Elements Of Intuitionism

This is a long-awaited new edition of one of the best known Oxford Logic Guides. The book gives an informal but thorough introduction to intuitionistic mathematics, leading the reader gently through the fundamental mathematical and philosophical concepts. The treatment of various topics has been completely revised for this second edition. Brouwer's proof of the Bar Theorem has been reworked, the account of valuation systems simplified, and the treatment of generalized Beth Trees and the completeness of intuitionistic first-order logic rewritten. Readers are assumed to have some knowledge of classical formal logic and a general awareness of the history of intuitionism.

L. E. J. Brouwer, the founder of mathematical intuitionism, believed that mathematics and its objects must be humanly graspable. He initiated a program rebuilding modern mathematics according to that principle. This book introduces the reader to the mathematical core of intuitionism - from elementary number theory through to Brouwer's uniform continuity theorem - and to the two central topics of 'formalized intuitionism': formal intuitionistic logic, and formal systems for intuitionistic analysis. Building on that, the book proposes a systematic, philosophical foundation for intuitionism that weaves together doctrines about human grasp, mathematical objects and mathematical truth.

This debut novel by the two time Pulitzer Prize-winning author of The Underground Railroad and The Nickel Boys wowed critics and readers everywhere and marked the debut of an important American writer. Nominated as one of America’s best-loved novels by PBS’s The Great American Read. It is a time of calamity in a major metropolitan city's Department of Elevator Inspectors, and Lila Mae Watson, the first black female elevator inspector in the history of the department, is at the center of it. There are two warring factions within the department: the Empiricists, who work by the book and dutifully check for striations on the winch cable and such; and the Intuitionists, who are simply able to enter the elevator cab in question, meditate, and intuit any defects. Lila Mae is an Intuitionist and, it just so happens, has the highest accuracy rate in the entire department. But when an elevator in a new city building goes into total freefall on Lila Mae's watch, chaos ensues. It's an election year in the Elevator Guild, and the good-old-boy Empiricists would love nothing more than to assign the blame to an Intuitionist. But Lila Mae is never wrong. The sudden appearance of excerpts from the lost notebooks of Intuitionism's founder, James Fulton, has also caused quite a stir. The notebooks describe Fulton's work on the "black box," a perfect elevator that could reinvent the city as radically as the first passenger elevator did when patented by Elisha Otis in the nineteenth century. When Lila Mae goes underground to investigate the crash, she becomes involved in the search for the portions of the notebooks that are still missing and uncovers a secret that will change her life forever. Look for Colson Whitehead’s new novel, Crook Manifesto, coming soon!

Within the Western tradition, it was the philosophers Henri Bergson and Max Scheler who laid out and explored the nonrational power of "intuition" at work in human beings that plays a key role in orienting their thinking and action within the world. As author Adriana Alfaro Altamirano notes, Bergon's and Scheler's philosophical explorations, which paralleled similar developments by other modernist writers, artists, and political actors of the early twentieth century, can yield fruitful insights into the ideas and passions that animate politics in our own time. The Belief in Intuition shows that intuition (as Bergson and Scheler understood it) leads, first and foremost, to a conception of freedom that is especially suited for dealing with hierarchy, uncertainty, and alterity. Such a conception of freedom is grounded in a sense of individuality that remains true to its "inner multiplicity," thus providing a distinct contrast to and critique of the liberal notion of the self. Focusing on the complex inner lives that drive human action, as Bergson and Scheler did, leads us to appreciate the moral and empirical limits of liberal devices that mean to regulate our actions "from the outside." Such devices, like the law, may not only carry pernicious effects for freedom but, more troublingly, oftentimes "erase their traces," concealing the very ways in which they are detrimental to a richer experience of subjectivity. According to Alfaro Altamirano, Bergson's and Scheler's conception of intuition and personal authority puts contemporary discussions about populism in a different light: It shows that liberalism would only at its own peril deny the anthropological, moral, and political importance of the bearers of charismatic authority. Personal authority thus understood relies on a dense, but elusive, notion of personality, for which personal authority is not only consistent with freedom, but even contributes to it in decisive ways.

Luitzen Egburtus Jan Brouwer founded a school of thought whose aim was to include mathematics within the framework of intuitionistic philosophy; mathematics was to be regarded as an essentially free development of the human mind. What emerged diverged considerably at some points from tradition, but intuitionism has survived well the struggle between contending schools in the foundations of mathematics and exact philosophy. Originally published in 1981, this monograph contains a series of lectures dealing with most of the fundamental topics such as choice sequences, the continuum, the fan theorem, order and well-order. Brouwer's own powerful style is evident throughout the work.

Intuitionism is one of the main foundations for mathematics proposed in the twentieth century and its views on logic have also notably become important with the development of theoretical computer science. This book reviews and completes the historical account of intuitionism. It also presents recent philosophical work on intuitionism and gives examples of new technical advances and applications. It brings together 21 contributions from today's leading authors on intuitionism.