*Steven J. Osterlind*

- Published in print:
- 2019
- Published Online:
- January 2019
- ISBN:
- 9780198831600
- eISBN:
- 9780191869532
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198831600.001.0001
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy

The Error of Truth recounts the astonishing and unexpected tale of how quantitative thinking was invented and rose to primacy in our lives in the nineteenth and early twentieth centuries, bringing us ...
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The Error of Truth recounts the astonishing and unexpected tale of how quantitative thinking was invented and rose to primacy in our lives in the nineteenth and early twentieth centuries, bringing us to an entirely new perspective on what we know about the world and how we know it—even on what we each think about ourselves. Quantitative thinking is our inclination to view natural and everyday phenomena through a lens of measurable events, with forecasts, odds, predictions, and likelihood playing a dominant part. This worldview, or Weltanschauung, is unlike anything humankind had before, and it came about because of a momentous human achievement: namely, we had learned how to measure uncertainty. Probability as a science had been invented. Through probability theory, we now had correlations, reliable predictions, regressions, the bell-shaped curve for studying social phenomena, and the psychometrics of educational testing. Significantly, these developments in mathematics happened during a relatively short period in world history: roughly, the 130-year period from 1790 to 1920, from about the close of the Napoleonic era, through the Enlightenment and the Industrial Revolutions, to the end of World War I. Quantification is now everywhere in our daily lives, such as in the ubiquitous microchip in smartphones, cars, and appliances, in the Bayesian logic of artificial intelligence, and in applications in business, engineering, medicine, economics, and elsewhere. Probability is the foundation of our quantitative thinking. Here we see its story: when, why, and how it came to be and changed us forever.Less

*The Error of Truth* recounts the astonishing and unexpected tale of how quantitative thinking was invented and rose to primacy in our lives in the nineteenth and early twentieth centuries, bringing us to an entirely new perspective on what we know about the world and how we know it—even on what we each think about ourselves. Quantitative thinking is our inclination to view natural and everyday phenomena through a lens of measurable events, with forecasts, odds, predictions, and likelihood playing a dominant part. This worldview, or *Weltanschauung*, is unlike anything humankind had before, and it came about because of a momentous human achievement: namely, we had learned how to measure uncertainty. Probability as a science had been invented. Through probability theory, we now had correlations, reliable predictions, regressions, the bell-shaped curve for studying social phenomena, and the psychometrics of educational testing. Significantly, these developments in mathematics happened during a relatively short period in world history: roughly, the 130-year period from 1790 to 1920, from about the close of the Napoleonic era, through the Enlightenment and the Industrial Revolutions, to the end of World War I. Quantification is now everywhere in our daily lives, such as in the ubiquitous microchip in smartphones, cars, and appliances, in the Bayesian logic of artificial intelligence, and in applications in business, engineering, medicine, economics, and elsewhere. Probability is the foundation of our quantitative thinking. Here we see its story: *when, why*, and *how* it came to be and changed us forever.

*Christophe Reutenauer*

- Published in print:
- 2018
- Published Online:
- January 2019
- ISBN:
- 9780198827542
- eISBN:
- 9780191866418
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198827542.001.0001
- Subject:
- Mathematics, Pure Mathematics

Christoffel introduced in 1875 a special class of words on a binary alphabet, linked to continued fractions. Some years laterMarkoff published his famous theory, called nowMarkoff theory. It ...
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Christoffel introduced in 1875 a special class of words on a binary alphabet, linked to continued fractions. Some years laterMarkoff published his famous theory, called nowMarkoff theory. It characterizes certain quadratic forms, and certain real numbers by extremal inequalities. Both classes are constructed by using certain natural numbers, calledMarkoff numbers; they are characterized by a certain diophantine equality. More basically, they are constructed using certain words, essentially the Christoffel words. The link between Christoffelwords and the theory ofMarkoffwas noted by Frobenius.Motivated by this link, the book presents the classical theory of Markoff in its two aspects, based on the theory of Christoffel words. This is done in Part I of the book. Part II gives the more advanced and recent results of the theory of Christoffel words: palindromes (central words), periods, Lyndon words, Stern–Brocot tree, semi-convergents of rational numbers and finite continued fractions, geometric interpretations, conjugation, factors of Christoffel words, finite Sturmian words, free group on two generators, bases, inner automorphisms, Christoffel bases, Nielsen’s criterion, Sturmian morphisms, and positive automorphisms of this free group.Less

Christoffel introduced in 1875 a special class of words on a binary alphabet, linked to continued fractions. Some years laterMarkoff published his famous theory, called nowMarkoff theory. It characterizes certain quadratic forms, and certain real numbers by extremal inequalities. Both classes are constructed by using certain natural numbers, calledMarkoff numbers; they are characterized by a certain diophantine equality. More basically, they are constructed using certain words, essentially the Christoffel words. The link between Christoffelwords and the theory ofMarkoffwas noted by Frobenius.Motivated by this link, the book presents the classical theory of Markoff in its two aspects, based on the theory of Christoffel words. This is done in Part I of the book. Part II gives the more advanced and recent results of the theory of Christoffel words: palindromes (central words), periods, Lyndon words, Stern–Brocot tree, semi-convergents of rational numbers and finite continued fractions, geometric interpretations, conjugation, factors of Christoffel words, finite Sturmian words, free group on two generators, bases, inner automorphisms, Christoffel bases, Nielsen’s criterion, Sturmian morphisms, and positive automorphisms of this free group.

*Frank C. Zagare*

- Published in print:
- 2019
- Published Online:
- February 2019
- ISBN:
- 9780198831587
- eISBN:
- 9780191869525
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198831587.001.0001
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy, Applied Mathematics

The main purpose of this book is to demonstrate, by way of example, the several advantages of using a formal game-theoretic framework to explain complex events, diplomatic history, and contentious ...
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The main purpose of this book is to demonstrate, by way of example, the several advantages of using a formal game-theoretic framework to explain complex events, diplomatic history, and contentious interstate relationships, via causal mechanisms and rationality. Chapter 1 lays out the broad parameters and major concepts of the mathematical theory of games and its applications in the security studies literature. Chapter 2 explores a number of issues connected with the use of game-theoretic models to organize analytic narratives, both generally and specifically. Chapter 3 interprets the Moroccan crisis of 1905–6 in the context of an incomplete information game model. Chapter 4 surveys and evaluates several prominent attempts to use game theory to explain the strategic dynamic of the Cuban missile crisis of 1962. Chapter 5 offers a general explanation that answers all of the foundational questions associated with the Cuban crisis within the confines of a single, integrated, game-theoretic model with incomplete information. Chapter 6 uses the same game form to develop a logically consistent and empirically plausible explanation of the outbreak of war in Europe in early August 1914. Chapter 7 introduces perfect deterrence theory and contrasts it with the prevailing realist theory of interstate war prevention, and classical deterrence theory. Chapter 8 addresses the charge made by some behavioral economists (and many strategic analysts) that game theory is of limited utility for understanding interstate conflict behavior.Less

The main purpose of this book is to demonstrate, by way of example, the several advantages of using a formal game-theoretic framework to explain complex events, diplomatic history, and contentious interstate relationships, via causal mechanisms and rationality. Chapter 1 lays out the broad parameters and major concepts of the mathematical theory of games and its applications in the security studies literature. Chapter 2 explores a number of issues connected with the use of game-theoretic models to organize analytic narratives, both generally and specifically. Chapter 3 interprets the Moroccan crisis of 1905–6 in the context of an incomplete information game model. Chapter 4 surveys and evaluates several prominent attempts to use game theory to explain the strategic dynamic of the Cuban missile crisis of 1962. Chapter 5 offers a general explanation that answers all of the foundational questions associated with the Cuban crisis within the confines of a single, integrated, game-theoretic model with incomplete information. Chapter 6 uses the same game form to develop a logically consistent and empirically plausible explanation of the outbreak of war in Europe in early August 1914. Chapter 7 introduces perfect deterrence theory and contrasts it with the prevailing realist theory of interstate war prevention, and classical deterrence theory. Chapter 8 addresses the charge made by some behavioral economists (and many strategic analysts) that game theory is of limited utility for understanding interstate conflict behavior.