## Intuitionism as generalization (1990)

Venue: | Philosophia Math |

Citations: | 8 - 1 self |

### BibTeX

@ARTICLE{Richman90intuitionismas,

author = {Fred Richman},

title = {Intuitionism as generalization},

journal = {Philosophia Math},

year = {1990},

volume = {5},

pages = {124--128}

}

### OpenURL

### Abstract

Iwas inspired, not to say provoked, to write this note by Michel J. Blais's article A pragmatic analysis of mathematical realism and intuitionism [2]. Having spent the greater part of my career doing intuitionistic mathematics, while continuing to do classical mathematics, I have come to feel that most comparisons of these two approaches to mathematics miss the essential point: intuitionism, in its simplest form, is a generalization of classical mathematics that accomodates both classical and computational models. By intuitionism I mean the approach to mathematics based on intuitionistic logic, awell-de ned body of axioms and rules of inference [6] [3]. So, for example, my idea of intuitionism does not include the notion of a choice sequence [8], or the various continuity principles associated with intuitionism [9], and it does not refer to the more bizarre consequences that have been drawn from Brouwer's idea of a creating subject [3]. This lean version of intuitionistic mathematics is usually called constructive mathematics. Blais directs his comments in [2] at constructive mathematics rather than at the more esoteric varieties of intuitionistic mathematics.

### Citations

60 |
Foundations of Constructive Mathematics
- Bishop
- 1967
(Show Context)
Citation Context ...cs that accomodates both classical and computational models. By intuitionism I mean the approach to mathematics based on intuitionistic logic, a well-defined body of axioms and rules of inference [6] =-=[3]-=-. So, for example, my idea of intuitionism does not include the notion of a choice sequence [8], or the various continuity principles associated with intuitionism [9], and it does not refer to the mor... |

14 |
Choice sequences. A chapter of intuitionistic mathematics
- Troelstra
- 1977
(Show Context)
Citation Context ...h to mathematics based on intuitionistic logic, a well-defined body of axioms and rules of inference [6] [3]. So, for example, my idea of intuitionism does not include the notion of a choice sequence =-=[8]-=-, or the various continuity principles associated with intuitionism [9], and it does not refer to the more bizarre consequences that have been drawn from Brouwer's idea of a creating subject [3]. This... |

12 |
Choice implies excluded middle
- Goodman, Myhill
- 1978
(Show Context)
Citation Context ...ms to conflict with the fact that any realist proof of a statement A, not using the axiom of choice, is an intuitionistic proof of LEM ! A; and, as AC ! LEM is a theorem of intuitionistic mathematics =-=[5]-=-, any realist proof of A is an intuitionistic proof of AC ! A. It is a common misconception that intuitionistic mathematics deals with a special class of mathematical objects that are, in some sense, ... |

1 |
A pragmatic analysis of mathematical realism and intuitionism
- Blais
(Show Context)
Citation Context ...niversity Philosophia Mathematica, 5(1990), 124--128 I was inspired, not to say provoked, to write this note by Michel J. Blais's article A pragmatic analysis of mathematical realism and intuitionism =-=[2]-=-. Having spent the greater part of my career doing intuitionistic mathematics, while continuing to do classical mathematics, I have come to feel that most comparisons of these two approaches to mathem... |

1 | Cambridge lectures on intuitionism, edited by D - Brouwer - 1981 |

1 |
A course in constructive algebra, Springer-Verlag
- Mines, Richman, et al.
- 1988
(Show Context)
Citation Context ...t's proved, which sounds much like a realist's view also. It is important to be aware, as background, that a substantial body of mathematics has recently been developed along intuitionistic lines [1] =-=[7]-=-. The scope and depth of this work gives the lie to the misconception of mathematics outside of the realist framework as some sort of crippled stepchild. Any analysis of realism versus intuitionism in... |

1 |
of constructive mathematics, Mathematical logic, edited by Jon Barwise, North-Holland
- Aspects
- 1977
(Show Context)
Citation Context ...axioms and rules of inference [6] [3]. So, for example, my idea of intuitionism does not include the notion of a choice sequence [8], or the various continuity principles associated with intuitionism =-=[9]-=-, and it does not refer to the more bizarre consequences that have been drawn from Brouwer's idea of a creating subject [3]. This lean version of intuitionistic mathematics is usually called construct... |

1 | Cambridge lectures on intuitionism, edited by D.van Dalen - Brouwer - 1981 |