The Society of Mathematics

Mannequin in front of summation formula

This post concerns the Association for Mathematical Research, or AMR. A number of people are upset by its existence. I am not exactly one of them, but am suspicious, mainly because I do not know why a new organization would be needed, when we already have

The Twitter account of the AMR is dated to April, 2021. The website of the AMR supplies a list of founding members, but no account of when, how, or why they became founders. The site has a brief mission statement:


Are those other organizations not doing a good job? According to the MAA website, “Collaboration between AMS and MAA has been strong throughout the last century,” the latter organization having been formed after a resolution of the Council of the AMS in 1914, whereby

It is deemed unwise for the American Mathematical Society to enter into the activities of the special field now covered by the American Mathematical Monthly.

Possibly the failure of the AMR to explain itself is itself explained in “New Math Research Group Reflects a Schism in the Field” (Scientific American, January 11, 2022), where Rachel Crowell refers to

a growing division between

  • researchers who want to keep scientific and mathematical pursuits separate from social issues that they see as irrelevant to research and
  • those who say even pure mathematics cannot be considered separately from the racism and sexism in its culture.

(I have added the bullets.) If the founders of the AMR explained themselves, this itself might be taking up the social issues that they want to avoid.

Diagram of diffraction grating

They cannot avoid social issues. Mathematics itself is a social activity. There is a good account, or example, of how this is so in the dialogue by Ulf Persson called “A conversation with Reuben Hersh,” published recently by one of the established mathematical societies listed above (EMS Magazine 2021 / No. 121, pp. 20–35). According to the character called Hersh (who is based on the actual human being),

in practice truth is agreed on by a process of social confirmation. I can give you a specific concrete example …

The example is of a theorem in linear partial differential equations with variable coefficients by Heinz-Otto Kreiss, generalizing Hersh’s work with constant coefficients. Hersh could never understand all of Kreiss’s proof, but Peter Lax somehow decided it was correct, and everybody else accepted Lax’s word.

Persson supplies the additional examples of Hironaka’s resolution of singularities and the limitation to twenty-six of the number of sporadic groups. We may well accept as true what others tell us, without actually checking for ourselves.

Whose word will we accept then? Are there mathematicians whom we do not trust? Are there classes of people from whom we do not expect trustworthy mathematicians to come?

We may try to avoid such social questions, and there is indeed a sense in which mathematics is not social. As the character of Persson responds to Hersh,

I think that there is something beyond the practice of mathematics, beyond the human fallible way of doing mathematics. Outside of mathematics, socially accepted truths may be successfully challenged. And even in mathematics, if there is a counter-example to a previously authorized theorem, that will surely trump.

Mathematics puts us in touch with something beyond ourselves. One can try to say that there isn’t really any such thing; but I don’t know how one can believe it. In any case, perhaps the AMR arises from a wish to get in touch more directly with what is beyond.

If so, why is the AMR trying to gain members? I might sign up for an AMR newsletter; but what we are invited to sign up for is membership. “There are no membership fees or dues,” we are told. What does it mean to be a member then? Membership seems like a political statement. It is a way to associate yourself with the founding members, of whom the AMR website names 268 (I counted them by cutting and pasting into a spreadsheet).

In her Scientific American article, Rachel Crowell mentions Louigi Addario-Berry, who writes something to my mind on his blog:

if [the AMR] aim to be a professional organization, advocating on behalf of the mathematical research community, then I want to know who they think they are advocating for.

There is that question again: whom do we accept as a mathematician? Apparently Addario-Berry thinks the AMR will be too restrictive here; for he says of the founders and directors,

I view the actions of at least two of them as actively harmful to the effort to build an inclusive mathematical community.

Those two persons are Robion Kirby and Abigail Thompson, and as far as I can tell, they are to be considered “actively harmful” for having done some controversial writing.

Addario-Berry may be engaging in conceptual overreach. I take that term from philosopher John Tasioulas, who writes in “The inflation of concepts” (Aeon, 29 January 2021),

‘conceptual overreach’ … occurs when a particular concept undergoes a process of expansion or inflation in which it absorbs ideas and demands that are foreign to it. In its most extreme manifestation, conceptual overreach morphs into a totalising ‘all in one’ dogma. A single concept – say, human rights or the rule of law – is taken to offer a comprehensive political ideology, as opposed to picking out one among many elements upon which our political thinking needs to draw and hold in balance when arriving at justified responses to the problems of our time.

It seems to me that actively causing harm is only a species – albeit the worst – of not being perfectly good. That which is not good is not necessarily harmful. Speech in particular may be bad, without being harmful: I think one is supposed to learn this in childhood, from chants such as

Sticks and stones may break my bones,
but names can never hurt me.

Names can hurt, and yet we have got to learn to deal with this. I have seen the plausible suggestion (I wish I knew where) that bad speech is called harmful, in hopes that it will lose protection as free speech. I do not think well of going down that road. It is the road being taken by reactionary activists in the United States, who want to protect the feelings of white children by not teaching about slavery in school.

Mathematics itself can be done only in freedom: freedom to take chances with one’s own thoughts, and freedom to question the assertions of others.

That freedom comes with a responsibility to justify one’s own assertions.

Addario-Berry says of Kirby and Thompson respectively:

  • “He is not someone I want mathematical advocacy from.”
  • “I view Thompson’s position as explicitly exclusionary.”

It seems to me, if you can’t say simply that Thompson’s position is explicitly exclusionary, but you have to add the qualifier “I view,” then while her position may be somehow exclusionary, it must not be explicitly so. You should then show how it is so.

Certain kinds of exclusion are desirable: exclusion of bad arguments and of the people who continually make them in bad faith.

On his website, Rob Kirby has an important piece of advocacy: an essay in the Notices of the AMS for February, 2004, called “Fleeced,” about the profits that commercial publishers make from us:

We mathematicians simply give away our work (together with copyright) to commercial journals who turn around and sell it back to our institutions at a magnificent profit. Why? Apparently because we think of them as our journals and enjoy the prestige and honor of publishing, refereeing, and editing for them …

What can mathematicians do? … A possibility is this: one could post one’s papers (including the final version) at the arXiv and other websites and refuse to give away the copyright.

Perhaps this is all well understood by now; but somebody had to figure it out, and we may praise Kirby for being such a person.

Nonetheless, Kirby also says,

People who say that women can’t do math as well as men are often called sexist, but it is worth remembering that some evidence exists and the topic is a [legitimate] one, although Miss Manners might not endorse it.

This in an essay that Kirby posted on his website in 1998, not having been able to get the Notices to print it as he wanted. One can be annoyed or horrified by Kirby’s words, and this may be why Rachel Crowell quotes them in her Scientific American article. According to Kirby, the evidence for women’s mathematical inferiority is of three kinds:

  1. Women have different brains.
  2. Women test differently, as for example on the SAT and GRE.
  3. Most famous mathematicians are men.

One can talk about such evidence, and people do in fact talk about it. However, what is it evidence for? What does it even mean to propose that “women can’t do math as well as men”? According to Kirby,

Most likely the phrase usually means that at any given time, speaking statistically, a woman who chooses to do math is less likely to do it as well as some statistical comparison group of men.

Kirby still does not say what it means to do math well; he just assumes it is something that can be measured. I question the assumption, because mathematics has something in common with art, and there is no measure for judging whether somebody is doing art well. We can look at which artists get their work in museums, or are winning awards, or are making money. None of this tells us whether their art is any good; for that, we have to make our own judgment. Even then, the artists will have their own judgment, which we cannot contradict.

The same is somehow true in mathematics. The satisfaction that we derive from our subject is for us alone to assess. Nonetheless, in mathematics, we have something more than personal satisfaction. We have correctness, which in principle must be shared. There are ways to measure the ability to get math right: we’ve got the tests that Kirby mentions, and we can count numbers of published papers. However, even these measurements are not, and cannot be, direct assessments of a person’s ability. In principle, for example, one has to check whether the papers are correct.

Colored lights behind mannequins

There are “replication crises” in science: see for example “A massive 8-year effort finds that much cancer research can’t be replicated,” by Tara Haelle (Science News, December 7, 2021). What would such an effort reveal, if applied to proofs in mathematical journals? We have looked at the dialogue of Hersh and Persson about social acceptance of difficult proofs. According to a recent preprint of Alexandre Borovik, Zoltan Kocsis, and Vladimir Kondratiev called “Mathematics and Mathematics Education in the 21st Century,”

the length and complexity of many very important proofs are now reaching the limits of human comprehension. We have to admit that mathematics faces an existential crisis. Indeed, what is the purpose of a mathematical proof that hardly anyone will ever be willing and able to read and check?

Without pivoting to a systematic use of computer-based proof assistants, and corresponding changes to the way mathematics is taught and the way mathematical research is published, the field will not be able to face its new challenges …

Can we hope to convert the difficult proofs to a form that a computer can verify? If we can do this, then, pace Hersh, according to Borovik et al., “proof is no longer a ‘process of social confirmation’, but a technological process.” Acceptance of the technological process is still a social process, it seems to me. The authors themselves go on to say,

What perhaps remains a social matter are possible attempts to reach a better understanding of what actually has happened inside of a computer-checked proof – which is likely to amount to construction of another, better structured, more transparent, better annotated, and ideally human-readable proof.

Writing in his essay on sexism in 1998, Rob Kirby is sanguine about our correctness:

We have pretty good habits because it is relatively easy to measure our worth as research mathematicians, and because it is hard to make unfairness pay off. For example, we have little temptation to fudge our experiments to make the data look better for we rarely experiment; and we don’t know how to fudge a proof to make it better.

Is that true? Can Kirby really not think of tricks to make it seem as if you are doing real mathematics? Introducing new notation is a simple one. There are also sham journals and sham conferences, set up to let you add a line to your CV, or win a prize from a government agency.

Regarding the putative evidence of a mathematical difference between the sexes, apparently Kirby thinks we could use it; we are just not there yet:

I think that most of us believe [the evidence] is not nearly strong enough for us to act on it; e.g. when considering candidates for admission, or an award, or a job, one has data about the candidates which far outweighs the sex of the candidate (when choosing tall people and picking between a 5’9″ man and a 5’10” woman, one does not pick the man because men are on the average taller). And even more so, most of us are agreed that one should not make public policy (favoring men) on the basis of the evidence described above.

That last sentence may mean that we should never try to discriminate between women and men. On the other hand, it might be argued that the discrimination was not to favor men, but to make sure that each person was in the place that she or he was best suited for. I believe such arguments are used to justify attempts to make racial distinctions among human beings: if some group of us are genetically inferior (says the argument), there may be social programs that can help that group.

Kirby’s rhetorical point about mathematical differences between the sexes is that, while evidence for them may exist in some technical sense, it is no good for anything – but then neither is the evidence of sexism in mathematics any good; for,

  • there is no statistical basis for assertions of sexism;
  • there are four anecdotes that suggest sexism in mathematics, but are distortions or misrepresentions of the truth.

Therefore, Kirby says, “My view of the math profession is that it is generally fair.” He goes on as I quoted earlier, about our “pretty good habits,” concluding,

It appears to me that mathematicians would genuinely prefer to see women do better in math …

Yet my view is, apparently, a minority one. The view that we often hear is represented by Cora Sadosky in the January, 1994, AWM Newsletter …

… In my view, the smaller number of women in math is not due to discrimination by men nor to any inherent inferiority in women, but rather is due to the simple fact that more men than women choose to enter mathematics. I agree that these choices are due to many factors, including the environment that girls and boys grow up in, but little of this has anything to do with our mathematical community, which strikes me as fair as any I know.

Kirby is making his own judgment. We are allowed and even obliged to do that in mathematics. In the strictest sense, a proposition is not mathematics unless it can be known to each of us individually, at least in principle. And yet each of us has to know the same thing about the proposition. Goldbach’s Conjecture cannot be true for you, and false for me, unless perhaps it should turn out to be independent of the first-order Peano axioms, and you and I assume contradictory additional axioms. Even then, we have got to agree on whether a proof based on given axioms is correct.

Kirby has judged what he has seen, in writing, about sexism. Has he ever talked with a woman about her personal experience as a mathematician? Since 1968, Kirby has apparently had fifty-three doctoral students. Among them are twelve students whom I cannot hazard to gender; among the rest (including two Turkish men), there seem to be only two women (one since 1998).

Perhaps most experiences of sexism are such as will not show up in statistics. I think of Alice Silverberg’s Adventures in Numberland, a collection of experiences, such as being asked to do things that a man would not: vacuum one’s office, or set out forks for a buffet in the department, or collect plates after a meal at a colleague’s house. The vignette that I remember best is “A conspiracy of women”; perhaps it was the first I read.

Kirby posted his essay almost twenty-four years ago; but it is on his personal website and is linked to from his homepage. Now he is on the board of directors of the Association for Mathematical Research.

A critic of the AMR called Tamara G. Golda has a blog post, dated November 16, 2021, called “Red Flags of the Association for Mathematical Research (AMR): Friends Don’t Let Friends Join the AMR.” I don’t know why it counts as a red flag that two of the eleven persons on the board of directors share a last name (they are married to one another).

“For the sake of the field of mathematics,” says Golda,

please do not encourage this organization in any way. Do not link to their web site, do not join, do not follow their Twitter feed, etc.

I have not joined the AMR. I do however follow its Twitter account, and I have linked to its website in this post.

In a post of this blog called “What Mathematics Is,” I mentioned instructions from fellow mathematicians to unfollow a particular account. I disobeyed the instructions, and this allowed me to learn about Euphemia Lofton Haynes and a lecture in which she said,

If we as teachers of mathematics do not break down at least in the academic world the traditional fallacies regarding the nature of mathematics and the delusions regarding the kind of activity in which the life of the science consists we are neglecting our obligations. Mathematics is no more the art of reckoning and computation than architecture is the art of making bricks, no more than painting is the art of mixing colors.

We can test for ability to reckon and compute, and perhaps this is what leads to the confusion that Lofton Haynes is trying to dispel. She wants to apply mathematics, but not in a crude way:

Now I would like to consider with you some of the contributions of this abstract science to the development of citizenship. Not unmindful of its many and valued applications, students of mathematics realize that, though most abstract in form, mathematics is a very real part of living. The problems of mathematics are in reality mere mathematizations of the problems of life. It is precisely the job of the teacher of mathematics to assist the pupil in, shall we say, mathematizing life’s problems.

I object to a certain kind of “mathematizing life’s problems”; this would be the mathematization that I think is alluded to by Louis Menand, professor of English, when he writes,

A class in social psychology can be as revelatory and inspiring as a class on the novel. The idea that students develop a greater capacity for empathy by reading books in literature classes about people who never existed than they can by taking classes in fields that study actual human behavior does not make a lot of sense.

That was in the New Yorker (December 9, 2021), and I happened to find reason to quote it in a post “On Reading Plato’s Republic.” In any case, by “mathematizing life’s problems,” Euphemia Lofton Haynes does not mean turning them into statistics. She says for example,

In the concept of invariance we have the concept of that which remains stable in midst of change. Here is the notion of value or relation that survives in an ever changing world. Reflection upon this doctrine of invariance shows why mathematics is listed with philosophy and theology among the spiritual enterprises of man. The human spirit craves invariant reality, it craves freedom, it craves peace, a sense of being in harmony with a divine Being infinite and eternal …

I might not have encountered these admirable thoughts, had I respected the command not to follow the account then called Great Women of Mathematics, on the basis that the account owner refused to declare that certain persons born male were actually women. (The account is now Your Daily Epsilon of Math.)

Some people write in their Twitter profiles that retweeting does not imply support. This ought to go without saying. Much less does following imply support. I try to follow accounts that I disagree with.

Diagram of diffraction grating

I said I had not joined that AMR. There’s no telling about the future, but free membership makes me less likely to join. An organization needs some resources to operate. I enjoyed watching the video that illustrated an AMR review of a preprint in dynamics, “The Birkhoff-Poritsky conjecture for centrally-symmetric billiard tables.” Maybe any needed funding for this and other activities of the AMR comes out of the creators’ pockets, like the fee that I pay WordPress not to show ads on this blog.

However, the AMR is inviting donations. Such donations may come from persons for whom controversy is politically useful. The controversy could be over who gets into a women’s toilet or prison, or what American children learn about slavery, or whether mathematics is being “inclusive.”

To me then, free membership in the AMR is a red flag. It is not one of Tamara Golda’s red flags, but like Louigi Addario-Berry, Golda does have a red flag that provides some backing for a call to boycott the AMR:

Two of the three AMR Officers have publicly expressed views that are antithetical to increasing diversity and promoting inclusion in mathematics …

  • Rob Kirby of UC Berkeley claims sexism does not exist in mathematics
  • Abigail Thompson of UC Davis equates diversity statements to McCarthy era loyalty oaths

I have already looked at Kirby’s essay. If you want to see for yourself whether Golda has fairly summarized the opinions of Kirby and Thompson, the help she gives you is:

These views and the reactions to them are not hard to find. Kirby’s can be found on his website, and Thompson’s piece appeared in AMS Notices.

Golda provides no links, “because I do not want to link to views that are so contradictory to my own.”

I myself continue to do as I have been doing: link to views of fellow mathematicians, particularly if I am going to be questioning them. A pdf file of Thompson’s statement in the Notices of the AMS (December 2019) is available. Pace Golda, Thompson does not equate diversity statements to loyalty oaths. She likens them, as Ethan Zell says in “A Defense of Diversity Statements in Hiring,” on the now-“retired” AMS Graduate Student Blog. Thompson would seem to have a carefully crafted argument:

Faculty at universities across the country are facing an echo of the loyalty oath, a mandatory “Diversity Statement” for job applicants. The professed purpose is to identify candidates who have the skills and experience to advance institutional diversity and equity goals. In reality it’s a political test, and it’s a political test with teeth.

What are the teeth? Nearly all University of California campuses require that job applicants submit a “contributions to diversity” statement as a part of their application. The campuses evaluate such statements using rubrics, a detailed scoring system. Several UC programs have used these diversity statements to screen out candidates early in the search process …

Why is it a political test? Politics are a reflection of how you believe society should be organized. Classical liberals aspire to treat every person as a unique individual, not as a representative of their gender or their ethnic group …

One can argue with this, as people have. I wish I could have a positive statement of Tamara Golda’s own views. They seem to include a belief in her own authority to issue strong condemnations without strong evidence.

Presumably Golda’s views include also the desirability of “increasing diversity and promoting inclusion in mathematics”; however, she has not shown that anybody disagrees. Kirby says, as in the quotation above, “mathematicians would genuinely prefer to see women do better in math.” Of course those are just words. Thompson is more explicit about the work to be done:

We should continue to do all we can to reduce barriers to participation in this most beautiful of fields … There are reasonable means to further this goal: encouraging students from all backgrounds to enter the mathematics pipeline, trying to ensure that talented mathematicians don’t leave the profession, creating family-friendly policies, and supporting junior faculty at the beginning of their careers, for example.

Thus wrote the person who is now secretary of the AMR. What is she leaving out?

I have talked about mathematics as a social activity. I call it both personal and universal, because:

  • we have the personal authority to decide for ourselves on the truth of a proposition that is of interest to us;
  • our decision cannot stand in contradiction to anybody else’s, but disagreements must be resolved amicably.

In practice there may be people we don’t get along with. This says nothing about their mathematics, which may be brilliant. It seems to me learning this ought to be a reason why we are taught mathematics in school.

In mathematics, we have the ideal of a truth that cannot be argued with. This bothers people like Malcolm X, who (in The Autobiography of Malcolm X) recalled seventh grade at Mason Junior High School in Michigan:

I’m sorry to say that the subject I most disliked was mathematics. I have thought about it. I think the reason was that mathematics leaves no room for argument. if you made a mistake, that was all there was to it.

He is right, as far as he goes. However, one can learn from one’s mistakes. Moreover, if you have not made a mistake, but have got something right, then everybody must respect this – at least everybody who matters.

Then again, if somebody offers a counterexample, you must listen. You cannot stop your ears. The people of the Association for Mathematical Research really should explain themselves. “We just want to do mathematics” is not good enough. They seem to want to stop their ears.

Colored lights behind mannequins

The photos are from a shop window display in Cevahir Mall, Istanbul, February 8, 2015

Copy edited, July 20, 2022.

Be it noted that the next post after this one is a response to it and a defense of the mission of the AMR by Sasha Borovik.

As of today, I have five times tweeted the article of John Tasioulas about the conceptual overreach. What Jessica Bennett wrote about in “If Everything Is ‘Trauma,’ Is Anything?” (New York Times, February 4, 2022) seems to be the kind of conceptual overreach that I looked at above:

There are plenty of horrible things going on in the world, and serious mental health crises that warrant such severe language. But when did we start using the language of harm to describe, well, everything?

I mentioned replication crises in science. In addition to the preprint of Borovik, Kocsis, and Kondratiev, I have now read Anthony Bordg, “A Replication Crisis in Mathematics?,” Math Intelligencer 43, 48–52 (2021),, detailing some of the issues with how

mathematicians rely partly on social processes, such as reputation and academic credentials, for verification, with the risk of incorrect results being unnoticed and then used in further developments.

Bordg makes the key point, “human peer review will remain necessary,” because

the peer review system has two functions,

  • not only to ensure the correctness of mathematical proofs,
  • but also to evaluate the mathematical value of new results.

I like Bordg’s account of what replication means in the first place:

a mathematical proof can be thought of as a set of instructions that allow readers to replicate the proof, hence convincing themselves of its correctness. This act of replication can be indirectly judged against an ideal personal proof that would unfold from theorems that are firmly held to be true by the reader.

Bordg follows this up in a curious fashion:

In other words, the replication of scientific results is as relevant in mathematics as it is in the natural sciences.

I’m not sure what the analogue of the “ideal personal proof” would be in the natural sciences. In aesthetics, it would seem to be the work of art, by the account of R. G. Collingwood, who says of the artist on page 139 of The Principles of Art (1938),

His [sic] business is not to produce an emotional effect in an audience, but, for example, to make a tune. This tune is already complete and perfect when it exists merely as a tune in his head, that is, an imaginary tune … The noises made by the performers, and heard by the audience, are not the music at all; they are only means by which the audience, if they listen intelligently (not otherwise), can reconstruct for themselves the imaginary tune that existed in the composer’s head.

I alluded to this passage when I posted “On Causation,” where I mentioned that the passage had appeared in the old Stay Free! magazine. Collingwood goes on to draw the parallel with science:

the listening which we have to do when we hear the noises made by musicians is in a way rather like the thinking we have to do when we hear the noises made, for example, by a person lecturing on a scientific subject … just as what we get out of the lecture is something other than the noises we hear proceeding from the lecturer’s mouth, so what we get out of the concert is something other than the noises made by the performers. In each case, what we get out of it is something which we have to reconstruct in our own minds, and by our own efforts.

Replication of a mathematical proof corresponds to reconstruction of thought. In the context of history as such, Collingwood called this “re-enactment,” and I took it up in a “2018 post,” revisiting the idea (because apparently it is difficult to understand) in “To Be Civilized.”

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