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# Topic: Foundations of mathematics

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 Foundations of mathematics - Wikipedia, the free encyclopedia Foundations of mathematics is a term sometimes used for certain fields of mathematics itself, namely for mathematical logic, axiomatic set theory, proof theory, model theory, and recursion theory. The truth of a mathematical statement, in this view, is then nothing but the claim that the statement can be derived from the axioms of set theory using the rules of formal logic. In mathematical realism, sometimes called Platonism, the existence of a world of mathematical objects independent of humans is postulated; the truths about these objects are discovered by humans. en.wikipedia.org /wiki/Foundations_of_mathematics   (497 words)

 Foundations of mathematics   (Site not responding. Last check: 2007-10-21) 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. Foundations of Mathematics This is a site for studying foundations of mathematics, or it can be used as a reference material. Practical Foundations of Mathematics An account of the foundations of mathematics (algebra) and theoretical computer science, from a modern constructive viewpoint by Paul Taylor. www.serebella.com /encyclopedia/article-Foundations_of_mathematics.html   (845 words)

 foundations of mathematics --  Encyclopædia Britannica Because mathematics has served as a model for rational inquiry in the West and is used extensively in the sciences, foundational studies have far-reaching consequences for the reliability and extensibility… Because mathematics has served as a model for rational inquiry in the West and is used extensively in the sciences, foundational studies have far-reaching consequences for the reliability and extensibility of rational... The entire scope of mathematics was enriched by the discovery of controversial areas of study such as non-Euclidean geometries and transfinite set theory. www.britannica.com /eb/article-9109826   (646 words)

 Foundations of Mathematics   (Site not responding. Last check: 2007-10-21) The struggle to development the mathematical instruments to describe quantum behaviour, and to resolve the epistemological problems which were generated by this work, raged throughout the inter-war period, and by the early years after World War Two, a settled interpretation of quantum physics was available which avoided subjectivist misconceptions. Mathematics provided a means to consistently describe quantum interactions, but any attempt to render the equations of quantum physics into the natural language referring to the objects of ordinary sensuous representation leads to contradictions. A quantum property, which is representable mathematically by a matrix, provides a substratum which allows of reification - that is, it may be deemed to adhere to a quantum object without leading to contradictions and inconsistency. www.marxists.org /reference/subject/philosophy/help/maths.htm   (3348 words)

 Mathematics Controversy over the foundations of mathematics - Chaitlin The mathematics of the twentieth century --- it's hard to write a history of mathematics from the year ten-thousand looking back because we're right here --- but the mathematics of the twentieth century you could almost say is set-theoretical, ``structural'' would be a way to describe it. If you believe that mathematics gives absolute truth, then it seems to me that Hilbert has got to be right, that there ought to have been a way to formalize once and for all all of mathematics. This should be a shocking idea, irreducible mathematical information, because the whole normal idea of mathematics, the Hilbertian idea, the classical idea of mathematics, is that all of mathematical truth can be reduced to a small set of axioms that we can all agree on, that are ``self-evident'' hopefully. www.gosai.com /science/mathematics-controversy.html   (8536 words)

 Foundations of Mathematics By David Hilbert (1927)   (Site not responding. Last check: 2007-10-21) And in mathematics, in particular, we consider is the concrete signs themselves, whose shape, according to the conception we have adopted, is immediately, clear and recognisable. To prohibit existence statements and the principle of excluded middle is tantamount to relinquishing the science of mathematics altogether. I am most astonished by the fact that even in mathematical circles the power of suggestion of a single man, however full of temperament and inventiveness, is capable of having the most improbable and eccentric effects. www.marxists.org /reference/subject/philosophy/works/ge/hilbert.htm   (4222 words)

 Practical Foundations of Mathematics Foundations have acquired a bad name amongst mathematicians, because of the reductionist claim analogous to saying that the atomic chemistry of carbon, hydrogen, oxygen and nitrogen is enough to understand biology. The category of contexts and substitutions (a phrase chosen to echo the mathematical examples) is generated by an elementary sketch (Sections 4.2-4.3) whose equations are defined by the substitution lemma in Section 1.1. The mathematical idea is to form the closure of the given system of equations to form a parsing congruence, but we observe that closure under the operations and transitivity is redundant. www.cs.man.ac.uk /~pt/Practical_Foundations/html/summary.html   (7546 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   (886 words)

 Foundations of Mathematics   (Site not responding. Last check: 2007-10-21) Mathematics, due to its absolute certitude, was for Kant the epitome of science. Hence it is worthwhile for the student of Kant to take a moment to consider his argument with regard to mathematics, for it should then shed light on his entire system of thought. Such an animal Kant never mentions, and rightly not, for it is totally contrary to the recognition we have of mathematics as an unconditioned necessity, i.e., where no exception is ever possible and so no need ever to even think of having to find an explanation for a deviation. www.mindspring.com /~kantwesley/Kant/SevenPlusFive_.html   (1666 words)

 Foundations of Mathematics   (Site not responding. Last check: 2007-10-21) Foundations of Abstract Mathematics -- A set of notes written for this course by the St. Mary's mathematics faculty; available in class. The course begins with an informal discussion of the logical foundations of mathematics. Kristine Roinestad, an upper-division mathematics major, will be teaching assistant for this class, and she is also available for help. www.smcm.edu /Users/rkstark/Courses/F02/Course3.html   (398 words)

 Jan Mycielski Conference   (Site not responding. Last check: 2007-10-21) Conference in Honor of Jan Mycielski During the first three days of June, 2002, the Mathematics Department hosted a conference in honor of Professor Jan Mycielski, on the occasions of his 70-th birthday and his recent retirement. Jan himself spoke on a theoretical (logical) view of applied mathematics. The conference received support from the Mathematics Department and from the College of Arts and Sciences. euclid.colorado.edu /~jmconf/main.html   (262 words)

 F. P. Ramsey Mathematics is therefore essentially extensional, and may be called a calculus of extensions, since its propositions assert between relations. The principal mathematical methods which appear to require the Axiom of Reducibility are mathematical induction and Dedekindian section, the essential foundations of arithmetic and analysis respectively. Mathematics then becomes hopeless because we cannot be sure that there is any class defined by a predicative function whose number is two; for things may all fall into triads which agree in every respect, in which case there would be in our system no unit classes and no two-member classes. www.hist-analytic.org /Ramsey.htm   (16707 words)

 Philosophy and Foundations of Mathematics   (Site not responding. Last check: 2007-10-21) Firstly, mathematics is the vintage example of a domain of knowledge whose truths are not (or do not seem to be) rooted in experience. Foundational doctrines are strongly informed by technical results in mathematical logic and, at times, they can even be carved out and made precise, opening in that way the possibility of their mathematical development and/or refutation. It is this sweet combination of mathematical rigour and philosophical reflection that makes the foundations and philosophy of mathematics such a fascinating subject. www.ciul.ul.pt /~ferferr/fundamentos_english.html   (445 words)

 The Foundations of Mathematics and Mathematica   (Site not responding. Last check: 2007-10-21) And that was the justification that was used for the kinds of constructs that were developed in mathematics, the ones that should be considered, and the ones that should not be considered. But in the axiom setup in mathematics, one is usually interested in asking whether one can find any sequence of transformation rules that will get one from one particular expression to another. It reminds me of a quote from Gödel, when asked about mathematics, what the role of a mathematician was, his idea was that the role of the mathematician was to work out what theorems are interesting. www.stephenwolfram.com /publications/talks/IMS/imstalk.html   (11940 words)

 Foundations of Mathematics   (Site not responding. Last check: 2007-10-21) To the extent that these consider particular mathematical topics, they border on other areas of the Mathematics Subject Classification; to the extent that these consider the nature of proof and of mathematical reality, they border on philosophy! 03: Mathematical logic, or Symbolic Logic, lies at the heart of the discipline, but a good understanding of the rules of logic came only after their first use. The apparent paradoxes of set theory (particularly from use of the Axiom of Choice) lead to foundational issues in topology and measure theory. www.math.niu.edu /~rusin/known-math/index/tour_fou.html   (402 words)

 Chaitin, Conversations with a Mathematician   (Site not responding. Last check: 2007-10-21) But if you want to be able to study mathematics, the power of mathematics, using mathematical methods, you have to ``desiccate'' it to ``crystallize out'' the meaning and just be left with an artificial language with completely precise rules, in fact, with one that has a mechanical proof-checking algorithm. He was going to formalize all of mathematics, and we were all going to agree that these were in fact the rules of the game. But a better way to say it, is that mathematical truth is an infinite amount of information, but any particular set of axioms just has a finite amount of information, because there are only going to be a finite number of principles that you've agreed on as the rules of the game. www.umcs.maine.edu /~chaitin/lowell.html   (10655 words)

 Simpson: Foundations   (Site not responding. Last check: 2007-10-21) My paper Logic and Mathematics is a survey of logic and foundations of mathematics, for the general reader. My current research project is Reverse Mathematics, an attempt to answer the foundational question: ``Which set existence axioms are needed to prove specific mathematical theorems?'' I study this question in the context of subsystems of second order arithmetic. MATH 558, Foundations of Mathematics I. This is an introductory graduate course on foundations of mathematics. www.math.psu.edu /simpson/Foundations.html   (212 words)

 Metamath Home Page   (Site not responding. Last check: 2007-10-21) Someone new to logic and set theory, who is still developing the mathematical maturity needed to follow informal textbook proofs, may find some reassurance in Metamath's step-by-step breakdown. And anyone who appreciates the stark beauty of mathematics for its own sake might enjoy just casually browsing through the proofs for their aesthetic appeal. It is intended to appeal to professional mathematicians and requires a certain mathematical maturity to be able to follow its proofs. metamath.planetmirror.com   (2802 words)

 FOM   (Site not responding. Last check: 2007-10-21) FOM is a closed, moderated, e-mail list for discussing From September 1997 through August 2002 the moderator of FOM was In September 2002 FOM moved to a new site at New York University, www.math.psu.edu /simpson/fom   (41 words)

 Ludwig Wittgenstein [Internet Encyclopedia of Philosophy]   (Site not responding. Last check: 2007-10-21) His interest in engineering led to an interest in mathematics which in turn got him thinking about philosophical questions about the foundations of mathematics. He was interested in questions of truth and falsehood, sense and reference (a distinction he made famous) and in the relation between objects and concepts, propositions and thoughts. His great contribution to logic was to introduce various mathematical elements into formal logic, including quantification, functions, arguments (in the mathematical sense of something substituted for a variable in a function) and the value of a function. www.utm.edu /research/iep/w/wittgens.htm   (6909 words)

 Meta-Mathematics and the Foundations of Mathematics   (Site not responding. Last check: 2007-10-21) 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. Then, as is pointed out in Turing's original paper (1936), and as was emphasized by Post in his American Mathematical Society Bulletin paper (1944), the set X of all theorems, consequences of the axioms, can be systematically generated by running through all possible proofs in size order and mechanically checking which ones are valid. It goes against the current paradigm of what mathematics is and how mathematics should be done, it goes against the current paradigm of the nature of the mathematical enterprise. www.umcs.maine.edu /~chaitin/italy.html   (4657 words)

 Foundations of Mathematics One can use this page to study the foundations of mathematics by reading topics following the links in their order or jumping over certain chapters. The Theory of The Foundations of Mathematics - 1870 to 1940 - by Mark Scheffer The Foundations of Mathematics: A Contribution to The Philosophy of Geometry - by Paul Carus sakharov.net /foundation_rt.html   (2708 words)

 Amazon.co.uk: The Foundations of Mathematics: Books   (Site not responding. Last check: 2007-10-21) This is a book for readers in transition from `school mathematics' to the fully fledged type of thinking used by professional mathematicians. 'The Foundations of Mathematics' is essential reading for anyone starting a mathematics degree. This book would also be suitable for anyone studying 'A' level maths who needed further inspiration on topics with which they were sruggling, or were contemplating studying maths at university. www.amazon.co.uk /exec/obidos/ASIN/0198531656   (576 words)

 Foundations Of Mathematics   (Site not responding. Last check: 2007-10-21) This course enables students to develop understanding of mathematical concepts related to introductory algebra, proportional reasoning, and measurement and geometry through investigation, the effective use of technology, and hands-on activities. Students will consolidate their mathematical skills as they solve problems and communicate their thinking. Learning through hands-on activities and the use of concrete examples is an important aspect of this course. calendar.tetraplex.com /MFM1P1.html   (145 words)

 Set Theory: Foundations of Mathematics   (Site not responding. Last check: 2007-10-21) An important part of Cantor's set theory, which forms the foundations of mathematics, is the concept of transfinite ordinals. Generalized continuum hypothesis is derived from the first axiom, and the infinitesimal is visualized using the latter axiom. The mathematical universe discussed here gives models of possible structures our physical universe can have. www.ece.rutgers.edu /~knambiar/intuitive_set_theory.html   (390 words)

 FOUNDATIONS OF COMPUTATIONAL MATHEMATICS This invaluable book contains 19 papers selected from those submitted to a conference held in Hong Kong in July 2000 to celebrate the 70th birthday of Professor Steve Smale. It may be regarded as a continuation of the proceedings of SMALEFEST 1990 ("From Topology to Computation") held in Berkeley, USA, 10 years before, but with the focus on the area in which Smale worked more intensively during the '90's, namely the foundations of computational mathematics. Readership: Researchers and graduate students interested in the computational aspects of mathematics. www.worldscibooks.com /mathematics/4883.html   (182 words)

 Practical Foundations of Mathematics   (Site not responding. Last check: 2007-10-21) Published by Cambridge University Press, as number 59 in their series Cambridge Studies in Advanced Mathematics (which includes books by Peter Johnstone and by Jim Lambek and Phil Scott). This is experimental, and only includes the narrative and simpler mathematical formulae, not the diagrams. The HTML that you see was generated by TTH, but this could only be made to work by writing a substantial program in Perl to simplify my LaTeX source. www.cs.man.ac.uk /~pt/Practical_Foundations   (791 words)

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