The Cyber-Cave

Reflections on the political, technological, cultural and economic trends of the world

L.E.J Brouwer

Mundus vult decipi”

L.E.J Brouwer (1881-1966): Dutch mathematician known for his insights into topology (such as the fixed point theorem) and his philosophy of mathematics (intuitionism) which eventually would make him clash against the formalists (such as Hilbert) and the logicians of his time (Russell).

Intuitionists only accept constructivist proofs, while rejecting the inferential conclusions of ordinary logic (such as the rule of the excluded middle). Some logical operators like the ‘OR’ (V) can lead to puzzling results such as the statement “1=1 V 1=2”- because of the ‘OR’ operator, the whole statement is considered to be true even though one part of the statement is clearly false.
He rejected much of Cantor’s set theory (like the theory of infinite cardinal numbers). He believed mathematicians should only treat potential infinity, not actual infinity.
He rejected Kant’s philosophy of geometry: because of the development of non-Euclidean geometry, he rejected the idea that knowledge of Euclidean geometry is synthetic a priori (actually it’s a posteriori). Though Brouwer acknowledges that due to our cognitive faculties it is natural for us to perceive space in a Euclidean way.
He supported Kant’s philosophy of arithmetic (synthetic a priori).
A general skepticism towards the use of language (sign and symbols) in language, since the core belief of intuitionism is that we can have in our mind languageless and valid mathematical constructions in our minds. Language inevitably distorts such mental constructions (according to the intuitionists).
He believed that mathematics is independent of the material world but not in a Platonic way: such independence only means that mathematics is about intuitive constructions existing just in our mind.

In 1898 he wrote a religious credo in order to be admitted to the Remonstrant Church (a Church for Dutch protestants). He states that his belief in God is not due to some sort of revelation that he had, but rather because of his sense of “intellectual powerlessness”. Then he adds that he can only be certain of his ego and of his representations (a solipsistic or Schopenhauerian-like position), however he feels that the origin of his ego and of his representations must be God. God is ‘a direct spontaneous emotion in me’.

Life, Art, Mysticism (1905)

A short essay where Brouwer explains his life philosophy. He cites Meister Eckart and Jacob Boehme to show his mystical view of God- the idea is that by turning-into-oneself and by forgetting material preoccupations, one’s consciousness may transcend physical causality. He also mentions the Karma and the Bhagavad Gita (he also quotes a passage from a Flaubert’s novel which deals with a gymnosophiste explaining Indian philosophy).
He blames the ‘intellect’ (the use of rationality as an end in itself and as responsible for creating the illusion of the ego and the outside world being separate) for detaching mankind from a harmony with nature.
He then claims men cannot communicate ‘soul to soul’ because linguistic concepts can only be equally grasped if there is mutual understanding on their meaning, but many words do not have the same semantic precision as of a word like ‘triangle’.
Linguistic structure may also inhibit our ability to think freely: “The belief in a reality, the same for all- existing outside and independent of them- made society foolishly attach great importance to “speaking the truth”… Yet… once someone is imprisoned in the belief in a logically coherent complex of externalities, which he calls ‘reality’, it becomes rather difficult to follow him in his folly and even more difficult to try to evoke in him a particular emotion or state of mind by means of words which he can only interpret in accordance with his reality”. [2] [Brouwer was part of a circle called ‘Significa’, which was about discussing semiotics].

SOURCES:
-[1] https://www.ucl.ac.uk/sts/staff/gillies/documents/1980a_Browers_P_of_M.pdf
[2] https://projecteuclid.org/download/pdf_1/euclid.ndjfl/1039886518

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