge of the mathematical literature.

kepler003

The third book is devoted to music theory proper and is Kepler’s contribution to a debate that was raging under music theorist, including Galileo’s father Vincenzo Galilei, about the intervals on the musical scale at the beginning of the seventeenth century. Galilei supported the so-called traditional Pythagorean intonation, whereas Kepler sided with Gioseffo Zarlino who favoured the ‘modern’ just intonation. Although of course Kepler justification for his stance where based on his geometrical arguments. Another later participant in this debate was Marin Mersenne.

kepler004

In the fourth book Kepler extends his new theory of harmony to, amongst other things, his astrology and his theory of the astrological aspects. Astrological aspects are when two or more planets are positioned on the zodiac or ecliptic at a significant angle to each other, for example 60° or 90°. In his Harmonice, Ptolemaeus, who in the Renaissance was regarded as the prime astrological authority, had already drawn a connection between musical theory and the astrological aspects; here Kepler replaces Ptolemaeus’ theory with his own, which sees the aspects are being derived directly from geometrical constructions. Interestingly Kepler, who had written and published quite extensively on astrology, rejected nearly the whole of traditional Greek astrology as humbug keeping only his theory of the astrological aspects as the only valid form of astrology. Kepler’s theory extended the number of influential aspects from the traditional five to twelve.

The fifth book brings all of the preceding material together in Kepler’s astronomical/cosmological harmonic theory. Kepler examines all of the mathematical aspects of the planetary orbits looking for ratios that fit with his definitions of the musical intervals. He finally has success with the angular velocities of the planets in their orbits at perihelion and aphelion. He then examines the relationships between the tones thus generated by the different plants, constructing musical scales in the process. What he in missing in all of this is a grand unifying concept and this lacuna if filled by his harmonic law, his third law of planetary motion, P12/P22=R13/R23.

kepler005

There is an appendix, which contains Kepler’s criticisms of part of Ptolemaeus’ Harmonice and Robert Fludd’s harmony theories. I blogged about the latter and the dispute that it triggered in an earlier post

With his book Kepler, who was a devoted Christian, was convinced that he had revealed the construction plan of his geometrical God’s cosmos. His grandiose theory became obsolete within less than fifty years of its publication, ironically pushed into obscurity by intellectual forces largely set into motion by Kepler in his Astronomia nova, his Epitome astronomiae Copernicanae and the Rudolphine Tables. All that has survived of his great project are his mathematical innovations in the first two books and the famous harmonic law. However if readers are prepared to put aside their modern perceptions and prejudices they can follow one of the great Renaissance minds on a fascinating intellectual journey into his vision of the cosmos.

(1) All of the illustration from the Harmonice mundi in this post are taken from the English translation The Harmy of the World by Johannes Kepler, Translated into English with an Introduction and Notes by E.J. Aston, A.M. Duncan and J.V. Field, American Philosophical Society, 1997

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Filed under Early Scientific Publishing, History of Astrology, History of Astronomy, History of Mathematics, History of science, Renaissance Science, Uncategorized

Christmas Trilogy 2017 Part 1: Isaac the Imperator

Isaac Newton came from a fairly humble although not poor background. His father was a yeoman farmer in Lincolnshire, who unfortunately died before he was born. A yeoman farmer owned his own land and in fact the Newton’s were the occupants of the manor house of Woolsthorpe-by-Colsterworth.

Woolsthorpe Manor, Woolsthorpe-by-Colsterworth, Lincolnshire, England. This house was the birthplace and the family home of Isaac Newton.
Source: Wikimedia Commons

Destined to become a farmer until he displayed little aptitude for life on the land, his mother was persuaded by the local grammar school master to let him complete his education and he was duly dispatched off to Cambridge University in 1661. Although anything but poor, when Newton inherited the family estates they generated an income of £600 per annum, at a time when the Astronomer Royal received an income of £100 per annum, his mother enrolled him at Cambridge as a subsizar, that is a student who earned his tuition by working as a servant. I personally think this reflects the family’s puritan background rather than any meanness on the mother’s part.

In 1664 Newton received a scholarship at Trinity and in 1667 he became a fellow of the college. In 1669 he was appointed Lucasian professor of mathematics. Cambridge was in those days a small market town and a bit of a backwater. The university did not enjoy a good reputation and the Lucasian professorship even less of one. Newton lived in chambers in Trinity College and it was certainly anything but a life of luxury.

Trinity College Great Court
Source: Wikimedia Commons

There is an amusing anecdote about David Hilbert writing to the authorities of Trinity at the beginning of the twentieth century to complain about the fact that Godfrey Hardy, whom he regarded as one of the greatest living mathematicians, was living in what he regarded as a squalid room without running water or adequate heating. What Hilbert didn’t realise was that Hardy would never give up this room because it was the one that Newton had inhabited.

Newton remained an obscure and withdrawn Cambridge don until he presented the Royal Society with his reflecting telescope and published his first paper on optics in 1672. Although it established his reputation, Newton was anything but happy about the negative reactions to his work and withdrew even further into his shell. He only re-emerged in 1687 and then with a real bombshell his Philosophiæ Naturalis Principia Mathematica, which effectively established him overnight as Europe’s leading natural philosopher, even if several of his major competitors rejected his gravitational hypothesis of action at a distance.

Having gained fame as a natural philosopher Newton, seemingly having tired of the provinces, began to crave more worldly recognition and started to petition his friends to help him find some sort of appropriate position in London. His lobbying efforts were rewarded in 1696 when his friend and ex-student, Charles Montagu, 1st Earl of Halifax, had him appointed to the political sinecure, Warden of the Mint.

Newton was no longer a mere university professor but occupant of one of the most important political sinecures in London. He was also a close friend of Charles Montagu one of the most influential political figures in England. By the time Montagu fell from grace Newton was so well established that it had little effect on his own standing. Although Montagu’s political opponents tried to bribe him to give up his, now, Mastership of the Mint he remained steadfast and his fame was such that there was nothing they could do to remove him from office. They wanted to give the post to one of their own. Newton ruled the Mint with an iron hand like a despot and it was not only here that the humble Lincolnshire farm lad had given way to man of a completely different nature.

As a scholar, Newton held court in the fashionable London coffee houses, surrounded by his acolytes, for whom the term Newtonians was originally minted, handing out unpublished manuscripts to the favoured few for their perusal and edification. Here he was king of the roost and all of London’s intellectual society knew it.

He became President of the Royal Society in 1703 and here with time his new personality came to the fore. When he became president the society had for many years been served by absentee presidents, office holders in name only, and the power in the society lay not with the president but with the secretary. When Newton was elected president, Hans Sloane was secretary and had already been so for ten years and he was not about to give up his power to Newton. There then followed a power struggle, mostly behind closed doors, until Newton succeeded in gaining power in about 1610 1710, Sloane, defeated resigned from office in 1613 1713 but got his revenge by being elected president on Newton’s death. Now Newton let himself be almost literally enthroned as ruler of the Royal Society.

Isaac Newton’s portrait as Royal Society President Charles Jervas 1717
Source: Royal Society

The president of the society sat at table on a raised platform and on 20 January 1711 the following Order of the Council was made and read to the members at the next meeting.

That no Body Sit at the Table but the President at the head and the two Secretaries towards the lower end one on the one Side and the other Except Some very Honoured Stranger, at the discretion of the President.

When the society was first given its royal charter in 1660, although Charles II gave them no money he did give them an old royal mace as a symbol of their royal status. Newton established the custom that the mace was only displayed on the table when the president was in the chair. When Sloane became president his first act was to decree that the mace was to be displayed at all meetings, whether the president was present or not. Newton ruled over the meetings with the same iron hand with which he ruled over the Mint. Meeting were conducted solemnly with no chit chat or other disturbances as William Stukeley put it:

Indeed his presence created a natural awe in the assembly; they appear’d truly as a venerable consessus Naturae Consliariorum without any levity or indecorum.

Perhaps Newton’s view of himself in his London years in best reflected in his private habitat. Having lived the life of a bachelor scholar in college chambers for twenty odd years he now obtained a town house in London. He installed his niece Catherine Barton, who became a famous society beauty, as his housekeeper and lived the life of a London gentleman, albeit a fairly quiet one. However his personal furnishings seem to me to speak volumes about how he now viewed himself. When he died an inventory of his personal possessions was made for the purpose of valuation, as part of his testament. On the whole his household goods were ordinary enough with one notable exception. He possessed crimson draperies, a crimson mohair bed with crimson curtains, crimson hangings, a crimson settee. Crimson was the only colour mentioned in the inventory. He lived in an atmosphere of crimson. Crimson is of course the colour of emperors, of kings, of potentates and of cardinals. Did the good Isaac see himself as an imperator in his later life?

 

All the quotes in this post are taken from Richard S, Westfall’s excellent Newton biography Never at Rest.

 

 

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Filed under History of Astronomy, History of Mathematics, History of Optics, History of Physics, History of science, Newton

Men of Mathematics

This is something that I wrote this morning as a response on the History of Astronomy mailing list; having written it I have decided to cross post it here.

John Briggs is the second person in two days, who has recommended Eric Temple Bell’s “Men of Mathematics”. I can’t remember who the first one was, as I only registered it in passing, and it might not even have been on this particular mailing list. Immediately after John Briggs recommended it Rudi Lindner endorsed that recommendation. This series of recommendations has led me to say something about the role that book played in my own life and my view of it now.

“Men of Mathematics” was the first book on the history of science and/or mathematics that I ever read. I was deeply passionate fan of maths at school and my father gave me Bell’s book to read when I was sixteen years old. My other great passion was history and I had been reading history books since I taught myself to read at the age of three. Here was a book that magically combined my two great passions. I devoured it. Bell has a fluid narrative style and the book is easy to read and very stimulating.

Bell showed me that the calculus, that I had recently fallen in love with, had been invented/discovered (choose the verb that best fits your philosophy of maths), something I had never even considered before. Not only that but it was done independently by two of the greatest names in the history of science, Newton and Leibniz, and that this led to one of the most embittered priority and plagiarism disputes in intellectual history. He introduced me to George Boole, whom I had never heard of before and whose work and its reception in the 19th century I would seriously study many years later in a long-year research project into the history of formal or mathematical logic, my apprenticeship as a historian of science.

Bell’s tome ignited a burning passion for the history of mathematics in my soul, which rapidly developed into a passion for the whole of the history of science; a passion that is still burning brightly fifty years later. So would I join the chorus of those warmly recommending “Men of Mathematics”? No, actually I wouldn’t.

Why, if as I say Bell’s book played such a decisive role in my own development as a historian of mathematics/science, do I reject it now? Bell’s florid narrative writing style is very seductive but it is unfortunately also very misleading. Bell is always more than prepared to sacrifice truth and historical accuracy for a good story. The result is that his potted biographies are hagiographic, mythologizing and historically inaccurate, often to a painful degree. I spent a lot of time and effort unlearning a lot of what I had learnt from Bell. His is exactly the type of sloppy historiography against which I have taken up my crusade on my blog and in my public lectures in my later life. Sorry but, although it inspired me in my youth, I think Bell’s book should be laid to rest and not recommended to new generations.

 

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Filed under Book Reviews, History of Logic, History of Mathematics, History of science, Myths of Science

Can we please stop (mis)quoting Albert on Emmy, it’s demeaning?

Emmy Noether, whom I’ve blogged about a couple of times in the past, is without any doubt one of the greats in the history of mathematics, as is well documented by the testimonials written by some of the greatest contemporary mathematicians and physicists and collected in Auguste Dick’s slim but well research biography, Emmy Noether: 1882–1935.

Emmy Noether c. 1930
Source:Wikimedia Commons

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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Filed under History of Mathematics, Ladies of Science

A Lady Logician

Today George Boole is regarded as one of the founders of the computer age that now dominates our culture.

George Boole
Source: Wikimedia Commons

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.

Charles Sanders Peirce
Source: Wikimedia Commons

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.

James Joseph Sylvester
Source: Wikimedia Commons

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

George Elias Muller
Source: Wikimedia Commons

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

Hermannvon Helmholtz in 1848
Source: Wikimedia Commons

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.

Portrait of Arthur Konig from Pokorny, J.
Source: Wikimedia Commons

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.

Emile Berliner
Source: Wikimedia Commons

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.

Edward Bradford Titchener
Source: Wikimedia Commons

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.

Christine Ladd-Franklin
(1847–1930)
Source: Wikimedia Commons

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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Filed under History of Logic, History of Mathematics, History of science, Ladies of Science, Uncategorized

The history of mathematics is not that simplistic.

The Conversation recently posted an article with the title, Five ways ancient India changed the world – with maths, which to be honest left much to be desired as a piece of mathematical history. First off, if you are going to write about #histSTM then a piece of good advice is avoid BuzzFeed style lists, history should never be presented as a collection of bullet points; such an approach is bound to produce dubious and inaccurate claims and statements, as in this case.

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 14th to 16th centuries and did some quite remarkable work on infinite series, anticipating work that was first done in Europe in the 17th 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 x2 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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Hyping the history of mathematics

A while back the Internet was full of reports about a sensational discovery in the history of mathematics. Two researchers had apparently proved that a well know Babylonian cuneiform clay tablet (Plimpton 322), which contains a list of Pythagorean triples, is in fact a proof that the Babylonians had developed trigonometry one thousand years before the Greeks and it was even a superior and more accurate system than that of the Greeks. My first reaction was that the reports contained considerably more hype than substance, a reaction that was largely confirmed by an excellent blog post on the topic by Evelyn Lamb.

Plimpton 322, Babylonian tablet listing pythagorean triples
Source: Wikimedia Commons

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.

A leaf from the Bakhshali Manuscript, showing off Indian mathematical genius. A zero symbol has been highlighted in the image.
Courtesy of the Bodleian Library

 

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

Bodleian Library Carbon dating finds Bakhshali manuscript contains oldest recorded origins of the symbol ‘zero’

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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Filed under History of Mathematics, History of science, Myths of Science