Meta- and Galois

First, some extradiegetic comments. I did not post much last year. The main reason is obvious – for the first half of the year, I was very busy finishing the proof of the ternary Goldbach conjecture; then, during the summer, I had many speaking engagements – and I also spent a non-trivial amount of time writing a popular account of the proof for this blog. I spent the October-December rewriting the second, longer half of the proof – so that the proof is now in three papers, each of them a little over 70 pages in length. (I was also looking for new topics, doing some more travelling and lecturing, and retaking Russian and Classical Greek, among other things.)

At the same time, even though I had many reasons for staying away for blogging, it is a bit of a pity that I did not have time to blog precisely at a time when there started to be many more potential readers for it. Of course, some of them (you) are probably still around.

So far, I have written here mostly on mathematics, cinema and my travels. I would also like to touch a bit more frequently on a few serious subjects, not strictly mathematical, though sometimes related to mathematics and academia.


There is an idea going around my head since I read T. Rothman’s Genius and Biographers: The Fictionalization of Evariste Galois. This well-received essay from 1982 has had a definite influence on how Galois is seen nowadays; see a summary by a popular science writer, or the references in the introduction to the third edition of I. Stewart’s Galois-theory textbook. It has, to a great extent, replaced the account in E. T. Bell’s rather dated Men of Mathematics (1937). (E. T. Bell’s book is a collection of short biographies that inspired generations of mathematicians, while being famously imprecise and slanted, to say the least.)

It’s easy to correct Men of Mathematics on just about anything; at some point, before its general lack of accuracy was known, it may also have been necessary and worthwhile. What is, then, genuinely bothersome – or simply wrong-headed – about Rothman’s article?

First, it comes across as an effort, not just to defictionalize Galois, but to deromanticize him. The two concepts are not identical. Romanticization, strictly speaking, consists in the projection of a sensibility or the incorporation into a narrative, rather than in the practice of playing fast and loose with the facts. More importantly, there is no projection here, in the sense of imposing a sensibility alien to the subject. Galois lived in a romantic age; to understand his behavior, we must accept that hero worship, the search of sacrifice and martyrdom, the simultaneous identification with the people on the part of progressive sectors of bourgeois youth – combined with claims on a previously aristocratic concept of honour – were all concepts that pervaded the climate, rather than parts of a later grid. Rothman states under the heading “Harsher words” that “[Infeld] intends to make Galois a hero of the people”; this is a rather odd indictment, given that Galois was a member of an illegal organization called La societé des amis du peuple. We can discuss to what extent Infeld’s (socialist) conception of le peuple differed from that of Galois’s friends, or mention, as others have done, that (both in Galois’s and in Infeld’s time!) many of those self-identifying as amis du peuple were of middle-class extraction; still, what cannot be nullified is Galois’s self-inscription into a developing collective narrative.

Is it healthy for a biographical article to be written in a way that is so out of tune with its subject’s sensibility? This is not necessarily a disqualification; what is odd is to view accounts closer to an emic perspective as imposing an alien narrative. An etic perspective may differ sharply from an emic perspective; an etic account can draw attention to this fact without invalidating itself — and, conversely, it should not confuse the presence of a particularly large difference with objectivity.

To give an extreme analogue: an atheist may of course write a biography of Joan of Arc — and a historical essay on her should certainly be written differently than whatever account of her purported miracles was used at the Vatican for her canonization. Still, if a biographer charged another with “intending to make Joan of Arc into a religious figure”, something would be seriously amiss. We would also nowadays be careful not to rush to pathologize whatever seems to us unusual in some of her narrated experiences, given that, say, to state that one had visions was seen as acceptable and in consonance with the sensibility of a sector of society at the time. We would try to contextualize matters, even though we can probably agree that it is easier to make a case for a pathology in her case than in that of Galois, who had no visions and scaled no walls. Calling either self-destructive (not Rothman’s term) is to both say a triviality and to miss the point; it was neither’s intention to maximize his or her chances at survival.

Evariste Galois, drawn at 18.

Evariste Galois at 18, as drawn by a classmate.


There is another point to make, one about tone. Perhaps conscious of his subject’s identification with the sort of narrative that he dislikes, Rothman comes across as an adversary not just of E. T. Bell or Infeld, but of Galois; at times, his text comes across as a speech for the defense of an accused establishment. Since Rothman makes some remarks (in “Harsher words”) on the motivations of previous biographers, it seems fair to place his own habits of thought and language within a certain tradition still alive in contemporary academia. What we are dealing with is a discourse often used to defend academic hierarchy; if the speaker is fortunate, he will defend a hierarchy elsewhere, at a different time, and affecting people other than himself. Thus goes Rothman:

But Galois’s troubles were not yet over. A few days later, he failed his examination to l’Ecole Polytechnique for the second and final time. Legend has it that Galois, who worked almost entirely in his head and who was poor at presenting his ideas verbally, became so enraged at the stupidity of his examiner that he hurled an eraser at him. Bell records this as a fact but according to the little-known study of Joseph Bertrand the tradition is false. Bertrand, who appears to have detailed information about the event, records that Galois, while expounding on the properties of logarithmic series, refused to prove his statements to the examiner M. Dinet and, in response to Dinet’s questions, replied merely that the answer was completely obvious. So was the result. [my highlight]

Rothman does not mention a version mentioned by Stewart (Galois theory, “Historical introduction”, 1973):

A variant asserts that Dinet asked Galois to outline the theory of “arithmetical logarithms.” Galois informed him, no doubt with characteristic bluntness, that there were no arithmetical logarithms. Dinet failed him.

(Stewart then goes into an interesting digression, stating that it is possible that Dinet might have been referring to the index modulo m. This seems unlikely at first sight: the term “discrete logarithm” for what Gauss himself calls the index sounds like much later nomenclature — and, since Disquisitiones Arithmeticae was still relatively recent and Dinet has passed to posterity mostly for failing Galois, it does not seem plausible that Dinet would have been thinking of this. I will gladly stand corrected on this, however; where can one find what was expected from a candidate to admission at École Polytechnique at the time?)

At any rate, the meaning is clear. Rothman (quite reasonably) rejects Bell’s claims of eraser-hurling (which, according to Stewart, go back to Dupuy); then, he gives a version in which there is no misbehavior on Galois’s part, but simply some impatience at an imprecise remark. It is not extraordinary that a second-rate individual would have perceived Galois’s attitude as petulant. What was perceived as extraordinary even by Galois’s contemporaries is that such an individual would have then taken this as a sufficient reason to fail the candidate.

Rothman very nearly comes across as taking the opposite view. “So was the result” does not just make it seem as if the outcome should have been expected (by a seventeen-year-old candidate); it is a kind of statement that, by taking a response on the part of a individual up in the hierarchy as if he were a force of nature, manages to condemn the person down in the hierarchy, while pretending not to pass judgement. This kind of shorthand should be familiar enough to all readers; we are Rothman’s contemporaries. (Rothman quotes Galois himself as stating “Hierarchy is a means for the inferior”; he seems to have little time for such sentiments.) Rothman later says that “[he] do[es] not wish to suggest Galois should have been failed.”; if he had not already come close to suggesting as much, such a disclaimer would not have been necessary.

There is an entire theme to be developed here: neither Rothman nor romantic “historians” are indulging in anachronism – rather, Rothman would seem to sympathize with the hierarchical sensibility that condemned Galois, and that still exists in some weakened and modified form to this day. (An actual continuity here may be a point for debate; it may be simply a case of one hierarchy’s sympathy for another, with which it identifies.) What would have been impossible in the early 19th century is something else, namely, Rothman’s amateur psychologizing. This and the defense of academic hierarchy are related, however, in that the sort of superficial and conventional “psychology” in which Rothman’s essay engages is precisely the sort that is used nowadays (or, in many places, a generation or two ago) to defend a hierarchy, while implying that only an individual can be unreasonable.

Rothman’s main imputation is — no surprises here — is that Galois had “developed not a little paranoia”. At some point, academic paranoia will be understood by all to mean something rather different from the common kind – that is, it will be a set phrase imposed by force of habit. In the meantime, however, paranoia is still a clinical diagnosis, and, unless it is meant as an insult (much like, say, the originally medical term “idiot”), it has to be supported when used to describe a scholar much as in any non-academic context.

Some of the evidence adduced is decidedly odd. A shot was fired from a guard’s garret into a cell that Galois shared with several other prisoners. This was interpreted by Infeld as an attempt on Galois’s life. Rothman says he has “tried to present this episode in as neutral a tone as possible”; apparently, the way to do this is to seem to go to some length to attempt to defend the decision to throw Galois into a dungeon (“evidently because he had insulted the superintendent”) together with the man who was actually shot – something that was considered by other political prisoners to be unusual and completely out of line. At any rate, we are given no evidence that Galois believed that the shot had been aimed at him; the mere belief that a shot fired into a prison cell may have been intentional is enough.

The other evidence is that Galois took rejection letters badly. There is also another little matter – at least one of his manuscripts got lost after submission to the Academy. (Whether a second memoire got lost or neglected by Cauchy is something that seems unproved in either direction; Rothman cites R. Taton’s case against this – and also gives references that suggest that the suspicion of intention on Cauchy’s part was solely Infeld’s, and not Galois’s.)

Rothman’s essay ends in a sardonic note:

The underlying assumption is apparent: Galois was persecuted because he was a genius and all scientists, to a greater or lesser degree, understand that genius is not tolerated by mediocrity. A genius must be recognized as such even when standing drunk at a banquet table with a dagger in his hand. [...] This is a presumption of the highest arrogance.

In fact, some of the material there and elsewhere gives a picture of a young man who was generally known to be, at the least, very talented; word of this had got around in academic circles, and also beyond that – he was “our little scholar” to other prisoners. The likely reasons for his sad and brief life and career can be multiple – but we cannot say that he had somehow managed to make his talent unrecognizable.

Évariste Galois, as drawn sixteen years after his death (1848) by his brother Alfred.

Évariste Galois, as drawn sixteen years after his death (1848) by his brother Alfred.


In the end, a popular revolutionary, a romantic hero and a difficult young person are not three different characters, nor even three distinct, incompatible views of the same person. Neither do these categories match poorly with the view of Galois as a richly gifted mathematician frustrated by pedantry on the part of the incompetent and fumbling on the part of those who were usually more than competent. Of course one may argue that Galois was ill-equiped to deal with such a situation; almost all adolescents would have been, even those whose fathers had not been pushed to suicide by local Jesuits. The way that Galois responded – namely, by a sharpening of his conflict with authority as such – would have been within the bounds of what is normal in any era; more to the point, it was precisely what made sense in a young man of already formed republican convictions in an atmosphere of repression and stifled revolution – at, moreover, a time that exalted struggle and sentiment as much as it rewarded conformism. We can and should attempt to undo romantic legends, when they are legends; however, to deromanticize and depoliticize Galois is to misunderstand him.


Rothman’s essay was brought back to my mind by an essay by M. Duchin (marked as juvenilia on her webpage). There, Rothman is paraphrased as having shown that E. T. Bell changed the chronology of events; it is also stated that Galois’s father’s suicide helps to explains the result of his (oral) examination at the École Polytechnique immediately thereafter. In all fairness, Rothman states that Bell’s main source does not make clear the chronology of events; moreover, Rothman can be interpreted to mean that Galois was in a particularly irritable mood (something that, in his view, makes claims of “the examiner’s stupidity” less valid), rather than to insinuate that Galois did badly in some objective sense.

In general, I found Duchin’s essay thought-provoking, and I certainly share her strong suspicion of the concept of “genius” itself. Still, I was unconvinced by her contention that there is something particularly male about genius-worship. I had also thought that there had been a transition at some point between the early nineteenth century, when one could speak of the genius of somebody (originally something close to a not entirely beneficent daemon), to popular usage in the twentieth century, when it became extremely common to say that somebody was himself (or sometimes herself) a genius. This arguably crucial shift is left unexplored.


Since I am travelling, here are the obligatory tourist photographs.

Yes, this is the sea.

The sea

IMG_0223

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About valuevar

I am a number theorist with side interests in combinatorics and group theory.
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One Response to Meta- and Galois

  1. valuevar says:

    Correction: Galois did once try to scale walls – namely, the Ecole Normale’s, during the July 1830 revolution. (He was unsuccessful.)

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