The Uses and Abuses of the History of Topos Theory

@article{McLarty1990TheUA,
  title={The Uses and Abuses of the History of Topos Theory},
  author={Colin McLarty},
  journal={The British Journal for the Philosophy of Science},
  year={1990},
  volume={41},
  pages={351 - 375}
}
  • C. McLarty
  • Published 1 September 1990
  • Mathematics
  • The British Journal for the Philosophy of Science
The view that toposes originated as generalized set theory is a figment of set theoretically educated common sense. This false history obstructs understanding of category theory and especially of categorical foundations for mathematics. Problems in geometry, topology, and related algebra led to categories and toposes. Elementary toposes arose when Lawvere's interest in the foundations of physics and Tierney's in the foundations of topology led both to study Grothendieck's foundations for… Expand
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References

SHOWING 1-10 OF 74 REFERENCES
Continuously Variable Sets; Algebraic Geometry = Geometric Logic
Publisher Summary The (elementary) theory of topoi is the basis for the study of continuously variable structures as classical set theory is the basis for the study of constant structures. Sheaves onExpand
Adjointness in Foundations
TLDR
This article sums up a stage of the development of the relationship between category theory and proof theory and shows how already in 1967 category theory had made explicit a number of conceptual advances that were entering into the everyday practice of mathematics. Expand
Toposes, Algebraic Geometry and Logic
Sheaf theory and the continuum hypothesis.- Classifying topos.- Deductive systems and categories III. Cartesian closed categories, intuitionist propositional calculus, and combinatory logic.- TheExpand
The Category of Categories as a Foundation for Mathematics
In the mathematical development of recent decades one sees clearly the rise of the conviction that the relevant properties of mathematical objects are those which can be stated in terms of theirExpand
Variable Quantities and Variable Structures in Topoi
Publisher Summary This chapter discusses variable quantities and variable structures in topoi. The chapter presents the conceptual basis for topoi in mathematical experience with variable sets andExpand
The Work of Samuel Eilenberg in Topology
Publisher Summary This chapter focuses on the work of Samuel Eilenberg in topology. Algebraic topology is growing and solving problems, but non-topologists are very skeptical. At Harvard, Tucker gaveExpand
The Consistency Problem for Set Theory: An Essay on the Cantorian Foundations of Mathematics (II)
  • J. Mayberry
  • Mathematics
  • The British Journal for the Philosophy of Science
  • 1977
2 The Problem of Infinity. 2.I The Problem of Infinity in Classical Mathematics. 2.2 Functionals and Continuity in Classical Set Theory. 2.3 A Formulation of the Consistency Problem for Set Theory. 3Expand
Defining sets as sets of points of spaces
  • C. McLarty
  • Mathematics, Computer Science
  • J. Philos. Log.
  • 1988
TLDR
This chapter discusses Cantor's definition of a set as "any collection" and the role of type theories and comprehension axioms in this conception of the continuum. Expand
An elementary theory of the category of topological spaces
An elementary system of axioms was given by F. W. Lawvere for the category of sets and mappings. The purpose of this paper is to provide a finite number of elementary axioms for the category ofExpand
Two Episodes in the Unification of Logic and Topology
  • E. Grosholz
  • The British Journal for the Philosophy of Science
  • 1985
The aims of this essay are twofold. In the first place I would like to exhibit a twentieth-century instance of an important pattern of mathematical reasoning which I have discussed elsewhere in itsExpand
...
1
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4
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