van Stigt (1.3.0) “Brouwer’s Intuitionist Programme” part 1.3.0, “[Introductory material to] The Nature of Pure Mathematics,” summary

 

by Corry Shores

 

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[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”

 

[1.3.0]

[Introductory material]

 

 

 

 

 

 

Brief summary:

(1.3.0.1) We previously noted (see section 1.2.2.4 and 1.2.2.5) the mathematical nature of causality (for causality is a mental construction resulting from calculating outcomes) and the Primordial Intuition of Time (by which we ascertain Two-ity and thus numerical multiplicity). These are the “central theses of Brouwer’s analysis of science and language” (7). Early on Brouwer takes these two theses as being closely related, and his notion of mathematics is also at this time “somewhat tainted by its association with ‘causal’ or ‘cunning acting’ ” (7). Later Brouwer comes to give pure mathematics “an independent and redeeming role” (7). Part of this line of thinking is Brouwer’s notion of the “Liberation of the Mind,” which, in the mathematical context, “refers to the elimination of all exterior, phenomenal elements and causal influences from the creative mathematical act. It allows the Primordial Intuition as an abstraction of pure time awareness, eliminating also the content of sensations, to be a pure and a priori basis of mathematics and its defining act” (7). The Primordial Intuition of Time {1} is “necessary and sufficient for the creation of two-ity,” {2} it “holds the continuum as ‘its inseparable complement’,” and {3} “contains the fundamental elements and tools from which and with which the whole of mathematics is to be constructed” (7). In fact, “mathematics is identified with the whole of the constructive thought-process on and with the elements of the Primordial Intuition alone. Brouwer’s preferred term is ‘building’ (Dutch: bouwen) rather than ‘construction,’ a building upwards from the ground, a time-bound process, beginning at some moment in the past, existing in the present, and having an open future ahead” (7).

 

 

 

 

 

Contents

 

1.3.0.1

[The Primordial Intuition of Time as Constructive of Mathematics]

 

 

 

 

 

Summary

 

1.3.0.1

[The Primordial Intuition of Time as Constructive of Mathematics]

 

[We previously noted (see section 1.2.2.4 and 1.2.2.5) the mathematical nature of causality (for causality is a mental construction resulting from calculating outcomes) and the Primordial Intuition of Time (by which we ascertain Two-ity and thus numerical multiplicity). These are the “central theses of Brouwer’s analysis of science and language” (7). Early on Brouwer takes these two theses as being closely related, and his notion of mathematics is also at this time “somewhat tainted by its association with ‘causal’ or ‘cunning acting’ ” (7). Later Brouwer comes to give pure mathematics “an independent and redeeming role” (7). Part of this line of thinking is Brouwer’s notion of the “Liberation of the Mind,” which, in the mathematical context, “refers to the elimination of all exterior, phenomenal elements and causal influences from the creative mathematical act. It allows the Primordial Intuition as an abstraction of pure time awareness, eliminating also the content of sensations, to be a pure and a priori basis of mathematics and its defining act” (7). The Primordial Intuition of Time {1} is “necessary and sufficient for the creation of two-ity,” {2} it “holds the continuum as ‘its inseparable complement’,” and {3} “contains the fundamental elements and tools from which and with which the whole of mathematics is to be constructed” (7). In fact, “mathematics is identified with the whole of the constructive thought-process on and with the elements of the Primordial Intuition alone. Brouwer’s preferred term is ‘building’ (Dutch: bouwen) rather than ‘construction,’ a building upwards from the ground, a time-bound process, beginning at some moment in the past, existing in the present, and having an open future ahead” (7).]

 

[ditto]

The mathematical nature of causality and the Primordial Intuition of Time as the fundamental creative act of mathematics are the central theses of Brouwer’s analsis of science and language. In Chapter 2 of The Foundations of Mathematics they are treated as closely related, and the “mathematical” appears somewhat tainted by its association with “causal” or “cunning acting.” There are, however, signs that as early as 1907 Brouwer had established an independent and redeeming role for pure mathematics. His Foundations ends with a summary that starts: “Mathematics is a free creation, independent of experience; it develops from one single a priori Primordial Intuition ...” (B1907, p. 179). In the original plan of the thesis, moreover, there is an additional chapter entitled “The Philosophical Significance of Mathematics,” in his Preparatory Notes referred to as “Mathematics and the Liberation of Mind.” The “Liberation of Mind” is a favorite theme of Life, Art and Mysticism. In the mathematical context it refers to the elimination of all exterior, phenomenal elements and causal influences from the creative mathematical act. It allows the Primordial Intuition as an abstraction of pure time awareness, eliminating also the content of sensations, to be a pure and a priori basis of mathematics and its defining act. The Primordial Intuition is not only necessary and sufficient for the creation of two-ity; it also holds the continuum as “its inseparable complement” and contains the fundamental elements and tools from which and with which the whole of mathematics is to be constructed. Indeed, mathematics is identified with the whole of the constructive thought-process on and with the elements of the Primordial Intuition alone. Brouwer’s preferred term is “building” (Dutch: bouwen) rather than “construction,” a building upwards from the ground, a time-bound process, beginning at some moment in the past, existing in the present, and having an open future ahead. Indeed, mathematics is the life of what Brouwer calls “the Subject,” “the Creating Subject,” or the “Idealized Mathematician.” Its characteristics are determined by the time-bound and individual nature of mind as the sole creator and seat of mathematical thought and by the limits of Intuition.

(7)

[contents]

 

 

 

 

 

 

 

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.