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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”
1.4.1
“Logic”
Brief summary:
(1.4.1.1) Brouwer has a notion of “theoretical logic.” It is an application of mathematics in which one uses a “mathematical viewing” of a given mathematical record to see some regularity in that symbolic representation. (1.4.1.2) What we consider classical laws or principles of logic are regularities that we discern secondarily after genuine (intuitive) mathematical constructions are created. (1.4.1.3) But the operation of mathematics in the verbal or symbolic domain in fact is operating outside mathematical reality. And logic itself cannot generate new mathematical truths. In fact, we cannot even apply all the principles of logic to mathematics. Brouwer’s famous example is the invalidity of the Principle of the Excluded Middle (PEM) in mathematical applications. He identifies the Principle of the Excluded Middle with the principle of the solvability of every mathematical problem. Brower takes a strict interpretation of affirmation and negation: “True mathematical statements, affirmative and negative, express the completion of a constructive proof; in particular the negative statement expresses what Brouwer calls ‘absurdity,’ the constructed incompatibility of two mathematical constructions represented, respectively, by the subject and predicate of the sentence” (9-10). But for infinite systems there is no absolute guarantee that a complete constructed affirmative proof or completed absurd construction can be formulated for them. (Thus, it can be that for such an infinite system, neither an affirmative nor a negative construction can be given for it. Now, the Principle of the Excluded Middle says that for any formula, it is either affirmative or negative, or understood another way, it is either true or false. But infinite systems are neither affirmable nor negatable.) Hence, “the Logical Principle of the Excluded Middle is not a reliable principle” (10).
[Theoretical Logic]
[The Secondariness of Principles or Laws of Logic to Mathematical Constructions]
[The Unreliability of the Principle of the Excluded Middle in Infinite Mathematical Systems]
Summary
[Theoretical Logic]
[Brouwer has a notion of “theoretical logic.” It is an application of mathematics in which one uses a “mathematical viewing” of a given mathematical record to see some regularity in that symbolic representation.]
[ditto]
The Foundations (B1907) defines “theoretical logic” as an application of mathematics, the result of the “mathematical viewing” of a given mathematical record, seeing a certain regularity in the symbolic representation: “People who want to view everything mathematically have done this also with the language of mathematics ... the resulting science is theoretical logic ... an empirical science and an application of mathematics ... to be classed under ethnography rather than psychology” (p. 129).
(9)
[The Secondariness of Principles or Laws of Logic to Mathematical Constructions]
[What we consider classical laws or principles of logic are regularities that we discern secondarily after genuine (intuitive) mathematical constructions are created.]
[ditto]
The classical laws or principles of logic are part of this observed regularity; they are derived from the post factum record of mathematical constructions. To interpret an instance of “law like behavior” in a genuine mathematical account as an application of logic or logical principles is “like considering the human body to be an application of the science of anatomy” (p. 130).
(9)
[The Unreliability of the Principle of the Excluded Middle in Infinite Mathematical Systems]
[But the operation of mathematics in the verbal or symbolic domain in fact is operating outside mathematical reality. And logic itself cannot generate new mathematical truths. In fact, we cannot even apply all the principles of logic to mathematics. Brouwer’s famous example is the invalidity of the Principle of the Excluded Middle (PEM) in mathematical applications. He identifies the Principle of the Excluded Middle with the principle of the solvability of every mathematical problem. Brower takes a strict interpretation of affirmation and negation: “True mathematical statements, affirmative and negative, express the completion of a constructive proof; in particular the negative statement expresses what Brouwer calls ‘absurdity,’ the constructed incompatibility of two mathematical constructions represented, respectively, by the subject and predicate of the sentence” (9-10). But for infinite systems there is no absolute guarantee that a complete constructed affirmative proof or completed absurd construction can be formulated for them. (Thus, it can be that for such an infinite system, neither an affirmative nor a negative construction can be given for it. Now, the Principle of the Excluded Middle says that for any formula, it is either affirmative or negative, or understood another way, it is either true or false. But infinite systems are neither affirmable nor negatable.) Hence, “the Logical Principle of the Excluded Middle is not a reliable principle” (10).]
[ditto]
The cunning application of such principles in the verbal or symbolic domain produces nothing but “verbal edifices” outside the mathematical reality:
Linguistic edifices, sequences of sentences which follow one another according to the laws of logic .... Even if it appears that these edifices can never show up the linguistic figure of a contradiction, they are only mathematics as linguistic constructions and have nothing to do with mathematics, which is outside this edifice. (B1907, p. 132)
Brouwer reiterates Descartes’ observation that logic cannot generate new mathematical truths. In his “Unreliability of the Principles of Logic” (B1908C) he goes one step further and questions the validity of the principles of logic when applied to mathematics. To prove his general point he singles out the Principle of the Excluded Middle (PEM) – identified with the principle of the solvability of every mathematical problem – as flawed and an obvious misstatement of fact. The argument here is based on the lack of guarantee of a solution for an infinite system and on his strict interpretation of affirmation and negation. True mathematical statements, affirmative and negative, express the completion of a constructive proof; in partic- | ular the negative statement expresses what Brouwer calls “absurdity,” the constructed incompatibility of two mathematical constructions represented, respectively, by the subject and predicate of the sentence. In B1908C Brouwer simply states that for an infinite system there is no guarantee that such a construction can be completed and “that for infinite systems the Principle of the Excluded Middle is not a reliable principle” (p. 157). His major campaign against the use of the PEM starts after the publication of “The Foundations of Set Theory Independent of the Logical Principle of the Excluded Middle” (B1918B and B1919A), when in a number of papers he challenges the proofs of certain classical theorems, analyzes the logic of negation, and tries to prove the invalidity of the PEM by means of his counterexamples of essentially unsolvable mathematical problems (see further Section 4.1).
(9-10)
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.
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