Meaning in Classical Mathematics: Is it at Odds with Intuitionism?

• Published in 2011
• Added on 2017-02-27

We examine the classical/intuitionist divide, and how it reflects on modern theories of infinitesimals. When leading intuitionist Heyting announced that "the creation of non-standard analysis is a standard model of important mathematical research", he was fully aware that he was breaking ranks with Brouwer. Was Errett Bishop faithful to either Kronecker or Brouwer? Through a comparative textual analysis of three of Bishop's texts, we analyze the ideological and/or pedagogical nature of his objections to infinitesimals a la Robinson. Bishop's famous "debasement" comment at the 1974 Boston workshop, published as part of his Crisis lecture, in reality was never uttered in front of an audience. We compare the realist and the anti-realist intuitionist narratives, and analyze the views of Dummett, Pourciau, Richman, Shapiro, and Tennant. Variational principles are important physical applications, currently lacking a constructive framework. We examine the case of the Hawking-Penrose singularity theorem, already analyzed by Hellman in the context of the Quine-Putnam indispensability thesis.

Links

• http://arxiv.org/abs/1110.5456v1
• http://arxiv.org/pdf/1110.5456v1

Other information

BibTeX entry

@article{MeaninginClassicalMathematicsIsitatOddswithIntuitionism,
	key = {MeaninginClassicalMathematicsIsitatOddswithIntuitionism},
	type = {article},
	title = {Meaning in Classical Mathematics: Is it at Odds with Intuitionism?},
	author = {Karin Usadi Katz and Mikhail G. Katz},
	abstract = {We examine the classical/intuitionist divide, and how it reflects on modern
theories of infinitesimals. When leading intuitionist Heyting announced that
"the creation of non-standard analysis is a standard model of important
mathematical research", he was fully aware that he was breaking ranks with
Brouwer. Was Errett Bishop faithful to either Kronecker or Brouwer? Through a
comparative textual analysis of three of Bishop's texts, we analyze the
ideological and/or pedagogical nature of his objections to infinitesimals a la
Robinson. Bishop's famous "debasement" comment at the 1974 Boston workshop,
published as part of his Crisis lecture, in reality was never uttered in front
of an audience. We compare the realist and the anti-realist intuitionist
narratives, and analyze the views of Dummett, Pourciau, Richman, Shapiro, and
Tennant. Variational principles are important physical applications, currently
lacking a constructive framework. We examine the case of the Hawking-Penrose
singularity theorem, already analyzed by Hellman in the context of the
Quine-Putnam indispensability thesis.},
	comment = {},
	date_added = {2017-02-27},
	date_published = {2011-12-07},
	urls = {http://arxiv.org/abs/1110.5456v1,http://arxiv.org/pdf/1110.5456v1},
	collections = {History,The act of doing maths},
	url = {http://arxiv.org/abs/1110.5456v1 http://arxiv.org/pdf/1110.5456v1},
	urldate = {2017-02-27},
	archivePrefix = {arXiv},
	eprint = {1110.5456},
	primaryClass = {math.LO},
	year = 2011
}