<p>The second half of the nineteenth century was an exciting time in European intellectual and social history. The period saw the growth of revolutionary activism, the rise of Darwinian evolu tionary biology, the emergence of women's rights movements, and other challenges to established ways of thinking. Progress and change were the key words of the day, and most members of the educated classes felt confident that the future would be bright. In Russia especially, the "intelligentsia" (an amorphous, pe culiarly Russian class of professors, writers, and thinkers) had the feeling that they were on the threshold of a great new age. Russia's ignominious defeat in the Crimean War in 1856 sig naled to many that wide-ranging political and social reforms were urgently needed. Most of the intelligentsia hoped that the defeat would be followed by the emancipation of the serfs, modernization of education, moves toward the equality of women, and other reforms. It was in this period that the Russian intelligentsia developed into an influential social group that concerned itself with much more than just cogitation and empty philosophizing. During the second half of the nineteenth century, Russian intellectuals increasingly assumed burdens that their counterparts in other European countries considered outside their province. The in telligentsia became the source of most political and social activ ism in Russia, the social conscience and often the sole voice of protest against autocratic and reactionary policies.</p>

A Convergence of Lives: Sofia Kovalevskaia: Scientist, Writer, Revolutionary

Biographies & Memoirs

KOBLITZ

Professionals & Academics

Birkhauser

A Course in Mathematical Logic for Mathematicians (Graduate Texts in Mathematics, 53)

<p>1. The ?rst edition of this book was published in 1977. The text has been well received and is still used, although it has been out of print for some time. In the intervening three decades, a lot of interesting things have happened to mathematical logic: (i) Model theory has shown that insights acquired in the study of formal languages could be used fruitfully in solving old problems of conventional mathematics. (ii) Mathematics has been and is moving with growing acceleration from the set-theoretic language of structures to the language and intuition of (higher) categories, leaving behind old concerns about in?nities: a new view of foundations is now emerging. (iii) Computer science, a no-nonsense child of the abstract computability theory, has been creatively dealing with old challenges and providing new ones, such as the P/NP problem. Planning additional chapters for this second edition, I have decided to focus onmodeltheory,the conspicuousabsenceofwhichinthe ?rsteditionwasnoted in several reviews, and the theory of computation, including its categorical and quantum aspects. The whole Part IV: Model Theory, is new. I am very grateful to Boris I. Zilber, who kindly agreed to write it. It may be read directly after Chapter II. The contents of the ?rst edition are basically reproduced here as Chapters I–VIII. Section IV.7, on the cardinality of the continuum, is completed by Section IV.7.3, discussing H. Woodin’s discovery.</p>

Politics & Social Sciences

Philosophy

Springer

. . . both Gauss and lesser mathematicians may be justified in rejoic ing that there is one science [number theory] at any rate, and that their own, whose very remoteness from ordinary human activities should keep it gentle and clean. - G. H. Hardy, A Mathematician's Apology, 1940 G. H. Hardy would have been surprised and probably displeased with the increasing interest in number theory for application to ordinary human activities such as information transmission (error-correcting codes) and cryptography (secret codes). Less than a half-century after Hardy wrote the words quoted above, it is no longer inconceivable (though it hasn't happened yet) that the N. S. A. (the agency for U. S. government work on cryptography) will demand prior review and clearance before publication of theoretical research papers on certain types of number theory. In part it is the dramatic increase in computer power and sophistica tion that has influenced some of the questions being studied by number theorists, giving rise to a new branch of the subject, called computational number theory. This book presumes almost no background in algebra or number the ory. Its purpose is to introduce the reader to arithmetic topics, both ancient and very modern, which have been at the center of interest in applications, especially in cryptography. For this reason we take an algorithmic approach, emphasizing estimates of the efficiency of the techniques that arise from the theory.

A Course in Number Theory and Cryptography, 2e

Science & Math

Mathematics

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