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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”
1.2
“Intuitionism and Brouwer’s Intuitionist Philosophy of Mathematics”
[1.2.2]
“Brouwer’s Outlook on Life and General Philosophy”
Brief summary:
(1.2.2.1) “Brouwer’s outlook on life and general philosophy can best be described as a blend of romantic pessimism and radical individualism” (5). Brouwer is critical of human industrialization and damage to nature and calls instead for a return to nature “and to mystic and solitary contemplation” (5). (1.2.2.2) In Foundations of Mathematics, Brouwer sees “the application of mathematics in experimental science and logic” as “the source of all evil,” because it superimposes “a mathematical regularity on the physical world” (5). In both Life, Art and Mysticism and Foundations of Mathematics, Brouwer expresses “his conviction of the opposition between mind and matter, the individual consciousness and the exterior world” (5). (1.2.2.3) Brouwer has a notion of personal identity, self, and subject as being a pure spiritual soul. “The life of the Soul is the complex of thought processes in response to its awareness of the world outside” (5). These processes are phases that deteriorate. (1.2.2.4) There is an original preperceptual phase of consciousness that is “stillness.” This is “followed by ‘the naive phase’ of receiving images through physical sensations and reacting spontaneously to them” (5). There is next a momentous event, called “the Primordial Happening” or “the Primordial Intuition of Time,” in which the subject links isolated sensations and becomes aware of time. This on the one hand “brings about a transformation of the Naive Consciousness to the rational ‘Mind’ ” while on the other hand it also “generates the fundamental concepts and tools of mathematics. The Primordial Intuition of Time is the fundamental single act of isolating and linking distinct moments in time, creating mathematical ‘Two-ity’ and the ordinal numbers as well as the continuum.” (6). This thereby gives us the “mathematical power to generate sequences,” which enables us to produce a human-made and mathematical interpretation of nature or the outside world. What we take to be “things,” “including other human beings, are no more than repeated sequences or sequences of sequences, manmade, as is indeed the so-called scientific or ‘causal’ coherence of the world” (6). Now, since these things are somehow fundamentally mathematical sequences that are thought up, “this universe of ‘things’ is wholly private,” and it is called “the Exterior World of the Subject” (6). “The scientific observation of regularity in Nature, linking things and events in time as sequences, is a creative, mathematical process of the individual Mind and is referred to as ‘mathematical viewing’ or ‘causal attention’,” and causality “is an artificial, mind-made structure, not inherent in Nature” (6). In fact, “Brouwer rejects any universal objectivity of things,” and he also rejects the idea that things are bound up by causality. Moreover, Brouwer “denies the existence of a collective or ‘plural’ mind” and instead favors “the essential individuality of thought and mind” (6). (1.2.2.5) The evolutionary movement of consciousness “enters a moral phase when man takes advantage of and acts upon his causal knowledge by setting in motion a causal sequence of events, selecting a first element of the sequence in order to achieve a later element, the desired ‘end’ ” (6). Given that there is an assessment of the causal sequence resulting from one’s actions, this is a “mathematical or causal acting” and it is “calculated” and “cunning”. As such, Brouwer condemns it “as ‘sinful’ and ‘not-beautiful’ ” on account of it being morally evil. (1.2.2.6) The next phase of conscious development, taking us even further away from consciousness’ “deepest home” is the “final phase of ‘social acting’ ” (6). Brouwer describes it as “ ‘the enforcement of will’ in social interaction and organization, in particular by the creation of language” (6). Brouwer thinks that we cannot communicate directly “soul-to-soul”. This creates the need for language, which Brouwer sees as the “ ‘imposition of will through sounds,’ forcing an other human being to act in pursuance of the end desired by the speaker” (6). Then, “As social interaction develops and grows more complex, language becomes more sophisticated, but its essence, as of all instruments, is determined by its purpose: the transmission of will” (7). But, since it is “Used as a means of communicating thought to others, language is bound to remain defective, given the essential privacy of thought and the nature of the ‘sign,’ the arbitrary association of a thought with a sound or visual object” (7). (1.2.2.7) But language is not restricted to the imposition of one’s will upon others. It also is implemented strictly within one’s own inner world as an aid to memory, “helping the Subject to recall his past thought” (7). But Brouwer, when discussing the limitations of real life mathematicians and when noting that even with the help of linguistic signs, memory is still fallible, he “introduces his notion of the ‘Idealized Mathematician’ ” (7).
[Brouwer’s Romanticism]
[Brouwer’s Critical View of the Application of Mathematics in Experimental Science; and the Opposition of Mind and Matter and of Consciousness and the Exterior World]
[The Self or Soul]
[The Conscious Movement to Numerical Plurality: The Intuition of Time and “Two-ity”]
[The Moral Turn in Conscious Evolution]
[Language as the Social Imposition of Will]
[Language as Memory Aid]
Summary
[Brouwer’s Romanticism]
[“Brouwer’s outlook on life and general philosophy can best be described as a blend of romantic pessimism and radical individualism” (5). Brouwer is critical of human industrialization and damage to nature and calls instead for a return to nature “and to mystic and solitary contemplation” (5).]
[ditto]
Brouwer’s outlook on life and general philosophy can best be described as a blend of romantic pessimism and radical individualism. In Life, Art and Mysticism (B1905) he rails against industrial pollution and man’s domination of nature through his intellect and against established social structures, and promotes a return to “Nature” and to mystic and solitary contemplation.
(5)
[Brouwer’s Critical View of the Application of Mathematics in Experimental Science; and the Opposition of Mind and Matter and of Consciousness and the Exterior World]
[In Foundations of Mathematics, Brouwer sees “the application of mathematics in experimental science and logic” as “the source of all evil,” because it superimposes “a mathematical regularity on the physical world” (5). In both Life, Art and Mysticism and Foundations of Mathematics, Brouwer expresses “his conviction of the opposition between mind and matter, the individual consciousness and the exterior world” (5).]
[ditto]
In his Foundations of Mathematics (B1907), especially its original version, it is the application of mathematics in experimental science and logic that is exposed as the source of all evil and analyzed as “the causal” or “cunning act,” superimposing a mathematical regularity on the physical world. Both works express his conviction of the opposition between mind and matter, the individual consciousness and the exterior world.
(5)
[The Self or Soul]
[Brouwer has a notion of personal identity, self, and subject as being a pure spiritual soul. “The life of the Soul is the complex of thought processes in response to its awareness of the world outside” (5). These processes are phases that deteriorate.]
[ditto]
Reflecting on the nature of man, Brouwer identifies personal identity, the “Self” or “the Subject,” with the pure-spiritual “Soul” in his later work referred to as “Consciousness in its deepest home” (“Consciousness, Philosophy and Mathematics,” B1948C). The life of the Soul is the complex of thought processes in response to its awareness of the world outside. They are analyzed as distinct mental states, “phases of consciousness” in a process of evolution, each resulting from a definite “happening” and each producing its characteristic form of knowledge and human activity. It is a “deteriorative” process moving consciousness further and further away “in its exodus from its deepest home” on a sliding scale from “beautiful,” that is, good, to evil.
(5)
[The Conscious Movement to Numerical Plurality: The Intuition of Time and “Two-ity”]
[There is an original preperceptual phase of consciousness that is “stillness.” This is “followed by ‘the naive phase’ of receiving images through physical sensations and reacting spontaneously to them” (5). There is next a momentous event, called “the Primordial Happening” or “the Primordial Intuition of Time,” in which the subject links isolated sensations and becomes aware of time. This on the one hand “brings about a transformation of the Naive Consciousness to the rational ‘Mind’ ” while on the other hand it also “generates the fundamental concepts and tools of mathematics. The Primordial Intuition of Time is the fundamental single act of isolating and linking distinct moments in time, creating mathematical ‘Two-ity’ and the ordinal numbers as well as the continuum.” (6). This thereby gives us the “mathematical power to generate sequences,” which enables us to produce a human-made and mathematical interpretation of nature or the outside world. What we take to be “things,” “including other human beings, are no more than repeated sequences or sequences of sequences, manmade, as is indeed the so-called scientific or ‘causal’ coherence of the world” (6). Now, since these things are somehow fundamentally mathematical sequences that are thought up, “this universe of ‘things’ is wholly private,” and it is called “the Exterior World of the Subject” (6). “The scientific observation of regularity in Nature, linking things and events in time as sequences, is a creative, mathematical process of the individual Mind and is referred to as ‘mathematical viewing’ or ‘causal attention’,” and causality “is an artificial, mind-made structure, not inherent in Nature” (6). In fact, “Brouwer rejects any universal objectivity of things,” and he also rejects the idea that things are bound up by causality. Moreover, Brouwer “denies the existence of a collective or ‘plural’ mind” and instead favors “the essential individuality of thought and mind” (6).]
[ditto]
The original preperceptional stage of “stillness” is followed by “the naive phase” of receiving images through physical sensations and reacting spontaneously to them. | The momentous event of the Subject linking isolated sensations, becoming aware of time, referred to by Brouwer as “the Primordial Happening” or “the Primordial Intuition of Time, “brings about a transformation of the Naive Consciousness to the rational “Mind” and at the same time generates the fundamental concepts and tools of mathematics. The Primordial Intuition of Time is the fundamental single act of isolating and linking distinct moments in time, creating mathematical “Two-ity” and the ordinal numbers as well as the continuum. It is first mentioned in Brouwer’s analysis of science, Chapter 2 of his Foundations, where it is used to show the priority of mathematics with respect to science and expose the ideal nature of science, no more than man’s mathematical interpretation of the world. The mathematical power to generate sequences enables man to create in his individual thought-world an interpretation of “Nature,” the outside world, which is manmade and mathematical. “Things,” including other human beings, are no more than repeated sequences or sequences of sequences, manmade, as is indeed the so-called scientific or “causal” coherence of the world. And because of the individual nature of human thought, this universe of “things” is wholly private. Brouwer refers to it as “the Exterior World of the Subject.” The scientific observation of regularity in Nature, linking things and events in time as sequences, is a creative, mathematical process of the individual Mind and is referred to as “mathematical viewing” or “causal attention.” Causality is an artificial, mind-made structure, not inherent in Nature. Indeed, Brouwer rejects any universal objectivity of things as well as their “causal coherence,” basing his argument on the essential individuality of thought and mind. In “Consciousness, Philosophy and Mathematics” (B1948C) he emphatically denies the existence of a collective or “plural” mind.
(5-6)
[The Moral Turn in Conscious Evolution]
[The evolutionary movement of consciousness “enters a moral phase when man takes advantage of and acts upon his causal knowledge by setting in motion a causal sequence of events, selecting a first element of the sequence in order to achieve a later element, the desired ‘end’ ” (6). Given that there is an assessment of the causal sequence resulting from one’s actions, this is a “mathematical or causal acting” and it is “calculated” and “cunning”. As such, Brouwer condemns it “as ‘sinful’ and ‘not-beautiful’ ” on account of it being morally evil.]
[ditto]
The Brouwerian evolutionary “exodus from its deepest home of consciousness” enters a moral phase when man takes advantage of and acts upon his causal knowledge by setting in motion a causal sequence of events, selecting a first element of the sequence in order to achieve a later element, the desired “end.” Such mathematical or causal acting is “calculated” and “cunning,” condemned as “sinful” and “not-beautiful,” that is, morally evil.
(6)
[Language as the Social Imposition of Will]
[The next phase of conscious development, taking us even further away from consciousness’ “deepest home” is the “final phase of ‘social acting’ ” (6). Brouwer describes it as “ ‘the enforcement of will’ in social interaction and organization, in particular by the creation of language” (6). Brouwer thinks that we cannot communicate directly “soul-to-soul”. This creates the need for language, which Brouwer sees as the “ ‘imposition of will through sounds,’ forcing an other human being to act in pursuance of the end desired by the speaker” (6). Then, “As social interaction develops and grows more complex, language becomes more sophisticated, but its essence, as of all instruments, is determined by its purpose: the transmission of will” (6). But, since it is “Used as a means of communicating thought to others, language is bound to remain defective, given the essential privacy of thought and the nature of the ‘sign,’ the arbitrary association of a thought with a sound or visual object” (7).]
[ditto]
Even more remote from the “deepest home of consciousness” is the next and final phase of “social acting,” described as “the enforcement of will” in social interaction and organization, in particular by the creation of language. Brouwer’s philosophy of language starts from the conviction that direct communication between human beings is impossible. His chapter on “Language” in Life, Art and Mysticism (B1905) starts as follows: “From Life in the Mind follows the impossibility of communicating directly with others ... never has anyone been able to communicate directly with others soul-to-soul.” (p. 37). The privacy of mind and thought and the hypothetical existence of minds in other human beings, who are no more than the Subject’s mind-creations, “things in the exterior world of the Subject” rule out “any exchange of thought” (B1948C, p.1240). In line with his “genetic” principle of ontological analysis, Brouwer searches for the nature of language in the process that brought it into being. He traces the origin of language to a particular form of cunning or mathematical acting, the “imposition of will through sounds,” forcing an other human being to act in pursuance of the end desired by the speaker: “At the most primitive stages of civilization ... the transmission of will to induce labour or | servitude is brought about by simple gestures of all kinds especially and predominantly the emotive natural sounds of the human voice” (B1933, p. 51). As social interaction develops and grows more complex, language becomes more sophisticated, but its essence, as of all instruments, is determined by its purpose: the transmission of will. Used as a means of communicating thought to others, language is bound to remain defective, given the essential privacy of thought and the nature of the “sign,” the arbitrary association of a thought with a sound or visual object.
(6-7)
[Language as Memory Aid]
[But language is not restricted to the imposition of one’s will upon others. It also is implemented strictly within one’s own inner world as an aid to memory, “helping the Subject to recall his past thought” (7). But Brouwer, when discussing the limitations of real life mathematicians and when noting that even with the help of linguistic signs, memory is still fallible, he “introduces his notion of the ‘Idealized Mathematician’” (7).]
[ditto]
Within the private world of the Subject, language may have a function as “an aid to memory,” helping the Subject to recall his past thought. In “Will, Knowledge and Speech” (B1933), when Brouwer had to accept human frailty, the limitations of the flesh-and-blood mathematician, even the stability of such private language, was called into question: “The human power of memory ... is by its very nature limited and fallible” (p. 58), even when it calls in the help of linguistic signs. It is at this point that Brouwer introduces his notion of the “Idealized Mathematician.”
(7)
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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