by Corry Shores

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[Mathematics, Calculus, Geometry, Entry Directory]

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[Walter P. van Stigt, entry directory]

[Stigt, “Brouwer’s Intuitionist Programme,” entry directory]

[The following is summary. I am not a mathematician, so please consult the original text instead of trusting my summarizations. Bracketed comments are my own. Proofreading is incomplete, so please forgive my mistakes.]

Summary of

Walter P. van Stigt

“Brouwer’s Intuitionist Programme”

in

*From Brouwer to Hilbert:*

*The Debate on the Foundations of Mathematics in the 1920’s*

Part I.

L.E.J. Brouwer

Ch1.:

“Brouwer’s Intuitionist Programme”

1.4

“Mathematics, Language, and Logic”

[1.4.0]

[Introductory material]

Brief summary:

(1.4.0.1) The objective of Brouwer’s first intuitionist campaign was the freeing of “mathematics from its traditional reliance on language and logic” (8). (1.4.0.2) The subject when generating mathematics in their mind may make no use of any aspect of language. At best, the mathematician may record their constructions in symbols to aid their memory. However, this symbolization cannot be a part of the mathematical process itself. (1.4.0.3) In fact, symbolization cannot be relied upon for communicating mathematical constructions, because {a} we cannot be sure that other subjects, who are mere things in our created exterior world, have minds, and {b} even if other minds do exist, we cannot be sure that commonly shared words will “represent the same thought-construction in the private worlds of different individuals” (9). (1.4.0.4) Thus “Brouwer’s ‘mathematical language’ is the report, the record of a completed mathematical construction; its truth and noncontradictority are due solely to the constructions it represents” (9).

[Mathematics without Language]

[Restrictions Against Symbolization for Mathematical Construction]

[Language as Unable to Communicate Mathematical Constructions]

[Language as Record of Construction]

Summary

[Mathematics without Language]

[The objective of Brouwer’s first intuitionist campaign was the freeing of “mathematics from its traditional reliance on language and logic” (8).]

[*ditto*]

Within the Brouwerian conception of mathematics as pure, individual thought construction on the basis of the Primordial Intuition alone there is clearly no place for language in any form. The emphatic “languageless” in nearly all his definitions of mathematics reflects the need for express denial of the role language plays in almost all alternative philosophies of mathematics, even that of his favorite “preintuitionist,” Poincaré. Freeing mathematics from its traditional reliance on language and logic was the objective of Brouwer’s first Intuitionist campaign, the “First Act of Intuitionism,” in his historical surveys described as “the uncompromising separation of mathematics and mathematical language and thereby of the phenomena described by theoretical logic.”

(8)

[Restrictions Against Symbolization for Mathematical Construction]

[The subject when generating mathematics in their mind may make no use of any aspect of language. At best, the mathematician may record their constructions in symbols to aid their memory. However, this symbolization cannot be a part of the mathematical process itself.]

[*ditto*]

In the genesis of mathematics, wholly confined to the private thought-world of the Subject, no use is made of any aspect of language, either as the carrier of common “objective” concepts or as primitive symbols with no meaning. For the sake of “aiding the memory” the flesh-and-blood mathematician may resort to recording | his constructions in symbols, linking a thought-construction to a name, “an aural or visual thing”; such recording, however, is a posteriori and not part of the mathematical process itself. Moreover, it is essentially private language since both the thought-construction and the assignation to a particular symbol are exclusively acts of the individual mathematician.

(8-9)

[Language as Unable to Communicate Mathematical Constructions]

[In fact, symbolization cannot be relied upon for communicating mathematical constructions, because {a} we cannot be sure that other subjects, who are mere things in our created exterior world, have minds, and {b} even if other minds do exist, we cannot be sure that commonly shared words will “represent the same thought-construction in the private worlds of different individuals” (9).]

[*ditto*]

As to language as a means of communicating mathematics to other individuals, there is no basis for agreement between the constructive thought-processes of different individuals represented in a “common language.” To the Subject other individuals are “things,” creations of his Exterior World, and the existence of other minds similar to his own “mere hypothesis.” And even if the existence of other minds were to be conceded, there is no guarantee that common words would represent the same thought-construction in the private worlds of different individuals.

(9)

[Language as Record of Construction]

[Thus “Brouwer’s ‘mathematical language’ is the report, the record of a completed mathematical construction; its truth and noncontradictority are due solely to the constructions it represents” (9).]

[*ditto*]

Brouwer’s “mathematical language” is the report, the record of a completed mathematical construction; its truth and noncontradictority are due solely to the constructions it represents.

(9)

From:

Stigt, Walter P. van. (1989). “Brouwer’s Intuitionist Programme” In: *From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920’s*, edited by Paolo Mancosu. Oxford: Oxford University.