The Principles of Pythagorean Mathematics Applied to Ethics/Actions
We will be discussing how mathematics, in particular how Pythagorean geometry, provides the principles for our actions; in other words, how mathematics determines our ethics. In doing so, we shall examine some considerations of the differences between what is called calculative thinking and what is called contemplative thinking.
The Greeks initially thought that mathematics is “what can be learned and what can be taught”. What we call mathematics is the axiomatical theoretical viewing and projection of the world which establishes the surety and certainty of the world through calculation. Axioms provide the principles or the archai, which in turn determine the laws which, in turn, lead to the calculative results. Calculative thinking based on the principle of reason (nothing is without reason) determines that the things of the world are disposables and are to be used by human beings in their various dispositions towards the world. This commandeering challenging of the world and the beings in it to demand of them to give us their reasons for being as they are is what we have come to call knowledge. This under-standing is that upon which all of our actions are based although we attempt to separate theory from practice. This surety or certainty that beings are as we say they are through calculation arises through the viewing and use of algebraic calculation in the modern world, the mathematical projection. The results of what is and has been achieved through this calculative thinking are what we have come to determine what knowledge is in our day and what is best to be known and how it is to be known. The axiomatic viewing or theory of mathematics, what provides the theory, is not separate or distinct from the praxis required to carry it out. The theory is not something left behind as the journey for results/knowledge becomes undertaken. The theory permeates each and every stage of the journey or search, what we call the methodology.
Ethics are based on what Aristotle called phronesis: our careful deliberation over what best actions will ultimately bring about the best end result. We call this end result our happiness or what Aristotle called our eudaimonia, our “good spirits”. But how can happiness be the end result of what is, essentially, a hubristic way of viewing and being in the world?
We shall reflect on this question by examining the passage below from Shakespeare’s King Lear.
We are not the first
Who, with best meaning, have incurr’d the worst.
For thee, oppressed king, am I cast down;
Myself could else out-frown false fortune’s frown.
Shall we not see these daughters and these
No, no, no, no! Come, let’s away to prison:
We two alone will sing like birds i’ the cage:
When thou dost ask me blessing, I’ll kneel down,
And ask of thee forgiveness: so we’ll live,
And pray, and sing, and tell old tales, and laugh
At gilded butterflies, and hear poor rogues
Talk of court news; and we’ll talk with them too,
Who loses and who wins; who’s in, who’s out;
And take upon’s the mystery of things,
As if we were God’s spies: and we’ll wear out,
In a wall’d prison, packs and sects of great ones,
That ebb and flow by the moon.
Take them away.
Upon such sacrifices, my Cordelia,
The gods themselves throw incense. Have I caught thee?
He that parts us shall bring a brand from heaven,
And fire us hence like foxes. Wipe thine eyes;
The good-years shall devour them, flesh and fell,
Ere they shall make us weep: we’ll see ’em starve
Explication of the Passage from King Lear
To attempt a summary and explication of the whole of the greatest work in the English language is impertinent. But a brief introduction is necessary to provide some context with which to understand the play as it appears in the scene above.
At this point in the play, Lear and Cordelia, supported by French troops, have lost the civil war for Britain to Edmund’s forces. Lear, as King, has been ultimately responsible for this civil war. At the beginning of the play, he has disowned his ‘truthful’ daughter, Cordelia, and fallen victim to the flattery and machinations of his two eldest daughters, Goneril and Regan. He has divided the kingdom in two giving each sister control of half, his intention being to avert future strife. Lear wishes to retain the appurtenances of a king, the appearances of a king, while retaining none of the responsibility: Lear is satisfied with the appearances rather than the realities of things. It is this satisfaction with the appearance of things that leaves Lear open to the machinations of his two daughters, Goneril and Regan.
Lear’s responsibility is, chiefly, a moral one. Goneril and Regan soon work together to remove from Lear the power and possessions that he once held. Lear becomes an O, a nothing, and there is a pun here indicating both his reality and his suffering. In his nothingness, Lear becomes mad and rages against the ingratitude shown by his daughters and the injustice that he sees in the nature of things and in the created world as it is. This scene quoted above is Lear’s anagnorisis or moment of enlightenment, the moment in tragedies when all tragic heroes recognize the errors of their ways and the consequences of their hubris which is at the root of their previous actions.
Lear ends up houseless and homeless and wanders on a heath in the heart of a terrible storm, the inner and the outward man caught up in a terrible disunity. Lear’s physical, mental and spiritual sufferings soon drive him mad. The storm’s effect is a purification of Lear: Lear removes his clothing; his ego, his “I”, is destroyed in the madness; he no longer focuses on himself but is able to see the ‘otherness’ of human beings and to feel compassion and pity for them (in the characters of Edgar as Poor Tom and the Fool) because he sees himself and his humanity in them. Lear’s life as a king has been one of a tyrant for whom all sense of ‘otherness’ has been forgotten. Edgar, too, has become a ‘nothing’ due to the machinations of his bastard brother Edmund in the parallel plot of the play. Lear has gone from King to nothing and he is ready for re-birth. His ego has blinded him to understanding what his true relationship to his god is: initially he looked upon this god and his power as being something which he, Lear, himself possessed. Lear believed that only he himself possessed the truth and that that truth lay in his power.
The play King Lear is a play about the consequences of not knowing who we truly are, as individuals and as a species. Lear, focused as he is on his ego, his Self, is willingly duped by machination in the play; he is willingly duped by flattery as this flattery is recognition of his social prestige. His suffering and madness bring him to a true understanding of his relation to the god and to other human beings, and this relationship is Love. Love is, as Plato describes it, “fire catching fire”. It is recognition that in the most important things, all human beings are equal in that all are capable of the capacity for Love. It is not without reason that in our art and poetry, Love has been described as a homeless, houseless beggar.
Many critics suggest this play is atheistic; Lear has lost his faith in God. The above passage suggests that such is not the case: what Lear has come to understand is his true relationship to God, the true relationship of all human beings to God. Lear has lost the illusion of what he had once understood as God and what his relationship was to that God. It is this illusion that is the trap cast for those who believe that they are in possession of the truth or that truth is of their own creation or doing, their power. The God in King Lear is absent: He will not perform some miracle preventing the hanging of Cordelia by the Captain later in the play. Cordelia has been a symbol of truth throughout the play, and in her death we see the literal destruction of truth among human beings. Good does not triumph over the evil of human actions in this play and we, too, by our very silence, are made complicit in the death of Cordelia.
The play King Lear shows that the purpose of suffering is to allow the de-creation of our selves, the de-struction of ourselves. For the Pythagoreans, the study of geometry served an identical purpose: the purification of our selves through a contemplative understanding of the things that are. For the Pythagoreans, mathematics was a religious activity. When we stand on the outside of the sphere above (the circumference) and are subject to its spinning, we suffer the ups and downs of Fate. There is a wheel of fortune motif that runs throughout the play: Fortune is personified in the passage through the alliteration and personification of ‘false fortune’s frown’ to illustrate that it is, in this case, one of human making: even with the best of intentions one can incur the worst: good does not triumph over evil in this sphere but is subject to the same necessity as are rocks and stones. To decreate one’s Self is to have the Self replaced by an assimilation into the divine; it is to become one of ‘God’s spies’, to see all with God’s eyes and to see all for God. When a human being sacrifices the Self, his most treasured possession, for assimilation in God, “the gods themselves throw incense” upon this sacrifice. We believe our Self to be our most precious possession; the renouncing of this possession is not easy, nor is it pleasant: it is done through suffering.
The centre of the sphere is simultaneously both in time and space and out of time and space. The Self as center here is indifferent to the size of the prison, the size of the circle, what can be encountered through ‘experience’ and novelty. For Lear, imprisonment is a liberation, not a restriction.”Suffering (affliction), when it is consented to and accepted and loved, is truly a baptism” (Simone Weil, “The Love of God and Affliction”). Baptism is a spiritual re-birth. The spiritual rebirth for Lear is clear from this passage as well as from Act III onwards in the play. The attempted suicide of Gloucester due to his suffering is a counterpoint to this: suicide is a sin against the gods because we falsely believe that our self is our own and of our own making and that we can do away with it as we would with any other possessions that we may have. Gloucester’s realization of this results in his finding Edgar again and having ‘his heart burst smilingly’ when he meets his final end. Contrary to our view, in the world of Shakespeare some kinds of suffering have a purpose.
Our personal knowledge is our ‘sphere of influence’ on our world and on the other human beings who inhabit our world. It is what we have come to call our empowerment. That sphere should be seen as composed of wheels within wheels with our actions the spokes of the wheels. The spokes reach out to the circumferences of the wheels: from the diameter, the right angled triangle cannot exceed that circumference. The sphere created by the circumferences may be large or small; most of our lives are spent in our attempts to enlarge this sphere. In it we are ’empowered’ to carry out our activities, but the prison of ourselves is still a ‘prison’ beginning with our bodies and our egos which are placated by the social prestige and recognition which comes from this fulfillment. We become the ‘poor rogues’ and ‘gilded butterflies’ that Lear and Cordelia will chat with, the partisans and politicians of the court. The outer edges of the sphere in its spinning indicate the fates of those who are ruled by Fortune: ‘who loses and who wins; who’s in, who’s out’. It is the fate of all of us who are dominated by the wish for social prestige, recognition. This fate and our desire for this fate is part of the ‘mystery of things’: to see this we must remain at the centre of the sphere where we are not moved by the wheel’s or the sphere’s spinnings, nor are our desires dominated by the wish for social prestige and recognition.
Lear, through his madness and suffering, has been re-born (see other sections of the play particularly Lear’s awakening when he sees Cordelia as an angel, a mediator, and in the play she is, from the beginning, representative of truth). His self, ego, I has been destroyed. In this scene, Lear demonstrates the friendship that is the love between two unequal yet equal beings. Lear’s ‘kneeling down’ when asked for his blessing in order to ask for forgiveness is the recognition of this equality. It is no longer the view of the Lear who said “I am a man more sinned against than sinning”, a false view of Lear’s at the moment of its occurrence in the play for it is the view of most of us with regard to our own sufferings.
It is with a great and terrible irony that after this speech of Lear’s, the following occurs:
Come hither, captain; hark.
Take thou this note;
Giving a paper
Go follow them to prison:
One step I have advanced thee; if thou dost
As this instructs thee, thou dost make thy way
To noble fortunes: know thou this, that men
Are as the time is: to be tender-minded
Does not become a sword: thy great employment
Will not bear question; either say thou’lt do ‘t,
Or thrive by other means.
I’ll do ‘t, my lord.
About it; and write happy when thou hast done.
Mark, I say, instantly; and carry it so
As I have set it down.
I cannot draw a cart, nor eat dried oats;
If it be man’s work, I’ll do ‘t.
The Captain’s final words are a statement for all of us motivated by social prestige. Human crime or neglect is the cause of most suffering. On the orders of superiors, we carry out acts that we believe are “man’s work” i.e. they are not the work of Nature but we ascribe the moral necessity for our actions to Nature: “I cannot draw a cart, nor eat dried oats”. We believe that we are compelled to commit immoral actions because we believe Nature imposes its necessities upon us; and, at times, Nature does indeed do so. But if we live with a thoughtful recognition that there are simply acts which we cannot and must not do, we are capable of staying within these limits imposed by the order of the world upon our actions. Such words as the Captain’s have been used by human beings to justify to themselves and to others the reasons for their actions from the committing of petty crimes to genocides. They see their crimes as performing a duty, just following orders.
The root of all crimes is, perhaps, the desire for social prestige whether that is achieved through position, money or recognition. For the Captain, it is Edmund who will determine what ‘happy’ will become for him by his giving to the Captain ‘noble fortunes’; and the Captain believes it. He will achieve his noble fortunes through the committing of an ignoble act. One would need to look far across the breadth and depth of English literature to find two more contrasting views of humanity in a work than that which is presented here in these two brief, sequential scenes from King Lear. Human beings are capable and culpable of both forms of action: we have an infinite capacity for Love and forgiveness as well as a finite capacity for committing the most heinous crimes; only Love is both beyond and within the circle, and all human action is done within the circle (or the realm of Necessity).
Contemplation and Calculative Thinking: Living in the Technological World
The passages from King Lear give us an entry to understanding a practical alternative way of being-in-the-world to the current conditioning or ‘hard-wiring’ of our way of being under the technological world-view operating as it does under the principle of reason. This alternative way involves contemplative thinking as opposed to calculative thinking. This contemplative thinking is open to all human beings: it is not a special mental activity for the few. It is an attitude toward things as a whole and a general way of being-in- the-world. It is the attitude that Lear proposes for himself and Cordelia on how they will spend their time in prison: while they will still be in the world, they will not be of the world. While they will be involved with the “poor rogues” and “gilded butterflies”, the world of these rogues and butterflies will not be their world.
What does this mean for us? It suggests that we are in the technological world, but not of this technological world; we are here in body but not in spirit. We avail ourselves of technological things but we place our hearts and souls elsewhere. This detachment involves both a being-in and a withdrawal-from. Like Lear and Cordelia, we let the things of the technological world go by, but we also let them go on. Like Lear and Cordelia, the detachment is both a “no” to the social and its machinations, but it is also a “yes” to it in that it lets that world go on in their entertaining of it.
What is Calculative Thinking:
Calculative thinking is how we plan, research, organize, operate and act within our everyday world. This thinking is interested in results and it views things and people as means to an end. It is our everyday practical attitude towards things. Contemplative thinking is detached from ordinary practical interests.
Calculative thinking is not computational. It does not require computers or calculators and it is not necessarily scientific or sophisticated. It would be better understood in the sense of how we call a person “calculating”. When we say this we do not mean that the person is gifted in mathematics. We mean that the person is designing; he uses others to further his own self-interests. Such a person is not sincere: there is an ulterior motive, a self-interested purpose behind all his actions and relations. He is engaged with others only for what he can get out of them. He is an “operator” and his doings are machinations. His being-in-the-world may be said to rest on the principle attributed to H. L. Mencken: “No one ever went broke underestimating the intelligence of the American public.”
Calculative thinking is, then, more of a general outlook on things, our ‘way of life’. It is an attitude and approach that the things are there for what we can get out of them. People and things are there for us to exploit. This general outlook is determined by the disclosive looking of technology and its impositional attitude toward things.
There is no lack of calculative thinking in our world today: never has there been so much planning, so much problem-solving, so much research, so many machinations. TOK itself is a branch and flowering of this calculative thinking. But in this calculative thought, human beings are in flight from thinking. The thinking that we are in flight from is contemplative thinking, the very essence of our being human. In this flight, we are very much like Oedipus who, after hearing the omen from the oracle at Delphi and its prophecy, rashly flees in the hope that he can escape his destiny. As with Oedipus we, too, are blind and unable to see in our flight from thinking and our rash attempts to “change the world”.
What is Contemplative Thinking:
Contemplative thinking, on the other hand, is the attention to what is closet to us. It pays attention to the meaning of things, the essence of things. It does not have a practical interest and does not view things as a means to an end but, much like Lear and Cordelia, dwells on the things for the sake of disclosing what makes them be what they are. Contemplative thinking allows us to take upon ourselves “the mystery of things”, to be “God’s spies” in the two-way theoretical looking of Being upon us and of ourselves upon Being. To be God’s spies we must remove our own seeing and our own looking, that looking and seeing that we have inherited as our “shared knowledge”, and allow Being to look through us. This seeing and looking is not a redemption that is easily achieved. The pain-filled ascent in the release from the enchainment within the Cave to the freedom outside of the Cave or Lear’s suffering and de-struction on the heath in the storm are indications of just the kinds of exertions that are required. King Lear in his anagnorisis has arrived at the truth of what it means to be, as such, and of his place in that Being. Contemplative thinking is a paying attention to what makes beings be beings at all, but it is not a redemption which can be cheaply bought.
The word “con-templation” indicates that activity which is carried out in a “temple”. It is a communing with the divine. The temple is where those who gather receive messages from the divine. Lear and Cordelia’s prison is, as such, a “temple” to Lear. Within a temple, one receives auguries. An augury is an omen, a being which bears a divine message which must be heard by those to whom it is spoken. In and through this hearing one is given to see the essence of things and to “give back” those essences to Being. Contemplation is the observing of beings just as they exist and is an attending to their essence. It is a reserved, detached mode of disclosing that expresses itself in gratitude, the giving of thanks. This attention is available to all human beings who through their love, like Lear and Cordelia, are open to the otherness of beings without viewing those beings as serving any other purpose than their own being. For human beings, it is the highest form of action directed by what the essence of human being is. As the highest form of human being itself, it must be available to all since it is our very nature as human beings.