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Cambridge, at the University Press, 1903. Royal 8vo. Original blue full cloth binding, all edges uncut. Capitals and upper front hinge with a bit of wear and corners a little bumped. But otherwise a very nice copy. Internally fresh and clean. XXIX, (1), 534 pp. The uncommon first edition of Russell's landmark work in mathematical logic, in which theory of logicism is put forth and in which Russell introduces that which is now known as "Russell's Paradox". The work constitutes the forerunner of Russell and Whitehead's monumental "Principia Mathematica", and it seminally influenced logical thought and theories of the foundations of mathematics at this most crucial time for the development of modern mathematical and philosophical logic."The present work has two main objects. One of these, the proof that all pure mathematics deals exclusively with concepts definable in terms of a very small number of fundamental logical concepts, and that its propositions are deducible from a very small number of fundamental logical principles, is undertaken in Parts II. - VI. Of this Volume, and will be established by strict symbolic reasoning in Volume II. . The other object of the work, which occupies Part I., is the explanation of the fundamental concepts which mathematics accepts as indefinable. ." (Russell, Preface, p. (III)).At the age of 27, in 1898, Russell began working on the book that became "The Principles of Mathematics". He originally set out to investigate the contradiction that is inherent in the nature of number, and he originally imagined doing this from a Hegelian standpoint. However, after having read Whitehead's "Universal Algebra", Russell gave up his Hegelian approach and began working on a book that was to be entitled "An Analysis of Mathematical Reasoning". This book never appeared, as he gave it up in 1900, but much of it is what lies at the foundation of "The Principles of Mathematics". After having attended a congress in Paris in 1899, where Peano was present, Russell began rewriting large parts of the work, now with the aim of proving that all of mathematics could be reduced to a few logical concepts, that that which is called mathematics is in reality nothing but later deductions from logical premises. And thus he had developed his landmark thesis that mathematics and logic are identical" a thesis that came to have a profound influence on logic and the foundations of mathematics throughout the 20th century.Since the congress, Russell had worked with the greatest of enthusiasm, and he finished the manuscript on the 31st of December 1900. However, in the spring of 1901, he discovered "The Contradiction", or as it is now called, "Russell's Paradox". Russell had been studying Cantor's proof, and in his own words, the paradox emerged thus: "Before taking leave of fundamental questions, it is necessary to examine more in detail the singular contradiction, already mentioned, with regard to predicates not predictable of themselves. Before attempting to solve this puzzle, it will be well to make some deductions connected with it, and to state it in various different forms. I may mention that I was led to endeavour to reconcile Cantor's proof that there can be no greatest cardinal number with the very plausible supposition that the class of all termes (which we have seen to be essential to all formal propositions) has necessarily the greatest possible number of members." (p. 101). The class of all classes that are not members of themselves, is this class a member of itself or not? The question was unanswerable (if it is, then it isn't, and if it isn't, then it is) and thus a paradox, and not just any paradox, this was a paradox of the greatest importance. Since, when using classical logic, all sentences are entailed by contradiction, this discovery naturally sparked a huge number of works within logic, set theory, foundations of mathematics, philosophy of mathematics, etc. Russell's own solution to the problem was his "theory.
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