Answers to this rephrased question, some of which Slade lists, with all the attendant symbols and terminology, fill the next six notes; presumably these and like notes form the bulk of the calculus. How he arrived at some of these solutions would, presumably, have been discussed in the two, undelivered lectures. Fortunately, Slade’s students from BPR-57-c have been able to fill in much here, as this is exactly what Slade had been wrestling with during the original work sessions for three years. Some of their papers will appear in future issues.
Since Liebniz, or even Aristotle, the boundaries between mathematics and logic, and between logic and philosophy, have always been strangely fuzzy. Try to define them too carefully, and they disappear. Change your position only a fraction of a degree, and they seem clearly present once more. From this new angle we begin to define them again—and the process repeats. Is it, then, just the maverick statements that our third critic would claim Slade has simply scattered through his discussion of the logic of models that tempt us to take what seems essentially a discussion of the foundations of a limited, mathematical discipline and call it a philosophy? Your editor does not think so; we feel that for all its eccentricity of presentation, Slade’s work is philosophically significant—though already (a situation which has existed about Slade’s wofk since the publication of the Summa) articles have appeared which claim otherwise. The emblem of a philosophy is not that it contains a set of specific thoughts, but that it generates a way of thinking. Because a way of thinking is just that, it cannot be completely defined. And because Slade’s lecture is incomplete, we cannot know if he would have attempted even a partial description. Your editor feels that the parameters for a way of thinking have, in the extant notes of Shadows, been at least partially generated. Rather than try to describe it, we think it is best to close this limited exegesis with an example of it from Slade’s lecture. The note we end on—note seven—along with note twenty-two, completes the clearest nonmathematical explanation of the calculus Slade was trying to describe. (In note six, Slade talks about the efficiency of multiple modeling systems, or parallel models, over linear, or series models: his use of pictures, in note seven, to distinguish between words about reality and the real itself is a self-evident example of what he discusses in six. Slade drew the pictures hastily on his blackboard with blue chalk and pointed to them when they came up again in the flow of his talk.) Here is note seven:
There are situations in the world. And there are words—which are, to put it circularly, what we use to talk about them with. What makes it circular is that the existence of words, and their relationship to meanings, and the interrelationships among them all, are also situations. When we talk about how words do what they do, we are apt to get into trouble because we are maneuvering through a complex house of mirrors, and there is almost no way to avoid that trouble, short of resorting to pictures—which I am not above doing.
Many situations in the world have aspects that can be talked of as directed binary relationships. Some examples of talk about these situations which highlight the directed binary relationship are:
“Vivian loves the Taj Mahal.”
“Alicia built a house.”
“Chang threw the ball.”
“Sad means unhappy.”
“The hammer hit a nail.”
Let us take the last sentence, “The hammer hit a nail,” and consider it and the situation it might commonly be used in, and explore the modeling process that is occurring in some detail. First, we have a thing, the phrase the hammer, standing for a thing, c=^ . In that phrase, we have a thing, the word the, standing for an attitude toward •=4 , and we have another thing, the word hammer, standing for the object n^ itself. Next, we have a thing, the verb hit, standing for a relationship. After that, we have still another thing, the phrase a nail, standing for another thing, T . As in the first phrase, in the second we have a thing, the word a, standing for an attitude toward the object T different from the attitude modeled by the word the. And, as in the phrase the hammer, we have a thing, the word nail, which stands for the object T itself. Also, we have a relationship, composed of which thing (i.e., word) is put before the verb and which thing (i.e., word) is put after it, that stands for an aspect of the relationship c==A not completely subsumed by the verb hit alone, i.e., which object is the comparatively active one and which is the comparatively receptive one—or what can be talked of as “the direction of the binary relationship.” Now the direction of the relationship is, itself, a relationship; so here we have a relationship, between noun, verb, and noun, standing for an aspect of the relationship c^A .
Now there are other notable relationships in the sentence “The hammer hit a nail,” to attract our attention. In the phrase the hammer, for instance, which we have said consists of two things, the word the and the word hammer, it is necessary that the things appear in just that order. Likewise, the phrase a nail must preserve its order, if the sentence is to strike us as proper. What are these particular relationships necessary for? What would be wrong with the sentence “Hammer the hit nail a,” or “Hammer hit nail a the,” or “Hammer a the hit nail,” or “The a hammer hit nail”? In all of these, we still have the things in the sentence which stand for the things in the situation, and in all of them the relation between hammer, hit, and nail, which models the direction of the relation in the situation, is preserved. Is the relation between the and hammer, or a and nail, modeling anything in the situation which is suddenly lost or obscured if these relations are lost?
To the extent that our attitudes toward the objects in a relationship are not in that relationship, the simple answer is no. The relationship between the and hammer and between a and nail are necessary to preserve the integrity of the model itself; they are necessary if we are to recognize the model as a proper thing for modeling in the first place. But these relationships, between the and hammer and a and nail, do not model anything in the situation talked about by the sentence. To destroy them, however, may prevent other relationships (that may be modeling something in the situation, or may be preserving the integrity of the model) from standing forth clearly. This simple answer is, however, rather oversimplified.
What shows the situation to be more complicated than our discussion so far is that the same thing may be said about the relationship between, say, the three
But let us sum up what modeling is being done by the sentence The hammer hit a nail. We are modeling attitudes, objects, and various aspects of a relation between them; to do this job, we
are using, among a large group of things and relations, various of those things and relations to stand for the objects, attitudes, and relations we wish to model.
A last point more or less separates the place where the modular calculus separates off from the modular algebra: Suppose, by considering the sentence as a set of letters, we finally found a list of relationships that would completely describe it, such as:
1) Three a’$ must be separated by, respectively, seven letters and two spaces, and one letter and one space.
2) Two ra’s must be separated by no letters and no spaces.
3 ) One m must follow one a.
Et cetera ...
Even though, at the end, we have a list of relationships that completely describes the sentence (so that, say, a computer could translate our list into the matrix form of a flash-display, i.e., a list of numbers), still no relation, or even consecutive group of relations in our list, can be said to stand for any thing, attitude, or relation in the situation which the sentence models. Yet the sentence is completely described by this list.
Notice also: The list of numbers for the matrix display also completely describes the sentence. Yet here, some consecutive groups of numbers can be said to stand for things, attitudes, and relations in the situation—since certain groups of numbers each of which is given a number, then your minimum list of things and relations (minimum because some letters can be made in two forms:
stand for certain words and certain word groups. Notice as well that, while in this list there will be a consecutive group of numbers that stands for the relationship of the and hammer, and a and nail, there is no consecutive group that stands only for the relation of hammer, hit, and nail: because the numbers standing for the second a in the sentence will be in the way.
We can call the computer matrix display a modular description because it preserves some of the modular properties of the sentence in a list that describes the sentence.
We can call the list of letters in relation to each other a nonmodular description because it preserves none of the modular relations of the sentence in a list that describes the sentence.
As we have seen, with our computer example, complete descriptions of models can be translated from nonmodular descriptions into modular ones and back again and remain both complete and intact. The first useful thing the modular calculus yields us is the following information:
Consider language a list of relationships between sounds that model the various ways sounds may relate to one another—or, if you will, a list of sentences about how to put together sentences, i.e., a grammar. The modular calculus lets us know, in no uncertain terms, that even if such a list were complete, it would still be a nonmodular description. It has the same modular order (the proof is not difficult) as our description of the sentence The hammer hit a nail as a set of letters precisely spaced and divided.
The calculus also gives us tools to begin to translate such a list into a modular description.
Now the advantages of a modular description of either a modeling object, like a sentence, or a modeling process, like a language, are obvious vis-a-vis a nonmodular description. A modular description allows us reference routes back to the elements in the situation which is being modeled. A nonmodular description is nonmodular precisely because, complete or incomplete as it may be, it destroys those reference routes: it is, in effect, a cipher.
The problem that still remains to the calculus, despite my work, and that will be discussed in the later lectures, is the generation of formal algorithms for distinguishing incoherent modular descriptive systems from coherent modular descriptive systems. Indeed, the calculus has already given us partial descriptions of many such algorithms, as well as generating ones for determining completeness, partiality, coherence, and incoherence—processes which till now had to be considered, as in literature which so much of this at a distance resembles, matters of taste. But their discussion must be left for the last lecture.
About The Author
“Samuel R. Delany is the most interesting author of science fiction writing in English today,” said The New York Times Book Review. He was born in New York City in 1942. His acclaimed science fiction novels include Babel-17 and The Einstein Intersection, both winners of the Nebula Award for best science fiction novel. He has also written Nova, Triton, the best selling Dhalgren and the Neveryon fantasy series, Tales of Neveryon, Neverydna and Flight from Neveryon. Stars in My Pocket like Grains of Sand is the first volume of a science fiction diptych which will conclude with The Splendor and Misery of Bodies, of Cities, to be published by Bantam Spectra Books in 1986. Also a critic of science fiction, he has published two essay collections on the field, The Jewel-Hinged Jaw and Starboard Wine, as well as The American Shore, a book-length semiotic study of the science fiction short story “Angoul-eme” by Thomas M. Disch.
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