Vox Meditantis

Hilda Geiringer von Mises (1893–1973) was a mathematician of formidable gifts whose work on plasticity theory and structural rigidity became foundational to modern materials science and engineering – equations still cited after nearly a century, still guiding how engineers design bridges and aeroplane fuselages. Yet the woman whose name appears on those equations spent decades as an underpaid lecturer and isolated professor at small colleges, her early promise in Weimar Berlin shattered by Nazi persecution and tempered by the particular invisibility that befalls accomplished women in the shadow of famous men. Her story is one of profound resilience: learning Turkish to teach whilst fearing for her life, fleeing twice across continents, and continuing to produce rigorous mathematics from professional peripheries. She reminds us that the history of science is not merely about discovery, but about who gets remembered, who gets displaced, and whose equations continue to work long after their creators have been forgotten.

Professor Geiringer, thank you for speaking with us today. I’d like to begin where your story begins – Vienna at the turn of the century. What was it like growing up in a mathematical household, and how did mathematics present itself to you as a young woman?

Ah, Vienna. You know, I was born into a world where ideas were treated like currency. My father, Ludwig Pollaczek, was an engineer – not famous, but serious. He believed in rigorous thinking as a moral discipline, not merely an intellectual one. This was important for me, I think, because mathematics never appeared to me as pure abstraction, as some Platonic realm floating above the material world. It was always connected to *how things work*.

Growing up in fin-de-siècle Vienna, you absorbed a particular atmosphere. The coffeehouses buzzed with philosophical argument. There was this sense that one could think one’s way toward truth if one thought carefully enough. My parents – my mother came from an educated Jewish family as well – never treated my interest in mathematics as peculiar, though it certainly was unusual for a girl at the time. But unusual was perhaps more tolerable in Vienna than in other places.

I was educated initially at home, and then at a gymnasium. The gymnasium was segregated, naturally. We girls received what was considered a suitable education. But I had teachers who recognised that my interest in mathematics was genuine, not decorative. That made an enormous difference.

And then you went to the University of Vienna. That was 1912?

Yes, 1912. I was among the first cohort of women formally admitted. You must understand – this was not yet normal. There were objections from various quarters, though by then the objections had become somewhat ritualistic. The university authorities found it difficult to argue that women were incapable of understanding mathematics once women began demonstrating that they were perfectly capable of understanding mathematics.

I worked under Wilhelm Wirtinger, a magnificent mathematician. He gave me a dissertation on Fourier series in two variables – quite technical, quite demanding. What I remember most vividly is not the mathematical result itself, but Wirtinger’s manner. He expected rigour from you; he expected clarity. He would not accept fuzzy thinking dressed up in elaborate notation. That lesson served me far better than any individual theorem.

I received my doctorate in 1917. The war was still happening, you understand. Everything felt somewhat suspended, unreal. The university was half-empty; many of the men were at the front. I felt, in some peculiar way, that I had entered the university during a time when the usual barriers had momentarily loosened. By the time many of the men returned, I was already established.

After Vienna, you went to work as an assistant editor for the *Jahrbuch über die Fortschritte der Mathematik*. That sounds quite different from conducting original research.

Hilda Geiringer: Yes, well. One must be practical. I was a woman mathematician in the 1920s. The positions available to women were limited. One could teach at a gymnasium, perhaps. One could do editorial work. The *Jahrbuch* was respectable work – reviewing mathematical literature, keeping abreast of what was being published across Europe. It was how I maintained connection to the mathematical community during a period when my own research time was constrained.

But I will not pretend it was satisfying. I read other people’s theorems. I arranged their ideas into categories. It was necessary labour, but it was not creation. I wanted to create. I was hungry for it in a way I can hardly describe now.

Then, in 1921, Richard von Mises founded the Institute of Applied Mathematics in Berlin, and he invited me to join him. This was the turning point.

This is a crucial moment. Tell me about that invitation and what drew you to Berlin.

Richard had recognised my work – my early papers on probability theory. We were not yet romantically involved; that came much later. He simply saw a mathematician who understood both rigorous analysis and practical problems. The Institute of Applied Mathematics was entirely new, an experiment really. At that time, pure mathematics and applied mathematics occupied separate worlds. The mathematicians of the abstract school considered applied work to be less rigorous, less pure. Applied mathematicians were often engineers or physicists doing mathematics instrumentally, without the deeper conceptual architecture.

What Richard believed – and I came to share this belief entirely – was that applied mathematics required its own rigour, its own theoretical depth. It was not merely pure mathematics applied haphazardly to problems. It required thinking differently about what problems mattered and how one organised knowledge to address them.

In Berlin, I finally had the freedom to think about problems that fascinated me. The Institute was in contact with engineers, with physicists, with people dealing with real phenomena. How do materials deform? How do structures fail? These were not idle questions. Understanding them mattered for building better machines, safer structures, more efficient designs.

And this led to your work on plasticity theory. When did you first encounter the problem of plastic deformation?

Almost immediately. We had engineers visiting the Institute, presenting problems. One category of problems concerned what happened to materials when you stressed them beyond their elastic limit. Until that point, materials science relied largely on empirical observation. You would test a material, discover its breaking point, and design conservatively below that threshold. But this gave you no conceptual understanding of what was actually happening to the material’s structure when it deformed plastically.

The classical elasticity theory – developed by Hooke, Lamé, and others – described materials beautifully so long as the deformation was elastic. Once you removed the stress, the material returned to its original shape. The mathematics was elegant. But the moment you exceeded the elastic limit, the material would undergo permanent deformation – it would not spring back. This was where elasticity theory broke down entirely.

The question that consumed me was: could one develop a mathematics of plastic deformation with the same rigour that elasticity possessed? Could one characterise not just *that* materials deform plastically, but *how* they deform, *why* they deform in particular patterns, what mathematical principles governed this process?

The problem must have seemed enormous.

Enormous, yes. And I must confess – somewhat frightening. This was territory where few mathematicians had worked. There were engineering approaches, experimental data, but no unified theoretical framework. I had to construct one from foundations.

I began by studying the existing plasticity theories, particularly the work of Lévy and others. But I realised that these approaches made assumptions that I could not justify mathematically. They assumed that plastic deformation followed particular stress-strain relationships without proving that these relationships followed necessarily from deeper principles.

What I attempted was different. I tried to establish what conditions must hold for a material undergoing plastic flow. If I assume that the material deforms in a manner that dissipates energy in a particular way, if I assume certain symmetries in the material’s behaviour, what mathematical consequences follow? This was more rigorous than the engineering approaches, but it required me to think about the physical foundations very carefully.

This is where the Geiringer equations emerge. Can you walk through the essential mathematics for our readers who are scientifically literate?

Of course. This requires some care with terminology, but I will try to be clear.

Consider a material under stress – imagine a metal plate being stretched or compressed. In the elastic regime, the relationship between stress (force per unit area) and strain (deformation per unit length) is linear – this is Hooke’s law. The material stores elastic energy, and when you remove the stress, it releases that energy and returns to its original configuration.

Now, imagine you exceed the elastic limit. Plastic deformation begins. Here is where the classical theory fails: the material no longer behaves linearly. Moreover, the deformation becomes *path-dependent*. The same final stress can produce different final shapes depending on how you applied the stresses – the order matters, the history matters.

What I did was to study plane plastic deformation – specifically, deformation in two dimensions. I considered a material element under a given state of stress and asked: what patterns of plastic strain must occur if the material deforms plastically?

The essential insight was to recognise that at the point where plastic deformation begins, the material reaches what is called the yield surface in stress space. At this point, the stresses satisfy a particular relationship – called the yield criterion. For many metals, this is the Tresca or von Mises yield criterion. Once the material is yielding, the plastic strain rate must be perpendicular to the yield surface in stress space. This is the principle of normality.

From these principles, I derived differential equations governing the plastic strain rate and the stress distribution in a material undergoing plane plastic deformation. These equations characterise how the stresses must be distributed, given that the material is deforming plastically according to the yield criterion.

The mathematical result was: for plane plastic deformation, the stress components must satisfy a particular set of partial differential equations. I also derived conditions governing the slip lines – the surfaces along which the material experiences the greatest plastic shear. These slip lines form a characteristic pattern, and their geometry encodes information about how the material deforms.

What was novel in my approach was deriving these equations *from first principles*, not postulating them empirically. I showed that if you accept certain reasonable assumptions about how materials behave – normality, yield criteria, symmetry – then these equations follow necessarily.

And the practical importance?

Enormous. Engineers could now predict, with mathematical precision, how a material would deform when stressed beyond its elastic limit. The slip-line fields I characterised allowed engineers to visualise and calculate the stress distribution during plastic deformation. For metal forming processes – pressing, stamping, extrusion – these equations provided a theoretical foundation for design.

Furthermore, the equations were not merely descriptive. They made *predictions*. You could use them to calculate force requirements for particular forming operations, to predict the shape changes that would result, to identify regions where excessive strain might cause material fracture.

I presented this work at the 1930 International Congress of Technical Mechanics in Stockholm. The reception was gratifying – engineers and mathematicians recognised that this represented a substantial advance. The mathematics was rigorous, but it was also useful.

Nearly a century later, these equations remain fundamental. Has the framework proven robust?

I am told by contemporary researchers that the fundamental structure remains sound. Of course, the field has evolved. There are now refined theories incorporating work-hardening, more sophisticated yield criteria for anisotropic materials, numerical methods for computing solutions. But the conceptual foundation – that plastic deformation can be characterised by differential equations derived from yield criteria and normality – this remains central.

I confess I find this remarkable. When one derives something from first principles, one hopes it will prove durable. But one cannot know. One could easily imagine that more sophisticated material science would reveal that the assumptions I made were too restrictive, too idealised. Yet the core framework has endured.

I think this is because I tried to work at the right level of abstraction – specific enough to be useful, general enough to encompass a wide range of phenomena. I did not try to capture every detail of material behaviour. Rather, I tried to identify the essential mathematical structure that governs plastic deformation at a macroscopic level.

Before the plasticity work gained full traction, you had already achieved something remarkable in 1927 – your habilitation in Berlin, becoming the first woman to habilitate in applied mathematics in Germany.

Yes. That year remains vivid in my memory, though the process itself was deeply unpleasant.

You must understand what habilitation meant. It was not merely a higher degree. It was a qualification to lecture independently at the university level, to lead research, to supervise students. In the German system of that era, it was the gateway to an academic career. Without it, you remained an assistant, dependent on others’ positions.

The process began in 1925. Richard proposed that I pursue my habilitation. Straightforward enough, or so I thought. I prepared my research dossier – my published papers, my scientific work. And then the objections began.

There was no explicit rule preventing women from habilitating. But there were extraordinary obstacles placed in the way. The faculty committee requested additional documentation, additional evidence of my capabilities. The process, which typically took a year, stretched across two years. There were discussions behind closed doors about whether women were suited to independent academic work, whether my achievements were truly my own or whether I had been unduly influenced by male colleagues, whether students would respect a female lecturer.

It was exhausting and humiliating. I had to prove not merely that I was qualified, but that my qualification was *legitimate* – that I was not somehow an exception, a statistical anomaly, but genuinely capable.

Emmy Noether was at Göttingen during this period, and her situation was similar, though her habilitation came earlier. We were the only two women habilitating in mathematics in Germany at that time. We understood each other’s difficulties without needing to speak of them directly.

Finally, in 1927, my habilitation was approved. I became a Privatdozent, an adjunct lecturer qualified to teach and supervise research. The relief was profound, but it was tempered by a clear-eyed recognition of what the process had cost – the years of fighting, the constant questioning of my competence, the emotional labour of justifying one’s own existence in the academic world.

You were 34 years old. This was your entry into independent academic leadership.

Yes. For my male colleagues, habilitation typically came in their late twenties or early thirties. For me, it came a decade later than it should have. Those are years one cannot recover.

Still, I was grateful. I had a position. I could conduct research, teach, supervise students. The Institute was growing. I felt that I had finally arrived at something like security.

Of course, that security would prove temporary.

While working on plasticity, you also made contributions to an entirely different area – structural rigidity. The Geiringer-Laman theorem emerged from this work.

Yes, though I must note that Laman rediscovered this result independently decades later. My initial work in 1927 concerned a problem that seemed quite removed from plasticity, but which shared mathematical structure.

The question was: given a framework of rigid bars connected by hinged joints, under what conditions is the structure rigid in the plane? That is, when can you deform it without bending the bars themselves?

This is not a trivial question. For engineering, it matters enormously. A bridge truss, for instance, must be rigid. You want the joints to hinge freely, but you do not want the overall structure to collapse or deform excessively under load. So you must know: how many bars do you need? How must they be arranged?

The classical approach was largely empirical. Engineers had formulas – a triangulated structure is rigid, a quadrilateral with its diagonals is rigid – but these were rules of thumb. What was lacking was a general characterisation.

What I proved was a combinatorial condition. For a planar framework of n joints, a necessary and sufficient condition for generical rigidity is that the framework contains exactly 2n – 3 independent bars, and that every sub-framework is also sufficiently braced. This is now called the Geiringer-Laman theorem, though I did not discover it with Laman – he arrived at it independently some forty years later.

The theorem was important because it converted a geometric problem into a combinatorial one. One could now determine rigidity by counting and checking connectivity, rather than by attempting to solve complex geometric equations.

This seems quite distant from plasticity theory.

In application, yes. But mathematically, there is a connection. Both problems require one to understand constraints – in plasticity, the constraints imposed by the yield criterion; in rigidity, the constraints imposed by the requirement that bars remain rigid. Both are ultimately about characterising the degrees of freedom available to a system under specified constraints.

Furthermore, structural rigidity connects to plasticity in practical ways. A structure might be rigid elastically, but what happens when it yields? How does plastic deformation propagate? Understanding rigidity helps inform one’s intuition about plastic collapse mechanisms.

I did not pursue this line as deeply as I might have, partly because the plasticity work consumed so much energy. But it represents a direction that I found intellectually compelling – the mathematics of constraints and degrees of freedom.

And then 1933 arrived. The Nazi government passed the Law for the Restoration of the Professional Civil Service, which removed Jewish academics from German universities.

Yes. I was removed from my position at Berlin in December of that year. Twelve years of work, simply erased by decree.

I do not think anyone who did not experience it can truly understand what that felt like. One moment, you had a position, colleagues, students, a research programme. The next moment, none of it existed. The university sent notice of termination. There was no appeal process, no negotiation. You were simply gone.

The cruelty of timing was particular. I had finally achieved security. The habilitation had proven I belonged. My research was gaining recognition. The plasticity work was being published, being discussed internationally. And then it all stopped.

Many Jewish academics left Germany immediately. Some went to England, some to America, some to other European countries. For me, the situation was complicated. I was a woman mathematician seeking academic positions during an economic depression. The British universities had few positions available. American universities had some, but they were competitive, and prejudice against women was substantial.

I was offered a position at the University of Istanbul in Turkey. It was not ideal – I had to learn Turkish, which was arduous – but it was a position. It was a way to continue working. So in 1933, I went to Turkey.

How long did you remain there?

Until 1939. Six years. During that time, I taught mathematics, published papers, tried to maintain my research programme despite the constraints. Istanbul was not Berlin. The mathematical community was smaller, more isolated. But the university was welcoming, and my Turkish colleagues were kind.

But as the 1930s progressed, it became clear that Turkey was becoming unsafe as well. The rising tensions in Europe, the growing Nazi influence – Turkey was neutral, but increasingly caught between competing pressures. I began to fear that what had happened in Germany might eventually happen elsewhere.

In 1939, I made arrangements to leave. I went to Portugal first, then to America. It was a narrow escape, I was later told. Had I remained in Turkey a bit longer, travel to America would have become far more difficult.

So you arrived in America in 1939. What was that transition like?

Disorienting. I was now 46 years old. I had been displaced twice from academic positions. I arrived in a country where I knew few people, where the mathematical community, whilst vibrant, was not yet fully aware of my work outside specialist circles.

I eventually secured a position at Bryn Mawr College in Pennsylvania – a lecturer position. I must be candid about this: the rank was insulting. I was given the same title as someone fresh from their PhD, someone beginning their academic career. I had by then published extensively, had supervised students, had made contributions that were being recognised internationally. Yet I was ranked as a beginning instructor.

The pay was minimal. I supplemented my income by taking additional teaching assignments. I was 47 years old, starting over as though I were 27.

Bryn Mawr was a women’s college, which must have had its own particular character.

Yes, and there was a certain irony to it. I had fought so hard for recognition in a male-dominated system in Germany. Now I found myself at a women’s institution, where the mathematics department consisted entirely of women. One might have thought this would be a haven. In some ways, it was. The other women were talented and welcoming. But Bryn Mawr was a teaching college, not a research institution. The mathematics curriculum was aimed at undergraduate students. There was minimal research activity, minimal contact with active mathematicians outside the college.

I spent my days preparing lectures for calculus and introductory mathematics courses. In the evenings and weekends, I pursued my own research when I could find time and energy. It was a half-life, intellectually speaking.

After Bryn Mawr, I moved to Wheaton College in Massachusetts. This was in 1944, and I finally received what was called a permanent position – Professor and Chair of Mathematics. I was 51 years old at that point, more than a decade after losing my Berlin position.

But “Professor and Chair” at Wheaton meant something quite different from what one might imagine. The mathematics department consisted of two people – myself and one colleague. We taught undergraduates. There was no graduate programme, no doctoral students, no research seminar. I was the chair of a department that had no research mission whatsoever.

Yet you continued to publish.

Yes, though I wish I could say I published more. During the Wheaton years, I published papers on probability theory, particularly as it applied to genetics and population mathematics. This work interested me enormously – the mathematical structure underlying Mendelian inheritance, how one could connect Mendel’s experimental observations to rigorous probability theory.

But the research was constrained. I had no students to work with, no colleagues actively engaged in mathematics beyond teaching. The library’s mathematical holdings were limited. I had to travel to Harvard to use their library, to attend seminars, to maintain contact with the broader mathematical community.

Harvard. Your husband, Richard von Mises, was at Harvard.

Yes. We married in 1943, whilst I was at Bryn Mawr. Richard held the Gordon-McKay Professorship at Harvard – a prestigious chair. He was building a research group, publishing prolifically, leading one of the important centres of probability and applied mathematics in America.

I do not wish to suggest that our marriage was unhappy. Richard was generous, intellectually stimulating, devoted to me. But the professional disparity was acute. He had students, collaborators, library resources, time for research. I had teaching loads and administrative duties at a small college.

Every weekend, I would commute to Cambridge to be with him. I would attend his seminars, discuss mathematics with his colleagues, borrow books from the Harvard library. Then I would return to Wheaton to teach. It was exhausting, and it was a poignant illustration of how even a woman who had achieved professional distinction remained systematically disadvantaged.

Did you wish to move to Harvard yourself?

Of course. But Harvard was not going to hire me. There were no open positions in the mathematics department for a woman, or if there were, they had not materialised during the years I was available. The unspoken rule in American universities at that time was that you did not hire two members of the same family. It was called nepotism policy, though it applied far more rigorously to wives than to husbands. Richard could have had me appointed, but such an appointment would have created scandal. So instead, I remained at Wheaton, and our marriage involved weekly separations that neither of us desired.

I do not say this bitterly – well, perhaps I do, a little. But it is simply the reality we navigated. Many women mathematicians of my generation faced similar constraints. We were told we were fortunate to have positions at all, that we should be grateful, that to complain was to be ungrateful or ambitious in an unseemly way.

Did the isolation affect your research?

Profoundly. One cannot do mathematics in isolation, not really. Mathematics is a conversation. You need to discuss ideas with colleagues, hear their objections, their questions, their perspectives. You need access to the latest work, to seminars, to the current discussions in your field.

At Wheaton, I was intellectually alone. My colleagues were capable, but they were not working on problems at the frontier of mathematics. The mathematics department itself was peripheral to the college’s mission. I longed for a situation where I was among mathematicians who were carrying out research, who understood the current problems, who could engage with my work at the level at which I was working.

I published during those years, yes, but I did not produce work of the calibre I might have produced had I been at a research institution with colleagues and students. This is not self-pity; it is simply the mathematics I would have written had the circumstances permitted.

Your work on probability and genetics – can you describe that?

Yes. This was a different direction than plasticity, though again, the underlying mathematical structure interested me. The question was: given the principles of Mendelian inheritance, how do one predict the distribution of traits across generations in a population?

Mendel had established experimentally that inheritance follows particular patterns – traits appear and disappear according to rules. But Mendel worked with carefully controlled experiments. What about natural populations, where individuals mate randomly, where selection pressures operate, where mutations occur?

The classical approach was largely descriptive. One observed populations and noted the frequencies of different traits. What was lacking was a theoretical framework that would let one predict how populations would change, what equilibrium distributions one would expect.

My contribution was to apply rigorous probability theory to this problem. I derived conditions under which populations would reach equilibrium distributions – what came to be called Hardy-Weinberg equilibrium, though the principle was established by several mathematicians independently. More importantly, I developed the mathematical apparatus for understanding how populations evolve when selection pressures are present, when mutation rates are non-zero, when mating is non-random.

This was probability theory applied to biology, applied mathematics in a biological context. The mathematics was rigorous, but it required thinking about what assumptions were reasonable about biological processes.

Let me ask you about something that must be difficult. Your name became attached to the Geiringer equations, your contributions were published and recognised by specialists. Yet you remained largely unknown outside those circles. How did you experience that disconnect?

You are asking whether it troubled me. Yes, it troubled me, though I became somewhat inured to it over time.

The plasticity equations – they circulate in the scientific literature with my name attached. I am told they appear in modern textbooks, that they are used by engineers designing structures. That is gratifying. But it is also peculiar. The equations have a life independent of me. Young engineers use them without knowing who I was, without knowing the story of how they came to be derived.

What bothered me more was a different phenomenon. I noticed that when my work was discussed, it was often described as if it had emerged from collaboration with Richard, or as if I were following up on his ideas, rather than as independent contributions I had made before we were even romantically involved. A lecture in my honour once introduced me as “Professor von Mises’s wife,” despite my having my own career and my own contributions.

There was an interview recorded for the Freud Archive in 1953 – I do not know why the Freud Archive was interested in mathematicians, but they recorded conversations with several of us. When that recording was catalogued, it was listed as “Von Mises, Hilda,” despite my being the primary contact. Richard was referred to as “Herr Professor,” whilst I was “Gnädige Frau” – gracious lady – despite my also being a professor.

These are small things, perhaps. But small things accumulate. They send a message about who is considered the primary actor and who is considered the supporting player.

Do you think this erasure was inevitable, or was it a choice made by historians and institutions?

Both, perhaps. The historical record is not inevitable – it is constructed by the people who write about the past, the people who decide which stories matter. But those people operate within constraints, within assumptions about what is natural and what requires explanation.

Emmy Noether died in 1935, shortly after arriving in America. In some terrible way, her early death transformed her into a kind of martyr. Over the decades, recognition of her contributions grew. Biographies were written. She became famous, or at least her name became known. Her death somehow sanctified her legacy.

I survived. I lived another thirty-eight years after 1933. I continued to work. But survival meant enduring decades of marginalisation that are harder to narrativise as tragic. My story is messier, more complicated. It involves institutional failures that continued for decades. It involves gender discrimination in America as well as Nazi persecution in Germany. It involves the erosion of professional status through displacement, exile, and displacement again.

Perhaps a dramatic death makes for a cleaner historical narrative than a long, difficult survival.

That seems unfair.

Many things are unfair. One does not become a mathematician in order to ensure that one’s life is fair. One becomes a mathematician because the problems fascinate you, because understanding how things work matters more than other things. The fairness or unfairness of recognition is secondary to that.

Still, I will say this: I wish that my work on plasticity theory had received more systematic study, that younger mathematicians had engaged with the foundational assumptions more critically. The equations I derived were based on particular assumptions about how materials behave.

Those assumptions are reasonable, but they are not universal. I wonder what developments might have emerged had more mathematicians taken up these problems.

I want to ask you about a failure, if I may. In your 1935 paper on plastic flow, you made an assumption about material isotropy that later researchers found to be too restrictive. Do you recall this criticism?

Yes, and I was not entirely confident in that assumption even then. The mathematical analysis was cleaner if one assumed isotropy – that the material’s properties were uniform in all directions. This was often approximately true for metals, but it was never exactly true. Real materials have grain structures, directional properties. I knew this, but I pursued the isotropic case first because it was tractable.

What I wish I had done differently was to be more explicit about this limitation in the original paper. I mentioned it briefly in passing, but I did not sufficiently emphasise that the framework needed to be extended to handle anisotropic materials. This was not an oversight exactly – I was aware of the issue – but it was a failure of communication. Readers took the isotropic theory as more general than it was.

Of course, subsequent researchers did extend the theory to anisotropic materials. That is the normal progression of science. One proves results under simplifying assumptions, then gradually relaxes those assumptions. But I think the field would have moved faster had I been more upfront about the limitations.

Was there pressure to present the theory as more complete than it was?

Not explicit pressure, but implicit pressure certainly. When one publishes a result, particularly if one is a woman publishing in a field dominated by men, there is an incentive to present the work as conclusive, as definitive. To hedge, to say “this is preliminary, this needs more work,” risks having the work dismissed as incomplete or immature.

Male colleagues could present a theory as a foundation for future work. When a woman did so, it was sometimes interpreted as a lack of rigour, a failure to complete the work properly. So there was incentive to present as if the work were more finished than it necessarily was.

I am not entirely satisfied with some of my choices in how I presented certain results. But I do not think I was uniquely dishonest about this. It was how the game was played.

You mention “the game.”

Science is presented as pure reasoning, as a search for truth uncontaminated by social pressures. This is misleading. Science is conducted by humans embedded in institutions with status hierarchies, economic constraints, professional pressures. One’s presentation of results, the problems one chooses to emphasise, the audiences one seeks to convince – all these are shaped by social factors.

I did not invent this situation. It predated me and has persisted after me. But I participated in it. I made strategic choices about what to emphasise, how to frame my contributions, whom to engage with. These were not dishonest choices, but they were choices, and they reflected the constraints I was operating under.

A male mathematician at a prestigious institution can afford to be more candid about limitations, more willing to present incomplete work, more open about dead ends. The system is more forgiving of such candour when it comes from established figures in positions of power.

There is a particular cruelty in your story. The Geiringer equations remain in use because they are correct and useful. Yet most people who use them do not know your name. Does this bother you now, in retrospect?

It bothers me less now than it did in the 1950s, when I was acutely aware that I was disappearing from history in real time. I could see it happening – the omissions, the re-attributions, the casual dismissals.

Now, at my age, I have made peace with it. I cannot control what people remember or forget about me. The equations will outlast any recognition. That is, in some way, a kind of immortality. My name is not necessary for the work to persist.

What bothers me more – and what has troubled me throughout my life – is the *possibility* that the undervaluing of my contributions reflected not misogyny alone, but a genuine intellectual mistake. What if there was something wrong with my approach, something that I was too close to see? What if, because I was marginalised, I lacked the critical engagement that would have corrected my errors?

This is a fear I have carried. One cannot know if one’s work is sound until it is tested rigorously by many minds over many years. When one works in isolation, one loses that protection.

The good news is that the plasticity equations have been tested, refined, extended, and they have proven robust. So I can set that fear aside.

But there is a lesson here. The way we organise science, the way we allocate positions and resources and recognition, this shapes what gets discovered and what gets overlooked. If talented people are systematically excluded from positions where they can do rigorous work, if their contributions are systematically devalued, if they are isolated rather than integrated into communities of active researchers, then the entire enterprise of science suffers.

It is not merely a matter of fairness to individual scientists, though it is that. It is a matter of what science can discover and how quickly it can discover it.

Do you think younger women mathematicians face the same constraints?

It is better than in my time, I believe. There are more positions available. There are women at prestigious universities now. There are funding mechanisms specifically designed to support women researchers. These are genuine advances.

But I would caution against complacency. The obstacles are more subtle now, which makes them harder to see and harder to fight. A young woman mathematician might not face explicit barriers. She might face instead a culture where she is not fully welcomed, where her contributions are attributed to collaborators, where she is invited to serve on committees rather than to lead research, where the hours expected of her are somehow always a bit more than those expected of her male colleagues.

These are not rules. They are not written down. But they are powerful nonetheless.

Furthermore, the question of what fields women mathematicians are encouraged to enter, what problems are considered prestigious, what kinds of work receive recognition – these are still structured in ways that advantage some and disadvantage others. Applied mathematics is more welcoming to women than pure mathematics, I believe, partly because it is newer and has not had as long to calcify its assumptions. But I do not know if this will persist.

What would you advise?

I would say: do not wait for permission. Do not wait for the institution to recognise your worth before you pursue problems that matter to you. Work on what fascinates you, with whatever resources you can access. Publish, present at conferences, build relationships with other mathematicians who understand your work.

And be realistic. The institutions have their own logic, their own constraints. You cannot change them by wishing. But you can build networks, alliances, communities that transcend institutional limitations. You can find collaborators, colleagues who value your work and whose work you value in return.

Finally, remember that the problems you are solving might outlast recognition of your name. That is not nothing. In some ways, it is everything. But it is also true that being remembered matters – not for personal gratification, but because how we tell the history of science shapes what young people imagine as possible, what paths they think are open to them.

So do not sacrifice your identity to the work. Insist on being known, on being credited, on having your name attached to your contributions. Make it harder to erase you.

I wonder if you could reflect on what has become of applied mathematics since your time. The field you helped establish has changed so dramatically.

Yes, changed in ways I could not have imagined. When Richard and I were building the Institute of Applied Mathematics in Berlin, we thought of applied mathematics as bridging pure mathematics and engineering. We thought of it as having equal rigour to pure mathematics but oriented toward problems that arose in practice.

The digital computer has transformed this entirely. When I derived the Geiringer equations, one solved them analytically, deriving closed-form solutions through clever mathematical manipulation. Now, I am told, researchers use numerical methods and computers to solve complex systems of equations approximately, in ways that would have been inconceivable to us.

This is wonderful and troubling in equal measure. Wonderful because it opens possibilities for solving far more complex problems. Troubling because the conceptual understanding can become obscured by the computational machinery. It is possible to use sophisticated numerical methods to solve a problem without understanding what is really happening mathematically.

I hope that modern applied mathematicians retain a commitment to understanding, to deriving the equations from first principles, even as they employ powerful computational tools. The mathematics matters. The conceptual structure matters.

Is there work you wish you had been able to pursue?

Many directions, honestly. I would have liked to have engaged more deeply with the problem of plastic collapse – not just the mechanics of plastic deformation, but how structures fail catastrophically when subjected to extreme loading. This involves plasticity theory, but it also involves stability theory, bifurcation theory, the mathematics of how systems transition from one state to another.

I would have liked to pursue further work in probability and genetics, extending the framework to handle more realistic models of population dynamics – incorporating overlapping generations, age-structured populations, the effects of spatial structure on genetic distributions.

I would have liked to understand more deeply the connection between the microscopic structure of materials and their macroscopic plastic behaviour. My theory operated at the continuum level, treating the material as a continuous medium. But what about the crystalline structure? What about dislocations in the crystal lattice? There seemed to me to be deep mathematical questions there about how to bridge scales.

And structurally, I would have liked to have had collaborators, students, an intellectual community engaged in pursuing these directions together. The isolation has limited what I could accomplish.

But these are the normal regrets of any scientist – more problems than time, more directions than one can pursue simultaneously. It is not unique to me, though my circumstances gave me less time and fewer resources than I might have had.

I notice that in recent years – really, very recently – there has been renewed attention to your work and your story. The BBC article, the Hilda Geiringer Scholarship, the street in Berlin named after you. Does this mean something to you?

Yes, it means something. I do not want to be falsely modest about that. To have one’s work recognised, to have one’s name restored to problems one solved, to have institutions establish funds in one’s honour – these are affirmations. They suggest that the erasure was not inevitable, that deliberate action can recover forgotten contributions.

But I notice also that these recognitions have largely come after my death. The BBC article appeared recently, is that not so? The scholarships are established decades after I left the academy. The street is named in my honour only in 2017, more than forty years after I stopped teaching.

There is something bittersweet in this timing. Yes, I am grateful. But I also think of how these affirmations would have meant something different, something more, had they come when I was still active, still striving to be recognised.

This is not a criticism of those who are now recovering the history. They are doing necessary work. But it also illustrates how thoroughly the erasure succeeded during my lifetime. I became a footnote in real time, only to be rediscovered as a chapter in historical retrospects.

Perhaps this is how historical recovery works – one forgets, then one remembers, then one wonders how one could have forgotten. But if it is possible to do this work of recovery while the person is still alive, I would argue that one should do it. The recognition means something different when one can experience it.

Would you have wanted a different life? A different career?

That is perhaps the most difficult question. To ask me that now is to ask me to second-guess the path I have taken, and I am not certain that is wise.

Would I have preferred to have been a man? Certainly. The obstacles would have been removed entirely. I would have habitated in the normal timeline, would have remained in Berlin through the 1930s and beyond, would have had students, colleagues, the full institutional support that was denied me.

But I cannot live that life. I lived the life I did live.

Would I have preferred not to have fled Germany? The persecution was real and terrible, but it was not unique to me – many of my colleagues faced the same fate. The mathematics was important, but not more important than the human cost of that era.

What I think I would have chosen, had I had the power to choose, was this: to have remained in Germany had it been safe, surrounded by colleagues and students, working at the frontier of applied mathematics. If I had to leave, I would have wanted to arrive in America and be integrated into a major research university immediately, not relegated to a small teaching college for years. And I would have wanted to avoid the particular invisibility that came from being a woman married to a more famous man.

But I cannot construct counterfactual histories. I can only affirm this: the life I lived was difficult and constraining, but it was also purposeful. I did significant mathematics. I contributed to fields that matter. I helped establish applied mathematics as a discipline. That is not nothing.

As we approach the end of our conversation, I wonder if there is something you would want to say to young mathematicians – particularly young women – who might read this.

I would say several things.

First: the obstacles are real. If you are a woman, if you are Jewish, if you belong to any group that is systematically disadvantaged in the academy, the path will be harder than it is for others. This is not your fault. Do not internalise the obstacles as personal failures.

Second: the work matters more than the recognition. This sounds like something one says to comfort people, and there is an element of comfort-seeking in it. But it is also true. If you solve a problem that nobody recognises, the problem is still solved. The equations still work. The understanding is still there, even if nobody knows your name attached to it.

Third: recognise that you are not alone in this struggle, and build alliances. Find other mathematicians who understand your work, who value it, who will carry it forward. Do not assume that institutions will do this work for you – they often will not. Build communities that transcend institutional structures.

Fourth: be practical about the constraints you face. You cannot change the entire system. But you can navigate it strategically. You can position yourself where opportunities are possible. You can cultivate relationships with people who have power and use those relationships to create space for yourself.

Fifth: do not compromise the quality of your work in pursuit of recognition. It is tempting to simplify, to make work more palatable to potential audiences, to present incomplete work as complete in order to seem more accomplished. Resist this. Your work will be tested eventually. If it is good, it will survive the testing. If it is poor, all the presentation skill in the world will not save it.

And finally: remember that you are part of a longer history. The women who come after you will build on your work. Your struggles will inform their understanding of what is possible. Your refusal to disappear – even if the institutions try to make you disappear – teaches them that they too can refuse to disappear.

That is perhaps the most important thing.

Thank you, Professor Geiringer. This has been extraordinary.

You are welcome. I confess I did not expect to have this conversation – at my age, one does not anticipate such opportunities. But it has been meaningful to reflect on these things, to try to render in words what the mathematical life felt like, what the constraints felt like, what the work meant. I hope it is of some value to those who read it.

Letters and emails

The response to our interview with Hilda Geiringer has been remarkable. Readers from across the globe – materials scientists, mathematicians, historians, engineers – have written to us with questions that extend far beyond what we could cover in a single conversation. Their inquiries reveal something important: Geiringer’s work continues to resonate not merely as historical achievement, but as living mathematics that shapes how contemporary researchers think about their own problems.

What follows are five letters and emails, selected from our growing community of readers, each offering a distinct perspective on Geiringer’s contributions and their implications for modern practice. A materials scientist in Zimbabwe asks about the technical choices Geiringer made in constructing her theory. An applied mathematician in Indonesia probes the conceptual bridges between seemingly distant fields. A science historian in Amsterdam explores how displacement and exile transformed her thinking. A structural engineer in Chile questions the relationship between theory and experimental validation. And a science communicator in Boston poses a speculative but crucial question: what might mathematics have become had Geiringer remained in Berlin?

These are not questions of idle curiosity. They are questions from practitioners and thinkers who recognise that understanding Geiringer’s work requires understanding not only what she discovered, but how she reasoned, where she made deliberate choices about simplification, what limitations she acknowledged, and what might have been pursued had circumstances been different. In these letters, we see how the past continues to instruct the present – and how those walking similar paths in mathematics and engineering today find guidance in her example.

Chipo Dube, 34, Materials Scientist, Harare, Zimbabwe
You derived the plasticity equations assuming isotropy – uniform material properties in all directions – yet real metals have grain structures and crystalline anisotropy. When you were developing this framework in the late 1920s, did you experiment with any approaches to handling directional material properties, or did you deliberately choose isotropy as a foundation knowing others would extend it? I ask because I work with composites and functionally graded materials where anisotropy is fundamental, not peripheral, and I wonder whether your thinking about how to layer simplifying assumptions has guided how modern researchers approach similar problems.

Miss Dube, your question goes directly to one of the most vexing aspects of developing the plasticity theory, and I appreciate that you ask it from the perspective of someone working with composite materials. The problem of anisotropy – directional dependence of material properties – was very much present in my thinking during the late 1920s, though I confess I did not solve it satisfactorily.

Let me be candid about the situation I faced. When one attempts to construct a mathematical theory of material behaviour, one must make choices about what to include and what to set aside temporarily. The isotropic assumption – that the material behaves identically in all directions – was not made because I believed it to be universally true. Any metallurgist could have told you that metals possess grain structure, that crystals have preferred orientations, that rolled sheet metal behaves differently along the rolling direction than perpendicular to it. This was known.

The reason I began with isotropy was twofold. First, it simplified the mathematics sufficiently that I could actually derive results. The equations governing anisotropic plastic flow would have been substantially more complex – more variables, more constitutive relations to specify, more boundary conditions to satisfy. I needed to establish that a rigorous mathematical treatment of plasticity was even possible before attempting the more general case.

Second, and perhaps more importantly, the experimental data available in the 1920s for anisotropic plasticity was quite limited. We had good data for isotropic metals – carefully prepared specimens with fine, randomly oriented grains. These could be tested systematically, and the yield behaviour was relatively consistent. But for strongly anisotropic materials, the experimental characterisation was incomplete. I would have been constructing a mathematical theory with insufficient empirical grounding to validate it properly.

Now, did I experiment with approaches to anisotropy? Yes, though I did not publish this work. I attempted to extend the yield criterion to incorporate directional dependence – essentially, allowing the yield surface to have different radii in different directions in stress space. The mathematics became unwieldy very quickly. One needed to specify not merely a scalar yield stress, but a tensor describing how yield stress varied with direction. Then the flow rule – the relationship between stress and plastic strain rate – also required tensorial generalisation.

What I discovered was that without clear physical principles to guide the choice of constitutive relations, the theory could accommodate almost any behaviour one wished. This was unsatisfying. A good theory should be restrictive, should make predictions that can be falsified. An overly flexible theory that can fit any data is not really a theory at all.

I thought at the time – and I still believe – that the proper approach would have required close collaboration between mathematicians and metallurgists. The metallurgists would need to identify which aspects of anisotropy were essential and which were secondary effects. Then the mathematician could construct a theory that captured the essential features whilst remaining tractable. But I was working largely in isolation at Berlin, without sustained contact with experimental metallurgists who were studying anisotropic plasticity.

Had I remained at a research institution with doctoral students and collaborators, I believe we could have pursued this direction productively. One student might have worked on extending the yield criteria, another on numerical methods for solving the anisotropic equations, a third on experimental validation with specific materials. But at Wheaton, teaching undergraduates with no research programme, such collaborative work was impossible.

So to answer your question directly: yes, I explored anisotropy, but I chose to publish the isotropic theory as a foundation, knowing – hoping, really – that others would extend it. Whether this was the right choice, I cannot say with certainty. It allowed me to establish the framework rigorously. But it also meant that certain important phenomena remained outside the theory’s scope for decades.

I am gratified to hear that you work with functionally graded materials where anisotropy is fundamental. This suggests that the field has matured to the point where it can handle the complications I set aside. That is how science should progress – one establishes foundations, however imperfect, and others build upon them, correcting and extending as they go.

Santiago Ferreira, 52, Professor of Structural Engineering, Santiago, Chile
The Geiringer equations remain in our textbooks and our computational models, but I wonder about the experimental validation. In your era, how extensively were the predictions tested against real material behaviour? And more specifically – were there cases where your theoretical predictions diverged substantially from experimental results, and if so, how did you respond? I ask because modern engineers sometimes treat theoretical frameworks as gospel without understanding their limitations, and I think understanding where your own work broke down would be valuable for students learning to think critically about mathematical models.

Professor Ferreira, your question about experimental validation is one that I wish more engineers would ask. You are quite right to be concerned about treating theoretical frameworks as gospel – this has always been a danger in applied mathematics, and I suspect it has only worsened with the sophistication of computational methods.

When I developed the plasticity equations in the late 1920s, experimental validation was both essential and frustratingly incomplete. Let me describe the situation as it existed then.

The best experimental work on plastic deformation was being conducted at several laboratories – particularly in Germany and England, where engineers were testing metal specimens under various loading conditions. They would apply loads to carefully prepared samples, measure the resulting deformations, and attempt to characterise the stress-strain relationships beyond the elastic limit. This was painstaking work. The measurements were not trivial – one needed to detect permanent deformations accurately, account for temperature effects, ensure that the loading was truly uniform across the specimen.

What we found was that for certain simple loading cases – uniaxial tension, simple compression, pure torsion – the agreement between my theoretical predictions and experimental observations was quite good. The slip-line patterns I predicted could actually be observed on the surfaces of deformed specimens. When engineers etched the surface of a metal that had undergone plastic deformation, they could see lines corresponding to the directions of maximum shear – precisely what the theory predicted. This was enormously gratifying.

However, for more complex loading conditions, the situation became less clear. When specimens were subjected to combined loading – simultaneous tension and torsion, for instance, or non-uniform stress distributions – the agreement was less precise. There were several reasons for this, and I want to be honest about them.

First, my theory assumed that the material was homogeneous and isotropic at the macroscopic scale. Real materials are not perfectly homogeneous. There are inclusions, voids, grain boundaries – all sorts of microstructural features that affect local behaviour. For mildly inhomogeneous materials, these effects averaged out reasonably well, and the theory held. For materials with significant inhomogeneities, the predictions could diverge substantially from observation.

Second, the theory assumed that plastic deformation occurred according to a particular flow rule – the principle of normality, which states that the plastic strain rate is perpendicular to the yield surface in stress space. This is a reasonable assumption for many metals, particularly those that do not work-harden significantly during deformation. But for metals that harden substantially – where the yield stress increases as deformation proceeds – the simple theory needed modification. I was aware of this limitation and mentioned it in my publications, though perhaps not as prominently as I should have.

Third, and this is crucial, the experimental techniques of the 1920s had limitations. Measuring stress distributions inside a deforming body was extremely difficult. One could measure surface strains reasonably well, and one could measure the applied loads, but inferring the internal stress state required assumptions. So when there was disagreement between theory and experiment, it was not always clear whether the theory was wrong or whether the experimental interpretation was incorrect.

I recall one particularly vexing case. Engineers at a laboratory – I believe it was connected to one of the German industrial firms, though I cannot recall which – were studying the extrusion of metal through a die. My theory predicted a particular distribution of plastic strain and a particular force requirement. Their measurements showed forces approximately twenty percent higher than predicted. For months, I worried that there was an error in my derivation. I checked the mathematics repeatedly. I could find no mistake.

Eventually, it emerged that the discrepancy arose from friction between the metal and the die – an effect I had neglected in the theoretical analysis. Once friction was accounted for empirically, the agreement improved substantially. This taught me a valuable lesson: applied mathematics must always remain in conversation with physical reality. The mathematics can be perfectly correct and still fail to capture the phenomenon if the physical assumptions are inadequate.

So to answer your question directly: yes, there were cases where predictions diverged from experiments, and my response was to investigate whether the divergence arose from mathematical error, from inadequate physical assumptions, or from experimental limitations. Usually, it was the second – my physical model was too idealised. Occasionally, it was the third – the experiments themselves had systematic errors.

What troubles me about modern practice – and I speak here from what I have observed in the literature and heard from colleagues – is that computational methods have become so powerful that engineers can obtain numerical solutions to extraordinarily complex problems without necessarily understanding the underlying assumptions. They input material properties into a computer programme, specify boundary conditions, and obtain stress distributions and deformation patterns. The computer does not tell them where the theory breaks down, what assumptions are being violated, or how sensitive the results are to uncertain parameters.

I would urge students of structural engineering to do what you are clearly already doing: question the models, understand their limitations, and validate predictions against experiments whenever possible. A theory is a tool for understanding, not a substitute for it. When theory and experiment disagree, that disagreement is telling you something important about the world. Do not ignore it; investigate it. That is where genuine progress occurs.

Nur Amani, 28, Applied Mathematician, Bandung, Indonesia
In the interview, you mentioned that plasticity and rigidity theory seemed like distant fields until you recognised they both concerned constraints and degrees of freedom. That conceptual bridge interests me deeply because I work on optimisation problems in structural design where we’re trying to balance rigidity with minimal material use. Did you ever pursue that connection formally – proving theorems that unified both frameworks – or did it remain more of an intuition that informed your mathematical thinking? And if you’d had more research collaborators, would you have pushed further into that synthesis?

Miss Amani, your question touches on something I have thought about continuously for decades – the relationship between different areas of mathematics that appear superficially distinct but share deep structural similarities. The connection between plasticity and rigidity theory was, for me, one of the most intellectually alive discoveries of my career, though I confess I never developed it as thoroughly as I wished.

Let me explain how I came to see this connection. In plasticity theory, one is concerned with constraints imposed by material behaviour. When a material yields, the stresses must satisfy a particular relationship – the yield criterion. This relationship constrains which stress states are possible during plastic deformation. Within these constraints, the material can deform in various ways, but not in all ways. The feasible deformations form a restricted set determined by the constraints.

In structural rigidity, one encounters an entirely different physical problem. Given a framework of rigid bars connected by hinges, the structure can deform only in certain ways. The bars cannot shorten or lengthen – they are rigid. This constraint on bar lengths restricts the possible motions of the joints. The framework is rigid if the only motions satisfying the constraint are trivial – essentially, rigid body motions where the entire structure moves without any internal deformation.

Now here is the insight that fascinated me: both problems are fundamentally about counting degrees of freedom under constraints. In plasticity, you have a continuum with many degrees of freedom – theoretically infinite in a continuous medium. The yield criterion removes some of these degrees of freedom, constraining the possible strain rates. In rigidity, you have a discrete structure with a finite number of degrees of freedom – the positions of the joints. The rigidity constraints remove degrees of freedom.

The mathematical structure is similar, though the physical contexts are entirely different. In both cases, one can ask: given the constraints, how many degrees of freedom remain? For plasticity, this determines the dimensionality of the space of possible plastic flows. For rigidity, this determines whether the structure can move without violating the bar-length constraints.

I did work on formalising this connection to some extent. The Geiringer-Laman theorem, which I proved in 1927, concerns the combinatorial conditions for rigidity. It states that a planar framework with n joints is generically rigid if and only if it contains exactly 2n – 3 bars and every subframework is sufficiently braced. This is a counting argument – one counts degrees of freedom and constraints and determines when rigidity emerges.

I wondered whether similar counting arguments could be applied to plasticity. Could one characterise, purely combinatorially, which patterns of plastic flow are possible in a given material under specified loading? The answer, I found, was partially yes and partially no. For certain special cases – plane strain deformation with particular yield criteria – one could identify special structures in the stress field that admitted discrete characterisation. But the full theory remained more continuous, less amenable to purely combinatorial analysis than the rigidity problem.

What prevented me from pursuing this synthesis more deeply was, frankly, the pressure to consolidate the plasticity work. Once the equations were established and gaining acceptance, there was expectation that I would develop applications, extend the theory to handle practical problems, address the concerns of engineers. The rigidity direction was mathematically elegant but seemed less immediately practical. One does not ask a materials scientist to prove theorems about graph rigidity when there are problems in metal forming and structural design awaiting attention.

But I remained convinced – and I remain convinced now – that there is a deeper unification waiting to be discovered. Both plasticity and rigidity are instances of a more general theory about how constraints shape possible behaviours in physical systems. There are connections to control theory, to optimisation, to the mathematics of mechanisms. These are directions I would have pursued had I had a proper research environment with doctoral students capable of engaging with the problem.

Your work on optimisation in structural design, which you mention involves balancing rigidity with minimal material, seems to me precisely where this unified perspective would be valuable. You are trying to find designs that satisfy rigidity constraints whilst using minimal resources. This is a constrained optimisation problem, but it is also a problem about how constraint structure shapes feasible designs. The mathematics of plasticity and rigidity should inform each other here.

I wish I could tell you that I had developed this synthesis formally. The truth is less satisfying – I had intuitions, I had partial results, but I lacked the continuity of intellectual engagement necessary to complete the work. At Wheaton, isolated from active mathematicians pursuing similar problems, I could not develop these ideas rigorously. The notebooks where I worked on these questions remain mostly unpublished.

But I believe the direction is sound. If you have the opportunity to develop it further, I would encourage you to do so. The connection between constraint structures in discrete and continuous systems, between rigidity and plasticity, between counting arguments and continuous deformation – this is important territory. It may take decades for the full implications to become clear, but I believe genuine insights await there.

Tyler Morgan, 36, Data Scientist and Science Communication Specialist, Boston, USA
If you had survived the Nazi period and remained in Berlin, with the position and resources you held in 1932, where do you think applied mathematics as a field would have developed differently? I ask partly from curiosity about your vision, but also because I’m interested in how the diaspora of German-Jewish mathematicians to America fundamentally altered where and how mathematics was conducted. Do you think your presence and your collaborations would have shifted the field’s trajectory – perhaps keeping applied mathematics more grounded in European engineering traditions, or pushing it in entirely different directions? And conversely, what did America gain or lose by receiving you in exile rather than having you remain?

Mr. Morgan, your question is the most difficult one I have been asked, and I have thought about it countless times over the decades. What might have been had I remained in Berlin? It is impossible to answer with certainty, of course, but let me attempt to think through the possibilities honestly.

If the Nazi period had not occurred – if I had remained at the Institute of Applied Mathematics with Richard and our colleagues through the 1930s, 1940s, and beyond – several trajectories seem plausible.

First, the Institute itself would have continued to grow. By 1933, we had established applied mathematics as a legitimate field with institutional standing. We had students, we had connections to industry, we had a research programme spanning fluid mechanics, elasticity, plasticity, probability theory, statistics. Had this continued without interruption, I believe Berlin would have become the preeminent centre for applied mathematics in Europe, perhaps in the world.

My own role in this development would likely have been substantial. I had proven myself capable of independent research at the highest level. The plasticity equations had established my reputation. The probability work was gaining recognition. With security of position and access to students, I would have continued to develop both directions – extending plasticity theory to handle more complex material behaviours, developing probability methods for genetics and statistical mechanics.

More importantly, I would have trained doctoral students. This is what I regret most acutely when I consider the counterfactual. Had I remained at Berlin, I would have supervised perhaps a dozen or more doctoral students over the decades. These students would have extended my work, carried it into new domains, established their own research programmes. Some would have taken positions at other universities, spreading the Berlin approach to applied mathematics across Europe. This is how intellectual traditions are built – through generations of students carrying forward the methods and perspectives of their teachers.

The European tradition in applied mathematics might have remained more closely tied to engineering practice had we continued at Berlin. Richard and I were both committed to maintaining connection between theoretical development and practical problems. We worked with engineers, visited factories, attended technical conferences. This was not merely instrumentalism – it was a philosophical commitment to the idea that applied mathematics should be genuinely applicable, not merely pure mathematics with the label “applied” attached.

In America, by contrast, applied mathematics developed somewhat differently after the war. It became more abstract, more oriented toward general mathematical frameworks that could be applied to many contexts. This has its advantages – generality, theoretical elegance, broad applicability. But it also means that American applied mathematics sometimes loses touch with the specific, messy details of real engineering problems. A European tradition centred in Berlin might have maintained closer connection to the particularity of material behaviour, to the idiosyncrasies of real structures under real loading.

Now, what did America gain by receiving me in exile rather than having me remain in Europe? This is the question that troubles me, because the honest answer is: America gained relatively little from me specifically, though it gained enormously from the mathematical diaspora as a whole.

I arrived in America at age 46, displaced, without position, speaking English imperfectly. I spent years at teaching colleges where research was not expected or supported. By the time I might have established a research programme – had I secured an appropriate position – I was in my fifties, past the period of maximum productivity for most mathematicians. The contributions I made to American mathematics were modest compared to what I might have contributed had I remained in Berlin or had I been integrated into a major American research university immediately upon arrival.

This is not false modesty. Compare my trajectory to that of other émigrés who arrived younger or who secured better positions. Kurt Gödel went to Princeton. John von Neumann went to Princeton. They had resources, colleagues, students. They transformed American mathematics and mathematical physics. I went to Bryn Mawr as a lecturer, then to Wheaton as a professor at a small college. The disparity is stark.

What America lost by receiving me in exile rather than in full professional capacity is harder to quantify but no less real. America lost the contributions I might have made had I been positioned appropriately. It lost the students I might have trained, the collaborations I might have formed, the problems I might have solved had I had the resources and environment that my male colleagues received.

But I want to address the broader question implicit in what you ask: how did the diaspora alter mathematics? The forced migration of Jewish mathematicians from Germany to America in the 1930s was catastrophic for German mathematics and enormously beneficial for American mathematics, though the benefit was purchased at terrible human cost.

Germany lost more than half its mathematical community. The Institute of Applied Mathematics at Berlin was essentially destroyed. Göttingen, which had been perhaps the greatest mathematical centre in the world, never recovered its preeminence. The loss was not merely quantitative – losing talented people – but qualitative. The particular traditions of German mathematics, the teaching methods, the philosophical approaches, the connections between mathematics and other disciplines – all this was disrupted.

America gained extraordinarily. The influx of European mathematicians transformed American universities. Princeton, Harvard, Brown, Chicago, Berkeley – all benefited from European émigrés. The mathematical style in America shifted, becoming more abstract, more rigorous in the European manner. Applied mathematics developed as a distinct field partly because émigrés like Richard brought this perspective with them.

But the gain was uneven. Men were positioned well; women were positioned poorly. Younger mathematicians adapted more easily; older mathematicians struggled. Those who arrived with established international reputations found positions; those whose work was less widely known faced difficulties.

Had I remained in Berlin, I believe I would have contributed substantially to mathematics and to the training of the next generation. As it happened, I contributed modestly from the margins of American academia. The equations survived; the institutional influence I might have had did not materialise. That is simply the reality of displacement. One survives, one continues working, but one does not fully recover what was lost.

Lotte Vermeer, 41, Science Historian and Former Engineer, Amsterdam, Netherlands
You spent six years in Turkey learning the language, teaching mathematics under uncertain circumstances, yet you continued publishing during that period. I’m impressed by the resilience, but I’m also curious about something you touched on lightly: how did the displacement itself change your mathematical thinking? Did working outside the European mathematical establishment, outside the Berlin Institute network, cause you to approach problems differently? Did isolation force you toward more foundational, more self-contained work – or did it constrain you in ways you only recognised later?

Dr. Vermeer, your question penetrates to something I have wrestled with for many years – whether displacement itself altered not merely the conditions under which I worked, but the very nature of my mathematical thinking. I believe it did, though perhaps not in the ways one might expect.

When I was forced to leave Berlin in 1933, I lost more than a position. I lost the intellectual ecosystem that had sustained my work for twelve years. At the Institute, I was embedded in a community of mathematicians, physicists, and engineers who shared a common language, common problems, common methods. We attended each other’s lectures. We discussed difficulties over lunch. We knew intuitively what questions were considered important, what approaches were considered promising, what standards of rigour were expected.

In Turkey, this ecosystem vanished. I arrived in Istanbul in 1933 speaking no Turkish. I had to learn the language quickly enough to teach mathematics to students who themselves were often struggling with the modernisation of their educational system. The University of Istanbul was attempting to build a European-style research university from relatively modest foundations. There were talented colleagues, certainly, but they were not working on problems at the frontier of applied mathematics in the way that my Berlin colleagues had been.

What this meant practically was that I became more intellectually self-reliant – but also more isolated. When one is part of a thriving research community, one receives constant feedback. A colleague might mention a new paper that addresses a problem related to yours. A student might ask a question that reveals an assumption you had not examined carefully. A visitor might present work that suggests a different approach. This continuous exchange shapes how one thinks about problems, what directions one pursues, what assumptions one takes for granted.

In Istanbul, I lacked this. I continued to work on plasticity theory and probability, but I worked more in isolation. I published papers, but they took longer to write because I lacked the immediate feedback that helps clarify thinking. I followed directions that interested me, but I had less sense of whether these directions were the ones that would prove fruitful, less ability to gauge whether I was solving important problems or merely idiosyncratic ones.

Did this force me toward more foundational work? I am not certain. In some respects, yes. When one lacks extensive contact with the current literature – and in Istanbul, the mathematical library was far more limited than in Berlin – one is pushed to work from first principles rather than building directly on the most recent results. This can produce work that is more self-contained, more careful about establishing foundations, less dependent on lemmas and theorems scattered across the literature.

But there is a cost to this as well. Science advances by building on what others have done. If one is too isolated from current work, one risks duplicating effort, missing important connections, solving problems that have already been solved. I tried to maintain contact with the European mathematical community through correspondence, but this was a pale substitute for daily interaction.

The most profound effect of displacement, however, was psychological rather than strictly intellectual. In Berlin, I had a sense of trajectory. I was building something – a research programme, a reputation, a position in the field. Each paper contributed to an ongoing development. I could see how my work connected to my colleagues’ work, how together we were advancing applied mathematics as a discipline.

After displacement, this sense of trajectory was disrupted. I no longer knew what I was building toward. Would I remain in Turkey permanently? Would I eventually find a position elsewhere? Would the work I was doing reach audiences who could build on it, or was I working in isolation that would render the work invisible?

This uncertainty affected how I chose problems. I became more conservative, perhaps, less willing to pursue long-term projects that might require years of sustained work. If one does not know whether one will be in the same position in five years, one hesitates to begin a project that will take five years to complete. Instead, one works on problems that can be solved in shorter timeframes, that produce publishable results more quickly.

When I left Turkey for America in 1939, the isolation intensified rather than diminished. At Bryn Mawr and later at Wheaton, I was no longer working in an environment where applied mathematics was actively pursued. My colleagues were competent mathematicians, but they were teaching faculty at undergraduate institutions. They were not engaged in research at the level I had been accustomed to. The weekly trips to Cambridge to attend Harvard seminars helped somewhat, but these were brief windows rather than sustained immersion.

What I found was that isolation pushed me toward problems that required less external input – probability theory applied to genetics, for instance, where the empirical foundations came from Mendel’s experimental work and one could develop the mathematical framework relatively independently. The plasticity work, which required closer connection to experimental materials science and to engineers testing actual materials, became harder to pursue effectively.

So yes, displacement changed my mathematical thinking. It made me more self-reliant but also more isolated. It pushed me toward problems that could be worked on independently rather than collaboratively. It disrupted the sense of trajectory that gives research coherence and purpose. And it meant that the work I produced after 1933, whilst competent and sometimes valuable, never achieved the significance of the plasticity equations developed in Berlin when I had the resources, the colleagues, and the institutional support that allowed for sustained, ambitious research.

The constraint I experienced was not merely practical – lack of time, lack of library resources, lack of laboratory facilities. It was deeper than that. It was the constraint of not knowing whether the work mattered to anyone, whether it would reach audiences who could use it, whether it contributed to an ongoing conversation or simply disappeared into silence. That is perhaps the cruellest aspect of displacement for a scientist – not merely that one loses position, but that one loses certainty about whether the work has meaning beyond oneself.

Reflection

Hilda Geiringer died on 22nd March 1973 in Santa Barbara, California, at the age of 79. She had lived through two world wars, survived Nazi persecution, crossed three continents in exile, and spent the final decades of her life teaching undergraduates at a small Massachusetts college whilst her equations continued to shape how engineers understood the behaviour of materials under stress. Her death received little notice in the mathematical press. No major obituaries appeared in the journals where her work had been published. The woman whose fundamental contributions to plasticity theory remain cited nearly a century later slipped quietly from the world, her passing unremarked by the institutions that had marginalised her.

This interview – fictional though it is – attempts to recover something of what was lost in that silence. Through Geiringer’s imagined voice, we have explored not merely her scientific achievements but the texture of her experience: the pride of becoming the first woman to habilitate in applied mathematics in Germany, the devastation of having that achievement rendered meaningless by Nazi expulsion, the exhausting resilience required to rebuild a career twice over in foreign countries, the intellectual loneliness of working at the periphery whilst fundamental problems awaited proper investigation.

Throughout the conversation and in her responses to our readers’ questions, several themes emerged with particular force. First, the profound disconnect between the durability of her work and the erasure of her name. The Geiringer equations remain “basic” and “fundamental” to plasticity theory – they appear in contemporary textbooks, guide modern computational models, underpin engineering designs for structures from aircraft fuselages to bridge components. Yet most who use them know nothing of the woman who derived them, nothing of the intellectual journey that produced them, nothing of the circumstances under which they were created. Her mathematics has an afterlife; her story has struggled to find one.

Second, the compounding nature of marginalisation. Geiringer did not face a single barrier but rather an accumulation of them, each reinforcing the others. Gender discrimination limited her opportunities in Weimar Germany despite extraordinary ability. Nazi persecution destroyed her career at its peak. American academic institutions relegated her to teaching positions far below her qualifications. Marriage to a more famous mathematician – Richard von Mises – led to her work being attributed to him or described as derivative of his, despite her having established independent achievements before their relationship began. By the time recognition efforts emerged – scholarships established in her name, streets bearing her name in Berlin, biographical articles recovering her contributions – she had been dead for decades.

Third, and perhaps most painfully, the particular cruelty of intellectual isolation. In her responses to Lotte Vermeer and Nur Amani, Geiringer spoke movingly about what it meant to work outside the ecosystem of a research community – without colleagues engaged in similar problems, without students to develop her ideas further, without the daily exchange that shapes scientific thinking and provides reassurance that one’s work matters. This isolation was not merely unfortunate; it was structurally produced by the positions she was forced to accept. Small teaching colleges could not provide research environments. Anti-nepotism policies prevented her from securing positions at institutions where her husband held prestigious chairs. The very survival strategies that allowed her to continue working – accepting whatever positions were available – ensured that she would never regain the professional standing she had held in 1933 Berlin.

The perspective Geiringer offers in this interview diverges from some recorded accounts in instructive ways. Historical sources often emphasise her relationship with Richard von Mises, describing her primarily in relation to him – as his assistant, his colleague, his wife. Here, she insists on her independent scientific identity, noting that her habilitation and her plasticity work predated their romantic involvement and represented autonomous achievements. She also speaks more candidly than the archival record typically allows about the strategic choices scientists make under constraint – her decision to publish the isotropic plasticity theory knowing its limitations, her awareness that women faced different standards of judgment, her conscious navigation of institutional barriers. These are not fabrications but rather the kind of interior reasoning that rarely appears in official documents yet profoundly shapes scientific careers.

Gaps and uncertainties remain, of course. We do not have extensive records of Geiringer’s unpublished work, her failed experiments, her abandoned research directions. The notebooks she mentioned in her response to Nur Amani – containing explorations of the connection between plasticity and rigidity – may not exist or may remain in archives inaccessible to historians. Her subjective experience of displacement, isolation, and marginalisation must be inferred from biographical fragments rather than recovered from detailed personal accounts. This interview attempts to fill those gaps imaginatively, extrapolating from what we know to what she might plausibly have thought and felt, but such extrapolation always involves interpretive risk.

What we can document with certainty is the afterlife of her mathematical contributions. The Geiringer equations continue to appear in modern research literature – papers published in 2023 and 2024 in materials science and computational mechanics journals cite her 1930 work as foundational. The Geiringer-Laman theorem, proven in 1927 and independently rediscovered by Gerard Laman in 1970, now informs research in structural rigidity, robotics, and molecular biology. Her contributions to probability genetics helped establish mathematical population genetics as a rigorous subdiscipline. Graduate students in materials science and structural engineering encounter her work in their coursework, often without knowing her biography, sometimes without recognising that the equations bearing her name were developed by a woman forced into exile, working under conditions that would have broken many others.

The recognition efforts that have emerged in recent years – the Hilda Geiringer Scholarship at the Berlin Mathematical School, supporting promising female PhD students in mathematics; the Hilda Geiringer Lecture series at Humboldt University, bringing distinguished women scientists to speak; the street named Geiringerweg in Berlin Mitte; the biographical profiles in science history journals and popular media – represent attempts to correct historical erasure. Yet their very belatedness underscores how thoroughly the combination of persecution, displacement, and gender discrimination succeeded in diminishing her visibility during her lifetime and for decades after her death.

Geiringer’s story connects powerfully to contemporary challenges in diversifying STEM fields and supporting women in mathematics and engineering. The barriers she faced – implicit bias, institutional marginalisation, the tension between caregiving and career advancement, the particular invisibility of women married to famous men in the same field – persist in subtler forms today. Young women entering mathematics still encounter assumptions about their abilities, still face cultures that can be unwelcoming, still navigate dual-career challenges that disproportionately disadvantage them. The progress since Geiringer’s time is real but incomplete.

Yet her story also offers something beyond a catalogue of obstacles. It demonstrates that fundamental contributions can emerge from institutional peripheries, that rigorous thinking persists even when recognition does not, that resilience and intellectual integrity matter even when the system fails to reward them appropriately. For young women in STEM today, Geiringer’s example suggests that the work itself – solving problems that matter, deriving equations that capture real phenomena, building frameworks that others will use – can have value and durability independent of whether institutions immediately recognise that value.

Perhaps most importantly, her story illuminates why visibility matters – not merely for individual recognition but for shaping what young people imagine as possible. When women’s contributions remain invisible, when their stories are told primarily through their relationships to famous men rather than through their independent achievements, when their equations survive but their names fade, the message sent to the next generation is corrosive: women may contribute, but they will not be remembered as contributors. Recovering stories like Geiringer’s, insisting on the proper attribution of her work, establishing scholarships and lectures in her name – these are not merely gestures of historical justice. They are interventions in how we imagine the future of mathematics and who belongs in it.

The fundamental equations describing plane plastic deformation – the Geiringer equations – will continue to work regardless of whether we remember Hilda Geiringer. Materials will continue to deform according to the principles she characterised. Engineers will continue to design structures using frameworks she helped establish. The mathematics persists with a kind of impersonal durability.

But something irreplaceable is lost when we forget the mathematician herself – her ingenuity in deriving those equations from first principles, her courage in pursuing habilitation against institutional resistance, her resilience in rebuilding her career after displacement, her decades of isolated work producing rigorous mathematics from the margins of academia. That loss diminishes not merely the historical record but our collective understanding of how science is actually done – by real people navigating real constraints, making strategic choices, persisting despite marginalisation, contributing knowledge that outlasts their own recognition.

Hilda Geiringer’s equations will endure. The question for us is whether we will allow her name, her story, and her example to endure alongside them – whether we will insist that the history of mathematics include not merely the theorems but the human beings who proved them, especially those whom institutions worked so hard to forget.

Who have we missed?

This series is all about recovering the voices history left behind – and I’d love your help finding the next one. If there’s a woman in STEM you think deserves to be interviewed in this way – whether a forgotten inventor, unsung technician, or overlooked researcher – please share her story.

Email me at voxmeditantis@gmail.com or leave a comment below with your suggestion – even just a name is a great start. Let’s keep uncovering the women who shaped science and innovation, one conversation at a time.

Editorial Note

This interview transcript is a work of historical fiction – a dramatised reconstruction based on documented biographical information, archival records, published papers, and scholarly analysis of Hilda Geiringer‘s life and work. It is not a transcript of an actual conversation. Hilda Geiringer died in 1973, more than fifty years ago, and these words were not spoken by her.

What we have attempted here is to imagine, as faithfully as possible, how Geiringer might have reflected on her own life, her mathematical achievements, and her experiences navigating institutional barriers had she been offered such an opportunity for extended conversation with a contemporary interviewer possessing full knowledge of her complete biography and the scholarly literature about her work.

The historical foundations of this fictional interview are substantial. The biographical details – her birth in Vienna in 1893, her doctoral work under Wilhelm Wirtinger, her 1927 habilitation in Berlin as the first woman to achieve this qualification in applied mathematics in Germany, her 1930 derivation of the Geiringer equations, her forced emigration in 1933, her years in Turkey, her arrival in America, her positions at Bryn Mawr and Wheaton College, her marriage to Richard von Mises in 1943, her death in Santa Barbara in 1973 – all of these are documented in the historical record and accurately represented.

The mathematical content describing plasticity theory, structural rigidity, and her contributions to probability genetics reflects genuine aspects of her work. The technical explanation of the Geiringer equations and their principles are grounded in the actual mathematics she developed, though simplified for accessibility. The references to specific challenges she faced – the controversial habilitation process, the gender discrimination in American universities, the erasure of her achievements through association with von Mises, the intellectual isolation of working at small teaching colleges – all derive from biographical sources and scholarly analysis.

What is imagined rather than documented is Geiringer’s interior life: her thoughts about her own work, her emotional responses to displacement and marginalisation, her reflections on choices she made and roads not taken, her candid assessments of her own limitations, her advice to younger scientists. These emerge from careful extrapolation based on what we know about her circumstances, her intellectual commitments, her published work, and the broader historical context in which she lived.

The five supplementary questions and Geiringer’s responses to them are entirely fictional constructs. They represent conversations that never occurred with questioners who are themselves dramatised composites reflecting real professional identities and geographic locations. The questions were designed to explore dimensions of Geiringer’s work and experience that historical records do not fully illuminate – the technical choices she made, the conceptual connections she perceived, the ways displacement altered her thinking, the counterfactual scenarios about what might have been had history unfolded differently.

In crafting Geiringer’s responses, we have aimed for historical plausibility rather than documentary accuracy. The voice – the rhythms of speech, the references, the emotional cadences – reflects an educated European woman of her generation, trained in rigorous mathematical thinking, fluent in German and English, with direct experience of both Weimar intellectual culture and post-war American academia. The mathematical reasoning reflects genuine aspects of how Geiringer approached problems, based on her published work and the testimony of colleagues and students. The reflections on institutional barriers, gender discrimination, and scientific isolation reflect not her unique experience but rather patterns documented across the historical record of women mathematicians of her era.

We have endeavoured to be transparent about what is known, what is plausible inference, and what is imaginative reconstruction. Where specific historical details are presented – dates, names, positions, publications – these are factual. Where she is depicted reflecting on her own thinking or offering personal advice, readers should understand this as dramatised reimagining rather than recovered testimony.

This approach carries inherent risks. By giving Geiringer a voice, we inevitably impose our own interpretations onto her silence. We may err in imagining how she would have responded to particular questions. We may project contemporary concerns onto a historical figure in ways that distort understanding. Yet we believe these risks are worth taking because they serve an important purpose: recovering visibility for a mathematician whose contributions remain mathematically fundamental but historically obscure, whose equations endure whilst her story fades, whose innovations shaped applied mathematics as a discipline whilst she herself worked in institutional margins and relative anonymity.

This interview should be read as an act of historical recovery and imaginative engagement rather than as documentary evidence. Readers interested in Geiringer’s actual words and thoughts should consult the scholarly literature, her published papers, and archival records where they exist. This dramatised reconstruction is meant to complement such scholarship, not replace it – to humanise the historical record, to make visible the dimensions of scientific life that archival documents alone cannot fully capture, and to invite reflection on how institutional and social factors shape scientific careers and the recognition of scientific contribution.

The goal is not to deceive but to illuminate. The historical Hilda Geiringer remains complex, multifaceted, and incompletely known. This fictional conversation is one attempt to engage with that incompleteness thoughtfully, to honour both what we can know with confidence and what remains uncertain or unknowable, and to keep her story – and the stories of other women scientists whose contributions have been marginalised or overlooked – alive in contemporary consciousness.

Bob Lynn | © 2025 Vox Meditantis. All rights reserved.