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Topic: Mathematical logic


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  Logic - Wikipedia, the free encyclopedia
Mathematical logic really refers to two distinct areas of research: the first is the application of the techniques of formal logic to mathematics and mathematical reasoning, and the second, in the other direction, the application of mathematical techniques to the representation and analysis of formal logic.
The boldest attempt to apply logic to mathematics was undoubtedly the logicism pioneered by philosopher-logicians such as Gottlob Frege and Bertrand Russell: the idea was that mathematical theories were logical tautologies, and the programme was to show this by means to a reduction of mathematics to logic.
Logic cut to the heart of computer science as it emerged as a discipline: Alan Turing's work on the Entscheidungsproblem followed from Kurt Gödel's work on the incompleteness theorems, and the notion of general purpose computer that came from this work was of fundamental importance to the designers of the computer machinery in the 1940s.
en.wikipedia.org /wiki/Logic   (3824 words)

  
 Mathematical logic - Wikipedia, the free encyclopedia
In essentials, it is still the logic of Aristotle, but from the point of view of notation it is written as a branch of abstract algebra.
Attempts to treat the operations of formal logic in a symbolic or algebraic way were made by some of the more philosophical mathematicians, such as Leibniz and Lambert; but their labors remained little known and isolated.
While the traditional development of logic (see list of topics in logic) put heavy emphasis on forms of arguments, the attitude of current mathematical logic might be summed up as the combinatorial study of content.
en.wikipedia.org /wiki/Mathematical_logic   (969 words)

  
 Facts about topic: (Mathematical logic)   (Site not responding. Last check: 2007-09-19)
Mathematical logic was the name given by Peano (additional info and facts about Peano) to what is also known as symbolic logic.
In essentials, it is still the logic of Aristotle (One of the greatest of the ancient Athenian philosophers; pupil of Plato; teacher of Alexander the Great (384-322 BC)), but from the point of view of notation it is written as a branch of abstract algebra (additional info and facts about abstract algebra).
While the traditional development of logic (see list of topics in logic (additional info and facts about list of topics in logic)) put heavy emphasis on forms of arguments, the attitude of current mathematical logic might be summed up as the combinatorial study of content.
www.absoluteastronomy.com /encyclopedia/m/ma/mathematical_logic.htm   (1316 words)

  
 Encyclopedia: Mathematical logic
Mathematical logic is a discipline within mathematics, studying formal systems in relation to the way they encode intuitive concepts of proof and computation as part of the foundations of mathematics.
The term foundations of mathematics is sometimes used for certain fields of mathematics itself, namely for mathematical logic, axiomatic set theory, proof theory, model theory, and recursion theory.
Earlier appellations were symbolic logic (as opposed to philosophical logic), and metamathematics, which is now restricted as a term to some aspects of proof theory.
www.nationmaster.com /encyclopedia/Mathematical-logic   (565 words)

  
 Theory of Knowledge by Bertrand Russell
In mathematical logic it is the conclusions which have the greatest degree of certainty: the nearer we get to the ultimate premises the more uncertainty and difficulty do we find.
Mathematics, therefore, is wholly composed of propositions which only contain variables and logical constants, that is to say, purely formal propositions-for the logical constants are those which constitute form.
Logic and mathematics force us, then, to admit a kind of realism in the scholastic sense, that is to say, to admit that there is a world of universals and of truths which do not bear directly on such and such a particular existence.
www.marxists.org /reference/subject/philosophy/works/en/russell.htm   (3645 words)

  
 Learn more about Mathematical logic in the online encyclopedia.   (Site not responding. Last check: 2007-09-19)
Mathematical logic is a discipline within mathematics, studying formal systems in relation to the way they encode intuitive concepts of proof and computation.
As a matter of history, it was developed to understand and present the work of Kurt Gödel on the foundations of mathematics.
As a result of studies in mathematical logic one can have a rational discussion of many of the issues in the foundations of mathematics, though it would be misleading to say that the controversies that were alive in the period 1900-1925 have all been settled.
www.onlineencyclopedia.org /m/ma/mathematical_logic.html   (543 words)

  
 Read about Mathematical logic at WorldVillage Encyclopedia. Research Mathematical logic and learn about Mathematical ...   (Site not responding. Last check: 2007-09-19)
philosophy of mathematics was greatly clarified by the 'new' logic.
list of topics in logic) put heavy emphasis on forms of arguments, the attitude of current mathematical logic might be summed up as the combinatorial study of content.
Computability logic (http://www.cis.upenn.edu/~giorgi/cl.html) A new direction in Mathematical Logic, turning it from a theory of truth into a theory of computability.
encyclopedia.worldvillage.com /s/b/Mathematical_logic   (870 words)

  
 03: Mathematical logic and foundations
Mathematical Logic is the study of the processes used in mathematical deduction.
In second-order logic, the quantifiers are allowed to apply to relations and functions -- to subsets as well as elements of a set.
Much of mathematical logic was developed in response to the questions surrounding the axiomatization of set theory.
www.math.niu.edu /~rusin/known-math/index/03-XX.html   (2050 words)

  
 Mathematical logic - Encyclopedia.WorldSearch   (Site not responding. Last check: 2007-09-19)
The main areas of mathematical logic include model theory, proof theory
relates to proof theory; intuitionistic logic and linear logic are significant here.
Computability logic A new direction in Mathematical Logic, turning it from a theory of truth into a theory of computability.
encyclopedia.worldsearch.com /mathematical_logic.htm   (709 words)

  
 Amazon.com: Mathematical Logic for Computer Science: Books: Mordechai Ben-Ari   (Site not responding. Last check: 2007-09-19)
Mathematical Logic for Computer Science is a mathematics textbook with theorems and proofs, but the choice of topics has been guided by the needs of computer science students.
Mathematics textbook specifically geared towards the topics most important to computer science, featuring theorems and proofs, as well as sound logic.
The study of logic was begun by the ancient Greeks whose educational system stressed competence in philosophy and rhetoric.
www.amazon.com /exec/obidos/tg/detail/-/1852333197?v=glance   (547 words)

  
 Classical Logic
Another view is that a formal language is a mathematical model of a natural language in roughly the same sense as, say, a collection of point masses is a model of a system of physical objects, and the Bohr construction is a model of an atom.
The purpose of mathematical models is to shed light on what they are models of, without claiming that the model is accurate in all respects or that the model should replace what it is a model of.
Logic books aimed at mathematicians are likely to contain function letters, probably due to the centrality of functions to mathematical discourse.
plato.stanford.edu /entries/logic-classical   (11934 words)

  
 MTH-3D23 : Mathematical Logic
It is a rigorous introduction to first-order logic.
Some degree of mathematical sophistication is called for and familiarity with (and a taste for) mathematical proofs, such as would be seen in a rigorous first-year analysis or algebra course, will be assumed.
This is what is needed to analyse `real’ mathematics and the extra ingredient is the use of quantifiers (`for all’ and `there exists’).
www.mth.uea.ac.uk /maths/syllabuses/0102/4D2301.html   (762 words)

  
 The Math Forum - Math Library - Logic/Foundations
From the Mathematical Logic Group, University of Bonn, and the Institute for Logic, University of Vienna.
Arché: Centre for the Philosophy of Logic, Language, Mathematics and Mind - Crispin Wright, Director; University of St. Andrews, U.K. Arché is a new research centre within the School of Philosophical and Anthropological Studies, St Andrews.
Aristotle and the Paradoxes of Logic - Gilbert Voeten
mathforum.org /library/topics/logic   (2198 words)

  
 Amazon.com: Mathematical Logic: Books: Joseph R. Shoenfield   (Site not responding. Last check: 2007-09-19)
Starting with the concept that mathematical logic is not a collection of vaguely related results, but a method of attacking some of the most interesting problems which face the mathematician, the author sets the tone for this classic introduction.
The basic concepts are presented in an unusually clear and accessible fashion, keeping in mind the original purpose of mathematical logic to build the foundations of this vast edifice of knowledge in a way that helps and intrigues the working mathematician as much as the philosophically minded student of logic.
Usually, of course, most work in mathematics doesn't require a deep knowledge of rigorous mathematical logic, but it's always a good thing to a serious mathematician to have some acquaintance with it, even if it's just to avoid boobytraps.
www.amazon.com /exec/obidos/tg/detail/-/1568811357?v=glance   (1113 words)

  
 Mathematical Logic Group
Logic seminar is on Fridays at 3:15pm in L3.
The Logic advanced class is on Thursdays at 11am in L3.
The Junior logic seminar is on Tuesdays at 12noon in SR1.
www.maths.ox.ac.uk /logic   (309 words)

  
 Amazon.co.uk: Introduction to Mathematical Logic: Books   (Site not responding. Last check: 2007-09-19)
This edition includes an extensive appendix on second-order logic, a section on set theory with urlements, and a section on the logic that results when we allow models with empty domains.
Logic and computability theory have also become indispensable tools in theoretical computer science, including artificial intelligence.
Introduction to Mathematical Logic covers these topics in a clear, reader-friendly style that will be valued by anyone working in computer science as well as lecturers and researchers in mathematics, philosophy, and related fields.
www.amazon.co.uk /exec/obidos/ASIN/0412808307   (419 words)

  
 Group in Logic and the Methodology of Science: Introduction
Ph.D. work in logic can also be carried out entirely within one of the departments of Mathematics, Philosophy, Electrical Engineering and Computer Sciences (see Graduate Study in Logic at UC Berkeley).
The program in Logic and the Methodology of Science is intended for students whose interests lie in more than one of these fields.
The program is administered by the Group in Logic and the Methodology of Science, an interdepartmental agency which cooperates closely with the Department of Mathematics, the Department of Philosophy, and the Department of Electrical Engineering and Computer Sciences.
logic.berkeley.edu   (416 words)

  
 Links for "Logical Systems"
The Modern Development of the Foundations of Mathematics in the Light of Philosophy.
Preprints in Mathematical Logic and Foundations of Mathematics.
Berkeley Group in Logic and the Methodology of Science.
www.earlham.edu /~peters/courses/logsys/lslinks.htm   (712 words)

  
 INI Programme LAA -
Theoretical Computer Science is broadly divided into disciplines dealing with logic, semantics and formal methods on the one hand, and algorithmics and computational complexity on the other.
Finite Model Theory: This draws on logic and combinatorial methods to study the expressive power of logical languages in the finite.
Proof Complexity: At the interface of logic and complexity theory, the study of proof complexity, both in terms of lengths of proofs and complexity of inference steps, provides powerful methods for complexity lower bounds.
www.newton.cam.ac.uk /programmes/LAA   (263 words)

  
 A K Peters, Ltd. - Mathematical Logic
This classic introduction to the main areas of mathematical logic provides the basis for a first graduate course in the subject.
It embodies the viewpoint that mathematical logic is not a collection of vaguely related results, but a coherent method of attacking some of the most interesting problems, which face the mathematician.
The author presents the basic concepts in an unusually clear and accessible fashion, concentrating on what he views as the central topics of mathematical logic: proof theory, model theory, recursion theory, axiomatic number theory, and set theory.
www.akpeters.com /product.asp?ProdCode=1357   (237 words)

  
 Mathematical Logic around the world   (Site not responding. Last check: 2007-09-19)
According to Google, this is the most authoritative source for mathematical logic on the web.
UC Irvine: Logic and Philosophy, Mathematics and Logic
Please note that all links should be relevant to mathematical logic rpt mathematical logic, it may not be the main focus, though.
www.uni-bonn.de /logic/world.html   (318 words)

  
 Research in Mathematical Logic - School of Mathematics   (Site not responding. Last check: 2007-09-19)
Turing was followed at Manchester by his former student Robin Gandy who made important contributions to the foundations of proof theory and constructive mathematics.
The logic group offers opportunities for PhD study in all the research areas mentioned below, as well as a one year taught MSc programme which currently enjoys funding from the EPSRC for about eight suitably qualified EU students.
The Logic Group is taking part of the Marie Curie - EST MATHLOGAPS consortium, providing funding for PhD studies (both towards the achievement of the degree and for short-term visits) to suitably qualified applicants primarily from the EU and associated States.
www.maths.man.ac.uk /Research/logic   (335 words)

  
 Mathematical Logic at Notre Dame   (Site not responding. Last check: 2007-09-19)
In addition to the basic graduate logic course and two undergraduate courses in mathematical logic, an advanced graduate topics class is offered almost every semester.
The logic group holds a logic picnic every year in the late summer at Warren dunes.
The research in mathematical logic at Notre Dame is mainly in two broad areas: computability theory and model theory.
www.nd.edu /~steve/logic   (656 words)

  
 Foundations of Mathematics. Mathematical Logic. By K.Podnieks
Mathematics is the part of science you could continue to do if you woke up tomorrow and discovered the universe was gone.
In fact, mathematics is a complicated system of interrelated theories each representing some significant mathematical structure (natural numbers, real numbers, sets, groups, fields, algebras, all kinds of spaces, graphs, categories, computability, all kinds of logic, etc.).
Maslov could have put it as follows: most of a mathematician's working time is spent along the first dimension (working in a fixed mathematical theory, on a fixed mathematical structure), but, sometimes, he/she needs also moving along the second dimension (changing his/her theories/structures or, inventing new ones).
www.ltn.lv /~podnieks   (856 words)

  
 Logic Resources   (Site not responding. Last check: 2007-09-19)
Nordic Journal of Philosophical Logic (NJPL) an international journal edited at the Department of Philosophy at the University of Oslo, Norway.
Chaitin is a prolific theoretical computer scientist who has done some significant work in computability, including an interesting new proof of the incompleteness theorem based on algorithmic information theory.
Mathematical Logic Group at the University of Bonn).
www-phil.tamu.edu /Philosophy/logic.html   (315 words)

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