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Iconic Arithmetic Volume I

Author : william bricken

Publisher : Unary Press

Page : 446 pages

File Size : 14,31 MB

Release : 2019-04-10

Category :

ISBN : 9781732485136

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Book Description

Arithmetic evolves. Iconic arithmetic is built from icons that look and feel like what they mean, rather than from strings of symbols that must be memorized. The book explores the formal structure of two types of postsymbolic boundary arithmetic. Ensemble arithmetic modernizes tallies to provide forms that add together by being placed together and multiply by being placed inside one another. James algebra defines the concepts and operations of arithmetic as different ways of arranging containers. Three simple axioms are sufficient. Features of iconic arithmetic include (1) a void with no representation and no properties instead of the symbol zero; (2) void-equivalent forms that can be freely deleted; (3) meaning based on existence of structure rather than truth or numerical value; (4) only one relation (containment) to represent all forms; and (5) construction and deletion to implement all transformations. Iconic forms and transformations can be represented as two and three dimensional structures that can be directly viewed, manipulated, and even inhabited. Many different spatial interactive dialects are described. The author explores this new kind of arithmetic from the perspectives of historical evolution, formal mathematics, computer science and mathematics education. The overall objective is to provide proof of principle that our current universal approach to the arithmetic of numbers is a design choice rather than a truth embedded in numbers themselves. Iconic Arithmetic recognizes that knowledge is embodied, multidimensional, sensual, simple. It helps us to transition into a postsymbolic world of interactive information.



Icons of Mathematics: An Exploration of Twenty Key Images

Author : Claudi Alsina

Publisher : American Mathematical Soc.

Page : 327 pages

File Size : 44,42 MB

Release : 2020-06-01

Category : Education

ISBN : 1470456168

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Book Description

The authors present twenty icons of mathematics, that is, geometrical shapes such as the right triangle, the Venn diagram, and the yang and yin symbol and explore mathematical results associated with them. As with their previous books (Charming Proofs, When Less is More, Math Made Visual) proofs are visual whenever possible. The results require no more than high-school mathematics to appreciate and many of them will be new even to experienced readers. Besides theorems and proofs, the book contains many illustrations and it gives connections of the icons to the world outside of mathematics. There are also problems at the end of each chapter, with solutions provided in an appendix. The book could be used by students in courses in problem solving, mathematical reasoning, or mathematics for the liberal arts. It could also be read with pleasure by professional mathematicians, as it was by the members of the Dolciani editorial board, who unanimously recommend its publication.



Iconic Arithmetic Volume III

Author : William Bricken

Publisher :

Page : 418 pages

File Size : 23,29 MB

Release : 2021-02-28

Category :

ISBN : 9781732485150

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Book Description

Volume III of this series applies the innovations of iconic arithmetic to several branches of elementary mathematics to yield new techniques and new insights that are not accessible to symbolic arithmetic. The rules of algebra are reduced to three simple patterns expressed as spatial containment relations. These patterns resurrect a discarded imaginary number that provides an additive foundation for the multiplicative i, replaces inverse operations by a single constant, and eliminates signed numbers by rendering polarity as an exponent. Postsymbolic arithmetic reduces trigonometry to reflection along a line, condenses calculus derivatives into a single generic pattern, and identifies and organizes infinite and indeterminate expressions. Other innovations include base-free logarithms, fractional polarity, bipolar numbers, and direct deletion of illusory structure introduced by typographic notation.The overall goal is to demonstrate a comprehensive formal system that can be interpreted as arithmetic but bears little resemblance to our current universally adopted symbolic arithmetic. Iconic Arithmetic provides a proof of concept that our understanding of arithmetic has been limited by the absence of formal techniques for sensory interaction with abstraction.



Psychology and Mathematics Education

Author : Gila Hanna

Publisher : Frontiers Media SA

Page : 552 pages

File Size : 39,53 MB

Release : 2023-09-05

Category : Science

ISBN : 2832529992

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Book Description

Modern Mathematics is constructed rigorously through proofs, based on truths, which are either axioms or previously proven theorems. Thus, it is par excellence a model of rational inquiry. Links between Cognitive Psychology and Mathematics Education have been particularly strong during the last decades. Indeed, the Enlightenment view of the rational human mind that reasons, makes decisions and solves problems based on logic and probabilities, was shaken during the second half of the twentieth century. Cognitive psychologists discovered that humans' thoughts and actions often deviate from rules imposed by strict normative theories of inference. Yet, these deviations should not be called "errors": as Cognitive Psychologists have demonstrated, these deviations may be either valid heuristics that succeed in the environments in which humans have evolved, or biases that are caused by a lack of adaptation to abstract information formats. Humans, as the cognitive psychologist and economist Herbert Simon claimed, do not usually optimize, but rather satisfice, even when solving problem. This Research Topic aims at demonstrating that these insights have had a decisive impact on Mathematics Education. We want to stress that we are concerned with the view of bounded rationality that is different from the one espoused by the heuristics-and-biases program. In Simon’s bounded rationality and its direct descendant ecological rationality, rationality is understood in terms of cognitive success in the world (correspondence) rather than in terms of conformity to content-free norms of coherence (e.g., transitivity).



Enthusiastic Mathematics

Author : Bernie Lewin

Publisher :

Page : 370 pages

File Size : 41,58 MB

Release : 2018-11-19

Category :

ISBN : 9781982930523

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Book Description

Fifty years ago, a small sparse book came out under the pretentious title Laws of Form. Its author might have once fallen in with the logical philosophers of Cambridge, including Russell and Wittgenstein. But only later, while designing primitive switching circuits for British Rail, did George Spencer Brown come upon the arithmetic underlying Boolean algebra. Laws of Form flips the reduction of mathematics to logic, revealing simple laws of being and knowing that only reflect what great mystics, East and West, have been saying all along. Enthusiastic Mathematics offers the first thorough, philosophical introduction to Laws of Form. With no presumption for logic or mathematics, the reader is delivered into its philosophical vision via a colourful journey through the history of science.



Finding Zero

Author : Amir D. Aczel

Publisher : Macmillan + ORM

Page : 199 pages

File Size : 19,33 MB

Release : 2015-01-06

Category : Mathematics

ISBN : 1466879106

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Book Description

The invention of numerals is perhaps the greatest abstraction the human mind has ever created. Virtually everything in our lives is digital, numerical, or quantified. The story of how and where we got these numerals, which we so depend on, has for thousands of years been shrouded in mystery. Finding Zero is an adventure filled saga of Amir Aczel's lifelong obsession: to find the original sources of our numerals. Aczel has doggedly crisscrossed the ancient world, scouring dusty, moldy texts, cross examining so-called scholars who offered wildly differing sets of facts, and ultimately penetrating deep into a Cambodian jungle to find a definitive proof. Here, he takes the reader along for the ride. The history begins with the early Babylonian cuneiform numbers, followed by the later Greek and Roman letter numerals. Then Aczel asks the key question: where do the numbers we use today, the so-called Hindu-Arabic numerals, come from? It is this search that leads him to explore uncharted territory, to go on a grand quest into India, Thailand, Laos, Vietnam, and ultimately into the wilds of Cambodia. There he is blown away to find the earliest zero—the keystone of our entire system of numbers—on a crumbling, vine-covered wall of a seventh-century temple adorned with eaten-away erotic sculptures. While on this odyssey, Aczel meets a host of fascinating characters: academics in search of truth, jungle trekkers looking for adventure, surprisingly honest politicians, shameless smugglers, and treacherous archaeological thieves—who finally reveal where our numbers come from.



Laws Of Form: A Fiftieth Anniversary

Author : Louis H Kauffman

Publisher : World Scientific

Page : 944 pages

File Size : 33,27 MB

Release : 2023-01-09

Category : Mathematics

ISBN : 9811247447

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Book Description

Laws of Form is a seminal work in foundations of logic, mathematics and philosophy published by G Spencer-Brown in 1969. The book provides a new point of view on form and the role of distinction, markedness and the absence of distinction (the unmarked state) in the construction of any universe. A conference was held August 8-10, 2019 at the Old Library, Liverpool University, 19 Abercromby Square, L697ZN, UK to celebrate the 50th anniversary of the publication of Laws of Form and to remember George Spencer-Brown, its author. The book is a collection of papers introducing and extending Laws of Form written primarily by people who attended the conference in 2019.



Number Theory, Analysis and Geometry

Author : Dorian Goldfeld

Publisher : Springer Science & Business Media

Page : 715 pages

File Size : 40,63 MB

Release : 2011-12-21

Category : Mathematics

ISBN : 1461412609

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Book Description

Serge Lang was an iconic figure in mathematics, both for his own important work and for the indelible impact he left on the field of mathematics, on his students, and on his colleagues. Over the course of his career, Lang traversed a tremendous amount of mathematical ground. As he moved from subject to subject, he found analogies that led to important questions in such areas as number theory, arithmetic geometry, and the theory of negatively curved spaces. Lang's conjectures will keep many mathematicians occupied far into the future. In the spirit of Lang’s vast contribution to mathematics, this memorial volume contains articles by prominent mathematicians in a variety of areas of the field, namely Number Theory, Analysis, and Geometry, representing Lang’s own breadth of interest and impact. A special introduction by John Tate includes a brief and fascinating account of the Serge Lang’s life. This volume's group of 6 editors are also highly prominent mathematicians and were close to Serge Lang, both academically and personally. The volume is suitable to research mathematicians in the areas of Number Theory, Analysis, and Geometry.



Science and the Founding Fathers

Author : I. Bernard Cohen

Publisher : W. W. Norton & Company

Page : 378 pages

File Size : 32,69 MB

Release : 1997

Category : Biography & Autobiography

ISBN : 9780393315103

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Book Description

Thomas Jefferson was the only president who could read and understand Newton's Principia. Benjamin Franklin is credited with establishing the science of electricity. John Adams had the finest education in science that the new country could provide, including "Pnewmaticks, Hydrostaticks, Mechanicks, Staticks, Opticks." James Madison, chief architect of the Constitution, peppered his Federalist Papers with references to physics, chemistry, and the life sciences. For these men science was an integral part of life--including political life. This is the story of their scientific education and of how they employed that knowledge in shaping the political issues of the day, incorporating scientific reasoning into the Constitution.



Benjamin Franklin's Numbers

Author : Paul C. Pasles

Publisher : Princeton University Press

Page : 272 pages

File Size : 49,40 MB

Release : 2007-11-04

Category : Mathematics

ISBN : 9780691129563

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Book Description

Few American lives have been as celebrated--or as closely scrutinized--as that of Benjamin Franklin. Yet until now Franklin's biographers have downplayed his interest in mathematics, at best portraying it as the idle musings of a brilliant and ever-restless mind. In Benjamin Franklin's Numbers, Paul Pasles reveals a side of the iconic statesman, scientist, and writer that few Americans know--his mathematical side. In fact, Franklin indulged in many areas of mathematics, including number theory, geometry, statistics, and economics. In this generously illustrated book, Pasles gives us the first mathematical biography of Benjamin Franklin. He draws upon previously unknown sources to illustrate Franklin's genius for numbers as never before. Magic squares and circles were a lifelong fascination of Franklin's. Here, for the first time, Pasles gathers every one of these marvelous creations together in one place. He explains the mathematics behind them and Franklin's hugely popular Poor Richard's Almanac, which featured such things as population estimates and a host of mathematical digressions. Pasles even includes optional math problems that challenge readers to match wits with the bespectacled Founding Father himself. Written for a general audience, this book assumes no technical skills beyond basic arithmetic. Benjamin Franklin's Numbers is a delightful blend of biography, history, and popular mathematics. If you think you already know Franklin's story, this entertaining and richly detailed book will make you think again.