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Elwes R: The Story of Huge Numbers

The Story of Huge Numbers

Elements in the Philosophy of Mathematics

This Element discusses the philosophical roles of definitions in the attainment of mathematical knowledge. It first focuses on the role of definitions in foundational programs, and then examines their major varieties, both as regards their origins, their potential epistemic roles, and their formal constraints. It examines explicit definitions, implicit definitions, and implicit definitions of primitive terms, these latter being further divided into axiomatic and abstractive. After discussing elucidations and explications, various ways in which definitions can yield mathematical knowledge are surveyed.

Definitions and Mathematical Knowledge

At first glance, the concept of equality in maths seems unambiguous. When we see the equality sign, we think of 'solving for x' or balancing two sides of an equation or maybe even the many famous equations that make use of this elegant, innocuous symbol. = But between those parallel lines lies a mathematical playground of choice and abstraction, leading to far greater insight than you could have dreamed. As it turns out, sameness and difference, equality and inequality, are not nearly as straightforward as they seem. =/= Unequal explores the rich and rewarding interplay between sameness and difference, from numbers to manifolds to category theory and beyond in a glorious celebration of mathematics that will change the way you look at maths - and the world around you - forever.

The Maths of When Things Do and Don’t Add Up

Unequal

Abstractionism

This Element looks at Quine's arguments against the concepts of analyticity. It mentions Boghossian's distinction between epistemic and metaphysical accounts of analyticity. It shows how analyticity is related to truth, necessity and questions of ontology. The relation of analyticity to the axiomatic method in mathematics is mentioned.

Mathematics is (mostly) Analytic

Discussing various versions of two medieval arguments for the impossibility of infinity, this Element sheds light on early stages of the evolution of the notion of INFINITIES OF DIFFERENT SIZES. The first argument is called 'the Equality Argument' and relies on the premise that all infinities are equal. The second argument is called 'the Mapping Argument' and relies on the assumption that if one thing is mapped/ superposed upon another thing and neither exceeds the other, the two things are equal to each other. Although these arguments were initially proposed in the context of discussions against the possibility of infinities, they have played pivotal roles in the historical evolution of the notion of INFINITIES OF DIFFERENT SIZES.

Medieval Finitism

?If you have ever wondered why an airliner bound from London to Los Angeles flies over Greenland, how we know what lies inside the Earth, or how computers correct errors, then this is for you' John Gribbin, author of Six Impossible Things How does a delivery driver distribute hundreds of packages in a single day? Why can maps help to win elections? Where should we look for serial killers? The answers lie in the crucial connection between maps and maths. In Mapmatics, mathematician Dr Paulina Rowinska embarks on a fascinating journey to discover the mathematical foundations of cartography and cartographical influences on mathematics. From a sixteenth-century map that remains an indispensable navigation tool on our phones to public transport maps that both guide and mislead passengers, she reveals how maps and maths shape not only our sense of space and time but also our worldview. In entertaining stories, surprising real-world examples, maps, illustrations and a cast of unforgettable characters, Mapmatics helps us to appreciate the mathematical methods and ideas behind maps - and leaves us better equipped to look after them. ?A wonderfully readable adventure' Edward Brooke-Hitching, author of The Phantom Atlas ?This book blew my mind, again and again' Roma Agrawal, author of Built ?Entertaining, illuminating . . . An engaging look at a fascinating subject' New Scientist ?An expansive journey through time and place' The Observer

How We Navigate the World Through Numbers

Mapmatics

"Galileo famously wrote that the book of nature is written in mathematical language. The Language of Mathematics is a wide-ranging and beautifully illustrated collection of short, colorful histories of the most commonly used symbols in mathematics, providing readers with an engaging introduction to the origins, evolution, and conceptual meaning of each one. In dozens of lively and informative entries, Raâul Rojas shows how today's mathematics stands on the shoulders of giants, mathematicians from around the world who developed mathematical notation through centuries of collective effort. He tells the stories of such figures as al-Khwarizmi, Renâe Descartes, Joseph-Louis Lagrange, Carl Friedrich Gauss, Augustin-Louis Cauchy, Karl Weierstrass, Sofia Kovalevskaya, David Hilbert, and Kenneth Iverson. Topics range from numbers and variables to sets and functions, constants, and combinatorics. Rojas describes the mathematical problems associated with different symbols and reveals how mathematical notation has sometimes been an accidental process. The entries are self-contained and can be read in any order, each one examining one or two symbols, their history, and the variants they may have had over time. An essential companion for math enthusiasts, The Language of Mathematics shows how mathematics is a living and evolving entity, forever searching for the best symbolism to express relationships between abstract concepts and to convey meaning"--

The Stories behind the Symbols

The Language of Mathematics

Much Ado About Numbers

WHAT DOES IT MEAN TO COUNT? WHY ARE HUMANS THE ONLY SPECIES ON EARTH THAT CAN DO IT? WHERE DID COUNTING COME FROM? HOW HAS IT SHAPED SOCIETIES ALONG THE WAY? AND WHY DOES IT MATTER?

Humans, History and the Infinite Lives of Numbers

Counting

A proof of a theorem can be said to be pure if it draws only on what is 'close' or 'intrinsic' to that theorem. In this Element we will investigate the apparent preference for pure proofs that has persisted in mathematics since antiquity, alongside a competing preference for impurity. In Section 1, we present two examples of purity, from geometry and number theory. In Section 2, we give a brief history of purity in mathematics. In Section 3, we discuss several different types of purity, based on different measures of distance between theorem and proof. In Section 4 we discuss reasons for preferring pure proofs, for the varieties of purity constraints presented in Section 3. In Section 5 we conclude by reflecting briefly on purity as a preference for the local and how issues of translation intersect with the considerations we have raised throughout this work.

Elements of Purity

This second edition continues to present meaningful and practical solutions for teaching mathematics and statistics online, focusing on the problems observed by mathematics instructors currently working in the field. With its set of standard practices, instructors will benefit from learning many new techniques and approaches.

Teaching and Learning Mathematics Online

Chapman & Hall/CRC Monographs and Research Notes in Mathematics

This book explores the underlying concepts, physics and mathematics of wave scattering and wave field structures and self-consistently introduces a range of topics in scalar acoustics and quantum and vector electromagnetic scattering.

Theory and Applications

Scattering of Waves

The Story of Huge Numbers

Informations bibliographiques

Mots-clés

Informations bibliographiques

Mots-clés

Autres titres sur ce thème