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TURING, A.M.

Systems of Logic based on Ordinals. [Received 31 May, 1938. - Read 16 June, 1938.]. [In: Proceedings of the London Mathematical Society. Second Series. Volume 45]. - [TURING'S PH.D.-THESIS]

Herman H. J. Lynge & Søn A/S
lyn54748
London, Hodgson & Son, 1939. Royal8vo. In a recent nice red full cloth binding with gilt lettering to spine. Entire volume 45 of "Proceedings of the London Mathematical Society. Second Series". Small white square paper label pasted on to lower part of spine, covering year of publication stating: "A Gift / From /Anna Wheeler". A very nice and clean copy without any institutional stamps. Pp. 161-240. [Entire volume: (4), 475 pp.].

The rare first printing of Turing's Ph.D.-thesis, which "opened new fields of investigation in mathematical logic". This seminal work constitutes the first systematic attempt to deal with the Gödelian incompleteness theorem as well as the introduction to the notion of relative computing. After having studied at King's College at Cambridge from 1931 to 1934 and having been elected a fellow here in 1935, Turing, in 1936 wrote a work that was to change the future of mathematics, namely his seminal "On Computable Numbers", in which he answered the famous "Entscheidungsproblem", came up with his "Universal Machine" and inaugurated mechanical and electronic methods in computing. This most famous theoretical paper in the history of computing caught the attention of Church, who was teaching at Princeton, and in fact he gave to the famous "Turing Machine" its name. It was during Church's work with Turing's paper that the "Church-Turing Thesis" was born. After this breakthrough work, Newman, under whom Turing had studied at Cambridge, urged him to spend a year studying with Church, and in September 1936 he went to Princeton. It is here at Princeton, under the guidance of Church, that Turing in 1938 finishes his thesis [the present paper] and later the same year is granted the Ph.D. on the basis of it. The thesis was published in "Proceedings of the London Mathematical Society" in 1939, and after the publication of it, Turing did no more on the topic, leaving the actual breakthroughs to other generations. In his extraordinary Ph.D.-thesis Turing provides an ingenious method of proof, in which a union of systems prove their own consistency, disproving, albeit shifting the problem to even more complicated matters, Gödel's incompleteness theorem. It would be many years before the ingenious arguments and striking partial completeness result that Turing obtained in the present paper would be thoroughly investigated and his line of research continued. The present thesis also presents other highly important proofs and hypotheses that came to influence several branches of mathematics. Most noteworthy of these is the idea that was later to change the face of the general theory of computation, namely the attempt to produce an arithmetical problem that is not number-theoretical (in his sense). Turing's result is his seminal "o-machines"; he here introduces the notion of relative computing and augments the "Turing Machines" with so-called oracles ("o"), which allowed for the study of problems that could not be solved by the Turing machine. Turing, however, made no further use of his seminal o-machine, but it is that which Emil Post used as the basis for his theory of "Degrees of Unsolvability", crediting Turing with the result that for any set of natural numbers there is another of higher degree of unsolvability. This transformed the notion of computability from an absolute notion into a relative one, which led to entirely new developments and in turn to vastly generalized forms of recursion theory. "In 1939 Turing published "Systems of Logic Based on Ordinals,"... This paper had a far-reaching influence; in 1942 E.L. Post drew upon it for one of his theories for classifying unsolvable problems, while in 1958 G. Kreisel suggested the use of ordinal logics in characterizing informal methods of proof. In the latter year S. Feferman also adapted Turing's ideas to use ordinal logics in predicative mathematics." (D.S.B. XIII:498). A part from these groundbreaking points, which Turing never returned to himself, he here also considers intuition versus technical ingenuity in mathematical reasoning, does so in an interesting and provocative manner and comes to present himself as one of the most important thinkers of modern mathematical as well as philosophical logic."Turing turned to the exploration of the uncomputable for his Princeton Ph.D. thesis (1938), which then appeared as "Systems of Logic based on Ordinals" (Turing 1939). It is generally the view, as expressed by Feferman (1988), that this work was a diversion from the main thrust of his work. But from another angle, as expressed in (Hodges 1997), one can see Turing's development as turning naturally from considering the mind when following a rule, to the action of the mind when not following a rule. In particular this 1938 work considered the mind when seeing the truth of one of Gödel's true but formally unprovable propositions, and hence going beyond rules based on the axioms of the system. As Turing expressed it (Turing 1939, p. 198), there are 'formulae, seen intuitively to be correct, but which the Gödel theorem shows are unprovable in the original system.' Turing's theory of 'ordinal logics' was an attempt to 'avoid as far as possible the effects of Gödel's theorem' by studying the effect of adding Gödel sentences as new axioms to create stronger and stronger logics. It did not reach a definitive conclusion.In his investigation, Turing introduced the idea of an 'oracle' capable of performing, as if by magic, an uncomputable operation. Turing's oracle cannot be considered as some 'black box' component of a new class of machines, to be put on a par with the primitive operations of reading single symbols, as has been suggested by (Copeland 1998). An oracle is infinitely more powerful than anything a modern computer can do, and nothing like an elementary component of a computer. Turing defined 'oracle-machines' as Turing machines with an additional configuration in which they 'call the oracle' so as to take an uncomputable step. But these oracle-machines are not purely mechanical. They are only partially mechanical, like Turing's choice-machines. Indeed the whole point of the oracle-machine is to explore the realm of what cannot be done by purely mechanical processes...Turing's oracle can be seen simply as a mathematical tool, useful for exploring the mathematics of the uncomputable. The idea of an oracle allows the formulation of questions of relative rather than absolute computability. Thus Turing opened new fields of investigation in mathematical logic. However, there is also a possible interpretation in terms of human cognitive capacity." (SEP).Following an oral examination in May, in which his performance was noted as "Excellent," Turing was granted his PhD in June 1938.
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(Stockholm, 1777). 8vo. As extracted from "Kungl. Svenska vetenskapsakademiens handlingar", uncut unopened. Fine and clean. Pp. 304-309.
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ALBERTI, MICHAEL.
Herman H. J. Lynge & Søn A/S
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Halle im Magdeburgischen, Hendel, 1721. 8vo. In contemporary full calf with four raise bands and richly gilt spine. Traces from old paper-label to upper part of spine. Leather on spine cracked, spine-ends slightly chipped. Internally nice and clean. (14), 620, (28) pp. First collected edition of Alberti’s essays. Alberti (1682–1757), professor of medicine and philosophy at Halle and later rector of the university, was a leading disciple of Georg Ernst Stahl who considered the soul as having control on the body. Therapies involved dealing with the internal senses and feelings.
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O Capital. (i.e. Portuguese:
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MARX, CARLOS [KARL] (+) GABRIELLE DEVILLE (+) [Translator:] ALBANO DE MORAES.
Herman H. J. Lynge & Søn A/S
lyn62679
Lisboa, De Francisco Luiz Goncalves, 1912. 8vo. In the original red printed cloth-binding with black and white lettering. Spine with loss of the white lettering. Paper-label pasted on to lower inner margin of front board. Very light wear to extremities, Internally very fine and clean. 240 pp. The exceedingly scarce first Portuguese edition of the most important abridged version of Marx's Capital ever to have appeared, published fifty-six years before the first full Portuguese (but published in Brazil) translation and whole sixy-two years before the first full translation published in Portugal. Curiously, two translations of the present work were made 1912 but the present translation seems to have priority (see Bastien, "Readings and Translations of Karl Marx in Portugal"). After the 1933 rise of Salazar's dictatorial Estado Novo regime, suppression of the relatively newly founded Communist party grew. Members were arrested, tortured, and executed and many were sent to the Tarrafal concentration camp in the Cape Verde Islands. Communist literature suffered an equally repressive fate, hence the rarity of the present work. Marxism and especially Marxist writing caught on comparatively late in Portugal: "As for the Socialist Party - supposed to be the main expression of Marxism -, it revealed itself unable to stimulate effective theoretical and doctrinal efforts. Its existence was an example of ambiguity and inconsequence. Its political programme went on mixing Marxian elements, associationist tradition and positivist thinking. Its strategy balanced continuously between an alliance with republican politicians and the maintenance of political autonomy. Its tatics balanced between electoral abstencionism and an involvement in election processes, that never led it to a relevant position in parliament. Even its international relations showed a lasting ambiguity: it had been created according to the instructions of the Marxist majority at the Hague Congress, when most of its members tended to support political abstencionism. When the formation of the Second lnternacional was taking place in Paris in 1889 Portuguese socialists tried to join the Marxist congress, after being present at the possibilist congress. In 1920 they decided to join the Third lnternacional (what was not accomplished), at the same time that an internal reformist turn was taking place." (Bastien, "Readings and Translations of Karl Marx in Portugal"). "The epitome, here translated, was published in Paris, in 1883, by Gabriel Deville, possibly the most brilliant writer among the French Marxians. It is the most successful attempt yet made to popularize Marx's scientific economics. It is by no means free from difficulties, for the subject is essentially a complex and difficult subject, but there are no difficulties that reasonable attention and patience will not enable the average reader to overcome. There is no attempt at originality. The very words in most cases are Marx's own words, and Capital is followed so closely that the first twenty-five chapters correspond in subject and treatment with the first twenty-five chapters of Capital. Chapter XXVI corresponds in the main with Chapter XXVI of Capital, but also contains portions of chapter XXX. The last three chapters-XXVII, XXVIII, and XXIX-correspond to the last three chapters-XXXI, XXXII, and XXXIII-of Capital." (ROBERT RIVES LA MONTE, Intruductory Note to the 1899 English translation). Capital de Marx also had a Portuguese edition at this time, or better, two different editions, both in 1912, but only in translation of the survey of Book I published in France by Gabriel Deville in 1883 (Marx, 1912a and Marx, 1912b). This version omitted material dealt with in at least four chapters of the original text and was not particularly appreciated by Engels. It was a simplified text, aimed at supporting the training of socialist militants and that made it possible for them to have access, indirect, to the work of Marx. The other summaries and anthologies of Capital, which, with a purpose similar to that of Deville, circulated in Europe during this period or ignored in Portugal, as was the case with Carlo Cafiero, or were only occasionally mentioned, as was the case with Paul Lafargue and Karl Kautsky, in its French versions. OCLC list two copies, both in the US.
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DARWIN, CHARLES.
Herman H. J. Lynge & Søn A/S
lyn62681
(Kharkiv), Derzhavne medychne vyd-vo (State Medical Publishing House), 1936. 8vo. In publisher's original grey cloth binding with black lettering to spine with Darwin's portrait embossed on front board. Wear to extremities, corner bumped and light spoling to back board. Inner hinges split and first 3 leaves partly detached. Last 20 ff. slighly creased due to dampstain, otherwise internally a nice and clean copy. 674 pp. + frontispiece, portrait of Darwin and 1 plate with genealogical tree. The exceedingly rare first Ukranian translation of Darwin's landmark 'Origin of Species'. OCLC only list two copies (Library of Congress and The Huntington Library, USA) Freeman F797.
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BOYE, JOHANNES.
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Fridericia, S. Elmenhoff, 1809. 8vo. In nice recent marbled paper covered boards with leather title-label with gilt lettering to spine. Ex-libris (Bent W. Dahlstrøm) to verso of front board. A nice and clean copy. 40 pp. Biblioteca Danica III, 572.
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VENTURA DA SILVA, JOAQUIM JOSE.
Herman H. J. Lynge & Søn A/S
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Lisboa, Officina de Simão Thaddeo Ferreira, 1803. Folio-oblong (365 x 255 mm). In contemporary half calf. Wear to extremies, upper part of spine with loss of leather. Ex-libris pasted on to pasted down front end-paper. With, primarily marginal, brownspotting throughout. Dampstain to inner margin and upper outer margin of last 10 ff. 32 ff. Rare first edition of the most celebrated Portuguese treatise on calligraphy. Joaquim José Ventura da Silva (1777–1849), regarded as one of Portugal’s finest calligraphers and teachers of writing, composed this methodological guide to handwriting in which he combines a historical survey of scripts used in Portugal with practical instruction for teaching and learning penmanship.Ventura da Silva is reffered to by Innocencio (Diccionario Bibliographico) as "one of the best Portuguese Calligraphers". A second edition was published in 1819, a third in 1841, and a facsimile was published in Porto in 1899.
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