by Corry Shores
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[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”
1.5
“Brouwer’s New Theory of Sets and the Continuum”
1.5.1
“The Brouwer Negation”
Brief summary:
(1.5.1.1) When we are constructing fundamental mathematical entities, as for instance the natural numbers, “there is no place nor immediate need for negation” (14). We only require negation when needing to determine elementhood for entities in some species S. “Such attempt may lead to ‘successful fitting in’; that is, a particular mathematical entity is established as an element of S. The alternatives to ‘successful fitting in’ are: (1) the constructed impossibility or ‘absurdity’ of fitting in; and (2) the simple absence of the construction of elementhood or of its absurdity. Only negation in the first sense, of constructed impossibility, meets Brouwer’s strict requirements and can claim to be an act of mathematical construction” (14). (1.5.1.2) By interpreting negation as such, it “immediately calls into question the use of double negation and the logical principle of the excluded ‘third’ or middle” (14). (1.5.1.3) To define Brouwer negation, we need to determine what constitutes “absurdity” or “constructed impossibility” (15). Brouwer’s definitions for negation remain vague, and often important concepts they employ, like “impossibility,” “incompatibility,” “difference,” and “contradiction”, appear to be defined circularly. For instance, he defines “contradiction” or “the impossibility of fitting in” (see section 1.5.1.1 above) in the following way: “I just observe that the construction does not go further [Dutch: gaat niet, that is, it does not work], that in the main edifice there is no room to be found for the posited structure” (15). (In other words, perhaps: something is a contradiction if it does not fit in, and the criterion for determining this is that it does not fit in. So we still need a more precise account for not fitting in, which is what we in fact sought in the first place.) And “impossibility” is defined as an “incompatibility;” but “incompatibility – latent and inherent in the structures concerned – is not sufficient by itself; he insists that negation is ‘a construction of incompatibility’ (B1954A, p. 3) or ‘the construction of the hitting upon the impossibility of the fitting in’ (B1908C, p. 3)” (15). (1.5.1.4) Post-Brouwer Intuitionism identifies the proof of the “absurdity of” or the incompatibility of two complex systems as a “reduction to a simple contradiction such as 1 = 0 or the logical p & ¬p” (15). But, even these sorts of contradictions and in fact all descriptions of “absurdity” “make use of some notion of negation or difference. Their absurdity can ultimately only be justified by some intuitive, primitive relation of distinctness, an element of the Primordial Intuition, the fundamental recognition of the Subject of distinct moments in time” (15). (So perhaps: negation is grounded in the intuition that one moment in time is not some other; but negation cannot be given a formal definition.) (1.5.1.5) There is a weaker form of negation which is not the proof of the absurdity of a formulation but rather is the fact that neither a proof affirming it nor a proof negating it has currently been found. It is recognized that in the future a proof could be found, so its negative status is not certain.
Brouwer also uses other, weaker forms of negation, in particular, where he moves outside the domain of mathematics proper into the realm of “mathematical language” and mathematical “assertions,” where, for example, he speaks of “unproven hypotheses,” “the case that α has neither been proved to be true nor to be absurd.” Negation in this case expresses the simple absence of proof, which in the world of mathematics as construction in time may well be reversed: Unsolved problems may one day become proven truth or absurdity. Moreover, “a mathematical entity is not necessarily predeterminate, and may, in its state of free growth, at some time acquire a property it did not possess before” (BMS59, p. 1), leading to further distinctions, in particular, between “cannot now” and “cannot now and ever,” the latter term frequently used by Brouwer in his later work as an alternative description of “absurdity.”
(15, boldface and underlining are mine)
[Negation for Determining Elementhood]
[The Restriction on Double Negation and Excluded Middle]
[Brouwer Negation as Insufficiently Defined]
[Post-Brouwer Intuitionism’s Definition of Negation. Negation’s Ultimate Grounding in the Temporal Intuition of the Distinctness of Different Moments in Time]
[Weaker Negation: Having Neither an Affirmative nor a Negational Proof]
Summary
[Negation for Determining Elementhood]
[When we are constructing fundamental mathematical entities, as for instance the natural numbers, “there is no place nor immediate need for negation” (14). We only require negation when needing to determine elementhood for entities in some species S. “Such attempt may lead to ‘successful fitting in’; that is, a particular mathematical entity is established as an element of S. The alternatives to ‘successful fitting in’ are: (1) the constructed impossibility or ‘absurdity’ of fitting in; and (2) the simple absence of the construction of elementhood or of its absurdity. Only negation in the first sense, of constructed impossibility, meets Brouwer’s strict requirements and can claim to be an act of mathematical construction” (14).]
[ditto]
In the generation of the fundamental “mathematical entities,” such as the natural numbers and the Brouwer set or spread and its elements, there is no place nor immediate need for negation. The question of negation only arises at the level of species construction, at the point where the Subject is attempting to establish elementhood of a species S over a given domain of existing mathematical entities. Such attempt may lead to “successful fitting in”; that is, a particular mathematical entity is established as an element of S. The alternatives to “successful fitting in” are: (1) the constructed impossibility or “absurdity” of fitting in; and (2) the simple absence of the construction of elementhood or of its absurdity. Only negation in the first sense, of constructed impossibility, meets Brouwer’s strict requirements and can claim to be an act of mathematical construction.
(14)
[The Restriction on Double Negation and Excluded Middle]
[By interpreting negation as such, it “immediately calls into question the use of double negation and the logical principle of the excluded ‘third’ or middle” (14).]
[ditto]
The implications of such strict interpretation of negation are far reaching. It immediately calls into question the use of double negation and the logical principle of the excluded “third” or middle. In Brouwer’s own reconstruction of set theory and mathematics the use of the Principle of the Excluded Middle is carefully and expressly avoided. In the foundational “debate,” however, it becomes the most contentious issue, the clearest manifestation of the fundamental differences between the | two opposing philosophies of mathematics (see further the introductions to Parts III and IV).
(14-15)
[Brouwer Negation as Insufficiently Defined]
[To define Brouwer negation, we need to determine what constitutes “absurdity” or “constructed impossibility” (15). Brouwer’s definitions for negation remain vague, and often important concepts they employ, like “impossibility,” “incompatibility,” “difference,” and “contradiction”, appear to be defined circularly. For instance, he defines “contradiction” or “the impossibility of fitting in” (see section 1.5.1.1 above) in the following way: “I just observe that the construction does not go further [Dutch: gaat niet, that is, it does not work], that in the main edifice there is no room to be found for the posited structure” (15). (In other words, perhaps: something is a contradiction if it does not fit in, and the criterion for determining this is that it does not fit in. So we still need a more precise account for not fitting in, which is what we in fact sought in the first place.) And “impossibility” is defined as an “incompatibility;” but “incompatibility – latent and inherent in the structures concerned – is not sufficient by itself; he insists that negation is ‘a construction of incompatibility’ (B1954A, p. 3) or ‘the construction of the hitting upon the impossibility of the fitting in’ (B1908C, p. 3)” (15).]
[ditto]
As to the Brouwer negation, the question still remains as to what constitutes “absurdity” or “constructed impossibility.” His definitions remain somewhat vague, and in their use of terms such as “impossibility,” “incompatibility,” “difference,” and “contradiction” they seem to be circular. “Contradiction” or “the impossibility of fitting in” is first described in B1907: “I just observe that the construction does not go further [Dutch: gaat niet, that is, it does not work], that in the main edifice there is no room to be found for the posited structure.” (p. 127). The impossibility is due to some “incompatibility,” a term Brouwer uses in his later work. But incompatibility – latent and inherent in the structures concerned – is not sufficient by itself; he insists that negation is “a construction of incompatibility” (B1954A, p. 3) or “the construction of the hitting upon the impossibility of the fitting in” (B1908C, p. 3).
(15)
[Post-Brouwer Intuitionism’s Definition of Negation. Negation’s Ultimate Grounding in the Temporal Intuition of the Distinctness of Different Moments in Time]
[Post-Brouwer Intuitionism identifies the proof of the “absurdity of” or the incompatibility of two complex systems as a “reduction to a simple contradiction such as 1 = 0 or the logical p & ¬p” (15). But, even these sorts of contradictions and in fact all descriptions of “absurdity” “make use of some notion of negation or difference. Their absurdity can ultimately only be justified by some intuitive, primitive relation of distinctness, an element of the Primordial Intuition, the fundamental recognition of the Subject of distinct moments in time” (15). (So perhaps: negation is grounded in the intuition that one moment in time is not some other; but negation cannot be given a formal definition.)]
[ditto]
Proof of the “absurdity of” or the incompatibility of two complex systems is identified, in particular, in the post-Brouwer Intuitionist tradition, with the reduction to a simple contradiction such as 1 = 0 or the logical p & ¬p. But these contradictions, as indeed all descriptions of “absurdity,” make use of some notion of negation or difference. Their absurdity can ultimately only be justified by some intuitive, primitive relation of distinctness, an element of the Primordial Intuition, the fundamental recognition of the Subject of distinct moments in time.
(15)
[Weaker Negation: Having Neither an Affirmative nor a Negational Proof]
[There is a weaker form of negation which is not the proof of the absurdity of a formulation but rather is the fact that neither a proof affirming it nor a proof negating it has currently been found. It is recognized that in the future a proof could be found, so its negative status is not certain.]
[ditto]
Brouwer also uses other, weaker forms of negation, in particular, where he moves outside the domain of mathematics proper into the realm of “mathematical language” and mathematical “assertions,” where, for example, he speaks of “unproven hypotheses,” “the case that α has neither been proved to be true nor to be absurd.” Negation in this case expresses the simple absence of proof, which in the world of mathematics as construction in time may well be reversed: Unsolved problems may one day become proven truth or absurdity. Moreover, “a mathematical entity is not necessarily predeterminate, and may, in its state of free growth, at some time acquire a property it did not possess before” (BMS59, p. 1), leading to further distinctions, in particular, between “cannot now” and “cannot now and ever,” the latter term frequently used by Brouwer in his later work as an alternative description of “absurdity.”
(15, boldface and underlining are mine)
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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