Editing Intuitionism (section)

==References==
*"Analysis." ''Encyclopædia Britannica''. 2006. [[Encyclopædia Britannica 2006 Ultimate Reference Suite DVD]] 15 June 2006, "[[Constructive analysis]]" ([[Ian Stewart (mathematician)|Ian Stewart]], author)
*[[W. S. Anglin]], ''[https://books.google.com/books?id=13dKav77vGsC&q=intuitionism Mathematics: A Concise history and Philosophy]'', Springer-Verlag, New York, 1994.
:In ''Chapter 39 Foundations'', with respect to the 20th century Anglin gives very precise, short descriptions of [[Platonism]] (with respect to Godel), [[Formalism (mathematics)|Formalism]] (with respect to Hilbert), and Intuitionism (with respect to Brouwer).

*[[Martin Davis (mathematician)|Martin Davis]] (ed.) (1965), ''The Undecidable'', Raven Press, Hewlett, NY. Compilation of original papers by Gödel, Church, Kleene, Turing, Rosser, and Post.  Republished as {{cite book | editor1-last = Davis | editor1-first = Martin | year=2004 | title = The Undecidable |publisher = Courier Dover Publications | isbn = 978-0-486-43228-1}}
* {{cite book|author = Martin Davis | year = 2000| title = Engines of Logic: Mathematicians and the origin of the Computer| publisher = W. W. Norton & Company, New York| edition = 1st | isbn = 0-393-32229-7| author-link = Martin Davis (mathematician)}}
*[[John W. Dawson]] Jr., ''Logical Dilemmas: The Life and Work of [[Kurt Gödel]]'', A. K. Peters, Wellesley, MA, 1997.
:Less readable than Goldstein but, in ''Chapter III Excursis'', Dawson gives an excellent "A Capsule History of the Development of Logic to 1928".

*[[Rebecca Goldstein]], ''Incompleteness: The Proof and Paradox of Kurt Godel'', Atlas Books, W.W. Norton, New York, 2005.
:In ''Chapter II Hilbert and the Formalists'' Goldstein gives further historical context. As a Platonist [[Gödel]] was reticent in the presence of the [[logical positivism]] of the Vienna Circle. Goldstein discusses [[Wittgenstein]]'s impact and the impact of the formalists. Goldstein notes that the intuitionists were even more opposed to [[Platonism]] than [[Formalism (mathematics)|Formalism]].

* {{in lang|fr}}  [[Jacques Hartong]] and [[Georges Reeb]], [https://web.archive.org/web/20071009030648/http://moire4.u-strasbg.fr/souv/Int84.htm ''Intuitionnisme 84''] (first published in ''La Mathématique Non-standard'', éditions du C.N.R.S.)
: A reevaluation of intuitionism, from the point of view (among others) of [[constructive mathematics]] and [[non-standard analysis]].

*{{cite book
|last=Frege
|first=Gottlob
|author-link=Gottlob Frege
|year=1893
|title=Grundgesetze der Arithmetik
|volume=1
|url=https://archive.org/details/bub_gb_LZ5tAAAAMAAJ
|language=de
|publisher=Hermann Pohle
|location=Jena
|quote=
Partial translation: Montgomery Furth, 1964. The Basic Laws of Arithmetic. Univ. of California Press. Translation of selected sections in {{harvtxt|Frege|1960}}. Complete translation of both volumes: Philip A. Ebert and Marcus Rossberg, 2013, Basic Laws of Arithmetic. Oxford University Press.
}}

*{{cite book
|last=Frege
|first=Gottlob
|author-link=Gottlob Frege
|year=1903
|title=Grundgesetze der Arithmetik
|volume=2
|language=de
|publisher=Hermann Pohle
|location=Jena
|quote=Translation of selected sections in {{harvtxt|Frege|1960}}. Complete translation of both volumes: Philip A. Ebert and Marcus Rossberg, 2013, Basic Laws of Arithmetic. Oxford University Press.
}}

*{{cite book
|last=Frege
|first=Gottlob
|author-link=Gottlob Frege
|year=1960
|orig-year=1893
|chapter=Grundgesetze der Arithmetik: Nachwort
|trans-chapter=Frege on Russell's Paradox
|editor1-last=Geach
|editor1-first=Peter
|editor2-last=Black
|editor2-first=Max
|edition=2
|title=Translations from the Philosophical Writings of Gottlob Frege
|publisher= Basil Blackwell
|location=Oxford
}}

* [[Jean van Heijenoort|van Heijenoort, J.]], ''From Frege to Gödel, A Source Book in Mathematical Logic, 1879–1931'', Harvard University Press, Cambridge, MA, 1967. Reprinted with corrections, 1977. The following papers appear in van Heijenoort:
:* [[L.E.J. Brouwer]], 1923, ''On the significance of the principle of excluded middle in mathematics, especially in function theory'' [reprinted with commentary, p. 334, van Heijenoort]
:* Andrei Nikolaevich [[Kolmogorov]], 1925, ''On the principle of excluded middle'', [reprinted with commentary, p. 414, van Heijenoort]
:* [[L.E.J. Brouwer]], 1927, ''On the domains of definitions of functions'', [reprinted with commentary, p. 446, van Heijenoort]
::Although not directly germane, in his (1923) Brouwer uses certain words defined in this paper.
:* [[L.E.J. Brouwer]], 1927(2), ''Intuitionistic reflections on formalism'', [reprinted with commentary, p. 490, van Heijenoort]
:* Jacques Herbrand, (1931b), "On the consistency of arithmetic", [reprinted with commentary, p. 618ff, van Heijenoort]
:: From van Heijenoort's commentary it is unclear whether or not Herbrand was a true "intuitionist"; Gödel (1963) asserted that indeed "...Herbrand was an intuitionist". But van Heijenoort says Herbrand's conception was "on the whole much closer to that of Hilbert's word 'finitary' ('finit') that to "intuitionistic" as applied to Brouwer's doctrine".

* {{cite book | first=Dennis E. | last=Hesseling | title=Gnomes in the Fog. The Reception of Brouwer's Intuitionism in the 1920s | publisher=Birkhäuser | year=2003 | isbn=3-7643-6536-6}}
* [[Arend Heyting]]: {{cite book | first=Arend | last=Heyting | title=Intuitionism: An Introduction | publisher=North-Holland Pub. Co | location=Amsterdam | orig-year=1956 | year=1971 | edition=3d rev. | isbn=0-7204-2239-6}}
* {{cite book | last=Kleene | first=Stephen C. | author-link=Kleene | year=1991 | orig-year=1952 | title=Introduction to metamathematics| publisher=North-Holland Pub. Co | location=Amsterdam NY| edition=Tenth impression 1991 | isbn=0-7204-2103-9}}
:In Chapter III ''A Critique of Mathematic Reasoning, §11. The paradoxes'', Kleene discusses Intuitionism and [[Formalism (mathematics)|Formalism]] in depth. Throughout the rest of the book he treats, and compares, both Formalist (classical) and Intuitionist logics with an emphasis on the former.

* [[Stephen Cole Kleene]] and [[Richard Eugene Vesley]], ''The Foundations of Intuitionistic Mathematics'', North-Holland Publishing Co. Amsterdam, 1965. The lead sentence tells it all "The constructive tendency in mathematics...". A text for specialists, but written in Kleene's wonderfully-clear style.
*{{cite book
| last=Lakatos
| first=Imre
| author-link=Imre Lakatos
| year=2015
| orig-year=1976
| title=Proofs and Refutations The Logic of Mathematical Discovery
| series=Cambridge Philosophy Classics
| isbn=978-1-107-11346-6
| publisher=Cambridge University Press}}

* [[A. A. Markov]] (1954) ''Theory of algorithms''. [Translated by Jacques J. Schorr-Kon and PST staff] Imprint Moscow, Academy of Sciences of the USSR, 1954 [i.e. Jerusalem, Israel Program for Scientific Translations, 1961; available from the Office of Technical Services, U.S. Dept. of Commerce, Washington] Description 444 p.&nbsp;28&nbsp;cm. Added t.p. in Russian Translation of Works of the Mathematical Institute, Academy of Sciences of the USSR, v. 42. Original title: Teoriya algorifmov. [QA248.M2943 Dartmouth College library. U.S. Dept. of Commerce, Office of Technical Services, number OTS 60–51085.] A secondary reference for specialists: Markov opined that "The entire significance for mathematics of rendering more precise the concept of algorithm emerges, however, in connection with the problem of ''a constructive foundation for mathematics''....[p. 3, italics added.] Markov believed that further applications of his work "merit a special book, which the author hopes to write in the future" (p.&nbsp;3). Sadly, said work apparently never appeared.
* [[Hilary Putnam]] and [[Paul Benacerraf]], ''Philosophy of Mathematics: Selected Readings'', Englewood Cliffs, N.J.: Prentice-Hall, 1964. 2nd ed., Cambridge: Cambridge University Press, 1983. {{ISBN|0-521-29648-X}}
: Part I. ''The foundation of mathematics'', ''Symposium on the foundations of mathematics''
:* [[Rudolf Carnap]], ''The logicist foundations of mathematics'', p. 41
:* [[Arend Heyting]], ''The intuitionist foundations of mathematics'', p. 52
:* [[John von Neumann|Johann von Neumann]], ''The formalist foundations of mathematics'', p. 61
:* Arend Heyting, ''Disputation'', p. 66
:* L. E. J. Brouwer, ''Intuitionnism and formalism'', p. 77
:* L. E. J. Brouwer, ''Consciousness, philosophy, and mathematics'', p. 90

* [[Constance Reid]], ''Hilbert'', Copernicus – Springer-Verlag, 1st edition 1970, 2nd edition 1996.
: Definitive biography of Hilbert places his "Program" in historical context together with the subsequent fighting, sometimes rancorous, between the Intuitionists and the Formalists.

* [[Paul Rosenbloom]], ''The Elements of Mathematical Logic'', Dover Publications Inc, Mineola, New York, 1950.
: In a style more of Principia Mathematica – many symbols, some antique, some from German script. Very good discussions of intuitionism in the following locations: pages 51–58 in Section 4 Many Valued Logics, Modal Logics, Intuitionism; pages 69–73 Chapter III The Logic of Propostional Functions Section 1 Informal Introduction; and p. 146-151 Section 7 the Axiom of Choice.

* {{cite conference
|last= Turing
|first= Alan M.
|author-link = Alan Turing
|year = 1939
|chapter = Systems of Logic Based on Ordinals
|title = Proceedings of the London Mathematical Society
|chapter-url= https://pure.mpg.de/rest/items/item_2403325_2/component/file_2403324/content
|series= 2
|volume= 45
|pages= 161–228
|access-date=17 January 2024
}}

*{{cite web
|last=Veldman
|first=Wim
|date=February 2021
|title=Intuitionism: An Inspiration?
|url=https://www.researchgate.net/publication/348973120
|doi=10.13140/RG.2.2.12313.54881
}}