Principia Mathematica.
Cambridge: at the University Press, 1910-12-13. First edition of all three volumes, and an association set: 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 closely to the framework of that dissertation. The work these margins interrogate crowned the logicist programme in the foundations of mathematics — in the words of the Oxford Dictionary of National Biography, “the greatest single contribution to symbolic logic”. The annotations appear to be the working notes of the scholar who set Wiener the agenda. That agenda was one of the earliest serious confrontations between the Peirce-Schröder algebraic tradition and the Frege-Russell logicism, conducted on precisely the ground that Tarski’s 1941 axiomatisation of the calculus of relations would later reclaim for the algebraic school. Volumes I and II each bear at the head of the title page the inscription “Mihailovič 7. IV. 1925”. The inscription marks the set’s transmission into Central European hands at precisely the moment when Schmidt was preparing his departure from Tufts for the chair of philosophy at Carleton — the pivot in which he would have been dispersing his working logic library. The signature on the front free endpaper of Volume I identifies a reader trained in the Marburg School, which met the logicist programme critically. Karl Schmidt 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, where he was publishing on the philosophy of logic by 1909. 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. The working supervision of Wiener’s doctorate was Schmidt’s. The Tufts Mathematics Department’s own account of Wiener’s education records that the Harvard thesis was obtained while working with Professor Karl Schmidt at Tufts, and the Mathematics Genealogy Project names Schmidt as the dissertation’s advisor; Josiah Royce, the senior Harvard philosopher co-listed in the record, had suffered a stroke in 1912, and the day-to-day direction of the project fell to Schmidt. Wiener had entered Tufts at the age of eleven and taken his degree in 1909, aged fourteen, in the years when Schmidt was on the Tufts faculty; for the doctoral project the prodigy returned to work under the German-trained logician across the Charles River from Harvard. The central piece of internal evidence for Schmidt’s authorship of the annotations is the 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) in mind throughout. 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. That a teenager unaided conceived an agenda requiring fluent mastery of both the German algebraic-logical tradition of the 1890s and Russell’s new logicist apparatus strains belief; 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 the economical explanation. Specific features of the marginalia tighten the case. An annotation at page 489 identifies proposition *73.88 as “the Schröder-Bernstein theorem” — the capstone of the Principia’s theory of similar classes, and the one landmark in the volume that carries Schröder’s own name; it is at this page, with this theorem, that the annotations stop — the natural terminus of a reading conducted against the algebraic tradition. The marginal note “cf. Problems of Phil.” at page 46 ties the reading to Russell’s The Problems of Philosophy, published in 1912, in the very window of the dissertation’s composition; Russell’s theory of description, the subject of that cross-reference, was among the points Wiener took up with Russell himself the following year. And the annotations critical of the Principia’s proof strategies — “bad! See…” with a reference to algebraic method — correspond to the dissertation’s central contention that Russell’s proofs are often needlessly complex beside Schröder’s, and to its conclusion that nothing sayable in the Principia’s theory of relatives is unsayable in the Peirce-Schröder calculus. In the Introduction’s informal exposition of the calculus of classes, the annotator polices the type discipline that the text itself leaves tacit, noting at page 29 that the x in a class-membership formula must be an individual, not a class, and again that “the x must be of suitable type”; across the spread at pages 34–35, on referents, relata, domains, and relative products — the heart of the theory of relations that the dissertation would set against Schröder’s — the notes turn symbolic, restating Russell’s arrow notations in the annotator’s own shorthand, with the sharp verdict on the symbolism entered in the margin in a third medium. At page 183, against *14’s doctrine that the word existence cannot be significantly applied to subjects immediately given, a long ink note sets the task of examining the relation between E!x, which the text declares meaningless, and the existential quantifier, which is not. At page 296 a five-line pencilled note probes *37.11, asking whether the proposition holds unconditionally or only when its symbols designate entities — precisely the existence-and-designation worry the descriptions theory was built to discharge. The underlinings in the Preface are equally pointed: they pick out the authors’ acknowledged debts to Frege, to Cantor, and to Peano — a genealogy in which Schröder figures only as a source of supplementary notation, precisely the limited acknowledgement that the Wiener dissertation would contest. The dissertation’s agenda was a combative one. Geraldine Brady, in From Peirce to Skolem (2000), has read Wiener’s thesis as marshalling evidence that Russell took his treatment of binary relations almost entirely from Schröder’s Algebra der Logik, “with a simple change of notation and without attribution” — and Grattan-Guinness has independently observed that Russell built his 1900 logic of relations largely ignoring the algebraic tradition of Peirce and Schröder. However the charge is finally weighed, so aggressive an agenda is a remarkable one for an eighteen-year-old to direct at Bertrand Russell, and it is most economically understood as the transmission of a framework supplied by an advisor with the standing, the expertise, and the philosophical conviction to mount it. 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” — records his own sense that the framework of the thesis had been inherited rather than independently conceived. The intellectual line runs from Natorp to Schmidt to Wiener, and this copy of Principia Mathematica sits at its middle link. 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. The work had been conceived as the formal second volume of Russell’s The Principles of Mathematics (1903), with the writing begun in 1900; as the authors record in the preface, the subject proved so much larger than they had supposed that the book became independent of its predecessor, and the collaboration consumed a decade. It was also, from the first, a famously unread monument. G. H. Hardy’s review of the first volume, published anonymously in the Times Literary Supplement of 7 September 1911, opens with the remark that “perhaps twenty or thirty people in England may be expected to read this book”; Russell himself reckoned in 1959 that he had known of only six people who had read its later parts — three of them Poles, the other three Texans. 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 by Whitehead 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. 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. Complete sets of the first edition are scarce. 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 work’s commercial prospects were felt even by its authors: the Press, facing a projected loss of £600 on the edition, absorbed £300 of it, the Royal Society granted £200, and Whitehead and Russell covered the remaining £100 between them — Russell’s rueful arithmetic in his Autobiography being that the two of them thus earned “minus £50 each by ten years’ work”. Provenance: Karl Schmidt (1874–1961), with his ownership signature on the front free endpaper of Volume I; a second owner, the bearer of the name “Mihailovič”, with the inscription “Mihailovič 7. IV. 1925” at the head of the titles of Volumes I and II; Volume III, married to the set at a later date, from the library of the University College of North Wales at Bangor, accessioned on 26 April 1920, shelved in the Science Library at QA91.W5 under accession number 32462, and formally withdrawn in modern times, its bookplate stamped cancelled. The Mihailovič inscriptions — written in Central European format (day, Roman-numeral month, year), with a diacritic on the final consonant — mark the set’s transmission into the second owner’s hands on 7 April 1925. The date falls 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, whose precise identification remains a matter for further research, belonged in all likelihood to the Czech, Slovak, or South Slavic academic milieu of interwar Central Europe, and the set’s eventual reappearance in the German book trade is consistent with a European second life after 1925. The consequences of the dissertation 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 of the Logic of Relations’ (Proceedings of the Cambridge Philosophical Society 17, pp. 387–390), in which he showed for the first time that ordered pairs can be defined in terms of elementary set theory. The Wiener pair, modified by Kuratowski in 1921 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 the Principia’s *30s essentially redundant: relations could be defined as classes of ordered pairs, 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, and it is canonised in van Heijenoort’s From Frege to Gödel (1967), the documentary anthology of modern logic’s formation. Nor did the question the dissertation posed ever close. When Tarski gave the Peirce-Schröder algebraic tradition its modern axiomatic foundation in ‘On the Calculus of Relations’ (1941), he opened by restoring the very lineage — De Morgan, Peirce, Schröder — whose displacement by the Principia Wiener’s thesis had been the first to interrogate; the algebra of relations that Tarski, Jónsson, and their successors carried across the twentieth century is the rehabilitation of the tradition this copy’s annotator read against Russell’s text. The thesis itself has remained an object of scholarly attention, from Grattan-Guinness’s study of it in the Annals of Science in 1975 to Brady’s monograph and Peckhaus’s survey of Schröder’s logic. References: Norman 1868 — Blackwell & Ruja, A Bibliography of Bertrand Russell, A9.1a — Church, Bibliography of Symbolic Logic, 194.1–3 (marked as of especial interest or importance) — Martin 101.01–03 — Grattan-Guinness, ‘A. N. Whitehead and Bertrand Russell, Principia mathematica’, in Landmark Writings in Western Mathematics 1640–1940 (Elsevier, 2005), ch. 61 — Grattan-Guinness, The Search for Mathematical Roots, 1870–1940 (Princeton, 2000) — Grattan-Guinness, ‘Wiener on the logics of Russell and Schröder’, Annals of Science 32 (1975), pp. 103–132 — [Hardy], ‘The New Symbolic Logic’, Times Literary Supplement, 7 September 1911 — Russell, My Philosophical Development (1959), p. 86 — Brady, From Peirce to Skolem: A Neglected Chapter in the History of Logic (Amsterdam, 2000) — Anellis, ‘How Peircean Was the Fregean Revolution in Logic?’, arXiv:1201.0353 — Peckhaus, ‘Schröder’s Logic’, in Gabbay & Woods (eds.), Handbook of the History of Logic, vol. 3 (Amsterdam, 2004), pp. 557–609 — Tarski, ‘On the Calculus of Relations’, Journal of Symbolic Logic 6 (1941), pp. 73–89 — Kneebone, Mathematical Logic and the Foundations of Mathematics (London, 1963), pp. 161ff. — Wiener, A Comparison Between the Treatment of the Algebra of Relatives by Schröder and that by Whitehead and Russell, Ph.D. diss., Harvard, 1913 (MIT Archives, MC 22) — Wiener, ‘A Simplification of the Logic of Relations’, Proc. Camb. Phil. Soc. 17 (1914), pp. 387–390; reprinted in van Heijenoort, From Frege to Gödel (Harvard, 1967), pp. 224–227 — Wiener, Ex-Prodigy (1953) and I Am a Mathematician (1956) — Schmidt, ‘Studies in the Structure of Systems’, Journal of Philosophy, Psychology and Scientific Methods 10, no. 3 (1913), pp. 64–75 — Schmidt, From Science to God: Prolegomena to a Future Theology (New York, 1944). 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, light fraying to the cloth at the spine ends (most pronounced at the head of Volume I), and shallow creasing to the cloth of the sides. Text-blocks clean, paper toned as usual with occasional light foxing, all hinges holding. Volume I with extensive marginal annotations through page 489 — in pencil, a greenish-teal ink, and a red-brown ink — in English, engaging critically with the text at proposition level, together with underlining in the Preface and the Introduction. Volume III with light traces of library handling but otherwise clean and sound.
Item #6628
Price: $145,000.00
