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Elements of Mathematics takes readers on a fascinating tour that begins in elementary mathematics-but, as John Stillwell shows, this subject is not as elementary or straightforward as one might think. Not all topics that are part of today's elementary mathematics were always considered as such, and great mathematical advances and discoveries had to occur in order for certain subjects to become "elementary." Stillwell examines elementary mathematics from a distinctive twenty-first-century viewpoint and describes not only the beauty and scope of the discipline, but also its limits.From Gaussian integers to propositional logic, Stillwell delves into arithmetic, computation, algebra, geometry, calculus, combinatorics, probability, and logic. He discusses how each area ties into more advanced topics to build mathematics as a whole. Through a rich collection of basic principles, vivid examples, and interesting problems, Stillwell demonstrates that elementary mathematics becomes advanced with the intervention of infinity. Infinity has been observed throughout mathematical history, but the recent development of "reverse mathematics" confirms that infinity is essential for proving well-known theorems, and helps to determine the nature, contours, and borders of elementary mathematics.Elements of Mathematics gives readers, from high school students to professional mathematicians, the highlights of elementary mathematics and glimpses of the parts of math beyond its boundaries.

Mathematics --- Math --- Science --- Study and teaching (Higher) --- Abstract algebra. --- Addition. --- Algebra. --- Algebraic equation. --- Algebraic number. --- Algorithm. --- Arbitrarily large. --- Arithmetic. --- Axiom. --- Binomial coefficient. --- Bolzano–Weierstrass theorem. --- Calculation. --- Cantor's diagonal argument. --- Church–Turing thesis. --- Closure (mathematics). --- Coefficient. --- Combination. --- Combinatorics. --- Commutative property. --- Complex number. --- Computable number. --- Computation. --- Constructible number. --- Continuous function (set theory). --- Continuous function. --- Continuum hypothesis. --- Dedekind cut. --- Dirichlet's approximation theorem. --- Divisibility rule. --- Elementary function. --- Elementary mathematics. --- Equation. --- Euclidean division. --- Euclidean geometry. --- Exponentiation. --- Extended Euclidean algorithm. --- Factorization. --- Fibonacci number. --- Floor and ceiling functions. --- Fundamental theorem of algebra. --- Fundamental theorem. --- Gaussian integer. --- Geometric series. --- Geometry. --- Gödel's incompleteness theorems. --- Halting problem. --- Infimum and supremum. --- Integer factorization. --- Integer. --- Least-upper-bound property. --- Line segment. --- Linear algebra. --- Logic. --- Mathematical induction. --- Mathematician. --- Mathematics. --- Method of exhaustion. --- Modular arithmetic. --- Natural number. --- Non-Euclidean geometry. --- Number theory. --- Pascal's triangle. --- Peano axioms. --- Pigeonhole principle. --- Polynomial. --- Predicate logic. --- Prime factor. --- Prime number. --- Probability theory. --- Probability. --- Projective line. --- Pure mathematics. --- Pythagorean theorem. --- Ramsey theory. --- Ramsey's theorem. --- Rational number. --- Real number. --- Real projective line. --- Rectangle. --- Reverse mathematics. --- Robinson arithmetic. --- Scientific notation. --- Series (mathematics). --- Set theory. --- Sign (mathematics). --- Significant figures. --- Special case. --- Sperner's lemma. --- Subset. --- Successor function. --- Summation. --- Symbolic computation. --- Theorem. --- Time complexity. --- Turing machine. --- Variable (mathematics). --- Vector space. --- Word problem (mathematics). --- Word problem for groups. --- Zermelo–Fraenkel set theory.

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A facsimile edition of Alan Turing's influential Princeton thesisBetween inventing the concept of a universal computer in 1936 and breaking the German Enigma code during World War II, Alan Turing (1912–1954), the British founder of computer science and artificial intelligence, came to Princeton University to study mathematical logic. Some of the greatest logicians in the world—including Alonzo Church, Kurt Gödel, John von Neumann, and Stephen Kleene—were at Princeton in the 1930s, and they were working on ideas that would lay the groundwork for what would become known as computer science. This book presents a facsimile of the original typescript of Turing's fascinating and influential 1938 Princeton PhD thesis, one of the key documents in the history of mathematics and computer science. The book also features essays by Andrew Appel and Solomon Feferman that explain the still-unfolding significance of the ideas Turing developed at Princeton.A work of philosophy as well as mathematics, Turing's thesis envisions a practical goal—a logical system to formalize mathematical proofs so they can be checked mechanically. If every step of a theorem could be verified mechanically, the burden on intuition would be limited to the axioms. Turing's point, as Appel writes, is that "mathematical reasoning can be done, and should be done, in mechanizable formal logic." Turing's vision of "constructive systems of logic for practical use" has become reality: in the twenty-first century, automated "formal methods" are now routine.Presented here in its original form, this fascinating thesis is one of the key documents in the history of mathematics and computer science.

Logic, Symbolic and mathematical. --- Turing, Alan, --- Alan Perlis. --- Alan Turing. --- Algorithm. --- Alonzo Church. --- Applicable mathematics. --- Automated theorem proving. --- Axiomatic system. --- Boolean algebra. --- Boolean satisfiability problem. --- C++. --- Calculus of constructions. --- Cantor's diagonal argument. --- Central limit theorem. --- Church–Turing thesis. --- Computability theory. --- Computability. --- Computable function. --- Computable number. --- Computation. --- Computer architecture. --- Computer program. --- Computer science. --- Computer scientist. --- Computer. --- Computing Machinery and Intelligence. --- Computing. --- Coq. --- Cryptography. --- Decision problem. --- Donald Gillies. --- EDVAC. --- ENIAC. --- Enigma machine. --- Entscheidungsproblem. --- Formal system. --- Foundations of mathematics. --- Georges Gonthier. --- Gödel's incompleteness theorems. --- Haskell Curry. --- Howard Aiken. --- Instance (computer science). --- Iteration. --- J. Barkley Rosser. --- John Tukey. --- John von Neumann. --- Kenneth Appel. --- Kepler conjecture. --- Konrad Zuse. --- Lecture. --- Lisp (programming language). --- Logic for Computable Functions. --- Logic in computer science. --- Logic. --- Logical framework. --- Marvin Minsky. --- Mathematica. --- Mathematical analysis. --- Mathematical logic. --- Mathematical proof. --- Mathematician. --- Mathematics. --- Model of computation. --- Monotonic function. --- Natural number. --- Notation. --- Number theory. --- Numerical analysis. --- Oswald Veblen. --- Parameter (computer programming). --- Peano axioms. --- Peter Landin. --- Presburger arithmetic. --- Probability theory. --- Processing (programming language). --- Programming language. --- Proof assistant. --- Quantifier (logic). --- Recursion (computer science). --- Recursion. --- Result. --- Rice's theorem. --- Riemann zeta function. --- Satisfiability modulo theories. --- Scientific notation. --- Simultaneous equations. --- Skewes' number. --- Solomon Feferman. --- Solomon Lefschetz. --- Systems of Logic Based on Ordinals. --- The Unreasonable Effectiveness of Mathematics in the Natural Sciences. --- Theorem. --- Theory of computation. --- Theory. --- Topology. --- Traditional mathematics. --- Turing Award. --- Turing machine. --- Turing's proof. --- Variable (computer science). --- Variable (mathematics).