L.E.J. Brouwer, A Mathematician on Self

Luitzen Egbertus Jan Brouwer (1881-1966) was a Dutch mathematician who emerged from academic obscurity to briefly challenge the scientific community of his day with a new theory of the foundation of mathematics that he called "Intuitionism." Unlike most thinkers whose ideas develop over the course of their career, Brouwer's ideas seemed to have matured early: in an adolescent "Profession of Faith" and in a lengthy essay composed when he was still a graduate student title "Life, Art, and Mysticism." The subject of the essay was unusual for a budding mathematician, but Brouwer was uniquely fascinated by philosophical issues. The work is a youthful apologia for solitude and the solitary life. The work remained the sentimental grounding for Brouwer's Intuitionism and was an inspiration to him throughout his life.


The life of Brouwer is easily summarized. His upbringing was entirely uneventful. He was an excellent student and quickly progressed through university studies. He married in 1904. As a graduate mathematics student in 1905, Brouwer composed "Life, Art, and Mysticism." About this time he also bought a plot of land in the forest outside of a Bohemian village (Laren) favored by artists and progressive thinkers, located some 20 miles from Amsterdam. A friend designed the cottage and Brouwer called it "de Hut." Here he worked on his 1097 doctoral dissertation (titled "On the Foundations of Mathematics") and would retreat the rest of his life to do his best work.

For five years, Brouwer addressed the field of topology in a flurry of lectures and over 40 scholarly articles. He became the foremost authority on topology in the Netherlands, culminating in appointment to a University of Amsterdam professorship. From this platform he launched his controversial Intuitionism, not only attacking the formalism of the mathematical thinking of his day but also proposing new and insightful views on set theory and negation.

As with most brilliant but unrelenting personalities, Brouwer provoked rivalries and enemies, so that by 1928 he was denied by the university all but mundane teaching (versus editorships, committees, conference travel, etc.). He spent the rest of his life in relative silence, seeking out occasional lectureships in Germany and Britain. He wrote to a friend:

All my life's work has been wrested from me and I am left in fear, shame, and mistrust, and suffering the persecution of my baiting torturers.

Brouwer's last years were spent in what commentator Stigt calls "social isolation." He was killed as a pedestrian by a car when he was 85 years old.

Art, Life, and Mysticism

The circumstances surrounding Brouwer's "Art, Life, and Mysticism" reveal much about his personality. Brouwer had just graduated, and in the Fall of 1904, an arch-Hegelian professor named Bolland had assumed the chair of philosophy at Leiden University. Bolland's pompous and arrogant views provoked young Brouwer into publicly responding to Bolland in a student magazine. Then Brouwer was invited to offer public comments in Delft, for which he composed lectures that became "Life, Art and Mysticism," printed in March 1905.

Stigt calls the essay "the ideological manifesto of one of the greatest mathematical philosophers of this [twentieth] century." "Life, Art, and Mysticism" is arranged into nine chapters:

  1. The Sad World
  2. Turning into Oneself
  3. Man's Downfall Caused by the Intellect
  4. Atonement
  5. Language
  6. Immanent Truth
  7. Transcendent Truth
  8. The Freed Life
  9. Economics


Chapter one opens with a reflection on how Brouwer's native Holland exists in the unnatural state of a perpetual "self-imposed burden," that of shoring itself against the sea. This duty is a living symbol of human decline from a simple and free state of natural existence. Brouwer offers this bit of postulated anthropology:

Originally man lived in isolation, supported by nature. Every individual sought to maintain his equilibrium between sinful temptations. This filled the whole of his life. There was no involvement with others , nor was there any worry about the future. As a result hard work did not exist, nor did sorrow, hatred, fear, or lust.

But the balance was lost when some desired control over others, a cascade of force, oppression, conspiracy, and consolidation of power.

We have now reached the point where everyone has power but at the same time suffers oppression. The old instinct of separation and isolation now lives only as pale envy and jealousy.

Brouwer quotes Meister Eckhart on how every person confuses acting and knowing, living for self-made illusions while insisting on knowledge and truth. By understanding is meant overthrowing and destroying and turning things upside down.


The remedy to this chaos is simply put:  "Look within yourself." At first there is resistance and inertia but the successful result of looking within is control of passions, the falling away of plurality and a sense of "joyful quiescence." Again, Brouwer quotes Eckhart and adds passages from the German mystic Jakob Boehme:

It is within you. If only you can be silent for one hour and forget all your desires and feelings, you will hear the unspeakable words of God.

When you are silent you are like God before the formed nature and creatures, including yours. You will then hear and see with what God saw and heard in you before your own willing, seeing, and hearing had begun.

From this complex field of the mind, its images and presentiments, the self finds context for understanding the world. The mind recognizes free will, causality and direction for self, contentment and equanimity. Looking at the "sad world," one feels thereafter a "pointless dissatisfaction."


In this section Brouwer develops the notion of intellect as the source of human captivity to desire and fear. The intellect links desire and image, object and passion, in a revolving and deepening circle of folly. As the intellect applies itself to society and culture, the circle of desire and fear widen to encompass greater human activity beyond the needs of physical nature. Most nefarious in this collective sphere is manufacturing, technology, and science, transforming the individual into a sickly cog.

In reaction to the cycle of desire and fear, the intellect emerges as conscience, with intimations of happiness and nostalgia for a lost paradise. An individual must be moved by conscience to seek out something higher, but the regimen of society, labor, industry, and duty assure that such pangs of conscience will be well muted. Religion becomes a captive institution, and even art is controlled -- even those with potential to change the prevailing system.

Whoever plays a part in the mechanism of society and so helps to maintain its evil mass production are kept quiet and happy. In books and drama they are told about reformers, revolutionaries, and recluses, about contempt for law and order, self-denial, freely-chosen poverty and hunger, the free life, the Kingdom of God. Such people and their teachers are greatly admired, at least when presented in writings or on the stage: but when they appear in real life everyone is outraged and frightened, and they are locked up in prison or a lunatic asylum.

Brouwer repeats that conscience knows better if it can resist the corrupt world, that "every now and then conscience breaks away from the bonds of the sad world," discovering and admiring, especially when one is young, the purity of "dreamers, monks, and hermits."


Brouwer briefly describes how the corruption of the world easily reabsorbs the average individual into its prescribed ways:

You will be reconciled with your world and not try to change it. You will work, eat, sleep, and travel in your world, knowing it to be your inevitable karma.


Brower explores language as the companion of intellect and the prescribed social form of communication. Language channels the cycle of fear and desire into specific nuances and cross-purposes. Conversation is the intellect's great mask, concealing instincts and feelings in order to establish norms to control social parameters and behavior. But Brouwer argues these conclusions from anecdote and literary example, less than effective, and in a tone of disaffection.


Lingering discontent with the order and values of human society reveal "irruptions of truth." Here truth is sensed but not articulated; it is "immanent truth." It guides, enlightens, makes the nature of life and the world clearer. We see hints of immanent truths in art, for example, which "belies common sense, causality, and science everywhere."

To transcend worldly views, art must first be free of individual temperament -- where the created object is merely ingratiated with the superficialities of the world. Secondly, art must transcend historical materialism, wherein the created object is an intentional agent of the world. Brouwer criticizes comedy as a deliberate optimism that plays upon exaggeration and appearance. He criticizes tragedy as a deliberate accommodation to cruelty and fate. The visual arts are static, he claims, outside of time, offering greater potential for insight.

Brouwer digresses to a discussion of women that is characteristically misogynistic, representing his mindset and that of his era. Women are seen as shifting between tempestuous sirens and mere distractions (to men). Their passion is "free from the illusion of space" and their bodies and temperaments suited to patience and longevity of mind. Brouwer resents the notion of women's equality, linking the social role of women with the social degeneration of institutions and industry of the time. He sees the role of men as the actors and thinkers. Even negatively as "sins of male activity," women will inevitably inherit in a future equality a worse fate than the present fate to which women are doomed to "menial, ignoble tasks." If men's love for women is a "sad, blind passion," then a woman's love for a man is blindness.

These passages are particularly ungrateful and uncharitable aspects of Brouwer's personality and thought. As translator Stigt points out, Brouwer had just married a woman older than himself, well-established financially, and with a thriving business, while he in contrast was something of a penniless vagabond and a long-time student. A clear alliance of interests, if not love, propelled their successful and lifelong marriage, so that these remarks -- never repeated in subsequent writings -- reveal a passion or resentment in the youthful Brouwer. Such passions have often not escaped men, whether contemporary or classical, writing on the subject of women, from St. Jerome to Friedrich Nietzsche.


While immanent truths are suggested by perception, transcendent truth is first approached as breaking the dominant cycle of fear and desire. This negation of reality's harsh foundation suggests delusion and pretext, the salving of conscience. True enough that transcendent truth will not be found in the social order, nor in nearly any art, which is reduced to "titillation in times of prosperity, or ideal endorsement in times of strife and hardship."

Nor is immanent truth to be found in language, whether of philosophers, preachers, nor reformers (the latter being, according to Brouwer, vegetarians, theosophists, and socialists). Here Brower makes exaggerated attacks on these groups as if they dominated the society, culture, technology, and power structure that has brought modern culture to its present plight. Like Plato, Brouwer inveighs against poets as spinners of falsehood -- and likewise priests, politicians, and painters.

But Brouwer admits that "sometimes only the accompaniment of transcendent truth may be heard in life" by the humble-minded, even if truth is absent from that person's perceptions. Those who are "imprisoned in life" will call these flashes or insights mysticism. Those who understand will realize that mysticism is not simply a passing phantasm that does not interfere with life and duty. Rather, mysticism "denies that there is anything positive to be found in this life," that reality is a "pantheistic world" that envelops and accommodates transcendent truth.

Such a perception is an "irritation" to authorities. The shortcomings of Western "semi-mystics" in not developing their perceptions beyond the standard theological imagery has marked them as heretics on the one hand and instable and inadequate on the other.

Mysticism denies knowledge, where occultism promotes the intellect's thirst for it. Mysticism leaves specific issues to rational inquiry, to the mundane realm of human interaction. Mysticism does not promote contradictions but transcends them with irrefutable insights that only can be dismissed but not resolved. Brouwer offers passages from Boehme and the Bhagavad-Gita to illustrate the nature of mystical expression.


The "freed life" is the life of such insight, but it is not necessarily different outwardly, except that it will reveal humility, "disregard for pleasure, property, honor, and work -- except the tasks immediately before one." "Life will move toward absolute solitude," he writes.

Despite this humility and reticence, others will be uneasy, resentful, hostile to the point of violence and vengefulness towards that "freed" one. Such pressure to conform can only be resisted by constantly refreshing the intellect with the insights of immanent truth. Only this effort will "help his [the solitary's] patient move away from human society." The burdens of the body will grow fewer, needs will be eliminated, the environment of the mind cleansed, the path clearer.

He will go on and reach a state of ever greater solitude, poverty, and immobility. The last that society will see of him is when he disappears, a hermit seeking the barren heath over lush but dull vegetation, seeking the night rather than the insipid light of day. Often he will bathe in the ocean. He knows that he is destined for even greater poverty.

This is the poverty Meister Eckhart describes: "those who do not want anything, do not know anything, and do not have anything," as Brower quotes Eckhart in a long passage. Brouwer returns to the Bhagavad-Gita: "That one whose delight is but in self, whole pleasure is in self, whose satisfaction is in self alone, has no work that he must do." Brouwer ends the section with a colorful passage from Flaubert's Gymnosophists -- and, unfortunately, more misogynic rants to mar his placid air of self-confidence.


Brouwer excoriates those who work within society for its improvement. Injustice is intrinsic to society, to the play of power and abuse. Were it otherwise, society would not exist, would see its institutions and structures collapse in "self-correction." But Brouwer's attitude here lacks empathy. While it is true that the masses of humanity will never understand their self-inflicted misery and never come to any form of enlightenment, Brouwer's intemperate tone does not reflect a presumed appreciation of mysticism. Better to tear one's vision away from this sea of suffering and ignorance, Brouwer pleads -- and that is how he resolves the dilemma in the last possible paragraphs.

Look at this world, full of wretched people, who imagine that they have possessions, worried that they might lose them and ever toiling in the hope of acquiring more. Look at all these people, striving after luxury and wealth, those whose riches are secured, whose stocks and shares are safely deposited, and who now nurture an insatiable appetite for knowledge, power, health, glory and pleasure.

Only he who recognizes that he has nothing, that he cannot possess anything, that security is unattainable, only he who completely resigns himself and sacrifices all, who gives everything, who does not know anything, who does not want anything and does not want to know anything, who abandons and neglects all, he will receive all. The world of freedom is opened to him, the world of painless contemplation and -- of nothing.


"Life, Art, and Mysticism" draws its interest as the unexpected work of a youthful mathematician on the brink of intellectual and soul-searching discovery. Brouwer never again referred to this essay, though it forever crafted his philosophy of life and inspired his mathematical work in challenging logic and method with intuition. Though unrefined and crude in places, the essay redeems itself as a sweeping gesture in the style of the time, a heart-felt search for self-understanding and a person's place between society and solitude.



Luitzen Egbertus Jan Brouwer: "Life, Art, and Mysticism," translation and with an introduction by Walter P. van Stigt in Notre Dame Journal of Formal Logic, v. 37, no. 3, Summer 1996, p. 381-429. Dalen, Dirk van: Mystic, Geometer, and Intuitionist: The Life of L. E. J. Brouwer. New York: Oxford University Press, 1999-2005. (2 vols.); Walter P. van Stigt: Brouwer's Intuitionism. Amsterdam: North-Holland/New York: Elsevier Science, 1990.