Excerpts from LINCOS: Design of a Language for Cosmic Intercourse Hans Freudenthal North-Holland Publishing Company, Amsterdam, 1960 I: Mathematics | II: Time | III: Behaviour | IV: Space, Motion, Mass Chapter I: Mathematics 1 00 0. Pairs of signs # will enclose the printed image of a program text. A (metatextual) "and so on" after a text indicates that this text is an exemplary extract from the factual program. When carrying out the program we will replace this text by a large number of texts similar to the text we have printed. If the number of examples is large enough, we may expect that the receiver can generalize the program text. 1 01 0. # > | < | = | + | - | <> | <= | => | . | . . | . . . | . . . . . . . | 1 | 10 | 11 | 111 | a | b | c | -> | ? | /\ | \/ | <-> # Loose Lincos words are presented, without any context, in order to stress their individuality. So it will be somewhat easier for the receiver to recognize them when they occur in a certain context. The bold-faced strokes mean pauses. 1 01 1. # . . . . . > . . . # and so on. 1 01 2. # . . . < . . . . . # and so on. 1 01 3. # . . . . = . . . . # and so on. 1 01 4. # . . . . + . . = . . . . . . # and so on. 1 01 5. # . . . . + . . = . . . . . . # and so on. In these texts the Lincos phoneme that corresponds to the round dot is a short radio-signal (a peep). A Lincos word that consists of n successive phonemes of this kind, separated by short and equal intervals, is written as a group of n round dots. It both means and shows the natural number n. It is an ideophonetic word, which has the power of an image as well as that of a word. We also call it an ostensive numeral. The greater part of the Lincos vocabulary will be purely conventional; words may be permutated at pleasure. This is not true of ideophonetic words. Their essential features must not be changed. The Lincos word written >, <, =, +, -, and so on designate connectives with the usual meaning. The receiver should guess their meaning from the context. Therefore each of the first program texts contains one unknown word only. 1 02 1. # . = 1 . . = 10 . . . = 11 . . . . = 100 . . . . . = 101 . . . . . . = 110 . . . . . . . . . . . . . = 1101 # and so on. Ostensive numerals are superseded by algorithmic ones, composed of syllables written 0 and 1 according to the rules of the dyadic positional system. For the convenience of the terrestrial reader we shall sometimes use the decimal code, but as a matter of fact such occurrences should be translated into the dyadic code. 1 02 2. Text as those of 1 01 1 to 1 01 5 will be repeated, using algorithmic numerals instead of ostensive ones. 1 03 1. # 111 = 110 + 1 = 101 + 10 = 100 + 11 = 11 + 100 = 10 + 101 = 1 + 110 # and so on. 1 03 2. # 111 + 11 > 11 + 101 > 1 + 100 = 101 # and so on. ... 1 36 8. # ~ p /\ q <-> ~ p \/ ~ q ~ p \/ q <-> ~ p /\ ~ q p -> q <-> ~ p \/ q p <-> ~ ~ p # ... Chapter II: Time 2 00 0. Again we start with ostensive, ideophonetic signs, the so-called time-signals. They are even more ideophonetic than the peeps (written as dots) we used in Chapter I to introduce the natural numbers. While the peeps showed and meant arithmetical units, the new signs will not mean anything but themselves. So they can hardly be called words. 2 01 0. The new signs are radio-signals -- time-signals -- of various duration and wave-length. They are written as horizontal lines. 2 01 1. # Dur ____ = Sec a # and so on. The 'a' as it stands does not belong to our program text in the proper sense. It is a meta-text variable, used as a substitute for a Lincos constant. Eventually, this 'a' should be replaced by a Lincos word meaning a positive real number a such that the sentence "The duration of the factual time-signal indicated by the horizontal line is of a seconds" is true. The Lincos word written Dur (fL duratio = duration) means "duration". Syntactically it is to be handled as a function to the set of durations. The domain of this function is not exactly the set of time-signals. It is much broader, but at this stage it would be unwise and even impossible to circumscribe it in a too definite way. This is symptomatic of many functions we shall deal with. The Lincos word written Sec means the time unit second. Syntactically it behaves as a function from 'Pos' to the set of durations. So it is a paradigm of Lincos syntaxis for physical units. 'Cmt' and 'Gra' (centimeter and gramme) will occur as symbols for functions from 'Pos' to the set of lengths and the set of masses respectively. ... Chapter III: Behaviour 3 00 1. For the time being it would be premature to try to describe human behaviour by a system of general rules like the mathematical and chronometric rules of the preceding chapters and some of the mechanical laws of the next chapter. Instead we shall show behavior by quasi-regular examples, from which the receiver may derive as many general behaviour rules as he pleases. ... 3 00 2. As the program events are to display behaviour, it is necessary for at least part of them to be acts, i.e. caused by persons. Our Lincos vocabulary is still far from sufficient for introducing the bodies of the acting persons. So the only kind of act that can be displayed immediately is the act of speaking. The Lincos word that designates this activity, is written Inq (fL inquit = says). The terrestrial reader should guard against a too narrow interpretation of this 'Inq'. In the present chapter the physical background of the Inq-events, whether it be accoustical or optical or tactile or anything else, will remain undiscernable. ... 3 00 3. The names of the dramatis personae will be written Ha, Hb, Hc, and so on. In due course we will state that these persons are members of the set called Hom (fL homo = man) in written Lincos. 3 00 4. Our theatre is still incomplete. Besides persons and acts a third thing is needed. We have been able to build a vocabulary of mathematics without valuating our propositions. We had only to confine ourselves to true propositions. The falsehood of 1=2 could be formulated as 1<>2. Yet we cannot show behaviour by good actions only. We must stage bad ones too, if we wish to condemn them. We have to create a vocabulary that contains words meaning "good" and "bad" and intermediate valuations. To begin with, we shall stick to two values. Of course they cannot be 'Ver' and 'Fal', which are values of propositions. What we wish to valuate are acts, not propositions. (We are not here considering value-judgements of esthetics.) Our valuating words will be written Ben (fL bene = well) and Mal (fL male = badly). They mean "good" and "bad" respectively. ... Chapter IV: Space, Motion, Mass 4 00 0. So far the members of the class 'Hom' might be ghosts. The only extension we needed, was time. We shall now introduce space, motion, mass, and other notions of mechanics. We could do so by axioms, but such a procedure would be unsatisfactory. We prefer the behaviouristic approach. Afterwards the crude ideas we have acquired will be refined and settled with more precision by means of an axiomatic system. 4 01 1. Ha Inq Hb : \/ h : h @ Pos . /\ : Sec h . Pst . Hc Inq Hd p : Hd Ani : Utr . PAN Hc Inq Hd p . /\ . Sec h . Pst . Hd Inq Hc q : Hc Ani : Utr . PAN Hd Inq Hc q . ...
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