A Geometry of the Absolute

'Now let the centuple celves of my egourge as Micholas de Cusack calls them, - of all of whose I in my hereinafter of course by recourse demission me - by the coincidance of their contraries reamalgamerge in that indentity of indiscernible where the Baxters and the Fleshmans may they cease to bidivil uns and (but at this poingt though the iron thrust of his cockspurt start might have prepared us we are well-nigh stinkpotthered by the mustardpunge in the tailend this outlandin brown candlestock melt Nolan's into peese!'

- James Joyce, (1882 – 1941), 'Finnegans Wake'.

Nicholas of Cusa, (1401 – 64), is the philosopher referred to in this passage from 'Finnegans Wake'; mystic, theologian, a rather more dialogical than pedagogical religious thinker whose philosophical and political notions quietly reflected upon the uniting of ancient wisdom with that of the new, of Christian religious aspirations with that of Muslims, and upon the differences between nations and cultures; a humanist, whose humanism led him to highly regard the straightforward and artless articulation of the idiota, that is to say, the lay philosopher, over the excessively artful and involuted expressiveness, and the extensive erudition, of the highly trained scholar; author of 'On Learned Ignorance', 1440, in which he declares that contraries coincide, or are reconciled, in God; ('... the coincidance of their contraries reamalgamerge in that indentity of indiscernible', as Joyce puts it).

'On Learned Ignorance' may be summarised as follows:

1. God is the absolute maximum and also the absolute minimum; he is in all things and all things are in him.

2. If man makes his own ignorance the object of his desire for knowledge, he can acquire a learned ignorance; although God cannot be comprehended, some knowledge of him can be acquired by reflection on our limitations.

3. The absolute maximum (God) is absolute unity, for unity is the minimum (and God is the absolute minimum); God, as a unity excluding degrees of more or less, is infinite unity.

4. The visible world is a reflection of the invisible; man mirrors the eternal and the infinite by his conjectures.

5. God is best studied through the use of mathematical symbols.

6. In the providence of God contradictories are reconciled.

7. The world is the absolute effect of the absolute maximum; it is a relative unity.

8. Jesus is the maximum at once absolute and restricted; He is both God and man brought into perfection.

Salvador Dali, 'Crucifixion (Corpus Hypercubus)', 1954.

Cusanus was a mathematician who employed lines, circles, and triangles to illustrate infinity; his idea was that the limits of human knowing, not just as a finite end but as a pathway of inquiry, centred on the infinite; and the seeker after wisdom who would follow the pathway of Cusanus in the direction of wisdom has to walk that course afresh every day. The immanence in the world of the absolute maximum, God or the Absolute, is united with conjectural knowing in a new kind of synthesis; a bold thought, concerned as it is with the relationship between God and the world in terms of a logic of the coincidence of opposites. Cusanus investigated imaginatively and creatively the analogy between a God who creates a world endowed with beauty and the creativity and artistry of the human mind. Such a process of seeking out analogies between an inexpressible Absolute and an actual world that could be represented and explored through a topographical depiction yielded novel insights into aesthetics, linguistics, and the very act of creation.

And given his central theme of the infinite in his philosophy, the analogies provided by mathematics and especially by a geometry of the infinite were of particular importance for Cusanus. As he wrote in 'Complementary Theological Considerations':

'If we suppose an infinite circle, it is then necessary that the centre, the half-diameter and the circumference, should be completely equal. The centre of the infinite circle is also infinite, because one cannot hold that the infinite should be greater than the centre; for one cannot hold that that which cannot be smaller than the infinite and unlimited should be larger than the centre. For the centre is the end of the semi-diametric line, and the end of the infinite is itself also infinite. The centre of the infinite circle is therefore infinite. Just as its half-diameter and its circumference likewise. The equality between the centre, the half-diameter and the circumference of the infinite circle is thus complete. And since there cannot exist many infinites, for if so none of them would be infinite, the existence of many infinites implying a contradiction, the centre, the half-diameter and the circumference must therefore constitute one sole infinite'.

Joan Miró, '2 + 5= 7', 1965.

Through a utilization of the symbolism of mathematical figures Cusanus developed a method of theological figures as one aspect of his interposition into the infinite; demonstrated most markedly in the 'Complementary Theological Considerations'; a method whereby he reflects upon the investigation into mathematical figures from the perspective of how it may well permit a conjecturing of something of the divine form. In this passage he delineates the distinction between the contemplation of things by intelligence in truth from how they are perceived by the senses:

'The mind - which itself is not free of all otherness (not free, at least, of mental otherness) - sees [geometrical] figures as free of all otherness [when it contemplates the drawing of a triangle it abstracts from the thickness of the mark, its dimensions, etc.]. Therefore, it views them in their truth, but it does not view them beyond itself. For it views them, and this viewing cannot occur beyond itself. For the mind views [them] mentally and not beyond the mind - just as the senses, in attaining [them] perceptibly, do not attain [them] beyond the senses but [only] within the scope of the senses'.

Max Ernst, 'Euclid', 1945.

An example of such a difference that he gives in 'On Learned Ignorance' is that of an isosceles triangle ABC, whereby the sum of its angles adds up to two right angles; and if A at the top is brought close to BC, and the angle at A tends toward two right angles, the triangle then tends towards a straight line or segment that is at one and the same time one and three; there is thus identity, not for reason but for the intellect, of the straight line and of the triangle.

An Isosceles Triangle.

And so, when distinct geometrical figures, such as those of circles, triangles, straight lines, are taken to infinity, they coincide, and thus by some manner a coincidence of opposites is realized: the curve becomes straight, the triangle becomes rectilinear, etc. In reality finite figures are eradicated, and in the finite world, a world of quantity, such coincidences are unrealizable; but it is through the employment of quantity that some kind of anticipation of that which transpires at the level of the absolute maximum is achievable; and consequent upon the initial transposition, certain coincidences of opposites become apparent; coincidences which, in the subsequent transposition, will be related to divine attributes.

One might so put it that such coincidence is potential at the level of the finite and actual at the level of the infinite; an infinite figure is in deed everything that the corresponding finite figure is in potential. To conclude, it is via the analogy figure/form in God, an analogy in fact justified only by the ideal character of the unique mathematical figure that corresponds to a variety of empirical forms, that the mind, which knows mathematical figures perfectly, can know, or at least postulate that it knows, since in fact it does not have that knowledge, certain forms. But Cusanus goes further. Not only do mathematical figures permit the mind to perceive objects in the divine light through their essence, but in addition he grants to such figures as these a symbolic standing that permits the mind, regardless of the principle of learned ignorance, to conjecture something of the divine form itself.

This is his famous double transposition, though it may be objected that he equivocates between the meaning of the word light as the divine light that gives being and the light created by God which enlightens the mind and is a necessary prerequisite in its seeking after truth. It is by and in divine light that the mind apprehends material realities, and because mathematical figures cannot exist in the sensible world, where absolute equality or an absolute rectilinearity are unrealizable, but exist in God and in the mind, they permit the mind to apprehend certain objects. Mathematical figures are constructed by our reason; physical realities merely resemble them imperfectly, but the mind that possesses within it the ideal figures can thereby measure the empirical figures against them.

Thus the importance of mathematical science to the understanding, as the distinction between the object and the non-object is seen as well as the possibility of seeing the other transposed from one contemplative state to another. A problem however: we find ourself groping in an enigma here by means of symbolism (the groping metaphor may prove particular apt in what is about to follow). We may suspect we already have to possess a vision of the Absolute even supposing we understand what that term even means, as we can see were we to apply the method of Cusanus to another absolute enigma, such as we find in the 'Night Lessons' episode in 'Finnegans Wake', (though it is Dante Alighieri, (1265 – 1321), rather than Cusanus that haunts this episode); the enigma that besets one's boyhood concerning what lies beneath one's mother's skirts. This may be what underlies a particular scene in the Franz Kafka, (1883 - 1924), short story of 1912, 'The Judgement', which is actually about the relation between a man and his father, and being a Kafka story does of course concern an exaggerated sense of guilt, as Georg checks on his sick father in bed and tells him about his friend in Russia (whom his father doubts even exists), of his engagement to Frieda; in response to which his father accuses him of wanting him dead, of not being as affected so much by his mother's death as he is (he will eventually sentence his son to death by drowning after which Georg flees the home to throw himself of a bridge):

''But look at me', cried his father, and George ran, almost distracted, to the bed to take everything in, but he faltered half way. 'Because she hoisted up her skirts', the father began in an affected tone, 'because she hoisted up her skirts like this, the repulsive goose', and in order to imitate the action, he raised his shirt so high one could see the scar from his war years on his thigh, 'because she hoisted her dress up like this and this, you chatted her up, and that’s how you could satisfy yourself with her without being disturbed - you've disgraced our mother’s memory, betrayed your friend, and stuck your father in bed, so he can’t move. But he can move, can’t he?' And he stood completely unsupported and kicked his legs. He was radiant with insight'.

A Daintical Diagram, 'Finnegans Wake', p. 293.

The 'Night Lessons' episode in 'Finnegans Wake' features the lustful personality of Dolph who is similarly damned for his misdemeanours; he is presented to us as a teacher, instructing a young man called Kevin about sex. One assumes them to be Shem and Shaun, twin bothers and sons of Anna Livia Plurabelle, symbol of all rivers. 'Adelphos', presumably from where Dolph's name derives, is Greek, meaning 'brother', and they are described as a 'daintical pair of accomplasses', presumably 'dainty' = Dante, for the instrument he employs within his lesson, a diagram, relates to the structures of the Inferno, Purgatory and Paradise, and through their integration within the visual design of Dolph’s diagram the twins seek the mystery of creation through lifting the skirt of Anna Livia to see her 'muddy old triagonal delta'; shaped like the letter delta … a river mouth, and female pudenda. (I could insert Gustave Courbet's, (1819 – 1877), 'L'Origine du monde' ('The Origin of the World'), 1866, here, but I won't do that. You may Google it, or not, if genitalia offends).

Such is the free association within the dream there is a connection here to Dante's ultimate vision of paradise with its reference to the lore of geometry:

Oh eternal light!

Sole in thyself that dwellst; and of thyself

Sole understood, past, present, or to come!

Thou smiledst; on that circling, which in thee

Seem'd as reflected splendour, while I mused;

For I therein, methought, in its own hue

Beheld our image painted: steadfastly

I therefore poured upon the view. As one

Who versed in geometric lore, would fain

Measure the circle; and, though pondering long

And deeply, that beginning, which he needs,

Finds not; even such was I, intent to scan

The novel wonder, and trace out the form,

How to the circle fitted, and therein

How placed: but the flight was not for my wing;

Had not a flash darted athwart my mind,

And in the spleen unfolded what it sought.

Here vigour failed the towering fantasy:

But yet the will roll'd onward, like a wheel

In even motion, by the Love impelled,

That moves the sun in heaven and all the stars.

Sublimity in the middle ages was often attained through a blending of disparate and conflicting imagery and elements; for instance, the page in the 'Book of Kells', an illuminated manuscript of the gospels from around 800 BC, which presents two rats wrenching and ripping the Host (the holy bread eaten at C0mmunion, a Christian religious ceremony) from each other with their teeth.

Detail from 'The Book of Kells', showing two cats with mice or rats, and the Eucharist.

And of course there are the gargoyles and the chimeras watching over Notre-Dame cathedral in Paris. Joyce similarly employs such a calculated and determined disharmony in a re-imagining of Dante's vision of harmony to create the suggestion of an unresolved clash of incompatibles.

In his Elements Euclid, (mid 4th century BC – mid 3rd century BC), had provided a series of instructions following each of his diagrams, typically directions to draw lines or calculate and plot angles, and in a parody of Euclidian geometric construction Dolph provides Kevin with a series of instructions to recreate a diagram visually in five pictures. A figure is thus constructed via the geometric method; and its meaning? That may be gleaned from the encompassing narrative, but the design patently reproduces in biological completeness a vulva seen up close. The two circles have been defined as Anna Livia’s hips and buttocks, but the intersection of the circles would seem to provide the framework for an intricate image of the vulva. The two triangles in the interior form the outer structure of the vulva; the four half-elliptical shapes which connect together the triangles and the circles have the appearance of the folds of the labia that encompass it. The labia is diagrammatically represented three-dimensionally to frame the image of the vulva, allowing the clitoris, symbol π, to be included, along with the anus, placed opposite the clitoris, symbol P, a half circle shape above a vertical line.

The diagram is thus quite sexually explicit, in line with Dolph’s lustful appetites, and he is certainly pleased that Kevin has masturbated upon completion of the diagram, eliminating all ambiguity: 'Hissss!, Arrah, go on! Fin for fun! You've spat your shower like a son of Sibernia but let's have at it!'

Upon the completion of Dolph’s graphic descent into the Inferno, the pair return to the terrestrial plane, the instructor ejaculating ‘Prouf!, (that is to say, proof). The connection of the lustful figure of Dolph with Hell is reminiscent of William Shakespeare's, (1564 – 1616), 'Sonnet 129':

Th' expense of spirit in a waste of shame

Is lust in action; and till action, lust

Is perjured, murd'rous, bloody, full of blame,

Savage, extreme, rude, cruel, not to trust,

Enjoyed no sooner but despisèd straight,

Past reason hunted; and, no sooner had

Past reason hated as a swallowed bait

On purpose laid to make the taker mad;

Mad in pursuit and in possession so,

Had, having, and in quest to have, extreme;

A bliss in proof and proved, a very woe;

Before, a joy proposed; behind, a dream.

All this the world well knows; yet none knows well

To shun the heaven that leads men to this hell.

('Had, having, and in quest to have', with this line the poet is of course using language to mimic the sound of lustful panting).

'Shakespearean Equation, Twelfth Night', 1948, Man Ray.

Following on from Kevin's masturbation, Dolph endeavours towards further corruption of Kevin in the attempt at making him lift up Anna Livia’s skirt: 'Pisk! Outer serpumstances beiug ekewille, we carefully if she pleats. Lift by her seam hem and jabote at the spidsiest of her trickkikant (like thousands done before since fillies calpered. Ocone! Ocone!) the maidsapron of our A.L.P. fearfully!' Dolph's deviancy is thereby contrasted with Kevin's innocence. Kevin’s angry and disgusted and righteously indignant response is to punch Dolph in the face: 'And Kev was wreathed with his pother.... hit him where he lived'. Kevin is thereby connected to Purgatory, the mountain that penitent sinners have to climb to reach Paradise, his position at the summit expressed as 'The Turnpike under the great Ulm with Mearingstone in Foreground’. The ‘great Ulm’ is an Elm tree, (and of course Albert Einstein, (1879 – 1955), was born at Ulm), π at the top of the figure, the Tree of Knowledge of Good and Evil in the terrestrial paradise and that Dante and Beatrice Portinari, (1265 – 190), observed in their journey through Purgatory):

A tree we found, with goodly fruitage hung,

And pleasant to the smell: and as a fir

Upward from bough to bough less ample spreads,

So downward this less ample spread, that none.

Methinks, aloft may climb.

'All men by nature desire to know', said Aristotle, (384 BC - 322 BC). And Cusanus declared: 'Since the desire in us is not in vain, assuredly we desire to know what we do not know', and yet he differed from Aristotle in declining to set limits to the natural desire of a human being to use his or her innate capacities to recognize and achieve knowledge, and he posited a boundless and unquenchable desire in human beings that demonstrates the existence of a kind of dispensation in the order of things; but as we see with 'Finnegans Wake', making connections with the purpose of finding order and unity can takes us to some strange places of a kind Cusanus would not have envisioned, and is very much dependent upon knowledge already acquired; a knowledge that for the divine creator of signs is absolute. Cusanus incorporates the vision of God into a semiotic interpretation of reality; but in return given the likeness between divine and human creativity a vision of reality is incorporated into a semiotic interpretation of the Absolute. Dark and obscure as our knowledge of the divine intellect may be, it is symbolic; the narrator of ''Finnegans Wake' sports with metaphors, language, syntax, graphic imagery, knowledge of the world, and of God, and of course of geometry, that amounts to a Cusanian expression of the infinite array and diversity of an inexpressible Absolute.

Cusanus wrote, in 'Visio Dei':

'I have learnt that the place wherein Thou [God] art found unveiled is girt round with coincidence of contradictories, and this is the wall of Paradise wherein Thou dost abide.... Thou art there where speech, sight, hearing, taste, touch, reason, knowledge and understanding are the same. . . . Thine eternal Word cannot be manifold nor diverse... . Now and Then coincide in the circle of the wall of Paradise . . . it is beyond the Present and the Past that Thou dost exist and utter speech!... the wall of absurdity which is the coincidence of creating with being created.... While I imagine a Creator creating I am still on this side of the wall of Paradise. .. . I have not yet entered, but I am in the wall!'

A traveller puts his head under the skirts, so to speak, of the firmament; the Flammarion engraving, 1888. 

The human that desires to know is a creator of archetypal, paradigmatic signs, and thereby endeavours to represent not the thing as it is known in itself but the intention that lies behind the sign. It is only the human that knows that searches for a purely formal sign, one that can be abstracted completely from sensible signs; and the seeking after such a purely formal sign indicates the analogy between human and divine conception; the human knower creates knowledge out of signs and words, just as God creates the world out of things. The creative activity of the mind produces something new in a manner analogous to God’s creation of the world; the intention by which God creates necessarily is reflected in every level of the rational animal’s symbolic creativity.

For Cusanus the image of a human creator of signs and symbols recovers the living image of the divine creator, but a gap is always there between signs created by the human mind and signs situated in the world by the divine or Absolute artist, and which can never be crossed by a finite intellect; but to meditatively dwell upon the gap together with intellectually desiring to bridge it goads human creativity on a life-long pilgrimage in search for an image of the invisible God or Absolute precisely in the finite, semiotic world in which we happen to have discovered ourselves. For me, it is there in the barbarous sublimity of 'Finnegans Wake, of which Joyce said that anyone can read, provided they are prepared to return to it again and again, for the text has a 'significance completely above reality; transcending humans, things, senses, and entering the realm of complete abstraction'.

Notes to 'Finnegans Wake' quotation:

1. centuple = a hundred-fold.

2. celves = selves, pl. of self; and cells.

3. egourge = egoourgos, (Greek), worker for the self.

4. Micholas de Cusack = Nicolaus Cusanus, (1401-1464), cardinal, mathematician, scholar, experimental scientist, and philosopher; and Cusack, Michael, (1847-1907), founder of the Gaelic Athletic Association in 1884, the 'Citizen' of Ulysses.

5. hereinafter = after this, in the following part of this writing or document.

6. recourse = a periodical recurrence of something, repeated visiting.

7. demission = the act of resigning or giving up, dismissal, abasement.

8. amalgamerge = amalgamate, to unite together (classes, races, societies, ideas, etc.) so as to form a homogeneous or harmonious whole.

9. undiscernible = not visible or perceptible.

10. Baxters = baxter, baker; and nursery rhyme, 'the butcher, the baker, the candlestickmaker'.

11. Fleshmans = Marthe Fleischmann, a young Swiss woman with whom Joyce was enamoured in 1919 (a model for Gerty MacDowell and Martha Clifford in 'Ulysses').

12. bidivil = (Anglo-Irish Pronunciation), bedevil.

13. uns = (German), us.

14. poingt = point; and poing (French), fist.

15. cockspurt = cock sure, quite safe, of certain outcome, marked by certainty and conviction; and Hotspur (in William Shakespeare's 'King Henry IV', Part 1 and 'King Henry IV', Part 2).

16. well-nigh = well nigh, very nearly, almost.

17. stinkpotthered = stinkpot, an earthen jar with materials of an suffocating smell sometimes thrown upon an enemy's deck.

18. punge = to prick, pierce; to affect pungently.

19. tailend= the concluding part of an action, period of time, etc.

20. outlandin = out and in, in and out, outside and inside, out of the place and in again.

21. candlestock = candlestick, a support for a candle; and William Shakespeare: 'Macbeth', V. 5. 23: 'Out, out, brief candle!'

22. Nolan's = O'Nuallain (o'nulan) (Gaelic), descendant of Nuallan (diminutive of nuall, 'noble').

23. peese = peace.

'Polyhymnia', muse of Geometry, Giovanni Baglione, 1620.