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

by Amir Alexander

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.

Genres: Mathematics, Science, History, Nonfiction, History Of Science

320 pages, Hardcover

First published April 30, 2010

About the author

Amir Alexander teaches history at UCLA. He is the author of Geometrical Landscapes and Duel at Dawn. His work has been featured in Nature, the Guardian, among others. He lives in Los Angeles, California.

Reviews

[Adam · Rating 3 out of 5]

When I first read the description of this book on the dust jacket in the bookstore I realized I needed to buy the book immediately. When I got home, I started reading right away. I tore through the first parts of this book as the real stories of Galois and the authors premise came to light... but then my interest sort of stalled out after a while.

I think it had to do with the fact that this book wound up coming off as very repetitive. This book is extremely well researched and I loved finding out Galois' story in this new light which challenges a lot of typically told stories. However, I can only take the reiteration of Galois' and Cauchy's interactions so many times. Amidst the repetitions there might be a slight smattering of new detail, which was all the more frustrating to read about, because I'd wished it was in the section covering that topic. This happened with other mathematicians Amir covered as well.

I really thought the premise of this book was very interesting. It sets out to get an idea of mathematical tropes throughout history. Such as how would a mathematician be cast in a story in the 1700's, vs. 1800's and so on. He finds very different approaches to mathematics over the years and sees the great innovators of the 1800's and modern times more akin to artists, poets, and musicians than men of science, which is how it was previously looked upon in the 1700's. The book mostly looks at the giants of the field over the centuries, but with the shift of the 1700's "grand géometrès" being aligned with the doers of science, I wonder if the Applied Mathematician has simply transitioned into this role. Whereas the Purist stays rooted in the beauty and truth of mathematics as an artist would in approaching their craft. This question is never really addressed in the context of 20th century mathematicians and instead reference is only given to the tragic and misunderstood purists like Nash and Grothendiek. However, no mention is made of Erdös, who was no doubt an immense purist and giant of his field, yet he was never shunned in tragedy as the author tries to argue with his other examples. Erdös never really got disillusioned with the field even though his peculiarities easily could have led him in that direction. Instead the mathematical community seems to have embraced him and perhaps he is an example of the mathematical community learning from history, however, it seems the treatment of Perelman may have been a step backwards in some peoples opinion.

The other, far more minor, gripe I had with the book was the technical section in Chapter 7. Now, to Amir's credit he does point out that this section can be skipped for those not wishing to engage the technical work without any loss to the overall story. Now, I like doing and reading about mathematics as much as any mathematician, but I sort of felt that the amount of time spent discussing Euclid's 5th postulate was a bit overly verbose. I feel like the purpose was to show readers how rooted in reality mathematics had been for centuries. Then with the invention of Non-Euclidean geometries mathematicians were free of the shackles of reality and were able to consider all kinds of lofty worlds, whether they had any merit in reality or not. I am not sure this section was successful at this... while I was reading it, I never got this impression, but after considering it more, I feel that was the spirit of wanting to include the details. Unfortunately, in order to write it in a way that wouldn't lose the original thesis, the section is left dry, even though attempts are made to tie it into the overall story. This section just came off as forced in there, rather than enlightening.

In the end, I am glad I read this book. Very glad actually, for it has given me new insights into the history I cherish. It just wasn't executed in a flawless way. I feel like this book is about fifty pages too long and I do recommend future readers skip Chapter 7 as the author suggests and just focus on the story. I almost wish he had developed the conclusion's artistic criticism of portraits more. Rather than discussing that for a few pages then switching back to repeating most of the stories again for six pages more than was necessary. I think the book could have been easily fleshed out with other examples rather than only focusing on giants. The absence of female mathematicians is rather striking, and I can only imagine one can easily find tragic tales there.

In the conclusion he also touches on the possible new ways of approaching purists interests of proof. Now there are computers that can literally test all possible cases of a situation. I believe the 4-Color Theorem was found in this way. Much of the mathematical community shuns this "proof by computer" method, for the reasons he lists as its lack of elegance etc. But there is another major reason this is sort of shunned. In developing groundbreaking work, often times connections between fields of mathematics are made in the proof process that were previously unheard of. This gives us very deep insight into the structure of mathematics and brings us in whole new and creative directions. Andrew Wiles' proof of "Fermat's Last Theorem" brought together many fields of mathematics. So, personally, I don't think its just lack of elegance that we lose when a computer brute forces a proof for us... its those subtle connections between other branches of the discipline that are also lost.

I do agree with his ultimate thesis when it comes to the modern representation of mathematicians. They are often cast as misunderstood geniuses that just can't overcome the trappings of real world. Hence movies like "Proof" and "Goodwill Hunting" emphasize these kinds of tragic traits in some ways. I suppose this is a more favorable approach versus the misunderstood scientist whose only course of action seems to commit evil acts in todays typical stories. A very sad misrepresentation of science in the end...

[Alita · Rating 4 out of 5]

This was a terrific book, so long as the reader has an interest in either math or in modern intellectual history. The only way this book could have been improved is if the author had included some contributions from women mathematicians. They are conspicuous by their absence. I realize that 19th c. Europe was not swimming in female geometres, but it seems inconceivable that they played absolutely *zero* role in the developments chronicled in this book. That said, it was terribly interesting reading the true story of Galois. It would make a great movie, to be sure.

[Enes Ozel · Rating 4 out of 5]

Duel at Dawn was a book that my historian sister suggested a long time ago. I was looking forward to read it, but still I did not expect such a through study.

The book focuses on the transition in Mathematical studies from the Enlightenment to Romanticism, and identifies the duality that occurs in this process of change. Not only the way mathematicians understand and study mathematics changed: from a worldly science in which truth is intertwined with physical truth to a self-contained and rigor-bound, a more artistic study. But also the story of the main protagonists of the respective mathematical eras changed as well: from worldly men, who pursue to understand the world around them with the abstraction they produce using mathematics, who are also successful in their earthly pursuits and social lives, to romantic martyrs, whose eyes are fixed upon the realms of sublime truths and are mostly unfit to cope with the realities of the petty and obtuse society.

I found the book's language quite repetitive at points, which tends to be on the border of being boring. But as this was the only issue that bothered me and further the book presented a highly interesting thesis alongside strong arguments to support it, I was quite happy with my experience. I highly suggest it to anyone that has even the smallest interest in mathematics and/or history of sciences and arts.

[Fernando Pestana da Costa · Rating 5 out of 5]

This an absolutely stunning book on the history of Mathematics in the beginning of the 19th century, and of the huge changes in the perception of role of mathematics and of mathematicians, as well as their fitting in the wider world brought by the Romantic Era. The author examines the changes operating at a cultural and sociological level that led to the transformation of the image of the typical mathematician from the successful bourgeois civil servants, like d'Alembert, Lagrange, or Fourier, pursuing a "Natural Mathematics", to the young social misfits like Galois, a genial mathematician but also a republican firebrand with the "habit of insult", or the unfortunates Abel and Bolyai, or even (and somewhat more surprising) Cauchy. The character of the mathematics being developed was also changing from the "natural" questions arising from the physical world, for which that world would supposedly ensure consistency and correction, to a more "self-contained", pattern suffused mathematics that, later in the 19th century, progressed to the first two great triumphs of modern mathematics: the creation of non-Euclidean geometries (also discussed in the book), and the rigorization of Analysis. This is a very readable book about a central aspect of the history and sociology of Mathematics. A great read!

[William Bies · Rating 4 out of 5]

In view of man’s finitude, the discursiveness of his reason means that every science, and mathematics in particular, however cold and resplendent its finished form, is, after all, the product of a very human endeavor and not solely a timeless angelic contemplation. One need not go to the extreme of the sociology of scientific knowledge (the so-called SSK school) to take a lively interest in how the production of knowledge is conditioned by the personalities and intellectual commitments of its authors—for there is, indeed, no call for us to leap to its relativistic conclusion that scientific knowledge is nothing more than an ideological construct, much as, for Marx, the superstructure can be explained in toto as the outcome of prevailing economic relations. Rather, the contingent path along which knowledge unfolds is influenced but not determined by such factors, even though the truth in which it issues is objective. For this reason, the history of ideas offers often fascinating lessons on the nature of human being. (One wonders, what makes life interesting for an angel?)

The science writer Amir Alexander has established himself in recent years as one of the foremost contemporary practitioners of intellectual history, specializing in the field of mathematics. In the present volume, Duel at Dawn, he turns his gaze on one of the most storied episodes in mathematical culture, the rise of algebra in its modern guise at the hands of that youthful firebrand, Évariste Galois. But Alexander does not rest content with a stereotypical retelling of the exciting narrative along the lines of E.T. Bell’s memorable Men of Mathematics; instead, he situates it in the wider context of the arc of development of the overall field of mathematics from the eighteenth-century Enlightenment to twentieth-century modernism. For Alexander, it is an opportunity to engage in a character study of the mind of the mathematician.

The narrative begins in the eighteenth century, a period of consolidation of the revolutionary advances made in the previous century. The calculus was originally developed during the seventeenth-century scientific revolution with its applications to kinematics and dynamics of phenomena in the physical world very much in view. The great mathematicians of the Enlightenment era were not otherworldly dreamers, but honest, practical men who oriented their thoughts around what they wanted to do, which was, namely, to gain an understanding of and technical control over the forces of nature [in German, they would be called ‘Weltweise’]. Thus, Alexander: ‘For d’Alembert and his contemporaries, all mathematical relations, abstract though they were, were ultimately derived from the physical relations existing in the world around us and perceived by our senses. Conversely, they believed, general mathematical relations, derived at the highest levels of mathematical abstraction, informed us about the true realities of our physical world….This dual relationship between mathematical abstraction and the natural world was the defining characteristic of Enlightenment mathematics’ (p. 49). Typical problems of the day include the caternary and the brachistochrone (to find the configuration of a wire such that a bead sliding along it under the force of gravity will transit in the least possible time). Many of the redoubtable tools of analysis we still use today date to this heroic period, such as differential equations (both ordinary and partial) and the calculus of variations (leading to the Lagrangian reformulation of classical mechanics). As we see, the Bernoullis, Euler, Lagrange, d’Alembert and so on made considerable progress in the state of the art over what Descartes, Newton and Leibniz knew, but always stayed close to the applications in physics and celestial mechanics.

What changed in the nineteenth century? Alexander points to a crisis of ‘natural mathematics’ towards the end of the eighteenth century: there was an inner tension between the drive to ever-increasing abstraction and generalization, on one hand, and the physical roots of the subject on the other – too much abstraction threatened to undermine the very foundations of the field. Its resolution demanded a new type of man. From the worldly men of the eighteenth century, who enjoyed comfortable positions and high status in society, we go to tragic loners whose brilliance was scarcely recognized during their lifetime: Galois, Abel, Bolyai. The conservative royalist Cauchy figures as the villain in the story of Galois – yet in Alexander’s telling, he was not all that different; he also self-identified as a martyr. Indeed, given the ups and downs of revolution and restoration in French politics after Napoleon, Cauchy experienced great hardships at times in order to remain loyal to the royalist party and spent a good number of years in exile.

These men serve as role models for twentieth- and twenty-first-century mathematicians such as Gödel, Grothendieck and Perelman. They had a very different social standing than the Weltweise, but what about their mathematics per se? How was it different from what went before? Alexander summarizes by saying that they executed a Romanticist turn; he calls his heroes ‘mathematical poets’ (p. 162) and discusses the question, ‘why natural scientists are not poetic’ (p. 174). He illustrates what this all means with the contrast between Fourier and Jacobi (pp. 177-183) – utility, applicability to description of natural phenomena in the former versus purely internal standards in the latter. The following quotation spells out the significance of this change in orientation: ‘Simply put, whereas Enlightenment mathematics was concerned with this world, the new mathematics was focused on alternative universes, pure and beautiful, governed strictly by mathematical principles. Mathematicians, accordingly, are those who reach out to this better world and try to capture a glimpse of its glory. The most successful ones are those who have found their way to the mathematical realm and have observed its beauty and wonders firsthand….[Plato’s cave metaphor]….The vision of mathematics as a study of a pure realm of truth and beauty is therefore inseparable from the image of the mathematician as a tragic romantic figure’ (p. 182).

Again, in chapter seven, entitled ‘Purity and rigor: birth of modern mathematics’, Alexander asks how does ‘sublime mathematics’ differ from the ‘natural mathematics’ of the eighteenth century? Cauchy, more than anyone else the sponsor of the turn to rigor in the early decades of the nineteenth century, reappears; then the contrast between Lagrange (18th c.) and Galois is examined, summary on pp. 206-7. Alexander recapitulates his main point: ‘Cauchy, Abel, Galois and Jacobi shared the belief that the old mathematics had reached a dead end, and the field now required a new beginning. The new mathematics, they insisted, would no longer concern itself with specific numerical questions derived ultimately, even if remotely, from physical reality. Instead, mathematics would now be its own separate world, a realm of absolute truth governed solely by strict and rigorous reasoning….If the results ultimately corresponded to certain physical phenomena and could be useful in describing them, that was well and good. But it had no bearing on their truth, for the value of mathematical results would be judged by internal criteria alone’ (p. 210).

Chapter eight illustrates Alexander’s thesis with János Bolyai and the discovery of non-Euclidean geometry, summarized pp. 249-251. In conclusion: Alexander compares Galois, Abel and Bolyai with the canonical literary figures of Romanticism, Byron and Keats. They are certainly very different from natural scientists of the period like Kelvin or Helmholtz, who continued the tradition of the Bernoullis and d’Alembert – as is evident from the portraits he reproduces.

So much for the origin of modern mathematics and its cult of the solitary genius. Where do we stand today? Alexander’s suggestion: after discussing the role of computer-assisted proof in contemporary research, he raises the possibility of a symbiosis of hacker culture and the high romanticism of pure mathematicians, the representatives of ideal beauty, very much still alive today (cf. G.H. Hardy’s celebrated apology of the mathematical life as we moderns see it).

This reviewer’s take on today’s prospect: we should note the contrast between the new mathematical physics of string theory and so forth versus the old mathematical physics (represented by giants such as John von Neumann, Laurent Schwartz, Arthur Wightman, James Glimm, Arthur Jaffe, Rudolf Haag, Eliot Lieb). The relationship between mathematics and physics undergoes periodic swings, from close collaboration to a cool standoffishness. Many would say that recent decades have witnessed a renaissance of interdisciplinary cross-fertilization, after an interval of dissociation during the 1950’s to 1970’s. In some respects, trends in modern physics would justify such a view. Consider the central role played by symmetry principles in twentieth-century physics: Noether’s theorem, Einstein’s general covariance, Yang-Mills gauge theory in standard model. Thus, we can watch the pure mathematician’s ideal beauty propagating into the quotidian world of physics. Certainly, a lot of free invention according to what Alexander describes as Romanticist inspiration has become the norm in what goes under the name of mathematical physics nowadays. To be contrasted with the old-school mathematical physics adverted to above, which concerns itself with analysis and rigorous justification of problems arising from physics, mainly non-relativistic quantum mechanics and quantum field theory. The old school took its cue from concrete paradigmatic solutions rooted in experiment and sought to understand them with the techniques proper to mathematics, in order to supply a demonstrative foundation to already established theory; the new school seeks, rather, to discover novel theories of physics by applying some of the pure mathematics that our romantic geniuses have dreamt up without reference to physics at all. The movement bases itself to some degree on motivation originating in the perplexing outstanding problems with existing theories, but is conducted, in practice, in a manner largely independent of experiment (unlike the old school, as we saw).

Is there a problem here? Yes; without any check from experiment, one cannot know whether one is headed in the right direction. Sabine Hossenfelder contends in her Lost in math: How beauty leads physics astray that the turn to the new school of mathematical physics places us in a crisis, which we cannot get ourselves out of unless we revise our methodological standards of good scientific practice (q.v. this recensionist’s review of her provocative book). What would such a revision look like, from this reviewer’s perspective? Einstein distinguishes between theories of principle, which start from empirical propositions in the form of general principles and from these deduce the phenomena, versus constructive theories, which start from hypothetical microphysical constituents and derive how they would behave when acting together and are justified only by comparison with what does happen. Newtonian classical mechanics and Einstein’s special and general theories of relativity are theories of principle; the kinetic theory of gases and statistical thermodynamics, constructive. But today’s mathematical physics involves no principles arrived at by induction from experiment; it just jumps immediately into fancy mathematics for which, for the most part, the physics figures as merely a pretext – it is easy to see why this route is easier for a young scientist beginning his career, especially considering the pressure to publish or perish. All one has to do is to pick one’s favorite ‘hot topic’ and train up to the state of the art; no painstaking years-long groping in the dark such as Einstein endured from circa 1907 to 1915 required. But what do we get by following this procedure? A proliferation of Lakatosian degenerate research programs, nothing original. Consider how Roger Penrose, an otherwise quite able pure mathematician from whom Hawking cribbed, has been trapped for going on sixty years in his idea of twistors as a model of quantum fields, which to an outsider looks rather unpromising and ad hoc. At any rate, it incorporates scarcely any physical content. There are numerous pet theories in circulation analogous to Penrose’s, and to date none has achieved any striking empirical confirmation.

This reviewer will contend that we could use a reconnection to the physical world and a return to the ‘natural mathematics’ of the eighteenth century. Why? Isn’t there every reason to expect physical reality to be just as beautiful as our most inspired imaginings, if not more so? And the philosophical point: doesn’t poetic imagination ground itself in the seen outside world, whatever heights to which it may thereafter soar? (See Friedrich Schlegel’s Transzendentalphilosophie.) Hence, the overly dry and listless quality one perceives in much of current-day work at the research frontier in pure mathematics and the new mathematical physics, which for all its pretiose technical refinement has lost the requisite and life-giving tie to its roots in the physical world. Intuition informed by reflection on the empirical data is more likely to meet with success than genial spontaneity acting in a vacuum. Thus, if we are to make expeditious progress in the future, not just in mathematical physics but also in the purest of pure mathematics itself, we must come full circle with respect to the developments Alexander engagingly describes in the present study (without relinquishing what we have gained from the Romantics, or the enlivening tension between free invention and the insistence on rigor, yes, rigor at all costs!).

[Authorized · Rating 2 out of 5]

I'm no stranger to "difficult" academic texts, and I really, really wanted to like this book, but the only enjoyable thing I got out of this unending slog was the pleasure of writing this review.

Amir Alexander's Duel at Dawn starts out strong and looks to be an exciting treatise about the personalities behind modern mathematics; however, it soon becomes entrenched in the wash, rinse, repeat of descriptive and expository academic style.

Barring the final chapter (which, with previously unmentioned characters, feels unplanned and rushed), examples and events appear not only multiple times throughout the book, but multiple times within each section of each chapter, and with so little structural variation that the reader feels they have accidentally read backwards.

Dare I even mention the sections of mathematical formulae and accompanying figures? They appear so rarely that all they do for the lay reader is interrupt and distract from the narrative; they would be better served as appendices.

In conclusion (nudge nudge wink wink), the history is compelling, but this presentation would have benefited from a heavy-handed editor with a literary bent.

[Ibrahim Zaidan · Rating 4 out of 5]

3.5, if you are a historian with an interest in math you would give it a 5 for sure. I’m glad I read the book it makes you understand the real personalities of different mathematicians with historical accuracy. It corrected many of the myths I have read about Galois and it distinguishes between the facts and the added spices to the stories of amazing mathematicians to make them more mysterious. Also, the discussion on the quintic equation was pretty impressive.

[Nick Black · Rating 3 out of 5]

groundbreaking and meticulous research regarding the lives of galois, d'alambert, and abel (strongly refuting bell's the men of mathematics), but pretty repetitive after that, and lacking support for its central thesis.

the first book to have been read (for the first time, otherwise it'd be my cousin my gastroenterologist) in my new, permanent residence, hurrah! http://dank.qemfd.net/dankwiki/index.... w00t

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http://bnreview.barnesandnoble.com/t5... (linked to from http://www.aldaily.com)

[_ · Rating 4 out of 5]

I can think of a number of counterexamples to the author's claim that natural scientists can't be Byronic heroes. Victor Frankenstein, anyone? (I never said they had to be real people.) He's right about engineers, though.

[Elizabeth · Rating 3 out of 5]

Some really interesting points. But also super repetitive in places.