The problem of the invariance of dimension in the growth of modern topology, part II

Johnson, Dale

doi:10.1007/bf02116242

Cited by 46 publications

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“…The theorem shocked even its prover, for it refuted the usual understanding of dimensions; finding an alternative definition of it turned out to be a daunting task, not satisfactorily completed until the 1920s [Johnson 1981]. Whitehead surely was aware of this aspect of set theory, but he never mentioned it; however, his own theory of dimensions must be susceptible to it, for he used coordinates as set by kinetic axes.…”

Section: Some Limitations Of Whitehead's Enterprisementioning

confidence: 94%

“…(2) Cantor had proved that an isomorphism could be found between the points in a square and those of any one of its sides [Johnson 1979]. The theorem shocked even its prover, for it refuted the usual understanding of dimensions; finding an alternative definition of it turned out to be a daunting task, not satisfactorily completed until the 1920s [Johnson 1981].…”

Section: Some Limitations Of Whitehead's Enterprisementioning

confidence: 98%

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Algebras, Projective Geometry, Mathematical Logic, and Constructing the World: Intersections in the Philosophy of Mathematics of A. N. Whitehead

Grattan‐Guinness

2002

Historia Mathematica

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A. N. Whitehead (1861Whitehead ( -1947 contributed notably to the foundations of pure and applied mathematics, especially from the late 1890s to the mid 1920s. An algebraist by mathematical tendency, he surveyed several algebras in his book Universal Algebra (1898). Then in the 1900s he joined Bertrand Russell in an attempt to ground many parts of mathematics in the newly developing mathematical logic. In this connection he published in 1906 a long paper on geometry, space and time, and matter. The main outcome of the collaboration was a three-volume work, Principia Mathematica (1910Mathematica ( -1913: he was supposed to write a fourth volume on parts of geometries, but he abandoned it after much of it was done. By then his interests had switched to educational issues, and especially to space and time and relativity theory, where his earlier dependence upon logic was extended to an ontology of events and to a general notion of "process," especially in human experience. These innovations led to somewhat revised conceptions of logic and of the philosophy of mathematics. C 2002 Elsevier Science (USA) A. N. Whitehead (1861-1947) contribuiu de forma marcante para os Fundamentos da Matemática Pura e Aplicada, especialmente entre o fim da década de 1890 e meados da década de 1920. Sendo um algebrista na sua vertente matemática, fez um levantamento de diversasálgebras no seu livro Universal Algebra (1898). Pouco depois de 1900 juntou-se a Bertrand Russell numa tentativa para basear várias partes da matemática sobre a lógica matemática, que se começava então a desenvolver. Nesseâmbito publicou em 1906 um longo artigo sobre geometria, espaço e tempo, e matéria. O principal resultado da colaboração foi um trabalho em três volumes, Principia Mathematica (1910Mathematica ( -1913: estava previsto que Whitehead escrevesse um quarto volume sobre aspectos das geometrias, mas abandonou-o depois de uma boa parte já estar escrita. Por essa altura os seus interesses tinham-se voltado para questões educacionais; especialmente para o espaço e o tempo e para a teoria da relatividade, onde a sua anterior dependência da lógica se estendeu a uma ontologia de acontecimentos e a uma noção geral de "processo" especialmente na experiência humana. Estas inovações levaram a concepções um pouco revistas da lógica e da filosofia da matemática. C 2002 Elsevier Science (USA) MSC 1991 subject classifications: 00A30; 01A60; 03-03; 03A05.

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Section: Some Limitations Of Whitehead's Enterprisementioning

confidence: 94%

Section: Some Limitations Of Whitehead's Enterprisementioning

confidence: 98%

Algebras, Projective Geometry, Mathematical Logic, and Constructing the World: Intersections in the Philosophy of Mathematics of A. N. Whitehead

Grattan‐Guinness

2002

Historia Mathematica

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“…An edition of his Collected works appeared in two volumes in the mid 1970s, with some Dutch items welcomely translated into English. His contributions to topology have benefited from historical study, most notably in [4]; and his intuitionism has been studied, especially in various papers and editions by the author (a distinguished logician) and also in [7]. His life has also been of interest; indeed, few great mathematicians of any period have had associated with them such a constellation of stories concerning their personality and behaviour.…”

Section: Imentioning

confidence: 99%

Book Review: Mystic, geometer, and intuitionist. The life of L. E. J. Brouwer. Volume 1: The dawning revolution

Grattan‐Guinness¹

1999

Bull. Amer. Math. Soc.

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“…This conclusion was accepted by most authors, particularly Helmholtz in his influential papers, until CANTOR showed in 1878 that points in an ndimensional manifold can be determined by means of a single coordinate (CANTOR 1878, 120-121). 41 Nevertheless, the theorem of the invariance of dimension under bicontinuous mappings was immediately suggested by DEDEKIND (see CANTOR & DEDEKIND 1937); although a correct proof of this result was only given by BROUWER in 1911, NETTO's and CANTOR's insufficient proofs seem to have satisfied everybody until the end of the century (JOHNSON 1979/81, DAUBEN 1979.…”

Section: Helmholtz and Betti And From Klein's Erlanger Programm Of 1mentioning

confidence: 99%

Traditional logic and the early history of sets, 1854?1908

Ferreirós

1996

Arch. Hist. Exact Sci.

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6.1. Pre-Russellian logicism.6.2. Effect of the antinomies. I. INTRODUCTIONA long-standing tradition considers Georg CANTOR as the 'creator' or founder of set theory.1 This view, correct when restricted to transfinite set theory, stands in need of revision within the context of a general account of the emergence of sets in mathematics. Such an account should face the broad questions: how did sets become important in the different branches of mathematics?how did they emerge as basic mathematical objects, as a fundamental language for mathematics? 2 Simultaneously, one should try to establish how and why some mathematicians began to think of sets as the foundation for mathematics. In fact, within the context of the early history of sets during the second half of the nineteenth century, it is convenient to differentiate three aspects of our picture of set theory: the language of sets, the theory of sets, and the idea that sets form the foundation of mathematics. These aspects are so intertwined today that it may seem artificial to distinguish them, but in my opinion the distinction is useful, and even necessary, if we want to clarify the history of sets. During the second half of the nineteenth century, the interaction between the three aspects was quite complex.Along these lines, a discussion of the work of Bernhard RIEMANN (1826-66) and Richard DEDEKIND (1831DEDEKIND ( -1916 forms the core of this paper. In my view, both mathematicians offer the most clear examples of the use of set language in mathematics, and the elaboration of foundational views based on the notion of set, prior to the emergence of CANTOR's work. This is why they should figure prominently in a general account of the emergence of sets. The triad RIEMANN/DEDEKIND/CANTOR constitutes a quite closed net offering basic clues for a satisfactory explanation of the ascent of sets. (In a detailed description, subsidiary authors, like GAUSS, DIRICHLET, WEIERSTRASS, and HANKEL, would also be of great importance.) RIEMANN, DEDEKIND, and CANTOR give the main examples of set-theoretical and set-1 For early and authoritative instances, cf. the dedication of HAUSDORFF 1914, ZERMELO's words in the preface to CANTOR 1932, and several passages of HILBERT 1930. But see ZERMELO's crucial paper on the axiomatization of set theory, where he said that it had been "created by CANTOR and DEDEKIND" (ZERMELO 1908a, 200).2 Around 1870, the language of sets was being applied in geometry, elementary arithmetic, algebra, and even complex function theory. During the last two decades, much has been done to consider these developments, at least for the cases of algebra and real analysis. For the emergence of point-sets within real analysis cf. HAWKINS 1975, DAUBEN 1979, COOKE 1992. DEDEKIND's structural view of algebra is studied in DUGAC 1976and 1981, EDWARDS 1980, SCHARLAU 1981, EDWARDS, NEUMANN & PURKERT 1982. A wealth of material on both topics can be found in MOORE 1982, and an attempt to synthesize the foregoing studies in FERREIRÓS 1993a.Viewed in the light of such prev...

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The problem of the invariance of dimension in the growth of modern topology, part II

Cited by 46 publications

References 101 publications

Algebras, Projective Geometry, Mathematical Logic, and Constructing the World: Intersections in the Philosophy of Mathematics of A. N. Whitehead

Algebras, Projective Geometry, Mathematical Logic, and Constructing the World: Intersections in the Philosophy of Mathematics of A. N. Whitehead

Book Review: Mystic, geometer, and intuitionist. The life of L. E. J. Brouwer. Volume 1: The dawning revolution

Traditional logic and the early history of sets, 1854?1908

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