 The Practice of Logical Frameworks (1996)  Frank Pfenning
A survey article discussing the history of work on logical frameworks and their current open problems. Deductive systems, given via axioms and rules of inference, are a common conceptual tool in mathematical logic and computer science. They are used to
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 Preuve Proof Prueba
A newsletter in French, Spanish, and English, the theme of which is teaching and learning mathematical proofs. Includes an extensive bibliography consisting of books, chapters in books, journal articles, and theses.
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 Principia Cybernetica Web  Turchin, Joslyn, Heylighen, Bollen, eds.
Principia Cybernetica Project (PCP) is an international organization that tries to tackle ageold philosophical questions with the help of the most recent cybernetic theories and technologies. The Project's aim is the computersupported collaborative
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 Principia Mathematica  A.D.Irvine; Stanford Encyclopedia of Philosophy
The philosophy encyclopedia's entry on Principia Mathematica, the landmark work on mathematical logic and the foundations of mathematics written by Alfred North Whitehead and Bertrand Russell. Next to Aristotle's Organon, it is the most influential book
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 Principia Mathematica II  Michael Meyling
An open source project attempting to present mathematical knowledge in a formal, correct form. It includes a proof verifier which checks a mathematical proof written in a certain formal language and an HTML converter for making mathematical HTML pages.
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 The Prisoner Problem  Anne Arundel Community College
A classic logic problem about three prisoners, each of whom tries to determine what color his own hat is in order to be set free.
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 A Problem Course in Mathematical Logic  Stefan Bilaniuk
A freeware mathematics text, available by anonymous ftp in LaTeX, PostScript, and PDF formats, for an introduction to mathematical logic for undergraduates with some mathematical sophistication. It supplies definitions, statements of results, and problems,
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 Problem Solving in Mathematics  Jim Wilson; Univ. of Georgia
Problems in algebra, geometry, conversion, cryptarithm, "mean," trigonometry, and a mixture of all, many with comments and solutions, from a course given in the Dept. of Mathematics Education at the University of Georgia. The site also provides links
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 Problem Solving Island  Samuel A. Rebelsky
Challenging openended word problems on the topics of games, such as Blockade, Finger Multiplication, and The Word Pyramid; puzzles such as Error Correction, Error Detection, The Fake Coin, Friday the 13th, and Leap Birthdays; and problems based on figures,
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 Problems with a Point  Education Development Center, Inc. (EDC)
A collection of problems designed to help students in grades 612 learn new mathematical ideas by building on old ones. Varying in difficulty and approaches, these problems are useful for teachers, students, parents, math clubs, homeschoolers, and others.
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 Progetto Rhoda  Emilio Brengio
"Numeri con qualità." Software (.exe) per la didattica della matematica nella scuola elementare: problemi; operazioni; geometria; logica; procedure e verifiche; poligoni, angoli, perimetri, e aree; e giochi.
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 Projet Folium
Folium est un ensemble organisé d'articles sur les mathématiques, du cours aux biographies de mathématiciens célèbres, en passant par les exos les plus classiques et les théorèmes les plus puissants. Articles
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 Proof  Jim Loy
Brief descriptions of different types of proof, both mathematical and not.
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 Proof  Mariotti, MariaAlessandra; and Pedemonte, Bettina, editors
Archives of the quarterly "international newsletter on the teaching and learning of mathematical proof." Articles have included "The effect of using Transparent PseudoProofs in Linear Algebra," "Beliefs and beyond: hows and whys in the teaching of proof,"
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 Proof, Mathematical (Encarta Encyclopedia 2000)  Microsoft Encarta Online
An argument that is used to show the truth of a mathematical assertion. In modern mathematics, a proof begins with one or more statements called premises and demonstrates, using the rules of logic, that if the premises are true then a particular conclusion
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 Proofs in Mathematics  Interactive Mathematics Miscellany and Puzzles, Alexander Bogomolny
A discussion of the value of proofs in the classroom, followed by a collection of proofs classified as either simple or charming. Simple proofs: Number of vowels in a Lewis Carroll game; Pigeonhole Principle; Number of X's and O's; Coloring points in
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 Proofs  James Garner
An essay on different approaches using mathematical proofs: inductive reasoning, reasoning by analogy, deductive reasoning, direct method, contrapositive approach, etc.
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 Proofs Without Words  Cut the Knot!, Alexander Bogomolny
Martin Gardner wrote, "There is no more effective aid in understanding certain algebraic identities than a good diagram. One should, of course, know how to manipulate algebraic symbols to obtain proofs, but in many cases a dull proof can be supplemented
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 Proof Techniques (Analysis Webnotes)  John Lindsay Orr; Dept. of Mathematics & Statistics, Univ. of NebraskaLincoln
Techniques of writing proofs for: Showing two sets are equal; Showing a set is closed; How to show a sequence converges; and How to use a convergent sequence (to prove other things).
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 Proof: The Sum of Prime Twins is Divisible by 12  Mathematics Dept., College of Marin
Prime twins are prime numbers that are two numbers apart. For example 41 and 43 are prime twins. Can you PROVE that the sum of prime twins is divisible by 12?
(This proof doesn't work for the prime twins 3 and 5.)
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 The ProvenMath Project  Michal Ryszard Wojcik and Michal Stanislaw Wojcik
Proofs of the Fundamental Theorem of Algebra, the theorem about integration by change of variables for realvalued functions of nvariables, a derivation of Zorn's Lemma, and others. The site renders axioms and derives proofs using only ASCII characters.
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 Puzzlers  Bryan Clair, University of Chicago
A collection of Clair's favorite puzzles (no discussion or answers are provided): The Nine Dots; Weighing Coins; The Dollar Hotel, Planning an Orchard; The Math Department; Tea and Cream; Bugs on a Square; River Crossing; More Crawling Bugs; The Unexpected
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 Puzzles  Martin Watson
Blogs, comments, musings, design sketches, and photographs of mechanical puzzles by an avid collector and craftsman. Puzzle news, the author's own burrs (Watson's wormhole, digigrams, Dutch cubist), indepth explorations of manipulative puzzles (Puzzles
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 The Puzzle TOAD  Tom Bohman, Oleg Pikhurko, Alan Frieze, and Danny Sleator, School of Computer Science, Carnegie Mellon University
Send the authors your solutions to their challenging graph theory, number theory, geometry, discrete mathematics, logic, and other puzzles. Revisit periodically to see the latest question, and download PDFs of solutions of past problems.
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 Puzzlies.net  Michał Barciś
Play sudoku and griddler logic puzzles online. Log in to save game play and compete in the rankings.
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 The Puzzling World of Barry R. Clarke  Barry R. Clarke
Brain teasers from Clarke's column in The Daily Telegraph (UK) and from his various puzzle books: liar problems, logic teasers, number conundrums, matchstick puzzles, lateral mindbenders, and more. With links to other puzzles sites and an article on
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 Query Interface to the Contents of the Mathematics Archives  University of Tennessee, Knoxville (UTK)
Search software, abstracts and reviews; teaching materials, etc. by keyword.
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 Quiz Hub  Schmidel & Wojcik: Educational Web Weavers
A collection of online interactive learning quizzes that help K12 students enhance their core knowledge of English, languages, math, science, and social studies. Subscription required.
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 Raise Smart Kids  Inna Shapiro
Download Powerpoint presentations of math problems, with answers: logic, basic arithmetic operations, natural numbers, pigeonhole principle, fractions, pouring, cutting, weighing, divisibility, and more. For Croatian translations of some of Shapiro's
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 Recent Papers in Category Theory  Prof. George F. Nassopoulos
Recent papers in Category theory and its applications to functional analysis, global analysis, and abstract differential geometry.
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 The rec.puzzles Archive
The index of the rec.puzzles archive. Puzzles are categorized by subject area and include solutions compiled from various sources and supposed to be definitive. Subjects include: analysis, arithmetic, combinatorics, competition, decision, geometry, group,
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 The rec.puzzles Archive  Massachusetts Institute of Technology (M.I.T.)
A list of puzzles, categorized by subject area, from the newsgroup rec.puzzles. Puzzles include solutions compiled from various sources, supposed to be definitive. Subjects include analysis, arithmetic, combinatorics, competition, decision, geometry,
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 rec.puzzles FAQs by Newsgroup  Dept. of Computer Science, Universiteit Utrecht
Information on the following topics: crosswords, analysis, arithmetic, combinatorics, competition, cryptology, decision, geometry, group, induction, Instructions, language, logic, physics, Pickover (Cliff), probability, reallife, references, series,
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 Reference Notes  Department of Mathematics, Cal. State U. San Bernardino (CSUSB)
Notes on set theory, symbolic logic, methods of proof, basic proofs, functions, relations, binary operations, and groups. Also, Math Tools for practicing modular arithmetic and truth tables, or for solving Diophantine equations. Search the Notes, or scan
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 Reflections on Deep Blue  Keith Devlin (Devlin's Angle)
We generally assume that the ability to play a good game of chess requires intelligence when it comes to people. Why not for chessplaying machines as well? Well, here is one argument that shows that things are not so simple. One aspect of human intelligence,
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 Resource Material  Spencer, Chan; Univ. of Toronto Mathematics Network
A collection of resource material for use in classes or extracurricular activities in high schools: answers, explanations, and expositions of some common but deep math questions. In the process of answering three basic questions, this area delves into
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 Revathi Narasimhan  Department of Mathematics, Kean University
College Math, College Algebra, and Calculus course resources and links include Narasimhan's own Interactive Calculus tutorial, which requires the LiveMath plugin. See also Using Excel in Finite Math and Applied Calculus, freely downloadable Excel workbooks
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 Robbins Algebras are Boolean  William McCune, Automated Deduction Group, Argonne National Laboratory
The Robbins problem  are all Robbins algebras Boolean?  has been solved: Every Robbins algebra is Boolean. This theorem was proved automatically by EQP, a theorem proving program developed at Argonne National Laboratory. Information on the solution;
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 The Roman Press, Inc.  Steven Roman
Home page of a California mathematics professor, including research articles, books and textbooks, and software.
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 Rubik's Cube Information  Michael Reid
All about Rubik's Cube  even a mailing list. Site contents include: the Nederlandse Kubus Club; Rubik's cube pages on the Web; Cube books; Sources of cubelike puzzles; the author's notes, including an optimal cube solver; other Cube solutions on the
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 Rubik's Cube Java Applet  Michael Schubart
To twist the cube, press the mouse button over a cubelet face that belongs to the slice you want to move. Keeping the button pressed, drag the mouse in the desired direction; then release the button. Moving the whole cube is done in the same way  just
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 Rubix Corner  Duane Cash
The site includes information regarding the Duane Cash Cube Collection now featured at Hessport's Rubikshop in their "collectibles" section. New puzzles include the Rubik's Triamese, Fusion, Fusion+, SpeedFusion, 5x3, and 4x3 Cubes, which give those interested
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 Science, Math, and Technology Tutorial (SMATT)  ThinkQuest 1996
Topics: Math History (biography of Descartes); Logic (how to construct logical arguments and put them into proofs); Math Tutorial (Basic Algebra with problems and solutions); Environmental Science (ozone in the atmosphere); Astronomy (implications of
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 Sciences exactes / Mathématiques  Encyclopédie Hachette Multimédia En Ligne
Recherche Mathématiques par thème: Mathématiciens et logiciens; Mathématiques générales; Logique; Algèbre; Topologie; Analyse; Géométrie; Cinématique; Statistiques; Probabilités;
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 Search Ask Dr. Math Archives  The Math Forum
Search by keyword K12 math questions answered by volunteer 'math doctors' in the U.S. and abroad.
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 Search for Math on the Internet  The Math Forum
Keyword search of all searchable archives, Newsgroup archives, Dr. Math archives, Problems of the Week, Internet Math Library, or any combination thereof.
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 SelfReference and Apparent SelfReference  Interactive Mathematics Miscellany and Puzzles, Alexander Bogomolny
Various paradoxes, real and imagined, including verbal, logical, and mathematical examples, and links to: Blinking page with errors; Apparent paradox; Set of all subsets; An Impossible Page; Russell's paradox; and An Impossible Machine.
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 Shady Puzzles  LambdaMu Pte Ltd
A logic and elimination (disjunctive syllogism) puzzle: the column and row headers identify the number and spacing of shaded squares within a gridplace shaded squares where they belong according to those guides. The site includes a history of the puzzles,
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 Sites with Problems Administered by Others  Math Forum
Problems of the week or month: a page of annotated links to weekly/monthly problem challenges and archives hosted at the Math Forum but administered by others, and to problems and archives elsewhere on the Web, colorcoded for the level(s) of the problems
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 Sketchpad Activities for Young Learners: Grades 35  Nathalie Sinclair
This collection of activities for grades 35 offers examples of the ways in which Sketchpad can be used across the range of mathematical subject areas in the elementary curriculum.
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