stuff I don't understand: the story of e

Pi (π) and e are numbers. Strange and ethereal numbers that cannot be quantified. Because the phrase "be quantified" denotes the perfect tense, which implies a perfected action. Done. Complete. Finito.

Well, actually, finito means finite. But it still captures the anomalous nature of the situation because these numbers (π and e), though less than 4 and 3 respectively, cannot be said ever to end. It makes one wonder how anyone ever computed π, since it's defined (in words) as the ratio of a circle's circumference to its diameter. A ratio is by definition rational, is it not? Someone measured the length of a circle's circumference and its diameter and divided the first by the second. What's irrational about that? A rational number, after all, is defined as one that can be expressed as a ratio of two integers. Yet π is nothing if not irrational.

But this blog isn't about π. That's not the book I read. This blog is about e.

Let's say you have a curve that if the value of the function that defines it is checked at any given point of x (that is, what's y equal to at any given x?), you find out that that value is the same as the slope of the curve at that point. And what if this is true no matter what point you pick on the curve? This represents an exponential function--but not just any exponential function. It's the exponential function. The one that's related to e. The one we'll have to come back to shortly.

Now, we all know that functions are usually expressed in terms of x. Such as x plus two (x+2) or x squared (x²) or x cubed (x³). Usually if you work out these formulas by putting in a numerical value for x, you'll get a numerical answer back. Since this answer changes for every value of x, we usually just call it y till we know what it is for sure. Example: x+2 = y. But exponential functions are different. They have the variable (the x-value) in the exponential position. Like this: three to the x (3^x) or four to the x (4^x) or anything to the x (a^x).

I wrote above about the slope of a curve at any given point (2 paragraphs up). It might seem strange to think of a curve having a slope at just one point, because the direction of the curve is always changing. (Actually, mathematicians usually call even straight lines curves, but let's think of the really curvaceous ones right now). Think of the slope as the general direction that the curve happens to be taking at that point. What if it stopped turning right then? Which way would it be headed? It's something like that. And it turns out that if you know a formula for a given curve, there's a way use it to find a new formula that will give you the slope at any time. (Remember, this e curve I described above is special. No other kind of curve would ever have given us the value of the slope at a point simply by finding out y. In fact, for a long time, no one even knew it existed.)

But exponential functions in general are kind of special too. When you follow the method for finding the slope-at-any-given-point (which math people call the derivative), you just end up with the function you began with multiplied by some constant. The value of that constant would depend on the function we started with in the first place. So someone brilliant thought, what if we could find that number that when it's at the base of the exponential function (as 3 is at the base of this exponential function: 3^x) the constant that comes back to multiply it in the derivative is 1? What if we could find that number? That would mean that we found a function whose derivative (slope-at-any-given-point) is itself (since anything times 1 is itself). This is a HUGE deal, because that would make it so much easier to work out a bunch of high-level mathematical things that I don't even know about yet.

It turns out that number does exist somewhere too. And since the constant that comes back to multiply 2^x is about 0.693 and the one that comes back with 3^x is about 1.099, then they knew the number they sought had to be between 2 and 3. Trying a whole bunch of different numbers in between brought back the answer 2.7182818284 (continuing forever), which they shortened for mortality sake to "e". So e^x is our special function whose derivative is e^x.

This was my first introduction to the number e.

a pertinent idea

The idea expressed below demonstrates the spirit of life which, like Henry James, does routinely include tragedy and doesn't offer happy endings "at any cost." Interestingly, this passage was clipped from a blog entitled The Art of Fiction by Mauricio Salvador:

"Como diría el narrador de Muerte en la tarde, de Hemingway: 'Señora, todas las historias, si continúan lo suficiente, terminan en la muerte, y no es un auténtico narrador de historias quien se lo oculta.' Y Philip Roth, sin duda, es un auténtico narrador de historias. "

A crude translation: "Like the narrator of Hemingway's Death in the afternoon said: 'Madam, all stories, if they go on long enough, end in death, and he who tries to hide that is no true storyteller.' And Philip Roth, without doubt, is a true storyteller."

I've left only to add: Y Henry James, sin duda, es un auténtico narrador de historias.

The Art of Fiction, pt. 2

My goal with this blog is to revel in the literature rather than criticise to death. It's not always easy, especially when the subject matter is itself critical—as in The Art of Fiction. I know I've already been guilty in several other posts, but I'm going to be especially careful from now on.
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James writes that the work of fiction is to catch "the very note and tick, the strange irregular rhythm of life." Yet these irregularities do often seem to be beneath the surface, because the plots of his novels rarely contain anything too far out of the ordinary, as far as I've seen. I think earlier we touched on the idea of "the happy ending at any cost." James avoids this—but life often does too, so his plots don't strike me as being too strange.

Here's a humorous selection of James' own thoughts on those who consider art in terms of rainbows:

For many people art means rose-coloured windowpanes, and selection means picking a bouquet for Mrs. Grundy. They will tell you glibly that artistic considerations have nothing to do with the disagreeable, with the ugly; they will rattle off shallow commonplaces about the province of art and the limits of art till you are moved to some wonder in return as to the province and limits of ignorance.

James a sample... (read: James is classic...)

To me, Henry James' charm is that he fills in the interstices between conversation tags. According to Annie Dillard, "there are no events but thoughts." Therefore, it is in the mind that any occurrence becomes an event. James echoes this himself in his preface to The Portrait of a Lady when he writes, "but what is truer than that on one side—the side of their independence of flood and field, of the moving accident, of battle and murder and sudden death—her adventures are to be mild? Without her sense of them, her sense for them, they are next to nothing at all; but isn't the beauty and the difficulty just in showing their mystic conversion [...] into the stuff of drama?" This "sense for them" is what transforms the incident into the event, and it is just this sense that James succeeds in illuminating.

I’ve certainly always considered Henry James to be the most dramatic of all writers. But when I’m able to get outside myself, I believe that I’m able to see how others might consider him boring. I’ve actually heard it expressed. As an undergraduate while in the dorm computer lab I once overheard someone say, “Jane Austen’s books are so-o boring.” And while I was in the middle of feeling as though a stake were being run through my heart, her companion replied, “Yeah. And have you read Henry James? He’s even worse!” Needless to say I needed oxygen by that point. I didn’t get up to note the faces of the heretics; hatred is unchristian. But after thinking about it—and at times even while reading James, I distinctly get the impression of how a person could come to such a perfidious conclusion. If it weren’t for my peculiar interest in a subject’s sense for his experiences—however mundane—I’d never be interested in the text. So while it is inaccurate to say nothing happens in the stories, the truth is that the events occur so slowly that it is no doubt an interest outside of the plot that propels me as reader.

he is holy; he is firm, pt. 4

Everything I see [...] looks brittle and unreal, a skin of colors painted on glass, which if you prodded it with a finger would powder and fall. A blank sky, perfectly blended with all other sky, has sealed over the crack in the world where the plane fell, and the air has hushed the matter up.

Clearly, something that had to do with eternity took place here where the plane fell. Something ripped time for a brief moment, a split second. To "fall from the dark" is, I suppose, to wake up. Our narrator fell from the obscurant light of the stars--visible light, which by itself is darkness--and she found the truer light of knowledge. But knowledge of what? Of good and evil? Knowledge of pain? Does pain split illusion like the plane split the sky? The millstones of time and space are illusory, yet the pain of their churning is "undeniably real." But if it's true that "only the good is real," what then does that say of pain?

But there's more here. She writes, "The pain [...] is real, for our love [...] is real," and in so doing almost equates pain with love. Yet we detect the even greater depth of this equation's significance in the subsequent revelation that those seraphs who are "aflame with love for God" are higher than those who have "perfect knowledge of him." The love they possess consumes them again and again. And here we have a culmination of the paradox we felt coming.

The flames that issue from the throne of God are themselves the conduit through which the seraphs are transported to their world. Presumably, the flames are also the material from which these angels are made, as well as the substance that destroys and returns them again to their source. All is one is God. For you, O Lord, are a sun and shield--the sun that scorches; the shield that guards. From him we have both everything and nothing to fear. Love and hate commingle. You know, we could contract ourselves to a point running to and from him at the same time.

It would take immeasurable faith to believe in a favourable view of things, because God seems so absent from this illusory present. His love is manifest in his abandonment of humans to the shackling laws of time and space. And the pain of it seems so much more real than the love--which we're told are one and we are confused. He treats us with pain as with love, and we seek to love him back in kind. So our love is a forgetting and, of necessity, a misinterpretation of everything in the universe. For now that creation, as predicate, has been slashed "loose at base from any roots in the real," it is effectively divided from its subject, and all is gibberish. We shuffle objects to make sense of it all.

Faith would be, in short, that God has any willful connection with time whatsover, and with us. For I know it as given that God is all good. And I take it also as given that whatever he touches has meaning, if only in his mysterious terms, the which I readily grant. The question is, then, whether God touches anything.

In creating the laws of the universe and in requiring that they be kept, does God also restrict his own power? Well... without messing with his omnipotence, she describes this restriction as "a contraction of the scope of his will." If that's the case, then the separation is complete, for wouldn't his of all wills be the most unbending? Even Christ's advent might be seen as succumbing to the laws of gravity, as he descends to his suffering, to his death. And he might have conquered the material experience on earth, but he can't have liked it any better than we, for he left again--abandoning us twice. Yet she (as I) searches for a trail he must have left, some significance in the ceremony of his returning at all.

Is there no link at the base of things, some kernel or air deep in the matrix of matter from which the universe furls like a ribbon twined into time?

Yes he might have saved our souls, but what of all this matter? Is there nothing that extends this redemption to the temporal, to our breakable bones and decaying teeth?

he is holy; he is firm, pt. 3

Didn't I fall from the dark of the stars to these senselit and noisesome days?

Our days are illuminated by our senses, for it is through them that we are connected to all that we experience. We see, feel, hear, taste, and smell our universe, and are therefore tethered to the notion that this is what's real. But "by what freak chance does the skin of illusion ever split, and reveal to us the real, which seems to know us by name?" And how does it know us, and why do we know that it does?

In The Writing Life Dillard briefly revisits this problem when she asks, “Why does death so take us by surprise?” The reality of being one day finished-- being "over"-- jolts us every time we look at it because there's something about us that simply expects to continue forever. We take immortality for granted, and are shocked every day to find out that it isn't granted. Even time's passage is enigmatic; we hardly even feel it. Of course, physics itself considers this arrow of time to be strange and perhaps illusory. In The Fabric of the Cosmos, Brian Greene writes this:

All the physical laws that we hold dear fully support what is known as time-reversal symmetry. This is the statement that if some sequence of events can unfold in one temporal order [...] then these events can also unfold in reverse. [...] Not only do known laws fail to tell us why we see events unfold in only one order, they also tell us that, in theory, events can unfold in reverse order (p. 145).

This passage isn't exactly referring to time travel, but in such a universe--this universe--the uncharring of Julie's face should be a definite possibility.

Still, what is this thing we call time and what are these its constraints? We feel we should be able to splash about in it as we do in space. Of course, space does have its own problems and it is spacetime itself that distracts us from what we intuitively feel to be real.

My niece at four inquired, awed, if I knew that everybody was going to die.

The Art of Fiction, pt. 1

No blog of mine could be complete (or even properly begun) without a discussion of Henry James. This will be an on-going project.

In The Art of Fiction, James takes an opportunity (afforded by the writer and critic Walter Besant) to speak on the subject to which he devoted most of his time: fiction writing. The introductory paragraphs demonstrate his belief that fiction written in the English language was at that time undergoing a change from the naïveté of such authors as Thackeray and Dickens to a more complex form. It appears that people began to require more of fiction. Prior to this shift in sentiment, James writes that "there was a comfortable, good-humoured feeling abroad that a novel is a novel, as a pudding is a pudding, and that our only business with it could be to swallow it."

He writes later: "That, I think, represents the manner in which the latent thought of many people who read novels as an exercise in skipping would explain itself if it were to become articulate." This is reminiscent of Ezra Pound's reference to readers of "habitually slack attention" who desire fiction that can easily be lapped up.

Yet James identifies the fact that no longer were people just "swallowing" novels, and this appetite for masticable prose apparently gave rise to such authors as James himself, whose fiction represents a psychological exploration. Other authors of this persuasion imbue their fiction with philosophy and thereby make the work "discutable."

James goes on to imply that this group of persons who demand more of their fiction might still represent a minority. He alludes to an idea that fiction was once even considered "wicked" in England--though wicked perhaps in the sense of being jocular and inconsequential. I believe that impression to be correct, because I remember reading such works by Jane Austen and perhaps the Brontës in which people would remark that they were only reading a novel. This activity seemed (chauvinistically) relegated to women, as if only a "feeble" mind would even consider engaging in such a mawkish activity as reading a novel.

Yet, he laments this pervasive idea that all novels should admit to being "only a joke." He also laments "certain established novelists [who] have a habit of giving themselves away which must often bring tears to the eyes of people who take their fiction seriously." He identifies Anthony Trollope who is often in the habit of reassuring his readers parenthetically that the events of his novel are only make believe. "He admits that the events he narrates have not really happened, and that he can give his narrative any turn the reader may like best." James considers this a "terrible crime." I do too.

In fact, it seems to me that any serious story does lead the writer as much as it takes its readers on a journey. According to my experiences, a writer is little more than a midwife who delivers each sentence as it is begot by its predecessor. A major difference is that the writer too feels the pangs of birth. Therefore, any writer who could be in such charge of the events of a story that he/she can make it take any turn desired has to be guilty of some kind of literary crime. Their stories usually end up lacking unity. No one could accuse Henry James of this.

He goes on to speak of the parallels between the novel and the painting, as both attempt to reflect life.

The artistic novelist is often spurned by the "public" who consider the goodness of a novel to lie in its ability to divert--perhaps through containing lots of action. Other such persons would denote such "goodness" as being dependent on a "'happy ending,' on a distribution at the last of prizes, pensions, husbands, wives, babies, millions, appended paragraphs, and cheerful remarks." lol...

He continues: "The 'ending' of a novel is, for many persons, like that of a good dinner, a course of dessert and ices, and the artist in fiction is regarded as a sort of meddlesome doctor who forbids agreeable aftertastes." So true. I am not strictly opposed to happy endings myself, but unity and artistic integrity dictate that the author go where the work takes him. Or her. That seems a bit more important and noble than writing-the-happy-ending-at-any-cost.

he is holy; he is firm, pt. 2

The corner where eternity clips time must be the point at which life has any meaning at all, and if she has touched it, then hasn't she somehow touched God?

The sailor who helped her draw a key to the islands spoke with authority like Christ. Yet she comes upon a new thing, a new island whose name she does not know. She labels it "Waiting for Sailor," as (for some reason) a name is necessary to a thing. Why? Again, it was Walker Percy who brought up this question in his essay Metaphor as Mistake. He spoke of the charm he found in some inadvertent misnomers of his childhood. Whenever he found some new spectacle, he had to know what it was called--something about not being able to truly appreciate the innerness of the thing without knowing its name. I don't know about all that; I think you could more or less still appreciate a thing without having a name for it. But, something still compels you to pin it down. I think it might be the fact that names are containers that allow you to hold (and express) all that you do know and feel about a thing. It aids communication. I admire those Ents in The Lord of the Rings whose "words" were as long as the entire history of the thing to which they referred.

In our text, the significance of the word itself is in the way it concretizes idea, as though it had been a word--the same word for which she awaits the sailor--that had called forth her islands from the sea.

Yet we suspect the idea must be realer than these diversions: stones, trees, sky. So she, as I, suspects fraud in all this material--"the world, a dream forced into my ear and sent round my body on ropes of hot blood." She tries looking past the sky to see the real, as we all must in some way and at some point. Because we know there's something behind the facade, and it galls that the illusion forces itself so convincingly upon the senses and disguises the real when we try to see it. The mind betrays; it's in on the fraud, and pretends that the sky really is the blue canopy that it appears to be. But, avoiding our inquisitive eyes, the real peeps out behind our backs. Like wave-particles, they resist the probe.

The earth, she writes, "fizzes up in trees." Yes, trees are a visible frenzy. Yet Julie Norwich's father must have overlooked the one that felled their plane. This god--let's call him November 19--was a cruel day. How can we wish for God when he stands back and watches us suffer--or worse, has a hand in our suffering? Who would want to have anything to do with a god like that? She calls him as she sees him: a "hounding and terrorist" god.

But God is good. We have to take it on faith, don't we?

Time itself can trick us into thinking all is well here. Even our bodies join the illusion and betray; for some days are fun days governed by "heyday" gods. But when we hurt, the body mutinies by translating all the pain to us and taking for itself none of the agony. It is we who bear all. So it turns out we are periodically disillusioned--we who are "connected by cells right down to our feet, and those feet likely to fell us over a tree root or jam us on a stone." And it is then that we are reminded that we don't really belong here.