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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.

The mathematical nature of the play

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 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: 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.

Mathematics in the age of Shakespeare

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

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.

Much Ado

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:

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:

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.

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:

Such speech is frequent in Shakespeare plays. For example looking almost at random into Twelfth Night we find:

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:

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:

MESSENGER

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

MESSENGER

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.

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.

Good Signior Leonato, are you come to meet your
trouble: the fashion of the world is to avoid
cost, and you encounter it.

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.

Thus primed for some upset, Leonato cannot resist a wry jest at his own uncertainty:

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:

And his reasoning behind this fully recursive observation is couched in terms of what are fast becoming self-sustaining processes:

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.

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.

Paradox and Contradiction

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 either raining or I am a Dutchman, then it is either not raining or I am a Dutchman.

That Shakespeare feared for the effects of contradiction on his characters is clearly shown by this, which quickly follows:

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

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:

At times the watch seem unable to distinguish who is the criminal and who the policeman:

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.

To Do is to Be

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.

Shakespeare as mathematician

R.J.Bird

The mathematical nature of the play

Mathematics in the age of Shakespeare

Iteration and recursion

Much Ado

MESSENGER

MESSENGER

Paradox and Contradiction

To Do is to Be