Heidi and Sophie in Wonderland PDF

HEIDI AND SOPHIE IN WONDERLAND by Ernestus Cardona Padilla Sophie’s study room is protected like a fortress by bookshelves which literally shield the full heig...

HEIDI AND SOPHIE IN WONDERLAND by Ernestus Cardona Padilla Sophie’s study room is protected like a fortress by bookshelves which literally shield the full height and length of its three walls. She and her friend Heidi have just finished listening to her uncle’s song ‘Now I Wonder’ from a CD player lying on a table annexed to the remaining wall which displays Sophie’s 3’ by 4’ YOU Map. HEIDI: That’s a beautiful song. It reminds me of Martin Heidegger’s musings on wonder – thaumazein – in Basic Questions of Philosophy1. SOPHIE: Heidegger thrown out of his shell and projected with the ring of a bell, courtesy of my uncle I should say. HEIDI: No wonder I can still feel Heidegger vibrating in my body. SOPHIE: I guess Heidegger by way of the song resonates with the unquenchable thirst in each one of us for the truth about ourselves. As Cardinal John Henry Newman puts it – “Cor ad cor loquitur!”. HEIDI: Well, “Ex abundantia cordis os loquitur!”2 as the Good Book would have it. SOPHIE: That verse quite captures the link between being and language, if we let ‘heart’ stand in for ‘being’ and ‘mouth’ for ‘language’. HEIDI: Which brings us to the topic – ‘Ontology and the Word’. SOPHIE: Right on target – the entanglement between the World and the Word, or more precisely, as we are navigating the topology of existentialism, the entanglement between the human world and the human word. HEIDI: How about adopting a text as a port of entry to the world of the word? SOPHIE: That’s swell I think. Do you have any specific recommendation? HEIDI: I suppose it has to be a text about finding oneself, as you have mentioned a while ago about the unquenchable thirst for the truth about ourselves. SOPHIE: How about Dan Brown’s ‘The Da Vinci Code’? HEIDI: Sophie? I was expecting more from you than a penchant for propagating torrid tabloid tales. SOPHIE: Well it’s a text about a quest – the Quest for the Holy Grail. Sophie’s quest for the Holy Grail – for the truth about herself. And it’s replete with cracking codes – I mean language games to use a more contemporary phrase. Needless to say, it’s a good ‘port of entry’, to use your phrase, to the interface of being and language. 1 Translated by Richard Rojcewicz and André Schuwer (Indianapolis: Indiana University Press, 1994), pp. 131-156. 2 Mt. 12: 34. 1 HEIDI: You may have a point there. But as you broached the idea of the Quest, we might as well start from one of the earliest Holy Grail stories. Then later we can get back to your Dan Brown. So do you have a copy of ‘King Arthur and the Knights of the Round Table’? SOPHIE: I remember it once graced the walls of my fortress but someone borrowed it and since then it has not found its way back home. So I’m still waiting for the knight in shining armour to retrieve it. Sorry for ‘the hole in the wall’ if I may use a metaphor. But wait... I may still have what we need after all. (Sophie goes to the Ancient Texts Section and returns beamingly.) You’ll recognize it when you see it. (Heidi pores over the text and after a few lines looks towards Sophie to indicate recognition.) HEIDI: This is the story of Adam and Eve – “The Fall”, I can still recall. How come you tag it a Holy Grail story? SOPHIE: Why do you think Eve ate the fruit of the tree in the middle of the garden? HEIDI: Curiosity I suppose. SOPHIE: And did curiosity kill the cat? HEIDI: No one got killed that day Sophie, if you mean dead on the spot. Besides, a cat is reputed to have nine lives for a reason if you’re referring to Eve. SOPHIE: Don’t be silly Heidi. Didn’t YHWH expressly declare the eating of the fruit of the tree in the middle of the garden as forbidden under the pain of instant death? HEIDI: “You may eat from every tree in the garden. But from the tree of knowledge of good and bad: you shall not eat from it, because in the day you eat from it: you’ll die!” SOPHIE: YHWH blared “... in the day you eat from it: you’ll die!” He didn’t suggest “... tomorrow, or next week, or soon, or eventually!” So did Adam and Eve eat the fruit and drop dead that day? HEIDI: They ate the fruit but didn’t die that day. Oh my...! YHWH must have been prevaricating when He issued the commandment! SOPHIE: You’re almost calling YHWH a liar. Is it possible that the author of the text was depicting YHWH in the act of creating on the eighth day a New World – the World of Metaphor? Is it possible he was depicting the creation of the World of the Word? HEIDI: As if to say – the word is greater than the sum of its characters. As the Good Book puts it – “the letter kills, but the spirit gives life.”3 Sophie, you’re charitable to the writer. SOPHIE: I guess charitable to the speaker as well. Don’t you notice that the eating of the fruit was preceded by a conversation? HEIDI: Between Eve and the serpent? 3 2 Cor. 3: 6. 2 SOPHIE: Heidi, a while ago your fundamentalist stance sounded heretical. Now your fundamentalist stance is beginning to sound evangelical. HEIDI: But it’s right there in the text. SOPHIE: Of course I know it’s in the text. But we’re living in the age of science. Have you ever encountered speaking serpents? I’m referring to sneaky snakes found in bushes, marshes, holes, trees, forests, and mountains if you know what I mean. I’m not referring to salesmen with silver tongues. Not at all! I believe that the conversation between Eve and the serpent is a metaphor for... Eve being tempted to – or Eve attempting to enter into an intimate conversation with... herself. You know, women are good at it – talking to themselves. HEIDI: An intense inner conversation – a historical first. Eve was reflecting... or to sound more Heideggerian – Eve was thinking. She was weighing her options – To Eat or Not To Eat? SOPHIE: Nay, To Be or Not To Be? – a stark choice between life and death! She was a being in the jaws of nothing! HEIDI: To sound Heideggerian once again, it seems that Dasein is designed to design questions – essential questions. SOPHIE: But if we take the cue from Jean Paul Sartre (“existence precedes essence”), let me say – Dasein is designed to design existential questions. Dasein is the Grand Inquirer. What keeps Dasein going further and farther is – Questions! I can imagine in the best of possible worlds Socrates stripped by and stripped of his wife, and Heidegger stripped by and stripped of Elfriede Petri. But try as I might I could not imagine even for a moment the Greek gadfly or the German jargonizer stripped of his set of questions. Or philosophy stripped of its queries. Try waking up one morning in fiesta mood, yes you heard me right, you wake up one morning in fiesta mood, only to find out there’s no longer an interrogative mood in the ‘House of Being’ if I may use another Heideggerian creation. This can drive one insane in a world full of wonders. So may I ask you again – do you think Eve asked questions out of curiosity or out of wonder? HEIDI: At first glance it looks as if Eve was asking questions out of curiosity. Heidegger carefully identified the extraordinary or the unusual as the focus of amazement, admiration, and astonishment – the unholy trinity known commonly as curiosity. It looks as if Eve was engaged in idle talk about a curiosity – Why is the prime attraction of the garden declared in the same breath as off limits to the prime customers?. SOPHIE: Why plant it right in the middle of the garden and not in the dark recesses of a nondescript cave at the edge of the world? It seems that Eve’s eyes are piqued by the extraordinary – an Unexplainable Planted Object (UPO) – a curiosity. But come to think of it, she chooses to look through the eyes of wonder – to see the ordinary as extraordinary. HEIDI: Extraordinariness lies in the eyes of the wonderer. Eve reckons the tree as an anomaly in the very nature of things. It has no mind but it seems to offer wisdom. It is part of the earth but it seems to offer the heavens. It’s in-the-world but it seems to project something out-of-this-world, that is, divinity and damnation – being and nothingness. It is a being yet it seems to offer Being Itself. It is smAll but it seems to offer the All. To see the tree not as a mere tree but as tree-ing and to see a being not as a mere being but as be-ing marks the advent of wonder, or to be more precise the coming of age of 3 wonder-ing. I know in my heart of hearts that the tree offers more than a curiosity as far as Eve is concerned. SOPHIE: As far as the wonderer is concerned, everything is Being in disguise. And to question the most nondescript of all, the most taken for granted of all, the most ubiquitous resource of all, that is Being, and to make and bake it as your very own creation, let me call it Be(YOU)ing, marks the emergence of thinking, the birth of philosophy, and the rise of humanity. We must thank Eve for ushering in the Age of Dasein. She is the first Dasein – the first authentic homo sapiens. HEIDI: I love your term for authentic living – Be(YOU)ing – and I’m glad you called Eve the first authentic homo sapiens, or should I say the Be(YOU)er. SOPHIE: Sounds like the viewer – the one who sees! HEIDI: The one who sees in a different Light! SOPHIE: Eve refused to be thrown into the world like the rest of the present-at-hand and the ready-to- hand. She refused to be a mere Reactor. She refused to live in the-past-of-the-present. She reached out for the fruit of the tree and her eyes were opened to an omniverse which bathed her in light – reminiscent of the light that made the cosmos possible. She chose to live in the-future-of-the-present. She chose to be a Creator instead. HEIDI: Is it possible that the Light on the First Day is a metaphor for humanity whose identity and destiny is – as the Jewish Socrates in the New Testament later would have us believe – to be the Light of the World? SOPHIE: Whoa! What a radiant insight you’ve got there Heidi! You’re pushing the birth of metaphor farther back to the first moment of creation! HEIDI: The Book of Genesis is paved with metaphors from end to end. I’m even wondering if YHWH really meant what He said in the first prohibition. Perhaps He meant exactly the opposite. SOPHIE: You’re entertaining the possibility of YHWH sowing the seeds of irony in the Garden? You think it’s His way of letting the man and the woman in on His Open Secret? HEIDI: YHWH created them in His image and likeness – we all know that. If He wanted His children to get through to Him, why stand in the way instead of showing the way? SOPHIE: Unless... He was showing them the way by standing in the way. It’s called Reverse Psychology! He’s egging them on to think outside the box or their eggshells – to break out of their eggshells, as it were. HEIDI: And He raised the ante by dangling over their heads the spectre of Stillbirth – Being Six Feet Under! SOPHIE: Irony is all over the place! Geometrically speaking, aside from inscribing irony in the Garden, YHWH seems to have circumscribed the Garden with irony. It’s as if He’s asking them not merely to describe the Garden as they see it but to define the Garden as they see fit – “C’mon folks, dare to think! What is Eden for – Spoon-Feeding or Self-Learning?” HEIDI: What’s the Classroom for – Indoctrination or Education? 4 SOPHIE: What’s the Garden for – Sloth or Growth? HEIDI: What is Life about – Complacency or Audacity? SOPHIE: The irony is – YHWH made the prohibition sound like He was averse to risk-taking, when He was actually inviting the man and the woman to go beyond their comfort zones and acknowledge life’s ironies – the good and the bad. Call YHWH the Supreme Ironist! Remember how this Pioneer of Remote Learning, upon hearing of the coup in the Garden, enters His Classroom shooting a barrage of questions instead of parroting biblical verses. HEIDI: As if clueless and lost in the woods, He’s asking Questions – “Where are you? What is this that you have done? Have you eaten of the fruit that I forbade you to eat?”. YHWH’s picking their brains like Socrates quizzing Euthphro on piety, and Theaetetus on knowledge. Quite unlike the way He’s usually portrayed as a Know-It-All with Answers up His sleeve. SOPHIE: He seems to be paying tribute to the Grand Inquirer Eve. You know Uncle once wondered if the Padilla Question ever crossed Eve’s mind when she finally got the chance for the face to face modality with the Pioneer of Remote Learning. HEIDI: I’ve never heard of the Padilla Question before. SOPHIE: Well, Uncle self-effacingly calls it the God Question. HEIDI: You mean – the Big Question of Theodicy – how do we know that God exists? SOPHIE: No, that’s too medieval, and too metaphysical a question for Uncle’s taste. In fact, Uncle’s Question cedes precious ground to theists. He’s willing to give them the benefit of the doubt. HEIDI: So your Uncle grants, for the sake of argument, the basic assumption of theism – God exists. Why then is it called the God Question, if God’s existence is no longer the problem? SOPHIE: Well, Uncle wondered – Even assuming that God exists, how would humans know that it’s God who’s revealing Himself to them? For all you know, this could be an impostor! HEIDI: Well, the God of the Bible has a long chain of witnesses of His personal visitations in history – Adam and Eve, Noah, Abraham, Isaac, Jacob, Joseph, Moses, Gideon, Samuel, David, Solomon, Ezekiel, Zechariah, Jonah, Haggai, Elijah, Nathan, Isaiah, Jeremiah. I could go on and on Sophie. SOPHIE: Heidi, that’s what the Bible says. HEIDI: But it’s the Word of God! Therefore, it must be true! SOPHIE: And how do you know it’s the Word of God? HEIDI: Because the Bible says so! SOPHIE: Gotcha! – a glaring example of the fallacy of petitio principii – if ever there was one! That’s a circular form of reasoning! Extraordinary claims require extraordinary evidence. In the case of theophanies, I expect more from you Heidi than scriptural say-so. As I was saying, Uncle mused a lot on the God Question. This is exactly how he put it: “What if someone suddenly appears in the middle of the room at this very moment and claims to be God Himself. How do you know that it is God Himself and not a mere impostor?” 5 HEIDI: Well as you mentioned a while ago – extraordinary claims require extraordinary evidence. I’d ask Him for more evidence than His say-so. I’ll definitely be asking Him questions the answers to which I believe only God knows! SOPHIE: And what questions will you be asking this divine claimant? HEIDI: Existential questions like – What’s on my mind right now and why do I have these thoughts? I believe it’s only God who can divine my private thoughts at the moment. SOPHIE: What if He answers it right, does it necessarily mean He’s God. The only valid conclusion you can draw is that He can read what’s on your mind right now – that’s all. It doesn’t mean He knows everything in your mind. For all you know He’s equipped with a highly sophisticated technology that can read the brainwaves you’re currently emitting. HEIDI: Then I will ask Him what I had in my mind last night. For sure, those past brainwaves could not be recovered by Him at this very moment especially if I can’t remember those thoughts right now. But if He can remind me of those thoughts, how could He be other than God? SOPHIE: He could be an alien from a far advanced civilization that could read brainwaves of a human person at any time, but that does not prove He’s God. For Him to plumb what’s on your mind right now or last night does not sufficiently establish His omniscience – that He’s God! HEIDI: Then I’ll ask Him a series of questions whose answers are verifiable in one way or another by me. If He can rightly answer them all without batting an eyelash He must be God! SOPHIE: But if He could rightly answer all the questions you’ve asked so far – that’s still infinitely far from proving that He knows all the answers to all the questions that you can ask in the near or distant future. HEIDI: You’ve got a point there Sophie. Maybe, He just got lucky the nth time. SOPHIE: But He’s not an inch closer to being omniscient. If He can answer correctly the n questions that you have raised, it doesn’t necessarily follow that He can answer correctly the n + 1, n + 2, and n + 3 questions that you will be raising later. Indeed n + 1, n + 2, and n + 3 are still a long, long way from ∞. HEIDI: Assuming that He seems to know everything knowable, we still cannot discount the possibility that He’s just the contrivance of an omniscient creator. He seems to be God but is actually only His alter ego. He’s not God although He almost looks like God. Sophie, pursuing your Uncle’s line of reasoning, it dawned upon me – How can anyone ever jump to the conclusion that this divine claimant is absolutely omniscient? SOPHIE: You’ve caught Uncle’s drift after all! Indeed, how can anyone ever know that He knows not only googolplex things... but... all things that were known, all things that are known, all things that will be known, and all things that can ever be known – in a million years, in a billion years, in a trillion years... here, there, light-years away, and everywhere in the universe or multiverse? HEIDI: One has to be simultaneously located in all of space and all of time, and be able to simultaneously explore all of space and all of time, to arrive at the verdict that His knowledge encompasses practically everything in space and time. SOPHIE: It’s like saying one has to be God to know that He’s God. 6 HEIDI: I’m getting your Uncle’s drift, ain’t I? What if instead of ascertaining His omniscience which seems to be an impossible task, I would take issue with His omnipotence? SOPHIE: Whatever suits you Heidi. The ball is in your hands. How do you go about checking on His omnipotence? HEIDI: Well, I’ll ask Him to perform miracles that only God can do? SOPHIE: What kind of divine feat might that be? HEIDI: How about if I ask Him to bring back Lolo and Lola to life? SOPHIE: Suppose He does that to your satisfaction, does it prove His divine credentials? What if unbeknownst to you, He relied on a cloning mechanism which is a standard procedure in His galaxy? Future humans will be cloning their kind in the not too distant future, and they will be playing God. Some will be eliminating disease; some will be creating diseases; some will be deciding who will be born, who will be aborted, who will live, and who will die; some will be keeping the earth to themselves, others will be migrating to other planets; some will be resurrecting the past, others will be travelling into the future, or what have you, but does that make them God? HEIDI: Science and technology seem to be closing in on, but are still far from cloning God’s omniscience and omnipotence. What if I ask Him to create a new universe? Only God can create a universe, right? SOPHIE: How do you know that only God can create a universe? Where did you get that Heidi? Did you have a private audience with God Himself? Are you sure He’s the God Almighty? Can’t God, if indeed He’s God, possibly delegate the creation of a universe to somebody else? Wouldn’t He be more powerful than the God who could not delegate His creative powers? HEIDI: Where does your Uncle get all these questions Sophie? SOPHIE: I guess from his irrepressible sense of wonder. HEIDI: But should there not be an end to all this questioning? SOPHIE: All this questioning matters as long as it is part of a Quest. Remember, we’ve agreed that Eve asked questions, not out of sheer curiosity, but out of wonder? She ate the fruit because... ? What does the text say? HEIDI: Ah, here it is – “the woman saw that the tree was good for eating and that it was an attraction to the eyes, and the tree was desirable to bring about understanding” – I repeat – “the tree was desirable to bring about understanding.” SOPHIE: The JPS (Jewish Publication Society) Translation of the Tanakh renders the last line as “the tree was desirable as a source of wisdom”; the New American Bible renders it as “desirable for gaining wisdom”; and the New Jerusalem Bible as “enticing for the wisdom that it could give.” HEIDI: So Eve’s wonder was a wisdom-seeking wonder! SOPHIE: And the rest is... Enlightenment... and the history of Enlightenment! HEIDI: “... the eyes of the two of them were opened, and they knew that they were naked.” 7 SOPHIE: YHWH dared the man and the woman to break out of their eggshells, and it was the woman who first made it into the Light. Eve found it uncanny to live in a small corner, insufferable to remain in the dark even if it’s comforting as a cocoon. She had to issue her own Fiat Lux! in order to create a brave new world. She decided to step out of her comfort zone in order to get into a wide clearing and experience the living daylights of Being. Talk about ‘Ecstasy’, or ‘the Lightness of Being’. We must congratulate Eve for being the eve of a new day, the eighth day of the week. HEIDI: Notice that if we re-view the punishments meted out to the two risk-takers in the garden we are actually met by a dose of metaphors associated with creativity – pains of childbearing, desire for the object of one’s love, eking out a living, and even ‘Coming Home to Mother Earth’ to substitute a ‘live metaphor’ for a dusty one. SOPHIE: The journey of Be(YOU)ing is not a walk in the park. And as if to circumscribe the shock wave of realization that has come upon the adventurists, YHWH makes garments of skin to clothe them – a poignant act (a creative act as well) presaging the sartorial gesture of the father as he makes his prodigal son feel at home once again. YHWH sees Himself in their adventurism. He sees Himself in them. YHWH finally comes to see, to know Himself. He discovers Himself in humanity. And it all began as Eve chose to be the Light in the darkest of nights. We can portray this as the divinity’s journey from emptiness to wholeness. On the other hand, it can also be said that humanity discovers itself in the divine character. Or what amounts to the same thing, humanity discovers YHWH in the fruits of human creativity, in literary fiction for instance. This is humanity’s own journey from emptiness to wholeness. The future is in the fruit. The fruit is the future. I love the world of the word. It is language which makes us see the world differently. Can we now venture into another world – the world of my beloved Dan Brown? HEIDI: Sophie I’m just wondering what got you hooked on Dan Brown. SOPHIE: Like most Brownians, I came to know Dan through his Jesus/Mary Magdalene/Priory of Sion/Opus Dei/Da Vinci/Fibonacci/Robert Langdon/Sophie Neveu confectionery which inspired a movie featuring Tom Hanks and Audrey Tautou. Suspending disbelief as I do when I read fiction, I found myself up all night intrigued, enchanted, and riveted by a brilliantly conceived 24 hour action-packed odyssey of a symbologist and a cryptographer on a breathtaking quest that turns them from hunter to hunted as the truths they seek to unveil could undermine the foundations and change the course of Western civilization. I confess I could not help but empathize with one of the major characters who aside from being my namesake is caught in a drama whose resolution arguably entails the fundamental revision of truths about herself, her family, and her church. HEIDI: But don’t you think that ‘The Da Vinci Code’ despite its pretensions for accuracy is an extravaganza of debunked myths, discredited sources, despicable occultism, preposterous grand conspiracies, sensationalistic millenarian showdowns, and invidious propaganda? SOPHIE: Heidi I’m disinclined to dwell on matters I honestly feel incompetent to arrive at a final resolution. So don’t get me started on that slippery slope. But this much I can say in favour of the novel. The moment I saw a rabbit in the narrative I followed it through Wonderland. There was no turning back. The neurons in my brain got fired up and I was like a forest on fire. I felt like a burning bush. To tell the truth, I was never the same again. HEIDI: What rabbit are you talking about? Correct me if I’m wrong but I can’t remember having encountered any rabbits in DVC. 8 SOPHIE: I was speaking metaphorically of course. You were asking a while ago what got me hooked on Dan Brown? In the case of DVC, the cameo appearance of the Fibonacci sequence in the final communication of the murdered Louvre curator Jacques Sauniere in Chapter 8 served as my rabbit: “The message read: 13 – 3 – 2 – 21 – 1 – 1 – 8 – 5 O, Draconian devil! Oh, lame saint!” If we rearrange the numbers, we obtain the original Fibonacci sequence: 1 – 1 – 2 – 3 – 5 – 8 – 13 – 21. HEIDI: Why is it called the Fibonacci sequence? SOPHIE: The cryptographer Sophie explains to Captain Fache in Chapter 11 that it is a progression created by the thirteenth century mathematician Leonardo Fibonacci in which each number is the sum of the previous two. Notice that the third term, 2, is the sum of the first and second terms (1 + 1), the fourth term, 3, is the sum of the second and third terms (1 + 2), the fifth term, 5, is the sum of the third and fourth terms (2 + 3), and so on. It is interesting to note, although Sophie doesn’t mention it, that the sequence originally arose from a hypothetical problem involving rabbits increasing in number as time goes by. HEIDI: Are you then saying that the sequence can go on forever, that it doesn’t stop at 21? SOPHIE: Hypothetically yes. The succeeding terms to mention just a few are: 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584,.... It is possible in the world of numbers but not quite in the real world. Mathematics is a language with pragmatic provenance but it can rise above its humble beginnings. It is a being-in-the-world but its possibilities are endless. In fact there is a way of restating the Fibonacci sequence mathematically. It is crisp and elegant. It goes beyond enumerating succeeding terms. HEIDI: How then do you express it in math lingo? SOPHIE: 𝐹1 = 1, 𝐹2 = 1, 𝐹𝑛 = 𝐹𝑛−1 + 𝐹𝑛−2 HEIDI: And in English? SOPHIE: The first number in the sequence is 1, the second number in the sequence is also 1, and each succeeding number in the sequence is the sum of the two previous numbers in the sequence. Now 𝐹𝑛 = 𝐹𝑛−1 + 𝐹𝑛−2 can stand for 𝐹3 = 𝐹2 + 𝐹1 , 𝐹4 = 𝐹3 + 𝐹2 , 𝐹5 = 𝐹4 + 𝐹3 , and so on and on and on. It can stand for 𝐹1000 = 𝐹999 + 𝐹998 too. HEIDI: Wow your language conceals and unconceals a lot! SOPHIE: Now here comes the interesting part. I wanted to know whether there is a royal path, a short- cut, to knowing let’s say the 50th term in the sequence without going through all the previous sums of pairs in the sequence, that is to say, without eventually having to add the 48th and 49th terms. That is what we call in our world a problem which requires a solution. We can even generalize the problem to determining any term in the sequence without going through all the previous sums of pairs in the sequence. 9 HEIDI: In philosophy we call it a question which requires an answer. Now Heidegger intimates in ‘Being and Time’ that in asking a question, one asks for something (das Erfragte) about something (das Gefragte) from someone (das Befragte).4 SOPHIE: So if we fuse the two languages, 𝐹𝑛 is the Gefragte, the short-cut is the Erfragte, and the mathematical mind is the Befragte. Fortunately, I didn’t have to work out the answer myself. I was able to locate a book5 which provides the short-cut in the form of a formula. I might as well give it to you without much ado. ∅1 𝑛 − ∅2 𝑛 𝐹𝑛 = √5 HEIDI: Good Heavens, what are those aliens over the horizon? SOPHIE: Have no fear of the unknown this time. They’re just the powers of two known, though little known, numbers. In fact, although the symbol ∅1 does not appear in the novel, it is referred to by name and given an approximate value. In Chapter 20 Robert Langdon refers to PHI with the approximate value of 1.618 as the most beautiful number in the universe and Sophie Neveu refers to it as the Divine Proportion. HEIDI: Now I remember PHI and 1.618. Therefore I know ∅1. How about ∅2 ? SOPHIE: Well, ∅2 is just the negative reciprocal of ∅1 , and ∅1 , ∅2 are just the two solutions to the equation ∅2 − ∅ − 1 = 0. Do you still remember the Quadratic Formula from algebra? HEIDI: Of course! SOPHIE: Would you like to solve ∅2 − ∅ − 1 = 0? HEIDI: If I may? (Heidi plugs in the coefficients in the Formula and arrives at the two solutions.) Now check what I found: 1 + √5 1 − √5 ∅1 = , ∅2 = 2 2 SOPHIE: Bravo! Voila! You have the exact value of PHI right there! HEIDI: It is true that to know is to know how! Only one who knows how to use a language knows the language. SOPHIE: In mathematics, the medium is the message. The entanglement between the symbols and the objects they represent is there for all to see. In mathematics the train of thought is revealed in the written computation. It is in mathematics that the being of language is the language of being. Nowhere else than in mathematics does the Heideggerian adage “Language is the House of Being” acquire the ring of gospel truth. In a certain sense mathematical objects are cultural products. It would be hard to 4 Translated by John Macquarrie & Edward Robinson (New York: Harper & Row, 1962), p. 24. 5 ‘Excursions in Modern Mathematics’ by Peter Tannenbaum (New Jersey: Pearson Education , Inc., 2004), p. 361. 10 pin down ∅1 and ∅2 within the context of the Roman Numeral System. We have to be thankful to the Hindus and the Arabs for the facility with which we can navigate the labyrinth of rationals and irrationals. Indeed, the importance of the symbols used in the practice of one’s trade cannot be undervalued. Newton is no doubt a physicist without an equal during his time and although he and Leibniz are both credited for having invented differential and integral calculus I would have no second thoughts of choosing Leibniz over Newton to be my tutor in calculus had I lived in their time for the sheer advantage and power of Leibnizian over Newtonian notation. Mathematical objects are both beings-in-the-world and beings-in-time! HEIDI: You say that mathematical objects are subject to the vicissitudes of time. Are you saying then that luck sometimes enters into their constitution? SOPHIE: I think we’re on the same wavelength. I was about to say something about the role of luck in mathematical discovery. I’m glad you asked about it. You know, not long after I had my first encounter with the Fibonacci sequence in DVC, I tried to come up with my own sequence. But I stumbled upon it by a circuitous route. One day I was tinkering, better thinkering with a series of sums involving powers of ∅1 and ∅2. Here’s a portion of what I obtained. 1 + √5 1 1 − √5 1 ∅11 + ∅21 = ( ) +( ) =1 2 2 1 + √5 2 1 − √5 2 ∅1 2 + ∅2 2 = ( ) +( ) =3 2 2 1 + √5 3 1 − √5 3 ∅1 3 + ∅2 3 = ( ) +( ) =4 2 2 1 + √5 4 1 − √5 4 ∅1 4 + ∅2 4 = ( ) +( ) =7 2 2 1 + √5 5 1 − √5 5 ∅1 5 + ∅2 5 = ( ) +( ) = 11 2 2 1 + √5 6 1 − √5 6 ∅1 6 + ∅2 6 = ( ) +( ) = 18 2 2 1 + √5 7 1 − √5 7 ∅1 7 + ∅2 7 = ( ) +( ) = 29 2 2 1 + √5 8 1 − √5 8 ∅1 8 + ∅2 8 = ( ) +( ) = 47 2 2 1 + √5 9 1 − √5 9 ∅1 9 + ∅2 9 = ( ) +( ) = 76 2 2 1 + √5 10 1 − √5 10 ∅110 + ∅210 = ( ) +( ) = 123 2 2 11 At some point I told myself I need not go further for I had created my own sequence in the process. My sequence has 1 as the first term, 3 as the second term, and every succeeding term is the sum of the two previous terms. And I have a short-cut too for finding any term in the sequence without going through all the previous sums in the sequence. Look how elegant it appears in a formula: 𝑓1 = 1, 𝑓2 = 3, 𝑓𝑛 = 𝑓𝑛−1 + 𝑓𝑛−2 , 𝑓𝑛 = ∅1 𝑛 + ∅2 𝑛 HEIDI: You must have been ecstatic when you detected a pattern and arrived at the formula. SOPHIE: You can say I was beside myself when the Eureka moment came. Now I know how Mother must have felt when she found enlightenment under the tree in the middle of Eden. But the pursuit of the trail of the rabbit doesn’t end after a single triumph. When you look back you can even invent symmetries in formulae as when I came up with the following: ∅1 𝑛 + ∅2 𝑛 ∅1 𝑛 − ∅2 𝑛 𝑓𝑛 = 𝐹𝑛 = ∅1 + ∅2 ∅1 − ∅2 where the formula on the left gives the nth term of my own sequence (𝑓1 = 1, 𝑓2 = 3) and the formula on the right gives the nth term of the original Fibonacci sequence (𝐹1 = 1, 𝐹2 = 1). HEIDI: How sublime they appear to the eye. The formulae even look like soul-mates. But I suppose your armory of formulae keeps on building up. It’s like a living language that keeps on growing. SOPHIE: That’s very true. It was just a matter of time and logic that I was able to conjure formulae for the powers of ∅1 and ∅2 : 𝑓𝑛 + 𝐹𝑛 √5 ∅1 𝑛 = 2 𝑓𝑛 − 𝐹𝑛 √5 ∅2 𝑛 = 2 HEIDI: Wow! Who would have anticipated that within the powers of ∅1 and ∅2 are a hidden combination of the sequences of yours and Fibonacci’s? SOPHIE: Now there is a statement in DVC which got me wondering for a very long time. I said to myself this could be the hardest nut to crack in the DVC armory for it presupposes the concept of limits – a concept which properly belongs to calculus. Here is the nut: “... the number PHI was derived from the Fibonacci sequence – a progression famous not only because the sum of adjacent terms equalled the next term, but because the quotients of adjacent terms possessed the astonishing property of approaching the number 1.618 – PHI.” HEIDI: In a nutshell, the limit of the quotients of adjacent terms of a Fibonacci sequence is ∅1. SOPHIE: Exactly! Let me give you a limited litany of such quotients approaching 1.618 for both my sequence and Leonardo’s. 12  3 4 7 11 18 29 47 𝑓 Sequence: = 3.0000, = 1.3333, = 1.7500, = 1.5714, = 1.6363, = 1.6111, = 1 3 4 7 11 18 29 76 123 199 322 521 843 1.6206, 47 = 1.6170, 76 = 1.6184, 123 = 1.6178, 199 = 1.6180, 322 = 1.6180, 521 = 1.6180 1 2 3 5 8 13 21 𝐹 Sequence: 1 = 1.0000, 1 = 2.0000, 2 = 1.5000, 3 = 1.6666, 5 = 1.6000, 8 = 1.6250, 13 = 34 55 89 144 233 377 1.6153, 21 = 1.6190, 34 = 1.6176, 55 = 1.6181, 89 = 1.6179, 144 = 1.6180, 233 = 1.6180 But these are only two among an infinite number of Fibonacci sequences. The statement in DVC, on the other hand, applies to all Fibonacci sequences. I was asking myself how I can show that the limit of a Fibonacci sequence is ∅1 even if the first two terms of the sequence are not 1 and 3 (my sequence) nor 1 and 1 (Leonardo’s sequence). HEIDI: So you needed the proper Gefragte, Erfragte, and Befragte. SOPHIE: I needed to show that 𝑎𝑛 lim = ∅1 𝑛→∞ 𝑎𝑛−1 where 𝑎𝑛 and 𝑎𝑛−1 are adjacent terms of any Fibonacci sequence having as initial terms integers 𝑎1 and 𝑎2. The Gefragte is on the left while the Erfragte is on the right side of the equation. And the Befragte could be none other than myself – the inquirer in the instant case. Needless to say, I had to deal first with the challenge of creating formulae that give the nth and the (n – 1)th terms of any Fibonacci sequence whatsoever. As in any intricate math problem, I had to come up with a pilot test of a simpler problem, in this case – a general but controlled Fibonacci sequence. So I asked myself what if I stipulate 𝑎1 = 1 and 𝑎2 = 𝑘. HEIDI: And I suppose you were able to come up with no less than a general formula for the nth term of a sequence whose first term is 1 and whose second term is any integer? SOPHIE: If I were just curious about the problem I don’t think I would be able to dwell on it for a long time, much less til its final resolution. It was my resoluteness that ultimately paid off. I never allowed my attention to be diverted from the general contours of the prize though for the moment the prize lacked clarity. The disposition of wonder had to transcend the vicissitudes of the moment. I had to behold the future like a fruit in my mind. I had to guard it like the apple of my eye. After a lot of false starts and missteps I was able to pluck this priceless gem from the depths of my very own being: 𝑘− 3 𝑎𝑛 = ∅1 𝑛 + ∅2 𝑛 + (∅1 𝑛−1 − ∅2 𝑛−1 ) √5 = 𝑓𝑛 + (𝑘 − 3)𝐹𝑛−1 As you can see, ∅1 and ∅2 are lurking within Fibonacci sequences where 𝑎1 = 1 and 𝑎2 = 𝑘. 13 HEIDI: Let me guess. You were able to draw from the depths of your very own being the general formula for the nth term of any Fibonacci sequence whose first two terms are integers. And you were able to locate once again ∅1 and ∅2 in this new priceless gem of yours. SOPHIE: I recall that day. Oh girl, I felt transported to the seventh heaven. It was like navigating through the darkness of a labyrinthine underground river for an eternity and all of a sudden catching a breath of fresh air and a glimpse of the mouth of the cave and the rich kaleidoscope of the great blue yonder. It was like seeing the heavens part before my very eyes and being embraced by the light. It was like being born all over again and seeing the world for the very first time. Or coming back to life and seeing the world in a different light. Let me now share the formulae6 for the nth and the (n – 1)th terms of any Fibonacci sequence. Here are they: 𝑎1 − 1 𝑎2 − 3 𝑎𝑛 = ∅1 𝑛 + ∅2 𝑛 + (∅1 𝑛−2 − ∅2 𝑛−2 ) + (∅1 𝑛−1 − ∅2 𝑛−1 ) √5 √5 = 𝑓𝑛 + (𝑎1 − 1)𝐹𝑛−2 + (𝑎2 − 3)𝐹𝑛−1 𝑎1 − 1 𝑎2 − 3 𝑎𝑛−1 = ∅1 𝑛−1 + ∅2 𝑛−1 + (∅1 𝑛−3 − ∅2 𝑛−3 ) + (∅1 𝑛−2 − ∅2 𝑛−2 ) √5 √5 = 𝑓𝑛−1 + (𝑎1 − 1)𝐹𝑛−3 + (𝑎2 − 3)𝐹𝑛−2 6 After finishing the paper I was able to arrive at simpler formulae. I viewed the Fibonacci sequence in the following manner: 𝑎1 = 1𝑎1 𝑎2 = 1𝑎2 𝐹1 = 1 𝑎3 = 1𝑎1 + 1𝑎2 𝐹2 = 1 𝑎3 = 𝐹1 𝑎1 + 𝐹2 𝑎2 𝑎4 = 1𝑎1 + 2𝑎2 𝐹3 = 2 𝑎4 = 𝐹2 𝑎1 + 𝐹3 𝑎2 𝑎5 = 2𝑎1 + 3𝑎2 𝐹4 = 3 𝑎5 = 𝐹3 𝑎1 + 𝐹4 𝑎2 𝑎6 = 3𝑎1 + 5𝑎2 𝐹5 = 5 𝑎6 = 𝐹4 𝑎1 + 𝐹5 𝑎2 𝑎7 = 5𝑎1 + 8𝑎2 𝐹6 = 8 𝑎7 = 𝐹5 𝑎1 + 𝐹6 𝑎2 𝑎8 = 8𝑎1 + 13𝑎2 𝐹7 = 13 𝑎8 = 𝐹6 𝑎1 + 𝐹7 𝑎2 𝑎9 = 13𝑎1 + 21𝑎2 𝐹8 = 21 𝑎9 = 𝐹7 𝑎1 + 𝐹8 𝑎2 𝑎10 = 21𝑎1 + 34𝑎2 𝐹9 = 34 𝑎10 = 𝐹8 𝑎1 + 𝐹9 𝑎2 As you will notice, all coefficients of both initial terms 𝑎1 and 𝑎2 are none other than the original Fibonacci numbers. Moreover, in every term from 𝑎3 onwards each initial term’s coefficient is simply the sum of its coefficients in the two previous terms. This boils down to 𝑎𝑛 = 𝑎1 𝐹𝑛−2 + 𝑎2 𝐹𝑛−1 and 𝑎𝑛−1 = 𝑎1 𝐹𝑛−3 + 𝑎2 𝐹𝑛−2. ∅1 𝑛 − ∅2 𝑛 𝑎1 𝑎2 𝑎1 If 𝐹𝑛 = , then 𝑎𝑛 = (∅1 𝑛−2 − ∅2 𝑛−2 ) + (∅1 𝑛−1 − ∅2 𝑛−1 ) and 𝑎𝑛−1 = (∅1 𝑛−3 − ∅2 𝑛−3 ) + √5 √5 √5 √5 𝑎2 (∅1 𝑛−2 − ∅2 𝑛−2 ). So there you have the simpler formulae for the nth and (n – 1)th terms of any Fibonacci √5 sequence under the sun. 14 HEIDI: They’re sublime. I’d like to note that the first and second terms of your own sequence, that is, 1 and 3, are subtracted from the first and second terms, that is, 𝑎1 and 𝑎2 , of the new Fibonacci sequence. I suppose you already have all the ingredients you need to demonstrate that the limit of the quotients of the adjacent terms of any Fibonacci sequence is PHI. SOPHIE: This project was my Mount Everest and I needed the tools of calculus to scale it. The physicist Richard Feynman once said that calculus is the language of God. So I needed that language to scale my Mount Everest. Well, the calculus part was easy once I was able to create the preceding formulae. So here finally is my labor of love! Here are the fruits of my labor! Here I am! 𝑎𝑛 lim 𝑛→∞ 𝑎𝑛−1 𝑎1 − 1 𝑎 − 3 ∅1 𝑛 + ∅2 𝑛 + (∅1 𝑛−2 − ∅2 𝑛−2 ) + 2 (∅1 𝑛−1 − ∅2 𝑛−1 ) = lim √5 √5 𝑛→∞ 𝑎 − 1 𝑎 − 3 ∅1 𝑛−1 + ∅2 𝑛−1 + 1 (∅1 𝑛−3 − ∅2 𝑛−3 ) + 2 (∅1 𝑛−2 − ∅2 𝑛−2 ) √5 √5 𝑎1 − 1 𝑎 − 3 𝑛−1 𝑎 − 1 𝑎 − 3 𝑛−1 lim (∅1 𝑛 + ∅1 𝑛−2 + 2 ∅1 ) + lim (∅2 𝑛 − 1 ∅2 𝑛−2 − 2 ∅2 ) = 𝑛→∞ √5 √5 𝑛→∞ √5 √5 𝑎 − 1 𝑎 − 3 𝑛−2 𝑎 − 1 𝑎 − 3 𝑛−2 lim (∅1 𝑛−1 + 1 ∅1 𝑛−3 + 2 ∅1 ) + lim (∅2 𝑛−1 − 1 ∅2 𝑛−3 − 2 ∅2 ) 𝑛→∞ √5 √5 𝑛→∞ √5 √5 𝑎1 − 1 𝑎 − 3 𝑛−1 ∅1 𝑛 + ∅1 𝑛−2 + 2 ∅1 = lim √5 √5 𝑛→∞ 𝑎 − 1 𝑛−3 𝑎2 − 3 ∅1 𝑛−1 + 1 ∅1 + ∅1 𝑛−2 √5 √5 𝑎1 − 1 𝑎 − 3 𝑛−2 ∅1 (∅1 𝑛−1 + ∅1 𝑛−3 + 2 ∅1 ) = lim √5 √5 𝑛→∞ 𝑎 − 1 𝑛−3 𝑎2 − 3 ∅1 𝑛−1 + 1 ∅1 + ∅1 𝑛−2 √5 √5 = lim ∅1 𝑛→∞ = ∅1 In a nutshell: 𝑎𝑛 lim = ∅1 𝑛→∞ 𝑎𝑛−1 HEIDI: Thank you So-PHI-e for bringing me to the wuthering heights of Wonderland, for pushing the possibilities of the word to the limit, and for expanding the borders of my world. You brought me out of my shell and I shall never be the same again. You indeed remind me of Mother Eve. 15 SOPHIE: Well, I’d love to see her tribe increase coz there’s so much in our midst to wonder about. But Wonderland is within the wonderer! So guard the language of your heart, and your tongue, like the pupil of your eye! “A new species of philosophers is coming up: I venture to baptize them with a name that is not free of danger. As I unriddle them, insofar as they allow themselves to be unriddled – for it belongs to their nature to want to remain riddles at some point – these philosophers of the future may have a right – it might also be a wrong – to be called attempters. This name itself is in the end a mere attempt and, if you will, a temptation.” FRIEDRICH NIETZSCHE Beyond Good and Evil 16