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Duel at Dawn: Heroes, Martyrs, and the Rise of Modern Mathematics

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Overview

In the fog of a Paris dawn in 1832, Évariste Galois, the 20-year-old founder of modern algebra, was shot and killed in a duel. That gunshot, suggests Amir Alexander, marked the end of one era in mathematics and the beginning of another.
Arguing that not even the purest mathematics can be separated from its cultural background, Alexander shows how popular stories about mathematicians are really morality tales about their craft as it relates to the world. In the eighteenth ...
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Overview

In the fog of a Paris dawn in 1832, Évariste Galois, the 20-year-old founder of modern algebra, was shot and killed in a duel. That gunshot, suggests Amir Alexander, marked the end of one era in mathematics and the beginning of another.
Arguing that not even the purest mathematics can be separated from its cultural background, Alexander shows how popular stories about mathematicians are really morality tales about their craft as it relates to the world. In the eighteenth century, Alexander says, mathematicians were idealized as child-like, eternally curious, and uniquely suited to reveal the hidden harmonies of the world. But in the nineteenth century, brilliant mathematicians like Galois became Romantic heroes like poets, artists, and musicians. The ideal mathematician was now an alienated loner, driven to despondency by an uncomprehending world. A field that had been focused on the natural world now sought to create its own reality. Higher mathematics became a world unto itself—pure and governed solely by the laws of reason.
In this strikingly original book that takes us from Paris to St. Petersburg, Norway to Transylvania, Alexander introduces us to national heroes and outcasts, innocents, swindlers, and martyrs–all uncommonly gifted creators of modern mathematics.
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Editorial Reviews

Publishers Weekly
With tremendous attention to detail, historian Alexander examines the lives of 18th and 19th century mathematicians, finding much evidence to support his theory that the earlier geniuses of math (like Évariste Galois and Neils Henrik Abel) cultivated an artistic temperament, living short but fiery lives with little recognition, while the next generation (Jean le Rond d'Alembert, Leonhard Euler) pursued mathematics (and life) with purity and rigor, becoming "successful men of affairs who were the bright stars of their era and lived to a ripe old age." Though occasionally repetitive, Alexander's personable history of mathematics over two centuries (rounded out by a brief look at the present and future of the field) is filled with biographical details that will interest devoted mathematicians and historians of math or science; lay-readers may find Alexander's delivery too dry to stir their sympathies.
Copyright © Reed Business Information, a division of Reed Elsevier Inc. All rights reserved.
Peter Galison
Duel at Dawn is a delightful examination of the ways in which certain mathematicians have been made into mythical figures, and how the tropes of those canonical treatments have changed over the years. It's a fascinating and original book.
Jean-Michel Kantor
Does romantic mathematics exist? Romantic mathematicians do. Duel at Dawn reveals how the great mathematicians of the Enlightenment used geometry to study the earth and heavens, while their 19th century counterparts cherished internal beauty rather than practicality. Amir Alexander's original and convincing book opens a new path in the history of mathematics.
Forbes.com - Michael Patrick Brady
Through the life stories of three of the period's most controversial figures, Evariste Galois, Niels Henrik Abel and Janos Bolyai, Alexander reveals how their transgressive work changed mathematics and led to their lionization as Romantic heroes...Duel at Dawn neither talks over the head of its readers nor condescends, but instead ensures that the work of these Romantic mathematicians is not cloaked in obscurity. Of particular note is his breakdown of Hungarian mathematician Janos Bolyai's discovery of non-Euclidian geometry. Alexander does not shy away from the intricacies of the theory, nor the drawn out, convoluted history that underlies it. He takes readers through the process step by step, using plain language and clear diagrams to chart a course through the unknown. The larger narrative remains coherent without these more technical chapters, thanks to Alexander's ability to weave much of the mathematics into the fascinating lives of his subjects, but these in-depth studies of the math behind the men is very enriching. Mathematics need not be a scary, daunting subject, and Alexander does much to prove it.
Nature - Jascha Hoffman
Duel at Dawn suggests how preconceptions about the trappings of genius have radiated from art to maths. But its greater value lies in peeling back the layers of hagiography from figures such as Galois to reveal gloriously complicated men.
Times Higher Education - Tony Mann
This is a fascinating and provocative book. It is also extremely readable: the accounts of Galois, Abel, Cauchy and Bolyai and their posthumous reputations are engaging and entertaining, and along the way we meet many other fascinating personalities, including Guglielmo Libri, the aristocratic revolutionary, mathematician and stealer of rare books. Alexander's arguments are illuminating.
New Criterion - Martin Gardner
Alexander sees Galois's death as a turning point in the history of modern mathematics, a point at which math became less a study of nature than a purely abstract realm of its own, uncontaminated by the external world. He skillfully tells the story of this change, weaving it around the often tragic lives of the mathematicians most responsible for the change...[A] marvelous history.
Bookslut - Leigh Arber
Because it is such an engrossing story, it's easy to forget that the book's purpose also is to educate. Alexander conveys a general sense of who mathematicians were and how they fit in with society.
Booklist - Bryce Christensen
Though the Romantic ethos has persisted for a surprisingly long time among mathematicians, Alexander suspects that a cultural change is even now underway: reliance upon computers is replacing the mathematician-as-tragic-hero with the mathematician-as-skillful-nerd. Fascinating human faces peer out at the reader from behind seemingly sterile formulas.
Times Literary Supplement - Peter Pesic
[Alexander's] sensitive and thoughtful presentation illuminates the inner geometry of mathematical experience, leaving us to ponder whether its creators' parallel lives and works finally meet.
Forbes.com

Through the life stories of three of the period's most controversial figures, Evariste Galois, Niels Henrik Abel and Janos Bolyai, Alexander reveals how their transgressive work changed mathematics and led to their lionization as Romantic heroes...Duel at Dawn neither talks over the head of its readers nor condescends, but instead ensures that the work of these Romantic mathematicians is not cloaked in obscurity. Of particular note is his breakdown of Hungarian mathematician Janos Bolyai's discovery of non-Euclidian geometry. Alexander does not shy away from the intricacies of the theory, nor the drawn out, convoluted history that underlies it. He takes readers through the process step by step, using plain language and clear diagrams to chart a course through the unknown. The larger narrative remains coherent without these more technical chapters, thanks to Alexander's ability to weave much of the mathematics into the fascinating lives of his subjects, but these in-depth studies of the math behind the men is very enriching. Mathematics need not be a scary, daunting subject, and Alexander does much to prove it.
— Michael Patrick Brady

Nature

Duel at Dawn suggests how preconceptions about the trappings of genius have radiated from art to maths. But its greater value lies in peeling back the layers of hagiography from figures such as Galois to reveal gloriously complicated men.
— Jascha Hoffman

Times Higher Education

This is a fascinating and provocative book. It is also extremely readable: the accounts of Galois, Abel, Cauchy and Bolyai and their posthumous reputations are engaging and entertaining, and along the way we meet many other fascinating personalities, including Guglielmo Libri, the aristocratic revolutionary, mathematician and stealer of rare books. Alexander's arguments are illuminating.
— Tony Mann

New Criterion

Alexander sees Galois's death as a turning point in the history of modern mathematics, a point at which math became less a study of nature than a purely abstract realm of its own, uncontaminated by the external world. He skillfully tells the story of this change, weaving it around the often tragic lives of the mathematicians most responsible for the change...[A] marvelous history.
— Martin Gardner

Bookslut

Because it is such an engrossing story, it's easy to forget that the book's purpose also is to educate. Alexander conveys a general sense of who mathematicians were and how they fit in with society.
— Leigh Arber

Booklist

Though the Romantic ethos has persisted for a surprisingly long time among mathematicians, Alexander suspects that a cultural change is even now underway: reliance upon computers is replacing the mathematician-as-tragic-hero with the mathematician-as-skillful-nerd. Fascinating human faces peer out at the reader from behind seemingly sterile formulas.
— Bryce Christensen

Times Literary Supplement

[Alexander's] sensitive and thoughtful presentation illuminates the inner geometry of mathematical experience, leaving us to ponder whether its creators' parallel lives and works finally meet.
— Peter Pesic

The Barnes & Noble Review

In 1820, the Hungarian noble Farkas Bolyai wrote an impassioned cautionary letter to his son Janos:

"I know this way to the very end. I have traversed this bottomless night, which extinguished all light and joy in my life… It can deprive you of your leisure, your health, your peace of mind, and your entire happiness… I turned back when I saw that no man can reach the bottom of this night. I turned back unconsoled, pitying myself and all mankind. Learn from my example…"

Bolyai wasn't warning his son off gambling, or poetry, or a poorly chosen love affair. He was trying to keep him away from non-Euclidean geometry.

Staging an intervention to keep a child out of math trouble comes off as comic to the modern reader. But in the early nineteenth century, as Amir Alexander ably demonstrates in Duel at Dawn, mathematics was viewed as a passion on par with poetry -- an occupation that could lift a youth like Janos Bolyai to exalted heights, and just as quickly fling him into death or dissolution.

At the heart of Alexander's book is the story of Evariste Galois, the French mathematician who died at 20 in the titular duel. In his short life he managed to revolutionize algebra in a series of papers begun in his late teens -- with time left over to serve two prison sentences for political insurrection and to court the 16-year-old girl whose honor eventually demanded his presence at the dueling ground. His life was romantic, to be sure -- but not as romantic as the usual accounts have it. The story of Galois frantically writing his treatise on equations the night before his death, familiar to all aspiring mathematicians via E. T. Bell's account in Men of Mathematics, is a fabrication. And the extent of Galois's rejection by the mathematical grandees of the time was greatly exaggerated, not least by Galois himself. In prison for threatening the King's life, he wrote:

"I tell no one that I owe anything of value in my work to his advice or encouragement. I do not say so because it would be a lie. If I addressed anything to the important men of science or of the world (and I grant the distinction between the two [is] at times imperceptible) I swear it would not be thanks."

The quote encapsulates the character of Galois as Alexander presents him: an impetuous, peevish adolescent whose mathematical brilliance is matched only by his ability to make trouble and enemies. But in the posthumous mythology his co-revolutionaries erected around his memory, he was an innocent martyr whose flame burned too bright to last. That trajectory, Alexander argues, is characteristic -- almost obligatory -- for the genius in the romantic era of Keats and Byron. The genius gazes (ideally with smoldering eyes) into transcendent realms of truth and beauty invisible to lesser beings, and is inevitably destroyed by proximity to absolute truth and/or the jealousy of the dim establishment. Whether this accurately describes the genius's biography is beside the point; it was the story the romantics wanted to tell each other, and to hear.

As Alexander details, other great mathematicians of the era had their life stories pressed into the same romantic cookie-cutter, despite the wide range of their actual circumstances. In Duel at Dawn we meet the affable algebraist Niels Abel, who died poor and consumptive at 26 after having been turned down for an academic job in Norway; the politically ultraconservative but mathematically avant-garde Augustin-Louis Cauchy, fired from his professorship for refusing to teach what he considered a watered-down calculus course; and Janos Bolyai himself, who successfully got to the bottom of the problem that had bested his father, but retreated into bitter seclusion after the great Gauss failed to acknowledge his victory. All these mathematicians were viewed by their eulogists as martyrs to their own genius, and victims of a world too stupid to appreciate their brilliance.

The history of math is by now a ground well gone-over by popular biography; Alexander is to be praised for venturing beyond the usual "greatest hits" to cover underexposed mathematicians like Cauchy and d'Alembert. While Duel at Dawn is too long, revisiting the same interpretive claims over and over with minor changes, Alexander's broad range and ear for a pungent quote keeps it readable.

In the book's strongest sections, the author ties the new vision of the mathematical genius to a foundational shift in mathematical practice. Until around 1800, mathematics was yoked to physics -- the mathematician's role was to solve problems that arose, directly or indirectly, from material experience. Galois, Abel, Cauchy, and Bolyai, despite their disparate politics and biographies, had one thing in common: they represented a new notion of what mathematics was, in which the subject unhooked itself from the physical universe and held itself accountable only to its own logical rules.

In order to make this argument, Alexander needs to go past mathematical biography into mathematics itself. Accordingly, Duel at Dawn features brisk write-ups of the furious battle among the early modern Italians to solve the cubic equation; of the curve described by a hanging chain; of Cauchy's expulsion of infinitesimals from the calculus; and many more. The reader who is truly unwilling to encounter an equation may have to drape a cloth over these sections, but others will find much to gain from Alexander's clear and careful exposition. The history of non-Euclidean geometry is handled particularly well, offering many details absent from more casual treatments, and laying the ground for Bolyai's revolutionary advance.

Before Bolyai's era, Euclidean geometry was geometry. Euclid's so-called "parallel postulate" -- that every line admits a unique parallel through a given point not on that line -- was considered self-evidently true, though no one had been able to derive it logically from Euclid's other axioms. Bolyai took a different route, asking: what if it's not true? What if there were a whole spray of lines through the same point, all parallel to the same line? Mathematicians of an earlier era might have found the question incomprehensible; draw a picture on a napkin and you'll see there just aren't such lines. But Bolyai was different. He envisioned a geometry having nothing to do with the universe you can sketch on paper. And not only envisioned it -- he pinned it to the page, understood its rules and features.

"All I can say now," he wrote his father in triumph, "is that I have created a new and different world out of nothing." That's traditionally the artist's prerogative; when mathematicians took it on, Alexander writes, they became suitable, even inevitable subjects of the romantic myth. As G. H. Hardy observed, the age of Galois is the age in which mathematicians stopped asking what things were and started asking, instead, what they might be defined to be.

That might sound like a bad thing. Wouldn't it lead to a cloistered, inward-looking profession, making products of interest mostly to its own practitioners? The kind of profession, in other words, that contemporary art is routinely accused of being?

No. The irony, of course, is that yesterday's transcendent realm of pure thought is today's physics is tomorrow's basic engineering. After Poincaré and Einstein, we know that the geometry of the actual space we live in is non-Euclideanly weird. And the "group theory" pioneered by Galois, in its time an extravagant spin-out into abstraction, is by now a routine tool for any serious researcher in physics, chemistry, or signal processing. Once the idea of a square root of negative one was a passport into an alien universe; today we teach it in tenth grade.

If the romantic vision of the mathematician persists (see David Foster Wallace's 2000 essay "Rhetoric and the Math Melodrama" for a good rundown of recent examples in fiction), it is weakening. Recent popular portrayals of math like A Beautiful Mind and Proof feature mathematicians decoupled from ordinary reality, but this separation is presented as a hindrance to their research, not as its highest aim.

Maybe the reason is demographic. The romantic view of the self -- "nobody understands me, I'm made of higher stuff than the mundanes I have to live among" -- belongs, these days, to teenagers. And mathematics is no longer a teenager's game. It's too deep, the path to the frontier of research too long. We will likely never again encounter a figure like Galois, who can redirect an entire field of math while still young and emo enough to write, "I am disenchanted with all, even the love of glory. How can a world that I detest soil me?" The production of romantic mathematical heroes requires mathematicians who believe their own romantic hype, and that's not easy for a grown-up to do.

Alexander closes Duel at Dawn with a question: what will replace the romantic hero? He suggests that, as mathematics grows more computational, the mathematician of popular imagination will become a hypertechnical computer nerd. Unfortunately, I think a different outcome is likelier: the mathematician as neuroaberrant, like the narrator of Mark Haddon's blockbuster The Curious Incident of the Dog in the Night-Time, whose funny-peculiar brain is the cause, not the result, of his mathematical endeavors. We live in a world eager for medical etiology; "slightly elevated on the autism spectrum" goes down easy, while "touched the aether by sheer force of will" sounds antique. The mythology around Galois and Bolyai was a distortion of history and of mathematics, as Alexander amply demonstrates. But we might miss it when it's gone.

--Jordan Ellenberg

Jordan Ellenberg is Associate Professor of Mathematics at the University of Wisconsin-Madison, and the author of The Grasshopper King, a novel. He blogs at Quomodocumque.

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Product Details

Meet the Author

Amir Alexander is a historian and writer in Los Angeles.
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Table of Contents

  • Introduction: A Showdown in Paris
  • Part I: Natural Men
    • The Eternal Child
    • Natural Mathematics


  • Part II: Heroes and Martyrs
    • A Habit of Insult: The Short and Impertinent Life of Évariste Galois
    • The Exquisite Dance of the Blue Nymphs
    • A Martyr to Contempt


  • Part III: Romantic Mathematics
    • The Poetry of Mathematics
    • Purity and Rigor: The Birth of Modern Mathematics


  • Part IV: A New and Different World
    • The Gifted Swordsman


  • Conclusion: Portrait of a Mathematician
  • Notes
  • Acknowledgments
  • Index

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Customer Reviews

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Sort by: Showing all of 2 Customer Reviews
  • Posted July 10, 2014

    The Romantic side of mathematics is brought to life in this expl

    The Romantic side of mathematics is brought to life in this exploration of math history. Unexpected connections between mathematics, literature, and politics are illustrated by a short list of nineteenth century biographies. Prof. Alexander marvelously reminds us that mathematicians are more than the quiet, reclusive, socially inept nerds of the popular stereotype.

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  • Posted December 21, 2010

    Mixed Blessing

    Alexander has made an impressive effort to approach a change in the perception and function of mathematics, finding and using milestones (Galois, Abel, and Cauchy) to illustrate the events. Yet, in spite of a tremendous amount of research, Alexander fails to distinguish himself from other writers on the subject.

    What makes his book difficult to enjoy is the constant repetition of descriptive elements Alexander has identified for his main characters. Whilst revealing myths about Galois and his tragic death, his attempts to turn Cauchy from the "main culprit" hampering the advancement of Galois (and Abel) into a character that really deserves our sympathy because of his near revolutionary new approach towards the role and function of mathematics, are unpersuasive in that they fail to distinguish between Cauchy, the mathematician, and Cauchy, the individual.

    Whilst the first part of the book is virtually devoid of an explanation what contributions towards mathematics can actually be attributed to both Galois and Abel, the second part contains a collections of functions, the full significance of which Alexander neither explains nor fully understands. This leaves the reader stranded and wondering.

    This book lacks serious and constructive editing, and could have represented an impressive overview of "the time of change" if Alexander had drastically restricted himself to a narrative, and stripped out all repetitions and references to explanations, with which he lacks familiarity and comfort.

    There is enough substance in the book to warrant a second edition (if it were 150 pages shorter), but as it stands, it adds little to the wealth of writings that already exist on the subject.

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