Structure (mathematical logic)

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	Foundations of Mathematics. Mathematical Logic. By K.Podnieks
		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).
		I define mathematical theories as stable self-contained systems of reasoning, and formal theories - as mathematical models of such systems.
	www.ltn.lv /~podnieks (856 words)

	Tucson Unified School District
		Mathematical Structure/Logic - M6-P2 - PO 1 - Determine if the converse of a given statement is true or false #1
		Mathematical Structure/Logic - M6-P5 - PO 1 - Determine whether a given algebraic expression and a possible form are equivalent #1
		Mathematical Structure/Logic - M6-P3 - PO 1 - Construct a counterexample to show that a given invalid conjecture is false #1
	instech.tusd.k12.az.us /focus/twelve/gr12math6.htm (410 words)

	Kurt Gödel Research Center for Mathematical Logic
		Koepke, An Elementary Approach to the Fine Structure of L, Bulletin of Symbolic Logic 3 (1997), pp.
		(former Institute for Logic — vormals Institut für Formale Logik)
		July 2005: The research center received an excellent rating (text in German) in the final report of the 2005 evaluation of Austrian mathematics.
	www.logic.univie.ac.at (296 words)

	Logic and Foundations: 3D View of the Web
		Mathematical Logic and Foundations - Mathematical Logic and Foundations From The Mathematical Atlas, a resource of mathematics maintained by David Rusin.
		CEO Logic provides an organized structure for thinking about business, assisting ambitious businesspeople at all levels to develop power -- the power to get out of tough situations, the power to develop their careers, the power to make an organization perform, and the power to achieve true success.
		Mathematical Logic and Foundations - Mathematical Logic and Foundations Section of Math Guide.
	www.resolve3d.com /Science/Math/LogicandFoundations (1631 words)

	Mathematics - Wikipedia, the free encyclopedia
		Mathematical logic, which divides into recursion theory, model theory and proof theory, is now closely linked to computer science.
		Mathematics is often defined as the study of topics such as quantity, structure, space, and change.
		Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.
	en.wikipedia.org /wiki/Mathematics (2921 words)

	20th WCP: Semantic Realism: Why Mathematicians Mean What They Say
		We say that category theory is the language of mathematical theories and their relations because it allows us to talk about their structure in terms of "objects" and "functors", wherein such terms are, again, taken as syntactic assemblages waiting for an interpretation of the appropriate sort to give them formulas meaning.
		We say that category theory is the language of mathematical concepts and relations because it allows us to talk about their structure in terms of "objects" and "arrows", wherein such terms are taken as syntactic assemblages waiting for an interpretation of the appropriate sort to give them formulas meaning.
		In this paper I argue that if we distinguish between ontological realism (the claim that mathematical objects exist independently of their linguistic expression) and semantic realism (the claim that mathematical statements which talk about mathematical objects are meaningful), then we no longer have to choose between platonism and formalism.
	www.bu.edu /wcp/Papers/Math/MathLand.htm (3322 words)

	Foundations of Mathematics. Mathematical Logic. By K.Podnieks
		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).
		Bernays considers mathematical platonism as a method that can be - "taking certain precautions" - applied in mathematics.
	www.ltn.lv /~podnieks (1167 words)

	Difference to Inference Article
		I mean structure in the sense of logical structure, mathematical structure, or grammatical structure.
		I assume that this is because switching repeatedly forces users to learn higher-order logic that are common to the structures of all programs.
		It's not so much the specific menus and keystrokes that have to be relearned, rather it is a deeper level of how we organize our thinking in a way that fits with subtle underlying logic in the structure of the operating system which must be learned.
	www.psych.utah.edu /malloy/J-ITM_Difference_to_Inference00-09-26/JITM-Diff_to_Inf.htm (4821 words)

	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.
		Formal logic is the study of logical inference whose validity derives from its explicitly formal structure.
	en.wikipedia.org /wiki/Formal_logic (3434 words)

	Referativni Zhurnal Classification
		Groupoids, etc. 271.17.17.33.17 Special classes of groupoids 271.17.17.33.21 Groupoids with complemented structures 271.17.17.33.31 Quasigroups 271.17.17.33.31.17 Isotopies and homotopies of quasigroups 271.17.17.33.31.21 Identities and generalized identities on quasigroups 271.17.17.33.31.31 Loops 271.17.19 Rings and modules 271.17.19.15 Methods of mathematical logic in rings and modules 271.17.19.19 Associative rings and algebras 271.17.19.19.15 Structure of rings 271.17.19.19.15.17 Ideals in rings.
		Mathematical education 271.01.79.17 Popularization of the mathematical sciences 271.03 Foundations of mathematics, mathematical logic 271.03.15 Foundations of mathematics 271.03.15.15 General philosophical problems 271.03.15.17 Set theory 271.03.15.17.17 Naive set theory 271.03.15.17.19 Axiomatic set theory.
	www.ams.org /mathweb/Classif/RZhClassification.html (3434 words)

	Game Logic - Jumping
		And there you have it for mathematical, you will need to change the code slightly to include collisions and also include different directions (as the code above is just for jumping right, you'd need to modify it to include left and straight up), but that's the basic structure.
		The mathematical way of looking at jumping is applied into games such as the Dizzy saga for the spectrum and commodore C64 and Flimbo's Quest for the same computers.
		I will be looking at two viewpoints for jumping, the mechanical and the mathematical, the only difference between them is that the mechanical requires a little less mathematical knowledge, and is more free roaming.
	chompster0.tripod.com /gljumping.htm (357 words)

	Foundations of Mathematics. Mathematical Logic. By K.Podnieks
		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.).
		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).
		As a mathematical discipline travels far from its empirical source, or still more, if it is a second and third generation only indirectly inspired from ideas coming from 'reality', it is beset with very grave dangers.
	www.ltn.lv /~podnieks (357 words)

	Structuralism, Category Theory and Philosophy of Mathematics
		Category theory is the language best suited for this type of representation because it avoids the incommensurability problems which result from the Tarskian semantics essential to mathematical logic and model- theory for which satisfaction relations and truth definitions can only be defined for a specific language and the structure used to explicate the semantics.
		Structuralism holds that the "subject matter of mathematics consists of patterns or structures and not collections of mathematical objects." A mathematical structure can, perhaps, be similarly construed as the form of a possible system of related objects, ignoring the features of the objects that are not relevant to the interrelations.
		It is a significant theory that has much to offer for both the philosophy and foundations of mathematics, for category theory's primary concern is with the explication of mathematical structure.
	www.mmsysgrp.com /strctcat.htm (7237 words)

	Inductive Logic Programming
		The development of Inductive Logic Programming has been heavily formal (mathematical) in nature, because the major people in the field believe that this is the only way to progress and to show progress.
		ILP systems were used to determine rules for the mesh resolution of edges in the structure in terms of certain properties of the structure being modelled.
		As a final notation, it is important to remember that a logic program can contain just one Horn clause, and that the Horn clause could have no body, in which case the head of the clause is a known fact about the domain.
	www.doc.ic.ac.uk /~sgc/teaching/v231/lecture14.html (4223 words)

	20th WCP: Mathematical Models of Spacetime in Contemporary Physics and Essential Issues of the Ontology of Spacetime
		The Euclidean space and Minkowski's space are the intermediate structures between fundamental mathematical structure — complex space of spinors — and physical spacetime.
		The continuity and logic of evolution of physics of spacetime reconstructed by Heller is apparent, owing to the use of the language of contemporary mathematics.
		Logical development of the physics of spacetime in Heller's view is exposed through the use of the theory of fibre bundle and the theory of differential manifolds.
	www.bu.edu /wcp/Papers/Math/MathGos.htm (4223 words)

	Future Positive : Front Page
		Mathematical discoveries of the last few decades, culminating lately in the works of Whitehead, Russell, Keyser and Einstein, have made us conscious of the power of rigorous thought and have also disclosed the inner structure and working of this subtle instrument called human thought.
		Mathematics and logic have been proved to be one; a fact from which it seems to follow that mathematics may successfully deal with non-quantitative problems in a much broader sense than was suspected to be possible.
		Professor Keyser in his "Mathematical Philosophy" has done me the honor to devote a chapter to the new concept of man. I am frank to say that it is the best analysis of the concept in existence.
	futurepositive.synearth.net /2003/11/21 (6041 words)

	Learn more about Mathematics in the online encyclopedia.
		Philosophy of mathematics -- Mathematical intuitionism -- Mathematical constructivism -- Foundations of mathematics -- Set theory -- Symbolic logic -- Model theory -- Category theory -- Theorem-proving -- Logic -- Reverse Mathematics -- Table of mathematical symbols
		In order to clarify and investigate the foundations of mathematics, the fields of set theory, mathematical logic and model theory were developed.
		Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of 'figures and numbers'.
	www.onlineencyclopedia.org /m/ma/mathematics.html (1268 words)

	Mathematics
		Philosophy of mathematics -- Mathematical intuitionism -- Mathematical constructivism -- Foundations of mathematics -- Set theory -- Symbolic logic -- Model theory -- Category theory -- Theorem-proving -- Logic -- Reverse Mathematics -- Table of mathematical symbols
		In order to clarify and investigate the foundations of mathematics, the fields of set theory, mathematical logic and model theory were developed.
		Although mathematics itself is not usually considered a natural science, the specific structures that are investigated by mathematicians often have their origin in the natural sciences, most commonly in physics.
	www.1-free-software.com /en/wikipedia/m/ma/mathematics.html (1268 words)

	Modal logic - Wikipedia, the free encyclopedia
		Logical possibility is a form of alethic possibility; (4) makes a claim about whether it is possible for a mathematical truth to have been false, but (3) only makes a claim about whether it is possible that the mathematical claim turns out false, for all Jones knows, and so again Jones does not contradict himself.
		Significantly, modal logics can be developed to accommodate most of these idioms; it is the fact of their common logical structure (the use of "intensional" or non-truth-functional sentential operators) that make them all varieties of the same thing.
		Logics for handling a number of other ideas, such as eventually, formerly, can, could, might, may, must are by extension also called modal logics, since it turns out that these can be treated in similar ways.
	en.wikipedia.org /wiki/Modal_logic (3184 words)

	Logical argument - Wikipedia, the free encyclopedia
		In these cases, logic refers to the structure of the argument rather than to principles of pure logic that might be used in it.
		In ordinary language, people refer to the logic of an argument or use terminology that suggests that an argument is based on inference rules of formal logic.
		This is the case for arguments used in mathematical proofs.
	en.wikipedia.org /wiki/Logical_argument (3184 words)

	Modal logic - Wikipedia, the free encyclopedia
		Logical possibility is a form of alethic possibility; (4) makes a claim about whether it is possible for a mathematical truth to have been false, but (3) only makes a claim about whether it is possible that the mathematical claim turns out false, for all Jones knows, and so again Jones does not contradict himself.
		Significantly, modal logics can be developed to accommodate most of these idioms; it is the fact of their common logical structure (the use of "intensional" or non-truth-functional sentential operators) that make them all varieties of the same thing.
		A modal logic, or (less commonly) intensional logic, is a logic that deals with sentences that are qualified by modalities such as can, could, might, may, must, possibly, necessarily, eventually, etc. Modal logics are characterized by semantic intensionality: the truth value of a complex formula cannot be determined by the truth values of its subformulae.
	en.wikipedia.org /wiki/Modal_logic (1909 words)

	Mathematics - The Jiggies Reference Guide
		In the formalist view, it is the investigation of axiomatically defined abstract structures using logic and mathematical notation ; other views are described in Philosophy of mathematics.
		In order to clarify and investigate the foundations of mathematics, the fields of set theory, mathematical logic and model theory were developed.
		Mathematics is commonly defined as the study of patterns of structure, change, and space ; more informally, one might say it is the study of 'figures and numbers'.
	www.jiggies.com /reference/Mathematics (1909 words)

	Modal logic - Open Encyclopedia
		Logical possibility is a form of alethic possibility; (4) makes a claim about whether it is possible for a mathematical truth to have been false, but (3) only makes a claim about whether it is possible that the mathematical claim turns out false, for all Jones knows, and so again Jones does not contract himself.
		Significantly, modal logics can be developed to accommodate most of these idioms; it is the fact of their common logical structure (the use of "intensional" or non-truth-functional sentential operators) that make them all varieties of the same thing.
		Modal logic, or (less commonly) intensional logic is the branch of logic that deals with sentences that are qualified by modalities such as can, could, might, may, must, possibly, and necessarily, and others.
	open-encyclopedia.com /Modal_logic (1918 words)

	Uses and Misuses of Logic.
		Logical deduction, including mathematical logic, is the language with which we frame our theories of physics.
		Logic and mathematics are the cement that holds the scientific structure together, ensures its self-consistency, and helps us prevent errors of false inference.
		Because of this, some non-scientists think that mathematics and logic are used to "prove" scientific propositions, to deduce new laws and theories, and to establish laws and theories with mathematical certainty.
	www.lhup.edu /~dsimanek/logic.htm (5094 words)

	EMail Msg <9408241814.AA10711@ranch.poly.edu>
		Re sorted or typed logic: That is an issue that has been discussed many times in the past couple of years.
		Conceptual graphs have always been a typed logic, but we have agreed to adopt the KIF semantics, which would define the types in terms of monadic predicates.
		Therefore, the desire to support conventional mathematical notation should be considered as one of the formatting requirements -- we must be able to support a good external syntax for mathematical applications, but the syntax is a secondary issue.
	www-ksl.stanford.edu /email-archives/srkb.messages/307.html (5094 words)

	AG RVS - Explaining Failure with Tense Logic
		The extra twist with tense logic is that one must decide on a mathematical structure for time before one can determine which rules are valid even for just the mathematical semantics.
		An appropriate logical formalism could put accident reasoning on a firmer foundation: give it a form in which it is open to formal analysis and may be shown correct or incorrect, or complete or incomplete, by those with varying intuition about the subject matter.
		The use of a modal logic similar to deontic logic for expressing causality within a temporal logic framework was also considered in depth by von Wright (32).
	www.rvs.uni-bielefeld.de /publications/Reports/FailTemLog.html (5094 words)

	Home Page for Steven Lindell
		I study the logical definability of complexity classes given by models of computation whose resources are bounded by mathematical (or physical) limits.
		Computing monadic fixed-points in linear-time on doubly-linked data structures (Given June 24, 2005 at Seventh International workshop on Logic and Computational Complexity).
		I received my B.A. and M.A. in Mathematics at UCLA, and went on to receive a Ph.D. in theoretical Computer Science at UCLA under the supervision of Sheila Greibach and Yiannis Moschovakis (mathematical logic) in 1987.
	www.haverford.edu /cmsc/slindell (5094 words)

	Tickle: IQ and Personality Tests - Personalized IQ Report
		He could envision a future based on the patterns he saw in life, and used mathematical logic as a structure within which to present his philosophical arguments.
		You scored in the 90th percentile on the mathematical intelligence scale.This means that you scored higher than 80% - 90% of people who took the test and that 10% - 20% scored higher than you did.
		With that base he was able to use logic to formulate his theories.
	www.binarygazebo.com /gonainie/paidresult.jsp.html (8210 words)

	data.html
		Mathematics is difficult for many human minds to grasp because of its hierarchical structure: one thing builds on another and depends on it.
		So many mathematical conferences got held in such good restaurants that many of the finest minds of a generation died of obesity and heart failure and the science of math was put back by years.
		Not by logic, for logic has no existence independent of mathematics: it is only one phase of this multiplied necessity that we call mathematics.
	math.furman.edu /~mwoodard/data.html (17536 words)

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In the News (Sun 27 Dec 09)