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Why do we enjoy maths history misconceptions?

I don’t think I have come to a conclusion from my previous blog post about historical accuracy and popularisation, though there were some interesting points in the comments (relating less to my comments about the ‘errors which may be gently corrected’ and more to the ‘demands of the narrative’).

George Jelliss and Thony C. both read the famously inaccurate Men of Mathematics by E.T. Bell in their youth and were inspired to mathematical lives as a result.

Will Daniels suggests I should hold different standards for different people, so those writing historical research are held to a higher level of accuracy than those writing for a popular audience. I’m not sure this feels right. Thony asks a really interesting question:

is it possible to achieve the inspiration generated by Bell’s book and be historically accurate at the same time?

I think this is at the heart of the matter. If it is possible to inspire through popularisation while remaining completely accurate then I can safely hold everyone to this high standard. However, if inspiration requires a little showmanship, if telling a good tale means not getting lost in minor distractions and sub-clauses, then we have our double standard.

This brings me to a final, anonymous comment that includes the following statement:

I am a great believer in the wisdom of stories, regardless of their provenance. If some stories persist despite being disproven, there must be a reason.

My first reaction on reading this is that it is preposterous. If you start presenting stories you know to be disproven you are in the realm of historical fiction. Historical fiction is fine, but these are now just stories and have no place being presented as real accounts of historical mathematics and mathematicians. Then today I was struck by something relevant.

I was listening to Paul Dirac and the religion of mathematical beauty on the Royal Society Library podcast while wrangling with the washing machine. This recording, of a talk given in March 2011 by Graham Farmelo, covers the life of Paul Dirac. Farmelo talks about how Paul Dirac is considered to be the theoreticians’ theoretical physicist, yet he had a very practical schooling and took a practically-focused engineering degree. Farmelo says (15:40):

Let’s get one thing right, he was a very practically-minded person. Completely different from the image that he has among most theoretical physicists.

This is as so often the case; the established fact isn’t just slightly wrong but completely wrong. This is the case in the story that Einstein did poorly at school, a misconception that Thony C. tells me is not as well known as I thought it was when I used it as an example in my previous post.

Do we really want to believe Dirac is a theoretician with no practical sense, that Einstein was a terrible student made good? Is there really some “wisdom” in these stories that causes them to “persist despite being disproven”?

Rather than necessarily being wise, I think we are drawn to certain types of story. The Dirac perception reinforces the view of a flawed genius; a theoretical physicist with no sense of the real world. The Einstein story perhaps speaks to a desire for the plucky underdog to win out in the end.

Aren’t these classic Hollywood ideas? Do other common misconceptions fit into the Hollywood-style? (Galois’ heroic struggle against the odds to invent Galois theory in a single night before the dual springs to mind. What others?) Do, in fact, stories that deviate from historical record and persist deviate when the story fails to fit a certain sort of narrative?

Perhaps more importantly, are there correct historical stories which fit a classic Hollywood narrative? (I’m thinking, for example, of George Green teaching himself advanced mathematics “in the hours stolen from [his] sleep”.) Perhaps stories of this type are the key to achieving Bell-like inspiration while maintaining historical accuracy.

Apparently Gauss got in this bar fight with Hilbert…

The title is silly, of course, but is meant to refer to a problem with historical accuracy. I have had this blog post in draft for a long time and I am struggling to finish it. I would like to talk about an area in which I appear to have cognitive dissonance. I’m intending to ask a bunch of questions to which I do not have answers. I hope you will help me come to some.

I firmly believe that what is published on the history of mathematics should be correct. The history of mathematics is full of misconceptions and apocryphal stories and to propagate these is a terrible sin. Call this Principle A.

Now, from time to time I see someone who has had a good go at producing something on a historical topic which is mostly correct but repeats a few common errors. This work (or person) is then picked apart by those in the know, or the piece of work is roundly dismissed as entirely without merit. I’ve heard this in the case of very popular books – “it’s written well and tells a good story but it has this fact wrong so nobody should ever read it”.

I’m not talking about someone who copies wholesale from some website nobody has ever heard of without checking any of the facts. Nor am I talking about a serious academic history of mathematics work. Nor silly errors. I’m talking about cases where an enthusiastic amateur has put in the effort; they’ve read fourteen sources for a particular piece of information and when they publish it they are picked up for not having read the fifteenth – a recent research paper in a journal they can’t access – which debunks the fact.

I believe popularisation is good. Mathematicians would do well to know more of the history of their subject. I value the use of history in teaching as a way to engage students with the curriculum. I also believe history can be useful in outreach, the use of engaging stories to bring in more people to study of mathematics or its history. When I see someone having an honest attempt at telling some historical story, and they have done a reasonable level of research, I think it is bad to tear them apart or dismiss their effort. Instead we should encourage their keeness and perhaps gently steer them towards a better understanding (and they, in turn, should be pleased to learn). Sometimes this might mean you overlook a series of small errors to work, for now, on the major one. Pointing out everything that is wrong with a piece of work in minor detail can be very discouraging and, since popularisation and keeness are good, we hope to encourage this person not put them off from trying again. Call this Principle B.

You see the problem? Principle A tells me nothing should be produced with errors, but Principle B suggests work with minor errors should be taken in good faith. Both cannot hold. This is particularly a problem when I might be the person naively committing the sin (as I will be more often than the expert spotting the error). The fear of what might happen makes me feel very uncomfortable and hesitant to publish content on history.

There is another issue running along with this one. Perhaps the minor errors were not through ignorance but by choice, either due to restrictions of the format (word count or time available for a performance) or out of an attempt to keep the momentum of a story without getting sidetracked. This is like a piece of historical fiction where a character’s sister and cousin are amalgamated into one character because it would confuse the main thread to introduce a new minor character for some small interaction with the plot before they disappear. If the main story is basically being told correctly but a few peripheral details are being ignored or muddled to keep the momentum, is that a bad thing? We want an audience for our story, after all; is it possible that too much accuracy (or too many caveats) can make the story uninteresting? 

This puts me in mind of a piece of advice I was once given about writing popular mathematics. I was told that nobody should write a popular mathematics book unless they are a researcher in the topic of the book. I don’t agree with this at all. Sometimes the researchers are too close to the topic to explain it well, or to make it interesting, or perhaps there isn’t a talented writer researching a particular area but it should still be popularised. I wonder if people hold the same view – people should steer clear of history unless they are professional historians of mathematics? Won’t this lead to less history being told?

There are also cases where someone learns or remembers something, or builds confidence, as a result of a historical story. I can’t think of a better example right now but say for example I meet a twelve year old who was really struggling with mathematics when they were eight until a teacher told them that Einstein had failed mathematics in school and gone on to be a great physicist. A lot of ability in mathematics comes from perseverance which comes from confidence. Was the person who told the eight year old this story to boost their confidence wrong to do so? (There are surely cases where less decidedly wrong misconceptions apply to more nuanced situations but this will do as a placeholder; please don’t get too hung up on Einstein or my imagined twelve year old.)

I really don’t know the answer to these questions. I am asking them here in the hope that you might share your views. I really am interested to hear arguments either way.

Favourite popular mathematics books

I consider popular mathematics writing to be a good thing. I even tried a little myself and would be keen to try more. I am not, however, an expert in this genre. I certainly read popular maths and science books as a teenager and I remember fondly, along with a couple of physics books and biographies, the mathematical stories told in James Gleick’s Chaos, Ivars Peterson’s The Mathematical Tourist and Simon Singh’s Fermat’s Last Theorem. I’m not sure this is sufficient qualification to have a strong critical opinion. I have a copy of Alex Bellos’ Alex’s Adventures in Numberland that I was bought last birthday and, although it is on the top of my pile and I feel sure I will enjoy this when I get chance (perhaps someday I’ll spend a holiday not worrying about my PhD), I haven’t quite got around to reading it.

This week Guardian Books offered Ian Stewart’s top 10 popular mathematics books in which, the description promises, “the much-acclaimed author chooses the best guides to ‘the Cinderella science’ for general readers”. Why Cinderella you ask? Stewart means this in the sense at the start of the story, “undervalued, underestimated, and misunderstood”, and perhaps intends popular mathematics to take mathematics to the ball, saying:

Popular mathematics provides an entry route for non-specialists. It allows them to appreciate where mathematics came from, who created it, what it’s good for, and where it’s going, without getting tangled up in the technicalities. It’s like listening to music instead of composing it.

It will be no surprise, after the opening paragraph, if I admit that I neither own nor have I read any of Stewart’s choices. I’ve heard of several of them but by no means all. I was surprised by the inclusion of Newton’s Principia. In the back of my mind I have collected the ‘fact’ (citation needed) that Newton is a difficult read and I felt this made it a strange choice against the aim to bring “the best guides to ‘the Cinderella science’ for general readers” (though I’m aware the description will have been added later, possibly without Stewart’s knowledge). Stewart justifies its inclusion as “a great classic” saying that although this is “not popularisation in the strict sense”, this “slips in because it communicated to the world one of the very greatest ideas of all time: Nature has laws, and they can be expressed in the language of mathematics” and claims “no mathematical book has had more impact”.

On Twitter, Tony Mann confirmed my half-remembered notion that “Principia is hard, very hard. Even in English“. As to the claim of impact, Tony suggested Stewart should have chosen the Latin version as having more impact. Thony Christie agreed this is “a very hard book to read and comprehend“, though Christian Perfect suggested that he found the scans of Newton’s college notebooks which were recently made available online to be “quite readable“.

Reading what Stewart wrote about Newton’s Principia and its impact in the history of science, I wonder if the book was chosen more to tell the story in the article than out of a serious suggestion that it might be read. Christian Perfect makes this point more generally about the list over on my Google+ page:

I think he’s chosen 10 books about his favourite mathematical ideas rather than 10 books which most effectively communicate mathematical ideas to a member of the “populace”.

To include a classic, I wondered if something like Euler’s Elements of Algebra, which I had heard travels fairly well to a modern reader, might be a more appropriate choice. On my G+ page, Sarah Kavassalis suggested “one of Poincaré’s popular books instead though, for readability”.

I asked people for their thoughts on the list and what else they would include. It’s quite noticeable that several respondents report not having read many on the list (the same is true of the comments under the original article). Alex Bellos, on G+ expands on this:

I guess there are two types of “popular” – 1) something accessible for people who know no maths and 2) something fun for the math literate. I’d say Ian’s list is very much the latter. If a lay friend asked me for a maths book suggestion they might understand and enjoy, I would only recommend the first two on his list [Robert Kanigel’s The Man Who Knew Infinity and Douglas Hofstadter’s Gödel, Escher, Bach].

Given the popular medium and Stewart’s introduction to the article, in which he talks about popular mathematics as “an entry route for non-specialists”, it is strange to see the list being regarded in this way. There’s nothing wrong with a list of fun books for maths folks, with something to surprise us rather than just the obvious choices, but if that was what was intended then this probably should have said so. I worry about someone using this list to build a ‘must-read’ list and perhaps being put off popular mathematics as a result.

I also asked for your suggestions and these follow. It may not be fair but I have listed these in the order they were suggested. I’ve included descriptions, except where stated these are those given on Amazon UK.

Thank you to everyone who played along with this little game. We’ve got more than ten and I can’t vouch for which would suit “people who know no maths” or “the math literate”, but I’ve enjoyed looking through the suggestions. Further suggestions are, of course, welcome via the comments.

Alex Bellos’ Alex’s Adventures in Numberland (US title: Here’s Looking at Euclid)
Suggested by Vincent Knight and Singing Hedgehog on G+.

In this richly entertaining and accessible book, Alex Bellos explodes the myth that maths is best left to the geeks. Covering subjects from adding to algebra, from set theory to statistics, and from logarithms to logical paradoxes, he explains how mathematical ideas underpin just about everything in our lives.

Edwin Abbott’s Flatland: A Romance of Many Dimensions
Suggested by Sarah Kavassalis (“very different approach to popular mathematics”) and Singing Hedgehog (“strange since Ian Stewart wrote the follow up Flatterland!”) on G+.

How would a creature limited to two dimensions be able to grasp the possibility of a third? Edwin A. Abbott’s droll and delightful ‘romance of many dimensions’ explores this conundrum in the experiences of his protagonist, A Square, whose linear world is invaded by an emissary Sphere bringing the gospel of the third dimension on the eve of the new millennium. Part geometry lesson, part social satire, this classic work of science fiction brilliantly succeeds in enlarging all readers’ imaginations beyond the limits of our ‘respective dimensional prejudices’.

Ian Stewart’s Cabinet of Mathematical Curiosities and Hoard of Mathematical Treasures
Singing Hedgehog, on G+, recognises that Stewart can’t choose his own books for the list but would add Cabinet and Hoard, which he calls “fabulous repositories of interesting stuff”.

A book of mathematical oddities: games, puzzles, facts, numbers and delightful mathematical nibbles for the curious and adventurous mind.

A new trove of entrancing numbers and delightful mathematical nibbles for adventurous mind.

Clifford Pickover’s The Math Book
Suggested by Singing Hedgehog on G+.

Maths infinite mysteries and beauty unfold in this fascinating book. Beginning millions of years ago with ancient ‘ant odometers’ and moving through time to our modern-day quest for new dimensions, it covers 250 milestones in mathematical history.

Barry Mazur’s Imagining Numbers: (Particularly the Square Root of Minus Fifteen)
Suggested by Singing Hedgehog on G+.

The book shows how the art of mathematical imagining is not as mysterious as it seems. Drawing on a variety of artistic resources the author reveals how anyone can begin to visualize the enigmatic ‘imaginary numbers’ that first baffled mathematicians in the 16th century.

Florian Cajori’s A History of Mathematical Notations
Suggested by Singing Hedgehog on G+, who says this “covers the history of mathematics through the methods of writing it”.

Described even today as “unsurpassed,” this history of mathematical notation stretching back to the Babylonians and Egyptians is one of the most comprehensive written. In two impressive volumes–first published in 1928-9–distinguished mathematician Florian Cajori shows the origin, evolution, and dissemination of each symbol and the competition it faced in its rise to popularity or fall into obscurity.

Richard Elwes’ Maths 1001: Absolutely Everything That Matters in Mathematics
Susan Turnbull insists this mustn’t be forgotten over on G+.

Maths 1001 provides clear and concise explanations of the most fascinating and fundamental mathematical concepts. Distilled into 1001 bite-sized mini-essays arranged thematically, this unique reference book moves steadily from the basics through to the most advanced of ideas, making it the ideal guide for novices and mathematics enthusiasts.

William Poundstone’s The Recursive Universe: Cosmic Complexity and the Limits of Scientific Knowledge
Suggested by John Read on G+.

In The Recursive Universe, William Poundstone uses Conway’s Life as a vehicle to explore complexity theory and modern physics. Poundstone demonstrates how simple rules can produce complex results when applied recursively and suspects our own universe was created in a similar manner. (Description source)

Ivan Moscovich’s Super-games
Suggested by John Read on G+, but of which I cannot find a description.

Benoit Mandelbrot’s The Fractal Geometry of Nature
Suggested by John Read on G+.

“…a blend of erudition (fascinating and sometimes obscure historical minutiae abound), popularization (mathematical rigor is relegated to appendices) and exposition (the reader need have little knowledge of the fields involved) …and the illustrations include many superb examples of computer graphics that are works of art in their own right.” Nature

John Allen Paulos’ Innumeracy: Mathematical Illiteracy and Its Consequences
Suggested by John Read on G+.

Why do even well-educated people often understand so little about maths – or take a perverse pride in not being a ‘numbers person’?
In his now-classic book Innumeracy, John Allen Paulos answers questions such as: Why is following the stock market exactly like flipping a coin? How big is a trillion? How fast does human hair grow in mph? Can you calculate the chances that a party includes two people who have the same birthday? Paulos shows us that by arming yourself with some simple maths, you don’t have to let numbers get the better of you.

Martin Gardner’s Mathematical Puzzles and Diversions
Suggested by John Read on G+ who says this is “the first I bought and the one I go back to most” but I can’t find a cover blurb description of this.

Marcus Du Sautoy’s The Music of the Primes: Why an unsolved problem in mathematics matters
Suggested by John Read on G+.

In this breathtaking book, mathematician Marcus du Sautoy tells the story of the eccentric and brilliant men who have struggled to solve one of the biggest mysteries in science. It is a story of strange journeys, last-minute escapes from death and the unquenchable thirst for knowledge. Above all, it is a moving and awe-inspiring evocation of the mathematician’s world and the beauties and mysteries it contains.

Ian Stewart’s Game Set and Math: Enigmas and Conundrums
John Read on G+ says “I’d also pick an Ian Stewart – probably Game, Set and Math”. Again, I can’t find a description.

William Cook’s In Pursuit of the Traveling Salesman: Mathematics at the Limits of Computation
Mitch Keller on Twitter notes that only one book on Stewart’s list focuses on a specific problem and suggests this as another.

What is the shortest possible route for a traveling salesman seeking to visit each city on a list exactly once and return to his city of origin? It sounds simple enough, yet the traveling salesman problem is one of the most intensely studied puzzles in applied mathematics–and it has defied solution to this day. In this book, William Cook takes readers on a mathematical excursion, picking up the salesman’s trail in the 1800s when Irish mathematician W. R. Hamilton first defined the problem, and venturing to the furthest limits of today’s state-of-the-art attempts to solve it.

On G+ Alex Bellos recommended the following three for an accessible list for “people who know no maths”, saying “the challenge when writing a maths book is to find a strong narrative – and these three books do it better than any others”.

Simon Singh’s Fermat’s Last Theorem: The story of a riddle that confounded the world’s greatest minds for 358 years
Recommended by Alex Bellos on G+.

The extraordinary story of the solving of a puzzle that has confounded mathematicians since the 17th century… A remarkable story of human endeavour and intellectual brilliance over three centuries, Fermat’s Last Theorem will fascinate both specialist and general readers.

Apostolos Doxiadis and Christos Papadimitriou’s Logicomix: An Epic Search for Truth
Recommended by Alex Bellos on G+.

This brilliantly illustrated tale of reason, insanity, love and truth recounts the story of Bertrand Russell’s life… An insightful and complexly layered narrative, Logicomix reveals both Russell’s inner struggle and the quest for the foundations of logic. Narration by an older, wiser Russell, as well as asides from the author himself, make sense of the story’s heady and powerful ideas. At its heart, Logicomix is a story about the conflict between pure reason and the persistent flaws of reality, a narrative populated by great and august thinkers, young lovers, ghosts and insanity.

Apostolos Doxiadis’ Uncle Petros and Goldbach’s Conjecture
Recommended by Alex Bellos on G+.

Uncle Petros and Goldbach’s Conjecture is an inspiring novel of intellectual adventure, proud genius, the exhilaration of pure mathematics – and the rivalry and antagonism which torment those who pursue impossible goals.

For a list of “something fun for the math literate”, Alex recommended the following three.

Petr Beckmann’s A History of Pi
Recommended by Alex Bellos on G+.

The history of pi, says the author, though a small part of the history of mathematics, is nevertheless a mirror of the history of man. Petr Beckmann holds up this mirror, giving the background of the times when pi made progress — and also when it did not, because science was being stifled by militarism or religious fanaticism.

Tobias Dantzig’s Number: The Language of Numbers
Recommended by Alex Bellos on G+.

A new edition of the classic introduction to mathematics, first published in 1930 and revised in the 1950s, explains the history and tenets of mathematics, including the relationship of mathematics to the other sciences and profiles of the luminaries whose research expanded the human concept of number.

Paul Hoffman’s The Man Who Loved Only Numbers: The Story of Paul Erdös and the Search for Mathematical Truth
Recommended by Alex Bellos on G+.

The biography of a mathematical genius. Paul Erdos was the most prolific pure mathematician in history and, arguably, the strangest too.

For this group, Alex also recommends “the complete works of Martin Gardner”.

James Gleick’s Chaos and The Information
Recommended by Alex Bellos on G+. Alex says these are between the two lists as they are “both utterly brilliant but might lose the casual reader in parts”.

Chaos: This book brings together different work in the new field of physics called the chaos theory, an extension of classical mechanics, in which simple and complex causes are seen to interact. Mathematics may only be able to solve simple linear equations which experiment has pushed nature into obeying in a limited way, but now that computers can map the whole plane of solutions of non-linear equations a new vision of nature is revealed. The implications are staggeringly universal in all areas of scientific work and philosophical thought.

The Information: We live in the information age. But every era of history has had its own information revolution: the invention of writing, the composition of dictionaries, the creation of the charts that made navigation possible, the discovery of the electronic signal, the cracking of the genetic code.
In The Information James Gleick tells the story of how human beings use, transmit and keep what they know. From African talking drums to Wikipedia, from Morse code to the ‘bit’, it is a fascinating account of the modern age’s defining idea and a brilliant exploration of how information has revolutionised our lives.