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Validating Brouwer's continuity principle for numbers using named exceptions

2017
*
Mathematical Structures in Computer Science
*

*Using*these new features, we prove a version of

*Brouwer's*

*continuity*

*principle*

*for*

*numbers*. We also provide a simpler proof of a weaker version of this

*principle*that only

*uses*diverging terms. ...

*Using*

*continuity*and the fan theorem, we prove important results of Intuitionistic Mathematics:

*Brouwer's*

*continuity*theorem, bar induction on monotone bars and the negation of the law of excluded middle ... Constable, Rich Eaton, Evan Moran, and Ross Tate

*for*their helpful criticism. ...

##
###
Book Review: The foundations of intuitionistic mathematics

1965
*
Bulletin of the American Mathematical Society
*

*Brouwer's*

*principle*and the Bar Theorem are

*used*to show that every real-valued function on the interval [0, l] is uniformly

*continuous*, a famous result of Brouwer. ...

*Brouwer's*

*principle*roughly stated says that if ƒ is any function from the set of natural

*number*sequences to the natural

*numbers*then to every natural

*number*sequence {a n }=a corresponds an integer N ...

##
###
Exercising Nuprl's Open-Endedness
[chapter]

2016
*
Lecture Notes in Computer Science
*

versions of the axiom of choice and of

doi:10.1007/978-3-319-42432-3_3
fatcat:exbfyiolojditotrvifbhkprva
*Brouwer's*bar induction*principle*. ... We have recently exercised Nuprl's open-endedness by adding nominal features to Nuprl in order to prove a version of*Brouwer's**continuity**principle*, as well as choice sequences in order to prove truncated ...*For*that, following Longley's method [31] , we*used**named**exceptions*as a probing mechanism to compute the modulus of*continuity*of a function. (3) We have proved the*validity*of versions of*Brouwer's*...##
###
Book Review: L. E. J. Brouwer collected works, Volume I, Philosophy and foundations of mathematics

1977
*
Bulletin of the American Mathematical Society
*

(Quite generally,

doi:10.1090/s0002-9904-1977-14185-2
fatcat:ax5t5iepyncidgj6fk4wf6r4he
*Brouwer's*insistence on r.n.g. is appropriate when*continuity*is paramount: a*continuous*mapping from R into a discrete space is constant, but not*for*r.n.g.) ... (iii) In analysis, one asks about*Brouwer's*fixed point theorem,*for*the uniform convergence topology of mappings ƒ of, say S 2 H» S 2 : IS there a*continuous*J: ƒ H> X E S 2 such that ƒ[?(ƒ)] = ? ... Hilbert wanted to 'justify' the latter by*using*(i) formalizations <5 of*valid**principles**for*then-current mathematical concepts and (ii) proving the consistency of 5"(tacitly, constructively): he was ...##
###
At the Heart of Analysis: Intuitionism and Philosophy

2006
*
Philosophia Scientiæ
*

Acknowledgements This paper is dedicated to Dirk van Dalen in gratitude

doi:10.4000/philosophiascientiae.411
fatcat:bwpcdtglabeczfmqhpty4gwbpq
*for*years of inspiration and encouragement. ...*For*example,*Brouwer's*Theorem itself entails the failure of TND. Were TND generally*valid*, every real*number*would be rational or irrational. ... In consequence, the intuitionist feels herself under no more obligation to accept the*validity*of TND than she is to accept the*validity*of other plainly invalid*principles*,*for*example, p ∨ ¬q. ...##
###
Luitzen Egbertus Jan Brouwer. 1881-1966

1969
*
Biographical Memoirs of Fellows of the Royal Society
*

have a constructiv consistency proof

doi:10.1098/rsbm.1969.0002
fatcat:hzk643xinbazxdjggcg24ykea4
*for*the formal*principles*of analysis, and this would be a minimum requirement*for*a positive answer, since those*principles*are evidently*valid**for*the (non-constructive ...*For*sequences s taking a bounded*number*of values, p is uniformly con tinuous (pointwise*continuity*was assumed from the start). ... BIBLIOGRAPHY Titles of Dutch publications are followed, in brackets, by a free translation in modem terms,*for*example, in (112)*Brouwer's*'punt' is translated by 'point generator' and his 'kern', which ...##
###
Page 26 of Mathematical Reviews Vol. , Issue 83a
[page]

1983
*
Mathematical Reviews
*

Math. 24 (1978), no. 5, 427-436;
MR _ 80b:03096] to obtain a model of intuitionistic set theory
satisfying the Kripke schema,

*Brouwer’s**principle**for**numbers*,*Brouwer’s**principle**for*functions with the ... Another model satisfies Kripke’s schema and*Brouwer’s**principle**for**numbers*without parameters but not bar induction. The author stresses the topos-theoretic point of view. ...##
###
Bar induction: The good, the bad, and the ugly

2017
*
2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
*

We present an extension of the computation system and logic of the Nuprl proof assistant with intuitionistic

doi:10.1109/lics.2017.8005074
dblp:conf/lics/RahliBC17
fatcat:jilht2aeozh6zor7bhcqgx32se
*principles*,*namely*versions of*Brouwer's*bar induction*principle*, which is equivalent to transfinite ... We write B k*for*N N k , where k is a natural*number*and N k is the type of natural*numbers*strictly less than k. We*use*Π and Σ in lieu of the constructive logical quantifiers ∀ and ∃, respectively. ... Acknowledgements We thank David Guaspari and Evan Moran*for*their helpful criticism. ...##
###
Brouwer's Incomplete Objects

2010
*
History and Philosophy of Logic
*

The theory of the idealized mathematician has been developed to formalize a method that is characteristic

doi:10.1080/01445340903445071
fatcat:7zixwklkcrbmrd2st5k4p5jp5q
*for**Brouwer's*papers after 1945. ... We do not*use*an idealized mathematician. We claim that it is the systematic application of incomplete sequences, already introduced by Brouwer in 1918, that makes the method special. ... I thank the organization of the LIO*for*accepting the project. I thank Olivier Roy*for*his translation of the abstract. I am much indebted to ...##
###
Intuition, iteration, induction
[article]

2015
*
arXiv
*
pre-print

In Mathematical Thought and Its Objects, Charles Parsons argues that our knowledge of the iterability of functions on the natural

arXiv:1510.01094v1
fatcat:l7y4skiptzazbce6tmhkfnfb2i
*numbers*and of the*validity*of complete induction is not intuitive knowledge ... But I will try to make two points: (1)*Using*elements from Husserl and from Brouwer,*Brouwer's*claims can be justified in more detail than he has done; (2) There are certain elements in Parsons' discussion ... , Charles Parsons argues that our knowledge that these two general*principles*are*valid*is not intuitive knowledge. ...##
###
The Creating Subject, the Brouwer–Kripke Schema, and infinite proofs

2018
*
Indagationes mathematicae
*

Abstract Kripke's Schema (better the Brouwer-Kripke Schema) and the Kreisel-Troelstra Theory of the Creating Subject were introduced around the same time

doi:10.1016/j.indag.2018.06.005
fatcat:4qgl4bi5trd6leisflx5hoyy4q
*for*the same purpose, that of analysing*Brouwer's*... Then I discuss the original*use*of the Schema and the Theory, their justification from a Brouwerian perspective, and instances of the Schema that can in fact be found in*Brouwer's*own writings. ... WC-N can be strengthened to*Principle*25 (*Continuity**for**Numbers*). ...##
###
Intuitionistic Mathematics and Logic
[article]

2020
*
arXiv
*
pre-print

theories, and sketch his

arXiv:2003.01935v1
fatcat:ipbk2fmjxnd6zasjpcijdcqoxq
*use*of g\"odel*numbers*of recursive functions to realize sentences of intuitionistic arithmetic including a form of Church's Thesis. ...*for*logic and mathematics. ... Implicit in the*use*of weak counterexamples is*Brouwer's**continuity**principle*, to be explained below. The continuum cannot be ordered. ...##
###
The Creating Subject, the Brouwer-Kripke Schema, and infinite proofs
[article]

2018
*
arXiv
*
pre-print

Kripke's Schema (better the Brouwer-Kripke Schema) and the Kreisel-Troelstra Theory of the Creating Subject were introduced around the same time

arXiv:1805.00404v1
fatcat:mihkswwo5bhnjidf4mwdechwni
*for*the same purpose, that of analysing*Brouwer's*'Creating ... Then I discuss the original*use*of the Schema and the Theory, their justification from a Brouwerian perspective, and instances of the Schema that can in fact be found in*Brouwer's*own writings. ... WC-N can be strengthened to*Principle*25 (*Continuity**for**Numbers*). ...##
###
Arguments for the Continuity Principle

2002
*
Bulletin of Symbolic Logic
*

One, the

doi:10.2178/bsl/1182353892
fatcat:7yrrivv62vb3zezvcmsyseumz4
*principle*of bar induction, will not concern*us*here. The other, the*continuity**principle**for**numbers*, occurs*for*the first time in print in [4]. ... Let*us*note first that in one particular case the*principle*is obvious indeed,*namely*in the case of the lawless sequences. ... We wish to thank Andreas Blass, Michael Dummett, Hajime Ishihara, Anne Troelstra, Albert Visser and the referee*for*their comments, which led to several improvements. ...##
###
A nominal exploration of intuitionism

2016
*
Proceedings of the 5th ACM SIGPLAN Conference on Certified Programs and Proofs - CPP 2016
*

*Using*these new features, we prove a version of

*Brouwer's*

*Continuity*

*Principle*

*for*

*numbers*. We also provide a simpler proof of a weaker version of this

*principle*that only

*uses*diverging terms. ...

*Using*

*continuity*and the fan theorem we prove important results of Intuitionistic Mathematics:

*Brouwer's*

*continuity*theorem and bar induction on monotone bars. ... Constable, Ross Tate, Rich Eaton, Abhishek Anand, and Evan Moran

*for*their helpful criticism. ...

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