Pseudonymous mathematician Nicolas Bourbaki and his lesser-known counterpart E.S. Pondiczery, devised respectively in France and in Princeton in the mid-1930s, together index a pivotal moment in the history of modern mathematics, marked by international infrastructures and institutions that depended on mathematicians’ willingness to play along with mediated personifications. By pushing these norms and practices of personification to their farcical limits, Bourbaki’s and Pondiczery’s impersonators underscored the consensual social foundations of legitimate participation in a scientific community and the symmetric fictional character of both fraud and integrity in scientific authorship. To understand authorial identity and legitimacy, individual authors’ conduct and practices matter less than the collective interpersonal relations of authorial assertion and authentication that take place within disciplinary institutions. There is a general pattern to becoming a modern mathematical author. First comes a person who grapples with the discipline’s foundations, enters professional communities, finds collaborators, and gains wider recognition through publications followed by citations. With luck and wit, the author may travel the world and live on in the literature as an eponym or legend.1Nicolas Bourbaki’s story ran more or less in reverse. His surname, borrowed from a military personage, was in the air at the elite French École normale supérieur when, circa 1923, a student manifested it in an obscure accent from behind a false beard in a prank lecture as an eponym: Bourbaki’s theorem.Another student in the audience, André Weil, went on to obtain a doctorate in mathematics and found his first professional appointment in Aligarh, India.2 Kosambi gave Bourbaki a first initial (D), a nationality (Russian), a cause of death , and an unpublished Nachlass reputedly kept in the Leningrad Academy. In lieu of details of Bourbaki’s mathematics, he referred to a recent work then available only in Russian, thanking Weil for furnishing a reprint. A footnote anticipated both the  Russian paper’s and the  Bourbaki manuscripts’ hopeful translation and adjoined further details of Bourbaki’s death by way of a joke about the Cyrillic alphabet.3 The implication was that Kosambi’s countrymen could not tell a genuine obscure Russian mathematician and theorem from a fake, or were not worldly enough to care.There, Weil met the Harvard-educated Indian mathematician Damodar D. Kosambi, who, with Weil’s encouragement, worked the eponym into the title and discussion of his second ever mathematical paper.4 They adopted the collective pseudonym Bourbaki, christened Nicolas by Eveline de Possel, who would divorce Bourbaki cofounder René de Possel and marry Weil in 1937. Over the years, the critical and contrarian Bourbaki acquired a distinctive mathematical style and a fragmentary and sometimes-contradictory body of biographical lore, and became the only non-deceased figure included in the Dictionary of Scientific Biography.5None of these biographical details outlived Kosambi’s article, but the name Bourbaki went on to adorn articles and textbooks that would fundamentally reshape the practices, ideologies, cultures, and politics of modern mathematics. Beginning in late-1934, Weil joined with a small band of fellow upstarts in an ambitious project of curricular and conceptual reform that would continue for decades hence.6Because Bourbaki could not travel, write, or speak for himself, his personal manifestations relied upon impersonations. The self-styled “collaborators” of Bourbaki jointly prepared authoritative texts and joined with others to represent Bourbaki in a variety of contexts, often teasing or transgressing disciplinary norms as they did so. Relishing myth and misdirection, Bourbaki’s collaborators propounded his pseudonymous authorship as an open secret around which to establish or evade various forms of credit and responsibility in and beyond the mathematics profession. Their text-based personification reflected through parody how a widening world of mathematicians with strange names and biographies came to know and interact with each other, and dramatized the challenges and opportunities of an emerging global authorial order.7 Early modern conceptions of scientific authorship focused on assigning legal responsibility for textual claims rather than giving credit for the creative acts that underwrote them, and the tension between credit and responsibility defined competing economies of authorship over the ensuing centuries.8 Conceptions of fraud and integrity, in this tradition, have correspondingly focused on the relationships between individual named authors and associated facets of conduct and accountability, a focus often ill-matched to the structures and geographies of modern and contemporary research.9 Latter-day regimes of secrecy and bureaucratic control of authorship that subordinate individual labor under the name of a sacralized figurehead have been glossed as “the ‘Bourbakification’ of science,” in reference to the Bourbaki collective’s concerted obfuscation, the collaborative ideal this supported (in principle), and the havoc it made for attribution.10The problems of trust and verification are endemic to modern science, with the social problem of knowing and trusting people crucially mediating the epistemic problem of knowing and trusting the facts of the world.11 Because the evidence for a mathematical claim is ideally contained in the exposition itself rather than produced in a laboratory or other setting inaccessible to a reader, conventions of mathematical authorship and attribution might be expected to emphasize credit over responsibility. Indeed, to the extent responsibility continued to matter in the history analyzed here, it is significant that charges of deception related to authorial identity were not connected to questions about the authenticity or validity of the mathematics beneath the byline. Authentication and integrity mattered outside of the usual social and philosophical matrix linking scientific evidence to scientific subjectivity – a disjunction evident in more recent efforts, such as the ORCID program of persistent digital identification, to authenticate scientific identities.12Such subordination may have a special character for mathematical texts written with a demonstrative posture based on an ideal author and ideal reader joined through mutually accessible technical relations, putatively  free from relying on privileged authorial testimony.13With the story of Bourbaki and E.S. Pondiczery, a contemporary Bourbaki-inspired pseudonym, I shall advance two claims about authorship, fraud, and integrity in modern science. First, I claim that authorial integrity is a product, not a foundation, of the inevitably fictional co-construction of would-be authors and their would-be communities. This work of personification necessarily entails projection and always risks imposture. Scientific communities embrace or reject participants in turn by defining the bounds of acceptable representation. The primacy of these communal dynamics can be seen as well, in their absence, in a later attempt at pseudonymous projection by Kosambi, to whom this article returns in conclusion. Kosambi’s abortive personification of a mysterious figure, Sven Ducray, faltered for its lack of a consensual community to play along with the farce. For the two more successful pseudonyms of Bourbaki and Pondiczery, such collective representational labor constituted and authenticated legitimate – with Bourbaki, even monumental – authorial contributions to the mathematics discipline. Parodies defined communities of insiders who, by willingly or unwittingly playing along, made common cause with embodied and pseudonymous colleagues alike.unwilling to play along by means just as fictional as those by which the pseudonyms’ supporters asserted their integrity. Authorial fictions, here, could not be exposed by reference to some regulatory truth, but only by constructing competing fictions. Fraud was a negotiated condition, dependent on local sociotechnical constructions of valid or invalid attempts to contribute to or profit from disciplinary formations.14 The boundaries between deception, parody, prank, and fraud were drawn and redrawn according to the social, institutional, and intellectual aims and values of different parties. The success or failure of each pseudonymous provocation hinged on dueling conceptions of the bounds of legitimate dissimulation, and the corresponding limits of who could be in on (or the butt of) a joke.Second, if authorial integrity is a fictive co-construction, so too is its converse of fraud. Faced with pseudonymous impostures, some mathematicians responded by defining their own communities of those Following the layered representations and misrepresentations that animated Bourbaki and Pondiczery suggests that trust and verification may hinge less on faithful representation than on the socially constructed license to deceive fruitfully. Such deceptions are integral to modern scientific institutions, and can no more be rooted out than can authorship itself. Conventions of biography and bibliography, of validation and veneration, combined to support mathematicians’ participation in collective disciplinary formations that left ample space for play. Authorship was not an individual act but the outcome of communal relations of assertion and authentication – relations of a sort that continue to define legitimate participation in globally distributed scientific communities today. Here, integrity and fraud emerge as symmetric products of a common project of disciplinary fiction, defining both the individuals and communities that made up modern mathematics.That he did not generally speak for himself hardly made Bourbaki unique among famous names of mid-century mathematics. Secretaries routinely transcribed, composed, typed, and sent letters in mathematicians’ names. Correspondence was rife with excerpts, carbon copies, reported speech, and rumors. Most publications attributed a single author, but co-authorship was not unusual, and was understood to subsume unequal individual and shared efforts under a common attribution. However many authors adorned the byline, behind every paper lay a bevy of interlocutors and editors shaping words, formulations, and concepts.15 In this regard, establishing integrity in the sense of projecting the author as a coherent whole was always in tension with establishing integrity in the sense of transparently representing an author’s individual action. Projections of integral authorial identity were highly unstable, subject to refraction and revision according to changing social and intellectual goals and contexts.16 If Bourbaki’s integrity as a mathematician was an especially patent fiction, it depended on the consensual fictionality of every mathematician’s mediated identity.The higher one’s status in the discipline, the more one’s name ultimately reflected such impersonations, which nucleated distributed collectives of scientific activity under a unifying fiction of personal agency.17 Promising mathematicians from outside the disciplinary metropoles might round out their training abroad, but (with some exceptions) even elite mathematicians had few needs or opportunities to travel extensively once established. Interwar philanthropy facilitated young mathematicians’ mobility to and between major centers, reinforcing those centers’ hegemonic position while extending their reach more decisively beyond national borders.18Long segmented into relatively self-contained national and regional communities concentrated around a small number of dominant cities and institutions in Europe, mathematical research was by the early twentieth century only beginning to take a self-consciously international orientation, with periodicals leading the way.19 These allowed mathematicians to form and sustain scholarly communities with prohibitively distant colleagues, many of whom they would know primarily or exclusively by text. Large-scale projects of scientific bibliography dating to the latter half of the nineteenth century, including systematic international bibliographies in mathematics, raised the status and evaluative importance of articles and authorship.20 In 1931, the German publisher Springer launched an explicitly internationalist abstracting journal, the Zentralblatt für Mathematik, joined in 1940 by the American Mathematical Society’s Mathematical Reviews and in the 1950s by the Soviet Institute of Scientific and Technical Information’s Referativny Zhurnal.21With new opportunities to be interested in far-flung mathematicians came new challenges to keep up with their work, as well as new infrastructures to aid this pursuit.22 global communications infrastructures and the international institutions they supported and depended upon meant that mathematicians’ names traveled dramatically farther still. Bylines ran far ahead of bodies, and knowing people personally was, of necessity, supplemented and even displaced by knowing names. Each displacement demanded a cascading representation, a juncture for integrity or fraud that would test mathematicians’ emerging global institutions, norms, and practices.Together, these periodicals helped mathematicians across the globe to imagine something like an entire disciplinary literature to which presses and institutions around the world could contribute, unified despite the gaps and idiosyncrasies in individual library collections. Through reviews and citations, bylines multiplied across this expanded literature. As Cold War funding, jet travel, and other changes to mathematicians’ professional conditions let them personally travel much greater distances than their forebears,In the customary apparatus of academic publishing, cover letters bridge the personal, professional, and scholarly, allowing authors to assert identities, conflicts, and contexts that would not ordinarily appear in a formal academic text. Cover letters also, by this virtue, give cover to those who write them, offering a protected genre of quasi-confidential disclosure.Comptes Rendus in 1935.23 According to Weil, Cartan was a witting accomplice, going so far as to consult with fellow academicians at a regular lunch group before endorsing the article Weil attributed to Bourbaki.24 The cover letter made a loophole out of a good-faith assumption, exposing the potential for absurdity in a system that relied on its constituents to play along. In Weil’s telling, he stressed to Cartan that the Academy member who communicated a paper was expected to vouch for its scientific merit but had no obligation to certify the authenticity of the paper’s author.Under cover of cover letter, André Weil enlisted Élie Cartan, member of the Academy of Sciences and father of Bourbaki collaborator Henri Cartan, to communicate Bourbaki’s first mathematical publication to the Academy’s Comptes Rendus. In principle, such gatekeepers protected the content of the mathematical literature – much as peer reviewers have been said to do in other scientific contexts – but in practice this system of personal referral was as much (or more) about endorsing authors as legitimate contributors to that literature. Communicating an article declared that its author ought to be heard – in the pages of the Comptes Rendus and implicitly beyond, including in bibliographic apparatuses such as the Zentralblatt or Mathematical Reviews that relied on each journal’s authorial authentication – not that its communicator agreed with every line. Weil invoked a specious ideal of scientific validation to license a subversive variation on its routine companion practice of personal validation.Unestablished scholars routinely drew on personal connections to figures like the elder Cartan to gain access to establishment journals like the Sending up the presumed familiarity shared within a scholarly community, Weil introduced Bourbaki in the cover letter with a knowing “Vous n’ignorez pas que. . .” – “You are not unaware,” or “As you know.” He situated the wayward professor at a café in the Paris suburb of Clichy, where he would fit stereotypically as a refugee from the vaguely Eastern-European invented country of Poldavia, recently “wiped off the map of Europe” by world events. Those very world events, combined with a longer history of Franco-Russian mathematical exchange, meant that a strange Slavic migrant was hardly an unlikely source of noteworthy results. Though no longer an active mathematician, Weil’s Bourbaki had previously lectured at the Royal University of Besse in Poldavia, a reference to Besse-en-Chandesse, France, site of the gathering where the Bourbaki collaborators conceived of the article. Bourbaki reportedly allowed Weil to browse his unpublished papers, whence the proffered communication.25 This Poldavia began as a farcical government-in-exile whose members allegedly induced Paris sympathizers  to discuss the plight of the unfortunate nation at an open meeting, where they were ridiculed for their gullibility. That origin story may itself have been a retrospective cover story for Poldavia’s earliest historically verifiable appearance in the pun-dense 1929 missives by far-right-wing journalist Alain Mellet to dismiss soft-hearted rivals on the left.26 Mellet’s Poldavia had many imitators, including Belgian cartoonist Hergé, who dressed his hero Tintin in a false beard as Poldavian consul in a volume serialized just as Weil and collaborators were concocting their project under Bourbaki’s banner.The wider story of Poldavia, only partially evident in Weil’s letter, suggests further aspects of the cover letter genre’s work of disclosure, origin-making, and social justification. In his autobiography, Weil attributed Poldavia to another normalien prank, a decade before the parody that gave name to Bourbaki.27 than from the suggestion that the audience was susceptible to be duped. The only escape from this bind was to be in on the joke, to sympathize with the pranksters.The cruel logic of Poldavian attribution could serve normaliens, the Action Française, or the Bourbaki conspirators just as well: in a political environment and academic discipline built on solidarity across imagined ties to unfamiliar people in strange-sounding places, the role of hardscrabble migrant could belong to anyone. To do anything other than take a pitiable biography at face value would be to invite charges of callousness, even inhumanity; to take it at face value, conversely, was to be exposed as naïve, credulous, provincial. The imposture’s credibility mattered little; its force derived less from the prospect of a successful duping28 As Bourbaki became better known and the circle of those who knew not to take his biography seriously grew, biographical posturing became even more an inward-facing game than an outward-facing prank, a contest of artful allusions and clever inversions that affirmed membership in an in-group of consequence.Here, in distilled form, was the politics of playing along. Through all manner of referrals and introductions, mathematicians formed ties to strangers on the basis of chains of trusting association, augmenting by compounded credulity. Parodying this genre served similar ends, propagating a joke to create a group of mathematicians that got it. Neither Weil, nor Kosambi before him, aimed with their transparently farcical biographies to admit Bourbaki into the mathematics profession; both aimed to assert their own places to friendly audiences by creating a joke they could share with people who mattered at the expense of people who did not.29 When Smithies and then Boas parted from Princeton, their and Tukey’s triangular correspondence formed a veritable trove of Poldaviana.30When Weil visited Princeton in the first half of 1937, he inducted some of the younger mathematicians there into the Poldavian ruse, while also promoting Bourbaki’s mathematical ambitions. Among the newly minted Poldavians were Institute for Advanced Study postdoctoral scholar Frank Smithies, Princeton graduate-turned-postdoc Ralph Boas, and Princeton graduate student John Tukey.31 Inspired by Bourbaki’s entry in the Comptes Rendus, the Princeton conspirators decided to submit a compilation of their favorite methods pseudonymously to a leading American journal, the American Mathematical Monthly.32 Since the article was plainly farce, they attributed it to a plainly farcical author, H. Pétard – a reference to Shakespeare’s ‘hoisting’ line in Hamlet.Like their Bourbaki counterparts from France, the young Princeton mathematicians maintained an intense social life rife with in-jokes and wordplay. One favorite dinner-table punning game involved transmuting common mathematical turns-of-phrase into means of locating and caging a lion, for instance by inverting a spherical cage or traversing a space-filling curve.33 This latter pseudonym was a multi-layered tribute to Bourbaki, distorting the French-Indian city of Pondicherry with a Slavic spelling fit for a fellow Poldavian expatriate. Pondiczery’s initials came from the group’s casual, sardonic interest in extra-sensory perception, with the ‘S’ for Stanislaus and the ‘E’ later specified as Ersatz. To establish Pondiczery’s identity, the Princeton collaborators first contributed a half-page note in his name to the relatively unestablished-author-friendly collegiate “Questions, Discussions, and Notes” section of the American Mathematical Monthly, evidently without the editors’ awareness of Pondiczery’s pseudonymity.34They explained the Pétard pseudonym in a cover letter signed with another pseudonym, E.S. Pondiczery.35 By the time the article appeared, however, Boas had decamped to Cambridge, England, to continue his postdoctoral studies. He left the job of “general forwarding agent” to his sister, Marie, who handled payments and reprints on Pondiczery’s behalf.36 Informing Tukey of the arrangement, Marie speculated that “if he [Pondiczery] continues to receive mail,” it might be better to route it to her brother by “send[ing] him [Pondiczery] back to Poldavia, address unknown, but stopping in Cambridge to see Ralph.” On matters of copyright for the lion-hunting article, the conspirators cut out the middleman and conducted correspondence as Pétard rather than Pondiczery.37The business of fronting a pseudonym with another pseudonym required some ongoing misdirection for Pondiczery’s correspondence. Boas was Pondiczery’s principal ghostwriter for editorial matters.38 When Tukey conveyed a word of praise for the article, Boas mused: “Since Pétard owes his existence to his friends, I don’t see how they can avoid accepting some of the responsibility for his works, alas.”39 In the two years following his debut, Pondiczery made several contributions to the “Gleanings Near and Far” column of mathematical miscellany in The Mathematical Gazette, while Pétard added to a Cambridge student society’s parodies.40 At the start of 1939, Boas proposed a new mathematical result to attribute to Pondiczery, but Smithies advised it was “too good for E.S.P.”41Social credit for Pondiczery and Pétard’s initial and ongoing work raised further considerations. Boas reported from Cambridge that “Pétard’s paper is attracting attention here,” generating “subdued chuckles . . . in the Philosophical Library.”42 Boas replied that the officer’s “reactions indicate that the use of pen names was a very good thing. It is as well not to have even lunatics thinking you are crazy.”43 Paul Dirac, Boas gossiped, was not amused by Pétard’s “Dirac method.” The three gleefully reported on mistaken speculations about the article’s authorship.44The pen name also helped the conspirators avoid credit. One Tukey correspondent showed a Pétard reprint to a Rockefeller Foundation officer, who “didn’t see why they should publish such trash and [asked] did I know this man Pétard and, if so, was he as crazy as the stuff he wrote.”Mathematical Reviews in 1940, Boas contributed three reviews to each of its first two issues, and a total of twenty-eight that year, thirty in 1941, and thirty-four in 1942.45 In April 1942, fearing conscription after the U.S. entry into World War II, Boas found a job teaching mathematics to naval aviators in North Carolina.46 After discovering that his commanding officer did not want his staff publishing mathematics, Boas proposed to tell the Mathematical Reviews editor “that I have arranged for E.S.P. to take over my reviewing” and to give Tukey’s address so that the latter could forward materials to Boas.47Using pseudonymity to subvert some norms allowed the pseudonymers to sustain others. For Boas, one of the most important professional norms was engagement with the disciplinary literature as an author, referee, and reviewer. When the American Mathematical Society launched 48 Over the first three issues of 1943, Boas’s reviews faded out and Pondiczery’s faded in, reversing again in the October issue. Pondiczery completed fourteen reviews that year to Boas’s twelve in 1943, and Boas roared back into form with forty entries under his own name in the 1944 volume.On the strength of Tukey’s recommendation, Boas mused that the editor “either suspects everything or nothing” and was “apparently going to play ball.”Duke Mathematical Journal’s J.M. Thomas, who confessed his puzzlement on learning that the S. Nooten whose manuscript Tukey had favorably reviewed was in fact the mathematician Warren Ambrose (himself an Institute for Advanced Study postdoc from 1939 to 1941 and Princeton instructor from 1941 to 1943).49 Though Thomas corresponded with Nooten at his given address in Princeton and was reassured by Tukey’s favorable referee report, Ambrose-qua-Nooten had raised in the editor “a suspicion that someone might be pulling our legs.”The politics of playing along did not always require knowing who else was in the game. An editor who suspected everything or who suspected nothing would accommodate a pseudonym equally well; only one who suspected something but also had his doubts might raise trouble. So it was that Tukey found himself counseling the 50 In cases like that of Bourbaki, “The use of a pseudonym by a continuing group personality is regarded as requiring no justification,” provided an editor at the journal knew “at least one member of the group” and could “sponsor the non-mathematical content of the papers.” For isolated papers, a group pseudonym “may or may not be advisable,” and was a matter for editorial discretion. Tukey did not give any examples, Pétard or otherwise, for this case. Finally, as with W.S. Gossett’s pseudonym Student behind the “Student’s t-test” in statistics, an individual might use a pseudonym with the approval of a fully witting senior editor who knows the actual author and circumstances. Indeed, Gossett’s 1907–9 circumstances were not so different from Boas’s wartime situation, both using pseudonyms to circumvent an employer’s prohibition on publication – although Gossett-qua-Student did so with the consent of his employer, the Guinness brewery. With both Bourbaki and Student, Tukey averred, “pseudonymous publication has, it seems to me, definitely advanced mathematics.” Ambrose-qua-Nooten’s paper, however, ultimately appeared under Ambrose’s name.Tukey pulled hard in the other direction. In reply to Thomas, he laid out three “principles concerning pseudonymous mathematical papers,” without revealing that he himself had first-hand experience with the genre.American Mathematical Monthly – after a short 1939 note in the journal of the Indian Mathematical Society and a four-year wartime lull – leading to his lone appearance in Mathematical Reviews as an author rather than reviewer. The journal’s Leonard Carlitz addressed a letter to Pondiczery, and though the letter itself does not appear to have been preserved, Tukey’s reaction indicates that it raised the matter of pseudonymity explicitly. Accordingly, Tukey sent Carlitz two replies, one in Pondiczery’s name – advising Carlitz that Pondiczery would like the article to appear under Pondiczery’s own name and that he had shared the letter with Tukey, who might write separately – and one under Tukey’s own name.51The very next month, editors at the same journal discussed what to do with a submission by one E.S. Pondiczery. Tukey had found his mark, and the Princeton Poldavians picked the Duke journal for Pondiczery’s first American research publication outside of the I am prepared to say that Pondiczery is a continuing group personality, that I have seen some of his articles in other journals, and that I know some members of the group well enough to be very willing to stand sponsor for the non-mathematical content of any papers he may offer for publication.Tukey’s autograph letter referred Carlitz to his advice to Thomas about Nooten. In accordance with that advice, and without revealing his part in the Pondiczery scheme, Tukey wrote:52Adding “I doubt that he will become another Bourbaki,” Tukey suggested Pondiczery was “much more likely to benefit mathematics than to hinder it.” Unlike Nooten’s, Pondiczery’s paper appeared later that year under the pseudonym.whether they could be a boon to mathematics, but under what circumstances. Names, in a publishing profession, indexed credit, reputation, and responsibility alongside social, professional, and intellectual ties. Permissible misrepresentations were governed by higher principles of participation, collegiality, solidarity, and insight, to be adjudicated with confidential assurances shared among credentialed elites. Deceit could be virtuous if committed honestly.While pseudonyms clearly gave editors pause when they knew enough to spot them, the prevailing question was not Mathematical Reviews, were now not just available but imposed on his disciplinary field of view.53 Boas took over the editorship of Mathematical Reviews at the start of a postwar flood, not just from the Soviet Union: after a norm of around 2,000 total papers annually, he gasped in February 1946 at a delivery of 400 new papers from Japan alone.54 That October, he averred to Smithies that “material is accumulating at a frightening rate.”55To an unwitting reader – that is, at first, to all but the conspirators themselves and the small circle with whom they shared their joke – Pondiczery and Bourbaki blended seamlessly into the flood of strange names that inundated mathematical libraries wherever professional mathematicians could be found. As early as 1941, Boas noted to Tukey the mass of obscure Soviet journals that, thanks to the new Mathematical Reviews, cards recorded both the mathematical literature and the addresses and competencies of prospective reviewers. During Boas’s tenure as editor, he used the latter reviewer file to assign a paper in Gaelic to the lone reviewer who declared competence in the language, whereupon he learned that the paper’s byline was a Gaelicized name from the presumptive reviewer.56 While not strictly a pseudonym, the author’s Gaelic name may as well have been one, and the same bureaucratic and attributory practices that accommodated such variants also accommodated proper pseudonyms like Bourbaki and Pondiczery. Editors habituated themselves to dealing with unfamiliar addresses, trusting in the institutional machinery of international mathematics to hold everything together. Where editors were in the know, such apparatus need not extend to colleagues’ fabrications: Bourbaki’s fictionalized addresses, like the portmanteau Nancago of Nancy and Chicago, did not filter into Mathematical Reviews, nor did Bourbaki ever join Pondiczery as a reviewer.57To keep up, editors and researchers redoubled their reliance on a framework of attribution and circulation driven by authors’ bylines, supported in libraries and editorial offices by the powerful indexical technology of the card catalog. At Mathematical Reviews, and Boas continued to promote Bourbaki in reviews after his tenure.58 As a reviewer himself, however, he did not treat Bourbaki as an ordinary author, twice in 1949 outing the name as a pseudonym. The first was in an annual review of mathematics for the Encyclopaedia Britannica, over which Bourbaki lodged a complaint with the encyclopedia’s editors, who did not know to treat it as a joke and asked Boas to support his attribution. The second was in a Mathematical Reviews entry quoting André Delachet’s description of Bourbaki as “the polycephalic mathematician” – a quotation Boas found in the course of defending his Britannica entry from Bourbaki’s protests and presumably determined ought to be shared more widely.59Under Boas’s postwar editorship, Bourbaki’s textbooks drew prominent and sympathetic treatment in Mathematical Reviews, Bourbaki’s name appeared a total of twenty-nine times: eleven as an author and the rest in citations in other reviews. Reviews for five of the first six volumes of Bourbaki’s textbook fell to Samuel Eilenberg, who went on to join the Bourbaki collaboration in 1950. For the most part, Eilenberg treated Bourbaki as an ordinary name in the mathematical literature, marking it as a pseudonym only in his first review. As citation, Bourbaki filtered into Mathematical Reviews when reviewers noted prominent citations from the articles under consideration – articles themselves typically written by Bourbaki collaborators who featured Bourbaki as a means of sideways self-promotion. In six reviews, Bourbaki collaborator Jean Dieudonné insisted that the pseudonym’s primacy or priority had been neglected in the reviewed work.In the first decade of Where Pondiczery was most often a pseudonym of convenience, with his biography an ongoing social game, the comparatively systematic Bourbaki project placed greater organizational importance on the pseudonym’s byline while drawing attention away from biographical details that were incidental to the project’s ongoing operation. In practice, Bourbaki’s impersonators treated their collective as an open secret, breaking in and out of character where needed, for instance, in a grant application or career move. Behind the scenes, as opposed to in the mathematical literature, Bourbaki’s personhood was often fleeting. Though more famous for his outlandish biography, his bibliography mattered most.American Men of Science.60 The line between such a venture and article-oriented indices like Mathematical Reviews was thin but significant: both were guides to who and what mattered in the discipline, but the respective emphases on the men or the papers made the difference between earnest participation in a collective publishing enterprise and gnomic parody of indexers’ indifference to their subjects’ humanity or reality.Playing along with Bourbaki or Pondiczery in the mathematical literature entailed different demands and called upon different norms and practices from playing along with the pseudonyms as biographical figures. These latter exigencies, invoked outside the matrix of principles that legitimated routine pseudonymity in publication, let the pseudonyms nucleate challenges to the institutional orders of international mathematics. One glimpses this transgressive potential in Boas and Tukey’s 1942 speculation about getting Pondiczery into the reference volume 61 The 1948 application arrived from the University of Chicago not long after Weil joined its mathematics faculty, claiming Bourbaki as a ‘nominee’ covered by the department’s institutional membership and bearing a signature closely resembling two specimens attributable to Weil from a decade prior.62 The rest of the application, calling for various biographical attestations, mixed the vaguely plausible with the plainly parodic. Bourbaki was currently employed as a “Fellow of the Rockefeller Foundation” and could be reached at the Institut de Mathématique at the University of Nancy, a lightly fictionalized address that adopted Bourbaki’s textbooks’ characteristic singular variation on mathematic(s). From 1910 to 1919, he was “Professor, Royal Poldavian University,” and was born in Cucutemi, Poldavia, a play on the real village of Cucuteni, Moldavia, on the non-leap year date of 29 February 1885. His textbooks appeared by indirect reference to his employment as “Scientific Advisor, Hermann Publishing Co., 1934-.” AMS Secretary John Kline did not allow the application to be taken seriously.63First in 1948 and again in 1949, Bourbaki applied for membership in the American Mathematical Society (AMS), and was rebuffed both times.64 Though no historian has unearthed the paperwork,65 there is no reason to doubt that Bourbaki had entered the rolls of the Société Mathématique de France (SMF) earlier that year, and so was eligible under a 1946 reciprocity agreement with the AMS to join the American society as well.66 The new application was entirely handwritten, and the handwriting appears to support Beaulieu’s attribution to Jean Dieudonné, of Nancy.67 Dieudonné’s Bourbaki was born in Cucuteni, Moldavia, on 12 December 1886, elected to the Royal Academy of Poldavia in 1917 and the SMF in 1949, and presently employed at Dieudonné’s institute in Nancy. His 1910 doctorate came from Kharkov University, a real institution but sufficiently remote as to be unverifiable. Previous employers included the real Dorpat University, in Estonia, and the fictional Zorngahr College, an apparent portmanteau of German algebraist Max Zorn and Weil’s first employer in Aligarh.Bourbaki’s second application, dated 15 December 1949, would be harder to dismiss.68 His response is telling for its procedural assumptions. That Bourbaki was a pseudonym was well-known at this point to a great many AMS members , but Kline did not simply do as he had done the first time and rule Bourbaki ineligible on those grounds. Rather, he expended a special effort to demonstrate Bourbaki’s application as not just fictional but fraudulent.Kline took offense at once again being the butt of a Bourbaki prank, considering it an affront to the AMS’s dignity.70This subtle distinction reflected the bureaucratic logic of the far-reaching international ties Kline had, by then, been building for nearly a decade as AMS Secretary. Because he had to take unverifiable biographies at face value in the daily operation of a society that now embraced personally unknown mathematicians from far away, any principle used to reject Bourbaki must also derive from taking his application at face value. The primacy of face-value representations in international mathematical institutions and infrastructures had, by this point, become axiomatic. Indeed, as one of the mathematicians who initially received and cosigned the application observed, to deviate at all from the “essentially clerical” role of processing the applications as submitted, and especially to deny a member ostensibly in good standing of the SMF the benefits of the reciprocity agreement, might invite “international complications.”71 The very fluidity of Bourbaki’s biography, in other contexts, was part of the joke: few biographies of legendary figures are fixed with rigid certainty, and inconsistent rumors, if anything, added verisimilitude to a past shrouded in mystery. This conceit only worked, however, for biographies framed as second-hand reports, precisely the biographical genres that dominated the discourse of communities of insiders built around a shared joke. On an application form, Kline could take the biographical details as personal attestations coming from Bourbaki himself, and so deduce – while still taking everything at face value – that the attesting person named on both forms must have lied on at least one of them.Because Dieudonné could not consult Weil’s prior application and relied instead on their shared but fragmentary stock of Bourbaki lore for the second application’s particulars, significant discrepancies could hardly be avoided. Kline promptly had a secretary compose a two-column table comparing the biographical details.72 Weaver endorsed Kline’s evaluation of the prank as “quite childish” and noted that his foundation had supported Bourbaki but not, of course, as an individual fellow. Aware of Weil’s role in the collaboration but not of his involvement in the application, Weaver supposed that as a “responsible member of the group” Weil could broker a more reasonable approach.73 As it did for Tukey in his exchange with the Duke Mathematical Journal, the selective anonymity of Weil’s pseudonym-building let him maintain a reputation as a responsible broker while avoiding responsibility for potentially less-reputable impostures.Most of the biographical claims were, in practice, unverifiable. Kline could neither write to far-off universities or possibly defunct academies, nor assert with incontestable authority that they did not exist – notwithstanding his justified certainty that the claims were invented. Bourbaki’s Rockefeller Fellowship, however, could be easily assayed from within Kline’s own professional network. He wrote to the foundation’s Warren Weaver on the pretense of checking Bourbaki’s fellowship history, but really as a chance to vent his frustration over the undignified behavior of the French upstarts. Even Bourbaki’s discrepant signatures earned a remark, having deteriorated from that “of a determined man” in the first application to a “cramped” and “infantile” one in the second.74 Only Einar Hille, recent AMS president and a supporter of the Bourbaki enterprise, found Kline’s approach unconvincing.75 Hille advised that “a formalistic attitude” played right into the Bourbaki provocation, “and standing on our dignity will not get us anywhere and may expose us to ridicule.” Favoring the kinds of disembodied contributions to mathematics reflected in journals, reviews, and lore, Hille asserted that, ultimately, “a good fictitious character lives more intensely and a good deal longer,” and he had no doubt “that N. Bourbaki has made a stronger imprint on present day mathematics and his fame will last longer than that of most of present members of our Society and it behooves us of taking cognizance of this fact.” Hille thought the AMS should welcome the chance to join in the joke, but Kline stuck to his formalism and recommended the AMS Council reject the application.76Kline’s strategy of undermining the application at face value proved effective for most members of the AMS Executive Council whom he consulted.77 The AMS had to pretend to take the application as a sincere attempt to gain membership from an individual in order to challenge the premise that Bourbaki should be eligible. That is, they rejected the application because they knew it to be a farce, but their procedure for this rejection depended on an elaborate performance of believing the application to be genuine. Demonstrating fraud required fictions just as disingenuous and implausible as the alleged fraud itself.The Council carried out the recommendation in December 1950, writing that “it is well known that Bourbaki is not an individual but rather a group of distinguished mathematicians” and offering the group an institutional membership.78 Where the AMS tried to deflate the prank by taking it seriously, Dieudonné insisted that the Americans must have been joking, and that Bourbaki had taken their (otherwise implicitly offensive) rejoinder in that spirit. To the offer of institutional membership, Dieudonné countered by suggesting institutional membership for the University of Nancago. If AMS officers thought they could win by playing along, Dieudonné would change the game.Their reliance on formal criteria provoked a straightforward response from Dieudonné on Bourbaki’s behalf, reminding the AMS that “it is the clear meaning of the first paragraph of the reciprocity agreement . . . that the A.M.S. had no right to scrutinize” the application.Dukker (Marathi for pig), with the first name from Svana (Sanskrit for dog).79 Ducray began writing and publishing at the tail end of Kosambi’s once-bright mathematical career, as Kosambi tried and failed to convince friends and colleagues at home and abroad that he had proven the Riemann Hypothesis, to this day one of the most important open problems in mathematics and a repeated subject of claimed proofs from established and fringe mathematicians alike. In 1958, as Kosambi began circulating his proofs, Weil pled with him to keep them to himself and questioned his mathematical acumen.80It was only a matter of time before Kosambi, who brought Bourbaki-the-mathematician into the world, birthed his own pseudonym. Like most of the Bourbaki story, this part runs backward: Nicolas Bourbaki’s history was an elaborate and consequential farce, repeated in Kosambi’s pseudonym Sven Ducray as tragedy. The name recalls the mathematician’s affectionate nickname for his family’s overfed dog: 81Little is known about Sven Ducray except that he emerged as a number-theorist during long convalescence in India after a voyage over the backlands of China and Burma. The constantly changing, unpredictable date-line of his rare letters, from places right off the map, indicate neither fixed address nor connection with any academic institution. Though our author travelled around the world on some west European passport, he neither claims nor disclaims kinship with the Ducray whose name appears briefly in the annals of the French Revolution.A narrow slip of paper tucked among a collection of Kosambi and Ducray offprints in Kosambi’s archive, now preserved in New Delhi, gives the pseudonym’s only biography of any detail:82In the mathematical literature, Ducray teased with circumlocutions, such as the acknowledgment that “This paper would not have been possible without the constant labour of Prof. D. D. Kosambi” and, elsewhere, that “The full extent of my debt to Prof. D. D. Kosambi will some day be revealed, as well as the reason for not making better acknowledgement here.”Mathematical Reviews and the Zentralblatt für Mathematik, with reviews signaling apparent errors.83 Representing himself as Ducray’s mentor, Kosambi corresponded with at least one publisher on Ducray’s behalf and traded on his own prestige to see Ducray’s paper to print.84 In addition to publishing as Ducray, Kosambi circulated unpublished drafts of Ducray’s work to other mathematicians, including some with whom Kosambi had longstanding friendly correspondence of his own, without ostensibly revealing Ducray’s pseudonymity.85 In epistolary global mathematics, only Kosambi knew Ducray was a dog.86Ducray’s papers detailed arguments that Kosambi advanced and developed in the wake of 1958 through an extensive correspondence with leading mathematicians in Europe and North America, whose initial incredulity wore into exasperation or exhaustion. Helped along by Kosambi’s promotion, the papers received ordinary notice in both Ostracized in his national mathematical community and courting discredit abroad, Kosambi concocted Ducray in his refusal to play along. Without a community of co-conspirators identified with Kosambi’s quixotic assault on the Riemann Hypothesis, Ducray floated through letters, papers, and reviews as an unremarkable name unable to inspire the winking solidarity that marked the likes of Bourbaki for greatness but equally indistinguishable from the masses of faceless entrants to a burgeoning global profession. The pseudonyms of modern mathematics have been most powerful at their least deceptive. For the ever-widening circle who knew that Bourbaki was a pseudonym, playing along was a deliberate decision about one’s mathematical sociability. Younger mathematicians who thrilled to Bourbaki’s postwar textbooks alike embraced Bourbaki-the-byline as one more way to align themselves with the mathematical vanguard. Making up people made up communities, movements, and generations.Mid-century mathematicians’ impersonations followed the channels and infrastructures by which non-pseudonymous entrants to the profession established themselves: in short articles advanced by allies, in letters and reviews, in professional associations. Impersonation was a serious matter, defining the conditions of participation in a discipline where people depended on long-distance personal connections made and maintained multifariously. Defending those conditions from farce and imposture was also a serious matter, requiring fictions of its own. Neither legitimate nor fraudulent contributions to mathematics can be understood solely (or even primarily) in terms of the actions of individual perpetrators. Authenticity and fraud were both the products of collective fiction within mathematicians’ social institutions, reliant on their changing infrastructures, available alike to man and myth.


Aim: Analyse factors affecting retirement decisions comparing employees expecting to retire before, at, and after the state pension age. Methods: In the SeniorWorkingLife cohort, 12,269 workers aged +50 replied to questions about expected retirement age, reasons for leaving, and reasons for staying longer. Results: For all groups, poor health, wish for more leisure, and economy were the most salient expected reasons for retiring. Many would stay longer if there were better possibilities for more leisure time, more work flexibility, and economic benefits. Those expecting to retire before state pension age were more likely to point at desire for more leisure time as expected reason for retiring, and less likely to point at economic reasons, and more likely to point at health and work demands as possible reasons for prolonging working life. Those expecting to retire after state pension age were more likely to point at external factors and recognition from the management. Conclusion: Factors influencing retirement decisions are similar across the groups. Initiatives for better work–life balance, healthy lifestyle, and economic incentives to continue working may help prevent early retirement and motivate prolonging work life beyond retirement. Results also indicate that less strenuous work is particularly relevant to prevent early retirement. As a consequence of the ageing population in many western countries, the proportion of older workers and, thus, workers potentially leaving for retirement is increasing. To deal with the potential lack of qualified workers and potential economic strain caused by these demographic changes, there is strong political interest in prolonging work life in most western countries. Thus, knowledge on factors that can prevent early retirement and contribute to prolonged labour market participation is becoming increasingly salient. Various factors influence workers’ retirement intentions and behaviour ,2 and thA large body of research on the topic already exists and many factors have been identified influencing workers’ retirement intentions and behaviour. In the present study, we analyse and compare the self-reported salience of central factors underlying Push, Pull, Jump, and Stay among three different retirement groups: workers expecting to retire before, at, or after the state pension age, respectively. Below we provide a brief overview of research on these central factors.Poor health has been identified as the strongest Push factor. Health status is associated with and predicts time of retirement ,5,6 and Work demands may likewise push the worker out of the labour market. A scoping review identified physical and cognitive demands at work as barriers for prolonged work participation after pensionable age , these fFinally, interviews with people who retired early revealed that feeling that their knowledge was not being used in the organisation contributed to pushing them towards early retirement. However, some also reported not wanting to invest in new skills and knowledge and, therefore, retired early to “escape” attending courses and training . Other factors motivate workers to leave the labour market rather than pushing them out involuntarily. Research indicates that the decision to retire is influenced by the spouse . MarriedEconomy may also work as a Pull factor. Friis, Ekholm  found thThe desire for more leisure time appears to be a strong Jump factor leading older workers towards retirement. Interviewees reported the desire to “enjoy life now”, “wish to do other things outside work”, “wanting more flexibility”, and “to spend more time with or taking care of family members or friends” as motivations for retiring early . A scopiWhile the abovementioned factors are likely to lead the worker towards retirement, other factors may contribute to delaying retirement. Although ill health has been identified as a strong Push factor, health considerations can also work as a Stay factor ,12. DuriFurthermore, research indicates that influence and recognition may also work as stay factors motivating employees to continue working. High work time control has been identified as a key factor for extended employment into old age . In addiMoreover, skill development and challenges at work have been identified as Stay factors. Employees who report greater focus on development of skills and knowledge have been found to be less likely to retire early , and chaThe financial situation can also work as a Stay factor . For exaFinally, just as having a non-working spouse may pull the worker out of the labour market, qualitative findings show that not wanting to sit home alone while the spouse is working and being single are motivations for continuing work .Lastly, there are strong social norms regarding retirement age and these norms may influence retirement timing , this isAs documented in the literature, the retirement decision is often complex and influenced by many factors pulling towards either early or delayed retirement. However, less is known about differences and similarities in regard to Push, Pull, Jump, and Stay factors between workers who intend to leave the labour market before and after the state pension age. Are these the same factors that workers expecting to retire early perceive as important for the retirement decision, as those who expect to work beyond state pension age? Or the same factors that motivate prolonging work life even further? Knowledge on similarities and differences between these two groups of workers may inform targeted interventions to prevent early retirement as well as interventions to motivate workers, already working beyond the retirement age, to prolong their labour market engagement even further. Therefore, using survey data from a large representative sample of workers aged 55 or older, the present study analyses and compares the self-reported salience of the factors for conditioning retirement intentions presented above, among three different retirement groups: workers expecting to retire before, at, or after the state pension age, respectively. Further, in the same way, the study also analyses and compares factors that could potentially change such retirement decisions and prolong working life.This study is a part of the SeniorWorkingLife project that investigates Push and Stay mechanisms for labour market participation among workers aged 50+ years in Denmark. The study is registered as a cohort study in ClinicalTrials.gov (identifier: NCT03634410) and the open-access protocol is published elsewhere . In the For the present study, a total of 30,000 Danes aged 50 years or more  were drawn as a probability sample within each specified strata by Statistics Denmark and invited with a personal questionnaire-link via e-Boks  to participate. We only include employed individuals in the current article. Among those who were employed, the response rate to the entire questionnaire was 56%, but for the present analyses, those replying only partly were included as well, yielding a total sample size of 12,269 employed individuals. For the purpose of the present article, we divided the sample into three retirement groups (before (n = 6101), at (n = 3766), and after (n = 2402) the state pension age) based on their answer to the question “at which age do you expect to leave the labour market permanently”? Those who expected to retire at least one year before and after, respectively, the official state pension age were defined as ‘before’ and ‘after’ the state pension age. Those expecting to retire within one year before or after were defined as ‘at’ the state pension age.The questions were inspired by The Danish Longitudinal Study of Ageing  and the The SurveyFreq procedure of SAS (version 9.4) was used to estimate prevalence and 95% confidence intervals. For the different questions, the SurveyLogistic procedure was used to estimate odds ratios (ORs) and 95% confidence intervals for those intending to leave before and after state pension age, respectively, compared to those leaving at state pension age. The analyses were controlled for sex, age, and highest obtained education. Model-assisted weights were included in the procedures, based on information from high-quality national registers at Statistics Denmark, and took into account sex, age, occupational industry, highest completed education, family income, family type, and origin .As shown in Comparing participants expecting to retire at the state pension age with those who expect to retire before and after this age, respectively, the results show that those expecting to retire early were a bit, but significantly, more likely to point towards the two factors related to leisure as conditioning their decision to retire (OR: 1.26 (1.14–1.39), 1.14 (1.03–1.26)).Looking at health, work demands and well-being, there are no significant differences between participants expecting to retire at the state pension age and those expecting to retire early or late, respectively. Regarding economy and retirement considerations, not surprisingly, participants expecting to retire early were much more likely to point towards the possibility of receiving voluntary early retirement pension. Likewise, participants expecting to retire at the state pension age were more likely to point towards the possibility of receiving state pension than both of the other retirement groups. In addition, the participants expecting to retire early were less likely than participants expecting to retire at the state pension age to point at economic considerations as conditioning their decision to retire (OR: 0.71 (0.62–0.80)).When looking at norms, those expecting to retire at the state pension age were more likely than both of the other groups, to point towards the factor that it is common to retire at that age as conditioning their decision to retire.more likely to point towards the three external factors (OR: 1.53 (1.21–1.94), 1.97 (1.52–2.57), 1.84 (1.31–2.60)), and those expecting to retire earlier were less likely to point towards the external factors except for “wish from spouse” (OR: 0.64 (0.50–0.83), 0.50 (0.36–0.71). Although, in general, only a small proportion of the participants pointed towards external factors as expected reasons for retiring, when compared with participants expecting to retire at the state pension age, participants expecting to retire later were Finally, the participants expecting to retire after the state pension age were more likely than those expecting to retire at the state pension age, to choose the ”none of the above” factors as conditioning their decision to retire (OR: 3.74 (2.75–5.08)).When comparing the retirement groups, the results show that those expecting to retire later were less likely to point towards the two factors related to leisure as possible reasons for them staying in the labour market for longer (OR: 0.59 (0.51–0.68), 0.69 (0.59–0.80)). Regarding flexibility, participants expecting to retire at the state pension age were more likely to point towards “if the working time was better organised according to your needs” than both of the other retirement groups. more likely to choose the factors “if your health had been better” (OR: 1.35 (1.17–1.56)) and “if the work was less mentally strenuous” (OR: 1.30 (1.13–1.50)), while those expecting to retire later were less likely to point towards the work being less both physically and mentally strenuous as possible reasons for them staying in the labour market longer (OR: 0.55 (0.45–0.69), 0.59 (0.47–0.73)).When looking at health and work demands and comparing with participants expecting to retire at the state pension age. Participants expecting to retire early were more likely to point towards “if the management wanted you to stay longer” than those expecting to retire early and significantly less likely than those expecting to retire later. Moreover, those expecting to retire early were a bit, but significantly, more likely to point towards “if your work was appreciated to a greater extent” than those expecting to retire at the state pension age. Regarding recognition and influence at work, participants expecting to retire at the state pension age were significantly more likely to point towards “if there were less demands for adaptation and change” (OR: 1.18 (1.01–1.39)), while participants expecting to retire later, were less likely to point at this factor (OR: 0.62 (0.49–0.79)). In addition, those expecting to retire later, were more likely to point at “if there were greater professional challenges” as possible reason for them wanting to stay longer in the labour market (OR: 1.80 (1.36–2.39)).Looking at changes and challenges at work and comparing with participants expecting to retire at the state pension age, results show that participants expecting to retire early were a bit, but significantly, Regarding the factors related to education, there were no significant differences between the retirement groups.less likely (OR: 0.76 (0.61–0.95)) to point towards “if there was support from a spouse” and those expecting to retire later were more likely (OR: 1.63 (1.29–2.06)) to point towards this factor, than participants expecting to retire at the state pension age.When looking at the results regarding external factors, participants expecting to retire early were a little, but significantly, Finally, participants, expecting to retire after the state pension age, were more likely than participants, expecting to retire at the state pension age, to choose the “none of the above” factors as possible reasons for them wanting to stay longer in the labour market (OR: 1.47 (1.26–1.72)).The aim of the study was to explore factors conditioning retirement intentions and factors that could potentially prolong working life among workers expecting to retire before, at, and after the state pension age. In general, the results indicate that there are more similarities than differences between these three retirement groups. For all three groups, factors related to leisure, health, and economy were the most salient expected reasons for leaving the labour market and factors related to leisure, flexibility, and economy the most salient possible reasons for prolonging work participation. However, the results also found some differences between the groups. Those expecting to retire before the state pension age were more likely to point towards the desire for more leisure time as the expected reason for retiring, and less likely to point towards if it would pay off better as a possible reason for continuing working. At the same time, they were also more likely to point towards if their health had been better and if their work was less mentally strenuous as possible reasons for prolonging work engagement. Those expecting to retire after the state pension age were more likely to point towards external factors as influencing their retirement decisions.The most common factors conditioning the retirement decision (independent of retirement group) were related to more leisure time . When comparing the three groups, results show that participants expecting to retire early were a bit more likely to point towards leisure time as an expected reason for retiring. These findings are in line with previous literature showing that the desire for more leisure time is a motivation for early retirement . FurtherPoor physical health was a common factor across the three retirement groups as an expected reason for retiring, which is in line with previous literature ,10,11,12A large proportion of both workers expecting to retire before and at the age reported the possibility of receiving pension/early retirement pension as an expected reason for their decision to retire. This could reflect that this is a prerequisite to retire at all. Looking at possible reasons to prolong work life, “if it would pay off better economically” was one of the factors the largest proportion of the participants in all three retirement groups pointed at, supporting previous research indicating financial gain as an incentive to continue working ,10,11. T“Not being capable of doing your job” was also one of the factors the largest proportion of the participants in all three retirement groups expect to contribute to the decision to retire. There were no significant differences between the groups indicating that this is equally salient for all three groups. However, when looking at the factors “the work was less physically/mentally strenuous” as possible reasons for staying at work longer, participants expecting to retire later, were less likely to point at these. They also rated both the physical and psychological strain at work as lower indicating that they to a lesser extent have strenuous jobs. Physical and mental demands have been associated with early retirement ,13,14, iAlthough only a minority point at this, greater professional challenges appear to be a greater incentive to prolong working even further among those expecting to retire after the state pension age. These findings provide further support for research indicating that skill development and challenge motivate prolonging work life ,11,12,18Only a minority of the participants pointed at external factors such as “request from the workplace”, nevertheless, it appears to play a larger role in the decision to retire among employees planning to retire after the state pension age. This is further supported by the results that they were more likely to point at “if the management wanted you to stay longer” and “if there was support from my spouse”, as possible reasons for them staying longer, in other words, they are willing to continue working if encouraged to do so. The strengths of the study include the careful procedure to ensure that the estimates are representative of workers in Denmark aged 50 years or older. To produce the large representative sample used in the study, Statistics Denmark drew a probability sample among all eligible Danish residents age 50 years or older and combined this with model-assisted weights based on high-quality national registers. Furthermore, the electronic questionnaire was designed to reduce the risk of bias by randomising the order of the response options.The questions analysed in the present study were of a factual nature, i.e., the participants were asked to choose which of the listed factors they believed would contribute to their decision to retire as well as which factors they believed might motivate them to stay at work for longer. The listed factors were inspired from the literature. For these reasons, it is not relevant to validate the questionnaire. Nevertheless, one could argue that it is a limitation that we did not include an exhaustive list of possible factors, conditioning the retirement decision, and thus may have missed out potential important factors. However, firstly, as the list of factors included in the questionnaire was inspired from the rather extensive literature on the topic, we do believe that we have included the most important factors, secondly, we had to limit the number of factors included in the questionnaire to keep it at a reasonable length.A weakness of the study is that the results are based on cross-sectional data and intended behaviour and, therefore, do not reveal whether the identified factors relate to actual behaviour . Nevertheless, the purpose of the study was to investigate similarities and differences between the three retirement groups on factors they perceived as important for their retirement decisions. The data are therefore adequate for the purpose of the study. In the present study, we estimated ORs. However, it can be argued that the ORs overestimate the values found. Thus, a direct comparison with other studies using, e.g., prevalence ratios cannot be done. For interpretation of the present results, it is therefore also important to consider the respective prevalence (percentages in the first three columns of Finally, the applicability of the results to countries applying different arrangements of retirement benefits and labour market protection needs to be investigated in future research. For example, van der Wel, Dahl  concludeAll in all, the results indicate that the factors, that appear to be most salient when making the decision to retire, overall are similar for employees expecting to retire before, at, and after the state pension age, namely, leisure time, capability to do the job, health, and economy. Thus, initiatives to help ensure better work–life balance, enhance demand and resource fit, supporting healthy lifestyle, and that it pays off economically to continue working may help prevent early retirement and motivate prolonging work life beyond retirement age. However, the results also indicate that ensuring work–life balance and finding ways to make the work less strenuous are particularly relevant to prevent early retirement. The participants expecting to retire after the state pension age have already been prevented from leaving work early, so to say. To motivate this group of workers to prolong work life even further, in addition to focussing on work–life balance, healthy lifestyle, and economic incentives to continue working, the results indicate that professional challenges at work and being encouraged to continue working by the spouse and the workplace, may serve as motivation for at least some of these workers.


Entropy2018, 20, 103] presented a critique of several aspects of modern glass science, including various features of glass transition and relaxation, crystallization, and the definition of glass itself. We argue that these criticisms are at odds with well-accepted knowledge in the field from both theory and experiments. The objective of this short comment is to clarify several of these issues.In a recent article, Schmelzer and Tropin [ This short letter serves as a rebuttal to a recent article by Schmelzer and Tropin , which wGlass is a nonequilibrium, non-crystalline state of matter that appears solid on a short time scale but continuously relaxes towards the liquid state”. We also proposed an alternative, more detailed definition for use by advanced students and professionals in the field: “Glass is a nonequilibrium, non-crystalline condensed state of matter that exhibits a glass transition. The structure of glasses is similar to that of their parent supercooled liquids (SCL), and they spontaneously relax toward the SCL state. Their ultimate fate is to solidify, i.e., crystallize”.Much of the content of Schmelzer and Tropin’s paper is devoted to an attack on our recently proposed modern definitions of “glass” . The firWhile our definition stresses the hybrid nature of the glassy state, combining both solid-like and liquid-like features, Schmelzer and Tropin favor older definitions from the 1920s–1930s, where glass is treated as strictly a solid. As we explain in Ref. ,5,6,7, cSchmelzer and Tropin  continueRegarding glass flow and relaxation, Schmelzer and Tropin  write:flow with a perceptible rate on relevant time scales only in a certain temperature range and not beyond is well known in glass technology [24], as is clearly formulated also for example by Tammann  and is already given in the title of his well-known paper by Tool [53]. Of course, for certain applications, flow processes have to be taken into consideration as is well known already from the work of Kohlrausch, Weber, Williams & Watts, Adams & Williamson, Eyring & Tobolsky  starting around 1850. However, such possible flow processes under certain conditions have not been considered as essential by Tammann and Simon in their definition of glass. Such a point of view, that flow processes may be neglected in most applications for relevant times scales, has also been clearly expressed by one of the authors of Ref. [52] in Reference . For example, in Ref. [55] it is noted that “window glasses may flow at ambient temperature only over incredibly long times, which exceed the limits of human history”. The flow processes considered in Ref.  are primarily the response to external fields and are governed by viscosity. However, the viscosity and structural relaxation time are uniquely correlated as noted also in Ref. , where the analysis is performed widely in terms of relaxation times.”“The fact that predominantly glasses relaxation times —not the flow times under stress—of GeO2 glass and a commercial soda-lime-silica glass at room temperature, i.e., approximately 530 °C below their laboratory glass transition temperatures. The calculated relaxation times are indeed astronomical, but they are still finite!References  in   and spontaneously relax—even in the absence of external stress—towards the SCL state. Solids do not undergo spontaneous relaxation. They may creep or flow (by different mechanisms than glasses) under external stress.Schmelzer and Tropin  have alsglass is a rigid material obtained from freezing-in a supercooled liquid in a narrow temperature range”, they further state that “it is not clear if he intended to convey the same meaning we are using here (frozen = a temporary state)”. However, the meaning Simon assigned to his statement of freezing-in is clearly reflected in Ref. [32]. In free translation, it sounds as though freezing-in at gT does not imply that below gT relaxation processes are excluded. However, any such structural transformations proceed already slightly below gT with such large time scales that the suggestion of a permanent arrest of such structural changes is completely substantiated ; or, as stated by Davies and Jones , “Simon pointed out that as a glass is cooled through its transformation temperature the molecular diffusion which is necessary to effect the appropriate change in configuration is increasingly inhibited and finally becomes practically impossible”. This interpretation is fully in line with Ref.  but not with the revised definition of a glass given in Ref. [52]”.“In Ref. [52], Zanotto and Mauro discuss Simon’s definition of glass as a freezing-in process. Referring to Ref. [30] and presenting Simon’s point of view in the form that “Schmelzer and Tropin continue :“In RefgT the material is) and the entire thermal history . For small undercoolings below gT, the relaxation times (and flow under stress) will be only minutes or hours. However, if one is studying his/her glass well below the gT, the relaxation and flow times will be extremely long, as shown in Ref. [The relaxation times of a glass strongly depends upon the chemical composition , the temperature  was proposed in the 1920s as an empirical function to describe the viscosity of liquids and supercooled liquids . However to Ref. .Their ultimate fate, in the limit of infinite time, is to crystallize”. However, even if this statement would be true, it seems to us not to be reasonable to include such a statement into the definition, as it does not supply any additional information as to what glasses are. In addition, if at all, crystallization proceeds at a perceptible rate for states below the glass transition range also only at time scales exceeding the limits of human history.”“Further extending their modification of the definition of glass, Zanotto and Mauro propose to include into the definition of glass the statement, “Schmelzer and Tropin  argue aggT, these processes will take only minutes or hours; if one is working well below gT, say at room temperature for a typical oxide glass, crystallization will take longer than the age of the universe. To give a few examples, in the Zanotto laboratory we have crystallized lithium diborate glass and a diopside (CaO·MgO·2SiO2) glass, 30–50 °C below their respective gT in less than 3 months [However, glasses will always relax and then crystallize. Moreover, in certain cases, they can even crystallize before full relaxation . As for 3 months . Crystal3 months ,21.2O3 and albite, will crystallize below their gT with adequate heat treatments. The crystallization times are indeed very long well below gT, but they are finite. Even glasses that do not exhibit congruent crystallization will still crystallize incongruently in the limit of long time—in this case the primary devitrification phase corresponds to the crystal having the minimum Gibbs free energy in the phase diagram of the system. Finally, we should mention that atactic polymers do not crystallize because they are unstable artificial materials that degrade  upon treatment in a temperature range (somewhat above gT) where crystal nucleation and growth would be typically observed in laboratory time scales.As shown in Ref. , even thgT  also take place below the gT. It only takes longer as the temperature drops. In our opinion, this part of our new definition is very important because it reinforces the fact that glasses are not true solids: They only solidify at the end of their lives, when they crystallize. Our message is that all of the dynamic processes that take place above the laboratory Regarding the concepts of broken ergodicity and the configurational entropy of glass, Schmelzer and Tropin write :“Our conclusion is that the treatment of vitrification as a process of continuously breaking ergodicity with entropy loss and a residual entropy tending to zero in the limit of zero absolute temperature is in disagreement with the absolute majority of experimental and theoretical investigations of this process and the nature of the vitreous state. This conclusion is illustrated by model computations.”Unfortunately, their model does not account for broken ergodicity , which we argue is the single most fundamental, defining feature of the glassy state ,23,24. OWe have given arguments against several ideas promoted by Schmelzer and Tropin  regardinWe are thankful to Daniel R. Cassar, Marcio, L.F. Nascimento, and Raphael, M.C.V. Reis for their critical comments and suggestions.