Letters to the Editor

More on Galois Riemann’s relevance, see M. Fried, “Al- 2. About Galois’s death, I argue that Davide Bondoni (“Galois’s first mem- ternating groups and moduli space it was a suicide in disguise caused by oir”, Notices, May 2013) suggests to- lifting invariants”, arXiv #0611591v4, Galois’s inability to integrate himself day’s Galois theory is collective work, Israel J. Math. 179 (2010), 57–125, (DOI into a social group (be it academic, not purely Galois’s creation. He sug- 10.1007/s11856-010-0073-2), and my political, or other). Galois’s suicide, gests that we would best go indirectly website’s attached html file on modular masked as a duel, was the only way at the “master’s work”. towers and modular representations. he could affirm control of his life. Ap- Bondoni’s article may rightly react We gain much by looking back at the parent homicides were explained as to the recent C. Curtis review (Notices, optimism of Galois, Abel, and others, to suicides recently by Goeschel [Goe09] pursue results before they had a clue as December 2012) of P. Neumann’s book and by Emil Durkheim in his 1897 mas- to the outcome. Yes, newcomers would The Mathematical Writings of Evariste terwork [Dur13]. need help from those who have contin- Galois. Curtis restated the myth of ued such topics. References Galois’s death consequent on a duel. A variant on Rigatelli is in my Bul- Mathematicians have long accepted [Bon11] Davide Bondoni, Structural fea- letin London Math. Soc. 34 (2002), that was over a “tart” the night after he tures in Ernst Schröder’s work—Part I, 109–112, review of Matzat and Malle’s “created his theory of groups.” Bondoni Logic and Logical Philosophy 20 (2011), Inverse Galois Theory: http://www. refers only to Galois’s Theory erupting no. 4, 327–359. math.uci.edu/~mfried/booklist- [Bon12] ______, Structural features in overnight. vol/Matzat-MalleInvGal.pdf. Ernst Schröder’s work—Part II, Logic I address Galois’s supposed lack of and Logical philosophy 21 (2012), clarity. —Mike Fried, Emeritus no. 3, 271–315. 1. His work wasn’t just about groups University of California, Irvine [Dur13] Émile Durkheim, Le suicide, (as we might guess from Curtis’s re- [email protected] Presses Universitaires de France, Paris, view), or even groups delineating fields. 2013, reprint of the original 1897 2. Where Galois aimed remains still (Received May 6, 2013) edition. unreached. [Goe09] Christian Goeschel, Suicide L. T. Rigatelli, Evariste Galois: 1811– Reply to Mike Fried in Nazi Germany, Oxford University 1832, Italian-to-English translation by Press, Oxford, 2009. Let me ponder two questions present John Denton, Vita Math. 11 (Birkhäuser, [Sch74] Ernst Schröder, Über die for- Basel, 1996) remains strangely un- in Mike Fried’s letter. malen Elemente der Absoluten Algebra, acknowledged. Several use her cover 1. As a matter of fact, all math- E. Schweizerbart’sche Buchdruckerei picture of Galois while repeating myths ematicians do understand mathe- (E. Koch), Stuttgart, 1874. she debunks. Rigatelli documents that matical objects by relying on their [Sch12] ______, On the Formal Elements the girl in question was hardly a tart. own personal experience. We of the Absolute Algebra, LED Edizioni, Also, Galois was more likely a suicide must thank people like Klein for a Milan, 2012, English translation of structural reading of the concept than a duel victim. [Sch74] edited by Davide Bondoni. of group. While for Galois the con- Galois used Abel’s introduction of cept of group was only instrumen- modular curves we now call X (p). —Davide Bondoni 0 tal in proving the nonresolution of Thereby, Abel explained the smooth Independent scholar, Italy a fifth degree equation by radicals, variation of his famous elliptic curve [email protected] other mathematicians regarded function theorem. Galois’s unsolvability the concept of group from a meta- theorem showed these equations—ex- (Received June 28, 2013) mathematical perspective, i.e., as a cluding finitely many—were unsolvable concept of a theory useful for studying in j invariant radicals. Mathematics and Historical structured sets. Yet in 1874 the Ger- Galois deftly connected finite groups Chronology man mathematician Ernst Schröder in (even profinite by introducing the The Notices published in August 2013 a short pamphlet [Sch74] accomplished groups SL (pk)) and systems of analytic two letters related to my invited ar- 2 an analysis of such mathematical struc- spaces—before he was twenty-one and ticle “Mathematical methods in the tures, attaining the definitions of loops, without the best education to boot. study of historical chronology”, [No- semi-groups, and (commutative) groups, Spaces, especially profinite sys- tices, April 2013], which was duly starting from his personal structural tems, have components, cusps, dif- reviewed by seven anonymous referees. philosophy of mathematics.1 Accord- ferentials. These force you beyond This piece is based on my book The ing to such a point of view, an object in profinite thinking. (Contrary to what Lost Millennium—History’s Timetables itself does not exist; it exists only in a Curtis once insisted in my UCI of- Under Siege, an objective account of given context. Mathematics is not neu- fice.) Riemann, a certified genius, only the debate between historians and tral, but it is only context-dependent, as partially fulfilled over thirty years later revisionists of the established chronol- the various interpretations of Galois’s what Galois ended in 1832. ogy, the latter group known to include work testify. Even from Riemann—despite Gauss’ many cranks, such as Immanuel Ve- early help—much remains unfinished. I 1I had occasion to write on this topic ex- likovsky. Among other issues, I analyze still hear complaints about Riemann’s tensively (see [Bon11], [Bon12] and my there the controversial work of Anatoli clarity. For my meaning on finite introductory essay Algebra, what else? in Fomenko and his collaborators, which groups among profinite situations, and [Sch12, pp. ix–xli]). I divide into three categories: good

September 2013 Notices of the AMS 1015 Letters to the Editor

(follows established mathematical re- also played important roles in making it sults and has not been refuted, so far), firmly accepted. And the establishment mediocre or weak (unconventional or of a rigorous mathematical foundation questionable applications of otherwise of circular DNA topology further sup- sound mathematics), and blunders ports the helix structure. (conclusions drawn using false prem- Pauling and Corey proposed a triple ises or reasoning). In my Notices article helix DNA structure in 1953, which I focus on some mathematical methods turned out to be wrong. Watson and used in chronology that have already Crick had considered an incorrect DNA helped historians with dating events or model before the double helix; in their may do so in the future. But in the eyes earlier incorrect model, the negatively of the letter writers, Fomenko steals charged phosphate groups of two the show. strands of DNA interact by binding Claude LeBrun openly trashes Fo- with the magnesium ion between two menko’s work without providing any phosphates, and four bases (adenine evidence against my article, which he and thymine; guanine and cytosine) nevertheless dislikes, mostly for not have no interaction. having any mathematical content. Well, The double helix is intertwining and had he read what I wrote, he might have topologically constrained for circular learned that one of the issues I deal DNA. A big difficulty is the separation with is the motion of the moon, so it of strands during replication. There- embodies results in celestial mechanics fore the double helix was challenged (mainly differential equations) obtained even after Watson and Crick had been Proc. Natl. Acad. Sci. USA , Sept. 1976. by Newton, Lagrange, Laplace, Poisson, awarded the Nobel Prize in 1962, and [2] G. A. Rodley, R. S. Scobie, R. H. T. Jacobi, Poincaré, Hill, and others. My the side-by-side model of DNA was pro- article also refers to other branches of Bates and R. M. Lewitt, A possible posed (see accompanying figure) [2, 3]. conformation for double-stranded mathematics. The exclusion of the side-by-side model polynucleotides, Proc. Natl. Acad. Sci. The letter by Alex Emerenko and was recalled by Crick [4]: U.S.A. 73 (1976), 2959–2963. Victor Grinberg attempts, at least, to “At about this time Bill Pohl, a pure [3] W. F. Pohl and G. W. Roberts, Topo- argue against the publication of my mathematician, got into the act. He logical considerations in the theory article. But the way they do it reminds pointed out, quite correctly, that un- of replication of DNA, J. Math. Biol. 6 me of a joke about a man who was try- less something very special happened, (1978), 383–402. ing to find his keys around a lamppost, the most likely result of replicating a [4] F. Crick, What Mad Pursuit: A Per- though aware that he had lost them in piece of circular DNA would be two sonal View of Scientific Discovery, Basic the dark, a block away. Instead of dis- interlocked daughter circles rather Books, New York, USA, 1988, pp. 72–73. cussing the issues I present, they take than two separate ones. From this he [5] F. B. Fuller, The writhing number of a my statements out of context, apply space curve, Proc. Nat. Acad. Sci. USA deduced that the DNA chains could not them to all of Fomenko’s work, then 68 (1971), 815–819. be intertwined, as we had suggested, quote some of Fomenko’s blunders, and [6] J. H. White, Self-linking and the Gauss but had to lie side by side. finally let the reader decide whether integral in higher dimensions, Am. J. “Fortunately some brilliant work by my article is worth its salt. Arguments Math. 91 (1969), 693–728. Walter Keller and by Jim Wang on the to the point? Zero. If Emerenko and [7] F. H. C. Crick, Linking numbers and ‘linking number’ of circular DNA mol- Grinberg are so eager to defend the nucleosomes, Proc. Natl. Acad. Sci. USA ecules proved that all these side-by-side truth, they should consider sticking to 73 (1976), 2639–2643. models must be wrong.” the facts, abstain from distorting other In addition to William F. Pohl, the people’s statements, nurture some fair- —Min-Liang Wong ness, and use logic even when express- mathematical relationship of the link- Department of Veterinary Medicine ing number L of a circular DNA with ing an opinion outside their expertise. National Chung-Hsing University the twist T and the writhing number W Taichung, Taiwan was mainly accomplished by two math- [email protected] —Florin Diacu ematicians, F. Brock Fuller and James H. University of Victoria White [5, 6, 7]. The equation L = T + W is (Received June 12, 2013) [email protected] not a derivative of known mathematical or physical laws; it is unique for DNA (Received July 5, 2013) topology and geometry. This equation is one more example of Eugene Wigner’s The Incorrect DNA Structures “unreasonable effectiveness of math- and Mathematicians’ ematics in the natural sciences.” Contribution to DNA Topology 2013 marks the sixtieth anniversary References of Watson and Crick’s publication of [1] J. D. Watson and F. H. C. Crick, Mo- the double helix DNA structure [1]. It lecular structure of nucleic acids: A is worth emphasizing that arguments structure for deoxyribose nucleic acid, against the intertwined helical structure Nature 171 (1953), 737–738.

1016 Notices of the AMS Volume 60, Number 8