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”
1.3
“The Nature of Pure Mathematics”
[1.3.1]
“Mathematical Existence and Truth”
Brief summary:
(1.3.1.1) Mathematical existence means “having been constructed” and “remaining alive in the mind or memory” (8). Thus “The whole of the Subject’s constructive thought-activity, past and present, constitutes mathematical reality and mathematical truth: ‘Truth is only in reality, i.e. in the present and past experiences of consciousness’ ” (8). For Brouwer, we should identify mathematical entities “with the whole of their constructive pedigree, whether they be single concepts, such as, for example, an ordinal number, or more complex, such as a mathematical theorem, which combines various constructions” (8). (1.3.1.2) A (mathematical) construction consists of a sequence of constructive steps. Past, completed constructions would thus be finite sequences. But certain infinite values can have mathematical existence when an algorithm or law by which the number is uniquely determined can completely construct them. In such cases, the free power of the subject allows for the infinite values to be generated indefinitely. We see this sort of procedure for example in the intuition of the continuum, which is not for Brouwer a set of existing points but is rather “abstracted from the time interval, the mathematical ‘between’ that is never exhausted by division into subintervals” (8).
[Mathematical Existence as Construction]
[Mathematical Constructions of Infinity]
Summary
[Mathematical Existence as Construction]
[Mathematical existence means “having been constructed” and “remaining alive in the mind or memory” (8). Thus “The whole of the Subject’s constructive thought-activity, past and present, constitutes mathematical reality and mathematical truth: ‘Truth is only in reality, i.e. in the present and past experiences of consciousness’ ” (8). For Brouwer, we should identify mathematical entities “with the whole of their constructive pedigree, whether they be single concepts, such as, for example, an ordinal number, or more complex, such as a mathematical theorem, which combines various constructions” (8).]
[ditto]
Mathematical existence then in its strictest sense is “having been constructed” and remaining alive in the mind or memory. The whole of the Subject’s constructive thought-activity, past and present, constitutes mathematical reality and mathematical truth: “Truth is only in reality, i.e. in the present and past experiences of consciousness” (B1948C, p. 1243). Mathematical entities are identified with the whole of their constructive pedigree, whether they be single concepts, such as, for example, an ordinal number, or more complex, such as a mathematical theorem, which combines various constructions.
(8)
[Mathematical Constructions of Infinity]
[A (mathematical) construction consists of a sequence of constructive steps. Past, completed constructions would thus be finite sequences. But certain infinite values can have mathematical existence when an algorithm or law by which the number is uniquely determined can completely construct them. In such cases, the free power of the subject allows for the infinite values to be generated indefinitely. We see this sort of procedure for example in the intuition of the continuum, which is not for Brouwer a set of existing points but is rather “abstracted from the time interval, the mathematical ‘between’ that is never exhausted by division into subintervals” (8).]
[ditto]
Past, completed constructions consist of sequences of constructive steps and as such are finite. Mathematical existence can be claimed for “the infinite” within an interpretation that is based on completed constructions and the freedom of the Subject to proceed. In the case of a denumerably infinite sequence such as “the fundamental sequence” of ordinal numbers, the completed construction is the algorithm or “law” by which each element of the sequence is uniquely determined. The “free” power of the live Subject to proceed ensures that the elements be generated “indefinitely.” The essential active role of the Subject in constructing his procedure for determining elements and in the continued generation of these elements allows the possibility of extending the traditional notion of infinite sequence. Brouwer took this step in 1917, when he introduced the “free-choice sequence” and his new set concept as the procedure for generating “points on the continuum” (see further Section 1.5). The established concept of the continuum as a set, the totality of existing points, was rejected outright in The Foundations (B1907) and “On Possible Powers” (B1908 A). The Brouwer notion of the continuum-as-a-whole, “the Intuitive Continuum,” is a primitive concept generated in the Primordial Intuition of time. It is abstracted from the time interval, the mathematical “between” that is never exhausted by division into subintervals.
(8)
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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