## Principia Mathematica.

Cambridge: at the University Press, 1910; 1912; 1913.

First edition of this monumental work, one of the great rarities of modern science and mathematics, “the greatest single contribution to logic that has appeared in the two thousand years since Aristotle” (DNB). “In this monumental study of logic and set theory, Russell and Whitehead took up the task … of proving the logical basis of all mathematics by deducing the whole body of mathematical doctrine from a small number of primitive ideas and principles of logical inference. To do so, Russell and Whitehead devised a complex but precise system of symbols that enabled them to sidestep the ambiguities of ordinary language, and to give an exposition of sentential logic that has hardly been improved upon since” (Norman). Probably named after Isaac Newton’s great work, *Principia Mathematica* “has had an influence, direct and indirect, of near Newtonian proportions upon the spheres of its chief influence: mathematical logic, set theory, the foundations of mathematics, linguistic analysis and analytical philosophy” (Grattan-Guinness (1975), p. 89). “It also served as a major impetus for research in the foundations of mathematics throughout the twentieth century. Along with Aristotle’s *Organon* and Gottlob Frege’s *Grundgesetze der Arithmetik*, it remains one of the most influential books on logic ever written” (*Stanford Encyclopedia of Philosophy*). “Whether they know it or not, all modern logicians are the heirs of Whitehead and Russell” (*Palgrave*, p. 20). Complete sets of the first edition are very rare. Vol. I was printed in an edition of 750 copies and, due to disappointing sales, the publishers reduced the print run of Vols. II and III to 500 copies each, so that only 500 complete sets in first edition are possible. John Slater, Emeritus Professor of Philosophy at the University of Toronto and editor of *The Collected Papers of Bertrand Russell**,* suggests that there are probably fewer than 50 sets surviving in private hands.

“Gottlob Frege (1848-1925) had attempted to demonstrate logicism about arithmetic (though not geometry) in the period from 1879, when his first book, *Begriffsschrift*, was published, to 1903, when the second volume of his *Grundgesetze der Arithmetik* appeared. However, in 1902, as that second volume was in press, Russell (1872-1970) had written to him informing him of the contradiction that he had discovered in Frege’s system” (*Palgrave*, p. viii). This was ‘Russell’s paradox’, that the set of all sets that are not members of themselves is a member of itself if and only if it is not a member of itself. “Frege had attempted to respond to the contradiction … in a hastily written appendix, but he soon realized that his response was inadequate and abandoned his logicist project. It was left to Russell to find a solution to the paradox and to reconstruct the logicist program accordingly. The final result was Russell’s ramified theory of types and *Principia Mathematica* itself, but this theory and the logicist reconstruction in which it is embedded took a decade to develop” (*ibid.*).

“*Principia Mathematica* had its origins in Russell’s discovery of the work of [Giuseppe] Peano (1858-1932) at the International Congress of Philosophy held in Paris in the summer of 1900, which Peano and his supporters attended in force. To that time Russell had been working for several years attempting to develop a satisfactory philosophy of mathematics. Despite some philosophical successes … a satisfactory outcome had always eluded him. At the conference, however, he very quickly realized that the Peano school had a set of techniques of which he could make use, and on his return from the conference he immediately set about applying them. As a result, he quickly rewrote *The Principles of Mathematics*, which he had started in 1899, finishing the new version by the end of the year. It was published, after some delay and substantial revisions of Part I, in 1903, billed as the first of two volumes. It was intended as a philosophical introduction to, and defence of, the logicist program that all mathematical concepts could be defined in terms of logic and that all mathematical theorems could be derived from purely logical axioms. It was to be followed by a second volume, done in Peano’s notation, in which the logicist program would actually be carried out by providing the requisite definitions and proofs. At about the same time that Russell was finishing *The Principles of Mathematics*, he began the collaboration with his former teacher, Whitehead (1861-1947), that produced, many years later, *Principia Mathematica*.

“Whitehead in 1898 had published *A Treatise on Universal Algebra*, another first volume, in which a variety of symbolic systems were interpreted on a general, abstract conception of space. Again much detailed formal work was held over for the second volume. By September 1902 the two second volumes had merged, both authors having decided to unite in producing a joint second volume to each of their projects. This in turn grew until it constituted the three volumes of *Principia Mathematica*. The reason for the long delay in completing [*Principia Mathematica*] … was the difficulty in dealing with a paradox that Russell had discovered around May 1901 in the set-theoretic basis of the logicist system. The natural initial supposition of that system was that a class would correspond to each propositional function of the system, intuitively the class of terms which satisfied that propositional function. This being the case, there would be a class corresponding to the propositional function ‘*x* is not a member of itself’, and this class would be a member of itself if and only if it was not a member of itself. The problem of restricting the underlying logic so that this result could not arise while leaving it strong enough to support the mathematical superstructure that Russell and Whitehead wished to build on it absorbed many years of intense labour” (*ibid*., pp. xvi-xvii).

“*Principia Mathematica* proved to be remarkably influential in at least three ways. First, it popularized modern mathematical logic to an extent undreamt of by its authors. By using a notation superior to that used by Frege, Whitehead and Russell managed to convey the remarkable expressive power of modern predicate logic in a way that previous writers had been unable to achieve. Second, by exhibiting so clearly the deductive power of the new logic, Whitehead and Russell were able to show how powerful the idea of a modern formal system could be, thus opening up new work in what soon was to be called metalogic. Third, *Principia Mathematica* re-affirmed clear and interesting connections between logicism and two of the main branches of traditional philosophy, namely metaphysics and epistemology, thereby initiating new and interesting work in both of these areas.

“As a result, not only did *Principia* introduce a wide range of philosophically rich notions (including propositional function, logical construction, and type theory), it also set the stage for the discovery of crucial metatheoretic results (including those of Kurt Gödel, Alonzo Church, Alan Turing and others). Just as importantly, it initiated a tradition of common technical work in fields as diverse as philosophy, mathematics, linguistics, economics and computer science” (*Stanford Encyclopedia of Philosophy*).

“Although they divided up the initial responsibility for the various parts of the project, they checked each other’s work and in the end produced a truly collaborative text. ‘Whitehead and I make alternate recensions of the various parts of our book,’ Russell explained to P. E. B. Jourdain in a letter of March 1906, ‘each correcting the last recension made by the other.’ There were relatively few periods of extensive personal contact between them; indeed, during the period from the autumn of 1906 to the autumn of 1909, when most of *Principia Mathematica* was actually written, Whitehead lived in Cambridge and Russell in Oxford” (Grattan-Guinness (1975), p. 90). The final manuscript, of more than 4000 folios, was delivered to the printers on 19 October, 1909 (only a few leaves of the original manuscript are known to have survived). Publication was delayed by financial problems. In the end, Cambridge University Press contributed £300 to the cost of publication, a further £200 was obtained from the Royal Society, and the authors contributed £50 each (the latter contributions were refunded by the Press some 40 years later - but without interest!).

The first volume of *Principia Mathematica* was published in December 1910 in an edition of 750 copies, priced 25 shillings; volumes II and III had a print run of only 500 copies, and were priced at 30 shillings and 21 shillings, respectively. A fourth volume, dealing with the applications to geometry, was planned and much of it written, by Whitehead alone, but never published and the manuscript was destroyed shortly after his death. Complimentary copies were sent by the authors to the library of Trinity College, Cambridge (of which they both were or had been Fellows), R. G. Hawtrey (who checked over some of the text while it had been in preparation), G. G. Berry (a clerk at the Bodleian Library with remarkable abilities in mathematical logic), and Jourdain (this was the copy in the library of Haskell F. Norman). The Press sent copies to G. Peano, G. Frege, L. Couturat, J. Royce, W. E. Johnson (who had examined the manuscript for the Press), E. W. Hobson and A. R. Forsyth. We do not know the location of any of these copies, other than Jourdain’s.

The book was not a best seller. In his review of volume I in the *Times Literary Supplement*, Hardy wrote: “Perhaps twenty or thirty people in England may be expected to read this book,” but a few sentences he later qualified this estimate, writing “we should probably be over-sanguine if we suppose that there are half a dozen who will.” Erwin Schrödinger went further: he said that he didn’t believe that even Whitehead and Russell themselves had read all of it! Nevertheless, by the early 1920s the work was going out of print and a second edition, with three new appendices and a long new introduction, all written by Russell himself, was published in 1925-27.

After 1927 Russell withdrew from logic. “After the massive achievement of *Principia Mathematica*, nothing remained but a job of cleaning up the formal foundations, a task that was essentially completed by the early 1930s, as a result of the work of Ramsey, Gödel and Tarski” (Palgrave, p. 14).

Norman 1868; Blackwell & Ruja A9.1a; Church, *Bibliography of Symbolic Logic*, 194.1-1 (one of a handful of works marked by Church as being “of special interest or importance”); Martin 101.01-03; Kneebone, *Mathematical Logic* (1963), p. 161*ff*; *Landmark Writings in Western Mathematics 1640-1940*, Chapter 61; *The Palgrave Centenary Companion to Principia Mathematica*, N. Griffin & B. Linsky (eds.), Palgrave Macmillan, 2013; I. Grattan-Guinness, ‘The Royal Society’s financial support of the publication of Whitehead and Russell’s *Principia Mathematica*,’ *Notes and Records of the Royal Society of London*, Vol. 30 (1975), pp. 89-104.

Three vols., large 8vo (262 x 178 mm), pp. xiii, [3], 666, [2]; [ii], xxxiv, 772; [ii], x, 491, [1] (Liverpool Library ink library stamps and bookplates with ink withdrawn stamps to vols. II and III). Original dark blue cloth, spines lettered in gilt, cream endpapers (vols. II and III with Liverpool Library blind stamps to covers and lettering to foot of spines, spines with minor restoration). A very good set.

Item #5377

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Price:
$95,000.00
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