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.[1]
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.
For the time being, let us confine our discussion to alphabetical space
without numbers and focus on scholastical commentaries from the 13th century,
best typified perhaps by Petrus Lombardus' Sentence Commentaries undertook
the the task of explaining which speech act Jesus had actually executed
with the sacramental words of the last supper. While the vulgate could
simply and without commentary write "Hoc est corpus meum," theological
commentaries had to trace the reference structure of this speech and be
able to indicate whether the deictic expression "hoc" referred
to the bread as it had come from the baker of the Host, or rather to the
same bread that had been transformed into the body of Christ by the speech
act itself. This delicate question led to the hopelessly corrupt Latin
of Wilhelm von Auxerre's Summa aurea: "Sed queritur, cum dicitur hoc
est corpus meum, quid demonstret ibi hoc pronomen hoc."[2] "It
is now asked, however, to what, if it means that this is my body, the pronoun
this refers." Lacking any operators that could have set a distinction
between use of and reference to words, the impossibility of not using,
but simply referring to a distinct part of the quoted sentence could hardly
be more dramatic. Without plumbing any of the philosophical niceties at
stake in these matters, this move reduced the then-current dispute over
nominalism to the mere necessity of presenting the missing operators to
an essentially commentating culture. With the nominalist differentiation
between suppositio formalis and suppositio materialis, that is between
reference to the subject matter of words and reference to words as terms,
it became possible for the first time to comprehend the functional difference
between the two sentences "Angels have a nature," and "Angels
have five letters."
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."[3] 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."[4]
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.[5]
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.[6] 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!"[7] 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."[8] 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,[9] 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.[10] 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.[12] 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."[13]
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.
And that was perhaps not without reason. At the historical end of the monopoly
of the book, that is, in the age of telegraph[14] and telephone, the project
of dating the wonderful power of mathematical symbols and dashes back to
medieval chirography was a perfect misidentification of precisely the typography
that had set this power for mathematicians as well as writers.
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.[15]
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."[16]
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.[17]
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.[18]
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.[19] 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.[20]
[1]Cf. Eric Blondel, "Les guillemets de Nietzsche," in Nietzsche
aujourd'hui, vol. 2 (Paris 1973), pp. 153-82.
[2]Quoted in A.M. Landgraf, Dogmengeschichte der Frühscholastik, Part
1: Die Gnadenlehre (Regensburg, 1952), p. 22 (with thanks to Reinhold Glei).
[3]Quoted in Langraf, Dogmengeschichte, p. 23.
4Ibid., pp. 21-24.
[5]Cf. Erwin Arnold, "Zur Geschichte der Suppositionslehre,"
in Symposion, vol 3 (Freiburg/Brg., 1962).
[6]For first indications, cf. Johannes Lohmann, Philosophie und Sprachwissenschaft
(Berlin, 1965), pp. 44-46.
[7]Hans Magnus Erzenberger, Mausoleum. Siebenunddreißig Balladen
aus der Geschichte des Fortschritts (Frankfurt, 1975), p. 9.
[8]Quoted in Florian Cajori, A history of Mathematical Notations , vol.
1 (La Salle, Ill). p. 234.
[9]Cf. Joris Vorstius and Siegfied Joost, Grundzüge der Bibliotheksgeschichte,
7th ed. (Wiesbaden, 1977), p. 47.
[10]Cf. Cajori, Notations, vol. 2, p. 182f.
[11]Ibid., p. 184.
[12]Ibid., p. 182f.
[13]Quoted in Betsy Meyer, Conrad Ferdinand Meyer. In der Erinnerung seiner
Schwester (Berlin, 1903), p. 208f.
[14]Cf. Conrad Ferdinand Meyer, "Hohe Station," in Sämtliche
Werke, Hans Schmeer, ed. (Munich, 1903), p. 823.
[15]Quoted in Cajor, Notations, vol. 2, p. 184.
[16]Gauß, letter to Schumacher, September 1, 1850. Quoted in Hans
Wussing, Carl Friedrich Gauß, 2nd ed. (Leipzig, 1976), p. 65.
[17]Cf. Cajori, Notations, vol. 2, p. 130f.
[18]Cf. Andrew Hodges, Alan Turing: The Enigma (New York, 1983), pp. 96-110.
[19]Cf.Bernhard Dotzler, "Nachwort," in Alan Turing, Intelligence
Service. Ausgewählte Schriften, Bernhard Dotzler and Friedrich Kittler,
ed. (Berlin, 1987), p. 227.
[20]Cf. Turing, Intelligence Service, p. 15.