On The Take-Off of Operators
Let us say the function of literature and literary studies is to make transmittable the cohesiveness of the net in which everyday languages capture their subjects. And those to whom this designation sounds strange should first of all be reminded that, without communications-technical designations, there could hardly be any talk about literature and literary studies. Secondly, and somewhat more philosophically, Goethe's earth-spirit spoke of "weaving the living garment of divinity." But, as Faust's collapse at the theatrical entry of this spirit illustrated, such nets or reference totalities of an everyday language cannot themselves be captured. In order that everything is woven into a whole, a lid sits on the loom. With it, hopeful theoreticians of the Post-World War II era created- with an eye toward the threat to the intellect and its sciences posed by the formal languages of this century- the beautiful theorem that everyday languages are their own meta-languages and consequently capable of being viewed from behind and below.
But there are certainly opportunities for bringing the margins or edges of such nets to actuality without at the same time switching over to the side of formalization and thus sacrificing the communicability of everyday language. This approach toward the marginal values of language becomes more necessary the more consistently modern communications technologies, in forming compound media systems, knot together their closed and locked nets. The assurance that everyday language as its own meta-language is unterhintergehbar could soon offer little consolation in a situation where conversation, which is what we are with each other even according to Gadamer, says nothing about the factual way of the world or signal flow.
Of course, this situation appears new and disconsolate only under the humanistic premise that language is entirely subsumed by converstation, and conversation by the people who carry it out. Should one invert the premise for testing purposes, then the disappearance of the subject does not reside with language, as one might suppose, but instead has always proceeded via writing and media. An escape so boundless that the modern basis theorem of humanity as the master of language becomes questionable.
Everything that lies beyond the margin of everyday language can be related to the historical vectors of this escape. What in each case takes place are processes of lift-off, or take-off in the Americanized German of Peenemünde, since it has been experimentally proven that no return or landing must follow them. (Perhaps Patty Smith, when she sang "Landing," was for this reason so fervent.)
Concerning those take-offs that remain in, but also with, writing, Derrida is authoritative According to Derrida's analyses, all margins of a text- from the title to the motto to the footnote- are operations that simply cannot be made to speak. Quotation marks have proven to be the general pre-requisite of such instances of distancing, which must always be read in even when nothing is written. They are operators of writing to such an extent that it is customary, for intstance, in reading a paper at a solemn professional meeting to frame all quotes with the words "quote" and "unquote," a practice that routinely provokes laughter at the corner pub.
Derrida's deconstructions themselves, of course, operate in a domain where all operators of writing are readily available as typographical options. Thus his analyses are certainly able to withstand Nietzsche's philosophical critique, simply because Nietzsche himself, as the only philologist among the philosophers, raised quotation marks to the rank of a category. But with regard to old European texts, which did not even have the spacing of structuralists at their disposal, let alone meta-lingual operators, their inclusion, as anachronistic as it is systematic, threatens to transform analysis into over-interpretation.
Rather than making unconditional use of all the operators in current writing
practice, a more systematic approach would begin with their archaeology
and determine when and for what reason a particular operator was introduced,
and thus also when and for what reasons it did not exist.
But because it is equally true of thoughts and categories that nothing exists that is incapable of being switched, the nominalist knife needed a notational operator between the two ways of referring to words that had not existed in classical Latin and could not have existed. When Richard Fishacre returned to the problem of bread and wine, this and that, in his Sentence Commentaries, for example, the "hoc" that Wilhelm von Auxerre had been able to indicate only as "hoc pronomen hoc" was suddenly preceded by a sequence of letters, as inconspicuous as they were nonsensical, but which nonetheless eliminated Wilhelm's entire problem of formulation: "Sicut hic diceretur, quod li hoc non est demonstrativum, sed stat materialiter." In English: "Thus would be said here that the this is not demonstrative, but instead a suppositio materialis [that is, a reference to its own substance as a word] takes place."
Landgraf's comments on these examples in his History of Dogma exhibit charming innocence: "Those who occupy themselves with scholastics encounter at a certain moment the small word ly, which assumes the place of the article unknown to classical Latin. ... In the entire 12th century, this 'ly' is not yet encountered. The article 'li' thus entered the theological lecture hall from the streets of Paris and was able to sustain itself so well that it won the day over the Lombard 'lo'. Of course this is not surprising when one considers that Paris was of paramount importance during the period in which this transformation took place and had a major influence on the technology of school operations."
In truth, nothing is more surprising than the admission of an operator from vulgar speech into the technical language of middle-Latin theology, and nothing more banal than to explain it simply as an old French article, which surviving texts of this vernacular scarcely exhibit in place of pronouns or prepositions. As a definite article, ly would only have restored a possibility, which was known to Greek philosophers but lost in Latin, of creating any number of categories by making substantives out of verbs, prepositions, and other classes of words; in Aristotle's case, for example, the where, the that-for-the-sake-of-which, etc. Richard Fishacre or even Thomas Aquinus, whose summas probably did the most for propagating the use of ly , did not refer to a category of this sort but instead to the functioning of the word itself. Predicaments were not at issue, but rather predicables. While concepts in the Greek, and thus also categories in Aristotle, automatically conformed to the field of reference being discussed at the moment, that is, could deal with the world and the logos of that world equally well, reference to subject matter and reference to language were separated in the discourses of scholastics, if only for the reason that, according to a thesis of Johannes Lohmann, the texts were conceived in the vernacular but written in Latin. That the operator ly originated in a vulgar language and had to be grafted onto middle Latin is itself a symptom of the take-off that made languages at least conceptually manipulable if not yet technically. In other words, ly appeared in exactly the same place where our quotation marks would be inserted after the invention of printing, and thus the invention of titles, tables of contents, and word-addresses in general as well, but where, under the conditions of medieval hand-writing, there was a gaping typographical lacuna.
It is probably only the ineradicable familiarity with which readers look at books that prevents them from recognizing the invention of the ability to quote individual parts of a sentence in the 13th century as an historical rupture. "How this page," it read in Enzenberger's Gutenberg-poem, "resembles a thousand other pages, and how difficult it is to be amazed by that!" It is therefore only by looking at operators that are not readily available in every type composition box, since they do not belong to the basic alpha-numerical accouterments of school children, or these days even to word-processing programs, that one can make a plausible case that operators, far more than any battles or plagues, have made history. Those who tamper with the relationship of people to signifiers, according to Lacan, change the mooring of their being even and indeed precisely when the newly introduced operators are legible only to an elite or, in an extreme case, to machines. Compared to the take-off of numerical or even algebraic symbols, that of the alphabet is in any case merely a prelude.
In Greek there was clearly no possibility of writing the sentence "two and two is four" differently than it would be spoken. The operator 'plus' coincided with the 'and' of everyday language, which was fine and good only so long as no one wished to substitute intricate correlation commands for addition commands that can usually be done "in the head" (whatever that might be). Even when a symbol for subtraction of two numbers appeared in Diophantus' surviving works, it did not imply a corresponding sign for addition. Only in 1489, in Johann Widmann's Nimble and Handsome Calculations for Businessmen, were the two operators of cross and horizontal line explicitly used for functions of inversion. In doing this, however, Widmann apparently still found it necessary to provide his readers with a translation, just barely pronounceable, into their everyday business language: "the - that is minus and the + that is more." Once this translation could be completely forgotten, however, number columns could be manipulated independently of speech. It became historically unimportant whether Widmann's plus sign originated from the Latin et and his yet unexplained minus sign perhaps from Diophantus after all, simply because the two operators could henceforth prove their silent efficiency. Novalis' wish that "numbers and figures rule world history no more" was already obsolete at the moment of its formulation.
The actual take-off of operators, however, only takes place when operators issue from other operators, as if an avalanche had set off. Just like Widmann's innovations, the importation of the Arabic zero in the 13th century (which witnessed more indeed than merely the introduction of the ability to quote) probably did not take place according to plan. Without philosophers noticing any signs of trouble at all, small, innocent symbols revolutionized the business of bankers and trading agents. Nevertheless, or precisely for that reason, the algebraic operators of the early modern period constructed a consistent system, at the latest after Viete's cryograhic trick of inserting letters of the well-known alphabet for unknown numbers, which only needed feedback in order finally to allow operations via operators as well.
It was Leibniz who took this most important of all steps. Just as he had drawn the logical conclusion from Gutenberg's invention with his proposal for library catalogs, he also recognized nearly all of the implications of the historical accident of symbols such as zero. His correspondence with every important mathematician of the time from the two Bernoullis to Huygens and L'Hospital to Tschirnhausen asked all of his colleagues not only to introduce new operators for new operations, but also, "in the interest of the republic of scholars," as Leibniz wrote, to reconcile these innovations among themselves. When Tschirnhausen replied that the new terminology and new symbols made scholarship less comprehensible, Leibniz wrote back that one could have made the same objection even to the replacement of Roman letters with Arabic numerals or the introduction of zero.11 In other words: the mathematics of a Leibniz learned from the contingency of its own operators how to discover their power. Never before had anyone begun the experiment of manipulating not words or people, but instead bare and mute symbols. From Leibniz come not only the extremely technical symbols for integration or congruency, for example, but also symbols of such familiarity that it has become hard to tell just by looking at them that they are inventions at all.
There was, for example, no specific sign for division before Leibniz, only the familiar horizontal line that separates numerator and denominator. That might have been good enough under the conditions of medieval handwriting, but Leibniz explicitly criticized the fact that two or three-line expressions were an additional burden for the typesetter and no doubt for the reader's eyes as well. Thus Leibniz, and he was the first, replaced the fraction stroke with our colon, a success that was at least European-wide. With the feed-back between symbol and symbol, alphabet and algebra, mathematics reached the technical state of Gutenberg's print.
In the summer of 1891, Conrad Ferdinand Meyer made plans for a novella
about an early medieval monk who begins his career as a copyist of pious
parchments and ends it as a forger of juristically and economically relevant
parchments. To wit, pseudo-Isodor discovers while copying (to quote him)
"what wonderful power lies in these dashes and numbers! With a small
dot, with a thin line I change this number, and in so doing I change the
relations of property and force in the most distant regions."
Because of this fraudulence, which he gradually ceases to recognize, the
hero of Meyer's novella was, in the end, to go insane. Only instead of
the monk, who remained a fragment, the author himself unfortunately ended
up in the insane asylum because of his novella.
This misidentification resided in Europe's holiest concepts, however. Leibniz measured the operators, of which he invented more than anyone else, against a truth whose opposite could then oscillate between falsehood and fraud. Symbols, he wrote to Tschirnhausen, should represent the essence of a thing, indeed, should paint it, as it were, with just as much precision as conciseness. But even Gauß was still alarmed by his own insight that "the character of the mathematics of recent times [was] (in contrast to antiquity)," "that in our language of symbols we possess a device through which the most complex argumentation is reduced to a certain mechanism." In the good society of Goethe's time, Gauß warned against using "that device only mechanistically" and called instead for a "consciousness" "of the original determination" "in all applications of concepts."
In point of fact, all of this talk about essence or consciousness was simply an imprint of philosophy onto the operators, an imprint that only the mathematical contemporaries of Meyer were able to do away with all together. In 1849, Augustus deMorgan wrote of Euler's symbol ¡, which commonly designates the (imaginary) value of the square root of -1, that the repeated complaints about its "impossibility" fell by the wayside "as soon as one simply gets used to accepting symbols and laws of combination without giving them any meaning." Simply because mechanical calculations, even with complex equations, led to verifiable results, mathematics was able and allowed to use its operators for every possible kind of experiment. With this explicit departure from meanings, and therefore with the last remaining link to everyday language, a symbolic logic took off, which was able to leave behind experimentation in the technical sense even of deMorgan himself, which means it was able to move into silicon circuitry. The 1936 dissertation, in which Alan Turing presented the principle circuit of all possible computers, consequently no longer made the slightest distinction between paper machines and calculating machines, where the word "paper machine" was Turing's euphemism for mathematicians and himself.
In order to begin this final take-off, Turing and John von Neumann only had to remove a tiny, but sacred distinction that had still enjoyed inviolable authority in the time of deMorgan or Babbage: the distinction between data and addresses, operands and operators. When Babbage designed the first universal calculating machine in 1830, he shuddered at the thought of entering operations or commands into his machine in the same punch-card format that he had provided for arbitrary number values. Von Neumann's machines, on the other hand, write commands and data in the same format into the same, undifferentiated data bank; that is their stupidity and their power.
The take-off of operators does not complete a world-historical course of instruction that bring to maturity abstractions of ever higher levels. On the contrary, the differentiation between use and reference, signification and quotation, as it was introduced by the 'ly' of the 13th century, can and must once more implode in order to make operators so universal that they also operate on operators. To add a number with the binary value of the plus sign itself is no problem at all for Von Neumann's machines, but instead an- at least by the standards of everyday language- always lurking address error of the programming. No one can say in everyday language, however, whether such mistakes beyond human beings do not after all start up programs in the world that keep on running effectively and without crashing the system. For which reason Alan Turing, no sooner than he had gotten the first computer running, delivered the oracle that we should already now prepare ourselves for the take-over of machines.
Cf. Eric Blondel, "Les guillemets de Nietzsche," in Nietzsche aujourd'hui, vol. 2 (Paris 1973), pp. 153-82.
Quoted in A.M. Landgraf, Dogmengeschichte der Frühscholastik, Part 1: Die Gnadenlehre (Regensburg, 1952), p. 22 (with thanks to Reinhold Glei).
Quoted in Langraf, Dogmengeschichte, p. 23.
Cf. Erwin Arnold, "Zur Geschichte der Suppositionslehre," in Symposion, vol 3 (Freiburg/Brg., 1962).
For first indications, cf. Johannes Lohmann, Philosophie und Sprachwissenschaft (Berlin, 1965), pp. 44-46.
Hans Magnus Erzenberger, Mausoleum. Siebenunddreißig Balladen aus der Geschichte des Fortschritts (Frankfurt, 1975), p. 9.
Quoted in Florian Cajori, A history of Mathematical Notations , vol. 1 (La Salle, Ill). p. 234.
Cf. Joris Vorstius and Siegfied Joost, Grundzüge der Bibliotheksgeschichte, 7th ed. (Wiesbaden, 1977), p. 47.
Cf. Cajori, Notations, vol. 2, p. 182f.
Ibid., p. 184.
Ibid., p. 182f.
Quoted in Betsy Meyer, Conrad Ferdinand Meyer. In der Erinnerung seiner Schwester (Berlin, 1903), p. 208f.
Cf. Conrad Ferdinand Meyer, "Hohe Station," in Sämtliche Werke, Hans Schmeer, ed. (Munich, 1903), p. 823.
Quoted in Cajor, Notations, vol. 2, p. 184.
Gauß, letter to Schumacher, September 1, 1850. Quoted in Hans Wussing, Carl Friedrich Gauß, 2nd ed. (Leipzig, 1976), p. 65.
Cf. Cajori, Notations, vol. 2, p. 130f.
Cf. Andrew Hodges, Alan Turing: The Enigma (New York, 1983), pp. 96-110.
Cf.Bernhard Dotzler, "Nachwort," in Alan Turing, Intelligence Service. Ausgewählte Schriften, Bernhard Dotzler and Friedrich Kittler, ed. (Berlin, 1987), p. 227.
Cf. Turing, Intelligence Service, p. 15.