Whereas few European scientists escaped the politico-intellectual gnash of the 1930s unscathed, arguably none faced quite the looming combination of compound miseries thrust at a single, seemingly unbreakable individual of that era, Tatyana Ehrenfest-Afanassjewa (1876–1964). Displaced by systemic anti-Semitism, perpetually lacking stable work, shouldering the depression of her husband that led to his suicide and to the murder of one of her children, and bearing up against the freak death of another, Ehrenfest could easily have sunk under the weight of such dense misfortune, but amazingly she did not, and survived another quarter century after Europe’s unrelenting assault upon her person, and not only survived, but formed a wholly new approach to mathematical education and retained her position as a world expert on the mathematical processes underlying thermodynamic change.

Tatyana Afanassjewa was born on 19 November 1876 in Kiev, but at the age of 2, her railway engineer father, Alexey Afanassjew, was committed to a mental asylum following a breakdown, and her mother, Yekaterina Ivanova, took young Tatyana to live with her childless aunt and uncle in St Petersburg. The same year of her arrival in St Petersburg, 1878, saw the opening of the Higher Women Courses (otherwise known as the Women’s University) in that city, a concession by Tsar Alexander II, who wished to reverse the trend of Russian women going abroad in search of university degrees that he unwittingly set off with his 1862 prohibition on women’s higher education in Russia, and who noted the general success of his more targeted 1873 opening of the Women’s Medical Courses in the city.

*Tatiana and Paul Ehrenfest*

Whereas it would have made entire sense for a girl of Tatyana’s evident gifts to have attended the Higher Women Courses, her guardians decided to place her instead in a teacher training program, from which she graduated in 1897. Shortly thereafter, Tatyana’s uncle died, leaving her free to attend the Women’s University, where for the next three years she honed her skills in mathematics and physics. In 1902, she took the momentous step of transferring to the University of Göttingen, which we have met multiple times now as the mathematical mecca featuring the dual attraction of superstars David Hilbert and Felix Klein, and where a young man named Paul Ehrenfest strenuously argued that she ought to be allowed to join the students’ mathematical club in spite of her gender. Ehrenfest, who was Jewish, and Afanassjewa, who was Russian Orthodox, soon fell in love, and married in Vienna in 1904 after Ehrenfest received his doctorate from the University of Vienna.

The couple had to renounce their individual religions in order to marry, a fact that held them back professionally, especially in a time when lack of a professed religion was associated with radical movements like Communism and Anarchism. Paul and Tatyana moved to St Petersburg in 1907, where they struggled to find work, but where their home served as the base of an influential informal physics colloquium, dedicated to sharing and discussing the most recent developments in mathematics. In 1910, the couple’s second son, Anna, was born (the first, a future mathematician also named Tatyana, was born in 1905), adding extra financial pressure to the family, compelling Tatyana to take up a variety of odd jobs, including as a grammar school mathematics teacher, and as a member of the Pedagogical Museum of the Military Academy work group dedicated to formulating new approaches to mathematics education in Russia.

Though difficult financially, this era in their early joint work was fruitful intellectually, and in 1907 they published one of their most enduring early contributions to thermodynamics theory, the Ehrenfest Model, sometimes known as the ‘dog-flea model’. In the model, a container filled with gas molecules is connected to an empty vessel. The Ehrenfests investigated the long-term expected behaviour of this system, given different values for the probability q that, at any given moment, a particle should ‘choose’ to hop from one container to the other. What the model found was that, though choosing higher values of q causes the system to reach its equilibrium state more quickly, it does not impact what that equilibrium state ultimately is. An important result of this model is that it provides estimates for how long it takes the system, if it finds itself in a state that is not the equilibrium state, to return to equilibrium, and for how long you would have to wait for the system to randomly re-assume its starting configuration (spoilers: it is a long time, like twelve orders of magnitude greater than the current age of the universe long). This was important, because it tied into the era’s debates about the validity of Ludwig Boltzmann’s H-Theorem, which predicted that systems tend generally towards a certain optimal distribution of energy states over time, with deviations from that state brought back under control in a manner similar to the Ehrenfest Model’s predictable return to equilibrium.

The next gem of the early Ehrenfest partnership came in 1912 when Paul and Tatyana finally published the survey of statistical mechanics that Felix Klein had first requested of them in 1906 (largely on the strength of the Ehrenfest Model, which they had presented that year). Arising out of Boltzmann, Maxwell and Gibbs’s investigations into entropy and energy distributions over time, statistical mechanics is the branch of mathematics that uses the principles of statistics and probability to explain the behaviour of large collections of atoms, molecules, or microscopic substances. ‘The Conceptual Foundations of the Statistical Approach to Physics’ was a landmark and deeply influential publication that clearly set out the developments in the field thus far, laying out the merits of Boltzmann’s H-Theorem and warding off some of the more reckless objections to its results, while at the same time acknowledging the limitations in application caused by some of its more controversial assumptions (such as the ‘ergodic theorem’ that each particle in a gas visits every location in its container, given enough time to wander). Clear in its writing and comprehensive in its treatment, *Conceptual Foundations *laid out areas for further work and established the foundation for the theoretical study of irreversible processes.

*Conceptual Foundations *so impressed University of Leiden professor H.A. Lorentz that he decided, upon his retirement in 1912, his seat should go to Paul Ehrenfest, rather than the individual he had previously wanted to attract, Albert Einstein. This was a massive stroke of luck, as Ehrenfest’s Jewish ancestry and publicly confessed atheism had both steadily worked against him in professional academic circles for years. Tatyana, however, while recognising the financial necessity of the move, was frustrated that her work in Russian education reform was being uprooted just as she was set to receive the results of a massive survey sent to Russia’s university students asking them what their impressions of the strengths and weaknesses of their mathematical training in Russia were. While pedagogical theorists debated the relative merits of abstract proofs versus concrete examples, Tatyana had hit upon the radical notion of actually asking the students who went through the system what they thought about it, and the move to Leiden seemed likely to disrupt this promising work.

*Paul is seated far left, Tatiana at far right.*

While Paul was teaching in Leiden, Tatyana was continuing both her mathematical research and her studies of educational reform, working around the birth of two more children, Paul (1915), and Vassily (1918). Vassily was born with Down’s Syndrome, a fact which drove the depression-prone Paul further into increasingly erratic behaviour. Burdened with a large mortgage, the cost of Vassily’s care in Jena (where Tatyana brought him for specialised treatment in 1922), and his own sense of self-doubt about being worthy of his position, Paul’s emotional state spiralled persistently downwards, dragging the Ehrenfest’s marriage with it. In the 1920s, Tatyana regularly returned to Russia, where she cast down the gauntlet in 1924 to the entire Russian educational system, arguing that the proponents of both the abstract and practical pedagogical approaches were wrong, and that a new approach that allowed practical intuitions to develop over time into abstract realisations was what was needed to revitalise the nation’s mathematics programs. Paul remained in Leiden, isolated in his room, despairing of his declining importance, carrying on a desultory affair with a woman ten years his junior, and watching the disturbing rise of anti-Semitism in Europe. His general sense of hopeless despair worsened until, in 1933, in the waiting room of the Amsterdam medical facility where Vassily went for treatment after his transfer from Jena, something snapped in Paul, and he shot and killed Vassily, followed by himself.

After the death of her son and husband, Tatyana returned to Leiden, where she remained until her death in 1964. The Soviet Union that had attracted her in the 1920s as a country devoted to real and substantive change in how societies function repelled her in the 1930s as it gave itself increasingly over to the orchestrated brutalities of Stalinism. These were difficult years, with the tragedy of Paul and Vassily’s death compounded in 1939 by the loss of another son in an avalanche. In 1925, Tatyana had published a rigorous paper on thermodynamics that sought a clear distinction between the equilibrium state of a system and its irreversible progression over time, in many respects a far-sighted and revolutionary approach which fell entirely in between the cracks of an academic community obsessed with relativity and quantum physics, and this relative disinterest in her thermodynamic ideas would persist through the 1930s. She expanded her 1925 paper into a full book treatment of thermodynamics, *Die Grundlagen der Thermodynamik*, which she had difficulty finding a publisher for. Sending the manuscript to Einstein in 1947, she received the reply that the book as it stood was too concerned with minutely nailing down its fundamental axioms and definitions, and that the main thread of the story was lost in all the detail. He declined to help find it a publisher, and did not reply to Tatyana’s request for further advice.

*Die Grundlagen der Thermodynamik *was eventually published in 1956, but, as Einstein had predicted, her attempt to create a rigorous axiomatisation of thermodynamics failed to make many academic waves in spite of the several reigning paradoxes her system was able to resolve. A similar fate befell a 1958 paper on probabilistic motion for the *American Journal of Physics*. Her thoughts on education, and in particular about the early development of mathematical intuition as an important component of pedagogy, were more continually impactful, and in 1961 her essays on the subject were collected and published by Bruno Ernst. After a lifetime of tragedy and bad timing permeated by flecks of genius and rightful recognition, Tatyana Ehrenfest passed away in 1964 in Leiden. Her daughter, Tatiana Pavlovna Ehrenfest (1905–1984) became a mathematician in her own right, studying De Bruijn sequences and the BEST theorem.

FURTHER READING: For years and years, there was no single good source to go to for Tatyana Ehrenfest, and then suddenly, in 2021, *The Legacy of Tatjana Afanassjewa *was published by Springer, featuring articles about her life, her contributions to mathematical pedagogy, and her thermodynamics ideas. Since it is a Springer book it is, of course, absurdly expensive, coming in at over $80 for a *paperback *copy. The authors also made the strange choice to leave ‘Ehrenfest’ out of the title and to use a spelling of ‘Tatjana’ which is perhaps more representative of the right spelling, but is different from how you generally see her name spelt, so anybody looking for her most commonly given names of ‘Tatiana Ehrenfest’ or ‘Tatyana Ehrenfest’ on Amazon will simply not be able to find it, which only makes things more difficult for people trying to read more about her who haven’t happened to buy this book pointing them towards it. In any case, if you have the money, it is pretty much the book to get. If you don’t, I’d recommend getting the nice and inexpensive Dover edition of *Conceptual Foundations*, which features a reprint of the 1959 translation. Ehrenfest does the usual scientific widow thing that we saw in the last volume in the case of Margaret Huggins of writing herself out of the creation of the book in her introduction, but what follows is a beautiful tour through Boltzmann theory, clearly laid out and explained.

If you'd like to read more about women mathematicians like this one, check out my History of Women in Mathematics, launching in October 2023 from Pen and Sword Books!

## Comentarios