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# Topic: Metamathematics

###### In the News (Tue 25 Jun 19)

 Metamathematics - Wikipedia, the free encyclopedia In general, metamathematics or meta-mathematics is reflection about mathematics seen as an entity/object in human consciousness and culture. The working assumption of metamathematics is that mathematical content can be captured in a formal system, usually a first order theory or axiomatic set theory. Metamathematics is intimately connected to mathematical logic, so that the histories of the two fields largely overlap. en.wikipedia.org /wiki/Metamathematics   (298 words)

 Proof theory Proof theory, a form of metamathematics, studies the ways in which proofs are used in mathematics. In this strictly formal sense, proof theory is not necessarily a form of metamathematics, but can have immediate applications in artificial intelligence, where automated deduction plays an important role. Proof theory, model theory, axiomatic set theory, and recursion theory are the so-called "four theories" of the foundations of mathematics. www.ebroadcast.com.au /lookup/encyclopedia/pr/Proof_(logic).html   (322 words)

 [No title]   (Site not responding. Last check: 2007-10-21) This is not to say that metamathematics is useless; to the contrary, it can be used to formulate useful knowledge about mathematical systems themselves, including metatheorems about what is and what is not possible within a given mathematical system. However, the metamathematics itself presupposes the existence of appropriate universes of discourse in order to make such assertions, and as they are nonconstructive, cannot hold the same sort of validity that a proof in mathematics itself would hold. On "Absolute" structures: they are metamathematical in nature for the most part (especially if they are infinite), since they simultaneously attempt to encompass "all possible" viewpoints of an object, generally without giving a means of actually constructing the object in question. userpages.umbc.edu /~mmisam1/doc.philomath.txt   (1871 words)

 [No title]   (Site not responding. Last check: 2007-10-21) The original meaning of David Hilbert is closest to proof theory. The working assumption of metamathematics is that mathematical content can be captured in a formal system. On the other hand, quasi-empiricism in mathematics, the cognitive science of mathematics, and ethno-cultural studies of mathematics, which focus on scientific method, quasi-empirical methods or other empirical methods used to study mathematics and mathematical practice by which such ideas become accepted, are non-mathematical ways to study mathematics. wikiwhat.com /encyclopedia/m/me/metamathematics.html   (137 words)

 Ralph Dumain: "The Autodidact Project": "On the Dialectics of Metamathematics" (Excerpts) by Peter Vardy Metamathematics, the foundational science of mathematics, seeks to grasp the conditions under which a contradiction-free science of quantity is possible. Metamathematics seems to be confronted with the choice between an incomplete or contradictory self-foundation. The principle of circularity is, so to speak, a regulative principle for formal-mathematical objectivation also outside of type-hierarchies in which the prohibition of self-reference appears as such. www.autodidactproject.org /other/vardy2.html   (2450 words)

 Meta-epistemology   (Site not responding. Last check: 2007-10-21) Meta-epistemology is proposed as a mathematical theory in analogy to metamathematics. Metamathematics considers the mathematical properties of mathematical theories as objects. In particular model theory as a branch of metamathematics deals with the relation between theories in a language and interpretations of the non-logical symbols of the language. www-formal.stanford.edu /jmc/ailogic/node9.html   (574 words)

 [No title] Metamathematics advocates a clear distinction between logic-mathematical formalism (the so-called object language) and the metamathematical considerations. The formal theory is therefore opposed to a metalogic or metamathematics. Foot note 1_31 Metamathematical investigations, i.e., according to the Warsaw school, can then be carried out if the concept of statement and consequence is precise. www.sorites.org /Issue_16/padilla.htm   (3517 words)

 The Archimedeans - Greek Metamathematics This sounds, on a first hearing, as if it disposes of the contradiction, but the more one looks at it the less it seems to do so, and if one tries to rewrite Zeno's proof, one sees that the remarks have no bearing on the problem at all. Metamathematics has been developed almost entirely in the last 50 years or so, and has yielded many startling and important results. The really thrilling thing is to see how near to discovering metamathematics the Greeks were, and it is amusing to speculate what the trend of history would have been had they done so. www.archim.org.uk /eureka/27/metamathematics.html   (740 words)

 Chaitin, The Unknowable I'm going to tell you why the field of metamathematics was invented, and to summarize what it has achieved, and the light that it sheds—or doesn't—on the fundamental nature of the mathematical enterprise. Metamathematics was promoted, mostly by Hilbert, as a way of confirming the power of mathematics, as a way of perfecting the axiomatic method, as a way of eliminating all doubts. So in a sense, metamathematics was a fiasco, it only served to deepen the crisis that it was intended to resolve. www.umcs.maine.edu /~chaitin/unknowable/ch1.html   (9269 words)

 Metamathematics and Godel's Theorems The metamathematical part is the supposition that mathematics is a model of the sorts of things studied in formal logic. Or, you could (metamathematically) assert that the hypotheses of Gödel's theorems are true about mathematics, and then the rest of the proof would be a rigorous metamathematical proof. The reason to make it metamathematical is so that it can speak about mathematics, as opposed to the things that are studied by formal logic. www.physicsforums.com /showthread.php?p=947508   (1465 words)

 Metamathematics The theory of such things (known in the trade as metamathematics) would seem to be more like applied than pure mathematics. If a formal proof is written out, the written version is a physical thing: an assemblage, say, of ink-blobs spread out over the surface of one or more sheets of paper. Bottom line: Formal proofs, at least as regards the structural properties of interest to metamathematics, are objects in the ontology of the basic theories of mathematics. www.philosophy.unimelb.edu.au /staff/HazLu/Lu2.html   (897 words)

 The Crystallization of Concept Art in 1961 This is the simple basis which in spite of all philosophical controversies still unites the mathematicians all over the world into a family-like group which enjoys a perfect mutual understanding. I saw an analogy between the syntax which metamathematics arrived at, and the computational character, or derivational process character, of much new music. The syntactical conception of mathematics evolved by twentieth-century metamathematics is not necessarily a good analysis of the mathematical process in an "anthropological" sense. www.henryflynt.org /meta_tech/crystal.html   (3447 words)

 Oxford University Press Smullyan plays a significant role in the further development of mathematical logic and the elucidation of its relation to metamathematics. an interesting presentation of recursion theory from the point of view of its applications in metamathematics, indicating many interrelations between various notions and properties. The book deals mainly with those aspects of recursion theory that have applications to the metamathematics of incompleteness, undecidability, and related topics. www.oup.com /ca/isbn/0-19-508232-X   (304 words)

 A Short and Sweet Refutation of Gödel's Theorem Of course Gödel-numbers themselves belong to metamathematics, and not to mathematics, and may not validly be used in any mathematical formula. Any formula in which a Gödel-number appears must belong to metamathematics, not to mathematics; and thus if Gödel can actually prove that there is such a formula and that it is indeed undecidable, all he can possibly prove thereby is that it is his metamathematics that is incomplete... One can hardly argue that mathematics and metamathematics are essentially the same thing: for if they were, it should be possible to derive all of metamathematics from the axioms of mathematics alone (such as the Peano axioms, or the axioms of Zermelo and Fraenkel, later extended by John von Neumann). homepage.mac.com /ardeshir/S&SRefutationOfGoedel.html   (928 words)

 Metamathematics, Machines and Gödel's Proof - Cambridge University Press   (Site not responding. Last check: 2007-10-21) In practice, however, there are a number of sophisticated automated reasoning programs that are quite effective at checking mathematical proofs. Now in paperback, this book describes the use of a computer program to check the proofs of several celebrated theorems in metamathematics including Gödel's incompleteness theorem and the Church Rosser theorem. The mechanisation of metamathematics itself has important implications for automated reasoning since metatheorems can be applied by labour-saving devices to simplify proof construction. www.cambridge.org /catalogue/catalogue.asp?isbn=0521585333   (236 words)

 Watzlawick's Disciplinary Matric   (Site not responding. Last check: 2007-10-21) In analogy to metamathematics, this is called metacommunication. Compared with metamathematics, research in metacommunication is at two significant disadvantages. The second difficulty is closely related to the first: while metamathematics posses two languages (numbers and algebraic symbols to express mathematics, and natural language for the expressions of metamathematics), we are mainly restricted to natural language as a vehicle for both communications and metacommunications. www.gwu.edu /~asc/biographies/watzlawick/MATRIX/BMA/bma_15.html   (131 words)

 Meta-Mathematics and the Foundations of Mathematics The first part is retrospective, and presents a beautiful antique, Gödel's proof; the first modern incompleteness theorem, Turing's halting problem; and a piece of postmodern metamathematics, the halting probability Ω. The current point of departure for metamathematics is that you're doing mathematics using an artificial language and you pick a fixed set of axioms and rules of inference (deduction rules), and everything is done so precisely that there is a proof-checking algorithm. Classical metamathematics with its incompleteness theorems deals with a static view of mathematics, it considers a fixed formal axiomatic system. www.cs.umaine.edu /~chaitin/italy.html   (4657 words)

 Gödel's incompleteness theorems - Wikipedia, the free encyclopedia These arguments are philosophical in nature and are the subject of much debate; Lucas provides references to responses on his own website. Another notable work was done by Judson Webb in his 1968 paper "Metamathematics and the Philosophy of Mind." Webb claims that previous attempts have glossed over whether one truly can see that the Gödelian statement p pertaining to oneself, is true. Using a different formulation of Gödel's theorems, namely, that of Raymond Smullyan and Emil Post, Webb shows one can derive convincing arguments for oneself of both the truth and falsity of p. en.wikipedia.org /wiki/G%c3%b6del%27s_incompleteness_theorem   (4880 words)

 Publisher description for Library of Congress control number 94222366   (Site not responding. Last check: 2007-10-21) While Gödel's first incompleteness theorem showed that no computer program could automatically prove certain true theorems in mathematics, the advent of electronic computers and sophisticated software means in practice there are many quite effective systems for automated reasoning that can be used for checking mathematical proofs. This book describes the use of a computer program to check the proofs of several celebrated theorems in metamathematics including those of Gödel and Church-Rosser. The mechanization of metamathematics itself has important implications for automated reasoning, because metatheorems can be applied as labor-saving devices to simplify proof construction. www.loc.gov /catdir/description/cam032/94222366.html   (199 words)

 Infinite Ink: The Continuum Hypothesis by Nancy McGough In the metamathematics section I discuss CH in the absence of some of these assumptions. Metamathematics and CH The results of Gödel and Cohen about the consistency and independence of CH are metamathematical theorems. Questions that are not within the framework of standard mathematics, but are rather about the framework of mathematics, are part of metamathematics. ii.best.vwh.net /math/ch   (4563 words)

 Registration & Records - Course Catalog   (Site not responding. Last check: 2007-10-21) Advanced topics in logic and metamathematics: proof procedures, first-order theories, soundness and completeness theorems, recursive functions, the formalization of arithmetic, the Goedel Incompleteness Theorems. Emphasis on mathematical study of logic and mathematics. No one may receive credit for both LOG 435 and LOG 535. www2.acs.ncsu.edu /reg_records/crs_cat/LOG.html   (201 words)

 Amazon.ca: Recursion Theory for Metamathematics: Books   (Site not responding. Last check: 2007-10-21) This work is a sequel to the author's G"odel's Incompleteness Theorems, though it can be read independently by anyone familiar with G"odel's incompleteness theorem for Peano arithmetic. The book deals mainly with those aspects of recursion theory that have applications to the metamathematics of Be the first person to review this item. www.amazon.ca /exec/obidos/ASIN/019508232X   (299 words)

 Amazon.com: Introduction to metamathematics (The University series in higher mathematics): Books: Stephen Cole Kleene   (Site not responding. Last check: 2007-10-21) I own the rights to this title and would like to make it available again through Amazon. Kleene's book, having been written before this separation, is much more comprehensive than the modern textbooks. About the contents: it begins with a (very well) introduction explaining the meaning of Metamathematics. www.amazon.com /exec/obidos/tg/detail/-/B0007E0WRK?v=glance   (753 words)

 Metamathematics of First- Order Arithmetic - Price Comparison   (Site not responding. Last check: 2007-10-21) Metamathematics of First- Order Arithmetic - Price Comparison You are here: Books > Metamathematics of First- Order Arithmetic Prices and availability for this book was last updated: less than 1 day ago Get real-time prices books.compricer.com /3540506322   (53 words)

 Robinson (1986) Introduction to model theory and to the metamathematics of algebra Robinson (1986) Introduction to model theory and to the metamathematics of algebra Introduction to model theory and to the metamathematics of algebra To view the the latter's ratings, click on Chapters/Papers/Articles in the STATISTICS box, select a publication from the list that appears, and then click on either Quality or Interest in that publication's STATISTICS box. www.getcited.org /?PUB=102469218&showStat=Ratings   (105 words)

 [No title] The aim of this workshop is to bring together researchers working on the interrelations between proof theory of logical and mathematical systems on the one hand and computation and complexity on the other. Particular fields of interest are complexity of propositional and first-order logics, theories and proof systems; proof search procedures; proof theory of classical and non-classical logics; relation of proof theoretical systems to computation; generalization of proofs and Kreisel's Conjecture; metamathematics of first-order theories, in particular arithmetic and its fragments; bounded arithmetic. The workshop will consist of three and a half days of one-hour lectures. www.cs.cmu.edu /afs/cs.cmu.edu/project/ai-repository/ai/events/meetings/ptcm94   (406 words)

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