Principia Mathematica.
Cambridge: at the University Press, 1910-12-13. First edition of all three volumes of the monumental work that crowned the logicist programme in the foundations of mathematics — “the greatest single contribution to logic that has appeared” in two thousand years (DNB). The present set is of singular historical interest. Volume I carries the ownership signature of Karl Schmidt (1874–1961), the German-trained philosopher-mathematician who supervised Norbert Wiener’s 1913 Harvard doctoral dissertation comparing the algebra of relatives in Schröder with that in Whitehead and Russell, and is extensively annotated throughout in an English-language hand whose Schröderian orientation, critical perspective, and proposition-level engagement with the text correspond exactly to the framework of that dissertation. The annotations appear, in other words, to be the working notes of the scholar who set Wiener the agenda — one of the earliest serious confrontations between the Peirce-Schröder algebraic tradition and the Frege-Russell logicism, whose reverberations run through Tarski’s 1941 axiomatisation of the calculus of relations and into the architecture of modern mathematical logic. Volume II bears on its title page the inscription “Mihailovič 7. IV. 1925”, marking the set’s transmission into Central European hands at precisely the moment when Schmidt was preparing his move from mathematics at Tufts to the chair of philosophy at Carleton — the pivot in which he would have been dispersing his working logic library. Principia Mathematica represents the fullest realisation of the logicist thesis — the claim that mathematics is, at bottom, a branch of logic, its whole body deducible from a small number of primitive ideas and a small number of primitive principles of logical inference. The programme had its first major articulation in Frege’s Begriffsschrift (1879) and Grundgesetze der Arithmetik (1893–1903), but it was Russell’s 1901 discovery of the paradox that bears his name — that the class of all classes that are not members of themselves generates a contradiction — that forced a reconstruction of the logical foundations on a new basis. The theory of types devised by Russell and elaborated with Whitehead in the three massive volumes of the Principia was that reconstruction. It is, as Grattan-Guinness has written, one of the most impressive intellectual monuments of the twentieth century — and also, as its authors ruefully acknowledged, one of the least read. G. H. Hardy famously remarked in his Times Literary Supplement review that “perhaps twenty or thirty people in England may be expected to read this book”; Erwin Schrödinger went further, doubting that even Whitehead and Russell themselves had read all of it. Russell himself wrote in 1959 that he used to know of only six people who had read the later parts. Complete sets of the first edition are genuinely scarce. The Cambridge University Press printed 750 copies of Volume I; disappointing sales caused the publishers to reduce the print runs of Volumes II and III to 500 copies each, so that no more than 500 complete sets could ever have existed. Ex-library copies dominate the market; sets retaining their original publisher’s cloth, unrebacked and unsophisticated, are uncommon. The price the set commanded on its first appearance was felt even by leading mathematicians: the authors themselves had to contribute £50 each toward the cost of printing, a contribution refunded by the Press some forty years later, without interest. The intellectual character of the Principia rests on four linked achievements. First, the construction of a rigorous symbolic language — an extension and systematisation of Peano’s notation, informed by Frege’s Begriffsschrift — capable of expressing mathematical propositions without the ambiguities of natural language. The familiar signs of modern logic, many of them still in use today, make their decisive appearance here: the dot notation for conjunction and scope, the epsilon for set membership, the inverted iota for definite description, the horseshoe for material implication. Second, the theory of types: a grammatical stratification of the universe of discourse into individuals, properties of individuals, properties of properties, and so on, designed to block the vicious-circle paradoxes that had threatened naive set theory. Third, the systematic derivation of arithmetic from logic, following Frege’s definition of the number of a class as the class of classes equinumerous with it, and reaching by Volume II the Peano postulates as derived theorems rather than primitive axioms. And fourth, the extension of the programme through cardinal and ordinal numbers, real numbers, and — in the unfinished portions — toward geometry, for which a planned fourth volume was drafted but never published. The Principia’s achievements were partial and contested even at the moment of publication. Three additional axioms — reducibility, infinity, and choice — were required beyond the apparatus of type theory, and none of them seemed to be a truth of logic in any obvious sense. Frank Ramsey’s work in the 1920s, exploiting a purely extensional reading of higher-order objects, suggested that the ramified theory of types and the axiom of reducibility were dispensable; the changes were partially incorporated into the second edition of 1925–27. Kurt Gödel’s 1931 incompleteness theorems showed that no consistent system adequate for arithmetic could prove its own consistency, thereby placing a permanent ceiling on the logicist ambition. By the early 1930s the foundational work of Ramsey, Gödel, and Tarski had essentially completed the cleanup of the Principia’s formal apparatus, and Russell himself had withdrawn from logic. None of these developments diminishes the historical centrality of the work. Along with Aristotle’s Organon and Frege’s Grundgesetze, it remains one of the most influential books on logic ever written, and its impact on mathematical logic, set theory, the foundations of mathematics, linguistic analysis, and analytical philosophy is, in Grattan-Guinness’s phrase, of near-Newtonian proportions. The Modern Library placed it twenty-third in its list of the hundred most important English-language nonfiction books of the twentieth century. The signature on the front free endpaper of Volume I identifies a reader perfectly positioned to engage with this work at the deepest level. Karl Schmidt (1874–1961) studied at Marburg in 1893–94, Berlin in 1894–97, and again at Marburg in 1897–98, taking his doctorate from the Philipps-Universität in 1898 with a dissertation on the development of Kantian ethics (Beiträge zur Entwicklung der Kant’schen Ethik) supervised by Paul Gerhard Natorp (1854–1924), co-founder of the Marburg School of Neo-Kantianism. Natorp’s Die logischen Grundlagen der exakten Wissenschaften, published in 1910 in the same year as the first volume of the Principia, was a systematic engagement with the philosophy of logic, arithmetic, geometry, and physics that ran directly parallel to Russell and Whitehead’s project and, at key points, in critical opposition to it. The Marburg neo-Kantians endorsed logicism as a mathematical achievement but questioned the epistemological foundations that Russell attached to it, and a student of Natorp’s would have been trained to engage the Principia as an object of critical interrogation from the standpoint of an alternative tradition in the foundations of mathematics. Schmidt’s career matches the intellectual profile that the annotations demand. After his Marburg doctorate he served as First Assistant in the Physical Laboratory at Marburg in 1900–1901, then as Lecturer on Mathematics at Harvard in 1901–1903, Professor of Physics at Bates College in 1903–1904, and Professor of Mathematics and Astronomy at the University of Florida in 1904–1908, before taking up a professorship at Tufts College from c. 1908 into the 1920s. His 1913 paper ‘Studies in the Structure of Systems’ appeared in the Journal of Philosophy, Psychology and Scientific Methods — a title echoed almost verbatim in one of the book’s most philosophically pointed marginal notes, which observes that these are “Primitive ideas of a Structure of Systems rather than of Logic”. From 1928 until his retirement in 1946 Schmidt chaired the Department of Philosophy at Carleton College, returning at the end of his working life to the philosophy that, as he recalled in 1944, his Marburg professor had once warned him he was “wasting [his] time studying”. His 1944 book From Science to God: Prolegomena to a Future Theology (Harper & Brothers) marks the final articulation of a lifetime spent at the boundary of logic, mathematics, and philosophy. It was Schmidt, not Josiah Royce, who served as the principal advisor for Norbert Wiener’s 1913 Harvard doctoral dissertation — a fact recorded by the Mathematics Genealogy Project and corroborated by the Tufts Mathematics Department’s own account of Wiener’s education, which notes that the thesis was obtained while working with Schmidt at Tufts. Royce, whose name often appears as co-advisor, was at that time in rapidly declining health; the working supervision was Schmidt’s. Wiener had been Schmidt’s undergraduate student at Tufts from the age of eleven; the transmission from Cambridge, Massachusetts teacher to prodigy student ran across the Charles River for years. The decisive piece of internal evidence for Schmidt’s authorship of the annotations is the extraordinary thematic congruence between the marginalia and Wiener’s dissertation. The thesis — A Comparison Between the Treatment of the Algebra of Relatives by Schröder and that by Whitehead and Russell — is precisely what the annotations document: a systematic, proposition-by-proposition reading of Principia Mathematica carried out with Ernst Schröder’s Vorlesungen über die Algebra der Logik (1890–1905) open alongside. Someone read Principia this way; then someone wrote the dissertation. The reader and the writer are either the same person or, far more probably given Wiener’s age (seventeen when the work began, eighteen at its defence) and the topic’s technical demands, stand in a supervisory relation. The idea that a teenager unaided conceived an agenda requiring fluent mastery of both the German algebraic-logical tradition of the 1890s and Russell’s freshly minted logicist apparatus is not credible; the idea that his advisor — a Marburg-trained philosopher who had come of age at precisely the moment Schröder’s Algebra der Logik was the standard advanced logic text in Germany — set that agenda is overwhelming. Specific parallels tighten the case. An annotation underlines “This is the Schröder-Bernstein theorem” on page 489; the dissertation contains an extended treatment translating Principia’s proof of *73.88 into Schröder’s notation. The marginal note “cf. Problems of Phil.” at page 46 points forward to Chapter IV of the dissertation, which opens with Russell’s The Problems of Philosophy and his account of description. Annotations critical of Principia’s proof strategies — “bad! See …” with a reference to algebraic method — correspond to the dissertation’s repeated argument that Russell’s proofs are unnecessarily complex by comparison with Schröder’s. Most strikingly, the annotations stop at page 489, at precisely the proposition range (the *70s through the *90s) at which the dissertation’s comparative tables themselves trail off. The annotator and the dissertation author were working on the same terrain, and they stopped at the same place. The argument goes further still. Irving Anellis has noted that Wiener’s thesis marshalled convincing evidence to show that Russell had taken his treatment of binary relations in Principia Mathematica almost entirely from Schröder’s Algebra der Logik, with a simple change of notation and without attribution. That is an astonishingly aggressive claim for an eighteen-year-old to direct at Bertrand Russell; it makes sense only as the transmission of a framework supplied by an advisor with the standing, the expertise, and the philosophical conviction to mount such a charge. Wiener’s own later reflection — that on coming to study under Russell in England he found he “had missed almost every issue of true philosophical significance” — confirms that the framework of the thesis was inherited rather than independently conceived. The intellectual paternity runs from Natorp to Schmidt to Wiener, and this copy of Principia Mathematica is the physical trace of the middle link. The consequences of the dissertation Wiener wrote on Schmidt’s agenda were permanent. In the summer of 1913, shortly after the Harvard defence, Wiener travelled to Cambridge to show the thesis to Russell. Russell was displeased, complaining that his young American visitor had considered only “the more conventional parts of Principia Mathematica”. Out of that encounter — and out of the intensive study of relations that the dissertation had required — came Wiener’s 1914 paper ‘A Simplification in the Logic of Relations’ (Proceedings of the Cambridge Philosophical Society 13, pp. 387–390), in which he stated publicly for the first time that ordered pairs can be defined in terms of elementary set theory. The Wiener pair, modified by Kuratowski into the form now standard in every first course in set theory, rendered the whole axiomatic apparatus Russell and Whitehead had developed for relations in Principia’s *30s essentially redundant: relations could be defined as classes of ordered pairs, and ordered pairs could be defined in pure set theory, and so the theory of relations required no axioms or primitive notions beyond those of set theory. It was one of the foundational simplifications of twentieth-century mathematics. The intellectual genealogy of that insight runs back through Wiener’s dissertation to the annotations in the present volume. Tarski’s 1941 paper ‘On the Calculus of Relations’ — the paper in which the Peirce-Schröder algebraic tradition was at last given a modern axiomatic foundation — cites Wiener (1913) as a precursor. Volker Peckhaus, in his authoritative survey of Schröder’s logic, identifies Wiener’s dissertation as the earliest substantive engagement with the question of what the Principia’s treatment of relations owed to the algebraic tradition it displaced. The scholarly conversation that Tarski, Halmos, Jónsson, and others carried forward in the algebra of relations across the twentieth century begins, demonstrably, in this book, at Karl Schmidt’s elbow, in the years around 1910–1912. The second inscription in Volume II — “Mihailovič 7. IV. 1925”, written in Central European format (day, Roman-numeral month, year) and carrying the diacritical háček on the final consonant — marks a transmission of the set into a second owner’s hands on 7 April 1925. The date falls precisely in the window during which Schmidt, by then in his early fifties, was preparing to leave the Boston academic orbit for the Midwest: he took up the chairmanship of Philosophy at Carleton in 1928, and the years leading to that move were his pivot away from mathematical logic and back toward the philosophy that had been his original field. Dispersal of a working logic library — including a first edition of Principia Mathematica — is the natural correlate of such a transition. The bearer of the name “Mihailovič”, whose precise identification remains a matter for further research, belonged in all likelihood to the Czech, Slovak, or South Slavic academic milieu of Central Europe in the interwar period, and the set’s eventual reappearance in the German book trade is consistent with a European second life after 1925. Volume III, married to the set at a later date, came from the library of the University College of North Wales at Bangor, accessioned 26 April 1920, and bears the cancelled bookplate, accession number 32462, and shelfmark QA91.W5 of that institution. The present set is, in short, a working copy of Principia Mathematica from within the small and historically consequential community of its earliest serious readers. Its annotations are the trace of one of the first sustained critical engagements with Russell and Whitehead from outside the Anglophone logicist tradition, and the demonstrable connection between that engagement and Wiener’s 1913 dissertation — and through the dissertation to the Wiener pair, to Tarski’s algebra of relations, and to the deep structure of modern mathematical logic — gives the copy a position in the history of the discipline that no other surviving set can claim. Three volumes, large 8vo (260 × 170 mm), pp. xiii, [1], 666; xxxiv, [2], 772; x, [2], 491, [1]. Original publisher’s navy blue cloth, spines lettered in gilt with the Cambridge University Press arms stamped at the feet, teal head- and tail-bands. An unsophisticated set in original cloth, not rebacked, bindings sound throughout with light rubbing to the spine lettering and very minor fraying to the cloth at the extremities. Text-blocks clean, paper toned as usual with occasional light foxing, all hinges holding. Volume I with extensive marginal pencil and ink annotations through page 489, in English, engaging critically with the text at proposition level. Volume III with light traces of library handling but otherwise clean and sound.
Item #6628
Price: $145,000.00
