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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.

9 Responses to “Apparently Gauss got in this bar fight with Hilbert…”

  1. Avatar George Jelliss

    Eric Temple Bell’s book “Men of Mathematics” was an inspiration to me. However, I have subsequently read criticisms of his work, claiming that he overdramatised e.g. the stories of Galois (who died after a duel) and Cantor (who spent time in mental homes, and was prevented advancement by rivals). But some of this is surely a matter of judgment.

  2. Avatar George Jelliss

    As a devotee of knight’s tours, it annoys me when I read in many sources that Euler constructed the first magic knight’s tour. The diagram given is invariably the tour found by William Beverley in 1848. The trouble is that the authors, have not looked at the original sources. Once someone authoritative says it there is no eradicating the error!

  3. Avatar Thony C.

    Your example of the failing maths pupil is interesting in the context of your post because the story that Einstein was bad at or failed in maths is one of the biggest myths in the history of science. Einstein was always a straight A maths student.

    George Jellis: Bell’s book is possibly some of the worst history of maths that has ever been written but having said that, I read it as a teenager and it inspired me to become a maths historian. It’s a strange world.

  4. Avatar Peter Rowlett

    Thony: I chose Einstein because it’s such a well known example. I didn’t want to choose something more obscure – I considered someone who better remembers group theory after being told it was completely invented in one night before the dual by Galois – I wanted to be totally clear and simple to make the point. And that’s what I meant “please don’t get hung up on Einstein”.

    Both: I have heard several people say they were inspired to this life by Bell and I’ve also heard, as Thony says, how terrible the accuracy is. (I haven’t read Bell though I wouldn’t be surprised if some of his stories littered my undergraduate lectures, and therefore lurk in the back of my mind.) I think Bell’s influence inspiring young mathematicians is an interesting example – would the world be better off if MoM hadn’t been published?

  5. Avatar Thony C.

    I think the question should not be ‘Bell or not Bell’ but is it possible to achieve the inspiration generated by Bell’s book and be historically accurate at the same time.

    On your choice of Einstein in the last couple of months I have seen this old chestnut presented as the truth several times in the Internet and twice on TV so I’m not so sure that everybody is aware that it’s a myth ;)

  6. Avatar Will Daniels

    I think your problem is with Principle A. You may well appreciate and value perfection, but to insist upon it for yourself before each and every endeavour is surely as crippling as it is discouraging to get things wrong.

    Yes, Principle A and Principle B cannot hold for one person at the same time, so what is needed to know which to apply, and when, is a goal context. I think popularisation is clearly the better goal for a non-professional historian because that is something the professional is less likely to be able to achieve.

    Existing authoritative sources on the other hand should stick rigidly to Principle A. This way the two work together perfectly fine, you cannot be all things to all men and neither can the professional historian. There is a valuable purpose for both, and one tends to lead to the other.

  7. Avatar Anonymous

    I think the relevance of historical accuracy comes down to intent. Are you writing a history text book? Are you telling the stories as context for engaging students in maths? Is it background for explaining a particular principle?

    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.

  8. Avatar Anonymous

    An uneducated medical opinion “Sire, I cannot get the smallpox because I have had the cowpox” The uneducated aviator, the 1st balloonist “Clouds are made of smoke…” Me: Cantor was right when he thought he was wrong. Diagonal proof fallacious. Try it in binary, fractions in ascending order. the diagonal is .1 recurring, it is in the set, the last member. To put set 121 with whole numbers put mirror on decimal point.@davidfcox


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