I've been studying uses of quaternions to study various types of orbifolds. The important thing here though is that I came across an absolutely incredible result due to Vahlen in 1901, that apparently received no attention until Ahlfors revamped the idea in 1985 using more contemporary methods. The result (Vahlen matrices) is central to my research.

The problem with Vahlen is that he was about as much of a Nazi as a person could be. He was in the SA, the SS, accused brilliant Jews of plagiarizing Aryans, helped the Third Reich expunge Jews from the scientific community, etc. He even supported the Nazi party before its infamous rise to popularity.

My questions are:

  • What is the likelihood that Vahlen's results were neglected due to this political misfortune?
  • What are some similar instances, particularly in mathematics?
  • What is the etiquette of discussing a great result by a person who lead a morally reprehensible life?

Regarding the third question, Ahlfors simply writes about his math, which I think is the right way to go for a journal article. But what about a lecture or more general discourse among the community? I would feel weird if I were to give a lecture where I'm praising a guy who may have contributed to the extermination of an audience member's family. I have kind of a dark sense of humor, so I'm likely to mention the Naziism and poke fun at it ... maybe use it as an example for why you have to check everyone's results regardless of their other results. Personally I would find it funny to denote the Vahlen matrices by a swastika, but obviously this would be extremely inappropriate.

A more specific issue I'm having is that there is a lot of conflict about the notation for Vahlen's discovery, as other notation was developed for 84 years without it. It is an obstacle because we literally have different definitions for identical symbols all over the place. My feeling as that the clearest notation would be to use a $\mathcal{V}$ for the Vahlen matrices (which I have not seen in any papers thus far) and move on, but I wonder if there is a tendency in the academic community to avoid even that. I don't want even an iota of a suspicion of being sympathetic to this man's priorities.

An important note here is that Vahlen's matrices are in the domain of pure math. That means their development had nothing to do with any sort of experimentation, and are quite disjoint from these other aspects of Vahlen's life.

Follow-up: Without naming any names, some very established mathematicians I've been discussing my work with have unequivocally recommended omitting Vahlen's name from any constructions I use in my paper, specifically because of the political history. As in, don't call them "Vahlen matrices," call them "Ahlfors matrices," and things like that. Regardless of whether or not one agrees with that sentiment, to me it is concrete evidence of Vahlen's Naziism's effect on what credit he might receive for his mathematical work.

  • Comments are not for extended discussion; this conversation has been moved to chat.
    – ff524
    Commented Feb 10, 2017 at 4:44
  • Reminder that we can only move comments to chat once, so comments that are not requesting clarification or suggesting improvement will be lost. If you have an example of a "similar instance" that you would like to add, please add it as a bullet to the community wiki answer (note that this is a bit of an exceptional case; we do not generally take questions that ask for lists of things).
    – cag51
    Commented May 16, 2022 at 3:44
  • Could you call them Ahlfors matrices instead of Vahlen matrices?
    – Stef
    Commented Sep 5, 2022 at 10:04

5 Answers 5


I think there is no academic issue here.

The problem with Vahlen, as you may have guessed by the title, is that he was about as much of a Nazi as a person could be. He was in the SA, the SS, accused brilliant Jews of plagiarizing Aryans, helped the Third Reich expunge Jews from the scientific community, etc. He even supported the Nazi party before its infamous rise to popularity.

I looked into Theodor Vahlen, and I agree that he was a morally reprehensible individual. But this is irrelevant to a discussion of his work: he proved the theorems that he proved whether he was good, bad, or whatever. As is well known, the Unabomber is a published mathematician. His six papers have eight citations on MathSciNet, three of which came long after his capture, e.g. see here. (Though I do not know for sure that the author of this paper was aware that the T.J. Kaczynski whose work is cited is the Unabomber, it seems likely: the author is an American, and among Americans this name is well known to say the least.)

I was wondering what the likelihood is that his results were neglected due to this political misfortune. I was also wondering if people know of similar instances, particularly in mathematics. It seems like whenever I look up the politics of a respectable mathematician, they are either nonexistent, or equally respectable. I don't know if we tend to discredit people like this, or if people like this tend to not produce good results, or what.

In my opinion this is unlikely. There are very famous mathematicians who were, to lesser or greater extents, participaters in the Nazi movement. Perhaps Ludwig Bieberbach and Oswald Teichmuller are the two leading examples: these are household names in mathematics, and that they were intimately (and reprehensibly) involved in the Nazi movement is also very well known. (For that matter, I am not convinced that Vahlen's work is so little known. It is described in the wikipedia article linked to above, for instance.)

In my personal opinion, it is worth knowing when a mathematician (or other academic) has done reprehensible things in their personal life. Some years ago I compiled a list of "greatest mathematicians" on MathOverflow: I limited myself to choosing at most one mathematician born in a given year, but in some cases I included "honorable mentions." When I got to Teichmuller, I did not want to use the word "honorable" to describe him, so I wrote "dis/honorable mention." That was a personal decision (and by the way, I am partially of Jewish descent, so it is really not for me to forgive or forget such things). But I included him on the list anyway: that he was a terrible person does not influence his mathematics one way or another.

I'd like to just use a big-ol' script V for the Vahlen matrices (which I have not seen in any papers thus far) and move on,

There is no problem with this.

but I don't want even an iota of a suspicion of being sympathetic to this man's priorities.

We don't endorse the life of Teichmuller when we talk about Teichmuller space or Bieberbach when we talk about Bieberbach's Conjecture. We credit them with their work, as we must.

It would also be hilarious to just use a swastika, but that joke might not be funny to some people (understandably).

I am not laughing at all. Please do not do this.

Finally, you can read much more about mathematics in Nazi Germany from this book. Again, I think it is good to know these things.

  • 15
    Thanks for this. As far as using a swastika, of course I would not really do that. But I do want to emphasize my motivation for that idea, which is basically this: the last thing the Nazi's would want is for us to be laughing at them after their downfall. They would want their symbol, or anything, to continue to bring us down. As I said I have a dark sense of humor and personally it's gotten me through a lot of hardship, so please don't interpret that joke as a sign of disrespect.
    – j0equ1nn
    Commented Feb 9, 2017 at 23:25
  • 10
    @j0equ1nn Even if you indent to use the swastika in a humorous way, to make fun of it, it is still going to bring some people down. So, as stated in the answer, pleas do not do this. Some people will just not be laughing, others will be worse.
    – skymningen
    Commented Feb 10, 2017 at 16:08
  • 34
    @IWS In biomedical research, you get a moral dilemma that you don't get in mathematics: While a mathematician might have commited atrocities and advanced science, a physician may very well have commited atrocities in order to advance science, and if you take results from those atrocities and use them like any results at all, shaking your head sadly about the atrocities, you create a trade-off where people obsessed with advancing science may come to the conclusion that the advancement of science is well worth whatever personal retribution commiting atrocities will bring down on them.
    – sgf
    Commented Feb 11, 2017 at 19:54
  • 7
    @DanRomik I believe you misunderstand IWS's point. What he's trying to say, I believe, is not that fads in maths are engineered by social theorists, but that fads in maths are social theoretic entities and that therefore there is a social theoretic aspect to what happens in maths.
    – sgf
    Commented Feb 12, 2017 at 14:35
  • 5
    I generally agree with the answer, but not the very last part. Many spaces and conjectures are not referred to by the name of the person who introduced/worked on/conjectured something. Terminology is already established in some areas, but if one finds an interesting conjecture in the work of a reprehensible person, one might exercise creativity in naming the conjecture (of course, one still has to cite the originator). Commented Jan 2, 2018 at 0:47

I was wondering about the etiquette of discussing a result by such a person. Ahlfors simply writes about his math, which seems probably the right way to go. But it still feels weird if I were to give a lecture where I'm praising a guy who may have contributed to the extermination of an audience member's family.

You need to remind yourself why you are there giving a lecture to this audience member in the first place. What is the goal that your lecture is trying to achieve?

Is this a lecture about mathematics?


Is it a lecture about the history of Nazism?

If this is a lecture about mathematics, talk about mathematics. Think about it this way: if you invited a speaker to give an algebra seminar at your department and they instead went on a 10 minute digression about the history of automobiles and inserted icons of automobiles to stand for mathematical objects instead of Latin or Greek letters, would you think that was appropriate, or would you think that it detracted from the goal you had in mind when you invited them to give a talk, namely to learn about their work and mathematics related to it?

It's okay to tell little historical anecdotes about the people behind the mathematics in talks; we are humans, not robots. Nonetheless, I'm willing to bet that your hypothetical audience member whose family was exterminated by the Nazis is still there primarily to hear about interesting mathematics and not about historical anecdotes or to see cheap gimmicks ostensibly making fun of the Nazis, and ultimately doesn't care that much to have any of that stuff brought up. Your use of a swastika as a mathematical symbol may or may not offend him; will certainly not strike him as funny; and will probably just annoy him for being a distracting gimmick that undermines the primary goal of the talk.

To summarize, no one really cares what "feels weird" to you - that's your own private feeling; to the extent that that feeling is causing you to want to give an unfocused talk that goes on irrelevant tangents and annoys people, the correct thing to do is to ignore that feeling and instead give the best mathematical talk, which focuses -- as a mathematical talk should -- on the mathematics. (Hmm, and that's basically what Ahlfors is doing in his writing isn't it...?)

  • 9
    To summarize, no one really cares what "feels weird" to you — Best. Summary.. Ever. (+1)
    – Mad Jack
    Commented Feb 10, 2017 at 4:13
  • 7
    @MadJack Not really. The post mainly inquires about the history and current common practices, and their effect, regarding this type of information. My post was not to the effect of "I feel weird, does anyone care?"
    – j0equ1nn
    Commented Feb 10, 2017 at 21:11
  • 1
    @j0equ1nn you seem like a nice and quite interesting guy (I looked at your web page and academia.se history, some of which I feature in), but you take things a bit too literally for your own good. Lighten up, we are here to have fun and enlighten each other with interesting discussions and witty banter. You did mention feeling weird, and I was trying to be helpful by pointing out that in the context of the question, no one cares. Likewise MadJack was trying to be helpful to others by commenting on it in a humorous way that pokes a bit of fun at your expense. No one here wishes you any ill.
    – Dan Romik
    Commented Feb 10, 2017 at 21:49
  • 3
    @j0equ1nn ... The bottom line is that witty banter aside, you may notice that essentially everyone here actually took your question quite seriously and gave their best shot at addressing the concerns you brought to our attention. I hope you appreciate that.
    – Dan Romik
    Commented Feb 10, 2017 at 21:51
  • 1
    Find the integral of 2 Honda Saab squared delta Saab. Commented Jun 18, 2018 at 2:51

As @Rudiger says, we should tell the truth so far as we know it (and make an effort to know it!), and give credit where credit is due, regardless of the possible failings in the other parts of a person's life.

I disagree with @Rudiger's claim that mathematics has nothing to do with politics, at least insofar as all these things are mixed together and affect each other in the larger human enterprise. After all, Nazism affect mathematics profoundly, by affecting mathematicians profoundly (killing some, driving some to suicide, and driving others to the U.S. and other places far away from Germany).

So far as I know, in Vahlen's case, the lack of attention is not so much about his abhorrent politics but that people did not find his mathematical work terribly useful (whether or not you yourself do).

  • I appreciate this and respect your opinion. I'm coming at this as a new mathematical researcher in my first postdoc, encountering something like this for the first time, so I'm confused why anyone who would feel the need to insult me because of this post. Regarding usefulness or otherwise of Vahlen's work, I have noticed a lot of results from that time period that are worth another look (for me, especially due to the modern development of topology). And that does seem to happen independently of this Nazi issue so... I think you're right.
    – j0equ1nn
    Commented Feb 9, 2017 at 23:19
  • 5
    Yes, it is palpably true, though apparently not widely appreciated, that people 100 years ago were doing sophisticated things... not just proving basic theorems that are now in every textbook. :) There is a "Whiggish" tendency implicit in the over-simplifications suggested by, if not literally asserted in, many textbooks that leads people to believe that everyone nowadays is far more sophisticated than everyone in the past, in particular, people were dopier the farther back one looks. This simplistic outlook has an obvious appeal, but is crazily misleading... Commented Feb 9, 2017 at 23:23
  • 1
    Right on. My PhD thesis was germinated by taking a paper from 1900 (by Macfarlane), updating the notation and methodology, then seeing what I could do with it. The more research I conduct, the more important it seems to start by digging.
    – j0equ1nn
    Commented Feb 9, 2017 at 23:32

This community wiki answer was created from answers-in-the-comments.

What are some similar instances, particularly in mathematics?

  • At least according to Wikipedia, the non-subscription of "Deutsche Mathematik" outside Germany caused problems in the development of Teichmueller theory, as many of the papers of Teichmueller's were thus not available for researchers in the field.

  • Wagner's music was not publicly performed in Israel before the current century.

  • Alexandre Grothendieck's "political views were radical and pacifist", to the point of quitting academics, harassing people trying to publish his work, and living into reclusion. That did not prevent him from getting prizes (that he refused) nor to be abundantly quoted. He is often referred to as one of the founder of modern algebraic geometry. He was the opposite of a nazi, but actively involved in political activities that may shock / derange. If anything, this added to the awe that the mathematical community has for him.

  • Werner Heisenberg. Fortunately the uncertainty principle is no less amazing and cool because its discoverer was working to help the Nazis develop an atomic bomb.

  • Lev Pontryiagin

  • Johannes Stark

  • Philipp Lenard

  • Serge Lang had some pretty scary views about HIV/AIDS Commented May 16, 2022 at 6:05
  • Helmut Hasse. Prolific twentieth-century mathematician at the university of Göttingen. He was a nazi, which allowed him to survive the "great purge" that happened at Göttingen in 1933.
    – Stef
    Commented Sep 5, 2022 at 10:16

The fact that you would make a joke because of someone's political attitude shows that you lack professionalism. Mathematics has nothing to do with politics, and it is totally inappropriate to mix it with politics/ideology. Not giving credit to someone just because you don't agree with his personal opinion is also disingenuous. Richard Feynman and many other scientists worked on the atomic bomb; would you try to ignore them because of this?

"It would also be hilarious to just use a swastika [...]". This speaks for itself.

  • 4
    Okay, 3 things. 1- "Political attitude" feels insufficient to describe active Naziism. 2- Humor is important especially within topics that can be difficult and make listeners feel alienated. 3- I never said I wouldn't give the person credit, rather I asked if he may have been denied credit historically. Okay, 4- you seem to get upset easily and the insulting nature of your "answer" is unprofessional.
    – j0equ1nn
    Commented Feb 9, 2017 at 23:13
  • 6
    I agree with "Mathematics has nothing to do with politics" and "Not giving credit because you don't agree". The other arguments are not really appropriate. Whether a joke concerning political attitude is suitable or not has nothing to do with professionalism. In this particular case, the joke may be inappropriate, but this is not how the response has been phrased. And Feynman and others racing Germany on the bomb is really not the right comparison - these guys (as Feynman himself makes clear) were the guys who would be killed first if Nazism had won. They had a very strong reason to go for it. Commented Feb 10, 2017 at 0:20

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .