The search for mathematical roots : 1870-1940 : logics, set theories and the foundations of mathematics from Cantor through Russell to Gödel

You'll notice that although GG still lists Cantor's "paradox" in his index, in the text he doesn't quite bring himself to say that there is such a thing. Why not? Because he has read Garciadiego's BERTRAND RUSSELL AND THE ORIGINS OF THE SET-THEORETIC 'PARADOXES,' which shows quite clearly that there is no such thing as Cantor's paradox, or the Burali-Forti paradox, or Russell's for that matter. The so-called "set-theoretic paradoxes" were for the most part inventions of Russell, and not a single one from the period, comes out as anything but a meaningless formulation.The problem this creates for GG is that so-called "set theory" is nonsense, and not much worth wasting time on. Apart from Cantor's own pathetic inability ever to define what a set is, the history is a farce of the blind leading the blind--trying to "avoid" formulations which are not paradoxes or anything else. This is worth writing about? Worth listing 1900 items in a bibliography, about? It's sad, but a good study in how wastes of time and resources occur. So GG goes ahead and talks about these "paradoxes" as if they really were such, and about people's "responses" to them as if there was anything to respond to. GG still hasn't quite weaned himself from the "paradoxes," although he cites Garciadiego and should have known better. The gist of the book is that the "paradoxes" which led to Godel's argument (and those of the Intuitionists, the Logicists and Formalists as well as their successors), are not paradoxes at all--they are meaningless formulations. This undermines most, if not all, of twentieth-century mathematics, and in particular destroys Godel's very sloppy argument.Garciadiego cites Richard's own formulation of this "contradiction" (Richard's term) in a letter to Poincare. He also cites Richard reducing the argument to meaninglessness. What does this have to do with Godel? It's simple. For Godel, Richard's "paradox" means that truth in number theory cannot be defined in number theory. On this basis, he distinguishes truth from provability. He combines his idea of Richard's "paradox" with the idea that provability in number theory can be defined in number theory. He arrives at the conclusion that if all the provable formulae are true, there must be some true but unprovable formulae. However, since Richard's "paradox" is without meaning, since it has no logical content whatsoever and is simply letters pulled out of a bag, there is no basis in Godel's argument for distinguishing truth from provability. It turns out that there is no logical content in the idea that if all the provable formulae are true, there must be some true but unprovable formulae.People are having a hard time getting over the notion that Godel didn't do his homework, and has nothing to say, but really you have to grow up. Get over it. The problem is that Godel was a terrible scholar, and did not apply himself sufficiently to the details of the development of set theory.Garciadiego's book has implications for all twentieth-century mathematics. Here are just a few examples of horrendous errors which explain a lot about why mathematics today is regarded as the province of clowns. For example, Brouwer based the idea of an infinite ordinal number on the idea that Cantor had proved well-ordering of the ordinal numbers. But not only did Cantor never prove this, but also, he never said he had done so, and never used the term infinite ordinal number. Turing never examines the "paradoxes" in order to determine whether they are simply meaningless formulations. Thus, in an attempt to "prove that there is no general method for determining about a formula whether it is an ordinal formula, we use an argument akin tothat leading the Burali-Forti paradox, but the emphasis and the conclusion are different." As Garciadiego reveals, there is no Burali-Forti paradox. In the context of an attempt to prove the Trichotomy Law, START Start TRANSACTION WITH CONSISTENT SNAPSHOT; /\* 2152 = 4f915966ee24e2f1247693944d3ffdea While many books have been written about Bertrand Russell's philosophy and some on his logic, I. Grattan-Guinness has written the first comprehensive history of the mathematical background, content, and impact of the mathematical logic and philosophy of mathematics that Russell developed with A. N. Whitehead in their Principia mathematica (1910-1913). ? This definitive history of a critical period in mathematics includes detailed accounts of the two principal influences upon Russell around 1900: the set theory of Cantor and the mathematical logic of Peano and his followers. Substantial surveys are provided of many related topics and figures of the late nineteenth century: the foundations of mathematical analysis under Weierstrass; the creation of algebraic logic by De Morgan, Boole, Peirce, Schröder, and Jevons; the contributions of Dedekind and Frege; the phenomenology of Husserl; and the proof theory of Hilbert. The many-sided story of the reception is recorded up to 1940, including the rise of logic in Poland and the impact on Vienna Circle philosophers Carnap and Gödel. A strong American theme runs though the story, beginning with the mathematician E. H. Moore and the philosopher Josiah Royce, and stretching through the emergence of Church and Quine, and the 1930's immigration of Carnap and GödeI. Grattan-Guinness draws on around fifty manuscript collections, including the Russell Archives, as well as many original reviews. The bibliography comprises around 1,900 items, bringing to light a wealth of primary materials. Written for mathematicians, logicians, historians, and philosophers--especially those interested in the historical interaction between these disciplines--this authoritative account tells an important story from its most neglected point of view. Whitehead and Russell hoped to show that (much of) mathematics was expressible within their logic; they failed in various ways, but no definitive alternative position emerged then or since While many books have been written about Bertrand Russell's philosophy and some on his logic, I. Grattan-Guinness has written the first comprehensive history of the mathematical background, content, and impact of the mathematical logic and philosophy of mathematics that Russell developed with A. N. Whitehead in their Principia mathematica (1910-1913). ? This definitive history of a critical period in mathematics includes detailed accounts of the two principal influences upon Russell around 1900: the set theory of Cantor and the mathematical logic of Peano and his followers. Substantial surveys are provided of many related topics and figures of the late nineteenth century: the foundations of mathematical analysis under Weierstrass; the creation of algebraic logic by De Morgan, Boole, Peirce, Schroder, and Jevons; the contributions of Dedekind and Frege; the phenomenology of Husserl; and the proof theory of Hilbert. The many-sided story of the reception is recorded up to 1940, including the rise of logic in Poland and the impact on Vienna Circle philosophers Carnap and Gdel. A strong American theme runs though the story, beginning with the mathematician E. H. Moore and the philosopher Josiah Royce, and stretching through the emergence of Church and Quine, and the 1930s immigration of Carnap and GdeI. Grattan-Guinness draws on around fifty manuscript collections, including the Russell Archives, as well as many original reviews. The bibliography comprises around 1,900 items, bringing to light a wealth of primary materials. Written for mathematicians, logicians, historians, and philosophers--especially those interested in the historical interaction between these disciplines--this authoritative account tells an important story from its most neglected point of view. Whitehead and Russell hoped to show that (much of) mathematics was expressible within their logic; they failed in various ways, but no definitive alternative position emerged then or since. While many books have been written about Bertrand Russell's philosophy and some on his logic, Ivor Grattan-Guinness has written the first comprehensive history of the mathematical background, content, and impact of the mathematical logic and philosophy of mathematics that Russell developed with A. N. Whitehead in their Principia mathematica (1910-1913).This definitive history of a critical period in mathematics includes detailed accounts of the two principal influences upon Russell around 1900: the set theory of Cantor and the mathematical logic of Peano and his followers. Substantial surveys are provided of many related topics and figures of the late nineteenth century: the foundations of mathematical analysis under Weierstrass; the creation of algebraic logic by De Morgan, Boole, Peirce, Schroder, and Jevons; the contributions of Dedekind and Frege; the phenomenology of Husserl; and the proof theory of Hilbert. The many-sided story of the reception is recorded up to 1940, including the rise of logic in Poland and the impact on Vienna Circle philosophers Carnap and Godel. A strong American theme runs though the story, beginning with the mathematician E. H. Moore and stretching through the emergence of Church and Quine, and the 1930s immigration of Carnap and Godel.Grattan-Guiness draws on around fifty manuscript collections, including the Russell Archives, as well as many original reviews. The bibliography comprises around 1,900 items, bringing to light a wealth of primary materials.Written for mathematicians, logicians, historians, and philosophers -- especially those interested in the historical interaction between these disciplines -- this authoritative account tells animportant story from its most neglected point of view. Whitehead and Russell hoped to show that (much) mathematics was expressible within their logic; they failed in various ways, but no definitive alternative position emerged then or since.

I know of no comparably comprehensive treatment of the history of this important period in modern logic. There is a large body of historical literature that is in need of just the kind of synthesis and masterly overview that this work provides. Though most people recognize mathematics as a principal motivating force behind the development of modern logic, the influences on and from mathematics have been largely ignored or minimized. The Search for Mathematical Roots acts as a guide through that challenging mathematical thicket.—Albert C. Lewis, chief editor of The History of Mathematics from Antiquity to the Present

Ivor Grattan-Guinness provides a marvelous, comprehensive overview of the history of efforts to come to an understanding of mathematical logic and its relation to mathematics in the period 1870-1940. Given its rich detail and inclusion of under-appreciated figures who deserve to be better known, this is an especially important and useful book.—Joseph Dauben, author of George Cantor: His Mathematics and Philosophy of the Infinite

James W. Van Evra - Isis

Grattan-Guiness's uniformly interesting and valuable account of the interwoven development of logic and related fields of mathematics . . . between 1870 and 1940 presents a significantly revised analysis of the history of the period. . . . [His] book is important because it supplies what has been lacking: a full account of the period from a primary mathematical perspective.

Explanations -- Preludes : Algebraic Logic And Mathematical Analysis Up To 1870 -- Cantor : Mathematics As Mengenlehre -- Parallel Processes In Set Theory, Logics And Axiomatics, 1870s-1900s -- Peano : The Formulary Of Mathematics -- Russell's Way In : From Certainty To Paradoxes, 1895-1903 -- Russell And Whitehead Seek The Principia Mathematica, 1903-1913 -- The Influence And Place Of Logicism, 1910-1930 -- Postludes : Mathematical Logic And Logicism In The 1930s -- The Fate Of The Search -- Transcription Of Manuscripts. I. Grattan-guinness. Includes Bibliographical References (p. [594]-669) And Index.