• New mathematical solutions to an old problem in astronomy — ScienceDaily
    Mathematic

    New mathematical solutions to an old problem in astronomy — ScienceDaily

    For millennia, humanity has observed the changing phases of the Moon. The rise and fall of sunlight reflected off the Moon, as it presents its different faces to us, is known as a “phase curve.” Measuring phase curves of the Moon and Solar System planets is an ancient branch of astronomy that goes back at least a century. The shapes of these phase curves encode information on the surfaces and atmospheres of these celestial bodies. In modern times, astronomers have measured the phase curves of exoplanets using space telescopes such as Hubble, Spitzer, TESS and CHEOPS. These observations are compared with theoretical predictions. In order to do so, one needs a way of calculating these phase curves. It involves seeking a solution to a difficult mathematical problem concerning the physics of radiation.

    Approaches for the calculation of phase curves have existed since the 18th century. The oldest of these solutions goes back to the Swiss mathematician, physicist and astronomer, Johann Heinrich Lambert, who lived in the 18th century. “Lambert’s law of reflection” is attributed to him. The problem of calculating reflected light from Solar System planets was posed by the American astronomer Henry Norris Russell in an influential 1916 paper. Another well-known 1981 solution is attributed to the American lunar scientist Bruce Hapke, who built on the classic work of the Indian-American Nobel laureate Subrahmanyan Chandrasekhar in 1960. Hapke pioneered the study of the Moon using mathematical solutions of phase curves. The Soviet physicist Viktor Sobolev also made important contributions to the study of reflected light from celestial bodies in his influential 1975 textbook. Inspired by the work of these scientists, theoretical astrophysicist Kevin Heng of the Center for Space and Habitability CSH at the University of Bern has discovered an entire family of new mathematical solutions for calculating phase curves. The paper, authored by Kevin Heng in collaboration with Brett Morris from the National Center of Competence in Research NCCR PlanetS — which the University of Bern manages together with the University of Geneva — and Daniel Kitzmann from the CSH, has just been published in Nature Astronomy.

    Generally applicable solutions

    “I was fortunate that this rich body of work had already been done by these great scientists. Hapke had discovered a simpler way to write down the classic solution of Chandrasekhar, who famously solved the radiative transfer equation for isotropic scattering. Sobolev had realised that one can study the problem in at least two mathematical coordinate systems.” Sara Seager brought the problem to Heng’s attention by her summary of it in her 2010 textbook.

    By combining these insights, Heng was able to write down mathematical solutions for the strength of reflection (the albedo) and the shape of the phase curve, both completely on paper and without resorting to a computer. “The ground-breaking aspect of these solutions is that they are valid for any law of reflection, which means they can be used in very general ways. The defining moment came for me when I compared these pen-and-paper calculations to what other researchers had done using computer calculations. I was blown away by how well they matched,” said Heng.

    Successful analysis of the phase curve of Jupiter

    “What excites me is not just the discovery of new theory, but also its major implications for interpreting data,” says Heng. For example, the Cassini spacecraft measured phase curves of Jupiter in the early 2000s, but an in-depth analysis of the data had not previously been done, probably because the calculations were too computationally expensive. With this new family of solutions, Heng was able to analyze the Cassini phase curves and infer that the atmosphere of Jupiter is filled with clouds made up of large, irregular particles of different sizes. This parallel study has just been published by the Astrophysical Journal Letters, in collaboration with Cassini data expert and planetary scientist Liming Li of Houston University in Texas, U.S.A.

    New possibilities for the analysis of data from space telescopes

    “The ability to write down mathematical solutions for phase curves of reflected light on paper means that one can use them to analyze data in seconds,” said Heng. It opens up new ways of interpreting data that were previously infeasible. Heng is collaborating with Pierre Auclair-Desrotour (formerly CSH, currently at Paris Observatory) to further generalize these mathematical solutions. “Pierre Auclair-Desrotour is a more talented applied mathematician than I am, and we promise exciting results in the near future,” said Heng.

    In the Nature Astronomy paper, Heng and his co-authors demonstrated a novel way of analyzing the phase curve of the exoplanet Kepler-7b from the Kepler space telescope. Brett Morris led the data analysis part of the paper. “Brett Morris leads the data analysis for the CHEOPS mission in my research group, and his modern data science approach was critical for successfully applying the mathematical solutions to real data,” explained Heng. They are currently collaborating with scientists from the American-led TESS space telescope to analyze TESS phase curve data. Heng envisions that these new solutions will lead to novel ways of analyzing phase curve data from the upcoming, 10-billion-dollar James Webb Space Telescope, which is due to launch later in 2021. “What excites me most of all is that these mathematical solutions will remain valid long after I am gone, and will probably make their way into standard textbooks,” said Heng.

    Story Source:

    Materials provided by University of Bern. Note: Content may be edited for style and length.

  • Mathematical model predicts best way to build muscle
    Mathematic

    Mathematical model predicts best way to build muscle

    Mathematical model predicts best way to build muscle
    Figure 1. The “textbook” hierarchy in the anatomy of skeletal muscle. The overall muscle is characterized by its cross-sectional area (CSA), which contains a certain number (Nc) of muscle fibers (the muscle cells with multiple nuclei or multinucleate myocytes). A given muscle has a nearly fixed number of myocytes: between Nc ≈ 1000 for the tensor tympani and Nc > 1,000,000 for large muscles (gastrocnemius, temporalis, etc. Credit: DOI: 10.1016/j.bpj.2021.07.023

    Researchers have developed a mathematical model that can predict the optimum exercise regimen for building muscle.

    The researchers, from the University of Cambridge, used methods of theoretical biophysics to construct the model, which can tell how much a specific amount of exertion will cause a muscle to grow and how long it will take. The model could form the basis of a software product, where users could optimize their exercise regimens by entering a few details of their individual physiology.

    The model is based on earlier work by the same team, which found that a component of muscle called titin is responsible for generating the chemical signals which affect muscle growth.

    The results, reported in the Biophysical Journal, suggest that there is an optimal weight at which to do resistance training for each person and each muscle growth target. Muscles can only be near their maximal load for a very short time, and it is the load integrated over time which activates the cell signaling pathway that leads to synthesis of new muscle proteins. But below a certain value, the load is insufficient to cause much signaling, and exercise time would have to increase exponentially to compensate. The value of this critical load is likely to depend on the particular physiology of the individual.

    We all know that exercise builds muscle. Or do we? “Surprisingly, not very much is known about why or how exercise builds muscles: there’s a lot of anecdotal knowledge and acquired wisdom, but very little in the way of hard or proven data,” said Professor Eugene Terentjev from Cambridge’s Cavendish Laboratory, one of the paper’s authors.

    When exercising, the higher the load, the more repetitions or the greater the frequency, then the greater the increase in muscle size. However, even when looking at the whole muscle, why or how much this happens isn’t known. The answers to both questions get even trickier as the focus goes down to a single muscle or its individual fibers.

    Muscles are made up of individual filaments, which are only 2 micrometers long and less than a micrometer across, smaller than the size of the muscle cell. “Because of this, part of the explanation for muscle growth must be at the molecular scale,” said co-author Neil Ibata. “The interactions between the main structural molecules in muscle were only pieced together around 50 years ago. How the smaller, accessory proteins fit into the picture is still not fully clear.”

    This is because the data is very difficult to obtain: people differ greatly in their physiology and behavior, making it almost impossible to conduct a controlled experiment on muscle size changes in a real person. “You can extract muscle cells and look at those individually, but that then ignores other problems like oxygen and glucose levels during exercise,” said Terentjev. “It’s very hard to look at it all together.”

    Terentjev and his colleagues started looking at the mechanisms of mechanosensing—the ability of cells to sense mechanical cues in their environment—several years ago. The research was noticed by the English Institute of Sport, who were interested in whether it might relate to their observations in muscle rehabilitation. Together, they found that muscle hyper/atrophy was directly linked to the Cambridge work.

    In 2018, the Cambridge researchers started a project on how the proteins in muscle filaments change under force. They found that main muscle constituents, actin and myosin, lack binding sites for signaling molecules, so it had to be the third-most abundant muscle component—titin—that was responsible for signaling the changes in applied force.

    Whenever part of a molecule is under tension for a sufficiently long time, it toggles into a different state, exposing a previously hidden region. If this region can then bind to a small molecule involved in cell signaling, it activates that molecule, generating a chemical signal chain. Titin is a giant protein, a large part of which is extended when a muscle is stretched, but a small part of the molecule is also under tension during muscle contraction. This part of titin contains the so-called titin kinase domain, which is the one that generates the chemical signal that affects muscle growth.

    The molecule will be more likely to open if it is under more force, or when kept under the same force for longer. Both conditions will increase the number of activated signaling molecules. These molecules then induce the synthesis of more messenger RNA, leading to production of new muscle proteins, and the cross-section of the muscle cell increases.

    This realization led to the current work, started by Ibata, himself a keen athlete. “I was excited to gain a better understanding of both the why and how of muscle growth,” he said. “So much time and resources could be saved in avoiding low-productivity exercise regimens, and maximizing athletes’ potential with regular higher value sessions, given a specific volume that the athlete is capable of achieving.”

    Terentjev and Ibata set out to constrict a mathematical model that could give quantitative predictions on muscle growth. They started with a simple model that kept track of titin molecules opening under force and starting the signaling cascade. They used microscopy data to determine the force-dependent probability that a titin kinase unit would open or close under force and activate a signaling molecule.

    They then made the model more complex by including additional information, such as metabolic energy exchange, as well as repetition length and recovery. The model was validated using past long-term studies on muscle hypertrophy.

    “Our model offers a physiological basis for the idea that muscle growth mainly occurs at 70% of the maximum load, which is the idea behind resistance training,” said Terentjev. “Below that, the opening rate of titin kinase drops precipitously and precludes mechanosensitive signaling from taking place. Above that, rapid exhaustion prevents a good outcome, which our model has quantitatively predicted.”

    “One of the challenges in preparing elite athletes is the common requirement for maximizing adaptations while balancing associated trade-offs like energy costs,” said Fionn MacPartlin, Senior Strength & Conditioning Coach at the English Institute of Sport. “This work gives us more insight into the potential mechanisms of how muscles sense and respond to load, which can help us more specifically design interventions to meet these goals.”

    The model also addresses the problem of muscle atrophy, which occurs during long periods of bed rest or for astronauts in microgravity, showing both how long can a muscle afford to remain inactive before starting to deteriorate, and what the optimal recovery regimen could be.

    Eventually, the researchers hope to produce a user-friendly software-based application that could give individualized exercise regimens for specific goals. The researchers also hope to improve their model by extending their analysis with detailed data for both men and women, as many exercise studies are heavily biased towards male athletes.


    Body builders aren’t necessarily the strongest athletes


    More information:
    Neil Ibata et al, Why exercise builds muscles: titin mechanosensing controls skeletal muscle growth under load, Biophysical Journal (2021). DOI: 10.1016/j.bpj.2021.07.023

    Provided by
    University of Cambridge


    Citation:
    Mathematical model predicts best way to build muscle (2021, August 23)
    retrieved 19 September 2021
    from https://phys.org/news/2021-08-mathematical-muscle.html

    This document is subject to copyright. Apart from any fair dealing for the purpose of private study or research, no
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  • New mathematical solutions to an old problem in astronomy — ScienceDaily
    Mathematic

    Mathematical model predicts best way to build muscle — ScienceDaily

    Researchers have developed a mathematical model that can predict the optimum exercise regime for building muscle.

    The researchers, from the University of Cambridge, used methods of theoretical biophysics to construct the model, which can tell how much a specific amount of exertion will cause a muscle to grow and how long it will take. The model could form the basis of a software product, where users could optimise their exercise regimes by entering a few details of their individual physiology.

    The model is based on earlier work by the same team, which found that a component of muscle called titin is responsible for generating the chemical signals which affect muscle growth.

    The results, reported in the Biophysical Journal, suggest that there is an optimal weight at which to do resistance training for each person and each muscle growth target. Muscles can only be near their maximal load for a very short time, and it is the load integrated over time which activates the cell signalling pathway that leads to synthesis of new muscle proteins. But below a certain value, the load is insufficient to cause much signalling, and exercise time would have to increase exponentially to compensate. The value of this critical load is likely to depend on the particular physiology of the individual.

    We all know that exercise builds muscle. Or do we? “Surprisingly, not very much is known about why or how exercise builds muscles: there’s a lot of anecdotal knowledge and acquired wisdom, but very little in the way of hard or proven data,” said Professor Eugene Terentjev from Cambridge’s Cavendish Laboratory, one of the paper’s authors.

    When exercising, the higher the load, the more repetitions or the greater the frequency, then the greater the increase in muscle size. However, even when looking at the whole muscle, why or how much this happens isn’t known. The answers to both questions get even trickier as the focus goes down to a single muscle or its individual fibres.

    Muscles are made up of individual filaments, which are only 2 micrometres long and less than a micrometre across, smaller than the size of the muscle cell. “Because of this, part of the explanation for muscle growth must be at the molecular scale,” said co-author Neil Ibata. “The interactions between the main structural molecules in muscle were only pieced together around 50 years ago. How the smaller, accessory proteins fit into the picture is still not fully clear.”

    This is because the data is very difficult to obtain: people differ greatly in their physiology and behaviour, making it almost impossible to conduct a controlled experiment on muscle size changes in a real person. “You can extract muscle cells and look at those individually, but that then ignores other problems like oxygen and glucose levels during exercise,” said Terentjev. “It’s very hard to look at it all together.”

    Terentjev and his colleagues started looking at the mechanisms of mechanosensing — the ability of cells to sense mechanical cues in their environment — several years ago. The research was noticed by the English Institute of Sport, who were interested in whether it might relate to their observations in muscle rehabilitation. Together, they found that muscle hyper/atrophy was directly linked to the Cambridge work.

    In 2018, the Cambridge researchers started a project on how the proteins in muscle filaments change under force. They found that main muscle constituents, actin and myosin, lack binding sites for signalling molecules, so it had to be the third-most abundant muscle component — titin — that was responsible for signalling the changes in applied force.

    Whenever part of a molecule is under tension for a sufficiently long time, it toggles into a different state, exposing a previously hidden region. If this region can then bind to a small molecule involved in cell signalling, it activates that molecule, generating a chemical signal chain. Titin is a giant protein, a large part of which is extended when a muscle is stretched, but a small part of the molecule is also under tension during muscle contraction. This part of titin contains the so-called titin kinase domain, which is the one that generates the chemical signal that affects muscle growth.

    The molecule will be more likely to open if it is under more force, or when kept under the same force for longer. Both conditions will increase the number of activated signalling molecules. These molecules then induce the synthesis of more messenger RNA, leading to production of new muscle proteins, and the cross-section of the muscle cell increases.

    This realisation led to the current work, started by Ibata, himself a keen athlete. “I was excited to gain a better understanding of both the why and how of muscle growth,” he said. “So much time and resources could be saved in avoiding low-productivity exercise regimes, and maximising athletes’ potential with regular higher value sessions, given a specific volume that the athlete is capable of achieving.”

    Terentjev and Ibata set out to constrict a mathematical model that could give quantitative predictions on muscle growth. They started with a simple model that kept track of titin molecules opening under force and starting the signalling cascade. They used microscopy data to determine the force-dependent probability that a titin kinase unit would open or close under force and activate a signalling molecule.

    They then made the model more complex by including additional information, such as metabolic energy exchange, as well as repetition length and recovery. The model was validated using past long-term studies on muscle hypertrophy.

    “Our model offers a physiological basis for the idea that muscle growth mainly occurs at 70% of the maximum load, which is the idea behind resistance training,” said Terentjev. “Below that, the opening rate of titin kinase drops precipitously and precludes mechanosensitive signalling from taking place. Above that, rapid exhaustion prevents a good outcome, which our model has quantitatively predicted.”

    “One of the challenges in preparing elite athletes is the common requirement for maximising adaptations while balancing associated trade-offs like energy costs,” said Fionn MacPartlin, Senior Strength & Conditioning Coach at the English Institute of Sport. “This work gives us more insight into the potential mechanisms of how muscles sense and respond to load, which can help us more specifically design interventions to meet these goals.”

    The model also addresses the problem of muscle atrophy, which occurs during long periods of bed rest or for astronauts in microgravity, showing both how long can a muscle afford to remain inactive before starting to deteriorate, and what the optimal recovery regime could be.

    Eventually, the researchers hope to produce a user-friendly software-based application that could give individualised exercise regimes for specific goals. The researchers also hope to improve their model by extending their analysis with detailed data for both men and women, as many exercise studies are heavily biased towards male athletes.

  • New mathematical record: what’s the point of calculating pi? | Mathematics
    Mathematic

    New mathematical record: what’s the point of calculating pi? | Mathematics

    Swiss researchers have spent 108 days calculating pi to a new record accuracy of 62.8tn digits.

    Using a computer, their approximation beat the previous world record of 50tn decimal places, and was calculated 3.5 times as quickly. It’s an impressive and time-consuming feat that prompts the question: why?

    Pi is, of course, a mathematical constant defined as the ratio between a circle’s circumference and its diameter. The circumference of a circle, we learn at school, is 2πr, where r is the circle’s radius.

    It is a transcendental, irrational number: one with an infinite number of decimal places, and one that can’t be expressed as a fraction of two whole numbers.

    From ancient Babylonian times, humans have been trying to approximate the constant that begins 3.14159, with varying degrees of success.

    The amateur mathematician William Shanks, for example, calculated pi by hand to 707 figures in 1873 and died believing so, but decades later it was discovered he’d made a mistake at the 528th decimal place.

    In 1897, the Indiana Pi Bill in the US almost did away with fussy strings of decimals altogether. The bill, whose purpose claimed to be a method to square a circle – a mathematical impossibility – almost enshrined in law that π = 3.2.

    What is it good for? Absolutely everything

    Jan de Gier, a professor of mathematics and statistics at the University of Melbourne, says being able to approximate pi with some precision is important because the mathematical constant has many different practical applications.

    “Knowing pi to some approximation is incredibly important because it appears everywhere, from the general relativity of Einstein to corrections in your GPS to all sorts of engineering problems involving electronics,” de Gier says.

    In maths, pi pops up everywhere. “You can’t escape it,” says David Harvey, an associate professor at the University of New South Wales.

    For example, the solution to the Basel problem – the sum of the reciprocals of square numbers (1/12 + 1/22 + 1/32 and so on) – is π2/6. The constant appears in Euler’s identity, e+ 1 = 0, which has been described as “the single most beautiful equation in history” (and has also featured in a Simpsons episode).

    Pi is also crucial to something in mathematics called Fourier transforms, says Harvey. “When you’re playing an MP3 file or watching Blu-ray media, it’s using Fourier transforms all the time to compress the data.”

    Fourier analysis is also used in medical imaging technology, and to break down the components of sunlight into spectral lines, de Gier says.

    But, says Harvey, there’s a big difference between calculating pi to 10 decimal places and approximating it to 62.8tn digits.

    “I can’t imagine any real-life physical application where you would need any more than 15 decimal places,” he says.

    Mathematicians have estimated that an approximation of pi to 39 digits is sufficient for most cosmological calculations – accurate enough to calculate the circumference of the observable universe to within the diameter of a single hydrogen atom.

    62.8tn digit accuracy – what’s the point?

    Given that even calculating pi to 1,000 digits is practical overkill, why bother going to 62.8tn decimal places?

    De Gier compares the feat to the athletes at the Olympic Games. “World records: they’re not useful by themselves, but they set a benchmark and they teach us about what we can achieve and they motivate others.

    “This is a benchmarking exercise for computational hardware and software,” he says.

    Harvey agrees: “It’s a computational challenge – it is a really seriously difficult thing to do and it involves lots of mathematics and these days computer science.

    “There’s plenty of other interesting constants in mathematics: if you’re into chaos theory there’s Feigenbaum constants, if you’re into analytic number theory there’s Euler’s gamma constant.

    “There’s lots of other numbers you could try to calculate: e, the natural logarithm base, you could calculate the square root of 2. Why do you do pi? You do pi because everyone else has been doing pi,” he says. “That’s the particular mountain everyone’s decided to climb.”

  • The Mathematical Pranksters behind Nicolas Bourbaki
    Mathematic

    The Mathematical Pranksters behind Nicolas Bourbaki

    In 1935, one of France’s leading mathematicians, Élie Cartan, received a letter of introduction to Nicolas Bourbaki, along with an article submitted on Bourbaki’s behalf for publication in the journal Comptes rendus de l’Académie des Sciences (Proceedings of the French Academy of Sciences). The letter, written by fellow mathematician André Weil, described Bourbaki as a reclusive author passing his days playing cards in the Paris suburb of Clichy, without any pretense of overturning the foundations of all of mathematics (that more disruptive part of Bourbaki’s oeuvre and ambitions would come later). On the strength of Weil’s recommendation, Cartan helped launch what would become one of the most storied and notorious careers in the history of mathematics.

    Weil, for his part, was playing a prank, propagating an elaborate in-joke that continued among mathematicians for decades. The math was real. Nicolas Bourbaki was not.

    Bourbaki’s oeuvre, among the most cited and celebrated contributions to twentieth-century mathematics, was planned and written by a brazen collective of scholars. Initially based in France, with a somewhat shifting membership, the founding group was dominated by alumni of the École normale supérieure, a storied training ground for French academic and political elites. They were united by an international outlook, often cultivated in early scholarly experiences abroad, and a conviction that French mathematics was due for a forward-looking generational shift.

    As the collective imagined him, Bourbaki would be gnomic and mythical, impossible to pin down. His mathematics would be just the opposite, absolutely unified, unambiguous, free of human idiosyncrasy. Bourbaki’s Elements of Mathematic—a series of textbooks and programmatic writings first appearing in 1939—pointedly omitted the “s” from the end of “Mathematics” as a way of insisting on the fundamental unity and coherence of a dizzyingly variegated field.

    The method Bourbaki’s collaborators developed to bring unity to mathematics was a time-honored tactic for producing social cohesion: an elaborate inside joke. The tactic may at first seem out of place in cold, formal mathematics, where cohesion ought to come from austere and unequivocal deductions open to everyone. But even in mathematics, it’s more fun to be in a club with barriers to membership. In fanciful and pun-filled narratives shared among one another and alluded to in outward-facing writing, Bourbaki’s collaborators embedded him in an elaborate mathematical-political universe filled with the abstruse terminology and concepts of modern theories.

    In their world of mathematical wordplay, prank, and parody, earth-shaking ideas could come from anyone and anywhere, but tended in practice to come from rambunctious elite French men who could get the references and weave in their own. Bringing more and more mathematicians into the joke, they built an extended community dedicated to radically rethinking how mathematics should be done.

    via Wikimedia Commons

    To make rigorous mathematics out of a shared joke, Bourbaki’s Elements of Mathematic took a decades-old philosophical consensus and gave it a twist. The existing line of thought held that making mathematics unambiguously, absolutely, incontrovertibly solid was ultimately a problem of language. To produce rigorous mathematical proofs, one needed a surefire way to guarantee that the proofs’ language was free from the unaccountable variations of human thought and perception. If everyone could just absolutely and forever agree about what they were talking about, the rest of the math would safely sort itself out.

    Like those before him, Bourbaki insisted on setting mathematics in a “formalized language” with crystal-clear deductions based on strict formal rules. When Bertrand Russell and Alfred North Whitehead applied this approach at the turn of the twentieth century, they famously filled over 700 pages with formal symbols before establishing the proposition usually abbreviated as 1+1=2. Bourbaki’s formalism would dwarf even this, requiring some 4.5 trillion symbols just to define the number 1.

    The twist came from Bourbaki’s intention to “very quickly abandon formalized mathematics, but not before we have carefully traced the path that leads back to it,” as explained in the volume Theory of Sets. Through a careful process of cultivating what Bourbaki called “abuses of language,” mathematicians could avail themselves of the creative ferment of informal reasoning while staying true to the certainty of formal deduction. In other words, like prank-loving schoolboys, they could cleverly break the rules and have their fun while being sure not to get caught in a fallacy. Abuses of language would be common informal understandings that witting mathematicians could share freely and fruitfully, like an inside joke, provided they all knew not to take them too seriously. Everyday mathematics became a kind of logical deadpan.

    Mathematically, Bourbaki’s basic ingredients were “signs and assemblages”: arbitrary but unambiguous written symbols and clearly demarcated ways of putting them together to create new meanings. This process of making meaning could appear deflatingly mechanical and pedantic: write signs next to each other, join them together with lines, replace some signs with others, draw new links according to a fixed rule, rewrite, rewrite again, ad nauseum. The path from here to the creative brilliance of the most exciting mathematical theories was hard to picture and practically impossible to tread step by step. But by carefully marking how to walk this path and systematically building up provisional abbreviations and shorthand, one could claim the rigor of mind-numbing sign-and-assemblage work simply by imagining how it would go, if one had the time and patience to spare for it.

    Socially, Bourbaki’s practices were anything but clear, mechanical, or universally inclusive. Their boisterous and alcohol-fueled meetings regularly dissipated into shouting matches and involved a fair amount of hazing toward new recruits. These Bourbaki “congresses,” often in scenic locales in the French countryside, were notably exclusionary. Women, rarely allowed to join the mathematical sparring even informally, were occasionally lumped with townspeople and farm animals as “extras” in the meetings’ pun-filled and sometimes ribald official accounts.

    For those in the know, part of Bourbaki’s allure was the idea of a secret society committed to principles of radical collectivity in their thinking and writing, selflessly sacrificing their individuality to a standard-bearing pseudonym. This, too, was part of the joke. Many fans of Bourbaki were entirely ignorant of the particular mathematicians who wrote his texts, but it was not unusual for mathematicians to know at least some real names behind the fake one. Their mythologized writing process, based on searing interrogation and exacting egalitarian consensus, likewise, was understood to be more an ideal than a consistent practice.

    Bourbaki
    Bourbaki’s mathematics was based on precisely articulated rules for writing and rewriting formal expressions. Meticulously specifying how to write and transform mathematical formalisms made it possible mostly to ignore such excruciating details in everyday practice. Source: N. Bourbaki, Set Theory, from Elements of Mathematic, English ed., 17.

    Partial knowledge of alleged secrets about membership and methods brought a large community of adopters and advocates into the fold, helping them feel a part of a project bigger than themselves. Meanwhile, Bourbaki’s status as an open secret let the collaborators reap the prestige of being associated with an important foundational project while avoiding questions about who they included—and what right they as individuals had to stake such brash claims about mathematical truth and method.

    Bourbaki’s most enthusiastic fans took the joke and ran with it. Early on, before the Elements of Mathematic started to appear, Weil shared the Bourbaki collaborators’ ambitions and culture of games and wordplay with a group of young mathematicians in Princeton, New Jersey. Thrilled and inspired, they created their own pseudonym, E.S. Pondiczery, and attributed to him a number of articles and a large volume of reviews. In their zeal for pseudonyms, they gave Pondiczery a pseudonym of his own, H.W.O. Pétard, to whom they attributed their more explicitly parodic writing. The Bourbaki and Pondiczery communities crossed paths regularly and integrated their pseudonyms into shared worlds, even printing elaborate pun-filled invitations for a wedding between Pétard and Bourbaki’s daughter Betti (from the surname of a famous Italian geometer).

    Many did not get the joke, taking Bourbaki’s dense formalism too seriously and thereby missing the import of the intervention. For those who believed Bourbaki’s goal was to replace informal mathematical thinking with cold manipulations of signs and assemblages, Bourbaki’s aims certainly seemed preposterous, especially when extended beyond the walls of mathematics departments. Famously (or notoriously), the New Math reform of primary and secondary mathematics education in the 1960s, inspired by Bourbaki’s project, baffled and outraged parents and educators who didn’t see why mathematicians thought small children ought to learn the abstract theory of sets along with their sums and figures. Quite a few mathematicians understood the joke well enough, but disagreed with Bourbaki’s priorities, worrying that they undervalued applied mathematics or mainly offered window dressing to existing methods that were working just fine without them.

    Many more were never meant to be included in the joke. Bourbaki courted and depended on the support of selected senior mathematicians, such as Cartan, as well as International Mathematical Union founder Marshall Stone. But they mostly wrote off older generations as squares who lacked the youthful perspective, dexterity, and ambition to get with the program. The old guard wrote them off in turn, while recognizing the number of talented younger mathematicians around the world who seemed taken in by Bourbaki’s predilections.

    In a 1954 report, a senior figure of Argentine mathematics threw up his hands at the “young ‘bourbakistas’” who took flight “on wings of abstraction” in “painfully escalating” volumes explicating commonsensical concepts. In Latin America, India, and elsewhere, Bourbaki’s style and theories came to symbolize a generational divide in conflicts over how best to use resources that flowed to science and engineering in nations around the world after World War II.

    The problem with being a generational icon is that generations keep coming. Bourbaki left a lasting mark on the style, culture, philosophy, and values of postwar mathematics, and many of his collaborators made influential careers under their own names. By the 1970s, however, the mathematical and social thrill of Bourbaki’s in-joke was showing signs of waning. The widened circle of winking insiders was no longer so exotic and alluring, the imposture of Bourbaki’s mathematics no longer so exciting. The Bourbaki Seminar, established to live out the Bourbaki principle that any mathematics worth knowing was worth rewriting from a Bourbaki perspective, continues to hum along. It remains the premier setting for elite mathematicians to rewrite each other’s latest theories, often contributing new insights in the process.

    An inside joke dies when nobody gets it—or when everybody gets it. Bourbaki’s mathematics was a victim of its success. To be a mathematician in the decades that birthed Bourbaki was to struggle to find common ground in a world riven by war and conflict. To practice mathematics during Bourbaki’s prime was to feel that world coming back together, socially and conceptually. How it came together mattered: Bourbaki helped promote a particular kind of insider for a new era of international mathematics. No calculus of signs and assemblages can create a universally valid mathematics when it comes with a culture and style that still makes outsiders of so many. In that regard, perhaps the joke is on Bourbaki.


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