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Topic: Axiomatic method

Related Topics

Axiom

Euclidean geometry

Parallel Postulate

Euclid

Distance

275 BCE

	Axiomatic system - Biocrawler (Site not responding. Last check: 2007-11-04)
		In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems.
		An axiomatic system that is completely described is a special kind of formal system; usually though the effort towards complete formalisation brings diminishing returns in certainty, and a lack of readability for humans.
		A mathematical model for an axiomatic system is a well-defined set, which assigns meaning for the undefined terms presented in the system, in a manner that is correct with the relations defined in the system.
	www.biocrawler.com /encyclopedia/Axiomatic_method (661 words)

	20th WCP: The Heuristic Function of the Axiomatic Method
		The aim of the lecture is to argue for a new reading of the term "deductive method." The deductive method will be presented as an architectural scheme for the reconstruction of the processes of gaining reliable scientific knowledge.
		This analytic branch of the deductive method proceeds in a regressive manner from an instance to the general case.
		It furthermore models the relation between the analytical, critical and regressive method and the synthetical, dogmatic and constructive method in Kant, Jakob Friedrich Fries and Leonard Nelson.
	www.bu.edu /wcp/Papers/Scie/SciePeck.htm (1249 words)

	Springer Online Reference Works
		An axiomatic method that does not fix rigidly the applicable language and so does not fix the limits of a meaningful understanding of an object, but requires an axiomatic definition of all special concepts for the given object of study.
		The penetration of reasoning based on a meaningful understanding and common sense, and not on the axioms, into the axiomatic method stems from the non-fixed language in which the properties of axiomatically given systems of objects are stated and proved.
		Fixing the language leads to the notion of a formal axiomatic system (see Axiomatic method) and creates a material basis for the clarification and precise description of the admissible logical principles, and for the controlled usage of set-theoretical and other general concepts or of such concepts that are not special for the relevant domain.
	eom.springer.de /I/i051020.htm (758 words)

	[No title]
		In the first place, it has already been made clear that Hilbert's inter= est in the axiomatic method was closely connected with his awareness to the= constant changes that scientific theories undergo in the course of their h= istorical development.
		One of the aims of the axiomatic analysis of theories was for Hil= bert, the possibility of analyzing whether the adoption of new hypothesis i= nto existing theories would lead to contradiction with the existing body of= knowledge, a situation that in his view had been very frequent in the hist= ory of science.
		Hilbert's axiomatic analysis is part of an open-ended, flexibl= e and mainly empirically motivated process of knowledge-creation in mathema= tics, rather than the origin and justification of a rigidly conceived, and = a-priori determined course of evolution, that is realized by means of logic= al deduction alone.
	www.math.rutgers.edu /~zeilberg/mamarim/akherim/corry.txt (16241 words)

	More on Axiomatically
		A mathematical model for an axiomatic system is a well-defined set, which assigns meaning for the undefined terms presented in the system, in a manner that is correct with the relations defined in the system.
		An axiomatic system for which every model is isomorphic to another is called categorial, and the property of categoriality ensures the completeness of a system.
		The axiomatic method is often discussed as if it were a unitary approach, or uniform procedure.
	www.artilifes.com /axiomatically.htm (820 words)

	Feature Article - Building Better Vehicles via Axiomatic Design - 06/00
		Axiomatic design is not just a theory: it's another step in the design process that is finding a home in the automaker's relentless drive toward creating “good” designs.
		In the axiomatic design world, zigzagging between adjacent domains, that is between the “what” domain on the left and the “how” domain on the right, will lead to independent, uncoupled (or at least decoupled) design parameters— namely, “good” designs.
		Axiomatic design is not quite the Taguchi method, which is a specific application of Robust Design.
	www.autofieldguide.com /articles/060001.html (980 words)

	Lion Hunting
		Methods 1.1-1.9,2.1-2.4 and 3.1-3.3 were included in the original article; all articles reproduced without permission, but with due credit.
		The methods which we shall enumerate will easily be seen to be applicable, with obvious formal modifications, to other carnivores and to other portions of the globe.
		We remark that this method is obviously superior to the Good method, which only guarantees the capture of one lion, and which requires an application of the Weierkäfig Preparation Theorem.
	users.ox.ac.uk /~invar/lions.html (3352 words)

	The Axiomatic Matrix of Whitehead’s Process and Reality
		However, what is essential and more influential than the thesis itself is the method in which the authors went about their conviction.
		From his earlier work Universal Algebra throughout the Principia collaboration we find that Whitehead’s dominant interest was the very broad sense of mathematics as the study of pattern or relations in general; a definition easily applied to his metaphysical speculations where the patterns of relatedness express the character of anything and everything real.
		The impact of Principia Mathematica on Whitehead’s later thought centers on the issue that its method applied to a very general problem of deducing the whole of mathematics from a handful of elementary formal notions and axioms.
	www.religion-online.org /showarticle.asp?title=2576 (3772 words)

	Axiomatic system - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-11-04)
		Therefore discussion of axiomatic systems is normally only semi-formal.
		Therefore, there are at least three 'modes' of axiomatic method current in mathematics, and in the fields it influences.
		The first case is the classic deductive method.
	en.wikipedia.org /wiki/Axiomatic_method (790 words)

	Science and the Scientific Method
		Method, or methods, usually refers to procedures and prescriptions that are applied for finding solutions to new and unsolved problems.
		The scientific method encompasses both the mathematical and empirical sciences, and we briefly discuss why it is necessity that drives science and determines what we mean by the scientific method.
		The distinction between the intuitive mathematics of theoretical physics and the rigorous and axiomatic approach of pure mathematics can be used as a metaphor for the intuitive versus axiomatic approach in many of the subjects in the arts and social sciences.
	srikant.org /core/node2.html (3748 words)

	AxiomaticDesign.org
		Axiomatic Design is a system that applies to any kind of design activity, including planning, although it is more aptly applied to engineering design problems than to industrial design.
		Axiomatic design starts with the premise that good design is governed by axioms, or laws, like physics is governed by physical laws.
		He is a pioneer in developing effective teaching methods for applying axiomatic design and has developed a keen insight to the design theory and practice.
	www.axiomaticdesign.org /index.html (1045 words)

	Foundations of Mathematics (Site not responding. Last check: 2007-11-04)
		According to the axiomatic method, inductive logic is relegated to a purely informal use.
		In the formal axiomatic method, the role previously played by truth is now played by consistency, and consistency depends only on the syntactical form of the axioms with respect to their logical parts, not on their specific content (meaning) under a given interpretation.
		The historical origin of multiple interpretation and thus of the abstract axiomatic method was the appearance of non-Euclidean geometries in the nineteenth century.
	bahai-library.com /?file=hatcher_foundations_mathematics (14860 words)

	The Legacy of R. L. Moore - The Moore Method -- F. Burton Jones
		While one cannot say that the "Moore Method" of teaching mathematics has gained wide-spread acceptance in college and university circles, it has been and is being successfully used by enough people to attract attention -- even outside academia.
		Having selected the class he would tell them briefly his view of the axiomatic method: There were certain undefined terms (e.g., "point" and "region") which had meaning restricted (or controlled) by the axioms (e.g., a region is a point set).
		Of course, there is also a simple axiomatic approach to the topology of the line (which omits arithmetic) and Burgess has used it quite successfully.
	www.discovery.utexas.edu /rlm/reference/burton_jones.html (3266 words)

	21 Century Math Method - a Replacement of Axiomatic Method - math logic, algebra, math foundations (Site not responding. Last check: 2007-11-04)
		Axiomatic method (also often called 20st Century Math Method) was a new method of 20st century math.
		During my research in general topology I found that axiomatic method is not enough and developed this new method.
		Axiomatic loop™ (includes "axiomatic loop" as the proper name of a scheme, construct, or situation, excludes the cases when axiomatic is an adjective applied to the proper separate meaning of "loop" in other typical contexts of usage for "loop" such an axiomatic for topological loops).
	www.mathematics21.org /method-0.5.html (4573 words)

	Theorem 2.0
		The axiomatic method in mathematics and formal logic establishes axioms that consist of a set of symbols, a set of valid combinations of the symbols, and rules to transform valid combinations of symbols to other valid combinations of symbols.
		The rigorous theory of the axiomatic method took place as a development in the ongoing search for the foundations of mathematics that arose after the 19th century crisis.
		David Hilbert responded to the problem by shifting the focus to mathematical methodology and began the study of the axiomatic method itself, which came to be known as metamathematics.
	prizebudgetforboys.com /alfalfafalafel/theorem20.html (2052 words)

	Introduction to Extension-Definition Method (EDM) - replacement of axiomatic method - algebra, logics, math foundations (Site not responding. Last check: 2007-11-04)
		Axiomatic method was a new method of 20nd century math.
		We have a way back from EDM to axiomatic method that is having a (big) system of axioms which includes set theory.
		Before invention of EDM I developed AGT in axiomatic method which appeared to be inappropriate for this new theory as in axiomatic method it appears hard (or impossible) to write the most general cases of AGT axioms and the systems of axioms appear to be somehow
	www.mathematics21.org /old-method.html (2605 words)

	BOL \| Bücher: Intuition and the Axiomatic Method. The Western Ontario Series in Philosophy of Science, Band 70 von ...
		Several prominent mathematicians and physicists were convinced that the formal tools of modern logic, set theory and the axiomatic method are not sufficient for providing mathematics and physics with satisfactory foundations.
		In the present volume, various views of intuition and the axiomatic method are explored, beginning with Kant's own approach.
		Soft Axiomatisation: John von Neumann on Method and von Neumann's Method in the Physical Sciences.
	www.bol.de /shop/home/artikeldetails/intuition_and_the_axiomatic_method_the_western_ontario_series_in/e_carson/ISBN1-4020-4039-3/ID11409656.html (481 words)

	Computational Truth: Axioms and the Axiomatic Method
		That is, the reason that we have any standard axiomatizations of mathematics at all is so that mathematicians don't have to resolve all their disagreements about the philosophy of mathematics.
		It is the type of logic used which dictates methods not the particular choice of axioms.
		Of course in order for this to be true there would need to be certain axiomatizations which structuralists dislike and others that platonists dislike and so forth.
	computationaltruth.net /blog/2005/06/axioms_and_the_axiomatic_metho.html (3069 words)

	PhilSci Archive - How Metaphysical is "Deepening the Foundations"? - Hahn and Frank on Hilbert's Axiomatic ...
		At bottom of their neglect of Hilbert's axiomatic method stands their conviction that reconciling Ernst Mach's empiricist heritage with modern mathematics required to draw a rigid boundary between mathematics and physics and to subscribe to logicism, according to which mathematics consisted in tautologous logical transformations.
		In this way, they missed the substantial difference between the logical structure of a particular axiom system and the axiomatic method as a critical study of arbitrary axiom systems.
		To be sure, Logical Empiricists considered the goal of axiomatizing the sciences as an important task, but in the way how they set it up axiomatization became much closer tied to a success of the foundationalist program for all mathematics than Hilbert's axiomatic method ever was.
	philsci-archive.pitt.edu /archive/00000301 (282 words)

	Nineteenth Century Geometry (Stanford Encyclopedia of Philosophy)
		In fact, there is a logical gap already in Euclid I.1 (the solution of this problem rests on an unstated assumption of continuity) and it is not clear that Euclid regarded his postulates as self-evident (by calling them "requests" he suggested he did not).
		Riemann's basic scheme makes allowance for much greater generality than he actually reaches for; but, in his judgment, it should be enough for the time being to characterize the geometry of continuous manifolds in such a way that it agrees optimally with Euclidean geometry on a small neighborhood of each point.
		Riemann extends to n dimensions the methods employed by Gauss (1828) in his study of the intrinsic geometry of curved surfaces embedded in Euclidean space (called "intrinsic" because it describes the metric properties that the surfaces display by themselves, independently of the way they lie in space).
	plato.stanford.edu /entries/geometry-19th (4771 words)

	The Axiomatic Method (Site not responding. Last check: 2007-11-04)
		The precise formulation of first order logic reflects a modern view of the axiomatic method and also makes it easier to see how the method falls short of the needs of modern mathematics.
		The foundational approach involves a strict axiomatic treatment of the theory of sets, which establishes an ontological framework within which all other mathematical theories can be introduced by definitions rather than axioms.
		Some modern approaches to mathematics, notable the category theoretic approach, are not ideally supported by set theoretic foundations and suggest the need for further liberalisation of the cannons of demonstrative reasoning.
	www.rbjones.com /rbjpub/methods/fm/fm011.htm (421 words)

	Notes on the Symposium Report, The Role of Axiomatics and Problem Solving in Mathematics, published by the Conference ...
		In particular, Buck notes that in his time the axiomatic structure of mathematics was, in the schools, being elevated to a definition of what mathematical material ought to be taught.
		Dodes is addressing the question of axiomatics at the level of k-12, and structures his own paper carefully, with one section each for “structure”, “logic”, and “method”, intending to describe how much of each can or ought to be presented to students at varying stages of “mathematical maturity”.
		His paper for the conference on axiomatics (and problem solving) begins with the assumption that he is speaking about secondary students only, and at that he is limiting himself to those who will be able to understand the material he is recommending.
	www.math.rochester.edu /people/faculty/rarm/axiomatics.html (8542 words)

	Nolan Sandygren (Site not responding. Last check: 2007-11-04)
		There is no clear answer to either of these questions, but it is safe to say that the goal of the axiomatic method is to assume as little as possible and prove everything else.
		Euclid’s fifth postulate was superfluous when compared to the other four axioms, and had it been expressed in a plain and straightforward manner it probably would have been much more widely accepted.
		The axiomatic method has become the standard for all of “pure mathematics,” a type of reasoning that needs no physical experimentation to support its authenticity, because every step of the derivation can be traced back to the self-evident truths.
	community.middlebury.edu /~schar/Courses/fs023.F02/paper1/sandygren.htm (2034 words)

	6. Little Theories (Site not responding. Last check: 2007-11-04)
		The axiomatic method is commonly used both for encoding existing mathematics and for creating new mathematics.
		A chunk of mathematics is represented as an axiomatic theory consisting of a formal language plus a set of sentences in the language called axioms.
		The axiomatic method comes in two styles, both well established in modern mathematical practice.
	imps.mcmaster.ca /manual/node10.html (312 words)

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