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Mathematical Weaknesses
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Mathematical Logic
This textbook introduces first-order logic and its role in the foundations of mathematics by examining fundamental questions.What is a mathematical proof? How can mathematical proofs be justified? Are there limitations to provability? To what extent can machines carry out mathematical proofs?In answering these questions, this textbook explores the capabilities and limitations of algorithms and proof methods in mathematics and computer science. The chapters are carefully organized, featuring complete proofs and numerous examples throughout.Beginning with motivating examples, the book goes on to present the syntax and semantics of first-order logic.After providing a sequent calculus for this logic, a Henkin-type proof of the completeness theorem is given.These introductory chapters prepare the reader for the advanced topics that follow, such as Gödel's Incompleteness Theorems, Trakhtenbrot's undecidability theorem, Lindström's theorems on the maximality of first-order logic, and results linking logic with automata theory.This new edition features many modernizations, as well as two additional important results: The decidability of Presburger arithmetic, and the decidability of the weak monadic theory of the successor function. Mathematical Logic is ideal for students beginning their studies in logic and the foundations of mathematics.Although the primary audience for this textbook will be graduate students or advanced undergraduates in mathematics or computer science, in fact the book has few formal prerequisites.It demands of the reader only mathematical maturity and experience with basic abstract structures, such as those encountered in discrete mathematics or algebra.
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Mathematical Adventures
School mathematics sometimes excite and sometimes scare, and sadly, sometimes bore.In fact, elementary school arithmetic seems to get us through most situations.So why all the rest? This book aspires to give a glimpse into how things started and evolved, and how math can help us today, from simple measurements to navigating using mathematically simplified maps.The accompanying activities (some easier and some more challenging ones) aim to engage you with what was happening at a given time.Answers and explanations are in the end of the book.
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Mathematical Cavalcade
Mathematical Cavalcade follows the very successful Amazing Mathematical Amusement Arcade and The Mathematical Funfair by the same author.It contains a further 131 puzzles to challenge people of all ages.Hints and solutions are given in a commentary at the back of the book.
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What is currently trending or popular?
Currently, the trend of sustainability and eco-friendly living is gaining popularity. People are becoming more conscious of their environmental impact and are seeking out sustainable products and practices. Additionally, there is a growing interest in plant-based diets and veganism, as well as a focus on mental health and self-care. In the entertainment world, nostalgic content and throwback fashion from the 90s and early 2000s are also trending.
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What are mathematical prisms?
Mathematical prisms are three-dimensional shapes that have two parallel and congruent polygonal bases connected by rectangular faces. The bases can be any polygon, such as a triangle, square, or pentagon. The height of the prism is the perpendicular distance between the two bases. The volume of a prism is calculated by multiplying the area of the base by the height.
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What are mathematical terms?
Mathematical terms are words or phrases used to describe mathematical concepts, operations, or relationships. They are used to communicate specific ideas or instructions in the language of mathematics. Examples of mathematical terms include "addition," "subtraction," "equation," "variable," "function," and "theorem." Understanding mathematical terms is essential for effectively solving mathematical problems and communicating mathematical ideas.
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What are mathematical formulas?
Mathematical formulas are concise and specific representations of mathematical relationships or rules. They are used to express mathematical concepts, calculations, and relationships between variables in a clear and systematic way. Formulas often consist of symbols, numbers, and mathematical operations, and are used to solve equations, make predictions, and perform calculations in various fields of mathematics and science. They provide a standardized and efficient way to communicate mathematical concepts and principles.
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Mathematical Conundrums
Want to sharpen your mathematical wits? If so, then Mathematical Conundrums is for you. Daily Telegraph enigmatologist, Barry R. Clarke, presents over 120 fiendish problems that will test both your ingenuity and persistence.Between these covers are puzzles in geometry, arithmetic, and algebra (there is even a section for computer programmers). And, for the smartest readers who wish to stretch their mind to its limits, a selection of engaging logic and visual lateral puzzles is included.Although no puzzle requires a greater knowledge of mathematics than the high school curriculum, this collection will take you to the edge.But are you equal to the challenge? Features High-school level of mathematics is the only pre-requisiteVariety of algebraic, route-drawing, and geometrical conundrumsHints section for the lateral puzzlesWarm-up excercises to sharpen the witsFull solutions to every problemBarry R.Clarke has published over 1,500 puzzles in The Daily Telegraph and has contributed enigmas to New Scientist, The Sunday Times, Reader’s Digest, The Sunday Telegraph, and Prospect magazine.His book Challenging Logic Puzzles Mensa has sold over 100,000 copies.As well as a PhD in Shakespeare Studies, Barry has a master’s degree and academic publications in quantum physics.He is now working on a revised theory of the hydrogen atom.Other skills include mathematics tutor, filmmaker, comedy-sketch writer, cartoonist, computer programmer, and blues guitarist!For more information please visit http://barryispuzzled.com.
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Mathematical Logic
Assuming no previous study in logic, this informal yet rigorous text covers the material of a standard undergraduate first course in mathematical logic, using natural deduction and leading up to the completeness theorem for first-order logic.At each stage of the text, the reader is given an intuition based on standard mathematical practice, which is subsequently developed with clean formal mathematics.Alongside the practical examples, readers learn what can and can't be calculated; for example the correctness of a derivation proving a given sequent can be tested mechanically, but there is no general mechanical test for the existence of a derivation proving the given sequent.The undecidability results are proved rigorously in an optional final chapter, assuming Matiyasevich's theorem characterising the computably enumerable relations.Rigorous proofs of the adequacy and completeness proofs of the relevant logics are provided, with careful attention to the languages involved.Optional sections discuss the classification of mathematical structures by first-order theories; the required theory of cardinality is developed from scratch.Throughout the book there are notes on historical aspects of the material, and connections with linguistics and computer science, and the discussion of syntax and semantics is influenced by modern linguistic approaches.Two basic themes in recent cognitive science studies of actual human reasoning are also introduced.Including extensive exercises and selected solutions, this text is ideal for students in Logic, Mathematics, Philosophy, and Computer Science.
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Mathematical Physics
Mathematical Physics is an introduction to such basic mathematical structures as groups, vector spaces, topological spaces, measure spaces, and Hilbert space.Geroch uses category theory to emphasize both the interrelationships among different structures and the unity of mathematics.Perhaps the most valuable feature of the book is the illuminating intuitive discussion of the "whys" of proofs and of axioms and definitions.This book, based on Geroch's University of Chicago course, will be especially helpful to those working in theoretical physics, including such areas as relativity, particle physics, and astrophysics.
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Mathematical Meditations
Mathematical Meditations identifies, explores, and celebrates those aspects of mathematics that are good for you and your overall wellbeing.It is necessary for everyone to have a little time to think every so often: to contemplate, meditate, and try to understand where you are and what is going on around you.Mathematics can help you with all of that. The Meditations in this book are the product of thousands of years of mathematical discourse.As you read through the book and work through the various exercises, you will discover new mechanisms that allow you to contemplate and understand some complex mathematical principles.However, the focus will always be wider than a mere dry comprehension of theory, as you will be encouraged to meditate upon the deeper intrinsic beauty of mathematics and what it can reveal to us about the world around us. FeaturesAn original, engaging narrative format replete with novel exercises and examples. Could be used in a classroom setting for liberal arts students, mathematics undergraduates, or high school teachers. Accessible to anyone who wants to explore a different kind of perspective on mathematics.
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Are mathematical functions bounded?
Mathematical functions can be bounded or unbounded, depending on their behavior. A function is said to be bounded if its output values are limited within a certain range. For example, the sine function is bounded between -1 and 1. However, functions like the natural logarithm or the quadratic function are unbounded, as their output values can grow without limit. Therefore, whether a mathematical function is bounded or not depends on its specific properties and behavior.
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What are mathematical functions?
Mathematical functions are relationships between a set of inputs and a set of outputs, where each input is related to exactly one output. They are typically represented by an equation or a rule that describes how the input values are transformed into output values. Functions are fundamental in mathematics and are used to model various real-world phenomena, analyze data, and solve problems in a systematic way. They can take many forms, such as linear, quadratic, exponential, trigonometric, and logarithmic functions.
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What is mathematical optimization?
Mathematical optimization is the process of finding the best solution to a problem from a set of possible solutions. It involves maximizing or minimizing a certain objective function while satisfying a set of constraints. This can be applied to a wide range of fields, including engineering, economics, and computer science, to help make better decisions and improve efficiency. Optimization problems can be solved using various mathematical techniques such as linear programming, nonlinear programming, and integer programming.
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What are mathematical equations?
Mathematical equations are expressions that show the relationship between two or more quantities using mathematical symbols and operations. They typically consist of variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division. Equations are used to represent and solve various mathematical problems and are an essential tool in fields such as physics, engineering, and economics. They can be simple or complex, and their solutions provide valuable information about the relationships between different quantities.
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