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The Search for Mathematical Roots, 1870-1940 : Logics, Set Theories and the Foundations of Mathematics from Cantor through Russell to Goedel - Ivor Grattan-Guinness

The Search for Mathematical Roots, 1870-1940

Logics, Set Theories and the Foundations of Mathematics from Cantor through Russell to Goedel

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Published: 1st December 2000
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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).
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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 the philosopher Josiah Royce, and stretching through the emergence of Church and Quine, and the 1930s immigration of Carnap and GodeI.


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.

"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."--James W. Van Evra, Isis

Explanations
Salliesp. 3
Scope and limits of the bookp. 3
An outline historyp. 3
Mathematical aspectsp. 4
Historical presentationp. 6
Other logics, mathematics and philosophiesp. 7
Citations, terminology and notations
References and the bibliographyp. 9
Translations, quotations and notationsp. 10
Permissions and acknowledgementsp. 11
Preludes: Algebraic Logic and Mathematical Analysis up top. 1870
Plan of the chapterp. 14
`Logique'' and algebras in French mathematicsp. 14
The `logique'' and clarity of `ideologie''p. 14
Lagrange''s algebraic philosophyp. 15
The many senses of `analysis''p. 17
Two Lagrangian algebras: functional equations and differential operatorsp. 17
Autonomy for the new algebrasp. 19
Some English algebraists and logiciansp. 20
A Cambridge revival: the `Analytical Society, Lacroix, and the professing of algebrasp. 20
The advocacy of algebras by Babbage, Herschel and Peacockp. 20
An Oxford movement: Whately and the professing of logicp. 22
A London pioneer: De Morgan on algebras and logicp. 25
Summary of his lifep. 25
De Morgan''s philosophies of algebrap. 25
De Morgan''s logical careerp. 26
De Morgan''s contributions to the foundations of logicp. 27
Beyond the syllogismp. 29
Contretemps over `the quantification of the predicate''p. 30
The logic of two place relations, 1860p. 32
Analogies between logic and mathematicsp. 35
De Morgan''s theory of collectionsp. 36
A Lincoln outsider: Boole on logic as applied mathematicsp. 37
Summary of his careerp. 37
Boole''s `general method in analysis'' 1844p. 39
The mathematical analysis of logic, 1847. `elective symbols'' and lawsp. 40
`Nothing'' and the `Universe''p. 42
Propositions, expansion theorems, and solutionsp. 43
The laws of thought, 1854: modified principles and extended methodsp. 46
Boole''s new theory of propositionsp. 49
The character of Boole''s systemp. 50
Boole''s search for mathematical rootsp. 53
The semi-followers of Boolep. 54
Some initial reactions to Boole''s theoryp. 54
The reformulationp. 56
Jevons versus Boolep. 59
Followers of Boole and/or Jevonsp. 60
Cauchy, Weierstrass and the rise of mathematical analysisp. 63
Different traditions in the calculusp. 63
Cauchy and the Ecole Polytechniquep. 64
The gradual adoption and adaptation of Cauchy''s new traditionp. 67
The refinements of Weierstrass and his followersp. 68
Judgement and supplementp. 70
Mathematical analysis versus algebraic logicp. 70
The places of Kant and Bolzanop. 71
Cantor: Mathematics as Mengenlehre 3.1 Prefacesp. 75
Plan of the chapterp. 75
Cantor''s careerp. 75
The launching of the Mengenlehre, 1870-1883p. 79
Riemann''s thesis: the realm of discontinuous functionsp. 79
Heine on trigonometric series and the real line, 1870-1872p. 81
Cantor''s extension of Heine''s findings, 1870-1872p. 83
Dedekind on irrational numbers, 1872p. 85
Cantor on line and plane, 1874-1877p. 88
Infinite numbers and the topology of linear sets, 1878-1883p. 89
The Grundlagen, 1883: the construction of number-classesp. 92
The Grundlagen: the definition of continuityp. 95
The successor to the Grundlagen, 1884p. 96
Cantor''s Acta mathematica phase, 1883-1885p. 97
Mittag-Lefler and the French translations, 1883p. 97
Unpublished and published ''communications'' 1884-1885p. 98
Order-types and partial derivatives in the `communications''p. 100
Commentators on Cantor, 1883-1885p. 102
The extension of the Mengenlehre, 1886-1897p. 103
Dedekind''s developing set theory, 1888p. 103
Dedekind''s chains of integersp. 105
Dedekind''s philosophy of arithmeticp. 107
Cantor''s philosophy of the infinite, 1886-1888p. 109
Cantor''s new definitions of numbersp. 110
Cardinal exponentiation: Cantor''s diagonal argument, 1891p. 110
Transfinite cardinal arithmetic and simply ordered sets, 1895p. 112
Transfinite ordinal arithmetic and well-ordered sets, 1897p. 114
Open and hidden questions in Cantor''s Mengenlehrep. 114
Well-ordering and the axioms of choicep. 114
What was Cantor''s `Cantor''s continuum problem''?p. 116
"Paradoxes" and the absolute infinitep. 117
Cantor''s philosophy of mathematicsp. 119
A mixed positionp. 119
(No) logic and metamathematicsp. 120
The supposed impossibility of infinitesimalsp. 121
A contrast with Kroneckerp. 122
Concluding comments: the character of Cantor''s achievementsp. 124
Parallel Processes in Set Theory, Logics and Axiomatics, 1870s-1900s
Plans for the chapterp. 126
The splitting and selling of Cantor''s Mengenlehrep. 126
National and international supportp. 126
French initiatives, especially from Borelp. 127
Couturat outlining the infinite, 1896p. 129
German initiatives from Meinp. 130
German proofs of the Schroder-Bernstein theoremp. 132
Publicity from Hilbert, 1900p. 134
Integral equations and functional analysisp. 135
Kempe on `mathematical form''p. 137
Kempe-who?p. 139
American algebraic logic: Peirce and his followersp. 140
Peirce, published and unpublishedp. 141
Influences on Peirre''s logic: father''s algebrasp. 142
Peirce''s first phase: Boolean logic and the categories, 1867-1868p. 144
Peirce''s virtuoso theory of relatives, 1870p. 145
Peirce''s second phase, 1880: the propositional calculusp. 147
Peirre''s second phase, 1881: finite and infinitep. 149
Peirce''s students, 1883: duality, and ''Quantifying'' a propositionp. 150
Peirre on ''icons'' and the order of `quantifiers; 1885 153
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9780691058580
ISBN-10: 069105858X
Audience: Tertiary; University or College
Format: Paperback
Language: English
Number Of Pages: 624
Published: 1st December 2000
Country of Publication: US
Dimensions (cm): 23.5 x 15.88  x 3.81
Weight (kg): 0.98