Brouwer & the Gita

I think most non-Mathematicians have a vague idea that Brouwer was in the tradition of Poincare and Borel- i.e. critical of logicism- and intuitionistic in some special sense which pragmatism has had reason to value. It would be interesting to see how he is viewed by Dutch mathematicians. 

In a paper titled  L. E. J. Brouwer’s philosophical views and the Bhagavad Gita  Teun Koetsier writes-

 The philosophical views of the Dutch mathematician L. E. J. Brouwer had a crucial influence on his mathematical work.  His conviction that mathematics at heart consists of language-free and logic-free introspective constructions stems from his epistemological solipsism. This solipsism is embedded in an original world view based on Schopenhauerian ideas. In particular Brouwer’s references to the Bhagavad Gita are striking. 

I think a Hindu would say that Brouwer was a Samkhya dualist- the fundamental intuition for him is the 'two-ity' between what in Dutch is called ‘ikheid’ (‘the property of being me’ which corresponds to Purusha) &  the ‘images’ (‘voorstellingen’ which would correspond to Prakriti) appearing before the ego. In Samkhya the ikheid is a passive observer. Brouwer was an energetic 'constructivist' yet he longed for the restoration of a state of absolute purity and thus the 'Yogishvara'- Lord of Yoga- as depicted in the Gita appealed to him. This peculiarity of his makes it difficult to class him with the types of solipsism familiar to Western Philosophy. Instead, it would be more usual to see him as Schopenhauer style transcendental idealist. However, by the Thirties, it had become apparent that his notion of 'choice-sequences' (e.g in the hands of Turing) could do something Husserl's transcendental subject could not do. It seemed Brouwer, not Husserl or Wittgenstein, would be more philosophically fecund for the same reason he was so mathematically fecund. Alas, to my knowledge, what that philosophy might be yet eludes elucidation. 

Considering, as I do, the Mahabarata to have a mathematical structure (because it is careful to give each character or episode a dual so as to preserve symmetries such that karma and dharma, as principles of causality across time and space, are conserved, it would be interesting to see which parts of the Gita- considered as a Samkhya-Yoga text converging to the Advaita of the Chandogya- the great man found most evocative. 

In 1905 Brouwer published a remarkable booklet called Life, Art and Mysticism (LAM) in which he described science and technology as evil forces that lead mankind astray. The only way out of this sad world is via introspection. By turning-into-one-self liberation is possible.

Brouwer had joined the Remonstrant Church- a liberal Arminian (i.e. pre-destination means God knows who will come to truly believe and thus pre-destines them for Grace) sect which required converts to write a personal profession of faith. In other words, his mystical views were directly related to his hope of salvation. As with his mathematical work, he was serious and highly motivated. What made him different from ordinary people was his extraordinary mind and strong character. We may not understand exactly why Brouwer's work is helping create what appear to us to be magical new technologies, but such indeed appears to be the case.  

Brouwer quoted medieval mystics to illustrate his views. He also quoted from the Bhagavad Gita, for example, Book II, 62: “While contemplating the objects of the senses, a person develops attachment for them, and from such attachment lust develops, and from lust anger arises”, and Book II, 64: “But a person free from all attachment and aversion and able to control his senses through regulative principles of freedom can obtain the complete mercy of the Lord”. 

'regulative principles of freedom' is a stretch. Self-control and being sensible is all that is required.  

What is interesting is Brouwer's youthful conviction that '“In his life of lust and desire the intellect renders man the devilish service of linking two images of the imaginations as means and end.

we can imagine a virile young student needing to delink the image of a Dutch prostitute from a means, which. though pleasant enough, ends in Syphilis and a rapid decline in mental power. Also your nose falls off. Help me, Jesus! 

 Once in the grip of desire for one thing he is made to strive after another as a means to that end 

Hang on. Maybe the guy is talking about the misuse of mathematics! The lust for solvability may lead to striving after moonshine! Prior to Hilbert's formulation of his second problem, showing the consistency of axioms meant constructing a model. Brouwer thought the use of the law of the excluded middle (which gets us quickly to solvability)  was illicit for infinite sets. Hilbert countered that this was like not letting a boxer use his fists. This sounds like 'funktionlust'- the fist wanting to hit things because it confuses the reason it exists- which is precautionary- with its purpose in life- i.e. aggression.

It is certainly true that in any applied field, there must be strict scrupulousness or 'restricted comprehension'. This may not be the case for pure math. It may be that any and every 'maTam' (dogma) yields the same 'vigyaan' (science). Certainly, that is one way- an ecumenical way- to read the Gita. 

The act aimed at the means, however, always overshoots the mark […] Man’s blinkered view prevents him from recognizing the sometimes very detrimental effect of such action”

Indeed. The misuse of fixed point theorems have resulted in generations of economists and game theorists writing nonsense.  

Brouwer speaks directly about karma-

The fourth chapter is about life after one realizes the corruptness of the world: “You look on this life as the direction of your duty, and you live it as directed from within the self; in other words: you recognize that all these earthly bonds remain your inevitable karma until God releases you. No new desires will be able to deflect you from your path and you will not wantonly increase the burden of your karma” ([3], p. 400).

I suppose Brouwer was speaking metaphorically. He wasn't saying that his 'prarabdha karma' in this life was created in his past lives. His duty had been laid out for him by God's mysterious economy. Nothing wrong with that at all. What is interesting is that the Theosophists, at that time, were talking about how karma was like an arrow in flight. But a superior archer could deflect that arrow with an arrow of his own. Arjuna and other warriors in the Mahabharata did this all the time. I suppose one might think of the debate between the Intuitionists and the Formalists as an attempt to deflect the arrow of progress in their discipline.

 In this chapter the word ‘karma’ occurs for the first time. Brouwer uses it more than 40 times in the last six chapters of the book. He uses it to denote that what is inevitable in this world.

The Hindu view is that karma is a delusion (Maya). But if you truly lurve the Lord and want nothing more than to fulfil His Divine Plan, then you want Him to be the 'Mayin' or magician controlling you. This is Dualistic Theism and no one can say it is inferior to Mystical Monism. But, equally, there would be little point saying the reverse. In this sense Brouwer is right to condemn language devoted to such ends.  

 Chapter six is on immanent truth, that is the “Truth which in this world points to the inevitability of the karma of this world”. An important source of immanent truth is art. True art reveals the situation man is in. “Art which is real truth, belies common sense, causality, and science everywhere […] it sees the avenging of fate in everyone’s life, how the illusion, the hope, and the trust in the stability of this world is turned into misery” ([3], p. 405).

I suppose, 'Art' back then meant Wagner or Tolstoy rather than Disney Cartoons. 

The next chapter is on transcendent truth.

Which, for Hindus, is just Being- though it may be apprehended as Beauty or Bliss 

That is truth that guides a man to a life free from the shackles of fear and desire. In this chapter and the next Brouwer quotes rather extensively from a German translation of the Bhagavad Gita. He quotes from Book II, texts 60-66, from Book V, texts 20-24, Book VI, texts 20-23. Brouwer calls the quotations an example of “mystical sounds that hover above the previous chapters” ([3], p. 420).

For Christians, this may be the 'still small voice' or it might be as a great rushing wind or the cooing of doves etc.  

And in the next chapter on the liberated life he quotes Book III, 17-19, 27-28 and again from Book V, 7-12. Brouwer quotes Book III, 17: “But for one who takes pleasure in the self, whose human life is one of self-realization, and who is satisfied in the Self only, fully satiated – for him there is no duty”,

The Bhagvad Gita deals with the duty of the agent. Self-realization means you become a principal- like the Vyadha (butcher) of the Vyadha Gita who has attained the honeyed wisdom of the Chandogya while living an affluent life.  

and Book III, 18: “A self-realized man has no purpose to fulfill in the discharge of his prescribed duties, nor has he any reason not to perform such work. Nor has he any need to depend on any other living being”, Book III, 27: “The spirit soul bewildered by the influence of false ego thinks himself the doer of activities that are in actuality carried out by the three modes of material nature [goodness, passion, and ignorance – TK]”. In the German translation that Brouwer uses the ‘the three modes of material nature’ are simply called ‘the forces of nature’. And also Book V, 7: “One who works in devotion, who is a pure soul, and who controls his mind and senses is dear to everyone, and everyone is dear to him. Though always working, such a man is never entangled”, Book V, 8-9: “A person in the divine consciousness, although engaged in seeing, hearing, touching, smelling, eating, moving about, sleeping and breathing, always knows within himself that he actually does nothing at all. Because while speaking, evacuating, receiving, or opening or closing his eyes, he always knows that only the material senses are engaged with their objects and that he is aloof from them”, and Book V, 12: “The steadily devoted soul attains unadulterated peace because he offers the result of all activities to Me; whereas a person who is not in union with the Divine, who is greedy for the fruits of his labor, becomes entangled”.

The context is important. If you are a soldier you may have to kill people but you don't have to enjoy killing.  

The last chapter of LAM on economics contains a warning against attempts to improve society. Brouwer: “Trying to exert outside influence for the sake of a better world or one’s own power is vanity, blind folly, and lust for power.

Nothing wrong with trusting to the 'invisible hand' or 'mysterious economy' which holds at bay the Eschaton.  

The free man rather looks on his fellow men as burdensome hallucinations, luring him away from the right path and trying to make him join their ways because they cannot tolerate his freedom. The free man will carefully avoid them”.

Sound enough advise for a Christian who was also a Mathematician of genius. It may be that his faith sustained him in his work. What few could have appreciated at the time, was that his work would turn out to be so very useful in diverse fields.  But so would the work of Andre Weil- who knew Sanskrit- & Grothendieck's various 'Yogas' which seek to unify algebraic geometry.