by Corry Shores
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[Stigt, “Brouwer’s Intuitionist Programme,” entry directory]
[The following is summary. I am not a mathematician, so please consult the original text instead of trusting my summarizations. Bracketed comments are my own. Proofreading is incomplete, so please forgive my mistakes.]
Summary of
Walter P. van Stigt
“Brouwer’s Intuitionist Programme”
in
From Brouwer to Hilbert:
The Debate on the Foundations of Mathematics in the 1920’s
Part I.
L.E.J. Brouwer
Ch1.:
“Brouwer’s Intuitionist Programme”
1.2
“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).
[The Main Features of the Philosophy of Intuitionism]
[Precursors to Intuitionism]
[Descartes as the Father of Intuitionism]
[The French Heritage of Descartes’ Intuitionism]
[Intuitionism Being Made Mathematically Precise by Brouwer]
Summary
[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)
2. On the topic of this section, see also Largeault 1993b.
(20)
Largeault, J., 1993b, Intuition et Intuitionisme, Vrin, Paris.
(22)
[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)
[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)
[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.
[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)
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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