War Games

On April 6, 1916, the only war diary entry reads: “Life is a . . .” Under the date of the subsequent day, it goes on: “torture, from which one is only temporarily relieved so that one remains receptive for further agonies.”¹ What one cannot speak about, one must pass over in silence. “An exhausting march, a night of coughing, a society of drunks, a society of mean and stupid people.”² The gunner Ludwig Wittgenstein, declared “completely unfit”³ for service by the Austro-Hungarian Army, left behind his privileged life in the familial circle of friends, in which representatives of Viennese high finance met as well as artists. He left behind Cambridge’s elite mathematicians and finally his Norwegian hut, the refuge of his philosophical and homoerotic existence. Yet he never quite arrived in the war. First Wittgenstein stands at the searchlight of a captured Russian patrol boat and helps secure the Vistula on the Russian border. A first lieutenant who happens to hear of his mathematical and technical engineering education entrusts him initially with organizational tasks, then with the acting supervision of the artillery workshop of the Krakow fortress. For Wittgenstein that means “office work,”⁴ often extending into the night. But the implementation of the manifested expertise founders on the lacking power of command. Wittgenstein’s men refuse time and again to follow his orders. His superior then puts him in the uniform of a Landsturm engineer ⁵ until the war ministry intervenes, classifies the request for promotion as a presumption, and rejects it. After a transfer to the balloon section is also turned down,⁶ Wittgenstein requests to be sent to the front with the 4th battery of the 5th Field Howitzer Regiment. The men, however, “with few exceptions,” still hate him, the “volunteer.”⁷ Someone who, though he is exempt from service, nonetheless chooses war and not merely a military officer career—and who now gives orders, without having been ordered to serve in the war—appears suspect to his comrades. Wittgenstein has himself transferred again: “Tomorrow perhaps, at my request, I’ll get out to the scouts. Only then will the war begin for me. And maybe – life too! Perhaps the nearness of death will bring me the light of life.”⁸ Only now has Wittgenstein arrived in the war, at the front line. The constant complaining about his comrades, who make him think of demands for duels,⁹ now comes to an end in his diary. Instead, from that point on, only quick prayers are to be found. He is at a forward “observation post.”¹⁰

The consolidation of light field artillery and heavy howitzer battery under a central command, tactically integrated balloon sections and forward reconnaissance officers are all inventions of the First World War. In its initial stages, its artillery still resembled that of Napoleon, before it ultimately found its way to forms and standards that still prevail in Western armies today.¹¹ The European powers used the years of peace before the First World War to increase the quantity of their ordnance and its penetrating power. The tactics of the artillery nonetheless remained at the level of the previous battles still in memory. No one could imagine that the infantry would advance on the battlefield without, together with its own artillery, having its eye on the enemy. But when the massive firepower was revealed in the first battles, all that remained was the withdrawal of the batteries from the adversaries’ field of vision. The increased penetrating power of the shells promptly underwent a reevaluation: what now counted was the distance that the projectiles overcame. At 9,000 meters, it turned out to be considerably farther than ballistics experts had foreseen in their tables.¹² War now proves not to be the father of all things but the bastion of unforeseen and unrealized facts.

Of all places, Wittgenstein’s first operational area as a reconnaissance officer, north of the Carpathians, becomes a center of this bastion—that is, a tactical field of experimentation.¹³ What Wittgenstein’s war diary records from that point on can be read as the “double-entry bookkeeping . . .”¹⁴ of two experiments; one is the foray into a logic that “takes care of itself”¹⁵ and thereby breaks with Russell’s type theory, which must dodge onto a meta-mathematical level for its foundation, and the other is the record at a forward post, as it was only just established.¹⁶ Both experiments culminate in a self-abandonment that seeks on the one hand to escape compulsive suicidal thoughts in the writing of rules for one’s own life ¹⁷ and requires on the other hand meticulous compliance with tactical shooting rules so as to protect through covering fire above all one’s own warring units and ultimately also one’s own existence. Rules for reconnaissance officers do not, of course, assert a claim to eternity such as Alfred North Whitehead and Bertrand Russell already evoke in the title of their Principia Mathematica. Yet it seems worth the effort to reconstruct what Wittgenstein was toiling over during his two and a half years as a reconnaissance officer on the front and during several continued education endeavors, if not on Russell’s work—with whom he incidentally ceased all communication at this time.¹⁸

Since the Brusilov Offensive had decimated more than half of the Austro-Hungarian Army in the Bukovina in the summer of 1916, what remained of it—a remnant to which Wittgenstein belonged ¹⁹—stood effectively under Prussian command.²⁰ The Prussian supreme army command was itself advised in artillery tactics by Colonel Georg Heinrich Bruchmüller. His noteworthy second military career began on the eastern front and ended, honored with the Pour le Mérite, on the western front after Erich Ludendorff’s Spring Offensive. Perhaps Bruchmüller heard Eduard Kummer lecture on the calculations of shell trajectories in his last years at the Friedrich Wilhelm University. It is certain, in any case, that he gave up his studies of mathematics and physics quite quickly and chose instead to join a foot artillery company.²¹ His active period fell in post- and prewar years without his career ever developing on the fast track.²² After a fall from a horse and a subsequent nervous breakdown, Bruchmüller retired from active duty. Only when the front ranks were thinning out in the First World War was he conscripted again; he took over a command far below his rank and distant from the front. Bruchmüller’s moment had come in Przasnysz in the summer of 1915, when he tested for the first time an artillery fire that pushed toward the enemy positions in several phases and enabled one’s own infantry to move up “behind a mask of smoke and dust.”²³ The notorious creeping barrage, as it was called, would revolutionize positional warfare on both fronts. The “most precise preparation of infantry and artillery for this combat operation, as became the rule in the east from the beginning of 1916 and in the west probably from the beginning of 1918” deserves particular attention here too.²⁴

At Lake Naroch in White Russia in April of 1916, the creeping barrage achieved a breakthrough against a numerically superior Russian Army, and in Galician Tarnopol in 1917 it achieved, no less effectively, a counterattack. Ultimately, Bruchmüller’s central command in Riga—which coordinated the creeping barrage together with several divisions and various artillery and shell types—played no small part in the shattering of the tsar’s army. In Riga, Bruchmüller’s reputation preceded him and, for reasons of secrecy, required an elaborate journey from Galicia via Berlin to the fortress city.²⁵ His name was simply regarded as a synonym for breakthrough intentions: The army knew him only as “Durchbruchmüller” (Durchbruch is German for “breakthrough”)—a name that Ludendorff had carved in stone and that would stand on the wooden cross marking his grave in 1948. Still, Bruchmüller does not embody the esprit de corps of the elite. The history of the military elite in the First World War finds its high point with the development of the assault battalions (Sturmbataillone).²⁶ Nor, however, were assault battalions invented by the General Staff. They formed spontaneously in view of positional warfare.²⁷ The assault troops (Sturmtruppen) could be sure of their top position in the military hierarchy because their initiator Willy Rohr had created them from such a position.²⁸ If Bruchmüller brought together with his own artillery tactics in Riga what had emerged on the western front in the infantry tactics of the assault troops, it is not the supreme command of the military elite that should be discerned in that fact but the “unique result of an autopoiesis”²⁹ of the battlefields. Bruchmüller had given up his command in 1915 and was from that point on, strictly speaking, merely a retired colonel who temporarily occupied the position of an artillery adviser—even if crown prince Friedrich Wilhelm summarily instructed his army group during the Champagne-Marne offensive to equate Bruchmüller’s recommendations with the orders of the supreme army command.³⁰ And Bruchmüller inculcated in the artillerymen a perspective that valued the gratitude of the infantry more highly than medals and honors, “which only individuals can receive for the totality.”³¹

Perhaps only an adviser without high rank and command could propose detaching the various batteries from the autonomous responsibility of their commanders and coordinating them centrally from above divisions.³² But here too all commanders could do was give orders that issued from weeks of meticulously developed attack plans and were tailored only to the specific positions and weather conditions. More easily than commanders, however, plans can include the suspension of a centralized power of command when centrally coordinated actions—such as the artillery fire brought together into the creeping barrage—decompose into mere individual actions, for example, in order to secure conquered positions. Likewise, Bruchmüller’s plans included an interplay of command leadership between artillery and infantry, insofar as a specific situation demanded it. The greatest advantage of a form of warfare that consistently created facts initially on paper and not first on the battlefield clearly lies in its element of surprise. Instead of shooting and calibrating for hours or even days at known targets with shell trajectories that, due to prevailing influences, deviated to a greater or lesser extent from their standard values, and thereby possibly betraying one’s own positions and intentions to the enemy, Bruchmüller advocated the method of his captain Pulkowski, who suggested systematically investigating in advance the most diverse influence variables. From the beginning of the battle onward, the artillery commanders now shot not only at enemy positions that were hidden from view, but also by means of a method that instead of their empirical knowledge had recourse to a scientific system.³³

Because everything nonetheless depended on the execution of the plans, Bruchmüller introduced extensive briefings before every offensive in the infantry and artillery troops. Feedback thereby took place not only between soldiers in the ranks, who otherwise had to submit to a unidirectional command hierarchy, but was also intended for the battlefield, as soon as the infantry signaled to the artillery with flares that the creeping barrage should transition into the next phase. Even the horror experienced by the infantry would be applied through feedback to enemy positions as a psychological warfare measure. Efficiency no longer meant attaining a total physical annihilation through weeks of artillery fire, as had become common, and ultimately capturing only minimal strips of terrain. Rather, it now meant the neutralization of the enemy—its physical and psychological paralysis, while one’s own material and troop forces remained spared as much as possible. Bruchmüller preached the shock of the first wave of attack, instead of relying on the eventually stoically accepted barrage fire. In rapid alternation, he had enemy positions shelled, but with an arbitrariness that scarcely allowed the enemy infantry to come out of cover even during breaks in the firing. The contingency of a possible death was thereby given—along with the spatial scattering of the shells—a temporal dimension. In Riga, Bruchmüller ultimately used gas instead of explosive shells, which were growing scarce. The gas, only slightly heavier than air, even infiltrated underground positions and was mixed in such a way that tear gas penetrated behind the Russian gas masks and made them unwearable. Additional lethal gasses could then do their work unhindered. The use of gas led to more wounded and less dead, which was regarded as a tactical advantage, because the high number of victims that needed medical attention not only meant a loss of combat power, but also bound additional forces that were necessary to manage logistical problems, to say nothing of the moral dilemmas that they caused the enemy. The efficiency of neutralization as opposed to simple destruction consequently amounted to the delay of death—only, ultimately, to reign over it all the more powerfully. In the end, efficiency did not even stop at one’s own men: the gratitude paid to them did not come without the condition of putting their lives at stake in a calculated fashion, for the artillery could not simultaneously shell all of the enemy positions that the charging infantry battled, which is designated in the technical lingo as a “target-rich environment.”³⁴ On the basis of “plans produced at a particularly large scale, in which the individual time periods of the target combat are represented,” infantrymen therefore had to internalize a course of battle that had them charge enemy positions even when support by their own artillery was in the plan only at a later point in time.³⁵ But Bruchmüller’s tactics did not only demand of the infantry that they entrust their lives to the soundness of the operations on paper, but also that they subordinate their lives to the law of the large number. From the supra-individual perspective, an existential advantage was promised them: when in doubt, assault troops would be better off entering their own artillery fire from the creeping barrage, giving them cover, than in the sights of their enemy’s machine guns, who would be given time to disappear into the creeping barrage and to reappear in front of them.³⁶ Through plans of action, the infantry did not merely lead up spatially to a creeping barrage, but also charged toward a temporarily lethal zone, for the innovation of releasing a preliminary barrage of dispersed gas over the battlefield before the main barrage went back to Bruchmüller with his fondness for war gas. This was gas that only in the best-case scenario had completely evaporated when, shortly after the bombardment, assault troops had drawn level with the target area.³⁷

The eastern front has nearly fallen victim to collective forgetting, and only few specialist historians still remember that it became the first war theater of such tactical experiments and ultimately produced more corpses—and, above all, more nameless ones—than the western front.³⁸

Perhaps the experience of these real horrors explains why the personal diary entries of Wittgenstein do not maintain the cool distance of a Martin Heidegger. Friedrich Kittler has demonstrated that Heidegger must have thought about assault tactics from the safe distance of the weather station on the western front. In Being and Time, the creeping barrage seems, in any case, to have served as a model for the fundamental existential motion of Dasein: “Anticipation [Vorlaufen: literally, running or moving forward] reveals to existence its extreme possibility as self-abandonment and thus shatters any clinging to an attained existence.”³⁹

In the postwar period, Wittgenstein—much to his friends’ dismay—could imagine “what Heidegger means by being and anxiety. Man feels the urge to run up against the limits of language.”⁴⁰ It remains to be elucidated what language means against this background.

During the defense against the Brusilov offensive, there was not enough time for Wittgenstein to entrust to his war diary even one line—apart from a quick prayer. For his first deployment as a reconnaissance officer, it was his mission to direct his own artillery fire to enemy positions. After a month of “colossal exertions,” he has “thought much about all sorts of things, but strangely cannot make the connection to [his] mathematical thinking.”⁴¹ The next day he notes in his diary, “But the connection will be made! What cannot be said cannot be said!”⁴² Here—and only here—the “double-entry bookkeeping” of the diary is broken. His private writings, otherwise encoded entirely according to the rules for reconnaissance officers, extend only here in plain text over both pages. This one time, the confessions, vows, expressions of despair and war experiences intertwine with the unencrypted philosophemes on the right side—from which the Tractatus Logico-Philosophicus will emerge. Thomas Macho’s thesis is to be wholeheartedly embraced: “In many respects, the ‘Tractatus’ is the strangest war diary ever written.”⁴³ Wittgenstein’s later reflections also insist on the primacy of diagrammatic and cartographic constructs, on mathematical propositions, operations and orders, which are antecedent to thinking. The formation of his philosophemes is evidently characterized by the system of positional warfare—though this is much more fundamentally the case than Wittgenstein’s biographers acknowledge when they merely register the appearance of war metaphors in his writings. At most, they seek to document what has ostensibly always already been a brilliant mode of rumination in its disturbance through war. If Wittgenstein’s writings speak metaphorically of “laying siege”⁴⁴ to his mathematical and logical problems, of storming them and of “the blood” that he would rather pour before the fortress than “withdraw with nothing accomplished,”⁴⁵ the most striking aspect of this martial language is the anachronism—and Wittgenstein, before any contact with the front, was not the only one who stood under its spell. But the besieging and storming of fortresses and the heroic blood sacrifices are soon replaced by total sensual deprivation, death by asphyxiation, and trench systems in no man’s land. The war that Wittgenstein—after he left behind the fortress city of Krakow—saw approaching from the observation post can no longer be decided by the taking of fortresses. It is necessary above all to capture spaces, because only their conquest promises a power advantage. Meanwhile, the conquered terrain appears more inhospitable after every offensive. Once the conquest of a considerable portion of land is successful, as in the Spring Offensive of 1918, the defense of the expanded front can then be one’s undoing. If no metaphors of fortress structures can now be extracted from zones of visual deprivation, instruments of coordination can be—which Wittgenstein would find indispensable as professor of moral science at Cambridge University:

Language sets everyone the same traps; it is an immense network of well kept wrong turnings. And hence we see one person after another walking down the same paths & we know in advance the point at which they will branch off, at which they will walk straight on without noticing the turning, etc., etc. So what I should do is erect signposts at all the junctions where there are wrong turnings, to help people past the danger points.⁴⁶

The fact that Wittgenstein orients himself only on the surface by everyday language and an everyday situation, but is secretly drawing on his military practices, can be gleaned from his files in the Vienna war archives:

During the battles at Ldziany [Wittgenstein] carried out his duties as a reconnaissance officer in an exemplary fashion. Staying at his post under the heaviest artillery fire, it was possible only thus for the battery to direct the fire at threatened points that the battery commander could not see. In this fashion sensitive losses were inflicted on the enemy at decisive moments.⁴⁷

And ultimately the exact media transpositions that a reconnaissance officer had to perform are the basis of what will be called thinking in the most general sense in Cambridge’s analytic philosophy lectures:

We can substitute a plan for words. And a thought may be a wish or an order. Truth and falsehood then consist in obedience or disobedience to orders. Thinking means operating with plans. . . . How do we know that someone had understood a plan or order? He can only show his understanding by translating it into other symbols. He may understand without obeying. But if he obeys he is again translating—i.e., by coordinating his action with symbols.⁴⁸

The following description gives an impression of the coordinations (a word that was prevalent among artillery tacticians ⁴⁹), map operations, alphanumeric encryptions, and orders on the observation post: “Besides the captain, there is also a lieutenant there, a sergeant for operating the telephone and one for doing surveying work on the map, which is spread out over a board.” The “serious military work” amounts to announcing “rapid and brief phrases . . . almost always in numbers,” which remain in the memory of the civilian reporter merely as “an incomprehensible language.”⁵⁰ Unlike the exciting noise of the barrage fire of his own side as well as the enemy artillery, the memory of the constant orders that he had to relay through the field telephone would remain abhorrent to Wittgenstein.⁵¹ Nonetheless, on the observation post Wittgenstein finds himself for the first time in a position in which there is no question of the integrity of the chain of command. From the forward position, enemy trench mortars and artillery as well as “threatened points” of one’s own line by the charging enemy infantry must be sighted with “scissors telescopes,” coordinates must be determined on a battle map, and situation reports must be relayed to the battery commanders by telephone. Ultimately, the accurate fire by one’s own artillery must be observed and, if necessary, must be ensured by further instructions. In the mediatic rule system of orders by telephone, answers, and indexical battle maps, the chain of command becomes a circulus vitiosus. There is no longer a supreme command, but only feedback loops and differences between what must be said and what must be shown, interrupted by lost telephone connections, deafening enemy fire, or the silence of the cannons, which reveal their position neither through noise nor through muzzle flash.

On the observation post, all instruments and media are geared to the handling of signs and orders. They become the fundamental activities that, for the philosophy of the twentieth century, still give rise to thinking. One instrument, however, is completely missing: There is no longer any mention here of the use of weapons. Even if artillery observers advance about as far as the assault troops in their initial position, and are thereby closer to them than to their own battery, a fundamental difference exists between the two. It emerges clearly from the accounts by Ernst Jünger, an early assault troop officer.⁵² When Jünger is “given the job of observation officer”⁵³ due to a leg wound, he no longer has the front-line enemy positions in the sights of the assault rifle, but instead observes them with the telescope:

The observation post . . . was nothing more than a periscope through which I could view the familiar front. If the bombing was stepped up, there were coloured flares or anything else out of the ordinary, I was to inform the divisional command by telephone. . . .

The observation post was well camouflaged in the landscape. All that could be seen from outside was a narrow slit half hidden behind a grass knoll. Only chance shells ended up there, and, from my safe hiding-place, I was able to follow the activities of the individuals and units that I hadn't paid that much attention to when I myself had also been under fire. At times, and most of all at dawn and dusk, the landscape was not unlike a wide steppe inhabited by animals. Especially when floods of new arrivals were making for certain points that were regularly shelled, only suddenly to hurl themselves to the ground, or run away as fast as they could, I was put in mind of a natural scene. Such an impression was so strong because my function was a little like that of an antenna, I was a sort of advance sensory organ, detailed to observe calmly all that was happening before me, and inform the leadership. I really had little more to do than wait for the hour of the attack.⁵⁴

In the coordination and synchronization of individual arms of the service, Ernst Jünger now realizes as an “advance sensory organ” the autopoetic as well as operational closure of the military body. Along with enemy movements, he also follows the combat units of his own side with an equally great identificatory interest. Only from the observation post do they reveal what must remain hidden at the operational level even from an assault troop officer. The “advance” organic-mediatic extension also shows operational limitations and dependencies in the cooperation of the individual arms. The fact that the disabled assault troop officer in the function of the observation officer now limits his activity to waiting for the hour of the attack expresses the fragmentation based on the real of the battlefield.

At the Somme, Jünger, physically recovered, yields to a reflex that reverses the decoupling of shooting and observation:

Later that morning, I was strolling along my line when I saw Lieutenant Pfaffendorf at a sentry post, directing the fire of a trench mortar by means of a periscope. Stepping up beside him, I spotted a British soldier breaking cover behind the third enemy line, the khaki uniform clearly visible against the sky. I grabbed the nearest sentry’s rifle, set the sights to six hundred, aimed quickly, just in front of the man’s head, and fired. He took another three steps, then collapsed on to his back, as though his legs had been taken away from him, flapped his arms once or twice, and rolled into a shell-crater, where through the binoculars we could see his brown sleeves shining for a long time yet.⁵⁵

To carry the rifle on a loose strap ready to fire is a rule of the infantry tested in assault, to which Jünger himself no doubt contributed.⁵⁶ For the artillery observer, however, the exact opposite applies: he must avoid as much as possible any use of a weapon. The prime imperative is to maintain the secrecy of the observation post, because the enemy’s defensive fire threatened to aim at it.⁵⁷ To hit the observation post also meant rendering inoperative the battery connected to it by telephone. Observation posts, which are withdrawn from sight through camouflage and do not directly deploy weapons, at least do not run the risk, as batteries do, that light measurement techniques home in on and precisely locate their muzzle flash from various positions or calculate their firing sites with the help of sonic measurement techniques. Judged by the systems that positional warfare produced, Jünger’s shot is deplorable—for he endangers his own men more than the enemy’s side. His demonstration of an aimed shot in fact celebrates an outdated minimal model of war: the duel; because not only does a “British soldier [break] cover behind the third enemy line” in the scene, but Jünger himself also strolls so ostentatiously to the sentry post that he does not even need to come out of cover in the first place.

When Jünger later, during the Second World War, in the middle of the Caucasus mountains, spots a scattered Russian unit on the opposite mountain ridge, the gaze into the telescope transports him into a lunar landscape, which can only mean the memory of the battlefields of the positional war, riddled with shell craters. With somnambulistic precision, a thought haunts him: “During the First World War one would still have opened fire on them.”⁵⁸

“From single shot to creeping barrage”⁵⁹—thus the relevant literature sums up the irreversible course of history, propelled by a race between technology and tactics. The duel—which had been the principle to which, when in doubt, all complexities of war were still reduced up to the First World War—had thereby served its time. The Thirty Years’ War might have produced the monopoly of violence, but the world war realized it in its totality. The threat of the death penalty in Prussian law did not put an end to the duel. Nor did Kant’s appeal to reason, which argued that duelers by no means demonstrated the courage of the warrior, which was instrumental to states.⁶⁰ No state power was able to fight an institution that allowed the suffering and the exercise of violence for the restoration of honor. The duel is finally abandoned with the First World War, due to the intrinsic killing mechanisms of the world war, which revoke the equivalence with the duel. Even more than the right to exercise violence, the codex of putting one’s own life at stake was rewritten. Dramatists such as Kleist with the Prince of Homburg could still invoke the freedom of death-defying courage, as a result of which every insubordination is forgotten. The positional war largely did away with a rank consciousness exemplified by the long-serving regiments of the death’s-head hussars in favor of functional combat units, which answered to acronyms such as FEKA (Fernkampfartillerie, or “long range artillery”) and risked the existence of their own units for the safeguarding of another in a circular logic per se and not only in a state of emergency. The war machine disavows the possibility of the individual seeking death for the defense of his own honor. Instead, it exposes life to many life-threatening risks in a quite particular fashion and under a specific directive.

The deep cut that logic makes in existence still speaks out of Wittgenstein’s war diary: “If suicide is allowed then everything is allowed,”⁶¹ it reads, consistent with the argument of axiomatic mathematics, according to which a contradiction is to be ruled out not because something false or untrue can be found in it, but rather because otherwise there would be a threat of the total indifference of all proof processes. Conversely, it can scarcely be an accident that Wittgenstein chooses the picture of a duel for the illustration of mathematical and semiotic contexts: “A is fencing with B.”⁶² Wittgenstein returns to the example several times in his writings—before the demand for a duel ultimately becomes a real option for coping with life.⁶³ Pairs of fighting men, Wittgenstein elaborates, can be represented only by extrapolating from one fighting pair to further pairs.⁶⁴ Thus, a relation between signs no longer necessarily refers back to the signified, but inherits other semiotic structures with their own logical relations. Between sign and signified a “logical identity” is not necessary, if internal—that is, not sayable, but showable—logical relations bring them together. For the production of identity with the signified, signs and modes of signification have to enter a complex that aligns their logical properties with the logic of the situations of the world. Propositions thereby do not simply describe situations of the world, but rather recreate them: “In a proposition a world is as it were put together experimentally. (As when in the law-court in Paris a motor-car accident is represented by means of dolls, etc.)”⁶⁵ On the eastern front, whenever he has enough time to do so, Wittgenstein sets himself the task of finding the connection between models, pictures, and “the signs on paper”⁶⁶ on the one hand and a “situation outside in the world” on the other.⁶⁷ Even if not all situations can be turned into “pictures on paper,” Wittgenstein is certain that at least all “logical properties of situations” can be reproduced “in a two-dimensional script.”⁶⁸ In a world whose situations are reflected without assistance in relations between signs and whose logic has to take care of itself, subjects ultimately have no place: “There is no such thing as the subject that thinks or entertains ideas.”⁶⁹ It turns into an insurmountable limit, which cannot overcome itself in order to evaluate itself: “The Subject does not belong to the world: rather, it s a limit of the world.”⁷⁰ Wittgenstein thereby encapsulates Jünger’s “advance sense organ.” On the observation post, which is withdrawn from the enemy’s sight, and from which the battlefield experiences its limits through linguistic mediation, the “I of solipsism shrinks to an extensionless point and what remains is the reality co-ordinated with it.”⁷¹ In the solipsistic view, which according to a diary entry begins on the way into the firing position,⁷² nothing “in the visual field allows you to infer that it is seen by an eye. For the form of the visual field is surely not like this”⁷³ (figure 6.1).

Figure 6.1

Wittgenstein’s schema from his Tractatus.

Source: Wittgenstein 2001, 69.

Likewise, it cannot be concluded from the camouflaged battlefield that observing eyes are everywhere directed at it. The subject converges with a piece of paper and with a retina, which has abandoned thinking in duels but points the way for whole batteries.

Wittgenstein began his war diary two weeks after he entered the Austro-Hungarian Army as a volunteer with the sentence that “logic must take care of itself,”⁷⁴ so that “all we have to do is to look and see how it does it.”⁷⁵ In the end, after the dissolution of the dual monarchy and with it the Austro-Hungarian Army, his work has “extended from the foundations of logic to the nature of the world.”⁷⁶ If a problem is solved or a situation is managed, it loses its meaning. For Wittgenstein sentences are merely “ladders” that—as soon as they have proven their function— can be thrown away.⁷⁷ Jünger comes to the same conclusion in his study about the activity of the worker, whose type has emerged from the world war:

All of these concepts (Gestalt, type, organic construction, total) are notabene there by way of comprehension. We are not concerned with them as such. They can be forgotten or set aside without further ado after they have been used as magnitudes of work for the grasping of a definite reality which exists in spite of and beyond every concept; the reader has to see through the description as through an optical system.⁷⁸

In war, even the most decisive technologies and tactics ultimately betray their design to the enemy, however effective they initially were, and demand their own surpassing. For this logic of surpassing, the war of the twentieth century is not a last resort of political clarification. Rather, this logic delineates the insurmountable playing field of a war game that constantly creates new unspeakable and inconceivable facts.

“What is the difference between language (M) [as mathematics] and a game? You might say: It ceases to be a game when things begin to become serious, and here seriousness means application.”⁷⁹ Wittgenstein has returned to Vienna from Italian war captivity. He has again taken up his investigations of the foundation of mathematics—against the conclusion of his Tractatus, which declared the problem of logic solved once and for all and thereby showed how little that accomplished.⁸⁰

In Wittgenstein’s inquiries into the foundation of mathematics, war is again present. Now it has taken on the form of the war game, which outlasts every war and never runs out of material:

Think of the game of chess. Today we call it a game. Suppose, however, a war were waged in such a way that the troops fought one another on a field in the form of a chess-board and that whoever was mated had lost the war. The officers would be bending over a chessboard just as they now do over an ordnance map. Then chess would not be a game any longer; it would be a serious business.⁸¹

Wittgenstein does not recall a lived-through war, but puts himself on the level of the waging of it. The distance indicated by his analogy is double-edged. To the extent that the battlefield no longer constitutes the tactical basis of war, but rather an increasingly detached level of symbolic configurations, to maintain distance from the battlefield is the very meaning of waging war.

With the game, the war game shares to the point of indistinguishability a sphere that avoids as far as possible hindering a free unfolding through facts and circumstances. However, in contrast to the mere game, the war game reserves for itself their transformation.

Wittgenstein suggests that chess was perhaps not always regarded as a game; it may well be a construction of the nineteenth century that the game could enter—to some extent phylogenetically—into a fundamental opposition to seriousness in the first place. Viewed ontogenetically, it is becoming increasingly apparent to Indologists that chess emerged from a war game: North Indian rulers of the sixth century moved terracotta figures over the sandy ground whose configuration resembled the four branches of their army.⁸² Wittgenstein’s chess analogy thereby implies an assumption and raises a dual question: the fact is that officers also operate on ordnance survey maps at the current moment, though there is no war—as Wittgenstein’s use of the present tense makes clear. But what ensures that what is being conducted here is no mere game? And what ensures that in a game of chess no war is being waged? The answer to both questions is the same: nothing. It is precisely for this reason that Wittgenstein’s philosophy of language games will be impervious to a critique that states that its very name manifests its irrelevance. Language games may certainly prove to be irrelevant, but no definitional power can anticipate such proofs. The limits of the game can only be played out, and thus with every war’s end, the strategic and tactical playing through of a future one begins. If a game ends up in an application, then it leads to a serious case, and the application leads to a scientific application, as Wittgenstein shows by way of the same analogy of the game of chess and the war game: “If on Mars there were human beings and they waged war against each other in the way chessmen do on a board, then their headquarters would use the rules of chess for prophesying. Then it would be a scientific question whether checkmate can be reached in a certain constellation of the game, whether mate can be reached in three moves, and so forth.”⁸³ The difference between the application of signs and applications that derive from the application of signs is erased by Wittgenstein’s analogy: the use of signs can be exhausted in the game as much as it can predict whole courses of war. Both extremes can be subject to one and the same “system of game rules”⁸⁴—a system that does not fix the precise use of what it regulates.

On the basis of this dramatic indifference, Wittgenstein first approaches the subject of the game and links it to questions about the foundations of mathematics and its language. Commentators who regard the language game as an original philosopheme of Wittgenstein’s overlook prolonged discussions in which reference is habitually made to sign games for the elucidation of mathematical foundations. Most recently and perhaps most impressively, Hermann Weyl had applied Hilbert’s axiomatic-formalist proof procedure to concepts of the chess game and thereby provided Wittgenstein a basic schema for his language games.⁸⁵ Only when Wittgenstein transitions in his reflections from mathematically understood language games to general language games does his philosophy of mathematical language constructions experience its extension to a general ontology of grammatical rules.⁸⁶

Wittgenstein’s elaborations on the game, the war game, and mathematical formalizations were recorded by Friedrich Waismann so as to present them at the “Second Conference on the Epistemology of the Exact Sciences” in Königsberg.⁸⁷ At the conference, adherents of the logicist, formalist, and intuitionist schools convened to stake out once again their mathematical positions. At the beginning of the twentieth century, David Hilbert had committed mathematics to a formalism that, unlike Russell and Whitehead’s logicism, would not persist in the attempt to base mathematics solely on logical elements. His proof procedures united arithmetical and logical operators in order to establish an axiomatic basis. Above all, the Dutch mathematician Brouwer and, in the early 1920s, Hermann Weyl criticized Hilbert’s mathematical operations, arguing that their claimed existence and effect could not be ensured through any mathematical intuition or constructive procedure.

At the time of the Königsberg conference, the dispute over the foundation of mathematics had already passed its peak. Weyl had cast his lot once again with Hilbert, for however ontologically questionable formalism might have appeared to him, he deferred to the success that the application of Hilbert’s method within theoretical physics promised. Moreover, Kurt Gödel presented at the conference for the first time the basic features of his groundbreaking proof. Though this thwarted Hilbert’s dream of securing a consistent mathematics, it was nonetheless demonstrated solely with Hilbert’s formalist instruments and excluded Brouwer’s intuitionism.⁸⁸ Even if Hilbert’s long-term formalist objective of consistency and decidability proved to be unattainable, at least the formalist method had turned out due to Gödel’s work to be unrivalled in measuring its own limits.

Brouwer had last participated in the discourse two years earlier with two talks in Vienna. For a long time they would remain his last public appearances—the quarrel with Hilbert had escalated, and it had for a long time no longer been only about mathematical entities.⁸⁹ Among Brouwer’s listeners was Wittgenstein, who had first had to be persuaded by Waismann to attend the public event.⁹⁰ According to Herbert Feigl, who joined Waismann and Wittgenstein that evening, the lecture nonetheless impelled Wittgenstein to resume his philosophical ruminations.⁹¹

The question of whether Wittgenstein’s position should be regarded as a further mathematical approach was at least addressed at the conference.⁹² His standpoint was in any case understood to the effect that the meaning of a concept lies in its use.⁹³ It is thus clear that he proceeded primarily from Brouwer and Weyl’s earlier attacks on formalist mathematics, which culminated in the idea that mathematics was more activity than theory.⁹⁴

It is no longer the founders of the mathematical schools themselves, but meanwhile the generation of their successors who in Königsberg look back to the foundations of mathematics. Instead of Hilbert, John von Neumann spoke about formalism. Brouwer was represented by his pupil Arend Heyting, and Russell’s logicist position was elaborated by Rudolf Carnap.

If, rather then asking about the mathematical constructs disputed in the foundational debate, one asks about a referential system that, as a precondition of the foundational debate, needs no introduction and is not doubted by any side, then the answer is “the game.” Neumann, Heyting, and Gödel all took up at the conference in Königsberg the concept of the formula game; according to Heyting, the “word ‘mathematics’” for the intuitionist means “a mental construction,” for the formalist “a game with formulas,”⁹⁵ in which—to use Hilbert’s words—a “technique of our thinking”⁹⁶ is first constituted. Neumann underscored in his contribution to the formalist explanation of mathematics that, though “the content of a classical mathematical sentence cannot always (i.e., generally) be finitely verified, the formal way in which we arrive at the sentence can be.”⁹⁷ Therefore, it is less the statements themselves that should be investigated and more the methods of proof, which should be understood as a “combinatorial game played with the primitive symbols.”⁹⁸ In the run-up to the conference, Wittgenstein seems to comment in advance on Neumann’s elaborations:

Something in formalism is right and something is wrong. The truth in formalism is that every syntax can be understood as a system of game rules. I have thought about what Weyl may mean when he says that the formalist conceives of the axioms of mathematics as like the rules of a chess game. I want to say: Not only the axioms of mathematics but all syntax is arbitrary.⁹⁹

That axioms dispense with substantiation was mathematical consensus: if axioms are to constitute the basis of all derivations, they themselves escape every derivation. But Neumann and—under a different sign—Wittgenstein now see proof procedures too as springing from the arbitrariness of a semiotic game.¹⁰⁰ This viewpoint also dispels all the origin legends that ultimately regarded proof figures as emerging from the genius and the inspiration of mathematicians.

In any case, Neumann disagreed vehemently with Carnap and insisted that “it is actually meaningless symbols that are introduced. But for Hilbert the introduction of these meaningless symbols is not an end in itself.”¹⁰¹ Neumann was the first at the conference to recognize the consequences of Gödel’s proof, and he built a bridge for him with his statement. Gödel’s commentary in the conference publication, which once again discusses his proof of the impossibility of an irrevocable consistency in mathematics, takes up Neumann’s characterization of formalism and even adopts his conceptualizations. Formalism, according to Gödel, is about a “purely combinatorial property of certain sign systems and the ‘game rules’ that apply to them,” with which “combinatorial facts [can ultimately be] expressed in the symbols of mathematical systems.”¹⁰²

The game with formulas was thus by no means abolished with the waning of the foundational debate. On the contrary: there is no method that can demonstrate possible contradictions of combinatorial facts except for these themselves. Gödel could also declare the game with signs a combinatorial fact because it had long been common among mathematicians to plumb the foundation or the groundlessness of mathematics on this basis. However diverse the mathematical standpoints of Gottlob Frege, Weyl, Brouwer, Hilbert, or Bernays appeared in the foundational debate, what they all have in common is that they determine the nature of mathematics in the difference with or in the correspondence to the game.¹⁰³ Thus it comes as no surprise that the first historians of mathematics, such as Oskar Becker and Jean Dieudonné, described Hilbert’s formalism as a game with formulas.¹⁰⁴ If mathematicians thus began around 1930 to talk about the game, then it was no longer necessarily connected with the pejorative sense that had often been attached to the term in the past—for example, when Gauß had still spoken of the “meaningless formula game,”¹⁰⁵ or when even Hilbert himself still admonished in a lecture in 1919 that mathematics was “not like a game” in which “the tasks are determined by arbitrarily conceived rules.”¹⁰⁶ But by taking up the game as a sign system, the mathematical discourse experiences a radical expansion of its playing fields.

The first person after Leibniz to investigate games seriously in terms of their mathematical efficacy was Paul Du Bois-Reymond. Unlike his brother Émile, who more than almost anyone else was responsible for physiology’s rise to the status of a leading science in the late nineteenth century, Paul attended, alongside his medical studies, Dirichlet’s lectures on the integration of partial differential equations and ultimately held professorships in pure and applied mathematics.

The fact that a caesura within mathematics is nonetheless associated less with Paul Du Bois-Reymond and more with his brother is due to the altered position of mathematics with respect to other disciplines.

“Ignoramus et ignorabimus” were Émile Du Bois-Reymond’s concluding words at the Leibniz meeting at the Academy of Sciences in 1880: motive forces and consciousness are transcendentally unfathomable.¹⁰⁷ “In mathematics there is no ignorabimus!”¹⁰⁸—with these words Hilbert opened the Second International Congress of Mathematicians in Paris in 1900. Thirty years later, he still had not grown tired of repeating for radio listeners: “We must know, we shall know.”¹⁰⁹ When Hilbert presented at the congress the fundamental program of the dawning century, he rejected a scholarly class to which the brothers Du Bois-Reymond still belonged. Hilbert’s list of twenty-three problems that he first posed to the congress participants—and, to some degree, to mathematicians to this day—may have caused a stir in part because it broke with the humility of an auxiliary science that waited for the annual offering of a prize question from the academies. Hilbert replaced the monolithically embodied knowledge that academies spread out in their departments with the mathematical operation of the mathematical institute.¹¹⁰

His program was not merely interested in the general solvability of mathematical problems, but in the development of the axiomatic method to a procedure of knowledge that ultimately no science could circumvent.¹¹¹ To Émile Du Bois-Reymond’s Latin maxim, which emphasized limits of knowledge for the living metaphysical body, it was not even necessary for Hilbert’s positivism—propounded in East Prussian dialect and primarily directed toward sign systems—to respond. However, Du Bois-Reymond’s doubts were already coming from a level that had long been subject to the criteria of mathematics. He had vehemently advocated for the increased inclusion of mathematics in the curricula of the humanistic Gymnasium—if necessary, to the detriment of the ancient languages. The only answer his brother Paul Du Bois-Reymond and Hilbert had for the question of what mathematics could be applied to was the equally positive counterquestion: “What is not [applied] mathematics?”¹¹²

But the decisive, epochal caesura around 1900 is not to be found in different views as to how material foundations of force and of consciousness or life can be mathematically grasped. Rather, it is manifested in the struggle to trace mathematics back to foundations other than the Platonic heaven of ideas. Upon closer inspection, Hilbert’s program is therefore directed less against Émile Du Bois-Reymond than against his mathematician brother Paul, who was the first to proceed from the possibility of alternative foundations and not merely from systems in mathematics.¹¹³

In the course of this, Paul Du Bois-Reymond had in 1882 already called formalism by its name and declared it dead—long before Hilbert elevated a formalist mathematics based on axioms to a program. Du-Bois Reymond stated:

A purely formalist-literal structure of analysis, which is what the separation of number and sign from quantity amounts to, would ultimately degrade this science, which is in truth a natural science, even if it only draws the most general properties of the perceived into the domain of its research, to a mere game of signs, in which arbitrary meanings would be attached to the written signs as to chess figures and playing cards. As delightful as such a game can be, as useful for analytic purposes as the solution of the task of pursuing the rules between the signs, which emerged from the idea of quantities, to their ultimate formal consequences may even turn out to be, this literal mathematics, if it were completely detached from the ground on which it grew, would nonetheless soon enough exhaust itself in unfruitful sprouts, while the science that Gauss [italicized in original] so truly and profoundly called Grössenlehre [theory of quantities] possesses in the natural domain of human perception, which is always expanding, an inexhaustible source of new research objects and fruitful stimulations. Without question, with the help of so-called axioms, of conventions, ad hoc philosophemes, inconceivable extensions of originally clear concepts, one will be able to construct retroactively a system of arithmetic resembling in all respects that which emerged from the concept of quantity, in order to cordon off calculative mathematics, so to speak, through dogmas and defensive definitions from the psychological domain. An extraordinary acumen can even be applied to such constructions. Moreover, one would be able to think up in the same fashion other arithmetical systems, as has happened. The ordinary arithmetic is precisely the only one that corresponds to the linear concept of quantity, is so to speak its first registration, while analysis constitutes its highest development with the limit concept at the forefront. Even the difficulties of the limit concept, which we will readily confront fearlessly, one may believe that one can solve through symbolics. It will scarcely succeed. For every analyst who is more than a combinatorialist will want to pursue the origin of the game of signs, and thus find himself once again facing the circumvented problems.¹¹⁴

Du Bois-Reymond thus demands that rather than cordoning off the “psychological domain” through “dogmas and defensive definitions,” one must draw “real quantities” from a “domain of human perception, which is always expanding.” There are reasons external to mathematics for the fact that even a proof of the existence of a limit to continuous sequences, which proceeds with the help of discrete decimal fractions, does not become a paradox. It is not discrete signs that first segment the continuum into discrete quantitative sequences, but the “peculiarity of thinking” itself, which has its external sign in the “sight perception” of the “jerky turning of the eyeball.”¹¹⁵ Du Bois-Reymond, who initially made his mark with studies of the blind spot,¹¹⁶ placed mathematics on foundations that remain bound to human psychology and physiology. Furthermore, he distinguished real quantities from mathematical ones—each of which he differentiated in turn. Mathematical quantities, however, bring together combinatorial and “logical processes.”¹¹⁷ Du Bois-Reymond would refer them entirely to the domain of the mind, if another “combinatorial area” did not come to the fore—that of the game: “It cannot be denied that the knight’s move problems, but especially the so-called endgames of chess . . . exhibit the character of genuinely mathematical tasks, only within an extremely limited combinatorial area.”¹¹⁸ From a field that in the late nineteenth century was otherwise perceived only as mathematics for amusement, Du Bois-Reymond extracts a new dimension—only to reject it immediately. The “game quantities” have “unreality in common” with mathematical quantities—without, however, standing like the latter in “close relation” to reality.¹¹⁹

The reference to the game within mathematics, although or precisely because it appears peripherally, is the only one that persists as a “common platform of all discussions,” when Hilbert, as Weyl summed up laconically, “postulated his proof theory,” and toppled mathematics as a “system of substantial, meaningful, insightful truths.”¹²⁰ Though Du Bois-Reymond had found in “the game of signs” a “formalist-literal structure,” it did not come into serious consideration as a mathematical foundation due to its “limited combinatorial area.” Brouwer and Weyl’s critique of Hilbert’s program of formalization was diametrically opposed; formalism practices a formula game that goes beyond domains still accessible through acts of thinking in Brouwer’s sense.

Instead of proceeding merely from different phases in the confrontation over mathematical foundations,¹²¹ which have to manage the increasingly reciprocal references of their mathematical structures, it is the references themselves that must be interrogated. These cannot be constrained even by the rigid referential system of axiomatic postulations. It is here that the caesura and the fault line first become apparent that overtook the discourse of mathematical foundations. And this pertains above all to the mathematical sign, which is now taken as at once object and foundation:

No more than any other science can mathematics be founded by logic alone; rather, as a condition for the use of logical inferences and the performance of logical operations, something must already be given to us in our faculty of presentation [in der Vorstellung], certain extralogical concrete objects that are intuitively [anschaulich] present as immediate experience prior to all thought. If logical inference is to be reliable, it must be possible to survey these objects completely in all their parts, and the fact that they occur, that they differ from one another, and that they follow each other, or are concatenated, is immediately given intuitively, together with the objects, as something that neither can be reduced to anything else nor requires reduction. This is the basic philosophical position that I regard as requisite for mathematics and, in general, for all scientific thinking, understanding, and communication.

And in mathematics, in particular, what we consider is the concrete signs themselves, whose shape, according to the conception we have adopted, is immediately clear and recognizable. This is the very least that must be presupposed; no scientific thinker can dispense with it, and therefore everyone must maintain it, consciously or not.¹²²

What scientific thinkers have to envision consciously—and everyone else comprehend unconsciously—are signs, which do not point beyond themselves as referents. The assertion that “number signs [Zahlzeichen, or numerals], which are numbers and which completely make up numbers” nonetheless become the sole object of consideration, “but otherwise have no meaning at all,” provokes the first critical inquiries: “Can there be a sign without meaning?” Aloys Müller, who posed this question to Hilbert, also made an attempt to answer it himself:

A sign always signifies something that is different from the sign itself. Sign and signified object are assigned to each other. . . . If Herr Hilbert wants to insist that 1 and + are without meaning, then they are not signs, but in that case merely drawings [or] figures. . . . Is that a foundation for number theory? Certainly not. With the necessary imagination, one thus gets pretty moldings or wallpaper borders and for each a manufacturer’s trademark, but not mathematics.¹²³

Hilbert left the defense against the critique to Paul Bernays, who could not help making linguistic concessions. Instead of signs, it would be better to speak in the future of figures ¹²⁴—instead of numbers, numerals.¹²⁵ However, the Hilbert school proved to be firmer in the matter. Where the border ran between “meaningless figures”¹²⁶ and signs established by them remained subject to debate.

Hilbert seeks to solve the paradox of representing infinity and continuity with finite and discrete signs by declaring them solely a matter of axiomatic postulations and thus of sign systems. In contrast to Du Bois-Reymond and all mathematicians who, with Leibniz, proceeded from the assumption that the world makes no leaps, Hilbert excluded infinity from the physical world:

For everywhere there are only finite things. There is no infinite speed, and no force or effect that propagates itself infinitely fast. Moreover the effect itself is of a discrete nature and exists only in quanta. There is absolutely nothing continuous that can be divided infinitely often. Even light has atomic structure, just like the quanta of action. I firmly believe that even space is only of finite extent, and one day astronomers will be able to tell us how many kilometers long, high and broad it is. And although there are in reality often cases of very large numbers (for instance, the distance of the stars in kilometers, or the number of essentially different games of chess) nevertheless endlessness or infinity, because it is the negation of a condition that prevails everywhere, is a gigantic abstraction—practicable only through the conscious or unconscious application of the axiomatic method. The conception of the infinite, which I have grounded through detailed investigations, answers a number of important questions; in particular, it shows the baselessness of the Kantian antinomies of space and of the unlimited possibilities of division, and thus of the difficulties that crop up thereby.¹²⁷

Nothing remained of Du Bois-Reymond’s mathematical foundations, which in the final analysis were always perceived from limits and emanated from perceiving bodies. At most, gestalt-theoretical considerations still determine the discussion of the postwar years. In the foundational debate, bodies no longer matter, but space certainly matters everywhere.

That the body, understood psychologically, is excluded from a mathematical framework may still be traceable to a disciplinary differentiation. The legitimation strategy within mathematics to demonstrate its use for other disciplines and thereby its own necessity becomes dispensable. But Hilbert’s formalist mathematics itself still reaches beyond its own discourse by charging the general concepts of its choice in a specific fashion.

The concept of the body is not suited to that strategy. It can be the object of mathematical procedures that prove its calculability, but its investigation is left to other disciplines. Space is an entirely different matter—it stands at the center of the foundational debate. The mathematics of the postwar period asserted over all other disciplines the claim to a genuine concept of space and set out to provide means and techniques for the mastery of it. In this, it is crucial to free the Kantian “a priori theory” of the last “anthropomorphic dross.”¹²⁸ Hilbert’s formally cultivated mathematics as much as its intuitionist version demand an unmediated access to elements constitutive of space. While formalism dispenses with phenomenological interpretations ¹²⁹ and abandons itself to calculation as an object disseminated in signs, intuitionism subordinates a spatial continuum to the primacy of time. For Brouwer, mathematics therefore has to merge completely with its activity; for Hilbert, on the other hand, it exists entirely on paper: “The question of where scientific exactness exists is answered differently by the two parties: the intuitionist says: in the human intellect, the formalist: on paper.”¹³⁰ The debate over the foundation of mathematical operations has an underlying geopolitical subtext. Brouwer is chiefly responsible for introducing it into the discourse. Before foundational mathematical publications, he wrote about early mythical times and their disenchantment through techniques of land reclamation:

Holland was created and was kept in existence by the sedimentation of the great rivers. There was a natural balance of dunes and deltas, of tides and drainage. Temporary flooding of certain areas of the delta was a part of that balance. And in this land could live and thrive a strong branch of the human race.

But people were not satisfied; in order to regulate or prevent flooding they built dykes along the rivers; they changed the course of rivers to improve drainage or to facilitate travel by water, and they cut down forests. No wonder the subtle balance of Holland became disturbed; the Zuyder Zee was eaten away and the dunes slowly but relentlessly destroyed. No wonder that nowadays even stronger measures and ever more work are needed to save the country from total destruction.¹³¹

His dissertation on the foundation of mathematics soon ignited a quarrel with his doctoral adviser,¹³² who rejected large portions of his work and cut the sentence stating that science serves human beings solely in the struggle against their own kind and against nature—that it ultimately has only the value of a weapon.¹³³ Even astronomical models, he argued, were subject to the will of individuals. They knew how to read from the measuring instruments those values that lent themselves to the construction of theories. Brouwer therefore declared: “The laws of astronomy are no more than the laws of our measuring instruments.”¹³⁴ For Brouwer, mathematics—looking backward beyond the contrary positions of Kant and Leibniz—had to find its way back to a mystically charged primal intuition.¹³⁵

All the more surprising is the turn in Brouwer’s career when, against the will of the professor of applied mathematics, the latter actually became a weapon for him. Perhaps to avoid serving the Dutch Army once again as a reservist—his first period of service must have been traumatic—from 1915 on, Brouwer took the bull by the horns: he began to delve into photogrammetry and submitted a memorandum to the ministry for defense.¹³⁶ 1915 was the year he learned from Schönflies, during a visit to Göttingen, that many of the young mathematicians there were used by the military for measurements and transformation calculations of aerial photographs—and members of the Academy directly advised the General Staff.¹³⁷ But 1915 is also the year a new era dawned for photogrammetry in general. At that time, Oskar Messter registered for a patent for his “Method for Producing Photographic Images from an Aircraft.”¹³⁸ Aviatics and photography could thus enter a media-technological alliance that created maps out of survey photographs in a continuous technological processing chain—that is, with the exclusion of human perception. Brouwer calculated for the Dutch General Staff how, thanks to trigonometric methods, unavoidable angle differences in serial photographs of terrain could be reconciled and topographical maps with a larger scale than what was previously common could be created.¹³⁹ But the chief of the General Staff declined and declared the basic scale used at that time to be sufficient. The works on photogrammetry remained without any positive response and Brouwer fell into a depression.

When Weyl, after the war’s end, adopted Brouwer’s standpoint on mathematical foundations, geopolitical resonances were in play from the beginning. In this, Hilbert anticipated Weyl when, in a lecture on axiomatic thinking in Switzerland in 1917, he drew an analogy between the “life of science” and that of states, which “have to be well ordered” not only in themselves, but also in their relations to each other.¹⁴⁰ For Weyl, the situation culminated more drastically, resembling the “separation of the Occident from the Orient” at the time of the Persian wars, the tension and overcoming of which “became the driving motive of knowledge for the Greeks.”¹⁴¹ Against this background, the “antimonies of set theory” are

regarded as border conflicts that concern only the most remote provinces of the mathematical empire and can in no way endanger the inner solidity and security of the empire itself, its actual core areas. . . . Indeed: any serious and honest reflection must lead to the recognition that those detrimental effects in the border regions of mathematics must be judged as symptoms; in them comes to light what is hidden by the outwardly shining and frictionless operation in the center: the inner instability of the foundations on which the construction of the empire rests.¹⁴²

Ultimately, all that remains for Weyl is to “gain solid ground” in the face of the “looming dissolution of the polity of analysis” and to declare “Brouwer—that is the revolution!”¹⁴³ Subsequently, Hilbert accused Weyl and Brouwer of an “attempted coup,” a “dictatorship of prohibitions,” and “terror.”¹⁴⁴ There is no question that the use of a rhetoric for the continuation of war by verbal means may say as much about the time after 1918–1919 as about the intensity of the mathematical foundational debate. But the fact that, for all the metaphors, effective methods for the calculation of spaces and borders were flourishing should not be overlooked.

Just as proof figures come to the fore in the mathematical discourses, the metaphorical recedes. Bringing the measurement of a natural space under control is now less urgent than sketching spaces that arise from a sign-based apparatus and that are anything but mathematically secured.

From an operational point of view, fundamental questions of the establishment of a “mathematical apparatus”¹⁴⁵ and of a state apparatus are now on the same page. Brouwer, for one, recalls that it was “not just theoretical sciences like paleontology or cosmogony” that depended on trust in the principles of classical logic incriminated by him, “but also governmental institutions like the rules of procedure for a criminal trial.”¹⁴⁶ He calls into question the idea that a theorem or a formula should be regarded as true merely because a proof proceeding from the opposite assumption leads to a contradiction. Conversely, a theory may not be false merely because the assumption of an opposing proof reveals no contradiction. To lend his argument weight, he compares mathematical procedures with those of criminal law: “[An] incorrect theory remains incorrect even if it cannot be disproved by contradiction, in the same way that a criminal policy remains criminal even if it cannot be condemned and stopped by any legal process.”¹⁴⁷ Weyl too turns to the comparison between mathematical and police state methods. Thus, one should “beware of the idea that, when an infinite set is defined, its elements are, so to speak, merely spread out before one’s eyes, and that one need only go through them in succession, as a police officer goes through his register, in order to find out whether in the set an element of this or that sort exists. With respect to an infinite set, that is senseless.”¹⁴⁸ Brouwer and Weyl clearly no longer adhere to Hilbert’s textbook-like example sentences, which exclude any reference to the present, such as the still rather scholastic: “Aristides is corruptible.” In contrast, to demonstrate to formalized mathematics its lack of constructive methods for the securing of the infinite by finite means, Brouwer and Weyl adhere to state organs, the legal constitution of which was no less contentious at that time. Brouwer implies that when the criminal facts of a case apply to the executive himself, this does not challenge the legislative power. And Weyl proceeds from the assumption that the police operate with a register that encompasses a well-defined and clearly circumscribed set. Indeed, with their analogies they touch on areas of the state constitution that could not have been more controversial among constitutional law scholars of their time. Carl Schmitt, whose dominance in constitutional law discourse arose with the final days of Weimar, would not have agreed with Brouwer that the law is still in force when the executive is not in accordance with it. On the contrary, for Schmitt, “authority proves that to produce law it need not be based on law.”¹⁴⁹ The ultimately decisive exceptional case “makes relevant the subject of sovereignty, that is, the whole question of sovereignty. The precise details of an emergency cannot be anticipated, nor can one spell out [Aufzählen] what may take place in such a case, especially when it is truly a matter of extreme emergency and how it is to be eliminated.”¹⁵⁰ Jurists may still speak colloquially of enumeration (Aufzählen), but formalists have long required several words: they know how to differentiate countably (abzählbar) finite sets from uncountably (überabzählbar) infinite ones—and these, in turn, from counting to transfinite numbers.

In mathematics, it has been known since Georg Cantor’s set theory that there are infinite sets of different power or size (Mächtigkeit). At least the largest set of real numbers can therefore not be deterministically circumscribed by a system of order. The problem that the sovereign subject stands outside “the normally valid legal system” and “nevertheless belongs to it,” for it gives him the right to suspend it,¹⁵¹ doubles a situation that already applies to formal systems as such. And Weyl, as shall be shown in greater detail, proclaims a revolution for the very reason that he, with Brouwer, does not proceed solely from numerical sequences “determined by law” but also from those that emerge “from step to step through free acts of choice.”¹⁵²

These discourses of jurists and mathematicians correlate also with respect to the formation of their factions. Schmitt’s definition of the sovereign who decides on the state of exception opposes the purely formalist legal theory of his rival Hans Kelsen. Schmitt argues against Kelsen that the case that must first come to pass in order to be able to be judged and decided escapes the possibility of regulation by normative legislation. On the opposite end of the political spectrum, Evgeny Pashukanis—a leading representative of Marxist legal theory—ranked alongside Schmitt as one of Kelsen’s critics. He viewed Kelsen’s neo-Kantian legal positivism as a “legality of ought purified of all psychological and sociological ‘residues,’” which

neither has nor can have any rational definition at all. . . . On the level of the juristic ought there is only a transition from one norm to another on the rungs of a hierarchical ladder, on the top rung of which is the all-encompassing, highest norm-setting authority—a limit concept from which jurisprudence proceeds as from something given. . . . Such a general theory of law, which explains nothing, which turns its back from the outset on the facts of reality, that is, of social life, and busies itself with norms, without taking an interest in their origin (a meta-juridical question!) or in their relation to any material matters, can certainly at most lay claim to the name theory in the sense in which one, for example, customarily speaks of a theory of chess.¹⁵³

Pashukanis seems to have taken up the reservations that Brouwer and Weyl already expressed with respect to formalist mathematics and directed them against Kelsen’s formalist legal theory—not least of all, his choice of words and the chess analogy suggest as much. Weyl certainly lent a hand to the discursive leap by himself attaching a political, police function to simple quantifiers of predicate logic: “The ‘there is’ arrests us in being and law, the ‘every’ releases us into becoming and freedom.”¹⁵⁴

Since Cantor and Frege’s contributions to set theory, mathematical conceptions of space are no longer dominated by the limit value problem of analysis, but by possibilities of ordering that belong to the measuring and counting numerical sequences and sets themselves. Weyl shares this aim when he designs a continuum “into which the individual real numbers undoubtedly fall, but which by no means dissolves into a set of real numbers as finished beings,” but rather offers “a medium of free becoming.”¹⁵⁵

Du Bois-Reymond had already speculated about sequences that escaped every law, because they were either based on rolls of the dice or broke free of “human company” and continued “independently on the way into the endless,” through “a fixed rule” that was “given to them for the journey.” Ultimately, Du-Bois Reymond distanced himself with an empirical view of mathematics from “assuming and weaving into the mathematical thought process things of which we have and can have no conception.”¹⁵⁶

Weyl, on the other hand, inspired by Brouwer and supported by his own investigations of recursive sequences of dual fractions, was serious about the design of “judgment instructions” that are “self-sufficient” and “even contain at their core an infinite abundance of real judgments.” The clearly delineated judgment instructions “formulate,” according to Weyl, “the justification for all the singular judgments to be ‘redeemed’ from them.” The judgments themselves cannot be “redeemed” except through the execution of the process that produces them: “This happens insofar as we allow the emerging [werdende, i.e., becoming] sequence of choices at every step to produce a number or nothing or cause the breaking-off of the process, its own death and the annihilation of its previous production.”¹⁵⁷ The continuum, and with it space, now appears to Weyl to be “becoming toward within into the infinite.” He opens up a space that corresponds in fundamental respects to Carl Schmitt’s design of an “order of large spaces” (Großraumordnung) in international law. What is significant in Schmitt’s Völkerrechtliche Großraumordnung (his notorious text on the large spatial order) is first and foremost the supplanting of the juristic concept of the Reich—the realm or empire—by that of space. At the beginning and conclusion of the text, Schmitt does not neglect to deal with the concept “large space” (Großraum). Though Raum, or “space,” still has a mathematical-physical sense, Großraum encompasses more of a “technical-industrial-economic-organizational domain.”¹⁵⁸ It should no longer be regarded as mathematically neutral space, in the emptiness of which “the perceiving subject” inscribes “the objects of its perception,” in order to “localize” them.¹⁵⁹ With Großraum, Schmitt seeks to leave behind the “mathematical-scientific-neutral field of meaning” that still adheres to the concept of “space”: “Instead of an empty dimension of surfaces and depths in which physical objects move, the cohesive space of achievement (Leistungsraum) appears, as is proper for a history-filled and historic Reich, which brings along with it and bears within itself its own space, its inner measures and borders.”¹⁶⁰

Schmitt first comes to this conclusion in 1941. He formulates it in a chapter that he adds to his text on Großraumordnung, which appeared in 1939. It is in this chapter that Schmitt first sees the natural sciences open up to the Großraum through Max Planck’s “World Picture of New Physics” and Victor von Weizsäcker’s biological “space of achievement” (Leistungsraum). However, neither the experience of the Blitzkrieg nor the mediation through the natural sciences was necessary for his concept of space. In the medium of paper, which was itself subject to debate as the mathematical foundation, the mathematicians in Göttingen, Amsterdam, and Zürich had already expanded on orders and arrangements of space for a long time. For all the criticism of Schmitt in the evaluation of the consequences of his juristic and political engagements, the question of sources that he intentionally fails to acknowledge has remained unaddressed. His concept of “large space” is clearly not merely motivated by a political movement, but is also shaped by a mathematical discourse that had already developed on its own a political surplus. Thus Schmitt still clings to the existence of a sovereign subject that decides on a history-filled space, while mathematicians had long since moved on to studying the condition of possibility of a medium of free becoming, in which decision-making power emanates from the contingency of a game of signs.

Sign games, which set their own unfolding in motion, leave behind naturalistic and life-philosophical conceptions. Henri Bergson, for one, still found it completely unthinkable that mathematical symbols could designate duration and movement otherwise than only indirectly. They themselves are intrinsically immobile: “For the geometer all movement is relative: which signifies only, in our view, that none of our mathematical symbols can express the fact that it is the moving body which is in motion rather than the axes or the points to which it is referred.”¹⁶¹

For Bergson, the mathematical symbol that directly designates a body in motion would have to coincide with it. But the fact that symbols have always already been set in motion in games—such as the sign-bearing game pieces of the medieval Battle of Numbers, movable type in book printing, and the “types” in Reiswitz’s tactical war game, on instrument displays and in calculating machines—does not occur to Bergson. Most likely, he would scarcely have seen in these examples anything but exceptions in which the categories are mixed in a confused fashion.

In fact, mathematicians in the twentieth century began to design symbol systems with operations that, at least according to the fiction, are accompanied by movements. In this, it was not a matter of representing life-world phenomena of movement in imagined or actually constructed mobile sign configurations. Rather, it was the reverse: a matter of exploiting new possibilities of making-calculable through arrangements of signs set in motion.

If one disregards Adam Smith’s “imaginary machines,” which promised to transfer the principle of Newton’s celestial mechanics to state constitutions,¹⁶² then the eugenicist and statistician Francis Galton, toward the end of the nineteenth century, was the first to begin consistently representing inherently abstract mathematical models in machine models and thereby reversing the classical process of analysis. Galton initially furthered developments in representing statistics through graphic methods. Nonetheless, behind his efforts in the visualization of inherently abstract characteristics through mechanisms was an elaborate psychology of imagination that he called “mental imagery.”¹⁶³ This psychology would also enable the introspective design of “mechanical illustrations.”¹⁶⁴ There is no evidence that Galton ever mechanically implemented his illustrations as a model. Therefore it would be false to assume that his illustrations are construction drawings; rather, they must already be convincing as fictive machines. It is no longer necessarily the task of mathematics to calculate specific properties of mechanisms or machines. Rather, the latter now serve mathematics. They are therefore released from every other purpose. And the imagination too thus receives a different status. In contrast to the fictionalism of thought experiments, which also proceed from counterfactual assumptions, it is not compulsorily fictive. On the contrary—no imaginary surplus presses for the illumination of a domain that would pose a problem in its formal development. Rather, only what can easily be technically realized is imagined and transposed into the fictional. If the mechanical realization presented a problem, the fiction should not be maintained. If the realization is already established in the fiction, every effort of realization also ultimately becomes unnecessary. Galton’s mechanical illustrations therefore bring to light a new discursive figure with quite far-reaching consequences. In contrast to an only verbalized hypothesis, they show an effective way to implement what is said operationally.

Alongside game configurations, the invocation of fictive machines also plays an important role in the twentieth century during the mathematical foundational debate between the formalist and intuitionist camps.¹⁶⁵ In a controversial fashion, they reveal limits that can be symbolized and formalized on the one hand and imagined and thought on the other. Ultimately, the dispute was not least of all about whether in mathematics a shift onto a metalevel would be possible. While David Hilbert elevated this very shift into a program and hoped from a metalanguage that it could precisely describe an operationally occurring symbolic language, E. J. Brouwer raised considerable doubts. As an insuperable epistemic problem, he invoked temporality, which is proper and essential to a mathematical operation and which for him is realized through an act of thought. For Brouwer, this fundamental aspect is irretrievably lost in the attempt to refer to the operation in retrospect—not least of all because the reference is, in turn, subject to its own temporality.¹⁶⁶

In the invocation of fictive machines, there is however no discernible platform that bridges the splintering mathematical discourse of the foundational crisis and keeps it in productive motion. In contrast to a fixed and fixing metalanguage, fictive machines abolish the temporal-performative structure called for by the intuitionist Brouwer and viewed by the formalist Hilbert as a temporary operation that challenged the mathematician to catch up to it in thought. The foundations of mathematics conceived in the form of fictive machines and models and as sign games appear, however, without being introduced systematically or conceptualized rigorously. Still, fictive machines should by no means be regarded as mere examples. Rather, they form discursive levels that regularly come into play.

Someone who perceived from up close the “technization of formal-mathematical thinking”¹⁶⁷ was Edmund Husserl: “[A] technization takes over all other methods belonging to natural science. It is not only that these methods are later ‘mechanized.’ To the essence of all method belongs the tendency to superficialize itself in accord with technization.”¹⁶⁸ His Crisis of European Sciences—written in the 1930s, it should be recalled—does not primarily seek the reasons for “the crisis of our culture”¹⁶⁹ in the humanistic sciences—to question their scientific nature was not new. Rather, he discerns drastic “shifts in meaning”¹⁷⁰ specifically in the prosperous positive sciences, above all in pure mathematics.¹⁷¹ Husserl’s Crisis—in spite of his and the general political critical situation—evidently still stands under the spell of the mathematical foundational crisis.¹⁷² For him, science turns into techne not because thought first pushes toward mechanization but rather because it already thinks of itself as mechanics:

Are science and its method not like a machine, reliable in accomplishing obviously very useful things, a machine everyone can learn to operate correctly without in the least understanding the inner possibility and necessity of this sort of accomplishment? But was geometry, was science, capable of being designed in advance, like a machine, without an understanding which was, in a similar sense, complete—scientific? Does this not lead to a regressus in infinitum?¹⁷³

During Husserl’s lifetime, a text already appeared that would have caused his rhetorical questions to waver in a fundamental fashion—had this text initially received more than the attention of only a very small circle of mathematicians. In 1936, two years before his death and in the middle of the period in which he wrote Crisis, the British mathematician Alan Turing, at King’s College in Cambridge, produced his now famous text “On Computable Numbers with an Application to the Entscheidungsproblem.”¹⁷⁴ The fact that Turing in this text performed a decisive proof on the question of the formalizability of decidability by means of a fictive machine, a paper machine,¹⁷⁵ went beyond the previously prevailing, supplemental meaning of fictive machine constructs. Turing’s machine fiction serves in this text not as an exemplification of his proof; it is the proof.

For Husserl, this turn was not foreseeable; his gaze was still directed entirely at the praxis of a thoroughly axiomatized formal mathematics:

But now [only] those modes of thought, those types of clarity which are indispensable for a technique as such, are in action. One operates with letters and with signs for connections and relations (+, ×, =, etc.), according to the rules of the game for arranging them together in a way not essentially different, in fact, from a game of cards or chess.¹⁷⁶

Husserl denies an epistemologically open horizon to a formal mathematics conducted in this fashion and he grants it only the status of a “mere art.” ¹⁷⁷ The idealizations and constructions methodically practiced “intersubjectively in a community” can be used “habitually and can always be applied to something new—an infinite and yet self-enclosed world of ideal objects as a field for study.” Mathematical symbols are thus conceived as being

objectively knowable and available without requiring that the formulation of their meaning be repeatedly and explicitly renewed. On the basis of sensible embodiment, e.g., in speech and writing, they are simply apperceptively grasped and dealt with in our operations. Sensible “models” function in a similar way, including especially the drawings on paper which are constantly used during work.¹⁷⁸

Ultimately, mathematical symbols would therefore correspond to tools such as pliers and drills, created only for “mental manipulation.”¹⁷⁹ But by transferring the minimal definition for the characterization of mathematical work onto a machine so as to ensure that every step that it would execute could also be executed by a mathematician, Turing’s machine design was elevated into an episteme. It now provides the basis for judging in the first place what a mathematician and mathematics is capable of deciding. One could not have done away more radically with unquestioned operations and habitual sign games than with this intertwining of common references to machine fictions and semiotic games into a single construction: a theoretical machine that independently inscribes or overwrites a potentially endless tape divided into squares with a finite number of different signs according to a finite number of rules. Precisely because the operation with signs is elevated here into a mathematical object of investigation, Husserl’s critical question loses its validity. This is the question of the forgotten beginnings of a science, the concepts of which seem to follow a machine design while any meaningful derivation of their origins remains unaccomplished. And if Turing’s proof of calculable numbers also performs the self-referential operation whereby the sign operations of his machine simulate the sign operations of another one, and is ultimately able to simulate a universal machine of all these machines defined by Turing, then Husserl’s question of the “regressus ad infinitum” of a science subject to a machine design receives a negative answer, for ultimately the conclusion to which Turing comes is not positive. Turing rejected the decision problem (Entscheidungsproblem) that Hilbert gave up hope of solving—that every mathematical proposition of a formal system endowed with the power of arithmetic must prove through a procedure to be true or false. He did so by demonstrating the fundamental undecidability of the problem. Though a class of calculable numbers—that is, those that can be generated in an effective fashion—can be indicated, whether a real number in general is a calculable number escapes all calculability.

Turing’s text reveals still more clearly than Kurt Gödel’s work on formally undecidable propositions or Alonzo Church’s introduction of lambda calculus that mathematical practice encounters a limit in the medium of a fictive machine. Turing’s machine is a medium that does not itself possess ideal mathematical objectivity, but projects into the life-world. For that reason alone, the medium allows nonmathematical authorities in circumscribed contests with tool-like writing apparatuses their “mental manipulation” with “an infinite and yet self-enclosed world of ideal objects as a field for study.” Only a mathematics that is measured against this medium reveals that it is already condemned to performance for intrinsic reasons.

Turing’s discovery that if something is calculable, it can also be calculated by a machine, releases the subject from a cultural technique with a long history. But Turing’s text did not simply legitimize leaving calculation to machines, for the quite simple reason that there had been a long tradition of presenting such legitimations. The fact that his fictive machine would ultimately be realized in actuality as a real discrete sign-processing machine is not so much due to the possibility of carrying out calculations efficiently. What is much more decisive is that the making-calculable—and not primarily the calculation—is performed operationally and bound to discrete signs. With the Turing machine, a machine concept appears that is based on the strictest conceivable determinism in order to reveal by that very means an entirely new form of incalculability.¹⁸⁰ Because sign-processing procedures potentially prove only through their execution that they come to an end, it is not enough only to imagine the Turing machine—one must let it run.¹⁸¹

It should not be overlooked, however, that the Turing machine had long asserted itself as a fictive object during the Second World War. Logician Alonzo Church, in any case, who gave Turing’s fictive machine its name,¹⁸² in the course of a career that lasted from 1924 to 1995,¹⁸³ engaged only once with automata theory, whereas computer science cannot manage without his lambda calculus.¹⁸⁴ But the caesura thus occurs all the more violently in the Second World War when paper machines, of all things, unleash research offensives and matériel battles.

“Merely factual sciences make merely factual people,” Husserl still wrote in his Crisis.¹⁸⁵ Heinrich Scholz’s noteworthy scientific career, which was defined by a shift from theology to research into the foundations of logic in the first half of the twentieth century, is able to confirm Husserl’s assessment more than almost any other—but one should not therefore deny Scholz the consciousness of his own conversion. The protestant theologian was the first to respond to Oswald Spengler’s radical epochal conception, and it did not elude him that Spengler—despite all the prognosticated declines—had turned above all to the coming generation: “If, under the sway of this book, people of the new generation turn to technology instead of lyric poetry, the navy instead of painting, politics instead of critique of knowledge, then they will be doing what I wish. That is the meaning of the phrase ‘decline of the West.’”¹⁸⁶ Despite Scholz’s criticism of Spengler’s book, one must assume that Scholz had a Damascus experience shortly thereafter, when Whitehead and Russel’s Principia Mathematica fell into his hands.¹⁸⁷ He gave up his professorship in the philosophy of religion so as to establish the first chair of mathematical logic and foundational research in Münster.

Scholz joined a circle of mathematicians that had committed itself entirely to David Hilbert’s program of elucidating mathematical foundations. At the same time, Scholz was among the few who immediately recognized the significance of the fundamental work “On Computable Numbers” by the British mathematician Alan Turing. And he was the only one who asked Turing for an offprint.¹⁸⁸ If the Second World War had not intruded, Scholz would have ensured as early as in 1939 that Turing’s fundamental insights became common mathematical knowledge through an entry in the venerable Encyklopädie der mathematischen Wissenschaften.¹⁸⁹

Unlike Turing, who had already offered his mathematical skills to the British secret service for the investigation of cryptological methods before the outbreak of the Second World War, Scholz did not at first allow himself to be taken away from his studies of logistics, which he traced back from Gottlob Frege’s On Concept and Object to Leibniz.¹⁹⁰ But in 1944 he set off for Berlin at the invitation of the still completely unknown engineer Konrad Zuse to inspect his electromechanical Z4 computing machine. An air raid siren and the resulting forced stay together in a bomb shelter caused the exchange of ideas with Konrad Zuse to go on longer than planned.¹⁹¹ The connection between Zuse’s concrete electromechanical computing machine and Turing’s paper machine did not present itself automatically.¹⁹² In Scholz, Zuse nonetheless found an early and prominent advocate. The arms industry, however, had no use for his machine. Meanwhile, in Bletchley Park—Great Britain’s secret service establishment—Turing had managed to decrypt the German navy codes with computing machines christened “bombes,” which contributed decisively to the turn of the naval war to the Allies’ advantage.

Turing and his colleague Gordon Welchman had succeeded in reading self-reciprocal functions from the wiring of the German encryption and decryption machine for radio messages, the ENIGMA—a possibility that had not been foreseen on the German side. The encryption potential of the machine was thereby compromised. To avoid that would have been the task of Gisbert Hasenjäger, who had received his mathematical promotion from Scholz when he was just a high-school graduate, before he was drafted into the military and ultimately, with Scholz’s mediation, ended up in the department known as Referat IVa of the OKW code section, where it was his job to ensure and enhance the cryptological efficiency of Enigma.¹⁹³ But Hasenjäger and Scholz knew nothing of Turing’s cryptological operation on the British side.¹⁹⁴ Not until two decades after Scholz’s death did the British secret service captain Frederick William Winterbotham, after long hesitation, agree to report on the undertakings in Bletchley Park.¹⁹⁵

While their countries’ armies of millions were worn down in the Second World War, the rather small group of foundational mathematicians and logicians had been drawn into a game-like discursive formation that could not itself achieve a complete overview of itself. Thus the strategy, cultivated for quite a while in mathematical discourse, of reducing problems to game constellations, seems now to have encompassed those mathematicians themselves and assigned them each a limited function. The reductionism that characterizes the game concept thereby stands in a strangely reciprocal relation to an ever more unrestricted concept of calculability. An event in the spring of 1945 also makes this clear, when Zuse, in the composition of his plan calculus—which would later turn out to be the first design of a programming language—moves away from the strict concept of calculability and seeks, most likely still under the impact of the war, to grasp this concept in its whole scope in everyday usage. In a programming language, the concept of calculability must not remain limited to an arithmetical level. Rather, the word rechnen (reckoning, computing, calculating) must encompass the whole range of meaning that it has in German, and thus also the meaning in the following example: “I reckon [Ich rechne damit] that the enemy will withdraw when his supply line is cut and a breakthrough can be successfully averted.”¹⁹⁶ To exploit the possibilities of computing machines that go beyond the domain of the merely arithmetical, nothing seemed more appropriate than elevating the programming of games, and in particular that of chess, to a test case. Whenever military duty prevented Zuse from realizing electromechanical components as mathematical logics, he turned to the travel chess set as a substitute for developing logical calculus on a material basis.¹⁹⁷ When, in April 1945, not even a chessboard was available any longer, and Zuse kept his Z4 computer hidden from the advancing American troops in a hay shed in the Allgäu, he formulated the basic schema of today’s standard programming languages. His “plan calculus,” as he called the schema, was able to take up the game of chess along with its rules as “pure desk work.” Thus, it could deal with the framework of the game and its execution within the same “two-dimensional notation” on one and the same paper.¹⁹⁸ The fact that Alan Turing—and on the American side the communications scientist Claude Shannon—just as passionately programmed or built the first digital computers, which were capable of mastering chess and other games,¹⁹⁹ without having access to knowledge of Zuse’s hidden activity, awakens the suspicion that the affinity to the game arises from the medium of the computer itself. At the least, the game, as an application of the computer, illuminates the latter’s potential as a platform of the highest concretion and exemplary openness. Thus Shannon, on the occasion of receiving the highest distinction for engineers in the United States, claimed of his “game playing machines” that they could master all the other challenges that lay ahead in the time after the war.²⁰⁰ According to Shannon, programs that enabled a “general-purpose computer” to play chess also make it possible in the long run to translate languages, make strategic decisions in simple military operations, orchestrate a melody, or carry out logical deductions.²⁰¹

But it was someone else who realized the first step in taking the game seriously: John von Neumann. He grants the game a value that is markedly different from its common status during the foundational debate. If there the game helped formalism get beyond tautologies by way of signs, then knowledge of the game itself was never the central interest—not even when concrete games like chess were dealt with.²⁰² The great reductionist Neumann set to work in a quite different direction: he homed in on the game itself and could view himself with some justification as the founder of game theory. Just as Neumann would much later advise Norbert Wiener not to study the essence of life in the complexity of measured brain waves, but rather in its simplest conceivable form, the cell, the young Neumann turned away from chess and toward the most childishly easy games: the “even and odd” game or “rock, scissors, paper.” Neumann conceded that most games had more complicated rules and were, in addition, dominated by chance. Here one should consider card and dice games. Moves that are dependent on chance and the calculated steps of the players appear to demand a differentiation. Furthermore, one should assume that the interplay between the players requires an independent mathematical model: “[In particular] the consequences of the circumstance (so characteristic of all social events!) that every player has an influence on the results of all the others and at the same time is only interested in his own [must be taken into account].”²⁰³ Neumann introduces all these differentiations and variables only to discount them. The conscious steps or chance moves are determined by the rules of the game and delineated through random distributions. In Neumann’s theory of the game, there is no reason not to fix the strategy according to which one plays already before the game.²⁰⁴ What can be calculated can also be calculated into a “method of play”²⁰⁵ before the game. What cannot be calculated before the game also cannot be calculated in the game, and Neumann launches into an intricate proof that demonstrates the validity of his minimax theorem by means of a bilinear function. According to the theorem, methods of play exist that guarantee the highest possible wins, even if optimal methods of play are opposed to them. Only the assumption of an additional participant in a game for which Neumann succeeded in demonstrating the proof of the minimax theorem poses problems for which he can only begin to present a possible solution. His theory is unable to show whether and how minimax theorems can be calculated for the rules of random games—not even when he develops it into “game theory” in 1944 with the active participation of Oskar Morgenstern. In other words, selected games may have become calculable through Neumann; however, games as they are generally found have not.

Nonetheless, from an epistemic perspective, Neumann’s “Theory of Parlor Games” is highly significant, even if he himself first comes back to his early work in the priority debate with France’s great mathematician and sometime minister of the navy Émile Borel.²⁰⁶

Until Neumann developed his theory of the parlor game, he was a child prodigy who rather overtaxed his Berlin mathematics professor with a dissertation on the axiomatization of set theory.²⁰⁷ A closer exchange with Hilbert’s Göttingen school therefore seemed desirable. Scarcely three weeks after he had arrived in Göttingen as a Hungarian doctoral candidate on a Rockefeller fellowship,²⁰⁸ Neumann gave the first public talk of his life—before the renowned Göttingen Mathematical Society. The title of the lecture was “On the Theory of Parlor Games.”²⁰⁹ A reputation had preceded him for having advanced the axiomatization of Zermelo-Fraenkel set theory in a fashion that made the problematic axiom of choice dispensable. If Neumann had come to Göttingen as a young fellowship holder who introduced himself with his game theory, he ultimately left as a close colleague of Hilbert’s, who had made his so-called Hilbert spaces productive for the mathematical grasp of quantum mechanics. It is therefore plausible to suspect a not inconsequential strategy behind the fact that between these phases Neumann chose a “Theory of Parlor Games” for his first public appearance in Göttingen. Nonetheless, Philip Mirowski seems to be the only one thus far to attempt to unpack the epistemic content of the early game theory in the context of his other works without letting himself be led astray into mere speculations by the shimmering concepts of game theory.²¹⁰

For Neumann’s later companion and friend Stanislaw Ulam, the game is the medium that led to the theory of the computer, for Hilbert’s program of consistent axiomatization followed “the goal of treating mathematics as a finite game. Here one can divine the germ of Neumann’s future interest in computing machines and the ‘mechanization’ of proofs.”²¹¹ To be precise, it should be noted that Neumann did not immediately transition from formalist mathematics as a game with intrinsically meaningless signs to the mathematical theory of computing machines. Rather, he extracted from very concrete games mathematical problem situations that were subsumed neither in a pure formalism nor in a pure application. The fact that Neumann, after his lecture in Göttingen, devotes himself to current questions of physics and above all quantum mechanics is thus also rooted in his game theory. It made it possible first to pose questions of indeterminacies and the interplay of complex systems, and then to bring them to bear on quantum physics. With game theory as a consistent implementation of the axiomatization demanded by Hilbert—and indeed, for the first time, beyond the mathematical branches—a world picture also first appears that already reveals on a global scale what will first be proper to quantum mechanics on the smallest level according to the Copenhagen model: “That which is dependent on chance (‘random,’ ‘statistical’) is rooted so deeply in the nature of the game (if not in the nature of the world) that it is not even necessary to introduce it artificially through the game rules: even if in the formal game rules there is no trace of it, it establishes itself on its own.”²¹² Neumann thus sought to grasp the game theoretically at the moment when a theory of what mathematics is appeared most urgent and provided him the means for the formulation of his game theory. Neither an irrevocable determination of mathematics nor of games emerged from that. For that reason alone, Neumann’s “Theory of Parlor Games” does not offer an early mathematical model of social systems. His game theory did not bring to light a kernel of social and/or economic conduct, but a method that limited itself to sign operations and that, during the Second World War, brought into knowing and unknowing contact a small but influential circle of actors in nothing but zero-sum games. Shannon, in any case, designated as “zero-sum games” that level of the Second World War on which the warring parties attempted to crack the communications of the enemy and encrypt their own.²¹³

If mathematics itself nonetheless ultimately failed to produce an all-encompassing theory of the calculability of games reduced to sign operations, then it cannot be the goal of this study to present another attempt to define the game. It should, however, have become clear how games in particular, under high-tech conditions, served an ever-expanding tendency toward making-calculable as a conduit medium—and still do so today. Furthermore, it should have become clear what incalculabilities, from a historical perspective, were sometimes hazarded for that purpose.

Games, like mathematics itself, seem to be defined by a tautology that either includes or excludes everything. This unity in multiplicity is also evident in the fact that in German one speaks of die Mathematik in the singular, as if there were only the one, while in the English- or French-speaking world the pluralis tantum “mathematics” and “les mathématiques” invoke several modes of being.²¹⁴

Still more significant is the ontic dimension that the game shares with language: Heidegger knew all too well that expressions such as Der Raum räumt (“space spaces”) or Die Zeit zeitigt (“time times”) subvert customary linguistic usage.²¹⁵ In German, there are at least two exceptions: Spiele spielen and Sprachen sprechen mean to play games and to speak languages. In both cases, there is no way to phrase it without the tautology of noun and verb. This idiosyncrasy of German may serve as an illustration of a more general principle: games must be played and languages must be spoken in order to be what they are. A history of games—and a history of war games in particular—here reaches its limits.