Pre-script: This is absolutely not the moment for centering white perspectives in public discourse. That being said, everything I understand about the fight for equality and justice leads me to the conclusion that the responsibility to dismantle whiteness falls uniquely to white people. This post goes with this aim in mind, in solidarity with the uprising to end systemic anti-black racism.

1. Eponymy

Academia does not enjoy the fame of a position on the vanguard of social change. It is at once deflating and encouraging: the horror that must be witnessed, the massive public outcry it takes to incite our institutions to issue statements and undertake reflection that might to lead to action. On the other hand, the fact of these statements and the promise of action signal that our institutions, or the people that steer them, have tuned in, and it gives the opportunity to channel momentum to enact meaningful changes.

I have a proposal for a modest change. It may seem like something that doesn’t directly help the stakes of marginalized people in our community, but (1) it provides something everyone can do (2) it requires little effort and (3) I think it does have a direct impact on all of us. Further, there is a special role for the current generation of young mathematicians to play here, as we will soon become the primary bearers of mathematical culture. I’m talking about the monuments we have in mathematical language which appear to celebrate bigots and racists.

When a society names something for a person, whether it be a land, a town, a building, or an idea, the society is communicating its values. That person made some contribution which the society appreciates, and in return the society honors the individual with the transmortal fame of eponymy. By continuing to use the person’s name as the name of something more permanent, the culture celebrates that person and their legacy.

Over time, cultural values change, and a society may come to find that the once-sacred values a person stood for are no longer acceptable, let alone worthy of honor. Recently, this has become manifest in the efforts to remove Confederate monuments and symbols from public places and the names of Confederate generals and politicians from military bases, municipalities, and universities. The calculus here seems fairly simple. Slavery is abhorrent. We do not want to celebrate (or appear to celebrate) the history of slavery in this country. The psychic injury caused by the prominence of such monuments, not to mention the hypocrisy of their continued presence, outweighs any historic value they might tender. So they must come down. It is farcical to claim that the dislocation of inanimate sculpture amounts to an attack on heritage – heritage of the aggressor culture.

Academicians in general, and mathematicians in particular, have an obsession with eponymy. We also have what seems like a nervous compulsion to insist that as professionals, we are apolitical, and kindly stop bothering us trying to slip politics in with our morning theorem-precursor. This tenet of mathematical orthodoxy, always reminds me of Stephen Colbert, in character as Stephen Colbert, prefacing any discussion of race by making sure his interlocutor knows that he is “colorblind.” Nothing, of course, could be further from the truth. Mathematics was for many centuries the domain of the priest and the patrician – sophisticated leisure for the leisure class. Since then we have variously been court scientists to emperors, patronees of dukes and kings, and in more recent memory, mercenaries of Wall Street, surveillance capitalism, the modern warfare machine, and the retooling of this machine for local police forces to use on citizens with extreme prejudice. Proposition: Mathematicians are not political. Corollary: The pope is not catholic.

If we can dispense with that small issue and accept that everything we do is bound up in the power relations of our society, the political import of our habits and conventions can begin to come into relief. To avoid further pontification, let me just lay out some facts, and you see if you can spot the same issue I do.

2. Some American mathematicians

Benjamin Peirce (1809-1880)

Known as in:

• Peirce’s criterion: rule for eliminating outliers from a data set
• Peirce decomposition: decomposition of an algebra as eigenspaces of commuting idempotents
• Benjamin Peirce Fellow: what Harvard calls its math postdocs.

Less known:

• Defender of slavery, especially as it allowed an elite to pursue the sciences. (Josiah Lee Auspitz (1994) The Wasp Leaves the Bottle: Charles Sanders Peirce)
• “My constant text now is I have seen slavery and I believe in it.” (Peirce, quoted in: Louis Menand (2001) The Metaphysical Club, §7.3)
• “No man of the African Race has ever shewn [sic] himself capable of any advance in the mathematical sciences. If therefore we would insist upon it that the knowledge of God in the physical universe was the duty of all men and that this knowledge could only be acquired through mathematics, and that therefore any man of that race should be compelled to become a student of mathematical science we should labour in vain. We might as well hope to wash out his colour, as we should be attempting to prevent the order of God’s creation.” (Peirce, quoted in Menand, §7.3)

Charles Sanders Peirce (1839-1914) (son of Benjamin Peirce)

Known as in:

• Peirce’s law : ((P → Q) → P ) → P (axiom that implies the law of the excluded middle)
• the Peirce arrow: ↓ (symbol for NOR or “not or” in logic)
• the Peirce triangle: like Pascal’s triangle but counts set partitions

Less known:

• Shared his father’s views on race and slavery throughout his life. (Menand, §7.3)
• Fond of quoting racist syllogisms to illustrate the limitations of classical logic. (Menand, §7.3)

Robert Lee Moore (1882-1974)

Known as in:

• Moore space: a developable regular Hausdorff space
• the Moore plane: example of a completely regular Hausdorff space that is not normal
• the Moore method: teaching methodology in which students are only given definitions and theorems and must supply the proofs themselves
• Robert Lee Moore Hall: home of the mathematics department at UT-Austin

Less known:

• Strongly in favor of segregation, made many documented racist remarks. (Albert C. Lewis (2002) The Beginnings of the R. L. Moore School of Topology, p. 10)
• Refused to teach black students or listen to black mathematicians lecture. (Mac McCann (2015) Written in Stone: History of racism lives on in UT monuments )
• Frequently made anti-Semitic and misogynistic remarks. (Reuben Hersh & Vera John-Steiner (2011) Loving and Hating Mathematics: Challenging the Myths of Mathematical Life, p. 279)

The inclusion of Birkhoff and the individuals below is not intended to equate anti-Semitism, Nazism, or the Holocaust with systemic and anti-black racism in the US. Instead, I feel that the naming issue they present is too related to ignore. They also represent my point of entry to the more universal questions I wish to pose (see below), so I’m hoping an awareness of their biographies may function similarly for others.

George David Birkhoff (1884-1944)

Known as in:

• Birkhoff factorization: a decomposition for matrices with Laurent polynomial coefficients
• Birkhoff’s axioms: some postulates for Euclidean plane geometry
• Birkhoff interpolation: a method of polynomial interpolation of point sets
• Birkhoff’s theorem: (there are several)
• G. D. Birkhoff prize: given jointly by the AMS and SIAM for applied mathematics

Less known:

• Made consistent and documented anti-Semitic remarks (Reinhard Siegmund-Schultze (2001) Rockefeller and the Internationalization of Mathematics Between the Two World Wars: Documents and Studies for the Social History of Mathematics in the 20th Century,  p. 64)
• Opposed Lefschetz’ election to AMS presidency because he was Jewish, believed he would use the position “to work positively and strongly for his own race. They [Jews] are exceedingly confident of their own power and influence in the good old USA.” (Birkhoff, quoted in: Steve Nadis and Shing-Tung Yau (2013) A History in Sum: 150 Years of Mathematics at Harvard (1825-1975), p. 83)
• Accused by Jewish scientists and mathematicians (Einstein, Wiener) of anti-Semitic hiring practices while chair at Harvard. (1912-1944) (Nadis & Yau, p. 82)
• Speaking on his fear of a flood of immigrant scientists in the pre-war period, Birkhoff defended his purpose of protecting jobs for American mathematicians. (Nadis & Yau, p. 81)
• “[Birkhoff] speaks long and earnestly concerning the ‘Jewish question’ and the importation of Jewish scholars…. He is privately (and entirely confidentially) more or less sympathetic with the difficulties of Germany. He does not approve of their methods, but he is inclined to agree that the results were necessary.” (Letter of Warren Weaver (1934), quoted in Siegmund-Schultze, p. 200)

3. Some German mathematicians*

Ludwig Bieberbach (1886-1982)

Known as in:

• Bieberbach’s inequality and Bieberbach’s conjecture (now de Branges’ Theorem) on univalent holomorphic functions
• Fatou-Bieberbach domains, which are biholomorphically equivalent to $\mathbb{C}^n$

Less known:

• Nazi Party and Sturmabteilung (Nazi paramilitary group) member.
• Actively campaigned for the removal of Jewish colleagues from universities (e. g. Landau, Schur).
• Founded and promoted a nationalist Deutsche Mathematik which sought to racialize mathematical tendencies.

Oswald Teichmüller (1913-1943)

Known as in:

• Teichmüller spaces: moduli for complex/hyperbolic structures on a surface
• Teichmüller character: a kind of character of $(\mathbb{Z}/q\mathbb{Z})^\times$
• Teichmüller cocycle: a certain obstruction in Galois cohomology, named by Eilenberg-MacLane
• Inter-Universal Teichmüller Theory: Mochizuki’s name for his notorious work in arithmetic geometry

Less known:

• Was a “dedicated Nazi.” Joined Sturmabteilung (SA), Nazi Party (1931).
• Initiated boycotts of Courant, Landau, while a student at Göttingen.
• Collaborated with Bieberbach on the application of Nazi ideology to mathematical thinking.
• Participated in cryptographic work and the invasion of Norway for the Wehrmacht (Nazi armed forces).
• Killed in battle during German retreat from Soviet Union (Sep. 1943).

Erich Kähler (1906-2000)

Known as in:

• Kähler manifolds: complex manifolds with closed Hermitian 2-forms, named by Weil
• Kähler differentials: generalization of differential forms to schemes
• K3 surfaces: smooth complete surfaces with trivial canonical bundle (The other K’s are Kodaira and Kummer; also named by Weil, who at least had sense enough not to call them, well…)

Less known:

• Committed German nationalist, volunteered for military service 1935, served for all of WWII becoming a POW (1944-47).
• Defended the Reich for years afterward, keeping a Nazi navy flag in his office.
• Believed that the news of Auschwitz came from the Russians intending to defame Germany.

Ernst Witt (1911-1991)

Known as in:

• Witt vectors: provide a model for the p-adic integers
• Witt’s theorem: on quadratic forms, extending isometries
• Poincaré-Birkhoff-Witt theorem: gives a monomial basis for the universal enveloping algebra of a Lie algebra
• Hasse-Witt matrix: describes the Frobenius map on a curve over a finite field

Less known:

• Active Nazi Party and SA member, under the influence of Teichmüller.
• Worked for the cipher department of the Wehrmacht.
• Though not documented as outspokenly anti-Semitic, Witt took advantage of cooperation with Nazi administration for the benefit of his career.

Other mathematicans with known and documented Nazi affiliation:

• Wilhelm Blaschke (1885-1962): Blaschke product, Blaschke selection theorem, Blaschke conjecture
• Helmut Hasse (1898-1979): Hasse diagram, Hasse-Witt Matrix, Hasse principle, Hasse-Weil zeta function
• Gerhard Gentzen (1909-1945): Gentzen sequent calculus, Gentzen’s theorem

*… and more. See: Sanford L. Segal (2003) Mathematicians under the Nazis.

4. Reckoning

This issue first came to my attention early in graduate school near the beginning of a lecture by a mathematician and teacher I admired. The speaker, Jewish, and having to communicate a mathematical abstraction named for one of the men above (I think it was Hasse), pronounced the concept, defined it, stated the eponym and then inserted the unexpected phrase, “… who, by the way, was a Nazi…”  It need not be explained that white cis male identity, and especially of the WASP variety, acts to insulate the bearer against such small traumas. So imagine my wonder, realizing for the first time that it must be a very odd and troublesome thing to have in one’s livelihood, the constant necessity of naming and honoring individuals who willfully participated in, and even afterwards defended, a system that degraded, enslaved, and murdered your ancestors.¹

What do we do about this? What does it mean to honor the intellectual contributions of persons we find morally reprehensible? Is everybody just OK with it? Can we separate the mathematical from the political?

On the one hand, I think we can, but only in the following limited sense. I wouldn’t suggest that we suddenly stop studying some areas of mathematics just because they were touched by white supremacists. For instance, in my outsider’s understanding, Maryam Mirzakhani has some beautiful theorems in an area known as Teichmüller dynamics. Why would we let her forerunner’s Nazism debase the value of her work? Furthermore, as scholars concerned with our history, we should keep track of who contributed which ideas and when, and try not to let our political biases color the scientific record.

But we can do all of this without having these names constantly on our breath. Frankly, the usage is at times entirely senseless and gratuitous. Take the Peirce arrow – I mean, I’m sorry, but it’s a freaking arrow. Does it really matter if he was the first one to ascribe a particular logical meaning to it? Notation choices get trophies now?

I am tempted here again to draw the parallel with monuments to slavers, Confederate soldiers and politicians, and segregationists. But Our Problem is actually a little bit trickier than that: in the other context, the statues honor individuals precisely for their contribution to upholding a racist social order, making the moral imperative to remove them much clearer. In Ours, the conferred honor appears skew to any objectionable aspects of the honoree’s character, and so renders the orthodox agnosticism towards the content of that character defensible.

But here’s the thing: language is powerful. It can cause alienation and injury, and we ignore this to our collective peril. Consider the black (1) undergraduate who sits in combinatorics class listening to their white professor go on about the “stars and bars”; (2) graduate student who goes to work and study every day in Robert Lee Moore Hall; (3) postdoc who is lent the “honorific” Benjamin Pierce Fellow (people with this title on your CV – hi). Does it seem to them like mathematics is trying to become a more diverse and inclusive place?

So again, what do we do about this? As much as we might like to tear down the monuments to bigots and fascists and erect new ones for the heroes and martyrs of the moment, I don’t think this is the right answer for mathematics. We might easily be led down the path to witch-hunt and a new noxious puritanism, or alternatively earn ourselves the absurd task of trying to distinguish between “full-blown racists” and mere collaborators.

Instead, we can accept the idea that individual, flawed human beings might not be fit for the immortal ideals of our collective imagination, and stop using the names. But not just the names of racists, misogynists and the like: stop using all the names. We can construct better, more poetic and descriptive names from the bare elements of language as replacements for the myriad dead white men. In many cases, alternatives already exist, and we merely have to insist upon a preference. For example, a “Hasse diagram” is also called an “ordering diagram” (or also a “poset graph”), and the latter actually tells you what information the diagram communicates. The deployment of a name as argot seems here a deliberate attempt to alienate outsiders, or at least gate-keep by forcing indoctrination into the cult of white male worship upon those that wish to persist.

Coming up with descriptive names for some of the lemmas and theorems will require a bit more finesse, but we’ve done it before (here’s looking at you, snake lemma), and I bet we can do it again. We can also stop naming new things for people immediately, and this is where the younger generation can play a key role – if we build no more pipelines, we will eventually stop burning oil. If you discover/invent a new mathematical object, have the courage and scholarship to find a name that actually conveys some of its meaning. Don’t name it for your adviser, your hero, or the author of some paper you read. Your idea is more perfect than they can ever be.

Our use of surnames as jargon has truly reached levels of self-parody. You may have noticed that it was impossible for me to briefly mention the work of the mathematicians above without recourse to several other proper names. And scientific eponymy is fraught for reasons beyond the political. To cite just a few:

1. Concepts are often not named for (all of) their originators, adding confusion rather than clarity to historical record.
2. Full proper attribution in naming to all parties involved is much less wieldy, and (again) conveys much less information than succinct descriptive terminology (compare: the Albert-Brauer-Hasse-Noether theorem and the four color theorem).
3. Persons’ names become overburdened, compounding the confusion (as in, um, which Euler’s theorem? Which Borel?).
4. Bias which denies the contributions of women and other systematically excluded groups (*euphemism: hidden figures*) inevitably permeates the practice.

Moving away from our problematic love affair with eponymy seems like a pitiful first step in view of the challenges our society faces. But it isn’t nothing. I find it liberating because we also don’t have to ask permission from anyone in charge for this, unlike (it seems) so many other issues of injustice. It is rather an ancient tradition of youthful rebellion for a generation to differentiate itself in language (the “youthful” part is at this point, however, for many of us, debatable).

We can’t remake the history, and we can’t expect our elders to transform the system for us all at once – it is the system in which they, after years of their own struggles and pushes for change, now occupy a comfortable status. But we can transform the language, a little bit at a time. Onward, friends – to Wikipedia!

 [1] I do not know of any mathematicians that enabled, supported or vociferously deny the Armenian genocide. ↩︎

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