Consciousness, Literature and the Arts
Archive
Volume 6 Number 1, April 2005
___________________________________________________________________
–the
Use of Iteration and Recursion in Much Ado
About Nothing
University
of Northumbria
ABSTRACT
The nature of Shakespeare’s use of the logical and mathematical devices of iteration, recursion, contradiction and paradox in the play Much Ado About Nothing are considered, and related to the play’s objective which, it is proposed, is the establishment of consciousness. This consciousness is of two kinds; firstly, consciousness of the play itself and secondly, self-consciousness on the part of the watcher. It is suggested that an appreciation of this will not only enhance our understanding of the play but may also point the way to a solution of the so-called “hard” problem of consciousness.
In
this paper I shall try to show that the effectiveness of Shakespeare’s play Much
Ado About Nothing lies
in the use of two related logico-mathematical[1] devices, iteration
and recursion.
Both of these devices, I believe, would have been familiar to a large
section of Shakespeare’s audience and it seems likely that he relied on their
appreciation for the achievement of his highest effects.
Much
Ado is
sometimes reckoned as one of Shakespeare’s minor plays, even among the
comedies. As against this view, I
believe that the play contains in the working of its plot the key to an
understanding of much of Shakespeare’s work, and specifically an appreciation
of the method Shakespeare uses to heighten consciousness in his audience. I
shall also try to show that a study of this method suggests a clue to the
solution to the “hard” question of consciousness posed by, among others,
philosophers and neuroscientists, that is: what is the nature of consciousness
itself? [2]
Recursion
and iteration are used in the play to achieve two aims, one superficial land
dramatic, the other deeper and more philosophical in its implications. The
immediate dramatic aim is to create comedy, and these two devices, as I shall
show, are used in the play to produce two further consequences,
contradiction and paradox, on which a large part of the comedic effect depends.
However the deeper purpose of the play touches
on the question of consciousness itself, and here the role of recursion is
crucial, since consciousness involves an awareness of being oneself conscious,
and thus contains a necessary element of recursive self-reference.
The
consequences of self-reference
The
essence of the dramatic process at work in the play and implied by its
title—the creation of something from nothing—is self-reference, and this creation de nihilo is carried
out by means of the mathematical processes known as iteration and recursion[3]
leading to contradiction and to logical paradox. To see a stage performance of Much Ado is to experience the
consequences of this recursion at first hand: to read it on the printed page is
to see the mechanics of its recursive nature in detail. Almost on the first page
we are handed a mathematical joke, one moreover which foreshadows the whole
plot:
LEONATO:
How many gentlemen have you lost in this action?
MESSENGER:
But few of any sort and none of name.
LEONATO:
A victory is twice itself when the achiever brings home full numbers.
This
introduces the essence of self-reference which is used throughout the play.
If a victory can be twice itself, then the victory, or more precisely the
value of the victory, is defined in terms of itself.
The consequences of such self-referential definition are immediate and
dramatic. If a victory is twice itself when there are “full numbers”, then
the victory is doubled, and by the iterative[4]reapplication
of this rule, it is then redoubled and so on until it becomes as large as we
like[5].
The application of the rule leads to a number which may become as large
as we wish.
Interpreting
the meaning of “full” as “whole”, “round”, or “integral”[6],
we might rephrase this in mathematical language as: “the value of a function[7]
is twice itself when it returns an integer value.” If the value of a victory is a function of itself—itself
multiplied by two—then if the victory has any value at all, with the repeated
application of this rule (i.e. with iteration of the rule) its value
becomes infinitely large.
If
we consider the reverse (in mathematics the inverse) of this
doubling function, we get a function which divides by two instead of multiplying
by two. By the iterative
application of this function we can reduce any number to as nearly zero[8]
as we want—in effect to nothing—by repeatedly halving it.
These two inverse functions are exactly what the play is about: something
which can be manufactured out of (almost) nothing but which can also be reduced
to nothing when the basis of the iterative process supporting it collapses.
The
“thing” in question is of course love[9],
in this case the love of Beatrice and Benedick. The means by which it is made large is repeated suggestions
made by others, but it can also be reduced to nothing through suspicion, brought
about incidentally for a trivial reason[10].
We might add to Leonato’s remark: “A victory is half itself when it is
empty” and in the additional love-plot the temporary “defeat” of the love
of Hero and Claudio is empty both because it is baseless and because it is itself defeated.
If
this should seem to be leading towards an over-technical interpretation of the
play, we should recall that in Shakespeare’s time it was common for gentlemen
of society to be educated in mathematical ideas. John Aubrey in his Brief
Lives makes frequent reference to mathematics and mathematicians in terms
which show clearly that it was the stuff of everyday educated conversation.
“Hath he the Mathematiks?” was a question which could evidently be quite
seriously asked about a mutual acquaintance.
Mathematical
ideas would have been more familiar to the educated English man or woman of
Elizabethan or Jacobean times than they are to people of today, living in a
supposedly scientific age. It might at first seem surprising and even paradoxical that
four centuries of scientific progress has brought about a loss rather than a
gain in the general level of understanding of science and maths.
It has been plausibly argued[11]
however that the present age is one not so much of public education as of public
trivialization; unstructured and unrelated snippets of information taking the
place of learning and this is simply the consequence of the lack of system
in our approach to arranging facts. We live in the age of the
encyclopedia, a collection of known facts arranged alphabetically and with no
underlying order or connection between the different areas of knowledge it
contains.
Contrast
this atomic or atomized approach to knowing that of Shakespeare’s times which,
it should be remembered, were much closer to those when a coherent systematic
world view of organized knowledge existed.
Such systems of knowledge, which included basic elements like the months
of the year, the constellations and the names of trees are known to have existed
since prehistoric times[12].
Even the letters of the alphabet, reduced in an encyclopedic system to
simple ordinals, had their own natures and significance. One such knowledge
structure, the “Seven Pillars of Wisdom”, arranged the liberal arts—which
included mathematics—into a system which could be used as a basis for
organized thought[13].
This system of knowledge had persisted since the days of the Roman Empire
and was not to pass away fully until the Age of Reason.
As
a learned man of his times, Shakespeare was very probably as good a
mathematician as he was a lawyer, musician, statesman, botanist or physician and
his plays show his knowledge of these and many other branches of learning.
His audience was mixed, including everyone from the groundlings to the
gods but it is likely that a far higher proportion than today would have been
able to appreciate the wit of a play which used mathematical ideas, whether that
appreciation was at a conscious or merely a preconscious level.
Iteration
and recursion are concepts still unfamiliar enough to most people that it is
worth beginning with a brief description of their nature of each, how they are
related and how they differ. To iterate means to repeat.
However iteration should be distinguished from simple repetition, which
may produce nothing. In the process
of iteration a series of actions is carried out, the result of each action in
the series becoming the object of the same action the next time it is performed[14].
This produces a series of events moving forwards in time, such as
counting, building, growing or getting older.
The most powerful kinds of iterative process are those which sustain
themselves. In the modern world we
have an awe-inspiring example of an iterative self-sustaining process, the
nuclear explosion, which results from a so-called chain reaction of successive
atoms. A more ancient example is fire, which is equally a self-sustaining
process. So too, at another level,
is the reproductive nature of life, the underpinning of love itself.
Recursion
is related to iteration by a simple twist of definition.
Instead of the result of an action being the object of the repeated
action on the next occasion it is performed, as in iteration, in recursion the
object of the action becomes itself; in other words the action becomes a
self-referential one. There are
many examples of recursive processes, especially in speech.
Some of these must be as old as language itself: the paradox of the
Cretan Liar who says “All Cretans are liars” is self-referential and hence
recursive: if he is telling the truth he is lying and vice versa. Recursion
enters into our jokes too: To take an example of Jack Cohen’s, “the American
Dream” means “the right to pursue the American Dream”. The second world
war armed services song “We’re here because we’re here because we’re
here..” implies by its iterative form an underlying recursion in thought which
demonstrates the existence of self-reference at the most basic levels of
language and thought.
However
it is in mathematics and in its practical realization, computing, that the idea
of recursion comes into its own. For
example, the positive integers could be defined iteratively by the operation of
adding one to the previous integer. Alternatively they could be defined
recursively by saying that the value of an integer is one plus the value of the
next largest integer. The difference between the recursive and the iterative
definition of the integers may seem like hair-splitting, but is crucial.
Iteration takes place in a series of steps in time, while
recursion takes place outside time or rather embraces all times in its
definition[15]. Iteration
works forwards, by taking a value and then building on it to produce the next in
a series; recursion, on the other hand, takes a value and defines it in terms of
itself.
These
two processes are related, and indeed are sometimes regarded as identical,
because in computing the same result can frequently be produced by using either
iteration or recursion. Indeed so strongly do the two resemble each other that
one can find mathematical dictionaries which define iteration simply as
recursion; however there are very important differences as we can begin to see.
Recursion and iteration can be seen to be inverse processes rather like
multiplication and division, or addition and subtraction. What is inverted is time:
iteration takes place in time as it moves forward, while recursion involves an
apparent backward step in time, implied by the element of self-reference in its
definition.
Many
examples of recursion which we can easily think of either are or involve mental
concepts and the significance of this will become clear soon. Mathematics is one
of the highest faculties of our mental activity and functions at a level of
abstraction in which recursion naturally arises, but over and above this, there
is something about the nature of mental concepts which is itself recursive. We
have seen an example of recursion in Much
Ado in the form of self-referential, repeated multiplication: this could be
taken as a paradigm for thinking about recursion in the play.
I shall be arguing that the play is mathematical in its techniques, and
that the effect of this is the creation of a recursive process in the mind of
the audience, leading to a sense of awareness without which properly speaking
neither the play, nor indeed our own consciousness, would exist.
To
begin at the beginning, the title of the play itself is mathematical.
Not only does it implies a recursive process, but it reminds the
technologically-minded modern reader of a programming language: the title Much
Ado About Nothing is like a DO loop, with zero (nothing) as the initial
value of its argument, i.e. the value of the variable being iterated[16].
The
term “argument” is a mathematical one and means the variable which is the
input value to a function. Consider the crucial line by Benedick in his long
soliloquy in Act 2 scene 3:
I
do much wonder that one man, seeing how much
another man is a fool when he dedicates his
behaviors to love, will, after he hath laughed at
such shallow follies in others, become the argument
of his own scorn by failing in love:
This
of course is exactly what happens to him through falling in love and in this
lies the nub of the comedy. Shakespeare
however has picked on this precise mathematical formulation—to become one’s
own argument—to express the nature of the recursive relation which provides it
and has hence set the play on a firm logical-mathematical footing.
The
word “argument” occurs twice more in the play: another occurrence is its use
in the following exchange between Don Pedro and Benedick:
I
shall see thee, ere I die, look pale with love.
With
anger, with sickness, or with hunger, my lord,
not with love: prove that ever I lose more blood
with love than I will get again with drinking, pick
out mine eyes with a ballad-maker's pen and hang me
up at the door of a brothel-house for the sign of
blind Cupid.
Well,
if ever thou dost fall from this faith, thou
wilt prove a notable argument.
And
in the remaining example it is once again Benedick who is the referent of this
appropriately recursive word.
Iterative Wit
From
line 6 of the play onwards we are bombarded with a series of jokes, puns,
oxymorons, quiddities and paradoxes, all of which turn about the themes of
iteration and recursion. Going from
Act I Scene 1, line 14:
Better
bettered expectation (15.)
A
kind overflow of kindness (25.)
How
much better it is to weep at joy than joy at weeping! (26,27.)
A
lord to a lord, a man to a man (50.)
Such
speech is frequent in Shakespeare plays. For example looking almost at random into Twelfth Night
we find:
Better
a witty fool than a foolish wit (37.)
which
is a circular thought and therefore possibly leading to self-reference.
In this play however we are subjected to a barrage of such thoughts.
Their theme once established is not allow to go un-reinforced, for with
the use of repetition the author sets us in the right framework for what
follows:
MESSENGER
Don
Pedro: he hath borne himself beyond the
promise of his age, doing, in the figure of a lamb,
the feats of a lion: he hath indeed better
bettered expectation than you must expect of me to
tell you how.
Here
we have someone who has borne himself beyond himself; and who has in doing so
better bettered an expectation of his own self! This is self-reference taken to its sublimity of excess.
A
kind overflow of kindness: .... How much
better is it to weep at joy than to joy at weeping!
To
the idea of iteration there is now added the idea of circularity, whereby A
bears a certain relationship to B while B also bears the same relationship to A.
If A stands in the same relation to B as B does to A then A stands in a
certain relation to itself. The
circularity thus paves the way, via implied self-reference, for
recursion. It also gives us the basis of the interaction whether of love or of
mutual suspicion which is basic to the plot.
As
we go through the scene between Beatrice, who is an acknowledged wit, and the
messenger, we are given repeated examples of repetition:
And a good soldier too, lady
BEATRICE
And a good soldier too, a-lady
A
lord to a lord, a man to a man; stuffed with all
honourable virtues.
this
keeps up the “witty” mood but also keeps the idea of iteration fresh before
us. What is happening is a constant
feeling of happening appearing from nowhere, something building out of nothing,
almost by its own bootstraps. Shakespeare
is establishing a frame of mind in which we are thinking in terms of
relationships of this kind. A
little later we have
I
see, lady, the gentleman is not in your books.
Beatrice
No; and he were, I would burn my study
This
implies a more complicated construction, one which is crucial for the purpose of
the play. One person exists in the
mind of another, and can be eliminated from it by the drastic means of
self-destruction – “burning one’s study.” It contains also the seed of an implied recursion, since if one
exists in the mind of another, then the other also presumably exists in the mind
of the one, like mirror images. This
thought has to do with two things; first the plot of the play, turning as it
does upon mutual processes, whether of suspicion or reinforcement of affection,
and second, it reminds (literally, re-minds) us of ourselves, and the nature of
our own consciousness, even though in a roundabout way.
But
there is something beyond this half-conscious reference to the nature of
consciousness. The nature of theatre is not just that we enter into belief in
the play (or as is sometimes said, “suspend disbelief”) but also that we
become at times aware of our belief, and therefore temporarily alienated
from it, before regaining our belief with renewed strength.
Here Shakespeare is using one of his favorite devices; the raising of
awareness as a process working momentarily against, and therefore by contrast to
our former state again heightening our belief in, the play.
We
come up against this device in many plays, ranging from Henry V with its
deliberate mention at the start of the “wooden ‘O’ “ of the Globe
theatre, to “Jaques’ “seven ages” speech in As You Like It– a
play of special significance for the nature of theatre, since it was probably
the first to be performed in the Globe. From
“who keeps the gate here ho” to “we’ll try to please you every day”
Shakespeare uses the self-awareness of the play-goer as play-watcher to create
an acceptance of the play as a self-creating process.
In
Much Ado however we are treated to an additional device; that of the
cultivation of the recursive nature of awareness itself.
The
Abyss Opens
Of
all recursive processes the one which is most immediate for us is our own
consciousness. To be conscious is
to know that we are conscious. Thus
recursion, far from being merely an abstract mathematical idea is the very
ground of our awareness. In
possibly the highest state of awareness which we can achieve, meditation, the
object of the mind’s consciousness becomes simply consciousness itself.
Don
Pedro is in a mood for trouble:
Good
Signior Leonato, are you come to meet your
trouble: the fashion of the world is to avoid
cost, and you encounter it.
But
Leonato is anxious to disarm him:
Never
came trouble to my house in the likeness of
your grace: for trouble being gone, comfort should
remain; but when you depart from me, sorrow abides
and happiness takes his leave.
You
embrace your charge too willingly. I think this
is your daughter.
Thus
primed for some upset, Leonato cannot resist a wry jest at his own uncertainty:
Her
mother hath many times told me so.
Too
many evidently, for optimum credibility. Compare this to the Winter’s Tale
where Leontes (the similarity of name is more than coincidental) receives
stonily the assurances of the nurses as to the legitimacy of his child.
In that play the destructive force of doubt leads to tragedy; in this
present play tragedy is forestalled by wisdom.
However in both plays we are shown the recursive nature of suspicion and
jealousy; by this mental act the foundation is kicked from under the love of man
for woman, of father for child, or indeed of one for oneself.
Just as love can be made out of nothing by iteration so it can be reduced
to nothing by recursion. Shakespeare reminds us in these plays that the abyss is
always there.
But,
just as what can be done can, it is often said, be undone, so too what can be
undone can be done again: Don Pedro has a remarkable solution to this paternity
dilemma:
“Truly,
the lady fathers herself.”
And
his reasoning behind this fully recursive observation is couched in terms of
what are fast becoming self-sustaining processes:
Were
you in doubt, sir, that you asked her?
Signior
Benedick, no; for then were you a child.
You
have it full, Benedick: we may guess by this
what you are, being a man. Truly, the lady fathers
herself. Be happy, lady; for you are like an
honourable father.
And
Benedick, also a wit, cannot resist capping this with his own line
If
Signior Leonato be her father, she would not have his head on her shoulders for
all Messina, as like him as she is.
So we are now presented with the framework for a play which is, not the tragedy of doubt as is Othello or The Winter’s Tale, though it comes worryingly close at times, but a comedy which as a dramatic form has as much power as tragedy to question, and in the end to reaffirm, the basis of life and of love.
Following the appearance of recursion produced by iteration, we soon encounter the next logical device used to establish the consciousness which Shakespeare is aiming at in this play: that of contradiction. Contradiction in logic, just as falsehood in mathematics, is most feared and rightly, for from a contradiction (if one is allowed in the system) any other statement can be proved. I shall illustrate this in two ways: the first correct in logic, the second correct in mathematical intent and wit.
Suppose we allow that if it is raining it is not raining. This is a contradiction and from it I can prove for example that I am a Dutchman, as follows:
If
it is raining, then it is not raining (contradiction).
If
it is either raining or I am a Dutchman, then it is either not raining or I am a
Dutchman.
Therefore
I am a Dutchman[17].
That
Shakespeare feared for the effects of contradiction on his characters is clearly
shown by this, which quickly follows:
BENEDICK
Like
the old tale, my lord: 'it is not so, nor
'twas not so, but, indeed, God forbid it should be
so.'
The
corresponding bogeyman in mathematics to contradiction in logic is a false
statement because from a falsehood any other statement can be proved.
For example from the statement
we
can prove that
or
anything else we choose, such as that
and
in such a system productive mathematics becomes impossible.
This
was wonderfully exemplified by the great mathematician G H Hardy who was asked
by a layman to show how, if it was
true that in mathematics anything follows from a falsehood, it could be shown in
this way that Ramanujan[18]
was the Pope. Hardy reasoned thus:
“Let
us assume that two and two is equal to five.
That is false. Take three from each side.
It then follows that two is equal to one. Ramanujan and the Pope are two.
So they are one.”
Hardy's
joke works – almost. Of course it
was only mathematical statements that he meant, but this doesn't matter in the
joke or in the play for it stimulates the imagination by opening up new
possibilities, as does the wit of Carroll, Lear and the witty nonsense which is
talked by the many fools in Shakespeare’s plays.
“Fool” is an ambiguous term, for the role of the Fool is not to be
a fool but to fool others, just as the job of a cook is to cook food, not
himself. The role of the Fool in
Shakespeare is to expose the folly of the supposedly wise, often kings, nobles
and others of rank and position and this they do by highly meaningful nonsense.
An
idea closely linked to contradiction is that of paradox.
A paradox is a statement which is or appears to be both true and false at
the same time. Classical paradoxes of logic such as the Cretan Liar paradox
mentioned above have been known since antiquity.
Paradox can be produced from recursion by interpreting a statement which
is alternately true and false as being simultaneously true and false at the same
time[19].
In
the drama, paradox appears in the form of the putting together of apparently
contradictory propositions in a witty way. As Koestler[20]
has shown, much of the effect of humour is achieved through paradoxical jokes,
and arises when the tension between the opposing truth values is suddenly
released at the moment of realization (the “punch-line”).
In
Much Ado the fools are the police, a
traditional target for good-humored jibes whether appearing as Conan Doyle’s
Lestrade, Wodehouse’s country constables or, as in this play, the Watch.
The Watch proceed entirely by paradox and contradiction.
They enter with a flourish:
DOGBERRY
VERGES
Yea,
or else it were pity but they should suffer
salvation, body and soul.
DOGBERRY
Nay,
that were a punishment too good for them, if
they should have any allegiance in them, being
chosen for the prince's watch.
At times the watch seem unable to distinguish who is the criminal and who
the policeman:
DOGBERRY
Is
our whole dissembly appeared?
VERGES
O,
a stool and a cushion for the sexton.
SEXTON
DOGBERRY
Marry,
that am I and my partner.
VERGES
Nay,
that's certain; we have the exhibition to examine.
SEXTON
But
which are the offenders that are to be
examined? let them come before master constable.
Yet
in the end this bunch of Keystone Kops with its Cluezot-like Dogberry in
command, apprehend and charge the villains.
From their creative nonsense the right solution emerges: the Watch get
their man.
What
is the purpose of all this mathematical wit?
I suggest that it is twofold. In
part it has the same purpose as Shakespeare's other devices in many plays, to
reaffirm the basis of the play by setting up a relationship between its
recursive structure and that of the minds of the audience, a relationship which
literally creates the play in the watchers' minds.
The difference between a play's “working” and failing to work is
based solely on audience involvement, that is in their identification with, and
status in relation to, the play. Just
as in a detached state we stand back from reality and refuse to get involved, so
we can do the same with a play. If we are in unsympathetic mood it can seem,
like some unacceptable part of reality, unreal.
It is the task of the playwright to convince and involve us, and a method
which is often chosen and which Shakespeare chooses frequently, is to make us
aware of our observer status.
The
sudden triggering of self-awareness – “we are such stuff as dreams are made
on” –is used, as noted above, in many of Shakespeare’s
plays. This device, which
seems to have something in common with the alienation of Brecht[21], but predates it by many centuries, is however not the
only one which he uses here. The
use of recursion and of iterated contradiction in Much Ado induces a sense of
the basis not just of theatrical presence but of all consciousness.
Let us pause for a moment to reflect on where theatrical consciousness
lies, and see if there is a clue here to a wider and more intractable problem.
When
we see a successful play, one which is successful not only for us but for the
audience as a whole, there is a heightened consciousness, which I have equated
with the play “working”, but the locus of that consciousness is at first
hard to identify. It is not the consciousness which is in the minds of the
actors: they are thinking about their cues, their business or perhaps the drink
they so badly need. Nor is it the
consciousness which is in the mind of the director, who may also be thinking of
his own mundane concerns, or, if he is thinking of the play, is doing so in the
sense of its mechanics rather than its ideation. The consciousness which in a sense is the play is in
the mind of the watcher, but not wholly there, for there must be a collective
understanding for the play to fully succeed.
Somehow in the audience collectively there is a group consciousness,
which by observing the play brings it to life, and it was perhaps this analogy
which so fascinated Brecht and led him to see the theatre as a means of raising
social consciousness.
Here
we need the intermediate stage of the awareness of venue –“this wooden
O”—in order to make the next step towards full consciousness.
It is the same step as we make in reading Flatland, Alice in Wonderland,
or Gulliver’s Travels. The implication of these, or any similar “other-wordly”
text, is this: If we can as super-beings visit a lower world, then why should
not a higher being be able to visit ours? And
this is exactly what, so it was believed, theatre represented.
For just as we sit observing the play, so we in the world are observed by
the gods. O hardened, cynical
reader, forget for a moment, if you would understand this play, the context of
your skeptical, alienated and utterly untypical times. The gods look down: so it
was believed in Shakespeare's time and so it was believed long before him. And
indeed so it is believed still, though intellectuals pretend, with all the
insincere sophistication of worshippers at Samuel Butler's Erewhonian musical
banks, that it is not. The gods look down: as below, so above; we sit watching
the play and we are in turn observed at our play.
So
wherein lies our consciousness? This
burning scientific issue seems to be unique to our own times, yet I believe that
in the structure of this play Shakespeare was providing an answer which within
its own world-view is still valid. Just as with the play, our consciousness is not in the mind
of the “actor”, which we may identify with our body and its needs, who
thinks mostly of its lines or its uncomfortable boots. Nor is it in the mind of the “director”, our brain and
nervous system, with its vain concerns about the performance. There is only one possible source, and that is in the mind of
our observer—that is, of God. To
reflect on this is to see the play through the mind of a member of the audience
of Shakespeare's day, and so to be once again more truly a member of the
audience for whom he was writing. We
appreciate the play best when we see ourselves as if we were in a play, and it
is then that we understand more clearly what we are.
To see ourselves fully in the context of this play is not only to see
that our lives may often seem to be much Ado, but also what it is that this ado
is really about.
[1] The relationship between logic and mathematics has been much debated by philosophers and mathematicians alike. There have been many attempts, especially around the start of the twentieth century by Russell, Whitehead, Frege and others, to show that the two are in fact one and that the whole of mathematics can be shown to rest on the simplest of logical foundations. While it has been generally agreed that, so far at least, these attempts have failed, the relationship between mathematics and logic is so intimate that there is a region in which they virtually coincide. That is the region inhabited by the play we are considering.
[2] The so-called “easy” question is: what are the necessary and sufficient conditions for the occurrence of consciousness?
[3] A more formal definition of recursion will be given later, but a recursive function or value is one which is defined in terms of itself, i.e. by means of a self-referential definition.
[4] Iteration, the other main mathematical concept used here, is a repeated operation which uses as its input the output of the last cycle of the iterative process. An example of iteration in practical terms would be the positive whole numbers (integers), which can be generated by repeatedly adding 1 to the previous result thus:
[0] = 0
[1] = [0]+1
[2] = [1]+1
etc. where the numbers [0][1][2] etc are the positive whole numbers.
[5] The value of a victory is defined recursively by this rule as being a function of itself, specifically twice itself, This can be represented by the function definition
[6] This might be a misinterpretation of the word “full”, which may mean “ positive “ or even simply “large.” However, in whichever sense we take it, the same recursion results.
[7] A function in mathematics is a relationship between one set of values called the range, and another set of values called the domain. Examples of functions are which produces values oscillating between positive and negative values, approximated to by sound waves, and which is the so-called exponential function, characteristic of the growth of living systems in their initial stages.
[8] This is obtained by halving a number repeatedly and can be made as near to zero as desired is but never quite zero. It is the limit of the series 1, 0.5, 0.25, 0.125… Values arrived at in this way represent a controversial mathematical entity, called the infinitesimal, which has been alternately accepted and rejected by mathematicians for over four hundred years. The infinitesimal raises a philosophical issue “how can a value be both nothing and yet something at the same time?” and this remains an unsolved question. However for our purposes, such a rule reduces something to as near nothing as makes no practical difference.
[9] “Love” is also a score of nil at ancient (and modern) tennis and occurs in the expression “love all” from which it is only a small step to the vulgarism “fuck all.” We also use the expression “to make love” implying the deliberate creation of this (frequently evanescent) state.
[10] We could compare the motivation of Don John with that of Iago in “Othello” the reasons for whose actions seem decidedly trumped up.
[11] Neil Postman. Amusing Ourselves to Death. Methuen, London. 1987
[12] Robert Graves has explored this in The White Goddess. Faber and Faber, London. 1952
[13] Howlett, D Aldhelm and Irish Learning. [journal, vol., pages.]
[14] We can write this in the following way:
where the arrow symbol indicates the action in question.
In mathematical functional notation this can be written
Note that the order of the symbols has been reversed, the action is still moving forward in time. The first value of x produces the second and so on.
[15] In a sense iteration defines what time is. Most clocks embody an iterative process, whether it be the ticking of a watch, the dripping of a clepsydra or the changes of cesium atoms in an atomic clock.
[16] For I=1 to 100 DO; Make love; Next I. might be an object-oriented programming language statement which describes the play.
[17] If we allow a contradiction, such as “A implies not A” (if A is true it implies A is not true, written A => not A) then we can expand as follows:
1 We assume
A => not A
2 So attaching C to each side we get
A or C => not A or C
(where C is the proposition to be proved)
3 But A or not A
(one or the other of these must be the case)
4 Therefore C.
(C must be true either way)
[18] Srinivasa Ramanujan, the Indian mathematician and number theorist, whose arithmetic knowledge became the stuff of legend. He was jokingly said to be personally acquainted with the properties of every positive whole number.
[19] The propositional relationship
leads to an apparent contradiction. However if we recast this in functional logical form, (as in Spencer-Brown’s Laws of Form), we have
and the only values which satisfy this function are a series of alternating truth values.
[20] Koestler, A. The Act of Creation. Hutchinson & Co. London, 1964.
[21] It has been widely used as a device for heightening consciousness ever since, e.g. Jane Austen repeatedly draws the attention of her reader to the fact that s/he is reading. E.g. “The anxiety, which in this state of their attachment must be the portion of Henry and Catherine, and of all who loved either, as to its final event, can hardly extend, I fear, to the bosom of my readers, who will see in the tell-tale compression of the pages before them, that we are all hastening together to perfect felicity.” Northanger Abbey [edition?] p 209.