Kafka's Geometry  

(by David Steinsaltz)

(appeared in Seminar, November 1992)

 

 

In Kafka's fiction there are many journeys but few arrivals.  A typical example is the protagonist of "Ein Landarzt" riding naked through the blizzard, insisting, "Niemals komme ich so nach Hause."  The abrupt switch to present tense at this point, which suggests that the preceding story is past but the return journey continues even to the present moment, strengthens the impression that the journey truly is unending.  Yet this is the same ten mile distance that he had previously travelled in "only a moment" ["nur einen Augenblick"].  He expresses no concern about direction, only about the horses' sluggishness ("langsam wie alte Männer").  Common sense insists that if the wagon is advancing homeward, however slowly, it must eventually arrive.

This apparent paradox may be dismissed by saying that the account is to be interpreted symbolically, or mystically.  But this is not quite satisfactory, for it denies the intense physical realism of Kafka's narrative.  If it is symbolic truth that we are to read here, then we still ought not to gloss over the form, which is quite precise and realistic on the small scale, though contradictory on the large.  While vaguely described fantastic journeys are common in fairy tales and mystical writings (see, for instance, Jofen 32-33), Kafka's stories are exceptional for the minute care and tenacity with which they explore and flesh out a fairly consistent alternative geometry of distance which underlies the fantasy.

It might seem unwarranted to seek motifs of abstract geometry in Kafka's writings.  While "geometry" appears in the titles of important critical works on Kafka, Gerhard Stoltke's Franz Kafka: Eine Geometrie der Wahrheit and Henry Sussman's Franz Kafka: Geometrician of Metaphor, it is only a nebulous word for logical precision.  Kafka is supposed to have been a poor student of mathematics and theoretical science, and to have suffered "paralyzing incompetence in the presence of abstractions and abstract formulations" (Pawel 75).  But contemporary theories of mathematics and physics, particularly Cantor's transfinite set theory and Einstein's relativity theory, are known to have been popular among Kafka's friends (Wagenbach 174).  The new geometric concepts of fourth dimension and Non-Euclidean geometry were simultaneously engaging both the physicists and the mystically minded artists and philosophers of Europe (Henderson).

Kafka himself specifically mentions the geometric speculations of Zeno (Tagebuch, December 17, 1910) and Archimedes (in "Er"), and the relativity theory (Tagebuch, April 10, 1922), all in contexts which suggest a personal relationship to these issues.  Scientific writings on medicine, biology, and psychology are known to have influenced his work (Heller), and his affinities with Einstein (Kuna) and Max Planck (Greenberg) have been examined.  Walter Benjamin went so far as to say of a passage from Arthur Eddington's 1929 The Nature of the Physical World "Ich kenne in der Literatur keine Stelle, die im gleichen Grade den Kafkaschen Gestus aufweist" (Benjamin 199).

The general problem of space and time in Kafka's writings has been considered from various angles (e.g., Emrich 36, Karl, Sparks); but what is more remarkable is the number of stories and parables that raise questions of distance, and in which other elements of spaciality play no part.  Kafka shows a predilection for one-dimensional spaces, in which questions of altitude, direction, containment, and orientation are avoided, or shoved aside.  For instance, in the story "Das nächste Dorf", there is no suggestion of where this 'next village' is (or, for that matter, where the first village is), no mention of the direction from one to the other or the complexity of the route.  There is no suggestion that the young man undertaking the journey might lose his way.  We are led to believe that all complications have been removed by its being the 'nearest' village; nonetheless, the grandfather finds it inconceivable that one could expect to ride even this distance within the course of a single lifetime.  'Next,' like the German 'nächste,' has in addition to its idiomatic meaning of 'nearest' a literal signification which conjures up an image of towns strung out linearly like beads on a string, in orderly succession.  In "Ein Landarzt", the Doctor pointedly tells the stableboy, "Kutschieren werde aber ich, du kennst nicht den Weg."  Yet, in the event, there is no driver: the horses find their way unguided.  Similarly, the Doctor makes no attempt to guide the horses on the return;  the reins drag in the snow while the Doctor tries to speed the horses up, to increase the distance they will cover.  The direction is positively ignored.  There is no road, only a "snowy expanse" ["Schneewüste"] and an "open space" ["weiten Raum"] filled with snowdrifts.  The open space is effectively treated as being itself a path, or a line.  Space is linearized by seeing points as though they were ordered, with just a single 'degree of freedom' (to adopt a physicists' term) defining them, namely their distance.

This transformation is made more explicit in a pair of related stories, one titled "Der Aufbruch," the other untitled (Hochzeitsvorbereitungen 94).  In the former the following dialogue is related:  "Wohin reitest du, Herr?"  "Ich weiß es nicht, nur weg von hier, nur weg von hier.  Immerfort weg von hier, nur so kann ich mein Ziel erreichen [. . .]  'Weg-von-hier', das ist mein Ziel."  The latter begins, "Du willst fort von mir? [. . .]  Wohin aber willst du?  Wo ist das Fort-von-mir?  Auf dem Mond?  Nicht einmal dort ist es und so weit kommst du gar nicht."  The problem is to cover distance.  The direction is not merely arbitrary, these narratives hint; one may go without going in any direction at all.  This is logically absurd, unless one declines to observe spatial distinctions other than the single dimension of distance from a fixed starting point.  This is not as bizarre is it might sound at first.  After all, for most purposes we suppress the vertical dimension of our world.  We draw two-dimensional maps, and we follow them.  This procedure disturbs hardly anyone, because we are conditioned to ignore the (usually impossible) potential for motion above or below the two-dimensional surface of the earth.  In most of our movements, although our world is three-dimensional, we have only two degrees of freedom: we can walk forward or backward, left or right, or any direction in between; we walk up or down only as the ground slopes, not as an independent direction.  (Architects, on the other hand, often do require three-dimensional maps and diagrams.)  In these stories there is a still greater constriction and loss of freedom: the reduction of space to a single dimension, to a line.

Granted that this implicit reduction of space to a line does occur in Kafka's stories, the behavior of these lines, or of the distances they measure, is often bizarre.  Spatial position has been reduced to a simple question of distance, but distance turns out to be a genuine problem.  Distances should satisfy a few simple properties: they should be finite numbers such that the distance between two points is zero if and only if they are actually the same point, they should be temporally invariant (e.g., the distance from New York to London is the same this week as last week), and they should be reversible (e.g., the distance from New York to London is the same as the distance from London to New York.)  In Kafka's stories the distances between places defy all of these rules.  The distances are not only unstable, changing moment by moment, they even defy the commonsense requirement of finiteness: a span can shrink almost to nothingness, a distance which is passed in a moment, then immediately after expand to infinite length, a distance that can never conceivably be crossed.  These phenomena are linked: the path that is infinite at one moment, when traversed in one direction, is often vanishingly short when traversed in the other.  Here it is worth distinguishing among three distinct varieties of uncrossable distance, which are paired up with three distinct varieties of vanishing distance.  These types are not meant as mutually opposed or exclusive, but they are distinct enough in principle to justify the classification and to warrant trying to recognize them.  The three varieties of infinite distance will be termed 'Horizon,' 'Zeno's Racecourse,' and 'the River.'  Their complementary vanishing distances are 'the Mousetrap,' 'the Starting Mark,' and 'the Bridge.'

The simplest of these pairs, and perhaps the least characteristic of Kafka, is the Horizon/Mousetrap.  The Horizon is a tantalizing and terrifying vista which we can never reach because it ceaselessly recedes before us as we advance.  The Mousetrap is the condition of confinement and constriction, in which there are walls blocking the subject from the Horizon, walls which ultimately are as immobilizing as a mousetrap, in which the victim is clamped by the neck, unable to move even its head.  This dichotomy is presented most purely in the story titled "Kleine Fabel": "'Ach', sagte die Maus, 'die Welt wird enger mit jedem Tag.  Zuerst war sie so breit, daß ich Angst hatte'."  The Horizon is present here as a terrifying potential which is warded off by the comforting constraint of the walls.  The young mouse supposes that the artificial walls will serve her freedom as well as the true Horizon, but eventually any finite distance must be completed.  Her world grows shorter and shorter, until there is no longer any distance between her and the mousetrap in the final room.  The one-dimensional quality of the mouse's world is emphasized by the cat's admonition, "Du mußt nur die Laufrichtung ändern."  It is precisely this that is impossible, or, perhaps one might better say, this is impossible for the mouse alone; for the cat does change the mouse's direction and keep her from the trap, by devouring her.

Another example of the Horizon/Mousetrap is contained in the chapter "Belustigungen, oder Beweis dessen, daß es unmöglich ist zu leben" of "Beschreibung eines Kampfes."  In the segment titled "Ansprache an die Landschaft" the Dicke reproaches the mountain, "Berg, ich liebe dich nicht, denn du erinnerst mich an die Wolken, an die Abendröte und an den steigenden Himmel und das sind Dinge, die mich fast weinen machen, denn man kann sie niemals erreichen."  The mountain here represents to the speaker that which is terrifying because unreachable, the heavens and the horizon (which is the domain of 'Abendröte').  The narrator, too, had been comforted by the constraint of his surroundings.  "Erfreut über diesen Anblick legte ich mich nieder und dachte [. . .]  hier könnte ich zufrieden werden.  Denn hier ist es einsam und schön.  Es braucht nicht viel Mut, hier zu leben," he had said.  Yet while the Dicke fears the infinite, he simultaneously longs for it, knowing that the finite domain must inevitably be consumed.  (His great size implies that he needs enormous space in which to live)  The mountains are the image of the infinite: they remind him of that which is unbearable to imagine, and shut out any hope of the solace of attaining it: "Verdeckst du mir die Fernsicht, die mich erheitert, denn sie zeigt Erreichbares in schönem Überblick."  At last he cries out, "Jetzt aber -- ich bitte euch -- Berg, Blume, Gras, Buschwerk und Fluß, gebt mir ein wenig Raum, damit ich atmen kann."  These elements of the landscape press upon him, trap his unnaturally voluminous body.  He does, however, have hope of fording the river; this is the only direction open to him, and it turns out to be the final trap, for it drags him ineluctably over the waterfall to his death.

The second pair is 'Zeno's Racecourse/Starting Mark.'  The names refer to the most famous of the paradoxes formulated by the Greek philosopher Zeno of Elea and recounted by Aristotle, by which he intended to prove the impossibility of all motion.  This paradox, whimsically formulated in terms of a hypothetical race run by the hero Achilles (against a tortoise, according to the medieval sources through which it is best known), might be paraphrased as follows: In order for Achilles to run from one point to another, he must first run half the distance.  Next he must run half the remaining distance.  Then again half of what remains.  There are infinitely many steps, infinitely many points that must be passed, so the end can never be reached.  Thus all locomotion is merely an illusion (Salmon, 8).   In Kafka's stories 'Zeno's Racecourse' is the distance that is uncrossable because of the infinite number of points that must be crossed on the way.  With the dispelling of the illusion, the traveler finds he has never budged more than infinitesimally from the Starting Mark  (Compare Heller 46-50).  This is analogous to the characteristic of "contiguité des bureaux" which Deleuse and Guattari (92) recognize in Der Prozeß standing in place of an infinite hierarchy.  In this world, they point out, limited and infinite are linked qualities, opposed to boundless and finite (95, 123).

In the story "Eine Alltägliche Verwirrung" (whose phrasing, incidentally, is reminiscent of high school geometry text exercises) the character A sets out for the town H, a trip which took only ten minutes each way on the previous day.  On this occasion he spends ten hours on the way, and even so one may doubt that he has actually arrived, since he does not reach the man B who was his goal.  In fact, B is now at A's house.  One might suspect, then, that H and 'zu Hause' are the same place, which is to say precisely that A's movement is purely imaginary.  Certainly the repetition of the gratuitous "H" in the second sentence of the story is telling: "Er geht zur Vorsprechung nach H, legt den Hin- und Herweg in je zehn Minuten zurück und rühmt sich zu Hause dieser besonderen Schnelligkeit"  (emphasis added).  We do know definitely that A was painstakingly aware of all the insignificant details ("alle Nebenumstände") along his way.  These details which he must pass one by one as if on review correspond to the halfway point, the three-quarters point, and so forth, of Zeno's paradox; these  individually insignificant points in his path, when heaped together almost block it, without A even recognizing the fact.  The return voyage is exactly the converse: "Diesmal legt er den Weg, ohne besonders darauf zu achten, geradezu in einem Augenblick zurück."  He withholds his attention from the snagging details, and so returns home almost instantly, if indeed he ever left.  It is worth noticing here again the extreme emphasis on the linear nature of the journey between home and H.  When A arrives at H he is informed that B traveled in the opposite direction, so they must have encountered one another on the way ("sie sich eigentlich hätten treffen müssen").  Indeed, though the trip has been narrated from A's perspective without any meeting, it is logically necessary that they meet if they travel in opposite directions on a line.  That they did meet, at A's own front door in fact, is later confirmed, at the expense of the apparently more dispensable fact of A's self-identity across time.

The story "Beim Bau der chinesischen Mauer" is composed from many different strains of the Zeno's Racecourse principle.  In the section published separately under the title "Eine kaiserliche Botschaft," we find the Messenger unable to cross the vast distance which separates the emperor from the subject.  There are too many  obstacles.  The completion of the journey is a logical impossibility, not merely a practical difficulty; we are told repeatedly that the message is crucial, that the delivery has been facilitated by every possible means.  The Messenger is strong and tireless ("ein kräftiger, ein unermüdlicher Mann"), he has the emperor's mark on his breast which no one will resist: no obstacle can oppose him.  No one else could progress as rapidly as he does ("er kommt auch leicht vorwärts wie kein anderer").  But the obstacles are infinite, like the points on Zeno's Racecourse:  "Die Menge ist so groß; ihre Wohnstätten nehmen kein Ende."  It is stressed repeatedly that this is not merely a contingent difficulty imposed by the Messenger's immediate circumstances: "Immer noch zwängt er sich durch die Gemächer des innersten Palastes; niemals wird er sie überwinden; und gelänge ihm dies, nichts wäre gewonnen."  This subjunctive "nichts wäre gewonnen," recurs, jangling with irremediable hopelessness.  All forces, all circumstances, might be turned to the Messenger's aid, still nothing would be gained.  After the crowds there are courtyards, steps (a vivid symbol of the point-by-point stepwise progress which characterizes Zeno's paradox), palaces, more courtyards, more steps, houses, crowds, and so on.  Supposing he could reach open fields he might fly to his goal; "aber niemals, niemals kann es geschehen."

A deeper and more complex treatment of the Zeno theme is the account of the Wall itself, an audacious human endeavor to fill space, to construct an architectural artifact on a geologic scale.  (The hubristic scale is emphasized by the sight of "Wälder niederlegen, die zum Mauergerüst bestimmt waren, sahen Berge in Mauersteine zerhämmern.")  The Messenger's inability to arrive at his destination is suspect.  The narrator must insist repeatedly, in the most overbearing terms, that the Messenger can never complete his journey, that even millennia are insufficient for this undertaking.  The reader may be skeptical.  (Even within the context of the story this section is described as a fable.)  When, however, it becomes a matter of spanning thousands of miles not merely by passing through them, but by filling them with an elaborate wall, skepticism flips to the opposite side.  Zeno's paradox questions the possibility of locomotion by merely human, hence merely finite, processes, because it entails the passing of infinitely many points.  Our physical intuition balks at such a conclusion with respect to our everyday movements.  What Kafka has done is to increase the scale of Zeno's account enormously: the shrinking intervals that Achilles flashes by, magnified into palaces to be crossed and spectators to be pushed aside, here become back-breaking five-year construction of kilometer-long segments of the Wall.  Common sense, baffled by this vastness which exceeds all common experience, must be silent, allowing us to accept and to expect that the Wall cannot be completed, cannot even be imagined in its entirety.

The building procedure is described as follows:

 

Von Südosten und Südwesten wurde der Bau herangeführt und hier vereinigt.  Dieses System des Teilbaues wurde auch im Kleinen innerhalb der zwei großen Arbeitsheere, des Ost- und Westheeres, befolgt.  Es geschah das so, daß Gruppen von etwa zwanzig Arbeitern gebildet wurden, welche eine Teilmauer von etwa fünfhundert Metern Länge aufzuführen hatten, eine Nachbargruppe baute ihnen dann eine Mauer von gleicher Länge entgegen.

 

The process is to divide and conquer.  Troops are progressively divided in half, with each smaller group responsible for a smaller portion of the Wall.  The problem is that the five-hundred meter long segments that they are actually able to construct are infinitesimal points in comparison to the vast extent of the Wall.  Even the groups of twenty workers who build these segments are themselves subdivided into smaller groups of five, one of whom is a leader "der imstande war, bis in die Tiefe des Herzens mitzufühlen, worum es hier ging."  The question posed by Zeno is whether any number of points, even an infinite number, can ever fill a continuum.  In Kafka's account the continuum which is to be filled by the Wall is left incomplete, with vast gaps and isolated fragments of Wall.

 

It is acknowledged that the Wall would be more efficiently constructed as a unit ("zusammenhängend") but the workers would then lose all hope of ever reaching an end.  The distance is unbridgeable by the imagination; one cannot hope to reach one end from the other because of the infinity of points that must be filled between the two.  The most any individual can accomplish is to fill a single point, or isolated individual points.  The builders try to finesse the paradox by building everywhere at once, but the effort is doomed to fail because however large the population of workers which is mobilized, it is still finite.

The whole task is infinite, even though on the level of counting individual bricks it seems strictly finite.  Kafka shows us here the true infinity of human experience, which might be defined as 'that which is beyond human reckoning.'  If the number of segments were not infinite, but merely 21,000,000,000,000 so that the time required even to identify a segment would be longer than a hundred human lifetimes, what sense could there be to saying the process 'must end'?  Even if the number were just ten thousand, who has ever counted that high?  Who could comprehend not just the totality (which is easy) but the place of each element in the series? 

Even if we were to imagine an infinite number of workers, the construction might be doomed to fail.  We regularly seek to bring order to linear expanses by our familiar method of naming, or constructing, points on a line segment by infinite decimals.  This is a progressive subdivision into tenths, rather than the halves of Kafka's builders, but the process is perfectly analogous.  At any finite stage the points already constructed are still infinitely far from completing the continuum.  Even at an infinite stage with infinitely many workers and levels of hierarchy, completion is unattainable in the imagination.  As the great German mathematician Kurt Gödel, a generation younger than Kafka, pointed out, the real numbers (those points on a line segment which can be constructed by this process of decimal subdivision) only "form some kind of scaffold on the line" (Rucker 82).  John Conway has actually described a method for filling in some of the gaps between real numbers with surreal numbers (Conway 4).  The gaps in the Wall, which are the obstructions to all motion in Zeno's account, are thus found even in the purest abstraction of the problem.  They are an essential deficit of our imagination.  The Wall can never be completed; indeed its construction can hardly be said to have begun, since an incomplete and permeable wall is no wall at all: "Eine solche Mauer kann nicht nur nicht schützen, der Bau selbst ist in fortwährender Gefahr."

No filling of continuous space by motion through it is possible in imagination, according to Zeno, and no filling of continuous space by a wall is possible in Kafka's story. There will always be gaps in the Wall.  At last the Wall is declared to be complete despite the gaps, and the construction continues.  Eventually even the construction does stop, but there are still thought to be gaps: "es soll Lücken geben, die überhaupt nicht verbaut worden sind."  No one knows.  The only certain way to test the completeness of the Wall would be for a single person to perceive every point of the Wall's extent, which is impossible.  The gaps are "eine Behauptung [. . . die. . .] zu den vielen Legenden gehört, die um den Bau entstanden sind, und die, für den einzelnen Menschen wenigstens, mit eigenen Augen und eigenem Maßstab infolge der Ausdehnung des Baues unnachprüfbar sind."  Building in different places simultaneously was no help for the true problem of spanning the distance, because it does not allow a single consciousness to participate at every point.  Thus the Wall, which was intended to bring this huge span under human control, has, to the extent that it succeeded in filling the space, escaped comprehension.  This is why, despite its being a merely human artifact, a construct of hardheaded science and technical prowess, the Wall is surrounded by legends.

Another  clear exposition of the Zeno theme, one which is quite similar in many respects to "Eine kaiserliche Botschaft," is found at the beginning of the story "Die Abweisung."  In describing the distance from his hometown to the border the narrator comments, "Man wird müde, wenn man sich nur einen Teil des Weges vorstellt, und mehr als einen Teil kann man sich gar nicht vorstellen."  It is impossible even to imagine crossing the distance to the border.  Notice this important distinction: there is no difficulty in imagining the border itself, and the direction to the border is not in doubt.  But truly to imagine the way (for it is possible to believe erroneously that one has imagined it) is to conceive of physically crossing the intervening space, and this the narrator rules out.  There are inconceivably large cities on the way, so large that they are unavoidable (again, an expression of the one-dimensional nature of the distance problem), and so complex that one cannot pass through them (just as the imperial Messenger could not pass through the imperial city): "Verirrt man sich nicht auf dem Weg dorthin, so verirrt man sich in den Städten gewiß, und ihnen auszuweichen ist wegen ihrer Größe unmöglich."  An interesting gloss on the Zeno schema is the ensuing comparison between the distance to the border and the distance to the capital city.  The narrator informs us that it is farther to the capital than to the border, but he is unsure whether his claim makes sense, though it is factually and logically true.  "Es ist so, als wenn man sagte, ein dreihundertjähriger Mann ist älter als ein zweihundertjähriger," he cautions.  This is the classical paradox of comparing infinite quantities.  200 and 300 are numerically finite, but as ages they are practically infinite, just as the distance spanned by the wall is practically infinite because no one can experience its whole course.  There is an essential difference between an age and a number of years, hence it truly does seem nonsensical  to compare ages that are humanly unattainable.  This simile carries over the sense of infinitude and unattainability powerfully from the span of years to the span of miles.  (Note, though, that Kafka does not simply dash together paradoxes of time and space.)  Infinite spans are incommensurable.

The third pair of infinite/infinitesimal distances is the River/Bridge.  It is, in a sense, a reduction of Zeno's Racecourse to a single step.  The characteristic quality of the River is the utter impossibility of making even a single step.  The characteristic quality of the Bridge is crossing the impassable river in a single step.  On Zeno's Racecourse a character is unable to advance because of the infinity of trivial impediments which obstruct his path.  The River carries this plight to the extreme of Zeno's original argument: the infinity of impediments is arbitrarily close to the starting point, hence it is impossible even to begin to move.  Being on a line, there is only one direction in which to move, and that way is blocked by a river.  The traveler may attempt to ford the river, but he will only become mired or swept downstream.

This is precisely the fate of the Dicke and his bearers of "Beschreibung eines Kampfes."  The valley surrounding the river is Zeno's Racecourse.  The Dicke's fears that the mountains "würde stumm schrecklich kahle Wände mir vorschieben und meine Träger würden über die kleinen Steinchen am Wege stolpern."  These "walls" which block him are not solid walls; they are the myriad of pebbles over which one stumbles on the way, like the infinity of points which obstruct Achilles' advance.  But once the Dicke enters the river he is as helplessly immobile as a wooden statue.  He is described as "ein Götterbild aus hellem Holz, das überflüssig geworden war und das man daher in den Fluß geworfen hatte."  The trap into which the Dicke falls is not a simple mousetrap.  No, it first offers hope of salvation: "An dir aber Fluß habe ich so grosses Gefallen, daß ich mich durch dein biegsames Wasser werde tragen lassen."  (One might compare this "Gefallen" to the bit of cheese offered by the mousetrap to lure the victim.  Likewise, the "Wasserfall" which kills the Dicke is comparable to the "Falleisen" of the mousetrap.)  But as soon as he is in the water the prospect is immediately seen as hopeless.  His death is inevitable: the river will not be crossed.  "Versuchen Sie es nicht, mich zu retten.  Das ist die Rache des Wassers und des Windes; nun bin ich verloren."  Yet, when the victim has been dragged to his death, and the narrator utters his despairing plaint, the world abruptly shrinks, so that his body covers the entire countryside.  Notice that it is not the riverbank that contracts: "Dabei dehnten sich die Ufer dieses Flußes ohne Maß."  But the clouds, the mountains ("ich bin eine Lawine im Gebirge!"), the river, the entire landscape which entrapped the Dicke and the narrator himself, and which the narrator supposes himself actually to have constructed by his own effort.  He suffers for having bridged the comfortingly unfordable spaces which insulate him; by his own action, but against a part (at least) of his desire, he is set free to move, which demands far greater courage ("Mut") to live.  This plight is comparable to that of the protagonist of the story "Die Brücke," who is literally a bridge, spanning a chasm (or an "Abgrund").  She is prideful at first when someone comes to make use of her, but the reality of the crossing panics her, causing her to lose her grip and plunge to her death in the river below.

The story "Alexander der Grosse" (included as number 39a of Kafka's book of aphorisms, titled by Max Brod, "Betrachtungen über Sünde, Leid, Hoffnung, und den Wahren Weg"; Hochzeitsvorbereitungen 30-40) presents the River theme more succinctly.  The narrator expresses the belief that Alexander the Great could have found himself unable to cross the Hellespont to embark on his grand campaigns of conquest, despite all material support, strength of will, excellent prospects of success, simply because of "Erdenschwere."  Stated otherwise, the narrator recognizes that all worldly forces, physical strength, determination, and so forth, are altogether insufficient in themselves to allow a man to take a single step.  Thus, even with all the ingredients of historical inevitability pushing him forward, it is still remarkable that Alexander was able to cross the Hellespont.  This is comparable to the situation Kafka imagines for the biblical patriarch Abraham:

 

Ich könnte mir einen anderen Abraham denken [. . .] der die Forderung des Opfers sofort, bereitwillig wie ein Kellner, zu erfüllen bereit wäre, der das Opfer aber nicht fort kann, er ist unentbehrlich, die Wirtschaft benötigt ihn, immerfort ist noch etwas anzuordnen, das Haus ist nicht fertig, aber ohne daß sein Haus fertig ist, ohne diesen Rückhalt kann er nicht fort. (Briefe 333)

                                                          

 

Like Alexander, Abraham is imagined to be unable to take even a single step; if he could make the first step, he would be free to move on, but that first step is ruled out by the infinity of details that must be arranged beforehand.  This morass of petty tasks is an expanded version of Alexander's paralyzing "Erdenschwere."

The infinitesimal distance of the Bridge is expressed quite clearly in the journey to the patient's house of "Ein Landarzt."  The journey is completed in an "Augenblick."  It is suggested quite specifically that the space itself has contracted to the length of a single step through the Doctor's courtyard gate: "als öffne sich unmittelbar vor meinem Hoftor der Hof meines Kranken, bin ich schon dort."  The gate is a bridge: it transports him rapidly and effortlessly across the "Schneewüste," in which he will later be trapped, 'drowned' one might say, when he attempts to cross unaided.  (A gate allows effortless passage through an unscalable wall, just as a bridge allows effortless passage across an unfordable stream.  Each is a hole in an otherwise impenetrable barrier.)  Striking too is the Doctor's comment that he is carried along by the horses "wie Holz in die Strömung."  This is the same symbol of impotence which marks the Dicke's destruction; and here it suggests that even while the Doctor is being brought across one river to his destination, he is trapped in another that sweeps him inexorably to his perdition.

"Das nächste Dorf" is still more pointed.  To the grandfather, it is precisely the 'next' town which seems unreachable.  'Nächste' suggests a single leg of a journey, hence a single step.  The messenger of "Eine Kaiserliche Botschaft" (which, incidentally, immediately follows "Das nächste Dorf" in Kafka's carefully arranged volume of stories published under the title "Ein Landarzt") struggles with a neverending series of obstacles: another person to be shoved aside, another courtyard to cross, another flight of stairs to climb.  All that is eliminated from "Das nächste Dorf."  It is as though the messenger's bluff has been called, he has been granted surcease to the obstacles that he claimed were impeding his progress.  "Just cross this one courtyard," he is told, "and you will reach the open fields."  And yet, he finds himself unable to complete the transit, even in a thousand years.  The difficulty arises in the grandfather's attempt to apply the rationalizing yardstick of memory to the real world.  Memory is a Procrustean bed which compresses ("zusammendrängt") a life-span by cutting it to memory's own finite shape, eliminating the inherent residue of the infinite.  So reduced, the life-span cannot possibly suffice to cross even the smallest interval of physical distance, which we see simultaneously and infinitely present.  Zeno's paradox codifies this inability of rationality fully to comprehend the problem of motion.

In the second chapter of his "Essai sur les données immédiates de la conscience" (1888), Henri Bergson argues at great length that overreliance on spatial concepts leads us into gross philosophical errors.  He charges that we apply concepts of simultaneity, homogeneity, and decomposability to characteristics of the world, and of our own thought, to which they are inappropriate.  The most egregious misunderstandings arise when we transform time into a spatial concept.  Among these he includes Zeno's paradox, which he considers to be a simple result of the mistaken belief that we should be able to divide time into separate points, as we naturally divide space.  That we do think of time this way is undeniable.  We represent time as a line, on which we stand at a point called 'now,' facing toward the future, which rushes up to meet us, while the past recedes behind us.  Bergson writes "Nous projetons le temps dans l'espace, nous exprimons la durée en étendue." (Bergson 85)  His concern was amply justified by the ensuing fad for geometrization of time, spirit, and God.  Physicists soon accepted as a guiding principle, and by their growing influence persuaded most others to accept, this notion that Bergson had elaborated to condemn as a pernicious hidden assumption.

For Kafka, spatialization was not a solution: it was the crux of the problem.  Not only are questions about, for instance, God, not simplified or made more tractable by expressing them in terms of geometric distance, but that failure demonstrates how poorly understood was even that seemingly mundane concept.  Working obsessively to draw out the hidden complexities of his stories and drag them up to the light, an "unermüdliches Rechnen" (Born), Kafka found that the most familiar concepts of space, distance, and motion were unresolvable paradoxes inextricably tied to, even identical with, the most fundamental human mysteries.  Thus he wrote in his diary (Tagebuch, April 4,1922) "Wie weit ist der Weg von der innern Not [. . . ] und wie Kurz ist der Rückweg.  Und da man nun in der Heimat ist, kann man nicht mehr fort."  The "unermüdliches Rechnen" carried out in his own writing was itself just such a paradox of paralysis.  Like the Norse god Thor when he was tricked into trying to drink out a tankard which was magically filled from the ocean, Kafka could never come to an end because his "Rechnen" has roots deep in the infinite of reality, from which it draws.  "Chinesische Mauer" is, from its very beginning, engaged in questioning itself, doubting itself, analyzing itself.  Themes, motifs, problems, and confusions recur and are reimagined from story to story, as, for instance, the Tower of Babel theme from "Chinesische Mauer" is revised and deepened in "Der Stadtwappen."

Grasping at the infinite, the substantial reality that escapes words and consciousness, like seawater that flows between the strands of a net, is both Kafka's subject and his technique.  As Laura Quinney writes: "His subject is of such a "magnitude" that it generates a supra-cognitive excess, inundating and evading articulation" (Quinney 223).  Where convention would allow the smug satisfaction of completion, Kafka is driven onward into an "unablässiges Durchdenken aller Möglichkeiten," as Jürgen Born terms it.  "Zu einem Ergebnis aber können sie nicht führen" (Born 408).  Reasoning, like all constructs of words, is a barren skeletal frames, a discrete lattice, that may outline and interpenetrate, but is always infinitely far from filling space, like ideal grid lines on a sheet of graph paper.

Many of Kafka's stories examine and engage in this seemingly futile quest for the infinite.  The hollow precision of reason chases after the star of firm reality, never nearing the end, just as the construction of the Wall never comes any closer to completion, but only is declared complete.  This is the usual compromise, a concession (often unacknowledged) to blur the vision and so declare a problem "understood," even though the reality still lies all within the cracks in our understanding.  (A suggestive analogy is the procedure of 17th Century biologists who would 'create' living organisms by an elaborate recipe whose only function was to conceal from the scientist himself and others the fact that they were carelessly allowing eggs or spores to slip in.)

Kafka was too clearsighted and too self-conscious to allow such a resolution.  He craved to grasp reality itself: living, thick, and blooded, not a dry skeleton.  This unfulfillable longing is not only described repeatedly in the infinite geometry of distance, it is expressed in Kafka's compulsion to spin out his stories endlessly.  "Kafka's longer works, and short stories such as 'The Great Wall of China,' bear witness to his belief in the exigency of 'approximation'; obsessive and circular, these works play variations on the impasses they begin with, and never come to a climax or conclusion.  For all their discursive plenitude, they grind to a halt in quite literal incompleteness" (Quinney 223).  A story once begun could never be organically whole and complete.  One story he was moderately satisfied with was "Das Urteil," begun and finished in the demarcated term of a single night, developed with organic density in the womblike confines of his mind: solid, dense, alive, unlike any mere concatenation of words.  So he wrote in his diary: "Die Geschichte ist wie eine regelrechte Geburt mit Schmutz und Schleim bedeckt aus mir herausgekommen" (February 11, 1913) and "Nur so kann geschrieben werden, nur in einem solchen Zusammenhang, mit solcher vollständigen ˆñffnung des Leibes und der Seele" (September 23, 1912).  On the other hand, when he was contemptuous of himself and his work he wrote, "Alles erscheint mir als Konstruktion [. . .] Und sinnlos leer bin ich" (November 19, 1913).  We ought to consider this lament in light of David Lachterman's thesis in his recent book The Ethics of Geometry, that the essence of "modernism" is the belief in the mind as fundamentally an agent of construction (Lachterman, vii-xii, 1-24).  Kafka's stories explore the impossibility of constructing anything (a story perhaps most of all) by mere thought. 

This is one meaning of Walter Benjamin's claim that the "Kafkas Figur" reveals "Reinheit und [. . .] eigentümlicher Schönheit [. . .] von einem Gescheiterten."  "Nichts dengwürdiger als die Inbrunst, mit der Kafka sein Scheitern unterstrichen hat" (Benjamin 201-202).  Kafka himself described his life as a "Wüstenweg" (Tagebuch, January 28, 1922).  In an earlier entry (Tagebuch, October 19, 1922) he had explained "Das Wesen des Wüstenwegs" as follows:

 

Ein Mensch, der als Volksführer seines Organismus diesen Weg macht, mit einem Rest (mehr ist undenkbar) seines Bewußtseins dessen, was geschieht.  Die Witterung für Kanaan hat er sein Leben lang; daß er das Land erst vor seinem Tode sehen wollte, ist unglaubwürdig.  Diese letzte Aussicht kann nur den Sinn haben, darzustellen, ein wie unvollkommener Augenblick das menschliche Leben ist, unvollkommen, weil diese Art des Lebens endlos dauern könnte und doch wieder nichts ergeben würde als ein Augenblick.  Nicht weil sein Leben zu kurz war, kommt Moses nicht nach Kanaan, sondern weil es ein menschliches Leben war.  (emphasis added)

 

But Moses was by no means entirely a failure.  Though he never reached the Promised Land himself, he did glimpse it before his death, and he did lead his people there.  Neither was Kafka entirely a failure.  He recognized the bounds of his confinement, studied them, suffered because of them, and finally transcended them.  Kafka's wilderness was the supra-rational domain of paradox and contradiction, well described by Benjamin:

 

Man denke an die Parabel "Vor dem Gesetz."  Der Leser, der ihr im "Landarzt" begegnete, stieß vielleicht auf die wolkige Stelle in ihrem Innern.  Aber hätte er die nichtendenwollende Reihe von Erwägungen angestellt, die diesem Gleichnis dort entspringen, wo Kafka seine Auslegung unternimmt?  Das geschieht durch den Geistlichen im "Prozeß" -- und zwar an einer so ausgezeichneten Stelle, daß man vermuten könnte, der Roman sei nichts als die entfaltete Parabel.  Das Wort "entfaltet" ist aber doppelsinnig.  Entfaltet sich die Knospe zur Blüte, so entfaltet sich das aus Papier gekniffte Boot, das man Kindern zu machen beibringt, zum glatten Blatt.  Und diese zweite Art "Entfaltung" ist der Parabel eigentlich angemessen, des Lesers Vergnügen, sie zu glätten, so daß ihre Bedeutung auf der flachen Hand liegt.  Kafkas Parabeln entfalten sich aber im ersten Sinne; nämlich wie die Knospe zur Blüte wird.  (Benjamin165-166)

 

The "cloudy spot," the contradiction, the paradox unfolds like a blossom, growing more complex and inscrutable each moment.  While not as full and alive as God's creation, it escapes the flattening comprehension of the merely human.  It is not just any contradiction or inane non sequitur that will accomplish this feat.  Compare it to the problem of mapping the earth.  A flat map of the world is awash in contradictions: a given location may appear both on the extreme left and the extreme right, a single point may be stretched out across the whole top of the map, and so on.  Two maps drawn by different projections will disagree in most particulars: distances, areas, east-west orientation, etc.  From all these contradictions we may infer that the earth is not flat like the map, but has a higher dimension to it.  The collection of mutually and internally contradictory flat maps represents the sphere, as no single coherent map could.  That India is arrived at sailing east or west from Europe is a geographical paradox, and by those who were narrow-minded it was deemed impossible.  But an honest appraisal of these seemingly contradictory facts leads to a higher understanding.  As it happens, there is a simple model of the earth, the sphere, which explains all the contradictions; thus there no longer seems anything mysterious about them.  But if we lacked such a model the flat maps with their contradictions honestly drawn and accepted would point us toward it.  This is the procedure of modern geometers as well.  A higher-dimensional geometric space is constructed by defining a collection of flat, Euclidean sheets which are said to overlap, with contradictions.  The contradictions are rigorously defined so as to be logically consistent with one another.  There is usually no picture or model by which to imagine these spaces, but they may be manipulated and understood simply as particular sets of spatial contradictions.

Kafka manipulates logical, psychological, verbal, and other contradictions in a roughly analogous way.  It cannot be known whether he had a clear image of some higher aspect of the universe in which his paradoxes were resolved and ceased to be paradoxes, but their coherence and attendant persuasive power attest to such a vision.  The stories, mere words as they are, cannot contain the infinite of a living creation, just as a flat sheet of paper cannot contain a three-dimensional sphere.  Rather than flatten reality to fit the logical structure of words, he projected it by many contradictory methods, thus generating paradoxes and surreal effects.  The flat map is a means of transmitting information.  It is an invitation to reconstruct an implied spherical surface, using the three-dimensional world available to the user.  Likewise, Kafka's fictions invite us to reconstruct the world from an otherwise unknown (and indescribable) aspect.  Like the mapmaker, Kafka must distort what he sees to relate it: he must flatten  it into words and rational ideas; the contradictions, however, the "cloudy spots," invite the reader to reconstruct the higher reality being described.   Kafka's impulse to pursue these constructions interminably by logical means is the source of many an uncompletable, uncompleted story, and many a story with an arbitrary or inconclusive ending, such as "In der Strafkolonie," of which Kafka wrote, in a letter to his publisher (September 4, 1917): "Zwei oder drei Seiten kurz vor ihrem Ende sind Machwerk, ihr Vorhandensein deutet auf einen tieferen Mangel, es ist da irgendwo ein Wurm, der selbst das Volle der Geschichte hohl macht" (Cited by Emrich 224).  It is also the seed of the new world that Kafka imagined, the source of the new law of life and geometry that he struggled to expound, the fuel of his fiery imagination in which his "Beweis dessen, daß es unmöglich ist zu leben" was refined into a proof of life itself, and that "es ein menschliches Leben war."


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Bergson, Henri.  Essai sur les données immédiates de la conscience.   Geneva : Editions Albert Skira, 1911.

 

Born, Jürgen.  "Kafkas unermüdliche Rechner."  Euphorion 64 (1970): 404 - 13.

 

Conway, John.  On Numbers and Games.  New York : Academic Press, 1976.

 

Deleuse, Gilles and Félix Guattari.  Kafka: Pour une littérature mineure.  Paris : Les Éditions de Minuit, 1975.

 

Emrich, Wilhelm.  Franz Kafka.  Frankfurt am Main : Athenäum Verlag, 1961.

 

Greenberg, Valerie. Transgressive Readings: The Texts of Franz Kafka and Max Planck.  Ann Arbor : University of Michigan Press, 1990.

 

Heller, Paul.  Franz Kafka: Wissenschaft und Wissenschaftskritik.  Tübingen : Stauffenburg Verlag, 1989.

 

Henderson, Linda Dalrymple.  Fourth Dimension and Noneuclidean Geometry in Modern Art.   Princeton : Princeton University Press, 1983.

 

Jofen, Jean.  The Jewish Mystic in Kafka.  New York : Peter Lang Publishing, Inc., 1987.

 

Kafka, Franz.  Gesammelte Werke, vol.6: Hochzeitsvorbereitungen auf dem Lande, und andere Prosa aus dem Nachlaß and vol. 7: Tagebücher 1910-1923.  Edited by Max Brod.  Frankfurt : Fischer Taschenbuch Verlag, 1976.

----   Sämtliche Erzählungen.  Edited by Paul Raabe.  Frankfurt : Fischer Taschenbuch Verlag, 1970.

 

Karl, Frederick R.  "Space, Time, and Enclosure in The Trial and The Castle."  Journal of Modern Literature, 6 (1977): 389 - 397.

 

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Lachterman, David.  The Ethics of Geometry: A Genealogy of Modernity.  New York : Routledge, 1989.

 

Pawel, Ernst.  The Nightmare of Reason: A Life of Franz Kafka.  New York : Vintage Books, 1985.

 

Quinney, Laura.  "More Remote Than the Abyss."  In Modern Critical Views: Franz Kafka. Ed. Harold Bloom. New York : Chelsea House Publishers, 1986, 221-35.

 

Rucker, Rudy.  Infinity and the Mind.  Boston : Birkhäuser, 1982.

 

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Stolte, Gerhard.  Franz Kafka: Eine Geometrie der Wahrheit.  Frankfurt am Main : Peter Lang, 1979.

 

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Wagenbach, Klaus.  Franz Kafka: Eine Biographie seiner Jugend.  Bern : Francke Verlag, 1958.