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ChatGPT and history of maths misconceptions

By Peter Rowlett. Posted March 5, 2024



You know how loads of things in maths are named for the wrong person? In 1996, a fun quiz appeared in The Mathematical Gazette based on history of maths misconceptions. It contained a series of questions where the obvious answer is not correct, such as “Who discovered Cramer’s rule?”, “Did Pascal discover the Pascal triangle?” and “Who first published Simpson’s rule?”

I was looking for a demo to show my students that generative AI programs are not producing accurate knowledge when I thought of this quiz. I put its questions to ChatGPT to see how it did. The point of the exercise is that these systems just parrot back words from their training data without any concept of truth, so if the training data is full of misconceptions, so too will be the responses. But these are misconceptions from the 1990s, so how much influence will they have on the responses?

You can see how ChatGPT did when I gave it the quiz in a short, free to read, open access paper in the British Journal for the History of Mathematics: Generative AI and accuracy in the history of mathematics.

Mathematics: a culture of historical inaccuracy?

By Peter Rowlett. Posted July 31, 2012

Earlier this year, back when I somehow managed to find time to write blog posts (sorry!), I wrote a couple of pieces on incorrect but oft-repeated stories in history of mathematics, basically describing some issues and expressing my frustration. These were Apparently Gauss got in this bar fight with Hilbert… and Why do we enjoy maths history misconceptions?

Today Thony Christie wrote on Twitter (as @rmathematicus) with a link to this post by Dennis Des Chene (aka “Scaliger”): On bad anecdotes and good fun. As Thony points out, this is an “excellent piece of maths history myth busting” and I am writing this quick note to commend you to read it.

Why do we enjoy maths history misconceptions?

By Peter Rowlett. Posted February 2, 2012

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…

By Peter Rowlett. Posted January 27, 2012

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.