Elucidation of the secret words


Mysticism and mathematics:

Brouwer, Gödel, and the common core thesis

We would like to thank the editors for inviting us to contribute to this volume. In developing the ideas presented here, we have benefited from discussions with a number of people. In particular, we are grateful to John and Cheryl Dawson, Mitsu Hadeishi, Piet Hut, William Kallfelz, Juliette Kennedy, Rudy Rucker, Steven Tainer, and Olav Wiegand.Moreover, we are indebted to the Dawsons for kindly providing us with a catalogue of Gödel’s private library, and to the Sonnenberg family for creating excellent conditions for us to work together. One of us, van Atten, did his work under a Postdoctoral Fellowship from the Fund for Scientific Research-Flanders (Belgium), which is gratefully acknowledged.

Mark van Atten and Robert Tragesser

Now published in W. Deppert and M. Rahnfeld (eds.), Klarheit in Religionsdingen Leipzig: Leipziger Universitätsverlag 2003, pp.145160.

David Hilbert opened ‘Axiomatic Thought’ [15] with the observation that ‘the most important bearers of mathematical thought,’ for ‘the benefit of mathematics itself have always [. . . ] cultivated the relations to the domains of physics and the [philosophical] theory of knowledge.’ We have in L.E.J. Brouwer and Kurt Gödel two of those ‘most important bearers of mathematical thought’ who cultivated the relations to philosophy for the benefit of mathematics (though not only for that). And both went beyond philosophy, cultivating relations to mysticism for the benefit of mathematics (though not for that alone).
There is a basic conception of mysticism that is singularly relevant here. (’Mysticism’ labels that.) That corresponds to a basic conception of philosophy (’Philosophy’), also singularly relevant here. Both Mystic and Philosopher begin in a condition of seriously unpleasant, existential unease, and aim at a condition of abiding ease. For Mystic and Philosopher the way to that ease is through being enlightened about the real and true good of all things. Thus Mysticism and Philosophy are triply optimistic: there is a real, true good of all things, the Philosopher and Mystic can become enlightened about it, and being thus enlightened would give them ease.

There is an essential difference between the Mystic and the Philosopher and probably it can illustrated best by means of Plato’s “Allegory of the Cave.” The to the wall chained ones, unfortunately almost everybody in this world (except the little kids), don’t see and experience Reality but only see what they consider to be reality. There are people experiencing at some time in their lives a such unbearable way of living, that they little by little free themselves with great trouble from their chains and go the way out and behold as they all call it “The Light”, Reality. Also there are many people having experienced a so called Mystic Experience, provoked by drugs, extreme exhaustion, a Near Death Experience or straight away at once like a bolt from the blue. They throw, so to speak, although chained, a glimpse at the Light. It is described by Alfred Tennyson as follows: (from a letter by Tennyson to Mr. R.P. Blood): "... a kind of waking trance, I have frequently had, quite up from my boyhood, when I have been all alone. This has generally come upon me by repeating my own name two or three times to myself, silently, till all at once, as it were out of the intensity of consciousness of individuality, the individuality itself seemed to dissolve and fade away into boundless being; and this not a confused state, but the clearest of the clearest, the surest of the surest, the weirdest of the weirdest, utterly beyond words, where death was almost a laughable impossibility, the loss of personality (if so it were) seeming no extinction, but the only true life ..." It is like people wandering in the labyrinth sometimes throwing a glance outside across the hedge and afterwards continuing their ways. Between true and occasional mystics there is a essential discrepancy but there are similarities too. One of the similarities is that both of them are trying to put into words their experiences in vain, for that’s entirely impossible. It is as impossible as trying to explain to someone who never has tasted a strawberry what the taste of a strawberry is like, or explaining somebody who never has been seasick what it is feeling like. After all, words only have meaning for people sharing the same experience and then they are superfluous. That is why Ludwig Wittgenstein (Tractatus Logico-Philosophicus, 7), wrote that ‘what we cannot talk about we must pass over in silence’ and Lao Tzu (LVI) that ‘people that talk don’t know en people that know don’t talk.’ The Mystic who tries to put in words his experience is the in the cave returned one, unable to acclimatize again to darkness and apparent reality. The occasional mystic only has beheld Reality one single moment and recovers himself chained to the wall again. The Philosopher is and stays a chained one, able at best to describe the structure and quality and indissolubility of his chains and therefore has to conclude he never can and will behold Reality. Apart from that, the word philosophy means “to love wisdom” and originally pointed to a way of life. Consequently ‘philosophy of science’ is an impossible hybrid, a contradictio in terminis, just like philosophy of mathematics or philosophy of culture. What is called philosophy in this world is merely sophistry, building systems or thinking about what all those system builders may have meant.   

That Enlightenment sought comes from some sort of cognitive or intelligent engagement with what we will here call ‘the Good’.

Naming “The Good” for “Reality” is incorrect because “The Good” is an attribute attached to “Reality”. Moreover it implies the existence of “Evil” and so emphasizing duality. In “Jenseits von Gut unde Böse”, Nietzsche writes: “In  the ‘in itself’ are no ‘causal relations’, ‘necessity’, psychological constraint’, there does not follow ‘result after cause’ , there doesn’t rule a law. It are we and nobody else that have invented causes, the ‘one after another’, the ‘one before another’, relativity, compulsion, number, law, freedom, logical foundation, and aim.” Therefore in what exists there is no good or evil, over or under, in or out, winners nor losers, body nor mind, awareness of time or space, as it is strange to all little children.
The quest for Enlightenment has nothing to do with intelligent engagement with “The Good”, but “The Good” only can be ‘dis’-covered by removing all by what it is covered up. So the quest only can brought to an end per viam negationis. Virgin nature, so human nature too, only can brought to light by removing all unnature, all cultural artifacts, all human creations, all that is not innate, like you have to demolish sand castles to see the original beach again or can acquire a tabula rasa again by erasing all that is written on it. “The way up and down is the same”, Heraclites wrote (fragm. 57).   

Some use ‘the Absolute’ when it seems important to emphasize that ‘the Good’ is unconditioned—there is nothing behind it, nothing above it. Others use ‘the One’; still others, ‘God’.
It is natural to regard the Good as somehow mind-like, or like something (permanently) in mind.

But it is incorrect, because it would mean that existence would depend on what is present in the brains of men.

It should in either case be in some way homogeneous with, or in sympathy with, our minds, for the Good must attract and support the intelligent engagement of it by our minds. In that way it can enlighten us.

Reality or “The Good” cannot enlighten brain, on the contrary even. Thinking, conditionedness, just hinders beholding Reality, all that exists. Thinking darkens, makes blind and deaf, for it does people see what they think they see and hear what they think they hear. Knowledge is unnatural, because people first have to acquire it, next constructing with that a world picture and after that reflecting about all they did make their ‘own’. In fact consequently all knowledge is plagiarism, merely embroidering on others people thoughts. 

The distinction between Philosophy and Mysticism is a matter of degree. Philosophy is dominated by the intention to articulate and rationally proof all claims and insights. Mysticism is not so dominated. But nevertheless, perhaps at some point close to the Good, where every step so far has been rationally proofed, the Philosopher could well have the final and most sublime enlightenment, but find that it is beyond his power or interest to articulate and rationally proof the content of that.

The sophist (the reflecter) never can behold Reality and therefore never can reach Enlightenment, because he reflects and looks through his constructed thinking system, not realizing that. The Mystic, looking unbiased, can point out all obstacles hindering to see clearly and point out the way out and describe the chains by which his fellow men are chained, but as Plato wrote (The Republic VII, 514-520) “and if they find anybody trying to set free and enlighten one of their number, they will put him to death, if they can catch him”. It’s the fear for freedom, loss of control, pulling down the so laboriously constructed world picture, the fear for letting go. But the most important reason why people are so attached to their prison is that they have to admit of being mistaken their whole life, that all their drudging has been merely vanity and vexation of spirit. The more learned, the more foolish.

It could be beyond his interest because the massive insight is so bright and sharp, so ineluctably clear and certain, that any rational proof at its best could only yield something comparatively darker and less compelling. Gödel attributed such an experience to various philosophers:

I myself never had such an experience. For me there is no absolute knowledge: everything goes only by probability. Both Descartes and Schelling explicitly reported an experience of sudden illumination when they began to see everything in a different light. [22, p.170]

The Mystic Experience of Descartes and Schelling, Rousseau, Alan Watts, Brouwer, the Baghwan and thousands of other people, have been the starting point of their “Philosophies”, the failed attempts describing their experiences. They didn’t understood what and why they had met what they did experience. They all were presuming having discovered something quite new, whereas everybody always has known, what they tried to clarify and every little child could have tell them.

Where we encounter a Philosopher making claims from out of such a moment, but without any successful rational proofing of those claims, then we can regard what he has given us as Mysticism. In what follows, we will think of the practice of Mysticism as trying to find ways to experience this Good directly. The practice of Philosophy is the attempt to describe this Good intelligently.

Philosopher and Mystic have a totally different starting point. The Mystic has beheld Reality and tries to clarify to his fellow men the way they can do it too by describing for them the way leading to it, the Philosopher doesn’t know what he is looking for, he is a seeker not knowing the obstacles, not realizing he is wandering in a labyrinth he has woven and maintains himself. By thinking he tries to eliminate his thinking and that is an impossible Münchhausen trick.
Apart from that, labeling of someone as a Mystic usually is a successful attempt of the establishment making him inoffensive and harmless. Moreover a depressed, ill or unlucky mystic is a pseudo-mystic (hemi-mystic Brouwer called that)

Below, we will describe how Brouwer and Gödel each relate mysticism and mathematics, and make a comparison. On the basis of that, we then present a partial argument against what is known as ‘the common (or universal) core thesis’ (CCT). CCT says that the various mysticisms in the end all are just different ways to express the same core of truths. It seems to us that the common core thesis can be analyzed into two propositions:

(a)    Mysticism holds that Reality is Good. Mystical practice aims to perceive this Good.

The Mystic doesn’t state that Reality is Good, but he experiences that beholding and experiencing Reality, having an empty head, implies an imperturbable blessedness.

(b)    This Good is objective, i.e., the same for all varieties of mysticism.

Experiencing Reality, The Good, for every Mystic is identical. The only problem is they all followed a different way from an other starting point. Consequently only their terminology is different, deriving that from the culture they escaped from. Reality is all existing and so objective by definition, because Reality is disinterested and all attributes ascribed to it, exclusively is saying something about the attributes the prejudiced beholder has burdened himself with.

Of course (a) is only a minimal characterisation of mysticism. It leaves out most aspects of mysticism (e.g. feelings of bliss), but it seems extensionally adequate. We take it to be empirically adequate. ‘Reality’ with a capital ‘R’ has a different meaning than ‘all-inclusive reality’. The latter surely is one, in the tautological sense that there can be nothing outside of it. But the Good known from mystical traditions has more meaning attached to it.

Reality is identical with ‘all-inclusive reality’. Naming “Reality” “The Good” is ascribing an attribute, a quality of experiencing Reality, to Reality. The same applies for beautiful, perfect, glorious, blessed, free, all being no qualities of Reality, but merely an awkward attempt describing experiencing Reality.

What makes CCT prima facie implausible is that among the various mystical traditions, and even within each tradition, we find so much disagreement on explicit doctrine and methodology.

Doctrines and methodologies are something quite different from the awkward descriptions of Mystics of their mystical experience. People from all over the country describing the way to Amsterdam, don’t describe Amsterdam. The card isn’t the menu.

However, the interest of the common core thesis depends on the existence of such disagreements, for in the absence of that it would be almost trivially true.
There is a somewhat analogous case in the philosophy of science: scientific realists hold to a common core thesis with respect to scientific theories through the ages.

A scientific realist is a contradiction in terms. A scientist is a theorist, seeing Reality through his paradigm, his colored glasses, irrevocably distorted. For instance an evolutionist filters all he sees through his evolution glasses and a creationist through his Genesis glasses. They don’t see Reality, but see wh what they think about Reality. People adhering a theory wear colored glasses and tell nonsense by definition. There is no reason at all for taking seriously someone believing for example in the evolution theory or progress or getting cancer by smoking, but unfortunately believers are not amenable for reason.

These theories show massive disagreements; still, the realist holds, they all try to express the same objective reality.

Realists and Mystics realize they see the same indivisible Reality, trying to express not that, but only their experiences of that reality and that’s impossible.

One is reminded of a metaphor that Leibniz used: The same city may look very different depending on what direction you approach it from.

But no matter how, that town remains the same town. Indeed everyone looks from his “own” point of view and never can oversee the town in whole and all descriptions given by them consequently are reduced. Theoretically an infinite number of points of view is impossible and a complete description therefore would require an infinite number of words in an infinite number of books. Jorge Luis Borges gives a beautiful metaphor for that in his short story “The Library of Babel”, an endless library, “a sphere, being the right centre any hexagon, and the perimeter out of reach, containing all possible books,” remaining the town just the town and Reality just Reality. So there are as many points of view and world pictures, as there are people and from their points of view and world pictures they all are always right, but they don’t realize their point of view and world picture are fictitious.

Of course, neither the analogy to scientific realism nor Leibniz’s metaphor adds support to CCT. They just suggest how the thesis might be true in spite of prima facie evidence against it. The argument we want to suggest aims to weaken the case for CCT. It attempts to show that the references of Gödel’s and Brouwer’s terms for the Good cannot possibly be the same (Gödel speaks of ‘the Absolute’, Brouwer does not have a term but speaks of a return of consciousness to ‘its deepest home’).

Indeed Gödel merely talks about the for him unreachable vanishing point, Brouwer explicitly talks about the return, because he did experience that it is possible.

This leaves open the possibility that at least one of them even does not refer at all. The Good as conceived of by Brouwer may not exist, and the Good as conceived of by Gödel may not exist. One can intend, but not establish, reference to something that doesn’t exist. So an argument from the assumption that at least one of these does not exist to the conclusion that Gödel and Brouwer cannot be referring to the same thing is trivial.

By utterly consequently reasoning Gödel comes to the conclusion there is a magic border, behind which he suspects “the Good”, but never accessible for him, because it finally requires a magic jump, a human being never can make by thinking. It’s like endlessly diluting a salt solution. In that way you never can get perfectly pure water. As long as there is still one molecule NaCl in it, the water isn’t pure. A conditioned person can de-condition himself but never can reach the vanishing point of his own accord. “The jump to the being oneself and to freedom can be realized out of an extraordinary experience. This originates just in face of despair and out of the understanding of its impossibility: it is the experience of being given,” Karl Jaspers writes, and “Nihilism is an inevitability for the sincere man”, and therein he meets Brouwer.

The case that remains is to assume that both do exist and see if you can then also reach the conclusion that they cannot in fact be the same. Therefore we will consider the latter case.

1 Brouwer’s Mysticism

Brouwer thought (did experience) that there was a ‘deepest home’ of consciousness [5]. In the deepest home, our experience oscillates between stillness and having sensations.

In our deepest home, our true home, man’s cleavage is dissolved, the inside is like the outside, there are no  thoughts no more, no unrest or tension; there is a perfect contentment and man who became human again doesn’t feel standing himself opposite to all that “is”, but feels himself one with it. He doesn’t feel ‘his body’ no more, has no more emotions, but only experiences bliss. He is imperturbable. That’s what the Stoics understood by apatheia and all religions have their equivalents for.

There is no subject-object distinction there. This state Brouwer identifies with wisdom (compare [3, p.108] and [5, p.1240]). Our awareness of objects and other people arises in various stages on what he calls an ‘exodus’ of consciousness from the deepest home.

Because he has no knowledge no more and realizes he cannot know anything, because he has understood that all he did learn are merely awkward and darkening phantasms, he is wise. There are no more objects to him, no multitude of things, but an experiencing that all that “is” in one immense connection is inspired by: and here words fall short.

The first step of this exodus is the result of a free-will act that introduces an awareness of time.

Free will too is such a bizarre chimera, an obstinate illusion. In “Gödel, Escher, Bach” Hofstadter writes: ‘Just until a man has resigned himself and chooses his own desires (as well as the choice to choose his own desires and so on), one may state that he has an own will and can make choices.’ Men are slaves of their desires, their fears, their libido, their self-constructed worldviews, making do them what they do. Men are caught in a network of people influencing each other, all striving after maintaining the status quo. ‘Each family,’ as Samuel Johnson wrote, ‘is a small kingdom, torn by party rows and revolutions.’

In fact, time consciousness is a prerequisite for the awareness of objects and people (including oneself as an embodied person) and everything else in the exterior world.

No one child discerns things or people and it doesn’t experience time, because it has no knowledge of past and future. It all are little mystics, only being and experiencing wordless. So that is what we call childlike simplicity. But children have to adapt to grown-ups, who let determine their present by their past and their expectations and fears for the future. They draw children out of their paradisiacal state, out of their eternal life in the present, and once begun there is no end, so think on the end before you begin. Brouwer names that childlike and so experiencing mystically: ‘the original state wherein awareness softly floats there and back between complete silence and awareness’ and he makes that mistake by leaving out of account little children, for which Reality is anything but a chaos. The chaotic state is the state of grown-ups.

It is time awareness that introduces a distance between the experiencing ‘I’ and what is experienced. The latter recedes into the past, as a memory, while the former remains in the streaming ‘now’. This is the genesis of intentionality (a word that Brouwer does not use). Brouwer calls consciousness in so far as it exhibits intentionality ‘mind’ (we will, after the discussion of this particular aspect of Brouwer’s philosophy, not use that word in his technical sense). Once that is in place, the mind further develops with consciousness indulging itself in organising sensations into complexes, in particular into ‘causal chains’ and ‘things’: the former being vehicles for empowering the will to control the latter. Whatever hold all of these mind-particularizing contents have on the particular self, it is a hold that self maintains; the self could in principle and in practice free itself from that hold by as it were disclaiming all the relevant sensation complexes, for those complexes were adopted on the foundation of absolutely free will intrinsic to consciousness:

Everyone can have the inner experience, that he can at will dream himself to be without time awareness and without the separation of the I and the world of perceptions, or bring about this latter separation by his own effort [4, p.154] [our translation]

That is incorrect. You cannot evoke that condition by willing, but it happens to people. Meditating is an absolute insufficient method to bring about that and if it ever would happen for a moment, then thoughts come back irresistible and afterwards people simply turn to the order of the day again.

Mathematics, Brouwer says, is also built up from our experience of time, as in Kant—hence the name ‘intuitionism’ for Brouwer’s philosophy of mathematics, referring to the pure intuition of time.

Actually intuition means “the power or faculty of attaining to direct knowledge or cognition without evident rational thought and inference”(Merriam Webster Dictionary), so without thinking and without any hindrance by prejudices. Mathematicians makes use of thoughts and that is why intuitionism never can be put together with mathematics. Mathematics is a hindrance for intuition and that Brouwer has understood very well. But Brouwer evidently has tried to make the best out of two irreconcilable worlds and has yielded to that one world (for he had to keep the pot boiling too), and thus staying a wanderer between heaven and earth. It is not Both/And but Either/Or, it is Kierkegaard’s Enten/Eller.  No one can serve two masters unpunished.

The discrete (the natural numbers) arises from our awareness of successive ‘nows’, the continuous (e.g. the straight line) from our awareness that time is a flow and hence there is something ‘in between’ the discrete ‘nows’. In what Brouwer calls the unfolding of this basic intuition, all of mathematics is created. On this picture, mathematics is a creation of the individual mind. It does not describe an independent reality. It comes into being in an act of the will. In formalizing mathematics, on the other hand, any possible volitional elements are precisely shut out. This is why Brouwer kept clear from formalizing intuitionistic logic (as his student Heyting did), and from setting up a formal theory of the role of the subject in mathematics. As regards the latter endeavour, Stanley Rosen has aptly remarked that

[A]nalytical philosophy [. . . ] objectifies the subject, or overlooks the presence of the subject in the structure of the proposition [. . . ] This tendency is illustrated in the attempt by Kreisel and others to mathematize Brouwer’s conception of the creative subject as expressing the force of mathematics, a force that cannot itself be expressed in mathematical terms. [17, p.186]

According to Brouwer, if you look at it from the philosophical and not merely technical point of view, engaging in mathematics is one of the first things that lead consciousness out of its deepest home.

That is incorrect. When a little child has reached the age of one year, and actually far earlier, the mischief is done since a long time. Then it lives, by rewards and punishments, in time, in rituals, in patterns and in the rhythm of his educators. Like Ronald Laing (in: The Politics of Experience, 1967) described it so strikingly: “Long before a thermonuclear war can come about, we have had to lay waste to our own sanity. We begin with the children. It is imperative to catch them in time. Without the most thorough and rapid brainwashing their dirty minds would see through our dirty tricks. Children are not yet fools, but we shall turn them into imbeciles like ourselves, with high IQ's, if possible.
From the moment of birth, when the Stone Age baby confronts the twentieth-century mother, the baby is subjected to those forces of violence, called love, as its mother and father, and their parents and their parents before them, have been. These forces are mainly concerned with destroying most of its potentialities, and on the whole this enterprise is successful. By the time the new human being is fifteen or so, we are left with a being like ourselves, a half-crazed creature more or less adjusted to a mad world. This is normality in our present age.
Love and violence, properly speaking, are polar opposites. Love lets the other be, but with affection and concern. Violence attempts to constrain the other's freedom, to force him to act in the way we desire, but with ultimate lack of concern, with indifference to the other's own existence or destiny.
We are effectively destroying ourselves by violence masquerading as love.”

Consciousness builds up its world by starting an ‘exodus from the deepest home. We saw that he thinks of these building processes as operating on sequences of sensations, and that is where mathematics comes in. It is not only when doing technical work, but when just constructing 1 and 2 that you are on the wrong track, according to Brouwer! In his notebooks in which he conceived his 1907 thesis, there are many astonishing remarks on how destructive he thinks mathematics is. For example, one there finds gems such as ‘One could see as the goal of one’s life: abolition and delivery from all mathematics’ [8, p.83]. And he meant it: in writings all through his career, Brouwer comments on how mathematics (and, based on that, the natural sciences) introduces great unhappiness in our lives and keeps us away from attaining wisdom again (by returning to the deepest home). Without time awareness, there can be no mathematics.

And thus no scientist, minister, politician, carpenter, homosexual, Muslim, footballer, patient, poet, mystic, pigeon fancier, American, confectioner, soldier or other actor whatever. In other words, learning tricks requires time. Men only can be men, like all little children are. If you only take out one cog out of the machinery, the whole wheel work stops.

But to be free of mathematics is exactly what we should aim for in our pursuit of the deepest home. And there is even a chance of using language to indicate mystical experiences; but not in the form of analytical (i.e., mathematically structured) prose:

Perhaps the greatest merit of mysticism is its use of language independent of mathematical systems of human collusion, independent also of the direct animal emotions of fear and desire.

Mysticism has no merit at all, but it just is only disastrous, specious and absolute non-committal drivel. No more than the effusions of someone who climbed the top of the Mount Everest are of any use for the left behind, the stammering of the mystic, freed from his chains and having beheld the Light is of use for the still chained one.
Fear and desire are  no animal emotions, but results of living in time and so of abandoning the straight and simple path.

If it expresses itself in such a way that these two kinds of representations cannot be detected, then the contemplative thoughts—whose mathematical restriction appears as the only live element in the mathematical system—may perhaps again come through without obscurity, since there is no mathematical system that distorts them.[Our modification of Van Stigt’s translation [19, p.409]; original emphasis]

The mystical writer will even be careful to avoid anything that smacks of mathematics or logic: weak minds might otherwise be easily made to believe and act mathematically outside the domain where this is required either by the community or their own struggle for life and end up in all kinds of follies. [19, pp.409–410]

Nowhere in mysticism is there a thread or appropiate sequence; every sentence stands by itself and does not need another to precede or follow it [1, p.76] (trl. [19, p.122])

As examples of such language, Brouwer quotes, in 1905, from Meister Eckehart and Jacob Boehme [1]; and in 1948, from the Bhagavad-Gita [5]. The intellect has nothing to do with it. Access to the Good is only possible when the intellect is switched off.

Brouwer has a strange preference for the siren song of flipped mystics (actually all mystics are off the track) who always are trying utterly awkwardly putting into words ‘the unspeakable’. And wherever mystics are clear and direct, he repudiates them, because in his view they degrade what should sound elevated. In his view mysticism is somewhat for intellectuals, for literate people, while he on the other hand states the intellect is de great obstacle and evildoer. In that he is utterly contradictory.

In a review (1915) of a book called ‘Geometry and Mysticism’, Brouwer wrote:

As the making and observing of mathematical forms in the Anschauungsworld is a preparation for, and a consequence of, the intellectual self-preservation of man, and since theoretical mathematics can only be defined as the activity of the intellect in isolation, and since furthermore, mystical vision only begins after the intellect has gone to sleep, practical nor theoretical geometry can have anything to do with mysticism. [8, p.287] [original emphases]

We note that for Brouwer, mystical practice was a serious and solitary affair.

And so that is one of his mistakes, the same mistake made by pillar saints and hermits of all times. Like Flaubert, quoted by Brouwer, in The Temptation of St. Anthony makes saying Hilarion to Antonius: "Hypocrite! who plunges himself into solitude to free himself the better from the outbreaks of his lusts! You deprive yourself of meat, of wine, of stoves, of slaves, and of honors; but how you let your imagination offer you banquets, perfumes, naked women, and applauding crowds! Your chastity is but a more subtle kind of corruption, and your contempt for the world is but the impotence of your hatred against it! This is the reason that persons like you are so lugubrious, or perhaps it is because they lack faith. The possession of the truth gives joy.” That other Mystic said at least: “Be in the world, but not from the world.” It is his fictitious idea of free will, that Brouwer has given the idea that he has reached his insights by his own power, choice and free will, while he only discovered what every little child knows. That is the reason of his absolute lack of compassion with his fellow men. That is the fundamental difference too between Brouwer and his alter ego Carl Adama van Scheltema and for instance Gerrit Mannoury and Anton Pannekoek.

When visiting Krishnamurti, Brouwer said to a friend: ‘Oh my, this is the baby room of philosophy’. [9, p.324] [our trl.]
For Brouwer the intellect plays a negative role in spiritual life. Mathematics is a necessary step away from apprehending mystical truth to apprehending the outside world, one’s own body, one’s fellows.

There is no spiritual living, there is only living or being lived, experiencing unity or experiencing separateness, of freedom or lack of freedom. Neither there is a mystic truth. Something is true or not true, but never mystically true. And mathematics is not able at all to understand the outer world, not to mention the own body or your fellow-men. There is an essential difference between understanding and explaining. Explaining you do with help of theories, self-created systems of thinking, and Heraclitus tells about that: “human opinions are structures like little children make to play with” (quote by Iamblichus) Understanding you only can with your common sense. He who doesn’t understand himself, also cannot understand others, for he then beholds the mote in his brother’s eye, forgetting the beam in his own eye.

2 Gödel’s Mysticism

Rudy Rucker has reported on his conversations on mysticism with Gödel [18, pp.182–183]. Gödel’s philosophy of mathematics is called platonism. Philosophy of science is a contradiction in terminis He held that mathematical objects are part of an objective reality, and that what the mathematician has to do is perceive and describe them. Gödel once published some very brief remarks on how we have a perception of the abstract objects of mathematics in a way that is analogous to our perception of concrete objects [12].

Someone who perceives objects, experiences a shivered unity. He stands not in Reality, but opposite to Reality. He ‘has’ a hand, a gall-bladder, a stomach, and the same way he has divided himself, he has divided his outer world, by artificial limits. In oneness everything is connected with everything.

Rucker, seeking elucidation of these remarks, asked Gödel ‘how best to perceive pure abstract possibility’. Gödel says that, first, you have to close off the other senses, for instance, by lying down in a quiet place, and, second, you have to seek actively. Finally,

The ultimate goal of such thought, and of all philosophy, is the perception of the Absolute [. . . ] When Plato could fully perceive the Good, his philosophy ended.( The original incorrectly has ‘Plautus’ instead of ‘Plato’, but Rucker confirmed to us that this is a misprint.)

Therefore, according to Gödel, doing mathematics is one way to get into contact with that Absolute.

Doing mathematics is a sure way to prevent an entering in “the Absolute’ and Brouwer has understood that very well. The Theory of Everything is an unreachable utopia.

Not so much studying mathematics as such, but studying it in a particular frame of mind. This is how we interpret Gödel’s remark about Plato. There is, then, no break between mathematical and mystical practice.

So for mathematics there is an impregnable barrier to mystic experience. Like Kahlil Gibran said poetically: “(Faith is) an oasis in the heart which will never be reached by the caravan of thinking.”

The one is part of the other, and the good of mathematics is part of the Good. Gödel also talked about his interest in perceiving the Absolute with his Eckermann, Hao Wang. Wang reports:

One of Gödel’s recurrent themes was the importance of experiencing a sudden illumination—like a religious conversion—in philosophy. (This theme, by the way, reminds me of the teachings of Hui Neng’s ‘sudden school’ of Zen (Chan) Buddhism in China.) In particular, Gödel believed that Husserl had such an experience at some point during the transition between his early and later philosophy.’ [22, pp.169–170]

In the late 50’s, Gödel began to develop an interest in Edmund Husserl’s phenomenology. Besides a specific application of phenomenology to the foundations of mathematics, Gödel had a broader interest. This is again related to the Absolute. To Wang he said,

At some time between 1906 and 1910 Husserl had a psychological crisis. He doubted whether he had accomplished anything, and his wife was very sick. At some point in this period, everything suddenly became clear to Husserl, and he did arrive at some absolute knowledge. But one cannot transfer absolute knowledge to somebody else; therefore, one cannot publish it. A lecture on the nature of time also came from this period, when Husserl’s experience of seeing absolute knowledge took place. I myself never had such an experience. For me there is no absolute knowledge: everything goes only by probability. Both Descartes and Schelling explicitly reported an experience of sudden illumination when they began to see everything in a different light. [22, pp.169–170]

and, as we saw above,

Later, Husserl was more like Plato and Descartes. It is possible to attain a state of mind to see the world differently. One fundamental idea is this: true philosophy is [arrived at by] something like a religious conversion. [22, p.293]

Each new acquired and contracted notion, let a man see the world different. But that is not the issue. The issue is looking to yourself and the world open-minded, putting off your colored glasses and not replacing it by glasses with a different color. The ultimate paradigm is no paradigm, simply putting off your colored glasses.

It is likely that Gödel tried to experience such an illumination or conversion. In this connection, we mention that besides books on Christianity and Islam, introductions to Buddhism, Watchtower publications, works on theosophy and some on spiritism, Gödel’s personal library also contained

Wallace, R.K. (1973) The physiological effects of transcendental meditation, 3rd ed., MIU Press.

Rudy Rucker asked Gödel if he believed that there is a single Mind behind all the various appearances and activities of the world [18, 183]. Gödel assented: ‘yes, the Mind is the thing that is structured, but the Mind exists independently of its individual properties’.

If Mind is independent, it makes no sense saying that mind has individual properties too. Moreover naming Mind a structured thing, is a queer construction.

When Rucker then went on to ask Gödel if he believed that the Mind is everywhere, as opposed to being localized in the brains of people, Gödel again assented, saying, ‘Of course. This is the basic mystic teaching’.
Gödel was convinced that ‘the world is rational’, and that this rationality can be grasped by the mind: ‘There is a scientific (exact) philosophy and theology, which deals with concepts of the highest abstractness; an this is also most highly fruitful for science’ [22, p.316]. For Gödel, then, the intellect has a positive role to play in spiritual life.

Brouwer had understood that it is nonsense.

3 Comparison of Brouwer and Gödel: mathematics and the Good

We have seen that both Gödel and Brouwer were looking for mystical experiences, in which an openness of the mind to the Absolute is operative.

Brouwer was not looking for Mystic Experience. He experienced that, but could not find it again.

What is disclosed in such experiences has the air of being something imparted to the person. The imparting is preceded by a preparation or transformation of the person. The self must be brought into a condition to receive, support, and appreciate what is to be disclosed. This preparation we see mentioned by both Brouwer (the abandonment of mathematics) and Gödel (closing off the senses, etc.)

What has to be disclosed (you can call it common sense) is overgrown by intellect, knowledge, opinions and constructions and first that crust has to be demolished, before you can make use of it. You only can look and listen for yourself, if you are hindered no longer by the presumptuous brain, thinking it always knows better.

However, they made very different claims as to how what is disclosed in such experience is related to mathematics. What strikes us is how the bond between mathematics and mysticism is equally tight in Gödel and Brouwer, but that the signs are different so to speak. According to both, mathematics relates individual thought to ultimate reality, but Gödel thinks of a positive relation and Brouwer of a negative one. For Gödel, doing mathematics is a way of accessing the Absolute. For Brouwer, doing mathematics precisely prohibits access to the Absolute.

And that it is. But not only doing mathematics, but also being concerned with past and future, from what all human toiling proceeds, including mathematics.

Put differently, according to Gödel, mathematical experience reveals (part of) Reality; according to Brouwer, mathematical experience conceals Reality.
A mystical disclosure in the relevant sense has about it the phenomenological character of being a form of knowing or enlightened understanding; it discloses the Good, the significant, the important, fundamental values. Therefore, we may try to formulate the difference between Brouwer and Gödel in terms of the Good. What do they think about the relation of the Good and mathematics, and what do they think is the good of mathematics itself? (G.H. Hardy [14] aims at evaluating the good of mathematics and the good of mathematics in relation to himself. But he certainly is straining to avoid mystic ways. This gives a contrast between mystical and non-mystical evaluations.)
Let us call historical mathematics, mathematics as it is now and has been standardly practiced, H-mathematics. We can speak of the good of the good of such hammers as the one that we happen to find in our toolbox. But for that good, this hammer may not be the best we could make. Similarly, it could very well happen that the good of mathematics is not best served by any H-mathematics. That is to say, mathematics at its best (given what is the good of mathematics) may be rather different than H-mathematics.
Brouwer’s intuitionistic mathematics is often construed as merely an epistemological or semantical affair. But it might be better understood as a reform of H-mathematics in the direction of better serving its good. Brouwer tried to to realize a mathematics which is at once a creation of the free will for the sake of the fullest, most free, and most concrete exercise of the will. Our will and inner time are coeval, and inner time is where the will meets causality. Definitely controlling the structuring of time is the finest possible preparation for the will to exercise itself on causality through those temporal structures, which are the structures of intuitionistic mathematics. The good of mathematics, on this picture, is that it facilitates our will to power.

Nietzsches ‘Der Wille zur Macht,’ is a cultural artifact, born by fear. No one little child knows a craving for power, has not at all a need for power, but only can survive power and manipulations of its educators by revolting. In this by men created sado-masochistic universe the basic principle is dominion by one over an other. Every relationship between men is based on power and manipulation, but that is already said by Quohelet (8:9).

Brouwer’s program shows how a particular understanding of the good of mathematics can have a revisionary effect on mathematical practice.

It was not Brouwer’s intention to revise mathematics, but his eventual goal was to abandon mathematics. He wanted to show that the foundations of mathematics were shoddy, and that it therefore was a house build on quick sands, or like the Dutch nursery-rhyme: “once there was a man, he was not wise and build his house on ice….(and then the thaw set in, etc).” The fairy tale of the  “Snow Queen” is about the “icy game of reason”: “In the midst of its empty, endless hall of snow was a frozen lake, broken on its surface into a thousand forms; each piece resembled another, from being in itself perfect as a work of art, and in the centre of this lake sat the Snow Queen, when she was at home. She called the lake “The Mirror of Reason,” and said that it was the best, and indeed the only one in the world.
Little Kay was quite blue with cold, indeed almost black, but he did not feel it; for the Snow Queen had kissed away the icy shiverings, and his heart was already a lump of ice. He dragged some sharp, flat pieces of ice to and fro, and placed them together in all kinds of positions, as if he wished to make something out of them; just as we try to form various figures with little tablets of wood which we call “a Chinese puzzle.” Kay’s fingers were very artistic; it was the icy game of reason at which he played, and in his eyes the figures were very remarkable, and of the highest importance; this opinion was owing to the piece of glass still sticking in his eye. He composed many complete figures, forming different words, but there was one word he never could manage to form, although he wished it very much. It was the word “Eternity.” The Snow Queen had said to him, “When you can find out this, you shall be your own master, and I will give you the whole world and a new pair of skates.”

Notice also that we can speak of the good of something without that good being ultimately beneficial to us, and in that way not part of the Good. This is how Brouwer speaks of the good of mathematics.

At the same manner you endlessly can talk about “God”, “Reality”, “the Absolute”, honesty and a just world, without any result, but talking about something is something quite different than living according it.

It facilitates our will to power, but thereby collaborates in furthering our Fallen Condition, and in that sense is an evil. The good of mathematics does not coincide with any absolute good. As Brouwer wrote in the notebooks already mentioned, ‘That mathematics and its applications are sinful follows from the intuition of time, which is immediately felt as sinful’ [8, p.83]. What Brouwer means is that it is the move of time which leads to the way out of the deepest home; anything that keeps you from returning there is defined as ‘sinful’. However, Brouwer did acknowledge a more intrinsic yet limited or conditional goodness of parts of mathematics.

What he actually wanted to say, Brouwer has written in his Delftse Lectures and within a year afterwards he married, and that is in flat contradiction with all he wrote about women and relationships. Perhaps he remembered what Socrates had experienced himself: “By all means, marry. If you get a good wife, you'll become happy; if you get a bad one, you'll become a philosopher.” or what the Sufi-mystic (!) Rumi had said on marriage " God showed the Prophet a straight and hidden way. What is that way?
Marriage, so that we can endure the trials of living with the opposite sex, to listen to their demands, for them to ride roughshod over us, and so in this way to refine our own character. By enduring and putting up with the tyranny of your spouse it is as though you rub off your own impurity on them. Your character becomes good through forbearance; their character becomes bad through domineering and aggression. Once you have realized this, make yourself clean. Know that they are like a garment; in them you can cleanse your own impurities and become clean yourself. Rid yourself of pride, envy and jealousy, until you experience pleasure in struggling and enduring. Through their demands discover spiritual joy. After that, you will endure such struggles, and you will not run from oppression, since you will see the advantages they bring to you…….The way of the Prophet is this: It is necessary to endure pain to help rid ourselves of selfishness, jealousy and pride. To experience the pain of our spouses’ extravagant desires, the pain of unfair burdens, and a hundred thousand other pains beyond all bounds, so the spiritual path can become clear. The way of Jesus was wrestling with solitude and not gratifying lust. The way of Mohammed is to endure the oppression and agonies inflicted by men and women upon each other.
If you cannot go by the Mohammedan way, at least go by the way of Jesus, so you will not remain completely outside the spiritual path.” (Fihi Ma Fihi, chapt. 5)” But subsequently he had to keep the pot boiling and though he wanted to be the Trojan Horse in mathematics, he never has been, because he has denied his mystical insights. Moreover he was a very idle person and “idleness misleads,” and for the rest of his life he has rode two horses and Plato (Phaedrus 246 – 257) already has told us driving two horses is a laborious thing and that other Mystic had pointed out that you cannot serve two masters (Luc. 16:13).

For example, about classical logic he says,

Fortunately classical algebra of logic has its merits quite apart from the question of its applicability to mathematics. Not only as a formal image of the technique of common-sensical thinking has it reached a high degree of perfection, but also in itself, as an edifice of thought, it is a thing of exceptional harmony and beauty. Indeed, its successor, the sumptuous symbolic logic of the twentieth century which at present is continually raising the most captivating problems and making the most surprising and penetrating discoveries, likewise is for a great part cultivated for its own sake. [6, p.116]

In “Life, Art and Mysticism” Brouwer tells us profusely the way the original means became the end. He calls it the eternal “end-means-tumbling.” And thus in this world assistance in every way, medicine, religion and mathematics too, has become from means to end. “His whole life he was a mathematician (etc.), he  never became man again.” 

A yet higher beauty is that found in intuitionistic mathematics:

But the fullest constructional beauty is the introspective beauty of mathematics, where [. . . ] the basic intuition is left to free unfolding. This unfolding is not bound to the exterior world, and thereby to finiteness and responsibility; consequently its introspective harmonies can attain any degree of richness and clearness. [5, p.1239]

But neither classical nor intuitionistic mathematics has a share in the ultimate or highest beauty. The best would be to abandon logic and mathematics in order to return to the deepest home: ‘In wisdom, there is no logic’ [3] [trl. CW 110]. For Brouwer, the worth of philosophical investigation of mathematics is shown by its disclosing the relationships between the good of intuitionistic and classical mathematics, and between the good of mathematics and the Good. ‘[R]esearch in foundations of mathematics is inner inquiry with revealing and liberating consequences, also in non-mathematical domains of thought’ [5, p.1249].
We saw that for Gödel, on the other hand, the good of mathematics is part of the Good. This allows him to form, as projections from mathematical knowledge, expectations about the Good:

Merely expectations indeed and knowing that the matter would rest by expectations.

One uses inductive evidence. It is surprising that in some parts of mathematics we get complete developments (such as some work by Gauss in number theory). Mathematics has a form of perfection.
In mathematics one attains knowledge once for all. We may expect that the conceptual world is perfect and, furthermore, that objective reality is beautiful, good, and perfect. [22, p.316]

Such an induction would have been unacceptable to Brouwer.

Because he realized that it was impossible. Mathematics is human bungle and so anything but perfect.

In Brouwer, then, mathematics is action for volition, while in Gödel, mathematics is contemplation. Hence Brouwer’s disinterest in theoretical values in mathematics (values furthering contemplative knowledge, understanding), and hence Gödel’s obsession with the theoretical (contemplative) form of mathematics. (Incidentally, this distinction between views of mathematics pegged on contemplation and on volition also bears on the old issue whether mathematics is an ‘art’ or a ‘science’. The former correlates mathematics with activities, actions, controlled volitions, cunningly skilled doings; the latter correlates mathematics with demonstration, exhibition, insightful seeing and understanding.)  The contrast between the two attitudes is well illustrated by comparing this quote from Brouwer,

Strictly speaking the construction of intuitive mathematics in itself is an action and not a science; it only becomes a science [. . . ] in a mathematics of the second order, which consists of the mathematical consideration of mathematics or of the language of mathematics [2, p.99] [trl. CW p.61; original emphasis)

In other words: a game by means of a game, thinking about thinking, elucidating a construction with help of a construction, resulting finally only in endless circular reasoning, Möbius’ circle.

with Plato’s statement in the Republic, 527a6-b1,

They [i.e., geometers] speak in a way which is ridiculous and compulsory; for they are always talking about squaring and applying and adding as if they were doing things and were developing all their propositions for the sake of action; but, in fact, the whole subject is pursued for the sake of understanding.

Brouwer sees the good of mathematics in its power to facilitate the action of the will in ‘the world’, as only—given his conception of the world as something we have constructed from organized sensations, his ‘unbound by concept’ world, and on a higher level, one step up, mathematics reveals the freedom of the will and its power over all presumed logical apriori.

So according to Brouwer there are two worlds: the first and original is “Reality” (all that is), the second is that same “Reality” observed through a self constructed worldview, the beholding world, as he calls it; it is not Reality, but the image of Reality, seen by the in time captured man. It is the in things splintered reality, and those things he artificially separates from the all-enfolding coherence and names them.  He doesn’t see the whole, but trees, people, buildings, clouds and thousands other thinged ‘parts’ of the whole. He cannot see the wood for the trees no more. Only in a thinged world he can play mathematics, but the whole is more than the sum of the parts. Afterwards trying with help of that mathematics to make a whole again from that parts, you can compare with the vain attempt to anatomize a human being in parts and afterwards trying to construct a man again and then you only get Frankenstein’s monster. Brouwer is right by stating that mathematics is an expedient to facilitate the action of the will in society, for wielding power over other people, bending others to your will and fitting them in your worldview, but freedom of will, he is talking about, is what he means by ‘God’s Will’, the Will emanating creation, bringing into being Reality and disproving all logical apriori as chimera’s, unless he means they are coinciding with or are an other word for universal laws of nature. Apart from that, conferring a Will, - the world being a volition of that Will – to “the creating force” falls within talking about something unimaginable, and thus is unspeakable.

The free becoming of mathematics instructs us in the power of free will. It can shake the supposedly logic a priori, get around even such supposed universal and necessary laws. Brouwer’s debt to Schopenhauer is fully manifest [16]. For both, Will is prior to Intellect. The Will in its freedom can slay the ‘brain children’ of intellect, it can slay the very laws of logic.

First of all Brouwer is an adept

First of all Brouwer is an adept of Schopenhauer in his opinion about women, ‘a subordinate being,’….a sort of intermediate state between child and man, the last being the proper man as such.’ Life Art and Mysticism’ is soaked with it. And just like Schopenhauer he mixes up will and Will, with all consequences. He postulates that a human being is an action of will of itself and the world an action of the himself willing man and that is at least a bizarre idea.

Gödel sees the good of doing mathematics conceptually. It reveals the power of the logical apriori, its universality. It pervades all of Reality, and therefore Mind cannot free itself from it. In view of this, and in stark contrast to Brouwer, Gödel plays down the freedom of the will considerably, in the following sense.
To Rucker he said,

It should be possible to form a complete theory of human behavior, i.e., to predict from the hereditary and environmental givens what a person will do. However, if a mischievous person learns of this theory, he can act in such a way so as to negate it. Hence I conclude that such a theory exists, but that no mischievous person will learn of it. [. . . ] The a priori is greatly neglected. Logic is very powerful. [18, p.181]

Gödel here embraces Determinism. He starts from a new Predestination doctrine, in an other clothing, from hereditary qualities and inevitable influencing by environment.

This is most revealing about the depth of the Goodness of things. It means that even though there are in principle deep freedoms, they are kept from those who would use them for evil purposes or mischief.
Brouwer and Gödel would agree that the Good is to be sought; but they would disagree on the role that mathematics could play in that search.

So to speak Gödel starts from the idea it is possible catching a butterfly by running after it with a moth-net. Brouwer has understood that if you sit down quietly, the butterfly simply alights on your shoulder. Gödel thinks you have to open the door with a crowbar, Brouwer knows the door is open. Gödel supposes there is something behind that door, what he calls The Absolute, Brouwer knows what is behind that door, because he has ‘behold’ it, has experienced it, but never has found again.

4 A partial argument against CCT

We now suggest that, if you believe strongly enough in the stability of mathematics to recognize that in spite of their differences—the differences between classical and intuitionistic mathematics—, Gödel and Brouwer are both dealing with the same subject matter (i.e. mathematics), then their two cases taken together function as an argument against the common core thesis; for what is there left for a common core of truths if according to Gödel, mathematics leads you to the Absolute, whereas according to Brouwer, the same thing leads you away from it?

There is no “common core of truths.” There is a Reality, nothing can be told about and that is true. When that Mystic says (John. 8:31): “And ye shall know the truth, and the truth shall you make free”, he is not talking about Reality, neither about the “common core of truths,” but about the miserable way people are handling each other, every culture being a mistake, people being blind and deaf, worrying about tomorrow’s day, not living really, but being ‘asleep’ and ‘death,’ perceiving the mote in their fellow men’s eyes, but not the beam in their own eye. That is the Truth and it is utterly unpleasant.

What gives one access to the Good according to Gödel, denies this access according to Brouwer. But if both are speaking the truth, which we assumed for the sake of argument, then this must mean that by ‘the Good’ they mean different things. Therefore, Brouwer and Gödel cannot be referring to the same when they speak about the Good.
Note that the argument does not show that CCT is false; but, if correct, it shows the following: as long as we don’t know that Gödel’s or Brouwer’s position is false, there is no argument for CCT.

 From preceding remarks follows that Gödel’s point of view is incorrect.

So we hold that an argument for CCT would have to show that the positions of Gödel and Brouwer cannot both be true. To establish that would actually be a stronger conclusion than our one, which it implies, but is not implied by. However, it seems much simpler to point out a difference in methods of access such that it precludes sameness of reference, than directly to establish the truth or falsity of these mystical positions.
We are thus trying to say something about CCT while avoiding having to make doctrinal comments about what the Absolute is really like. We refrain from that (at the cost of not being able to say something about the truth of CCT directly) and focus on methods of access to the Absolute. Given that in history there has been much doctrinal as well as methodological disagreement, we see no reason why in general a method of access argument should fare better than a doctrinal argument. The relations between alternative (alleged) methods may be so unclear or loose as to yield no argument. What makes the Gödel/Brouwer case different is that their particular implicit disagreement on method admits formulation as a sharp antithesis, and what they are disagreeing about, mathematics, is itself something very stable.

5 Closing remarks

We would like to end by making the following two remarks.
First, of course one could, and usually does, engage in mathematics for its own sake, without any interest in relating it, be it positively or negatively, to mysticism.

There is no relationship between being engaged on mathematics and mysticism, as little as there is any relationship between being engaged in mathematics and life itself. Like Wittgenstein rightly observes in his Tractatus Logico-philosophicus (6.52): “We feel that even when all possible scientific questions have been answered, the problems of life remain completely untouched”. Being engaged on mathematics either serve society, either it is non-committal and endless Spielerei, that is building castles from parts of a fragmented and so reduced Reality, juggling on a from the infinity picked finite fowling-line, entertaining the illusion it ever may tell something about Reality, infinity, perfection, all and nothing. And that is just what all those stammering mystics are trying to express. In essence there is no difference between scholastic speculations about how many angels can dance on a needle point, and all those speculations about the demonstrability of the Riemann-hypothesis, while the world is burning and every three seconds somewhere in the world a child is starving to death.

From Gödel’s and Brouwer’s point of view, that would probably be not unlike the possibility to perform a hymn for its own sake, without any interest in the religious meaning it may have.
The second remark is related to the first. In spite of the incommensurability of Brouwer’s and Gödel’s positions, their respective motivations to take the mystical turn may have much in common. Both were disgruntled with the materialistic and formalistic philosophies prevalent at their times; both thought that these philosophies could not do justice to the Good.


[1] L.E.J. Brouwer. Leven, kunst en mystiek. J. Waltman Jr., Delft, 1905. English translation in Notre Dame Journal of Formal Logic, 37(3):381–429, 1996.
[2] L.E.J. Brouwer. Over de grondslagen der wiskunde. PhD thesis, Universiteit van Amsterdam, 1907. English translation in [7].
[3] L.E.J. Brouwer. De onbetrouwbaarheid der logische principes. Tijdschrift voor Wijsbegeerte, 2:152–158, 1908. English translation in [7].
[4] L.E.J. Brouwer. Mathematik, Wissenschaft und Sprache. Monatshefte für Mathematik und Physik, 36:153–164, 1929. Also in [7].
[5] L.E.J. Brouwer. Consciousness, philosophy and mathematics. Proceedings of the 10th International Congress of Philosophy, Amsterdam 1948, 3:1235–1249, 1948. Also in [7].
[6] L.E.J. Brouwer. The effect of intuitionism on classical algebra of logic. Proceedings of the Royal Irish Academy, 57:113–116, 1955. Also in [7].
[7] L.E.J. Brouwer. Collected works I. Philosophy and Foundations of Mathematics (ed. A. Heyting). North-Holland, Amsterdam, 1975.
[8] D. van Dalen. Mystic, geometer, and intuitionist. The life of L.E.J. Brouwer. 1: The dawning revolution. Clarendon Press, Oxford, 1999.
[9] D. van Dalen. L.E.J. Brouwer 1881–1966. Een biografie. Het heldere licht van de wiskunde. Bert Bakker, Amsterdam, 2001.
[10] J.W. Dawson, Jr. Logical dilemmas. The life and work of Kurt Gödel. A K Peters, Wellesley, 1997.
[11] W. Ewald. From Kant to Hilbert. Readings in the foundations of mathematics. Oxford University Press, Oxford, 1996.
[12] K. Gödel. What is Cantor’s continuum problem? In P. Benacerraf and H. Putnam, editors, Philosophy of mathematics: selected readings. (2nd ed.), pages 470–485. Cambridge University Press, Cambridge, 1983. Also [13], pp.254–270.
[13] K. Gödel. Collected Works. II. Publications 1938–1974 (ed. S. Feferman et al.). Oxford University Press, Oxford, 1990.
[14] G.H. Hardy. A mathematician’s apology. Cambridge University Press, Cambridge, 1940.
[15] D. Hilbert. Axiomatisches Denken. Mathematische Annalen, 78:405–415, 1918. English translation in [11].
[16] T. Koetsier. Arthur Schopenhauer and L.E.J. Brouwer, a comparison. In Combined Proceedings for the Sixth and Seventh Midwest History of Mathematics Conferences, pages 272–290. Department of Mathematics, University of Wisconsin-La Crosse, La Crosse, 1998.
[17] S. Rosen. The limits of analysis. Basic Books, New York, 1980. Reprint St. Augustine’s Press, South Bend, 2000.
[18] R. Rucker. Infinity and the mind. Birkhäuser, Basel, 1983.
[19] W.P. van Stigt. Brouwer’s intuitionism. North-Holland, Amsterdam, 1990.
[20] W.P. van Stigt. Introduction to ‘Life, art and mysticism’. Notre Dame Journal of Formal Logic, 37(3):381–387, 1996.
[21] H. Wang. Reflections on Kurt Gödel. MIT Press, Cambridge, MA, 1988.
[22] H. Wang. A logical journey. From Gödel to philosophy. MIT Press, Cambridge, MA, 1996.