Ludwig Wittgenstein and L. E. J. Brouwer

Brouwer and Wittgenstein were contemporaries, with the whole of Wittgenstein’s years contained within those of Brouwer’s (Wittgenstein lived 1889 to 1951 while Brouwer lived the longer life from 1881 to 1966). It is mildly ironic that even as Brouwer’s followers were playing down his mysticism and trying to extract only the mathematical content from his intuitionist philosophy (even the faithful Heyting distanced himself from Brouwer’s mysticism), Wittgenstein’s writings reached a much larger public which resulted in the mystical content of Wittgenstein’s works being played up and the early Wittgenstein himself, very much the logician following in the tradition of Frege and Russell, presented as a mystic.

Not only were Brouwer and Wittgenstein contemporaries, but we also know that Brouwer played a little-known role in Wittgenstein’s return to philosophy. After having written the Tractatus Logico-Philosophicus and then disappearing into the mountains of Austria to become a village schoolmaster in Trattenbach, some of those philosophers that continued to seek out Wittgenstein in his self-imposed exile convinced him to go to a lecture in Vienna in March 1928. The lecture was delivered by Brouwer (Brouwer gave two lectures; Wittgenstein is said to have attended one of them). Wittgenstein was said to have listened to the lecture with a surprised look on his face (sort of like G. E. Moore saying that Wittgenstein was the only student that looked puzzled at this lectures). So it may be the case that Brouwer played a pivotal role in the transition from the thought of the early Wittgenstein to the thought of the later Wittgenstein. (Matthieu Marion has argued this thesis.)

Wittgenstein’s distinction between saying and showing, a doctrine that dates from the Tractatus (cf. sections 4.113 and following), is often adduced in expositions of his alleged mysticism. According to Wittgenstein’s distinction, some things can be said but cannot be shown, while other things can be shown but cannot be said. While to my knowledge Wittgenstein never used the term “ineffable,” that which can be shown but cannot be said would seem to be a paradigm case of the ineffable. And since Wittgenstein identified a substantial portion of our experience as showable although unsayable, the ineffable seems then to play a central role in his thought. This puts Wittgenstein firmly in the company of figures like, say, St. Symeon the New Theologian (also, like Wittgenstein, an ascetic), which makes the case for his mysticism.

An extract from St. Symeon on the ineffable: “The grace of the all-holy spirit is given as earnest money of the souls pledged in marriage to Christ. Just as a woman without a pledge has no certainty that the union with the groom will occur within a certain length of time, so does the soul have no firm assurance that it will be re-united to its God and Master for all eternity. The soul cannot be certain that it will achieve this mystic, ineffable union nor that it will enjoy its inaccessible beauty if it does not have the pledge of His grace and does not consciously have that grace within.” (Krivocheine, Basil and Gythiel, Anthony P., In the Light of Christ: Saint Symeon, the New Theologian 949–1022, St. Vladimir’s Seminary Press, 1986, p. 367)

Brouwer was a bit more explicit in his doctrine of ineffability than was Wittgenstein, and he repeatedly asserted that the language of mathematics was a necessary evil that approximated but never fully captured the intuitive experience of mathematics, which he understood to be a free creation of the human mind. This comes across both in his early mystical treatise Life, Art, and Mysticism, which is pervaded by a sense of pessimism over the evils of the world (which include the evils of mathematical language), and his more technical papers offering an exposition of intuitionism as a philosophy of mathematics. But, like Wittgenstein, Brouwer does not (to my limited knowledge) actually use the term “ineffable.”

There is another ellipsis common to both Brouwer and Wittgenstein, and that is despite Brouwer’s openly professed intuitionism, which can be considered a species of constructivism (this latter is a point that needs to be separately argued, but I will only pass over it here with a single mention), and despite the strict finitism of the later Wittgenstein, which can also be considered a species of constructivism, neither Brouwer nor Wittgenstein employ Kantian language or Kantian formulations. No doubt there are implicit references to Kant in both, but I am not aware of any systematic references to Kant in the work of either philosopher. This is significant. Both Brouwer and Wittgenstein were philosophers of the European continent, where the influence of Kant remains strong even as his reputation waxes and wanes over the generations.

Kant was an early constructivist, or, rather, a constructivist before constructivism was explicitly formulated, and therefore sometimes called a proto-constructivist — although I have pointed out an obvious non-constructive dimension to Kant’s thought despite his proto-constructivism (which I do not deny, notwithstanding Kant’s non-constructive arguments in the first Critique). Kant’s classic proto-constructivist formulation is that the synthetic a priori truths of mathematics must be constructed, or “exhibited in intuition.” It is this latter idea, of a concept being exhibited in intuition, that has been particularly influential. But what does it mean? Obviously, a formulation like this has invited many interpretations.

The approaches of Brouwer and the later Wittgenstein could be considered different ways of exhibiting a concept in intuition. Brouwer, by casting out the law of the excluded middle from mathematics (at least in infinitistic contexts), assured that double negation was not equivalent to the truth simpliciter, so that even if you know that it is not the case that x is false, you still don’t know that x is true. (On the law of the excluded middle cf. P or not-P.) The later Wittgenstein’s insistence upon working out how a particular term is used and not merely settling for some schematic meaning (think of slogans like “don’t ask for the meaning, ask for the use” and “back to the rough ground”) similarly forces one to consider concrete instances rather than accepting (non-constructive) arguments for the way that things putatively must be, rather than how they are in actual fact. Both Wittgenstein’s finitism and Brouwer’s intuitionism would look with equal distaste upon, for example, proving that every set can be well-ordered without actually showing (i.e., exhibiting) such an order — also, the impossibility of exhaustively showing (i.e., exhibiting in intuition) that every set can be well-ordered if one acknowledges an infinity of sets.

I give this latter example because I think it was largely the perceived excesses of set theory and Cantor’s transfinite number theory that were essentially responsible for the reaction among some mathematicians that led to constructivism. Cantor was a great mathematical innovator, and his radical contributions to mathematics spurred foundationalists like Frege (who objected to Cantor’s methods but not his results) and Russell to attempt to construct philosophico-mathematical justifications, i.e., foundations, that would legitimize that which Cantor had wrought.

The reaction against infinitistic mathematics and foundationalism continues to the present day. Michael Dummett wrote in Elements of Intuitionism, a classic textbook on basic intuitionistic logic and mathematics, that:

“From an intuitionistic standpoint, mathematics, when correctly carried on, would not need any justification from without, a buttress from the side or a foundation from below: it would wear its own justification on its face.”

Dummett, Michael, Elements of Intuitionism, Oxford University Press, 1977, p. 2

In other words, mathematics would show its justification; in contrast, the foundationalist project to assure the legitimacy of the flights of non-constructive mathematics was wrong-headed in its very conception, because nothing that we say is going to change the fact that non-constructive thought that derives its force from proof, i.e., from what is said, does not show its justification on its face. Its justification must be established because it does not show itself. This is what “foundations” are for.

Note: There is also an element of intellectual ascesis in Dummett’s idea of a conservative extension of a theory, and this corresponds to the asceticism of Wittgenstein’s character, and, by extension, to the asceticism of Wittgenstein’s thought — asceticism being one of the clear continuities between the earlier and the later Wittgenstein — like the implicit development of constructivist themes.

But it was not only the later Wittgenstein who reacted with others against Cantor. It seems to me that the saying/showing distinction of the Tractatus is a distinction not only between that which can be said and that which can be shown, but also a distinction between that which is established by argument, possibly non-constructive argument, and that which is exhibited in intuition, i.e., constructed. If this is right, Wittgenstein showed an early sensitivity to the possibility of constructivist thought, and his later development might be understood as a development of the constructivist strand within his thinking, making Wittgenstein’s development more linear than is often recognized (though there are many scholars who argue for the unity of Wittgenstein’s development on different principles). The saying/showing distinction may be the acorn from which the oak tree of the Philosophical Investigations (and the subsequently published posthumous works) grew.

For the early Wittgenstein, the distinction between saying and showing was thoroughly integrated into his idea of logic, and while in the later sections of the Tractatus the mysticism of what which can only be shown but cannot be said becomes more evident, it is impossible to say whether it was the logical impulse that prevailed, and served as the inspiration for the mysticism, or whether it was the mystic impulse that prevailed, and served as the pretext for formulating the logical doctrines. But the logical doctrines are clearly present in the Tractatus, and serve as the exposition of Wittgenstein’s ideas, even up to the famous metaphor when Wittgenstein says that the propositions of the Tractatus are like a ladder than one must cast away after having climbed up and over it.

Just as there is a mathematical content to Brouwer’s mysticism, so too there is a logical content to Wittgenstein’s mysticism. It is, in fact, likely that Wittgenstein’s distinction between saying and showing was suggested to him by what is now called the “picture theory of meaning” given an exposition in the Tractatus. Few philosophers today defend Wittgenstein’s picture theory of meaning, but it is central to the metaphysics of the Tractatus. For Wittgenstein, the logical structure of a proposition can be shown but not said. Since for Wittgenstein in his Tractarian period, “The facts in logical space are the world” (1.13), and “In the proposition the thought is expressed perceptibly through the senses” (3.1) — i.e., the proposition literally exhibits its structure in sensory intuition — thus, “The proposition is a picture of reality.” (4.01) One might even say that a proposition exhibits the world in intuition.

Today these formulations strike us as a bit odd, because we think of anything that can be formulated in logical terms as a paradigm case of something that can be said, and very possibly also something that may not be showable. For us, logic is a language is among languages, and one way among many to express the world; for the early Wittgenstein, on the contrary, logic is the structure of the world. It shows itself because the world shows itself, and after showing itself there is nothing more to be said. The only appropriate response is silence.

As we all know from the final sentence of the Tractatus, whereof one cannot speak, thereof one must remain silent. According to the Wittgenstein of the Tractatus, all scientific questions can be asked and all scientific questions can be answered (shades of Hilbert’s “Wir müssen wissen. Wir werden wissen.” — which Per Martin-Löf has called Hilbert’s solvability axiom, and which is the very antithesis of Brouwer’s rejection of the law of the excluded middle), but even when we have answered all scientific questions, the problems of life remain untouched.

As implied by the early Wittgenstein’s insistence upon the solvability of all scientific questions, the metaphysics of Brouwer and Wittgenstein were very different. Their common constructivism does not prevent their having fundamental, I might even say foundational, differences. Also, while Wittgenstein comes across in a melancholic fashion (a lot like Plotinus, another philosophical mystic), he is not fixated on the evils of the world in the same way that Brouwer was. If both Brouwer and Wittgenstein can be called mystics, they are mystics belonging to different traditions. Brouwer was a choleric mystic while Wittgenstein was melancholic mystic.

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Grand Strategy Annex

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Plotinus according to the early modern imagination, from the Nuremburg Chronicle.

Plotinus is remembered as among the most otherworldly of philosophers, far more concerned with the eternal than the temporal. His biographer, Porphyry, famously said that he seemed embarrassed to have a body, and that, “So deeply rooted was this feeling that he could never be induced to tell of his ancestry, his parentage, or his birthplace.” But Porphyry also relates an interesting story about Plotinus’ early years:

“Despite his general reluctance to talk of his own life, some few details he did often relate to us in the course of conversation. Thus he told how, at the age of eight, when he was already going to school, he still clung about his nurse and loved to bare her breasts and take suck: one day he was told he was a ‘perverted imp’, and so was shamed out of the trick.”

Porphyry, Life of Plotinus, 3

So at the age of eight, Plotinus was still being breast fed. This is an odd detail to be preserved from a man who, “could never be induced to tell of his ancestry, his parentage, or his birthplace,” but perhaps a telling detail.

Freud opened his famous essay Civilization and its Discontents with a discussion of his correspondence with Romain Rolland about what Rolland thought that Freud had missed in his The Future of an Illusion. Rolland agreed with Freud on religion, but he still thinks that Freud has missed the point. For Rolland, the point that Freud missed is a feeling that Rolland has that he called the oceanic feeling, which Rolland identified as the “true source” of religion. Freud responded to this: “I cannot discover this ‘oceanic’ feeling in myself. It is not easy to deal scientifically with feelings.” (I discussed this previously in Algorithms of Ecstasy.)

Romain Rolland thought that Freud's The Future of an Illusion had missed the point, and that religion has its origins in an "experience of eternity" that he called the "oceanic experience".

Though Freud could not discover the oceanic feeling in himself, he made a brave effort at a psychoanalytical explanation of what Rolland described to him:

“An infant at the breast does not at yet distinguish his ego from the external world as the source of the sensations flowing in upon him. He gradually learns to do so, in response to various promptings. He must be very strongly impressed by the fact that some sources of excitation, which he will later recognize as his own bodily organs, can provide him with sensations at any moment, whereas other sources evade him from time to time — among them what he desires most of all, his mother’s breast — and only reappear as a result of his screaming for help. In this way there is for the first time set over against the ego an ‘object,’ in the form of something which exists ‘outside’ and which is only forced to appear by a special action. A further incentive to a disengagement of the ego from the general mass of sensations — that is, to the recognition of an ‘outside,’ an external world — is provided by the frequent, manifold and unavoidable sensations of pain and unpleasure the removal and avoidance of which is enjoined by the pleasure principle, in the exercise of its unrestricted domination.”

Freud, Civilization and its Discontents, pp. 13-14

Freud goes on like this for a page and a half — it is well worth it to consult the original text and read it all for yourself, but I didn’t feel like typing it all out right now — and comes to this speculative conclusion:

“Our present ego-feeling is, therefore, only a shrunken residue of a much more inclusive — indeed, an all-embracing — feeling which corresponded to a more intimate bond between the ego and the world about it… the ideational contents appropriate to it would be precisely those of limitlessness and of a bond with the universe — the same ideas with which my friend elucidated the ‘oceanic’ feeling.”

Ibid., p. 15

Thus Freud makes a connection between an infantile experience of absolute union with the world and a vaguely religious feeling of union with the world. I am not endorsing this view of Freud, which I think continues to miss the point which Rolland was trying to make to Freud, but this was Freud’s typical manner of proceeding: psychodynamic, reductionist, and positivistic. I wouldn’t have expected anything more or anything less from Freud.

But even if I don’t find Freud’s explanation of the oceanic feeling (which I, like Freud, cannot find in myself) to be adequate, it certainly is compelling, and it becomes even more compelling when placed in the context of Plotinus’ rather late weaning. If Plotinus did in fact continue to suckle for a rather longer period of life than most infants, well beyond infancy into childhood, one might speculate either that he felt this bond with limitlessness in suckling and was attracted if not fascinated by the feeling, or that his being accustomed to this pleasure with its component of being intimately connected to the nurturing body of the world, and its continuation into later childhood and its subsequent deprivation within a time continuous with memory into adulthood, may have prompted Plotinus to seek an alternative source of the same feeling.

There is no question that, as a philosopher, Plotinus was preoccupied with eternity, and Freud relates that Rolland also called the oceanic feeling “a sensation of ‘eternity’.” Plotinus’ Enneads are filled with reflections upon and even exhortations upon eternity and the eternal. Whatever the etiology of Plotinus’ sensation of eternity, it seems clear that it was vividly felt, and an important component of his experience that he felt called for philosophical explication.

The Third Ennead, Seventh Tractate, is devoted to the question of time and eternity. Plotinus defines eternity thus:

“That which neither has been nor will be, but simply possesses being; that which enjoys stable existence as neither in process of change nor having ever changed — that is Eternity. Thus we come to the definition: the Life — instantaneously entire, complete, at no point broken into period or part — which belongs to the Authentic Existent by its very existence, this is the thing we were probing for — this is Eternity.”

Plotinus, Enneads, 3.7.3

This definition of eternity is deeply embedded in Plotinian metaphysics, and no small gloss would be needed to adequately explicate its elements. But immediately before this definition, in the same section, we find this somewhat less metaphysically embedded passage on eternity:

“We know it as a Life changelessly motionless and ever holding the Universal content (time, space, and phenomena) in actual presence; not this now and now that other, but always all; not existing now in one mode and now in another, but a consummation without part or interval. All its content is in immediate concentration as at one point; nothing in it ever knows development: all remains identical within itself, knowing nothing of change, for ever in a Now since nothing of it has passed away or will come into being, but what it is now, that it is ever.


This passage has some affinities to Freud’s psychodynamic interpretation: clearly it evokes the limitlessness, that lack of barriers between self and world, that Plotinus, Rolland, and Freud alike sought to explain, after a fashion — consummation without part or interval. We may wonder if this “consummation,” which suggests other consummations of unions that are to become important later in life, but which Freud himself sought even in the earliest period of infancy, is the masculine counterpart of the penetrative mysticism that we find in Teresa of Avila.

Saint Teresa of Avila elaborated a uniquely penetrative mysticism.

A Freudian interpretation of Plotinus could be called a human, all-too-human form of eternity, except that I suspect that the experience that Freud describes is common to most large-brained mammals — in other words, this is something more than and beyond the human, all-too-human. It is animal, all-too-animal. That the human experience of eternity should be an expression of our animal nature coincides with the point that I attempted to argue in Nietzsche on Sexuality, that there is a unity of that which we have believed to be most bestial in our character and that which we have heretofore believed to be ideal and edifying, this tells us something about what we are. We are not divided between a bestial element and a celestial element; we are one and whole.

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Grand Strategy Annex

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Divine Penetration

21 October 2009


saint teresa

A few days ago I mentioned listening to a biography of Simone Weil. The last chapter had a number of extracts from Weil’s writings, and one in particular struck me, but I should try to give some context as to why it struck me.

Some months back I listened to a new translation of Saint Teresa of Ávila’s autobiography, and I had drafted a couple of posts about it that I never took the time to properly finish. I enjoy Saint Teresa’s writings about her mystical experiences and refer to them often, especially her detailed descriptions of the stages of prayer and her distinctions between different species of mystical experience.

Although Weil agonized over whether she could, in good conscience, join the Catholic church, her late thought is thoroughly assimilated to the Catholic paradigm, and she repeatedly self-identified in terms of the Catholic tradition. Weil is not so explicit as Saint Teresa of Ávila in describing her mystical experiences that brought her to the Catholic church (she seems to have been a deeply private person in this respect), but there are clear traces of her mystical experiences in her writings, though, like Saint Teresa of Ávila, it is a mysticism of a peculiar character:

“Above all our thought should be empty, waiting, not seeking anything, but ready to receive in its naked truth the object which is to penetrate it… We do not obtain the most precious gifts by going in search of them but by waiting for them… The soul empties itself of all its own contents in order to receive into itself the being it is looking at, just as he is, in all his truth.”

Simone Weil, Waiting on God, quoted in Simone Weil by Francine du Plessix Gray

There is an approximately parallel passage in the introduction to the writings posthumously collected as Gravity and Grace, a commentary on Weil’s thought by her friend Gustave Thibon:

“The hero wears armor, the saint is naked. Now armor, while keeping off blows, prevents any direct contact with reality and above all makes it impossible to enter the third dimension which is supernatural love. If things are really to exist for us they have to penetrate within us. Hence the necessity for being naked: nothing can enter into us while armour protects us both from wounds and from the depths which they open up.”

Thibon has here captured an essential aspect of Weil’s conception of the spirit and its function in the world, or, rather, its non-function defined in terms of passivity, in terms of waiting, in terms of attention. A similar concern with penetration and passive waiting is to be found in the famous passage of Saint Teresa of Ávila when she describes her vision of an angel penetrating her with a spear:

“In his hands I saw a long golden spear and at the end of the iron tip I seemed to see a point of fire. With this he seemed to pierce my heart several times so that it penetrated to my entrails. When he drew it out, I thought he was drawing them out with it and he left me completely afire with a great love for God. The pain was so sharp that it made me utter several moans; and so excessive was the sweetness caused me by this intense pain that one can never wish to lose it, nor will one’s soul be content with anything less than God.”

A secular commentary upon this penetrative feminine mysticism is to be found in Andrea Dworkin’s theoretical treatise Intercourse:

“By definition, as the God who does not exist made her, this lesser privacy, this lesser integrity, this lesser self, establishes her lesser significance: not just in the world of social policy by in the world of bare, true, real existence. She is defined by how she is made, that hole, which is synonymous with entry; and intercourse, the act fundamental to existence, has consequences to her being that may be intrinsic, not socially imposed. There is no analogue anywhere among subordinated groups of people to this experience of being made for intercourse: for penetration, entry, occupation.” (p. 155)

If Dworkin is right that being made for penetration is a unique existential condition, and that being such a being has ontological consequences, one should expect that the spiritual reality of such a being would differ from other beings, viz. beings for whom this unique existential condition does not apply.

One finds the account of such a being in Sartre’s Being and Nothingness, Sartre’s great treatise on phenomenological ontology. Sartre was a contemporary of Weil, and indeed Sartre’s long time companion Simone de Beauvoir sought out Simone Weil, wanting to meet her because of what she had heard about Weil breaking into tears upon news of a famine in China. But de Beauvoir and Weil did not hit it off, according to the brief account in de Beauvoir’s Memoirs of a Beautiful Daughter. After it became clear that Weil wanted to feed the world’s hungry while de Beauvoir wanted to help those who hungered for meaning, they apparently had nothing more to say for each other. But we note that both saw the world in terms of a hunger that demanded satiation — in other words, an emptiness in the world that needed to be filled. Enter Sartre.

The later portions of Sartre’s treatise elaborates what Sartre called existential psychoanalysis, which he proposes in contradistinction to Freudian psychoanalysis. One of the conclusions at which Sartre arrives in the course of this exposition is that human beings are hole-filling creatures.

“Here at the origin we grasp one of the most fundamental tendencies of human reality — the tendency to fill. We shall meet with this tendency again in the adolescent and in the adult. A good part of our life is passed in plugging up holes, in filling empty places, in realizing and symbolically establishing a plenitude.”

Jean-Paul Sartre, Being and Nothingness, Part Four, Chapter Two, III

Another conclusion, to put the matter bluntly, is that everyone wants to be God. In Sartre’s language this is formulated in terms of being-for-itself (i.e., consciousness) looking to the self-contained and self-sufficient character of being-in-itself and wanting to become being-in-itself. Being-for-itself, subjectivity, is vulnerable: when pierced by The Look it bleeds in the direction of the other. Being-in-itself possesses no such vulnerability (like the hero in armor described by Thibon).

“…the ideal of a consciousness which would be the foundation of its own being-in-itself by the pure consciousness which it would have of itself. It is this ideal which can be called God. Thus the best way to conceive of the fundamental project of human reality is to say that man is the being whose project is to be God.”


Now, not to put too fine a point on it, but if we all want to be God, and we are hole-filling creatures, then we need a hole to fill. God, according to Weil, may not need a hole to fill, but Weil made herself empty and waited to be filled by God, to be penetrated by God, to be taken by God. And there is a precedent for this: in the form of the Holy Spirit God came to Mary, Mother of God, and filled her so that Mary was with child by way of an Immaculate Conception. Saint Teresa was also penetrated by God, though she seemed to be more interested in what Dworkin called, “… the sensual reality of submission, violation, and being possessed.” (Intercourse, p. 196)

Is Sartre’s Being and Nothingness the secularized, masculine counterpart to the penetrative feminine mysticism of Saint Teresa and Simone Weil?

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