31 Dec 2018

van Stigt (1.4.0) “Brouwer’s Intuitionist Programme” part 1.4.0, “[Introductory material to] Mathematics, Language, and Logic” , summary

 

by Corry Shores

 

[Search Blog Here. Index-tags are found on the bottom of the left column.]

 

[Central Entry Directory]

[Mathematics, Calculus, Geometry, Entry Directory]

[Logic and Semantics, entry directory]

[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”

 

[1.4.0]

[Introductory material]

 

 

 

 

 

Brief summary:

(1.4.0.1) The objective of Brouwer’s first intuitionist campaign was the freeing of “mathematics from its traditional reliance on language and logic” (8). (1.4.0.2) The subject when generating mathematics in their mind may make no use of any aspect of language. At best, the mathematician may record their constructions in symbols to aid their memory. However, this symbolization cannot be a part of the mathematical process itself. (1.4.0.3) In fact, symbolization cannot be relied upon for communicating mathematical constructions, because {a} we cannot be sure that other subjects, who are mere things in our created exterior world, have minds, and {b} even if other minds do exist, we cannot be sure that commonly shared words will “represent the same thought-construction in the private worlds of different individuals” (9). (1.4.0.4) Thus “Brouwer’s ‘mathematical language’ is the report, the record of a completed mathematical construction; its truth and noncontradictority are due solely to the constructions it represents” (9).

 

 

 

 

 

 

Contents

 

1.4.0.1

[Mathematics without Language]

 

1.4.0.2

[Restrictions Against Symbolization for Mathematical Construction]

 

1.4.0.3

[Language as Unable to Communicate Mathematical Constructions]

 

1.4.0.4

[Language as Record of Construction]

 

 

 

 

 

Summary

 

1.4.0.1

[Mathematics without Language]

 

[The objective of Brouwer’s first intuitionist campaign was the freeing of “mathematics from its traditional reliance on language and logic” (8).]

 

[ditto]

Within the Brouwerian conception of mathematics as pure, individual thought­ construction on the basis of the Primordial Intuition alone there is clearly no place for language in any form. The emphatic “languageless” in nearly all his definitions of mathematics reflects the need for express denial of the role language plays in al­most all alternative philosophies of mathematics, even that of his favorite “preintuitionist,” Poincaré. Freeing mathematics from its traditional reliance on language and logic was the objective of Brouwer’s first Intuitionist campaign, the “First Act of Intuitionism,” in his historical surveys described as “the uncompromising separation of mathematics and mathematical language and thereby of the phenomena described by theoretical logic.”

(8)

[contents]

 

 

 

 

 

 

1.4.0.2

[Restrictions Against Symbolization for Mathematical Construction]

 

[The subject when generating mathematics in their mind may make no use of any aspect of language. At best, the mathematician may record their constructions in symbols to aid their memory. However, this symbolization cannot be a part of the mathematical process itself.]

 

[ditto]

In the genesis of mathematics, wholly confined to the private thought-world of the Subject, no use is made of any aspect of language, either as the carrier of common “objective” concepts or as primitive symbols with no meaning. For the sake of “aiding the memory” the flesh-and-blood mathematician may resort to recording | his constructions in symbols, linking a thought-construction to a name, “an aural or visual thing”; such recording, however, is a posteriori and not part of the mathematical process itself. Moreover, it is essentially private language since both the thought-construction and the assignation to a particular symbol are exclusively acts of the individual mathematician.

(8-9)

[contents]

 

 

 

 

 

 

1.4.0.3

[Language as Unable to Communicate Mathematical Constructions]

 

[In fact, symbolization cannot be relied upon for communicating mathematical constructions, because {a} we cannot be sure that other subjects, who are mere things in our created exterior world, have minds, and {b} even if other minds do exist, we cannot be sure that commonly shared words will “represent the same thought-construction in the private worlds of different individuals” (9).]

 

[ditto]

As to language as a means of communicating mathematics to other individuals, there is no basis for agreement between the constructive thought-processes of different individuals represented in a “common language.” To the Subject other individuals are “things,” creations of his Exterior World, and the existence of other minds similar to his own “mere hypothesis.” And even if the existence of other minds were to be conceded, there is no guarantee that common words would represent the same thought-construction in the private worlds of different individuals.

(9)

[contents]

 

 

 

 

 

 

1.4.0.4

[Language as Record of Construction]

 

[Thus “Brouwer’s ‘mathematical language’ is the report, the record of a completed mathematical construction; its truth and noncontradictority are due solely to the constructions it represents” (9).]

 

[ditto]

Brouwer’s “mathematical language” is the report, the record of a completed mathematical construction; its truth and noncontradictority are due solely to the constructions it represents.

(9)

[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.

 

 

 

 

van Stigt (1.3.1) “Brouwer’s Intuitionist Programme” part 1.3.1, “Mathematical Existence and Truth”, summary

 

by Corry Shores

 

[Search Blog Here. Index-tags are found on the bottom of the left column.]

 

[Central Entry Directory]

[Mathematics, Calculus, Geometry, Entry Directory]

[Logic and Semantics, entry directory]

[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”

 

1.3

“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).

 

 

 

 

 

Contents

 

1.3.1.1

[Mathematical Existence as Construction]

 

1.3.1.2

[Mathematical Constructions of Infinity]

 

 

 

 

 

 

Summary

 

1.3.1.1

[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)

[contents]

 

 

 

 

 

 

1.3.1.2

[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)

[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.

 

 

 

 

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

 

[Search Blog Here. Index-tags are found on the bottom of the left column.]

 

[Central Entry Directory]

[Mathematics, Calculus, Geometry, Entry Directory]

[Logic and Semantics, entry directory]

[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”

 

1.3

“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.

 

 

 

 

van Stigt (1.2.2) “Brouwer’s Intuitionist Programme” part 1.2.2, “Brouwer’s Outlook on Life and General Philosophy”, summary

 

by Corry Shores

 

[Search Blog Here. Index-tags are found on the bottom of the left column.]

 

[Central Entry Directory]

[Mathematics, Calculus, Geometry, Entry Directory]

[Logic and Semantics, entry directory]

[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).

 

 

 

 

 

Contents

 

1.2.2.1

[Brouwer’s Romanticism]

 

1.2.2.2

[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]

 

1.2.2.3

[The Self or Soul]

 

1.2.2.4

[The Conscious Movement to Numerical Plurality: The Intuition of Time and “Two-ity”]

 

1.2.2.5

[The Moral Turn in Conscious Evolution]

 

1.2.2.6

[Language as the Social Imposition of Will]

 

1.2.2.7

[Language as Memory Aid]

 

 

 

 

 

Summary

 

1.2.2.1

[Brouwer’s Romanticism]

 

[“Brouwers 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]

Brouwers 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)

[contents]

 

 

 

 

 

 

1.2.2.2

[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)

[contents]

 

 

 

 

 

 

1.2.2.3

[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)

[contents]

 

 

 

 

 

 

1.2.2.4

[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)

[contents]

 

 

 

 

 

 

1.2.2.5

[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)

[contents]

 

 

 

 

 

 

1.2.2.6

[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)

[contents]

 

 

 

 

 

 

1.2.2.7

[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)

[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.

 

 

 

 

van Stigt (1.2.1) “Brouwer’s Intuitionist Programme” part 1.2.1, “Intuitionism”, summary

 

by Corry Shores

 

[Search Blog Here. Index-tags are found on the bottom of the left column.]

 

[Central Entry Directory]

[Mathematics, Calculus, Geometry, Entry Directory]

[Logic and Semantics, entry directory]

[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”

 

[1.2.1]

Intuitionism

 

 

 

 

 

Brief summary:

(1.2.1.1) “Intuitionism is a philosophical trend that places the emphasis on the individual consciousness as the source and seat of all knowledge” (4). Intuitionism holds that the mind has not only a faculty and activity of reasoning but as well “a definite faculty and act of direct apprehension, intuition, as the necessary foundation of all knowledge, both in the grasping of first principles on which a system of deductive reasoning is built and as the critical link in every act of knowing between the knower and the object known” (4). Moreover, “Intuitionism stands in contrast to a more general rationalistic and deterministic trend that denies the possibility of knowing things and facts in themselves and restricts human knowledge to what can be deduced mechanically by analytical reasoning, ultimately from self-evident facts and principles that result from common sense or are based on the authority of collective wisdom” (4). (1.2.1.2) Elements of intuitionism can be found in previous philosophies. Aristotle’s νοῦς, for instance, is something like Brouwerian intuition; for, it is “a special faculty of direct apprehension, an active faculty that is indispensable in the creation of primary concepts and first principles as well as at every step of the thought process” (4). We can also find elements of intuitionism in “the systems of some of the modern German and English philosophers such as Kant, Hamilton, Whewell, and even Russell” (4). (1.2.1.3) Descartes can be considered the father of intuitionism, because for him, “every form of knowing ultimately requires an act of immediate mental apprehension, ‘intuition’” (5). (1.2.1.4) Descartes’ sort of intuitionism saw development in 19th century France by Maine de Biran, Ravaisson, Lachelier, and Boutroux. “It was developed into a full and comprehensive philosophy by Henri Bergson, who raised Intuition to the faculty of grasping the spiritual and changing reality, distinct from Reason, the analytical mind, which probes the material and static reality. Bergson’s living reality, however, did not include the mathematical universe; his concepts of number and the mathematical continuum are spatial, products of the analytical intellect” (5). (1.2.1.5) But the notion of intuition is vague in Descartes as well as with the French “New Intuitionists” Poincaré, Borel, and Lebesgue. It was not made mathematically precise until Brouwer “took Descartes’ intuitionist thesis to its radical subjective and constructive conclusion” (5).

 

 

 

 

 

 

Contents

 

1.2.1.1

[The Main Features of the Philosophy of Intuitionism]

 

1.2.1.2

[Precursors to Intuitionism]

 

1.2.1.3

[Descartes as the Father of Intuitionism]

 

1.2.1.4

[The French Heritage of Descartes’ Intuitionism]

 

1.2.1.5

[Intuitionism Being Made Mathematically Precise by Brouwer]

 

 

 

 

 

 

Summary

 

1.2.1.1

[The Main Features of the Philosophy of Intuitionism]

 

[“Intuitionism is a philosophical trend that places the emphasis on the individual consciousness as the source and seat of all knowledge” (4). Intuitionism holds that the mind has not only a faculty and activity of reasoning but as well “a definite faculty and act of direct apprehension, intuition, as the necessary foundation of all knowledge, both in the grasping of first principles on which a system of deductive reasoning is built and as the critical link in every act of knowing between the knower and the object known” (4). Moreover, “Intuitionism stands in contrast to a more general rationalistic and deterministic trend that denies the possibility of knowing things and facts in themselves and restricts human knowledge to what can be deduced mechanically by analytical reasoning, ultimately from self-evident facts and principles that result from common sense or are based on the authority of collective wisdom” (4).]

 

[ditto]

Intuitionism is a philosophical trend that places the emphasis on the individual consciousness as the source and seat of all knowledge.2 Besides the faculty and activity of reasoning, it recognizes in the individual mind a definite faculty and act of direct apprehension, intuition, as the necessary foundation of all knowledge, both in the grasping of first principles on which a system of deductive reasoning is built and as the critical link in every act of knowing between the knower and the object known. Intuitionism stands in contrast to a more general rationalistic and deterministic trend that denies the possibility of knowing things and facts in themselves and restricts human knowledge to what can be deduced mechanically by analytical reasoning, ultimately from self-evident facts and principles that result from common sense or are based on the authority of collective wisdom.

(4)

On the topic of this section, see also Largeault 1993b.

(20)

Largeault, J., 1993b, Intuition et Intuitionisme, Vrin, Paris.

(22)

[contents]

 

 

 

 

 

 

1.2.1.2

[Precursors to Intuitionism]

 

[Elements of intuitionism can be found in previous philosophies. Aristotle’s νοῦς, for instance, is something like Brouwerian intuition; for, it is “a special faculty of direct apprehension, an active faculty that is indispensable in the creation of primary concepts and first principles as well as at every step of the thought process” (4). We can also find elements of intuitionism in “the systems of some of the modern German and English philosophers such as Kant, Hamilton, Whewell, and even Russell” (4).]

 

[ditto]

Elements of Intuitionism can already be found in classical philosophies, for example, in the Aristotelian νοῦς, a special faculty of direct apprehension, an active faculty that is indispensable in the creation of primary concepts and first principles as well as at every step of the thought process. Elements of Intuitionism are also found in the systems of some of the modern German and English philosophers such as Kant, Hamilton, Whewell, and even Russell (for the Kantian roots of Brouwer’s philosophy of mathematics, see Posy 1974). But it is in the revolutionary and libertarian climate of Holland and France that Intuitionism took root and developed into a full and coherent philosophy.

(4)

[contents]

 

 

 

 

 

 

1.2.1.3

[Descartes as the Father of Intuitionism]

 

[Descartes can be considered the father of intuitionism, because for him, “every form of knowing ultimately requires an act of immediate mental apprehension, ‘intuition’” (5).]

 

[ditto]

Descartes, the father of modern philosophy, can rightly be claimed to be the father of modern Intuitionism. A Frenchman by birth, Descartes settled in Holland, “a country” – as he wrote to Balzac – “where complete liberty can be enjoyed.” His rebellion was the fundamental break with the traditional reliance on authority, religious and otherwise, as the ultimate source of truth and placing the origin and seat of knowledge firmly in the individual mind of man. He starts from “self-awareness” and distinguishes between various faculties in the process of acquiring knowledge, | but insists that every form of knowing ultimately requires an act of immediate mental apprehension, “intuition.” He insists on the need for rational argument and sets out rigorous rules of correct reasoning, but points out that logical deductive reasoning does not produce any new truths, that true knowledge comes from intuition.

(4-5)

[contents]

 

 

 

 

 

 

1.2.1.4

[The French Heritage of Descartes’ Intuitionism]

 

[Descartes’ sort of intuitionism saw development in 19th century France by Maine de Biran, Ravaisson, Lachelier, and Boutroux. “It was developed into a full and comprehensive philosophy by Henri Bergson, who raised Intuition to the faculty of grasping the spiritual and changing reality, distinct from Reason, the analytical mind, which probes the material and static reality. Bergson’s living reality, however, did not include the mathematical universe; his concepts of number and the mathematical continuum are spatial, products of the analytical intellect” (5).]

 

[ditto]

Descartes’ intuitionist lead was followed in the nineteenth century by a number of French philosophers such as Maine de Biran, Ravaisson, Lachelier, and Boutroux. It was developed into a full and comprehensive philosophy by Henri Bergson, who raised Intuition to the faculty of grasping the spiritual and changing reality, distinct from Reason, the analytical mind, which probes the material and static reality. Bergson’s living reality, however, did not include the mathematical universe; his concepts of number and the mathematical continuum are spatial, products of the analytical intellect.

[contents]

 

 

 

 

 

 

1.2.1.5

[Intuitionism Being Made Mathematically Precise by Brouwer]

 

[But the notion of intuition is vague in Descartes as well as with the French “New Intuitionists” Poincaré, Borel, and Lebesgue. It was not made mathematically precise until Brouwer “took Descartes’ intuitionist thesis to its radical subjective and constructive conclusion” (5).]

 

[ditto]

As to the precise nature of Intuition as the foundation of mathematics, Descartes remains somewhat vague: The fundamental mathematical truths are “indubitable” because they are “clearly and distinctly perceived” by the mind’s eye. Yet in his argument for the existence of God, for which he claims “the same level of certainty as the truths of mathematics,” he concludes that these truths, such as the essence and nature of the triangle, are “immutable and eternal and not invented by me nor dependent on my mind” (Descartes, 5th Meditation). Equally vague as to the nature of Intuition are the French “New Intuitionists” Poincaré, Borel, and Lebesgue.3 It was not until the beginning of the twentieth century that an attempt was made at a precise interpretation of mathematical Intuition, when Brouwer took Descartes’ intuitionist thesis to its radical subjective and constructive conclusion.

(5)

On the French Intuitionists, see references given in the introduction to Part 2 and Largeault 1993a and 1993b.

(20)

Largeault, J., 1993a, L’lntuitionisme des mathematiciens avant Brouwer, Archives de Philosophie 56, pp. 53-68.

Largeault, J., 1993b, Intuition et Intuitionisme, Vrin, Paris.

(22)

[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.

 

 

 

 

van Stigt (1.2.0) “Brouwer’s Intuitionist Programme” part 1.2.0, “[Intro Material for] Intuitionism and Brouwer’s Intuitionist Philosophy of Mathematics”, summary

 

by Corry Shores

 

[Search Blog Here. Index-tags are found on the bottom of the left column.]

 

[Central Entry Directory]

[Mathematics, Calculus, Geometry, Entry Directory]

[Logic and Semantics, entry directory]

[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”

 

1.2.0

[Introductory Material]

 

 

 

 

Brief summary:

(1.2.0.1) Brouwer’s intuitionism is primarily a philosophy of mathematics. (1.2.0.2) “Most of Brouwer’s philosophical views on life in general and on the nature of mathematics were formed during the years of undergraduate and doctoral studies, and they remained virtually the same throughout his life” (4). (1.2.0.3) “This section is a brief introduction to Philosophical Intuitionism and the main aspects of Brouwer’s philosophy as are relevant to his Intuitionist practice” (4).

 

 

 

 

 

Contents

 

1.2.0.1

[Intuitionism as Mathematics]

 

1.2.0.2

[Brouwer’s Early Philosophical Development]

 

1.2.0.3

[Announcing the Content of This Section]

 

 

 

 

 

 

 

Summary

 

1.2.0.1

[Intuitionism as Mathematics]

 

[Brouwer’s intuitionism is primarily a philosophy of mathematics.]

 

[ditto]

Brouwer’s Intuitionist reform of mathematics and his revolutionary views on the use of logic can only be fully understood in the context of his particular philosophy of mathematics. Indeed, his Intuitionism is first and foremost a philosophy of mathematics from which these new ideas emerge quite naturally.

(4)

[contents]

 

 

 

 

 

 

1.2.0.2

[Brouwer’s Early Philosophical Development]

 

[“Most of Brouwer’s philosophical views on life in general and on the nature of mathematics were formed during the years of undergraduate and doctoral studies, and they remained virtually the same throughout his life” (4).]

 

[ditto]

Most of Brouwer’s philosophical views on life in general and on the nature of mathematics were formed during the years of undergraduate and doctoral studies, and they remained virtually the same throughout his life. They are expressed most clearly in his early publications: his doctoral thesis On the Foundations of Mathematics (B1907) and Life, Art and Mysticism (B1905), in some of his post-1928 papers such as “Mathematics, Science and Language” (B1929), “Will, Knowledge and Speech” (B1933), and “Consciousness, Philosophy and Mathematics” (B1948C), and in unpublished papers.

(4)

[contents]

 

 

 

 

 

 

 

1.2.0.3

[Announcing the Content of This Section]

 

[“This section is a brief introduction to Philosophical Intuitionism and the main aspects of Brouwer’s philosophy as are relevant to his Intuitionist practice” (4).]

 

[ditto]

This section is a brief introduction to Philosophical Intuitionism and the main aspects of Brouwer’s philosophy as are relevant to his Intuitionist practice. A more detailed analysis is given in van Stigt 1990.

(4)

[contents]

 

 

 

 

 

 

1.2.0.4

[]

 

[]

 

[ditto]

x

[contents]

 

 

 

 

 

 

1.2.0.5

[]

 

[]

 

[ditto]

x

[contents]

 

 

 

 

 

 

1.2.0.6

[]

 

[]

 

[ditto]

x

[contents]

 

 

 

 

 

 

1.2.0.7

[]

 

[]

 

[ditto]

x

[contents]

 

 

 

 

 

 

1.2.0.8

[]

 

[]

 

[ditto]

x

[contents]

 

 

 

 

 

 

 

1.2.0.9

[]

 

[]

 

[ditto]

x

[contents]

 

 

 

 

 

 

1.2.0.10

[]

 

[]

 

[]

x

[contents]

 

 

 

 

 

 

1.2.0.11

[]

 

[]

 

[]

x

[contents]

 

 

 

 

 

 

1.2.0.12

[]

 

[]

 

[]

x

[contents]

 

 

 

 

 

1.2.0.13

[]

 

[]

 

[]

x

[contents]

 

 

 

 

 

1.2.0.14

[]

 

[]

 

[]

x

[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.

 

 

 

 

van Stigt (1.1) “Brouwer’s Intuitionist Programme” part 1.1, “The Intuitionist-Formalist Controversy”, summary

 

by Corry Shores

 

[Search Blog Here. Index-tags are found on the bottom of the left column.]

 

[Central Entry Directory]

[Mathematics, Calculus, Geometry, Entry Directory]

[Logic and Semantics, entry directory]

[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”

 

1.1

The Intuitionist-Formalist Controversy

 

 

 

 

 

Brief summary:

(1.1.1) Brouwer’s text “Foundations of Set Theory Independent of the Principle of Excluded Middle” diagnosed a crisis in the foundations of mathematics. This was noted by and had strong influence on Hermann Weyl. In his “Intuitionist Set Theory,” “Brouwer set out the consequences for established mathematics of his Intuitionist theses, in particular, his rejection of the logical Principle of the Excluded Middle: ‘the use of the Principle of the Excluded Middle is not permissible as part of a mathematical proof ... [it] has only scholastic and heuristic value, so that theorems which in their proof cannot avoid the use of this principle lack all mathematical content.’” (1.1.2) Hilbert saw Brouwer’s program as posing a threat to Cantorian set theory and to his own program, so he launched a counterattack in 1922. This begins the Intuitionist-Formalist debate of the 1920s. (1.1.3) The Intuitionist-Formalist debate revolved around two issues: {1} “The nature of mathematics: either human thought-construction or theory of formal structures” and {2} “The role of the Principle of the Excluded Middle in mathematics and Brouwer’s restrictive alternative logic” (2). (1.1.4) “Brouwer’s main concern was the nature of mathematics as pure, ‘languageless’ thought-construction” (2). His program aimed to convince the mathematical world of this view. His “Intuitionist Splitting of the Fundamental Notions of Mathematics” opened debate among logicians regarding an alternative “Brouwer Logic,” (including contributions by Kolmogorov, Borel, Wavre, Glivenko, and Heyting). But Brouwer was not concerned with logic, because he believed that “logic and formalization were ‘an unproductive, sterile exercise’ with no direct relevance to mathematics and its foundations” (2). (1.1.5) Hilbert’s program retains classical mathematics and bases “its validity on a proof of the consistency of its formalization” (3). Brouwer’s program had a constructive interpretation of mathematics. Both held some interest in the mathematical community, but Brouwer’s program failed to gain traction. (1.1.6) “The Brouwer-Hilbert debate grew increasingly bitter and turned into a personal feud” (3). Hilbert expels Brouwer from the board of the Mathematische Annalen. (1.1.7) Brouwer’s professional rejection led him to stop publicizing his program, even at the same time that Hilbert’s formalist program was shown to be fundamentally flawed.

 

 

 

 

 

 

Contents

 

1.1.1

[Brouwer’s Rejection of the Principle of Excluded Middle and Its Crisis for Mathematical Foundations]

 

1.1.2

[Weyl and Brouwer versus Hilbert in the Intuitionist-Formalist Debate]

 

1.1.3

[The Two Issues of the Debate]

 

1.1.4

[Brouwer’s Immediate Influence in Mathematics and Logic]

 

1.1.5

[The Immediate Reception of Hilbert’s and Brouwer’s programs]

 

1.1.6

[The Personal Side of the Brouwer-Hilbert Debate]

 

1.1.7

[The End of Brouwer’s Program]

 

 

 

 

 

 

Summary

 

1.1.1

[Brouwer’s Rejection of the Principle of Excluded Middle and Its Crisis for Mathematical Foundations]

 

[Brouwer’s text “Foundations of Set Theory Independent of the Principle of Excluded Middle” diagnosed a crisis in the foundations of mathematics. This was noted by and had strong influence on Hermann Weyl. In his “Intuitionist Set Theory,” “Brouwer set out the consequences for established mathematics of his Intuitionist theses, in particular, his rejection of the logical Principle of the Excluded Middle: ‘the use of the Principle of the Excluded Middle is not permissible as part of a mathematical proof ... [it] has only scholastic and heuristic value, so that theorems which in their proof cannot avoid the use of this principle lack all mathematical content.’” ]

 

[ditto]

In 1920 Hermann Weyl diagnosed “a new crisis in the foundations of mathematics” (Weyl 1921), sparked off by the publication of Brouwer’s “Foundations of Set Theory Independent of the Principle of Excluded Middle” (B1918B and B1919A). In a series of lectures at the Mathematical Colloquium of Zürich, he dramatically renounced his own Das Kontinuum and hailed Brouwer’s set theory and interpretation of the continuum as “the revolution”: “… und Brouwer – das ist die Revolution!” (Weyl 1921, p. 99), the one mathematician who at last had solved the problem of the continuum, which since ancient times had defeated even the greatest | minds. At the same time, in “Intuitionist Set Theory” (B 1919D) Brouwer set out the consequences for established mathematics of his Intuitionist theses, in particular, his rejection of the logical Principle of the Excluded Middle: “the use of the Principle of the Excluded Middle is not permissible as part of a mathematical proof ... [it] has only scholastic and heuristic value, so that theorems which in their proof cannot avoid the use of this principle lack all mathematical content.” (p. 23)

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1.1.2

[Weyl and Brouwer versus Hilbert in the Intuitionist-Formalist Debate]

 

[Hilbert saw Brouwer’s program as posing a threat to Cantorian set theory and to his own program, so he launched a counterattack in 1922. This begins the Intuitionist-Formalist debate of the 1920s.]

 

[ditto]

Both Brouwer’s challenge and Weyl’s support raised the alarm among the Cantorian and Formalist establishment of Gottingen. Hilbert, who had recognized Brouwer’s major contribution to topology and had welcomed him as a member of his inner circle, grew increasingly impatient with his old friend and alarmed by the implied threats to Cantorian set theory and his own programme. He launched a counterattack in 1922:

What Weyl and Brouwer do amounts in principle to following the erstwhile path of Kronecker: they seek to ground mathematics by throwing overboard all phenomena that make them uneasy ... if we follow such reformers, we run the danger of losing a large number of our most valuable treasures. (Hilbert 1922, p. 200)

The ensuing Intuitionist-Formalist “debate” dominated the foundational scene throughout the 1920s. Brouwer and Hilbert remained the main protagonists, each drawing support for his cause beyond national frontiers and an even greater audience of interested observers and commentators.

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1.1.3

[The Two Issues of the Debate]

 

[The Intuitionist-Formalist debate revolved around two issues: {1} “The nature of mathematics: either human thought-construction or theory of formal structures” and {2} “The role of the Principle of the Excluded Middle in mathematics and Brouwer’s restrictive alternative logic” (2).]

 

[ditto]

The debate centered on two different, though related, issues:

1. The nature of mathematics: either human thought-construction or theory of formal structures;

2. The role of the Principle of the Excluded Middle in mathematics and Brouwer’s restrictive alternative logic.

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1.1.4

[Brouwer’s Immediate Influence in Mathematics and Logic]

 

[“Brouwer’s main concern was the nature of mathematics as pure, ‘languageless’ thought-construction” (2). His program aimed to convince the mathematical world of this view. His “Intuitionist Splitting of the Fundamental Notions of Mathematics” opened debate among logicians regarding an alternative “Brouwer Logic,” (including contributions by Kolmogorov, Borel, Wavre, Glivenko, and Heyting). But Brouwer was not concerned with logic, because he believed that “logic and formalization were ‘an unproductive, sterile exercise’ with no direct relevance to mathematics and its foundations” (2).]

 

[ditto]

Brouwer’s main concern was the nature of mathematics as pure, “languageless” thought-construction. He had set himself the task of bringing the mathematical world around to his view, convincing them of the need for reform, and had started the programme of reconstructing mathematics on an Intuitionist basis. Most of his publications in the period 1918-1928 were part of this programme; only a few dealt directly with the “negative” aspects of his Intuitionist campaign: the misuse of logic, in particular the Principle of the Excluded Middle, and the flaws in the Formalist programme. Understandably these papers aroused greater interest and further controversy. His excursion into the field of logic (“Intuitionist Splitting of the Fundamental Notions of Mathematics,” Bl923C), in which he drew the immediate conclusions from his strict interpretation of negation and his rejection of the Principle of the Excluded Middle, created considerable excitement among logicians and started a debate about an alternative, “Brouwer Logic.” This debate was joined by Kolmogorov, Borel, Wavre, Glivenko, Heyting, and others (see Part IV). Brouwer himself did not take a further active part, remaining true to his conviction that logic and formalization were “an unproductive, sterile exercise” with no direct relevance to mathematics and its foundations.

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1.1.5

[The Immediate Reception of Hilbert’s and Brouwer’s programs]

 

[Hilbert’s program retains classical mathematics and bases “its validity on a proof of the consistency of its formalization” (3). Brouwer’s program had a constructive interpretation of mathematics. Both held some interest in the mathematical community, but Brouwer’s program failed to gain traction.]

 

[ditto]

The main Intuitionist-Formalist “debate” was a contest between the leaders of two opposing philosophies of mathematics, each with its own programme and competing for the support of the mathematical world. Apart from the occasional direct | exchange, each camp concentrated on its own programme. Hilbert’s Programme, retaining the “whole treasure of classical mathematics” and basing its validity on a proof of the consistency of its formalization, attracted widening support and an able team of collaborators. Brouwer’s constructive interpretation of mathematics, much in line with the natural outlook of the working mathematician, was enthusiastically received and raised early hopes. However, his austere programme of reconstruction within the Intuitionist constraints failed to gather momentum. His increasing isolation was partly due to his inability to work with others, but more important, Brouwer’s and Weyl’s hopes that the “natural” Intuitionist approach would lead to a simplification of reformed mathematics did not materialize. Indeed, it proved “un­bearably awkward” in comparison with traditional mathematics relying on the methods of classical logic. Even Weyl had to accept this with regrets:

Mathematics with Brouwer gains its highest intuitive clarity. He succeeds in developing the beginnings of analysis in a natural manner, all the time preserving the contact with Intuition much more closely than had been done before. It cannot be denied, however, that in advancing to higher and more general theories the inapplicability of the simple laws of classical logic eventually results in an almost un­ bearable awkwardness. And the mathematician watches with pain the larger part of his towering edifice, which he believed to be built of concrete blocks, dissolve into mist before his eyes. (Weyl 1949, p. 54)

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1.1.6

[The Personal Side of the Brouwer-Hilbert Debate]

 

[“The Brouwer-Hilbert debate grew increasingly bitter and turned into a personal feud” (3). Hilbert expels Brouwer from the board of the Mathematische Annalen.]

 

[ditto]

The Brouwer-Hilbert debate grew increasingly bitter and turned into a personal feud. The last episode was the “Annalenstreit,” or, to use Einstein’s words, “the frog-and-mouse battle.” It followed the unjustified and illegal dismissal of Brouwer from the editorial board of the Mathematische Annalen by Hilbert in 1928 and led to the disbanding of the old Annalen company and the emergence of a new Annalen under Hilbert’s sole command but without the support of its former chief editors, Einstein and Carathéodory.

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1.1.7

[The End of Brouwer’s Program]

 

[Brouwer’s professional rejection led him to stop publicizing his program, even at the same time that Hilbert’s formalist program was shown to be fundamentally flawed.]

 

[ditto]

For Brouwer it was the last straw. His failure to “simplify” Intuitionist methods and make Intuitionism the universally accepted mathematical practice had eroded his self-confidence. The conspiracy of his fellow Annalen editors and “lack of recognition” left him bitter and disillusioned. He abandoned his Intuitionist Programme and withdrew into silence just about the time when the Formalist Programme was shown to be fundamentally flawed. Some “books” were left uncompleted and unpublished. The 1928 “Vienna Lectures,” The Structure of the Continuum (B1930A) and “Mathematics, Science and Language” (B1929), and his paper “Intuitionist Reflections on Formalism” (B1928A2) mark the end of Brouwer’s creative life and his Intuitionist campaign. They reflect the stage his programme had reached and the mood of its founder at the time. The Structure of the Continuum summarizes his Intuitionist vision and analysis of the continuum. In “mathematics, Science and Language” he returns to the pessimism of his philosophy of science and language, which had inspired his Intuitionist rebellion. “Intuitionist Reflections on Formalism” is Brouwer’s final assessment of the state of play in the contest between Intuitionism and Formalism and an emotional outburst at the lack of recognition. It lists outstanding differences as well as “the Intuitionist Insights” adopted by Formalists “without proper mention of authorship,” such as the notion of meta­-mathematics.

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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.