Penelope Maddy
- Published in print:
- 1992
- Published Online:
- November 2003
- ISBN:
- 9780198240358
- eISBN:
- 9780191597978
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/019824035X.003.0004
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
Pursues the theoretical level of the two‐tiered epistemology of set theoretic realism, the level at which more abstract axioms can be justified by their consequences at more intuitive levels. I ...
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Pursues the theoretical level of the two‐tiered epistemology of set theoretic realism, the level at which more abstract axioms can be justified by their consequences at more intuitive levels. I outline the pre‐axiomatic development of set theory out of Cantor's researches, describe how axiomatization arose in the course of Zermelo's efforts to prove Cantor's Well‐ordering Theorem, and review the controversy over the Axiom of Choice. Cantor's Continuum Hypothesis and various questions of descriptive set theory were eventually shown to be independent of the standard axioms, and new axiom candidates—Gödel's Axiom of Constructibility, on the one hand, determinacy and large cardinal axioms, on the other—offer dramatically different solutions. This situation presents a new epistemic challenge to the set theoretic realist: on what grounds can we adjudicate between new axiom candidates?Less
Pursues the theoretical level of the two‐tiered epistemology of set theoretic realism, the level at which more abstract axioms can be justified by their consequences at more intuitive levels. I outline the pre‐axiomatic development of set theory out of Cantor's researches, describe how axiomatization arose in the course of Zermelo's efforts to prove Cantor's Well‐ordering Theorem, and review the controversy over the Axiom of Choice. Cantor's Continuum Hypothesis and various questions of descriptive set theory were eventually shown to be independent of the standard axioms, and new axiom candidates—Gödel's Axiom of Constructibility, on the one hand, determinacy and large cardinal axioms, on the other—offer dramatically different solutions. This situation presents a new epistemic challenge to the set theoretic realist: on what grounds can we adjudicate between new axiom candidates?
Joram Lindenstrauss, David Preiss, and Jaroslav Tier
- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153551
- eISBN:
- 9781400842698
- Item type:
- book
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153551.001.0001
- Subject:
- Mathematics, Analysis
This book makes a significant inroad into the unexpectedly difficult question of existence of Fréchet derivatives of Lipschitz maps of Banach spaces into higher dimensional spaces. Because the ...
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This book makes a significant inroad into the unexpectedly difficult question of existence of Fréchet derivatives of Lipschitz maps of Banach spaces into higher dimensional spaces. Because the question turns out to be closely related to porous sets in Banach spaces, it provides a bridge between descriptive set theory and the classical topic of existence of derivatives of vector-valued Lipschitz functions. The topic is relevant to classical analysis and descriptive set theory on Banach spaces. The book opens several new research directions in this area of geometric nonlinear functional analysis. The new methods developed here include a game approach to perturbational variational principles that is of independent interest. Detailed explanation of the underlying ideas and motivation behind the proofs of the new results on Fréchet differentiability of vector-valued functions should make these arguments accessible to a wider audience. The most important special case of the differentiability results, that Lipschitz mappings from a Hilbert space into the plane have points of Fréchet differentiability, is given its own chapter with a proof that is independent of much of the work done to prove more general results. The book raises several open questions concerning its two main topics.Less
This book makes a significant inroad into the unexpectedly difficult question of existence of Fréchet derivatives of Lipschitz maps of Banach spaces into higher dimensional spaces. Because the question turns out to be closely related to porous sets in Banach spaces, it provides a bridge between descriptive set theory and the classical topic of existence of derivatives of vector-valued Lipschitz functions. The topic is relevant to classical analysis and descriptive set theory on Banach spaces. The book opens several new research directions in this area of geometric nonlinear functional analysis. The new methods developed here include a game approach to perturbational variational principles that is of independent interest. Detailed explanation of the underlying ideas and motivation behind the proofs of the new results on Fréchet differentiability of vector-valued functions should make these arguments accessible to a wider audience. The most important special case of the differentiability results, that Lipschitz mappings from a Hilbert space into the plane have points of Fréchet differentiability, is given its own chapter with a proof that is independent of much of the work done to prove more general results. The book raises several open questions concerning its two main topics.
John P. Burgess and Gideon Rosen
- Published in print:
- 1999
- Published Online:
- November 2003
- ISBN:
- 9780198250128
- eISBN:
- 9780191597138
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198250126.003.0003
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
Develops in some detail a strategy of nominalistic interpretation that assumes that points of spacetime are legitimate, concrete, physical entities. What makes the strategy possible is the fact that ...
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Develops in some detail a strategy of nominalistic interpretation that assumes that points of spacetime are legitimate, concrete, physical entities. What makes the strategy possible is the fact that analytic geometry in the style of Descartes can be interpreted in synthetic geometry in the style of Euclid, using triples of points to represent real numbers (namely, as ratios of pairs of line segments connecting each of a pair of points with a third). This strategy can handle classical theories based on Euclidean space and special‐relativistic theories based on Minkowski space. The extension of the strategy to general relativity and quantum mechanics remains to be worked out, as does the treatment of the higher branch of geometry known as descriptive set theory.Less
Develops in some detail a strategy of nominalistic interpretation that assumes that points of spacetime are legitimate, concrete, physical entities. What makes the strategy possible is the fact that analytic geometry in the style of Descartes can be interpreted in synthetic geometry in the style of Euclid, using triples of points to represent real numbers (namely, as ratios of pairs of line segments connecting each of a pair of points with a third). This strategy can handle classical theories based on Euclidean space and special‐relativistic theories based on Minkowski space. The extension of the strategy to general relativity and quantum mechanics remains to be worked out, as does the treatment of the higher branch of geometry known as descriptive set theory.
