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Comprehensive account of constructive theory of first-order predicate calculus. Covers formal methods including algorithms and epi-theory, brief treatment of Markov’s approach to algorithms, elementary facts about lattices and similar algebraic systems, more. Philosophical and reflective as well as mathematical. Graduate-level course. 1963 ed. Exercises.
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Still an interesting read....
Those interested in mathematical logic will appreciate this book written by one of the main contributors to the field in the twentieth century. The technique of "currying" in higher order logic is named after the author, wherein unary functions can be used to emulate functions with many parameters. The book was first published in 1963, reprinted in 1977, and so is not a up-to-date treatment of mathematical logic, but it could still be used as an historical supplement to a course in this subject. The reader should be aware though the terminology employed by the author is very idiosyncratic and therefore it may not reflect what is currently used in the literature.

The first chapter of the book could be considered an introduction to the philosophy of logic and mathematics. The author though views "philosophical logic" as the study of the principles of valid reasoning, and this is to be distinguished from "mathematical logic", wherein mathematical systems are constructed to study (formally) the principles of valid reasoning. One can also according to the author view logic as a theory in itself, and many "models" of it can be studied, in much the same way as many different models of geometry can be considered. The author also discusses very succinctly the logical paradoxes, and the different schools of thought in mathematics, such as Platonism, intuitionism, and formalism. The author clearly advocates the formalist school of thought in this book.

In chapter 2, the author gets more into the details of formal reasoning, the field of semiotics is outlined, and the author first begins defining the grammar and symbols for the upcoming discussion. A theory is defined as a class of statements, and consistency and decidability of theories is defined. The idea of a deductive theory is also defined, and the author defines the notion of such a theory being complete. The notions of consistency, decidability, and completeness are the familiar ones now entrenched in current textbooks on mathematical logic. A formal system, according to the author, is a theory in which the parameters of the statements of the theory are introduced as unspecified objects, and the statements of the theory make assertions on the properties of the parameters and their relations. The author considers syntactical systems, wherein the formal objects are taken from some object language, and what he calls Ob systems, which are essentially the systems considered in modern mathematical logic.The author employs the familiar Godel numbering scheme to numerically represent formal objects. The notion of algorithm is brought in here as an effective procedure to manipulate the formal objects of a system.

The next chapter is basically an introduction to the analysis of what would now be called the metalanguage of a formal system. This analysis is done in terms of what the author calls epistatements and epitheorems. Examples of these epitheorems include the Godel incompleteness theorem and the Skolem-Lowenheim theorem. The author introduces and classifies variables, and defines free and bound variables. A brief introduction to the lambda calculus and combinatory logic is given.

Then in chapter 4, the author discusses logical systems which are relational but with no bound variables. These are called logical algebras by the author, and the reader will encounter the famous truth tables and lattices in this chapter. A discussion of the Heyting algebra is given in the notes to the chapter. The reader interested in the more exotic types of algebraic logic, such as quantum logic, could benefit greatly from the reading of this chapter.

The logic of propositional calculus in terms of algebraic logic is discussed in chapter 5. Called propositional algebras by the author, the author proves the deduction theorem for such systems in this chapter. Interestingly, the L systems introduced by Gentzen are also discussed in this chapter. Although there are much better overviews of Gentzen's work in the current literature, a reader may still profit from a perusing of this chapter. L-systems where negation is added is then the subject of the next chapter.

Quantification in formal systems is taken up in chapter 7, considered both in the usual predicate calculus and in L systems. Prenex normal forms, the Herbrand-Gentzen theorem, and the completeness theorem are discussed in fairly good detail, albeit with old-fashioned notation.

The last chapter covers the interesting concept of modal logic. First considered by Aristotle, the author discusses it in the context of L systems, with the presentation being the shortest in the book. ... Read more

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Outlines of an arbitrary philosophy of mathematics for which there are no arguments
Formalism is here openly adopted not because there are any good reasons for it but because it suits certain philosophical prejudices. Consider the rejection of intuitionism: "the resulting criterion of truth is vague at best, and depends on metaphysical assumptions from which mathematics, if it is to have the pre-philosophical character above mentioned, must be free" (p. 5). "It should be emphasized that the reason for this [rejection] is not that these assumptions are invalid. There is, in fact, much to be said for these assumptions." (pp. 6-7). But Curry doesn't like them so they are rejected simply for that reason and that reason alone. Instead, "The definition I advocate is briefly this: Mathematics is the science of formal systems" (p. 56). Intuitionism is faulted for not accounting for the "acceptability of classical analysis" which is an "empirical fact" (p. 61). But the obvious "empirical fact" that there has been much mathematics where there have been no formal systems is not mentioned. ... Read more

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Les achats comprennent une adhésion à l'essai gratuite au club de livres de l'éditeur, dans lequel vous pouvez choisir parmi plus d'un million d'ouvrages, sans frais. Le livre consiste d'articles Wikipedia sur : John Von Neumann, Alan Turing, Donald Knuth, Kurt Gödel, Claude Shannon, Haskell Curry, Seymour Papert, Richard Hamming, John Horton Conway, Alonzo Church, Doron Zeilberger, George Dantzig, Richard Karp, Joseph Sifakis, Marcel-Paul Schützenberger, Leslie Lamport, Giuseppe Longo, Robin Milner, Myron Tribus, Maurice Nivat, Charles Antony Richard Hoare, James Pustejovsky, Stephen Cook, Bernd Sturmfels, Juris Hartmanis, Emil Post, Dana S. Scott, Jean-Louis Krivine, Carl Adam Petri, Johan Håstad, Richard Stearns, David Gelernter, Pierre Lescanne, Corrado Böhm, Warren Weaver, Jean-Pierre Jouannaud, Jacques Mazoyer, Shmuel Winograd, Géraud Sénizergues, Rudolf Bayer. Non illustré. Mises à jour gratuites en ligne. Extrait : John von Neumann (né Neumann János, 1903-1957), mathématicien et physicien américain d'origine hongroise, a apporté d'importantes contributions tant en mécanique quantique, qu'en analyse fonctionnelle, en théorie des ensembles, en informatique, en sciences économiques ainsi que dans beaucoup d'autres domaines des mathématiques et de la physique. Il a de plus participé aux programmes militaires américains. Benjamin d'une fratrie de trois, il s'appelle tout d'abord Neumann János Lajos (les Hongrois placent les noms de famille en tête) à Budapest en Autriche-Hongrie. Il est le fils de Neumann Miksa (Max Neumann), un avocat-banquier, et de Kann Margit (Marguerite Kann). Il ne prête guère attention à ses origines juives, sinon pour son répertoire de blagues. János est un enfant prodige : à six ans, il converse avec son père en grec ancien et peut mentalement faire la division d'un nombre à huit chiffres. Une anecdote rapporte qu'à huit ans, il a déjà lu les quarante-quatre volumes de l'histoire universelle de la bi...http://booksllc.net/?l=fr ... Read more

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Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Chapters: Alonzo Church, Charles Sanders Peirce, Raymond Smullyan, Haskell Curry, George Boolos, Clarence Irving Lewis, John Corcoran, Ralph Johnson, Rohit Jivanlal Parikh, Bas Van Fraassen, Paul Benacerraf, Raphael M. Robinson, Hao Wang, Colin Mclarty, Jon Barwise, Richard Montague, Richard Jeffrey, Penelope Maddy, Gila Sher, Edward Nelson, Ronald Jensen, Alan Ross Anderson, Matthew Foreman, Tomek Bartoszyński, Hartley Rogers, Jr., John Etchemendy, Robert M. Solovay, J. Barkley Rosser, Leo Harrington, Nate Ackerman, James Earl Baumgartner, Jack Silver, W. Hugh Woodin, Harvey Friedman, Richard Shore, Yiannis N. Moschovakis, Solomon Feferman, Valentina Harizanov, Theodore Slaman, David Blitz, Peter B. Andrews, Alice Ambrose, Charles Parsons, Gerald Sacks, Irving Copi, Nuel Belnap, Frederic Brenton Fitch, Bob Meyer, Alexander S. Kechris, Donald A. Martin, John R. Steel, Isaac Malitz, John P. Burgess. Source: Wikipedia. Free updates online. Not illustrated. Excerpt: Charles Sanders Peirce (pronounced purse) (September 10, 1839 April 19, 1914) was an American philosopher, logician, mathematician, and scientist, born in Cambridge, Massachusetts. Peirce was educated as a chemist and employed as a scientist for 30 years. It is largely his contributions to logic, mathematics, philosophy, and semiotics (and his founding of pragmatism) that are appreciated today. In 1934, the philosopher Paul Weiss called Peirce "the most original and versatile of American philosophers and America's greatest logician". An innovator in many fields including philosophy of science, epistemology, metaphysics, mathematics, statistics, research methodology, and the design of experiments in astronomy, geophysics, and psychology Peirce considered himself a logician first and foremost. He made major contributions to logic, but...More: http://booksllc.net/?id=6117 ... Read more

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Chapters: Roger Penrose, Haskell Curry, Theodore Roethke, Sarvadaman Chowla, Ron Larson, Joseph Heller, Ivan Illich, Joe L. Kincheloe, Chris Carney, Gary N. Knoppers, Ernie Mccoy, Stanley Weintraub, Simon J. Bronner, Masatoshi Nei, William Tenn, Delano Meriwether, James Price Dillard, Henry Giroux, Ellen Dannin, Michael E. Mann, Bob Higgins, David Bressoud, Jorge Pullin, Calyampudi Radhakrishna Rao, George R. Milner, Ernest C. Pollard, James G. Neal, Bruce Weigl, Arthur M. Lesk, John Barth, Robin Becker, Richard Alley, Hamid Shirvani, Joseph Rothrock, George Tavard, Muzafer Sherif, France A. Córdova, Boris Weisfeiler, Paul Clark, Richard Herman, Philip Jenkins, Robert Mitinger, Robert N. Proctor, Stanley Rosen, Robert Titzer, Douglas M. Charles, Frederick A. de Armas, Erich Auerbach, Abhay Ashtekar, Rustum Roy, Aleksander Wolszczan, William S. Pierce, Frank Ritter, Akhlesh Lakhtakia, Hugh Llewellyn Keenleyside, Lee Giles, Steven L. Goldman, David Edwards, William H. Zimmerli, Frank C. Whitmore, David Shapiro, Michael Bérubé, Alberto Bressan, Nirmal Bose, Bruce Lipton, Rob Frieden, Charles Crodel, Michael Hecht, Nina Fedoroff, Samuel H. Smith, Elwood S. Moore, Jack Rabin, Neal Gabler, Michael Anesko, Augustin Banyaga, Paul Weiss, Thomas Christian Kavanagh, Velvet Brown, Donald B. Redford, Carl Mitcham, Webb Miller, Jackson J. Spielvogel, Adam Rome, Leon Sibul, Niel Brandt, Samuel Preston Bayard, Amit Schejter, Robert Trehy, John L. Lumley, George Andrews, Dean Mchenry, Susan Miller, Wilson Jeremiah Moses, Robert Plomin, C. S. Giscombe, Bohdan Kulakowski, John Curley, James Brasfield, Kevin Alexander Boon, Scott W. Williams, James Kasting, William C. Waterhouse, O. Richard Bundy, Maggie Anderson, Donald Byrne, Ping Li, Richard Felson, Kathleen L. Lodwick, Alan Walker, Josip Novakovich, Hendrik Tennekes, Christian M. M. Brady, Nora Newcombe, Neil Wallace, Robert T Oliver, Jainendra K. Jain, Jon Nese, Nathan Fine, Henry James ...More: http://booksllc.net/?id=22846547 ... Read more

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Chapters: Ludwig Wittgenstein, Bertrand Russell, George Lakoff, Gottlob Frege, Imre Lakatos, Haskell Curry, Hilary Putnam, Proclus, Luitzen Egbertus Jan Brouwer, Alain Badiou, Bertrand Russell's Views on Philosophy, Michael Dummett, Stephan Körner, John Lucas, Gerrit Mannoury, Paul Benacerraf, Stewart Shapiro, Crispin Wright, Jaakko Hintikka, Hao Wang, Colin Mclarty, James Franklin, Sofya Yanovskaya, Penelope Maddy, Horace Romano Harré, David Corfield, Michael Resnik, Donald A. Gillies, Jody Azzouni, Paul Ernest, Reuben Hersh, Hartry Field, Charles Parsons, Mark Steiner, Geoffrey Hellman, Stephen Yablo, James Robert Brown. Source: Wikipedia. Pages: 248. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: Ludwig Josef Johann Wittgenstein (26 April 1889 29 April 1951) was an Austrian-British philosopher who worked primarily in the areas of logic, philosophy of mathematics, philosophy of mind, and philosophy of language. Described by his mentor and colleague Bertrand Russell as "the most perfect example I have ever known of genius as traditionally conceived, passionate, profound, intense, and dominating," Wittgenstein is considered by many to be the greatest philosopher of the 20th century. Instrumental in inspiring two of the century's principal philosophical movements, logical positivism and ordinary language philosophy, he is considered one of the most important figures in analytic philosophy. According to an end of the century poll, professional philosophers rank both his Tractatus Logico-Philosophicus (1921) and Philosophical Investigations (1953) among the top five most important books in twentieth-century philosophy, the latter standing out as "the one crossover masterpiece in twentieth-century philosophy, appealing across diverse specializations and philosophical orientations." Wittgenstei...More: http://booksllc.net/?id=17741 ... Read more

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Kapitel: John Barth, Paul Davidson, Thomas Rogers, Haskell Brooks Curry, Boris Weisfeiler, Sarvadaman Chowla, Russell Mccormmach, Erwin Wilhelm Müller, Robert Charles Vaughan, Alan Walker, Stephen G. Simpson, George Andrews, Robert L. Burgess, William Alvin Howard, Jean-Luc Brylinski. Aus Wikipedia. Nicht dargestellt. Auszug: Haskell Brooks Curry (September 12, 1900 - September 1, 1982) was an American mathematician and logician. Curry is best known for his work in combinatory logic; while the initial concept of combinatory logic was based on a single paper by Moses Schönfinkel, much of the development was done by Curry. Curry is also known for Curry's paradox and the Curry-Howard correspondence. There are two important programming languages named after him, Haskell and Curry. Curry was born on September 12, 1900, in Millis, Massachusetts, to Samuel Silas Curry and Anna Baright Curry, who ran a school for elocution. He entered Harvard University in 1916 to study medicine but switched to mathematics before graduating in 1920. After two years of graduate work in electrical engineering at MIT, he returned to Harvard to study physics, earning a MA in 1924. Curry's interest in mathematical logic started during this period when he was introduced to the Principia Mathematica, the attempt by Alfred North Whitehead and Bertrand Russell to ground mathematics in symbolic logic. Remaining at Harvard, Curry pursued a Ph.D. in mathematics. While he was directed by George Birkhoff to work on differential equations, his interests continued to shift to logic. In 1927, while an instructor at Princeton University, he discovered the work of Moses Schönfinkel in combinatory logic. Schönfinkel's work had anticipated much of Curry's own research, and as a consequence, he moved to Göttingen where he could work with Heinrich Behmann and Paul Bernays, who were familiar with Schönfinkel's work. Curry was supervised by David Hilbert and worked closely w...http://booksllc.net/?l=de ... Read more

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Haskell Brooks Curry (September 12, 1900 – September 1, 1982) was an American mathematician and logician. Curry is best known for his work in combinatory logic; while the initial concept of combinatory logic was based on a single paper by Moses Schönfinkel, much of the development was done by Curry. Curry is also known for Curry's paradox and the Curry–Howard correspondence. There are two important programming languages named after him, Haskell and Curry. ... Read more