LINCOS

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	  Ostensive numerals are superseded by algorithmic ones, composed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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