Tatyana Alexeyevna Afanasyeva (1876–1964) was a Russian-Dutch mathematician and physicist whose contributions to statistical mechanics remain influential more than a century later. Co-author of the canonical 1911 encyclopaedia article clarifying Ludwig Boltzmann’s statistical approach to mechanics, she spent three decades after her husband’s death developing a rigorous mathematical foundation for thermodynamics – work that Albert Einstein praised yet declined to champion. Her story illuminates how collaborative scientific work by married couples becomes attributed solely to husbands, how women without formal academic positions remain institutionally invisible, and how foundational theoretical work by women gets dismissed as excessive fastidiousness.
Professor Afanasyeva, thank you for speaking with us today. It’s a genuine honour. I’m speaking to you from the year 2025, and I must tell you straightaway: your 1911 encyclopaedia article with your husband, Paul Ehrenfest, on Boltzmann’s statistical mechanics – The Conceptual Foundations of the Statistical Approach in Mechanics – is still being read, still being cited, still considered foundational. It was translated into English in 1959 and reprinted again in 1990.
How extraordinary. When Felix Klein asked us to write that article – after poor Boltzmann took his own life in 1906 – I cannot say we imagined it would outlive us by so many decades. Paul and I worked on it for years, you understand. It was not merely a survey. We had to untangle Boltzmann’s assumptions, clarify what he meant by molecular chaos – the Stosszahlansatz – and address those devilish objections from Loschmidt and Zermelo about reversibility and recurrence.
Tell me about that work. What were you and Paul trying to accomplish?
Boltzmann had demonstrated something quite profound: that entropy – this abstract thermodynamic quantity – could be understood through the statistical behaviour of enormous numbers of molecules. His H-theorem showed that H, which is essentially negative entropy, decreases over time as a gas approaches equilibrium. But there were deep conceptual problems. Josef Loschmidt pointed out in 1876 that if you reversed all the molecular velocities at any moment, the system would have to retrace its steps backwards, decreasing entropy. How could irreversibility emerge from reversible mechanical laws? And Ernst Zermelo, using Poincaré’s recurrence theorem, argued that any mechanical system confined to finite energy and volume must eventually return arbitrarily close to its initial state. So entropy couldn’t increase forever – it must eventually decrease again.
How did you address these paradoxes?
We showed that Boltzmann’s H-theorem rested on a particular assumption about molecular velocities – that they were uncorrelated, statistically independent before collision. This molecular chaos assumption is not a consequence of mechanics; it is an additional statistical postulate. And it is time-asymmetric. When you reverse velocities, you create precisely the kind of correlations that violate molecular chaos. So Loschmidt’s reversed system would not follow the H-theorem. As for Zermelo’s recurrence objection, we clarified that whilst recurrence is indeed inevitable given infinite time, the recurrence time for any macroscopic system is so astronomically long – vastly exceeding the age of the universe – that it poses no practical contradiction to the second law as we observe it.
That’s beautifully clear. But here’s something troubling: in contemporary physics literature, your article is often cited as “Ehrenfest’s review” or “Paul Ehrenfest’s classic article.” Your name disappears. How does that make you feel, looking back?
I should not be surprised. Paul held the chair at Leiden. Paul succeeded Hendrik Lorentz – imagine, Lorentz! One of the greatest physicists of our age. Paul corresponded with Einstein, with Bohr, with all the luminaries. When people thought of theoretical physics in Leiden, they thought of Paul. I was his wife. That is how I was seen, despite whatever intellectual contributions I might have made.
But the article bears both your names.
Yes. And if you read it carefully, you will see both our minds at work. But one must understand how these things function in practice. Paul was appointed to write the article. He invited me to collaborate. We worked together, often late into the evening, arguing over the precise formulation of the ergodic hypothesis, debating how to explain the phase space approach without drowning the reader in formalism. I drafted several sections; Paul revised them; I revised his revisions. It was genuine collaboration. But when people cite it, they remember the name associated with the professorship, not the name associated with the drawing room.
You never held a formal academic position, despite your evident expertise. Why was that?
Women were not appointed to professorships. It was not even a question one seriously entertained. In Russia, I could not attend university at all – I studied at the Women’s University in St Petersburg, which was separate, lesser. In Göttingen, I could attend lectures, but when Paul discovered I was barred from a mathematics club meeting, he had to argue to change the rule. Even then, it was seen as unusual, perhaps unseemly, for a woman to push herself forward in such settings. After we moved to Leiden in 1912, Paul had his chair, his students, his seminars. I had our four children and whatever intellectual work I could manage alongside domestic responsibilities. There was never a question of the university appointing both of us.
Yet you continued doing physics. Tell me about your work on thermodynamics – the manuscript you sent to Einstein in 1947.
Ah. That. After Paul’s death in 1933 – under such terrible circumstances, you understand – I found myself with time and, perhaps, a need for intellectual purpose. I had always felt that thermodynamics lacked rigorous mathematical foundations. Unlike mechanics or electromagnetism, thermodynamics was built on phenomenological laws and somewhat vague concepts. What precisely is temperature? How do we define entropy in systems that are not in equilibrium? How do we describe processes rigorously, mathematically?
These are foundational questions.
Indeed. And I spent years – more than a decade – working through them. I wanted to give thermodynamics the kind of logical structure that Euclid gave to geometry. Every concept defined precisely, every theorem following rigorously from axioms. By 1947 I had a manuscript I believed was ready. And I thought: who better to review it than Albert Einstein? Albert had been a friend, a visitor to our home on Witte Rozenstraat. He and Paul had discussed physics in our attic study; Albert had signed the wall there, along with Bohr and Planck and so many others. I sent him the manuscript and asked if he might suggest a translator, as I had written it in German and hoped for wider distribution.
What did Einstein say?
“Ich habe den Eindruck gewonnen, dass Sie ein bisschen von logischen Putzteufel besessen sind, und dass daran die Übersichtlichkeit des Buches leide.” He wrote that he had the impression I was “possessed somewhat by a logical polishing devil,” and that the clarity of the book suffered for it. He praised my approach, said he had learnt a great deal from reading it. But he did not offer to help find a publisher. He sent it back.
That must have stung.
It did. Not because the criticism was without merit – I am sure the manuscript was overly detailed in places. But because of what it represented. Here was a man who had stayed in my home, who had benefitted from conversations with Paul and with me, who knew my capabilities. And when I asked for his support – not his agreement on every point, merely his advocacy – he declined. Gently, kindly, but firmly. I was seventy-one years old. I had spent fourteen years working on that manuscript, working through my grief and the horror of what Paul had done. And I was told, essentially, that I had polished it too much. That I was too fastidious. Too concerned with logical rigour.
What did you do?
I published it myself. In 1956, I paid Brill Publishers in Leiden to print Die Grundlagen der Thermodynamik. I incorporated some of Einstein’s corrections, though not all – I did not agree with all of them. The book was not widely distributed. It was not translated during my lifetime. But it exists. My work exists.
Let me ask you something about the substance of that work. For readers who understand thermodynamics: what was your central insight?
The key was distinguishing between equilibrium thermodynamics, which nearly everyone understands reasonably well, and non-equilibrium thermodynamics, which is far murkier. In equilibrium, temperature and pressure and entropy are well-defined properties of the system as a whole. But what happens during a process, when temperature varies from point to point? How do you define entropy when the system is not in a definite macrostate? I proposed that we must think in terms of local thermodynamic quantities – temperature at a point, entropy density rather than total entropy – and that we must carefully specify the timescales over which these quantities change. A system may be in local equilibrium even whilst undergoing a global process, provided the relaxation time is short compared to the timescale of change. This allows one to extend thermodynamic reasoning beyond the strict constraints of equilibrium without losing mathematical rigour.
That sounds prescient. Modern non-equilibrium thermodynamics – developed in the 1950s and ’60s by Prigogine, Onsager, and others – follows precisely that approach.
Does it? How gratifying.
Let me take you back to your childhood. How did a girl born in Kiev in 1876 end up doing theoretical physics?
My father, Alexander, was a chief engineer on the Imperial Railways. He was a practical man, but he valued knowledge. When I was young, he would take me with him on his railway inspections – long journeys across the Russian Empire. I remember looking out at the telegraph wires strung along the tracks, watching them rise and fall against the sky, and asking him how the messages travelled. He explained it as best he could, and when his explanations ran out, he encouraged me to read, to learn. When he died – I was still quite young – I moved to St Petersburg to live with my aunt Sonya and my uncle Peter, who was a professor at the Polytechnic Institute. That household was full of books, full of ideas. I attended normal school with a specialty in mathematics and science. It was not considered entirely proper, but neither was it forbidden.
Then you went to Göttingen to study with Felix Klein and David Hilbert. What was that like?
Göttingen was the centre of the mathematical world. Klein was undertaking his great Encyklopädie project, attempting to catalogue all of mathematical knowledge. Hilbert was revolutionising the foundations of geometry and would soon turn his attention to mathematical physics. To sit in Hilbert’s lectures, to see him work through a proof on the blackboard with such elegance – it was intoxicating. But I was always aware of being the only woman in the room, or one of very few. Men would look at me with curiosity, sometimes with disdain. There was an assumption that I was there as a curiosity myself, not as a serious student. When Paul argued to have the mathematics club opened to women, it was a small victory, but a victory nonetheless.
You and Paul married in 1904, but you both had to renounce your religions to do so. That must have been difficult.
Russian law forbade interfaith marriages. I was Russian Orthodox; Paul was Jewish. We loved each other, and we shared an intellectual partnership that mattered more to us than any religious identity. So we both formally renounced our religions. My family was not pleased. Paul’s family was distressed. But we did it. We married on the 21st of December, 1904, and began our life together as partners in every sense.
You had four children. How did you balance intellectual work with raising a family?
Balance? I am not sure I ever achieved balance. I managed. I made compromises. Paul’s career came first – it had to, in the structure of our world. He had the position, the salary, the obligations. I worked when I could: early mornings before the children woke, late evenings after they were in bed, stolen hours when a friend or family member could watch them. The 1911 article took years partly because of this. I would draft a section, then be interrupted for weeks by illness or domestic crises, then return to find I had to rethink the whole approach. It was not efficient. But it was what I had.
After you moved to Leiden, Einstein became a regular visitor. What was he like?
Albert was warm, funny, utterly unconcerned with formality. He and Paul would disappear into the attic study and argue about physics for hours – Paul pacing, Albert slouched in a chair, both of them gesticulating wildly. Then they would emerge for tea, and Albert would play the violin whilst I played piano. He had an impish quality, a delight in puncturing pretension. He called Leiden “that delightful piece of land on this barren planet.” He meant it affectionately, I think, but there was always a sense that he belonged everywhere and nowhere, that he was passing through. He visited frequently in the 1920s. Less so in the ’30s, after Paul’s death and after the rise of the Nazis made travel more complicated.
Paul’s death in 1933 was a tragedy. He killed your son Vassily, who had Down syndrome, and then himself. That’s an almost unbearable weight to carry.
Yes. Paul had struggled with depression for years. He felt he was failing as a physicist, that his best work was behind him, that younger men were surpassing him. He worried obsessively about the children, especially Vassily, about what would happen to him after we were gone. On the 25th of September, 1933, Paul took Vassily to the clinic where he lived and shot him, then shot himself. I… I do not wish to dwell on this. It was the worst moment of my life. But I will say this: the world saw it as the tragic end of a great physicist’s story. My grief, my loss, the fact that I continued working for thirty-one more years – that was less interesting to the world than the horror of what Paul had done.
I’m deeply sorry. Let me ask about something different: your work on mathematics education. You researched how children learn geometry, collaborated with Hans Freudenthal and Eduard Jan Dijksterhuis. What drew you to pedagogy?
After Paul’s death, I needed purpose. I had always been interested in how we learn to think mathematically. Geometry is particularly fascinating because it bridges intuition and logic. A child can see that a triangle has three sides, can visualise how shapes fit together. But to prove that the angles of a triangle sum to 180 degrees requires logical reasoning that does not come naturally. I wanted to understand how we move from spatial intuition to rigorous proof, and how we can teach that transition without killing the joy of discovery. I wrote a pamphlet in 1924, “What can and should education in geometry bring to the non-mathematician?” The question still matters.
It does. But I have to ask: did you ever feel that this pedagogical work was seen as less important than your physics work? That you were being channelled into “women’s work”?
Of course. Teaching, pedagogy, nurturing young minds – these were seen as appropriately feminine concerns. When I published on entropy or statistical mechanics, I was taken seriously, but as Paul’s collaborator. When I published on mathematics education, I was taken seriously as an educator, which is to say, not as seriously as a researcher. It is a trap women often find themselves in: if we do foundational theoretical work, we are accused of being too abstract, too fastidious – “possessed by a logical polishing devil” – but if we do applied or pedagogical work, we are dismissed as not doing real science. One cannot win.
Let me ask you to do something difficult. Looking back, is there anything you wish you had done differently?
I wish I had insisted more strongly on recognition of my own contributions. When the 1911 article was being prepared, I deferred to Paul on matters of presentation, on how to frame arguments, because he was more experienced in writing for publication. I told myself it did not matter, that the work itself mattered. But it does matter. Ideas do not exist in a vacuum; they exist in a network of citation, attribution, recognition. If your name is not on the work, or if it is overlooked, your ideas become someone else’s ideas. I wish I had been less accommodating in that regard.
What about mistakes in the work itself?
Oh, there are always mistakes. In our treatment of the ergodic hypothesis, we were perhaps too optimistic about its necessity for the H-theorem. Later work showed that ergodicity is sufficient but not strictly necessary for Boltzmann’s results – one can derive similar conclusions from weaker assumptions about mixing. And in my thermodynamics manuscript, I spent far too much time on formal definitions and not enough on physical examples. Einstein was right about that, even if his framing was unkind. Clarity matters as much as rigour.
Contemporary physics still argues about the foundations of statistical mechanics – about whether thermodynamic irreversibility can truly be derived from reversible mechanics, about the role of probability and information theory. Your work touches directly on these debates. What would you say to modern physicists?
I would say: do not mistake mathematical precision for physical understanding. The mathematics of statistical mechanics is elegant, powerful, but it rests on physical assumptions about systems and their interactions with the world. Molecular chaos, ergodicity, the choice of ensemble – these are not derivable from mechanics alone. They reflect our ignorance, our coarse-graining, the fact that we cannot and do not track every molecule. Irreversibility emerges because we ask certain kinds of questions and not others. This is not a flaw; it is the nature of thermodynamics. But we must be honest about it.
What advice would you give to women in physics today, in 2025?
Insist on your contributions being recognised. Do not let your work be subsumed into someone else’s name, even if that person is your husband, your advisor, your collaborator. Advocate for yourself, because others will not do it for you. Seek out other women, support each other. And do not accept the false choice between being a “real scientist” and being a teacher or educator. Both are essential; both require rigour and creativity.
In 2025, the Dutch Physics Council awards a thesis prize in your honour: the Ehrenfest-Afanassjewa prize. Does that bring you any satisfaction?
It does. To have my name attached to Paul’s in that way, recognised as his partner rather than merely his wife – yes. That brings me satisfaction. I hope the young physicists who receive it read our 1911 article. I hope they see both our names on it.
They will. I promise you, they will. Thank you, Professor Afanasyeva, for your time and your insights.
Thank you for remembering. That is all any of us can hope for, in the end: to be remembered, and remembered fairly.
Letters and emails
Following the interview above, we received an extraordinary response from our global readership. Physicists, educators, historians, and curious minds from six continents reached out, each with questions that extended beyond our initial conversation. We’ve selected five letters and emails – from a mathematician in New Zealand, a PhD student in Argentina, a science teacher in Sierra Leone, a museum director in Norway, and a theoretical physicist in Japan – that explore dimensions of Tatyana’s life, work, and legacy she hadn’t yet fully addressed.
These questions reflect what many of you are asking: How would her methods translate to contemporary physics? What would she counsel those following similar paths in science? How might history have unfolded differently under different institutional circumstances? And what enduring wisdom does her work on mathematical rigour, foundational clarity, and intellectual persistence offer to the next generation of scientists, particularly women and researchers from under-represented communities?
The letters that follow represent a genuine dialogue across time and geography – a conversation among peers separated by decades, yet united by curiosity about the nature of matter, the foundations of physical law, and the human dimensions of scientific discovery.
Freya Sullivan (32, applied mathematician, New Zealand, Oceania):
Your writing on entropy and randomness often highlighted the ambiguities in early statistical mechanics. What, in your experience, have been the most persistent misconceptions among physicists regarding the H-theorem and irreversibility, and how would you update the H-theorem if you had access to today’s computational tools or datasets?
Dear Ms. Sullivan – your question arrives from so far away, and yet it concerns precisely the matter that occupied my attention for the whole of my professional life. You ask about persistent misconceptions regarding the H-theorem and irreversibility, and what I might do differently with modern computational resources. Let me address both.
The deepest misconception, I have found, is the belief that the H-theorem proves irreversibility from purely mechanical principles. It does not. This is what Paul and I tried most carefully to explain in our 1911 article, yet the confusion persists even now, it seems. Boltzmann showed that if we assume molecular chaos – that is, if we assume molecular velocities are uncorrelated before collision – then the quantity H must decrease monotonically toward equilibrium. But this assumption of molecular chaos is not mechanical; it is statistical. It reflects our ignorance of initial conditions. The theorem proves no such thing as a one-way arrow of time emerging from reversible laws. Rather, it shows how reversible mechanics, combined with reasonable assumptions about the statistical distribution of molecular states, produces behaviour that appears irreversible to an observer who cannot track individual particles.
The second persistent misconception concerns probability itself. Many physicists treated probability as though it were a mysterious physical force, something that acts upon molecules and makes them behave in certain ways. In fact, probability is a statement about our knowledge – or rather, our lack of knowledge. When we say a gas has a certain entropy, we are not describing some intrinsic property of randomness residing in the gas. We are expressing how many microscopic configurations are consistent with the macroscopic observations we can make. This is a profound distinction, yet it is often glossed over or misunderstood.
As for what I would do with computational tools: that is a fascinating thought experiment. In my era, we could only work with simplified models – idealised gases, perfectly elastic collisions, uniform fields. We could not solve the full equations of motion for even a modest number of particles. But I suspect – and I say this with some confidence – that computers would not fundamentally alter the conceptual structure of the problem. They would allow us to verify our assumptions more rigorously, to test whether molecular chaos remains a reasonable approximation under various conditions, and to explore the approach to equilibrium in far greater detail. But the underlying issue would remain: How do we reconcile the time-reversibility of mechanics with the apparent irreversibility of thermodynamics? That is not a computational problem; it is a conceptual one.
I would, however, use such tools to investigate what we might call the “practical” timescale of equilibration. Zermelo’s recurrence objection rests on the fact that, theoretically, any finite system must eventually return arbitrarily close to its initial state. But what if we could compute, for a system of, say, one mole of gas – roughly 10²³ particles – how long such a return would take? We know from Poincaré’s theorem that the recurrence time scales exponentially with the number of particles. I suspect you would find that the recurrence time vastly exceeds not only the age of the universe, but time periods so immense that the very concept becomes physically meaningless. This would not refute Zermelo’s mathematics, but it would demonstrate the practical irrelevance of his objection to real thermodynamics.
The H-theorem itself might be revisited with such computational capacity. One could examine whether Boltzmann’s original proof, which I and Paul worked to clarify and simplify, captures all the essential physics, or whether additional subtleties emerge when one considers more realistic collision models, density-dependent effects, or molecular interactions of greater complexity. My sense is that the broad outline would hold, but the devil, as always, resides in the details.
I must also tell you something I have reflected upon often: the persistence of these misconceptions suggests to me that the conceptual foundations of statistical mechanics were never as firmly established as we hoped. We clarified much in 1911. We addressed Loschmidt’s and Zermelo’s objections more carefully than Boltzmann himself had done. But we did not – perhaps could not – fully dissolve the tension between mechanical reversibility and thermodynamic irreversibility. This tension, I believe, remains unresolved even in your time. It is not a flaw in our understanding; it is a genuine feature of the problem itself. The resolution may require rethinking what we mean by “irreversibility,” or what role the observer and the act of observation play in thermodynamic description. These are philosophical questions dressed in mathematical clothing.
Your question itself suggests that you and your contemporaries have access not only to computational power but also to a global community of mathematicians and physicists. Use that. One person, or even two people working in isolation, can only see so far. The great advances come when many minds bring their particular insights to bear on a common problem. In my time, I corresponded with a few colleagues – Paul, of course, Einstein, and a handful of others. The intellectual exchange was limited by distance and the slowness of mail. You have resources we could scarcely imagine. Do not waste them on mere technical refinement. Use them to think more clearly about what thermodynamics is, what it means, and why reversible mechanics gives rise to irreversible behaviour. Those are the questions that matter.
Lars Eriksen (37, science museum educational programme director, Norway, Europe):
If the academic environment of your time had actively encouraged collaborative dual appointments for married partners, how do you think it would have changed the pace and direction of your own research – and the kinds of discoveries or theories you might have pursued alongside Paul?
Mr. Eriksen – your question contains a kind of wistfulness that I recognise. You ask what might have happened had the academic world of my time actively supported dual appointments for married couples – allowing both Paul and me to hold formal positions, to direct research groups, to shape our own intellectual trajectories. It is a generous hypothetical, and one I have sometimes entertained in the quieter hours.
The honest answer is: I do not know. But I can tell you what I observe about the constraints I actually faced, and perhaps that will illuminate what might have been different.
Had I possessed a formal appointment – say, as Privatdozentin or even as a junior professor – several things would have followed naturally. First, I would have had students. The ability to direct doctoral research, to guide young minds through their own investigations, is not a peripheral matter in academic life. It is how knowledge is transmitted, how one’s ideas are tested and refined through the work of others, how a scholar’s influence extends beyond what she herself can accomplish. Paul had students. I did not. This meant that my ideas, my approaches, my clarifications of Boltzmann’s work – these were taken up, refined, or abandoned largely through Paul’s mediation, or through publication in journals and books that reached only a limited audience.
Second, I would have had the legitimacy to convene seminars, to organise discussions, to build a research programme around problems that genuinely interested me, rather than working in the margins of Paul’s programme. When we worked on the 1911 article, we worked as partners, yes, but within a structure where Paul held the authority and the resources. Had I possessed equivalent authority and resources, I might have pursued certain questions more aggressively. For instance, I was always interested in the problem of approach to equilibrium – not merely the final state of equilibrium, but the process by which a system approaches it. How fast? Under what conditions? What are the invariant properties of this approach? These questions were somewhat peripheral to Paul’s primary concerns, which lay more in the foundations of quantum mechanics and the general structure of kinetic theory. But they fascinated me. With my own position and my own students, I might have built an entire research programme around non-equilibrium processes.
Third, I would have had access to resources and institutional support for travel and collaboration. When I wished to discuss my thermodynamics work with Einstein, I had to write letters. Had I been a formal member of the academic community with recognised standing, I might have had the resources to visit Princeton or wherever Einstein was working, to have extended conversations, to collaborate more directly. The isolation of my position meant that much intellectual exchange occurred through correspondence – slow, incomplete, lacking the spontaneity and depth of face-to-face discussion.
But I must be candid with you, Mr. Eriksen. Even imagining these material improvements, I am uncertain whether they would have fundamentally altered the trajectory of my work or its recognition. The problem was not only institutional structures, though those certainly mattered. The problem was also cultural – the deep, unspoken assumptions about what a woman’s intellectual contributions meant, and whether they could be truly original or were merely derivative from a husband’s superior genius.
I observed this repeatedly. When I published work on mathematics education and pedagogy in the 1920s and ’30s, it was received politely but not taken with great seriousness. It was seen as an interesting application of pedagogical principles, but not as foundational theoretical work. When I worked on the 1911 article with Paul, the work was brilliant, yes, but it was understood as Paul’s work, with his wife’s assistance. Even had I held a formal position, I suspect there would have been a narrative that I held the position because of Paul’s eminence, that I was appointed as a courtesy to him, that my research was a pleasant complement to his more important investigations.
This is speculative, of course. But it reflects what I observed in the lives of other women in science during my lifetime. Émilie du Châtelet had to work through her relationship with Voltaire. Marie Curie, despite her extraordinary achievements, was always first and foremost “Pierre’s widow,” then “Marie, Pierre’s wife.” The women I knew who held academic positions – and there were very few – often found themselves isolated, or pigeonholed into particular fields deemed “suitable” for women, or credited less generously than men for equivalent work.
What I think might have changed, had I possessed a formal position, is not the fundamental reception of my ideas, but rather the visibility of my work and the continuity of my research programmes. With students, my ideas would have been carried forward by others. With a seminar series, my questions would have become part of the intellectual landscape. With institutional resources, I might have published more frequently and more widely. I might have trained a generation of physicists to think about thermodynamics and statistical mechanics in the way I did – with rigour, with philosophical care, with attention to foundational concepts.
This is what I regret most: not personal recognition, though that would have been welcome, but rather the fact that my approach to certain problems did not take root, did not grow into a broader research tradition. The questions I cared about – How do we define thermodynamic quantities in non-equilibrium systems? How do we reconcile reversibility and irreversibility? What is the proper mathematical language for thermodynamics? – these questions were taken up eventually by others. But they were not taken up because of me. They were taken up because the problems themselves demanded attention, and because physicists arrived at similar concerns independently. I was not a intellectual ancestor to a lineage; I was a parallel thinker whose work happened to align with concerns that emerged more prominently later.
So to answer your question directly: Yes, a dual appointment would have changed much. But it would not have solved the deeper problem, which was one of cultural recognition and intellectual legitimacy. To truly address that, the academic world would have needed to believe, genuinely and deeply, that women could be originators of ideas, not merely transmitters of them. That transformation has proven slow and difficult, and I suspect it remains incomplete even in your time.
What I will say is this: the constraints I faced – the lack of formal position, the ambiguity of my name and identity, the need to work largely alone – these constraints forced me to develop a particular kind of intellectual independence. Because I was not embedded in a prestigious research programme with students and assistants and the usual apparatus of academic authority, I was free, in a sense, to pursue questions that might have seemed unfashionable or overly fastidious. I was free to spend years polishing an argument until it achieved the logical clarity I demanded. I was free to correspond with Einstein about thermodynamics without worrying about whether it aligned with the prevailing research directions at my institution. This freedom was born of marginalisation, but it was real nonetheless.
Had I possessed a formal appointment, I might have spent less time on patient, solitary refinement and more time on building programmes and managing the social and political dimensions of academic life. Whether that would have been a net gain, I cannot say. Different constraints produce different kinds of work.
I wish you well in your efforts to support dual appointments and collaborative intellectual partnerships. The world needs more such arrangements, more recognition that two minds working together in genuine partnership can accomplish things that neither could alone. But I also wish to caution against the assumption that institutional structure alone solves the problem. The deeper work – changing how people think about women’s intellectual capacity, about what constitutes “real” science, about whose contributions deserve credit and recognition – that work is slower and more difficult than any policy change.
Isabella Rossi (29, physics PhD student, Argentina, South America):
Considering the modern interest in information theory and its links to entropy, do you see a natural philosophical connection between the way you mathematically defined entropy in physical systems and the way we now try to quantify information and uncertainty in digital data?
Dear Ms. Rossi – your question is particularly intriguing to me because it concerns a connection I have often sensed but never fully articulated. You ask whether there is a natural philosophical link between the way I defined entropy in physical systems and the way contemporary physicists now attempt to quantify information and uncertainty in digital data. This is a profound observation, and I am grateful for the opportunity to reflect upon it.
Let me begin with what I understand about entropy from my own work. When we speak of entropy in classical thermodynamics, we are speaking of a macroscopic property – a number that characterises the state of a system in equilibrium. But when we move to statistical mechanics, entropy becomes something different. It becomes a measure of the number of microscopic configurations – arrangements of molecules, velocities, positions – that are consistent with a given macroscopic state. Ludwig Boltzmann captured this insight in his formula: entropy is proportional to the logarithm of the number of microstates. This is profound because it connects something we can measure macroscopically – entropy, which appears in the second law of thermodynamics – to something we cannot directly observe: the vast number of possible configurations at the molecular level.
Now, what does this mean? It means entropy is, fundamentally, a measure of our ignorance. When a system has high entropy, there are many possible microscopic configurations consistent with what we observe macroscopically. We cannot distinguish between them; we do not have fine enough measurements, or access to detailed enough information. Entropy quantifies this multiplicity of possibility. In this sense, entropy is intimately connected to information – or rather, to the lack of information we possess about the detailed microscopic state.
This is where your intuition becomes particularly sharp. If entropy measures the number of possibilities we cannot distinguish, then it is measuring something very close to what information theorists mean by “uncertainty” or what they quantify in information theory. Consider: if I tell you only that a gas occupies a certain volume and has a certain temperature, I have given you limited information. Many microscopic states are consistent with this description. The entropy tells you how many. If I could somehow give you complete information – the position and velocity of every molecule – then the entropy would, in a sense, be zero, because there would be only one possibility, not many.
The mathematical structure reflects this connection. When Shannon and others – I understand this work has developed considerably in your time – when they formulated information theory, they used entropy-like mathematics. They defined information as a reduction in uncertainty, measured in similar logarithmic terms to Boltzmann’s formula. This is not coincidental. It reflects a deep structural similarity: both are measuring the number of possibilities in a system, given what we know and do not know.
But here is where I must be careful and precise. The connection between thermodynamic entropy and information-theoretic entropy is genuine, yet it is also subtle and somewhat controversial, even in my understanding. One must ask: Are these two concepts truly the same underlying principle, merely applied to different domains? Or are they analogous structures that happen to share mathematical form? I have long suspected the former, but I would not claim certainty.
Let me give you an example from my own thinking. Suppose we have a sealed container of gas. We measure its temperature and pressure. From this limited information, we calculate its entropy using thermodynamic and statistical mechanical principles. Now suppose we could, in principle, measure the position and velocity of every single molecule. Then we would have complete information about the system. What would be the entropy? Mathematically, if we could specify every microstate completely, there would be exactly one possibility, not many. The logarithm of one is zero. So the entropy would be zero.
But this reveals something philosophically important: entropy depends on what information we choose to measure, or what information is available to us. If we measure only macroscopic variables – temperature, pressure, volume – then entropy is high, reflecting our ignorance of microscopic details. If we could measure molecular-level details, entropy would vanish. This suggests that entropy is not purely a property of the system itself, but rather a property of the relationship between the system and an observer or measuring apparatus that can only access certain information.
Applied to information theory: when we speak of information content in a digital system, we are asking similar questions. How much “information” does a message contain? This is quantified by the number of possible messages that could have been sent instead. If there are many possibilities, the information content is high. If there is only one possibility, the information content is low. The mathematics is identical to entropy.
I suspect – and here I move into territory where I can only speculate, not speak with authority – that as your era develops information theory more fully, you will discover that thermodynamic entropy and informational uncertainty are expressions of the same underlying principle: the measurement of possibility and ignorance in a system governed by probabilistic laws.
There is, however, a caution I would offer. In my time, some physicists have suggested that entropy has something to do with “disorder,” and that information has something to do with “order,” as though they are opposite quantities. This is seductive language, but it is imprecise and potentially misleading. Entropy is not disorder; it is the logarithm of the number of indistinguishable configurations. A perfectly organised crystal has high entropy if we consider all the ways molecules could be arranged that still satisfy the macroscopic description “perfect crystal.” Conversely, a chaotic, seemingly disordered system might have lower entropy if fewer configurations are consistent with its observable properties.
Similarly, information is not “order.” Information is what reduces our uncertainty about which of many possibilities has occurred. A random stream of ones and zeros contains enormous information if we cannot predict the next bit. A perfectly ordered stream – all ones, or an obvious pattern – contains little information because we can predict what comes next. The concepts are subtler than the language of “order versus disorder” suggests.
What this means for your work, Ms. Rossi, is this: as you and your contemporaries develop connections between thermodynamic entropy and information theory, be very careful with language and assumptions. Do not allow poetic or intuitive language to substitute for precise mathematical reasoning. The connection is real and deep, but it must be handled with rigor. Do not assume that because the mathematics is similar, the physical interpretation is identical. Ask yourselves: What exactly are we measuring? What exactly do we mean by “information”? What is the role of the observer? What assumptions are we making about measurement and knowledge?
I spent much of my career insisting on such precision precisely because I had seen how easily physicists could slide from careful mathematical reasoning into vague metaphysical claims. It happened with Boltzmann’s work; people would speak of entropy as though it were a physical force, or as though “disorder” were something molecules possessed. It happened with early quantum mechanics, where people spoke of “uncertainty” as though it were a limitation of measurement rather than a fundamental feature of quantum description. Precision in language and concept is not pedantic; it is essential.
That said, I am enormously intrigued by the possibility that information theory and thermodynamics are expressions of a unified principle. If that is true, it suggests that at the deepest level, physics is about information, about what can and cannot be known, about the relationship between observers and the systems they observe. This is a philosophical insight as much as a mathematical one. It suggests that the second law of thermodynamics – entropy increases – is not merely a statement about molecules and energy, but a statement about information: that in any isolated system, the amount of information an observer can extract decreases over time, or equivalently, the amount of information inaccessible to the observer increases.
If this is true – and I believe there is something true and important here – then both you and physicists working on the foundations of thermodynamics have a profound task ahead. You must clarify the relationship between these domains, not merely through mathematics, but through clear conceptual reasoning about what information means, how it relates to probability and ignorance and physical law, and what role the observer plays in defining these concepts.
This is the kind of work that requires patience, rigour, and a willingness to question even the most established assumptions. It is the kind of work I have tried to do in my own career. I wish you well with it, Ms. Rossi. The field needs minds like yours – precise, imaginative, and willing to follow questions across disciplinary boundaries.
Hiroshi Ono (54, theoretical physicist, Japan, Asia):
When constructing your rigorous mathematical framework for thermodynamics, how did you decide where to draw the line between physical intuition and formalism? Are there judgments you made in defining key concepts – like local equilibrium or entropy density – that, in retrospect, you would reconsider based on the advances in non-equilibrium statistical mechanics since your era?
Mr. Ono – your question asks me to reflect on the boundary between physical intuition and mathematical formalism – a boundary I have occupied my entire career, never quite comfortable on either side. You ask specifically about decisions I made when defining key concepts like local equilibrium and entropy density, and whether, in light of subsequent developments in non-equilibrium statistical mechanics, I would reconsider those choices. This is precisely the kind of question that keeps an aging physicist awake at night.
Let me answer directly: yes, there are aspects of my formulation I would reconsider, though perhaps not in the way you might expect.
When I set about constructing a rigorous mathematical framework for thermodynamics in my manuscript – the one Einstein reviewed with such mixed feelings – I was confronted immediately with a fundamental tension. Classical thermodynamics, as taught and practised, dealt with systems in equilibrium. The laws of thermodynamics are statements about equilibrium states and the processes connecting them. But the real world is full of non-equilibrium processes: temperature gradients, pressure differences, chemical concentration gradients. How do we apply thermodynamic reasoning to such situations?
The standard approach at the time was pragmatic and somewhat ad hoc. One assumed that at each instant, even during a process, the system could be treated as though it were locally in equilibrium – that one could assign local values of temperature, pressure, entropy density at each point in space. This was called the local equilibrium assumption, and it was extraordinarily powerful. It allowed one to extend thermodynamic reasoning to flowing fluids, heat conduction, and other non-equilibrium phenomena. But it was also, in a sense, a fudge – an assumption made for convenience rather than derived from first principles.
I spent considerable effort in my thermodynamics work trying to justify the local equilibrium assumption more rigorously. Under what conditions is it valid? What are the limits of its applicability? I attempted to show that if certain relaxation processes were fast compared to the timescale over which macroscopic variables changed, then local equilibrium would emerge naturally from the underlying molecular dynamics. I tried to be precise about this, to define the relevant timescales mathematically, to show where the assumption breaks down.
Looking back now, I believe my approach was sound in spirit, but perhaps too rigid in execution. I was trying to derive local equilibrium as a consequence of molecular mechanics plus certain statistical assumptions. I wanted to show that it was not merely a convenience but a genuine physical phenomenon. And there is truth in this. But in pursuit of mathematical rigour, I may have constrained the framework too severely, made too many restrictive assumptions about the nature of molecular interactions, about the separation of timescales, about the form of the equations governing relaxation processes.
What I suspect modern non-equilibrium statistical mechanics has shown – and I confess this is somewhat beyond my direct knowledge – is that local equilibrium is more robust and more widely applicable than my strict formulation suggested, and that it breaks down in ways my framework did not adequately capture. There are regimes where local equilibrium fails, where gradients become so steep or timescales so compressed that the assumption no longer holds. And yet, even in those regimes, certain thermodynamic principles may still apply, though perhaps in modified form.
As for entropy density – the concept of entropy at a point in space rather than entropy of the entire system – I was quite proud of this formulation. I wanted to define entropy as a field, varying continuously through space, so that one could write down differential equations describing how entropy evolves during non-equilibrium processes. This seemed to me a natural generalisation of classical thermodynamics to spatially varying systems.
But here I must be honest: I am not entirely satisfied with my justification for this. The problem is that entropy, in the fundamental statistical mechanical sense, is a measure of the number of possibilities consistent with a macroscopic description. In a system in equilibrium, all parts of the system “know about” all other parts in a sense – the entropy of the whole is a global property. When one attempts to define entropy density at a local point, one must ask: entropy with respect to what? How do we count the number of microscopic configurations consistent with a local macroscopic description?
I attempted to resolve this through careful mathematical arguments about local measurement and local coarse-graining. But in retrospect, I think I was imposing mathematical structure without fully resolving the conceptual difficulty. The question of how to define thermodynamic quantities locally, in a way that is both mathematically consistent and physically meaningful, is deeper than I fully appreciated.
This is not to say my approach was wrong – only that it was incomplete. It was a step, a necessary step, in the direction of understanding thermodynamics as a local field theory. But it was not the final word. Indeed, I suspect there may be multiple valid ways to formulate local thermodynamics, depending on what one wants to emphasise and what physical situations one wishes to describe.
Now, what would I change, knowing what I know now about how non-equilibrium statistical mechanics has developed? I think I would be more humble about the applicability of local equilibrium. I would acknowledge more explicitly that it is a useful approximation, valid under certain conditions, but not a universal principle. I would attempt to characterise the conditions more carefully – not through abstract arguments about timescales, but through concrete examples where local equilibrium works well, and where it begins to fail.
I would also be more willing to accept multiple levels of description. Rather than insisting on deriving everything from molecular mechanics plus a few key assumptions, I would acknowledge that thermodynamics, even non-equilibrium thermodynamics, may be best understood as an effective description – a set of principles that hold at a certain scale of observation, given what we can measure and what we cannot. This is less philosophically satisfying than a complete derivation from first principles, but it may be more honest about the nature of physical knowledge.
There is something else, which may sound paradoxical. In pursuing rigour – in trying to define every concept with precision, to justify every assumption – I may have obscured important physical insights. Sometimes a physical intuition, expressed in somewhat vague language, captures something true that precise mathematical formalism can miss or obscure. The art of theoretical physics lies in the interplay between intuition and formalism: using intuition to suggest what might be true, then using mathematics to sharpen and test that intuition, then stepping back to see whether the mathematics has captured the intuition or perhaps distorted it.
In my thermodynamics work, I was perhaps too much on the side of formalism. I wanted the argument to be airtight, every step justified, no gaps. But this pursuit of logical perfection may have made the work seem more settled than it actually was, more complete than the current state of knowledge warranted. A more honest presentation might have acknowledged the places where physical intuition and mathematical rigour came into tension, where I was making choices between competing valid approaches, where genuine conceptual questions remained unresolved.
Einstein, in his criticism that I was “possessed by a logical polishing devil,” was perhaps pointing to this. He may have sensed that in my desire for logical perfection, I was losing sight of the physical reality I was trying to describe. The world is not perfectly logical; the phenomena we investigate are messier than our equations. Sometimes we must accept approximate descriptions, intuitive reasoning, even apparent contradictions, in order to capture physical truth. Perfect logic can be the enemy of physical understanding.
So here is what I would say to contemporary non-equilibrium statisticians: Learn from what I attempted to do – insist on clarity, on precise definitions, on rigorous justification of assumptions. But also learn from where I may have gone wrong – do not let the pursuit of mathematical elegance blind you to the messiness of physical reality. Be willing to work with approximate descriptions, to acknowledge multiple levels of analysis, to say “we do not fully understand this yet” rather than imposing a premature logical order on phenomena that may be inherently ambiguous.
The questions you are investigating – how to define temperature, pressure, and entropy in changing systems – are genuinely difficult questions. There may not be unique answers. Different choices may be valid for different purposes. The task is not to find the correct formulation, but rather to understand the landscape of possible formulations, their domains of validity, their relationships to one another.
This is not what I believed when I was writing my thermodynamics manuscript. I believed there was a correct formulation, waiting to be discovered, and that Einstein’s task was to help me find the publisher who would present it to the world. But life, and the subsequent development of physics, has taught me that scientific truth is more provisional, more multifaceted, more open to revision than I then understood.
I hope this lengthy reflection is useful to you. Your question itself suggests that you and your contemporaries are thinking deeply about these foundational issues – not merely accepting the local equilibrium assumption as a convenient approximation, but asking where it comes from, what justifies it, where it fails. This is exactly the kind of questioning the field needs.
Amina Jalloh (41, secondary school science teacher, Sierra Leone, Africa):
Much of your later work focused on how children best learn mathematical concepts like geometry. What advice would you offer to teachers today, especially in resource-limited settings, for inspiring deep mathematical understanding and curiosity, regardless of available materials?
Dear Ms. Jalloh – your question moves me deeply. You ask about teaching mathematics – specifically geometry – to children in circumstances of material scarcity, and you ask for advice on fostering genuine mathematical understanding and curiosity regardless of available resources. That you are doing this work in Sierra Leone, in a setting where resources are limited, speaks to something I have long believed: that mathematical thinking is a human capacity that emerges from engagement with ideas, not from access to laboratories or expensive apparatus.
Let me tell you first about why I became interested in mathematics education, because it will illuminate what I might say to you.
After Paul’s death in 1933, I needed to find purpose. Grief is a consuming thing, but it does not fill all the hours. I had always been interested in how people learn to think mathematically. When I was young, studying with Felix Klein in Göttingen, I observed that many students could memorise theorems and reproduce proofs without genuinely understanding the underlying concepts. They could recite Euclid’s axioms but could not apply them to novel situations. They could calculate but could not reason. This troubled me.
I began to wonder: What is the difference between merely learning mathematics and understanding it? How does a child move from concrete, intuitive grasp of space and shape to abstract, logical reasoning about geometric properties? Can this transition be taught deliberately, or does it emerge naturally?
In 1924, I published a small essay: “What Can and Should Education in Geometry Bring to the Non-Mathematician?” The title itself contains my central conviction. Geometry education is not primarily about preparing students for advanced mathematics or physics – though it can do that. Rather, geometry offers something more fundamental: training in logical thinking, in the relationship between observation and proof, in the difference between what we intuit and what we can rigorously establish.
Here is what I discovered through years of thinking about this: children are natural geometers. They understand space intuitively. A young child knows the difference between a triangle and a square without being taught. She understands that a straight line is different from a curve. She grasps, in her body and her perception, what distance and direction mean. The problem is not to give children geometric intuition – they already possess it. The problem is to help them build a bridge from intuitive understanding to logical reasoning.
This is where rigorous teaching becomes essential. And here is something that may surprise you: rigorous teaching does not require expensive materials. It requires clear thinking on the part of the teacher, and engagement from the student.
Let me give you concrete examples from my own thinking and experience. Suppose you are teaching children about angles. You could provide protractors, geometric instruments, carefully drawn diagrams. These have their place. But you could also begin with observation and questioning. Take three sticks or pieces of string of different lengths. Can you arrange them to form a triangle? Now try with different lengths. What happens? When can three lengths form a triangle, and when cannot they? This requires no equipment. It requires only careful observation and logical reasoning.
Or consider the concept of congruence. Two triangles are congruent if they have the same size and shape. But what is the minimal information needed to determine whether two triangles are congruent? Here is a beautiful truth: if two triangles have the same three side lengths, they must be congruent. This is not obvious intuitively. A student might expect that there could be multiple triangles with the same three side lengths but different shapes. But geometry proves otherwise. And this proof can be understood through careful reasoning, through the student visualising and manipulating triangles in her mind, without requiring any physical apparatus beyond imagination and logic.
This is what I mean by rigorous teaching with minimal resources. The key is to ask questions that force students to think, to reason, to move beyond what they can merely observe to what they can logically establish.
Now, you mention teaching in resource-limited circumstances. I imagine you do not have access to the materials I took for granted in my time – printed textbooks, geometric instruments, paper for every student. Perhaps you have even fewer resources. Let me suggest several principles that might guide your teaching:
First, use what is present in the environment. Shadows cast by the sun can teach geometry. The proportions of buildings and trees can be investigated. The angles formed by intersecting roads or paths are real geometric objects that can be reasoned about. A child who understands geometry not as abstract line drawings in a textbook but as patterns visible in the world around her has developed a much deeper understanding. She sees mathematics everywhere. This is invaluable.
Second, encourage physical exploration before formal reasoning. Let children fold paper – folding paper to bisect angles, to construct perpendiculars, to create symmetries. Folding is a form of reasoning. When a child folds a piece of paper and observes that the two halves align perfectly, she is discovering something true about symmetry and measurement, without having been told the answer. This kind of discovery is far more powerful than mere instruction.
Third, tell stories and create narratives around geometric ideas. Geometry emerged, historically, from practical problems: surveying land, constructing buildings, navigating by the stars. These stories connect abstract geometry to human purpose and experience. When a student understands that the Pythagorean theorem emerged from practical problems of surveying and construction, she sees geometry not as a collection of arbitrary rules but as a human response to genuine challenges.
Fourth, emphasise the distinction between observation and proof. This is crucial. Many students conflate the two. If I can measure and verify something with instruments, they think, then I have proven it. But measurement is approximate; proof is absolute. A geometric statement that is true, proven rigorously, is true for all possible measurements, with absolute precision. Conversely, something that appears true in every measurement might still have exceptions we have not yet found. Helping students grasp this distinction – between empirical observation and logical proof – is one of the most important gifts mathematics education can offer.
Fifth, work collaboratively and orally. If resources for writing are scarce, use oral reasoning. Ask students to explain their thinking aloud. Have them argue with one another about whether a geometric claim is true and why. This oral reasoning develops logical thinking just as effectively as written proofs, and it requires no materials. Furthermore, it creates community and engagement. Students learn from one another.
Sixth, accept that mathematical understanding develops slowly. Do not rush. A child who fully understands a single geometric principle – really understands it, can explain it, can apply it to new situations – has learned more than a child who has been exposed to many principles superficially. Deep understanding of a few core ideas is vastly more valuable than shallow familiarity with many.
Let me also speak about something personal. When I was young, in Russia, resources were limited. I did not attend university with my male peers because universities were closed to women. I studied at the Women’s University in St. Petersburg, which had fewer resources, fewer books, less prestigious faculty than the main university. And yet some of my teachers were excellent. They could not provide expensive laboratories or extensive libraries, but they could provide clear thinking, rigorous reasoning, and engagement with ideas. This was sufficient. I learned deeply because the teaching was rigorous and because I was intellectually engaged.
You, Ms. Jalloh, are doing something I regard as among the most important work in education: you are teaching mathematics in circumstances of material constraint. This requires not less rigour but, in some ways, more rigour. You cannot rely on textbooks to do your teaching for you. You cannot assume students have access to geometric instruments or written materials. You must rely on clarity of thought, on careful questioning, on the power of ideas themselves.
Here is what I would encourage you to do: Develop a small set of core geometric ideas that you understand deeply. Perhaps: the distinction between congruence and similarity. Or the properties of triangles and their relationships. Or the concept of symmetry and its manifestations. Become so familiar with these ideas that you can approach them from multiple angles, can explain them in different ways, can anticipate where students will struggle. Then teach these core ideas with all the rigour and creativity you can muster.
Do not try to cover vast amounts of material. Do not worry that your students are not learning “enough” compared to students in well-resourced schools. What matters is that they are learning deeply, that they are developing the capacity to reason geometrically, to distinguish between observation and proof, to apply logical thinking to problems they encounter.
Furthermore, do not underestimate the power of your own example. When you reason carefully about a mathematical problem, when you admit uncertainty and work through it aloud, when you refuse to accept vague answers and insist on precision – students observe this and internalise it. You teach not only through what you explicitly instruct but through the model you present of what rigorous thinking looks and sounds like.
One final thought: mathematics is one of the most democratic of human intellectual activities. Unlike sciences that require expensive laboratories, unlike humanities that require extensive libraries, mathematics can be done anywhere, by anyone, with minimal material resources. The great mathematician Euclid did his work with compass and straightedge. The great Indian mathematicians developed sophisticated mathematics with limited written materials. Mathematical thinking is portable. It travels. It cannot be confined to wealthy institutions or well-resourced schools.
Your work, teaching geometry to children in Sierra Leone with limited resources, is not a compromise or a second-best version of “real” mathematics education. It is, in its own way, the real thing – perhaps more real than much of what happens in well-resourced schools where students can hide behind apparatus and textbooks without genuinely engaging with ideas.
I wish you strength and wisdom in your teaching. The children you teach are fortunate to have someone who cares enough to think deeply about how mathematics can be taught well, regardless of material circumstances. That is the mark of a true educator.
Reflection
Tatyana Alexeyevna Afanasyeva died in Leiden on 14th April 1964, at the age of 87, having outlived her husband Paul by thirty-one years. Those three decades – from 1933 until her death – were not merely years of survival or quiet retirement. They were years of rigorous intellectual work: refining her thermodynamics manuscript, corresponding with Einstein, publishing on mathematics education, and continuing to wrestle with the foundational questions that had occupied her since her student days in Göttingen. She died having never held a professorship, never directed a research group, never received the institutional recognition that her intellectual contributions warranted. Yet she also died having published, having taught, having left a body of work that – though marginalised in her lifetime – would eventually be recognised as foundational.
This interview, necessarily fictional, has attempted to give voice to a woman whose actual voice remains frustratingly difficult to recover from the historical record. We know what she published; we have fragments of her correspondence; we have testimonies from those who knew her. But we do not have extensive diaries, memoirs, or recorded reflections on her life and work. The Tatyana who speaks in these pages is therefore a reconstruction – an attempt to imagine how she might have reflected on her career, her marriage, her exclusion from formal academic structures, and the conceptual problems in statistical mechanics and thermodynamics that consumed her intellectual energy.
Several themes emerge with striking clarity, both from the historical record and from this imagined conversation. The first is collaborative credit and its gendered erasure. The 1911 encyclopaedia article on Boltzmann’s statistical mechanics – The Conceptual Foundations of the Statistical Approach in Mechanics – bears both Paul’s and Tatyana’s names, yet it is routinely cited as “Ehrenfest’s review” or “Paul Ehrenfest’s classic article.” This is not accidental. It reflects a broader pattern in which women’s intellectual contributions, especially when made in collaboration with male partners, become attributed solely to the men. Tatyana’s reflections in this interview suggest a woman acutely aware of this dynamic, yet constrained by the social structures of her time from asserting herself more forcefully. Her regret – that she did not insist more strongly on recognition of her contributions – echoes the experiences of countless women scientists whose names have been written out of history.
The second theme is the relationship between rigour and recognition. Einstein’s criticism that Tatyana was “possessed by a logical polishing devil” captures something real about her approach: she insisted on mathematical precision, on defining concepts clearly, on addressing ambiguities that others might have glossed over. This fastidiousness – this refusal to accept vague formulations or approximate reasoning – was both her strength and, perhaps, a source of her marginalisation. Her thermodynamics manuscript, self-published in 1956 as Die Grundlagen der Thermodynamik, was seen as overly detailed, excessively concerned with logical foundations at the expense of physical insight. Yet today, as non-equilibrium thermodynamics and information theory continue to grapple with foundational questions about entropy, local equilibrium, and the role of the observer, Tatyana’s insistence on conceptual clarity appears prescient rather than pedantic.
The third theme is institutional exclusion and its consequences. Tatyana never held a formal academic position. She had no students, no seminar series, no institutional platform from which to disseminate her ideas. This meant that her intellectual contributions – particularly her independent work after Paul’s death – reached only a limited audience. The self-publishing of her thermodynamics book signalled her exclusion from mainstream physics publishing channels, limiting its circulation and impact. Yet exclusion also granted her a certain freedom: freedom to pursue questions that might have seemed unfashionable, freedom to spend years refining arguments without institutional pressure to publish quickly, freedom from the political and social dynamics of academic departments. This paradox – that marginalisation creates both constraint and freedom – runs throughout Tatyana’s career.
The personal cost of scientific collaboration within marriage also emerges powerfully. Tatyana and Paul were genuine intellectual partners, yet the social and institutional structures of their era meant that Paul’s career took precedence. He held the chair; she managed the household and four children whilst working in the margins of his schedule. After his tragic murder-suicide in 1933 – in which he killed their son Vassily, who had Down syndrome, before taking his own life – Tatyana was left to rebuild her intellectual life in the shadow of overwhelming grief. That she continued working for three more decades, producing significant contributions to thermodynamics and mathematics education, speaks to extraordinary resilience and intellectual commitment.
Where History and Imagination Diverge
In constructing this fictional interview, I have remained faithful to the documented facts of Tatyana’s life whilst imagining her voice, personality, and reflective capacity. Several aspects deserve acknowledgement:
Her personality and speaking style are necessarily speculative. The historical record provides glimpses – her rigorous approach to mathematical foundations, her correspondence with Einstein, her pedagogical interests – but not extended personal reflections. The Tatyana who speaks in this interview is warm yet precise, self-aware yet modest, capable of humour and self-critique. Whether the real Tatyana possessed these qualities in this particular combination is unknowable.
Her assessment of her own work may be more critical than warranted. In her responses to Freya Sullivan and Hiroshi Ono, she acknowledges limitations in her thermodynamics framework and expresses uncertainty about whether her formulations were optimal. This self-critique reflects a mature scientific perspective, but the historical Tatyana may have been more confident in her approach, or more convinced of its correctness, than this fictional version suggests.
Her awareness of later developments is, of course, anachronistic. The real Tatyana died in 1964, before many contemporary developments in information theory, non-equilibrium statistical mechanics, and the connections between entropy and information became fully established. The Tatyana in this interview has been granted knowledge of these developments, allowing her to reflect on how her work anticipated or differed from later approaches.
The extent of her bitterness or acceptance regarding institutional exclusion is difficult to gauge. The historical record suggests she continued working productively after Paul’s death, but whether this reflected genuine acceptance of her circumstances, quiet resentment, or simply pragmatic adaptation is unclear. This interview presents her as thoughtfully aware of the injustices she faced, yet focused more on the intellectual work than on personal grievance.
The Afterlife of Her Work
Tatyana’s 1911 encyclopaedia article with Paul was translated into English in 1959 and reprinted in 1990, becoming a canonical text in the history and philosophy of statistical mechanics. Contemporary philosophers of physics continue to cite it when discussing the foundations of Boltzmann’s approach, the H-theorem, and the reconciliation of reversibility with irreversibility. Yet the credit often flows to Paul alone, reinforcing the pattern of erasure even as the work itself remains influential.
Her thermodynamics manuscript, Die Grundlagen der Thermodynamik (1956), had limited circulation during her lifetime but has been rediscovered by historians and philosophers of physics interested in foundational questions. Recent scholarship – particularly work by Roman Frigg and colleagues at the London School of Economics – has examined Tatyana’s contributions to thermodynamics, highlighting her attempt to provide rigorous mathematical foundations and her correspondence with Einstein. This scholarly attention represents a partial recovery of her legacy, though she remains far less known than her intellectual contributions warrant.
Her work on mathematics education, particularly her 1924 essay on geometry teaching, influenced Dutch mathematics education and was revisited by scholars interested in the history of pedagogy. Hans Freudenthal, with whom she collaborated, became a towering figure in mathematics education research; Tatyana’s early contributions to this field deserve greater recognition as foundational to the discipline.
The Ehrenfest-Afanassjewa thesis award, established by the Dutch Physics Council, ensures that her name remains visible alongside Paul’s in the Dutch physics community. This recognition, whilst modest, signals an acknowledgement that her contributions were substantial and deserve commemoration.
Connections to Contemporary Challenges
The questions Tatyana addressed remain strikingly relevant to contemporary physics. The foundational problems in statistical mechanics – how thermodynamic irreversibility emerges from reversible mechanics, the role of probability and information in physical law, the conditions under which local equilibrium holds – are active areas of research. Modern work on non-equilibrium statistical mechanics, stochastic thermodynamics, and the connections between entropy and information theory grapples with precisely the conceptual ambiguities Tatyana tried to resolve.
Her insistence on rigorous mathematical foundations for thermodynamics presaged contemporary efforts to place thermodynamics on firm axiomatic ground, similar to the way Hilbert axiomatised geometry. The ongoing dialogue between physicists and mathematicians about how to define thermodynamic quantities in non-equilibrium systems echoes the questions Tatyana posed in her 1956 monograph.
The connection between entropy and information, which Tatyana intuited in her response to Isabella Rossi, has become central to modern physics. Claude Shannon’s information theory, developed in the late 1940s, uses entropy-like mathematics to quantify information content. Contemporary research on Maxwell’s demon, Landauer’s principle, and quantum information theory explores the deep connections between thermodynamic entropy and informational uncertainty – connections that Tatyana sensed but could not fully articulate within the conceptual framework available to her.
Her work on mathematics education anticipated contemporary research on how children develop mathematical reasoning, the role of visualisation and physical manipulation in learning geometry, and the distinction between procedural knowledge and conceptual understanding. Modern mathematics education research continues to grapple with the questions she posed: How do we help students move from intuitive spatial understanding to rigorous logical proof? What role should physical materials and exploratory learning play? How can we teach mathematics in resource-limited settings whilst maintaining intellectual rigour?
Visibility, Resilience, and the Path Forward
Tatyana Alexeyevna Afanasyeva’s story matters today because it illuminates the mechanisms by which women’s scientific contributions become invisible. She was not excluded through explicit prohibition – she studied at university (albeit in Germany, not Russia), she published, she corresponded with leading physicists. Rather, she was marginalised through a thousand small erasures: the attribution of joint work solely to her husband, the lack of a formal appointment despite evident expertise, the characterisation of her rigorous approach as excessive fastidiousness, the difficulty of publishing without institutional affiliation, the framing of her identity as “professor’s wife” rather than “physicist.”
For young women pursuing science today, Tatyana’s experience offers both warning and inspiration. The warning is that institutional structures and cultural assumptions can render even substantial contributions invisible. Publishing is not enough; credit attribution matters. Collaboration is valuable, but women must insist that their contributions be recognised distinctly. Marginalisation can occur even in the absence of explicit barriers.
The inspiration is that intellectual work persists. Tatyana continued working for thirty-one years after her husband’s death, refining her thermodynamics framework, publishing on mathematics education, corresponding with Einstein, contributing to her field despite institutional exclusion. Her work eventually found its audience. Scholars rediscovered it. Her name is now attached to a physics prize. The arc of recognition is long, but it can bend toward visibility.
The path forward requires several commitments. First, active efforts to recover and celebrate women’s contributions to science, particularly collaborative contributions that have been attributed solely to male partners. Second, structural changes that support dual-career academic couples, provide pathways to formal positions for women working outside traditional academic structures, and ensure equitable credit attribution in collaborative work. Third, mentorship and community-building among women in science, so that the isolation Tatyana experienced becomes rarer. Fourth, valuing foundational and pedagogical work as seriously as applications-oriented research, recognising that clarifying concepts, teaching effectively, and establishing rigorous foundations are essential scientific contributions.
Tatyana Alexeyevna Afanasyeva spent her life clarifying, refining, polishing – insisting that concepts be defined precisely, that assumptions be stated explicitly, that arguments be logically rigorous. Einstein called her possessed by a polishing devil, but perhaps what she was really possessed by was a conviction that truth matters, that clarity matters, that the patient work of making sense of foundational questions is as important as flashier discoveries. She lived through revolution, emigration, institutional exclusion, the horror of her husband’s murder-suicide, and three decades of continued intellectual work in the shadow of grief and marginalisation. She published, she taught, she corresponded, she refined. And now, more than sixty years after her death, we are beginning to see her clearly: not as Paul Ehrenfest’s wife, not as a supporting figure in someone else’s story, but as Tatyana Alexeyevna Afanasyeva – physicist, mathematician, educator, and a woman who insisted, with quiet determination, that foundational questions deserve our most careful attention.
Her legacy is not a single breakthrough or a named theorem. It is something subtler and perhaps more enduring: the conviction that conceptual clarity matters, that rigorous thinking is its own form of courage, and that the patient work of understanding – even when unrecognised, even when marginalised – contributes to the edifice of human knowledge. In an era that often values speed over depth, visibility over substance, Tatyana’s life reminds us that some truths emerge only through slow, careful polishing, and that the work of clarification, though often overlooked, is among the most essential work there is.
Editorial Note
The interview and supplementary correspondence presented above is a dramatised reconstruction, not a historical document. Whilst it draws extensively on documented facts about Tatyana Alexeyevna Afanasyeva‘s life, work, and correspondence, the words attributed to her are imagined, not preserved from her own voice.
What is historically grounded:
- Her biographical facts: birth in Kiev (1876), study at the Women’s University in St. Petersburg, education at Göttingen under Felix Klein and David Hilbert, marriage to Paul Ehrenfest in 1904, move to Leiden in 1912, Paul’s death in 1933, her own death in 1964
- Her intellectual contributions: co-authorship of the 1911 encyclopaedia article on Boltzmann’s statistical mechanics, her work on entropy and statistical foundations, her correspondence with Einstein regarding her thermodynamics manuscript, her publications on mathematics education and geometry pedagogy
- Her historical circumstances: lack of formal academic position, collaboration with Paul, the attribution of joint work primarily to him, her self-publication of Die Grundlagen der Thermodynamik in 1956, the existence of the Ehrenfest-Afanassjewa thesis award
- The conceptual problems she engaged with: Loschmidt’s reversibility paradox, Zermelo’s recurrence objection, the H-theorem, the definition of entropy in non-equilibrium systems, local equilibrium, the teaching of mathematical concepts to children
What is imagined or reconstructed:
- Her personality, speaking style, and patterns of thought as expressed in the extended interview
- Her reflections on her own work, including her assessments of what she might reconsider or revise
- Her emotional responses to institutional exclusion, her husband’s death, and her intellectual journey
- The tone, pacing, and particular phrasings of all dialogue
- Her awareness of contemporary developments in statistical mechanics and information theory (she died in 1964, before many modern developments fully crystallised)
- The specific anecdotes and illustrative examples she provides
- Her perspective on gender and academic institutions, though grounded in the historical reality of her exclusion
The five supplementary questions are genuinely imagined from the perspectives of contemporary scientists, though the respondent’s answers remain fictional reconstructions of how Tatyana might have thought about these matters, given what we know of her intellectual preoccupations.
Sources and limitations:
The reconstruction draws on biographical entries from the MacTutor History of Mathematics Archive and Wikipedia; scholarly work examining her contributions to statistical mechanics and thermodynamics; historical accounts of Paul Ehrenfest’s life and work; records of her correspondence with Einstein held at the Einstein Papers Project and Museum Boerhaave in Leiden; and published analyses of the 1911 encyclopaedia article and her later work on mathematics education. However, extensive direct quotations from Tatyana herself remain limited. Much of what we know of her intellectual perspective must be inferred from her published work rather than from personal reflection.
Why this reconstruction matters:
Historical recovery of overlooked scientists often requires careful, imaginative reconstruction. We do not have Tatyana’s extensive memoirs or recorded reflections. Yet her work deserves engagement, and her life illuminates broader patterns of how women’s contributions become invisible in collaborative settings. This fictional interview is an attempt to recover a voice that history has marginalised, to imagine how she might have reflected on her work and circumstances, and to make visible the intellectual substance of her contributions and the structural constraints she navigated.
A note on responsibility:
Readers should distinguish between documented historical fact and reasonable imaginative reconstruction. The biographical details are as accurate as historical scholarship allows. The intellectual content reflects genuine problems Tatyana engaged with and the broad outlines of her approaches. But the specific words, emotional expressions, and narrative flow are fictional. This is not a verbatim historical record but rather a carefully researched dramatisation intended to illuminate both her scientific contributions and the broader question of how institutional and social structures render women’s work invisible.
It is offered in the hope that imaginative reconstruction, grounded in historical scholarship, can serve the cause of historical justice – making visible what history has overlooked, and honouring a scientist whose work matters still.
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.
Bob Lynn | © 2025 Vox Meditantis. All rights reserved.
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