SCRATCH SHEET

I’ve decided to publish here some of the many essays I’ve written on mathematical and philosophical subjects.  I realize that kind of stuff bores a lot of people (or, possibly, intimidates them–it often seems to me that so many people are afraid of math because they were taught to fear it–I think most people have a lot more ability at math than they give themselves credit for).  But, what the hey, this sort of thing occupies a large part of my thinking, and you might as well know about it.  You can always just skip them.

In fact, for a lot of them, that may be a good idea.  These essays are more or less the finished products.  The real fun has been on the scratch sheets, the rough and tumble of chasing down an idea and working it out, and the finished product is way more dense than the fun stuff.

And saying that gives me an excuse to copy in an old poem (written in a house backed by woods in Jasper, Texas).  I like to think the poem could apply equally well to mathematics or poetry itself.

THE EDGES, THE FRACTIONS, THE PIECES

When I used to work deferential equations

I had a neat sheet I kept track of it on

inking in ordered chains of tuneful logic

like dew collecting on a rusty screen

or sugar crumbling grain by grain to nectar

but when the next change stumped me as it did

more times than I’ll mention, out came Stubby

the friendly yellow # 2 chewed pencil

with round blunt lead and sweat-stained foreskin wood

long unwhittled in the sharpener’s whirling knives

and Eraser’s thin brass jacket bitten flat

to raise just one more day’s meniscus of correction

and out came Oscar, the mangy scratch-sheet

and all his haypile fat comedian friends

glad to see me as always and so we romp

we tussle in the briar-patch scratchy grass-fields

until a girl without an ounce fat on her

strides by in a pure white muscular gown

and I am dressed in wedding-white for church again

with an ink string tie the height of fashion

over the starchy ruffle of my beating chest

when I used to work D for any ol’ equations

THOUGHT

Thinking About Thought

Thought must consider thought itself.

Our consideration does not require an a priori definition of thought. In fact it is better if we do not have a definition to begin with. Useful definitions are usually arrived at as a result of well-managed consideration, and not in advance of the consideration. This is such a basic of good thinking that I am amazed so few take it into account. In particular, many recent discussions regarding consciousness, artificial intelligence, and the nature of reality appear to me to begin by presuming the fundamental character of the phenomenon they would ostensibly describe, and to proceed by means of argument, pro and con. This is not thinking, but the short-circuit of thinking.

The method of prior definition, argument, and counter-argument is the inevitable method of politics, and perhaps that is why it has become so prevalent in those fields that are supposed to value clear thinking: Perhaps it is because so much of our thinking has become politicized.

This method may show occasional partial successes, when the prior definition is a lucky hit, but it is an especially weak method for discovering the character of an uncharted phenomenon and one that is most probably polyvalent. Even simple logic, that limited hand-child of thought, teaches us that the disproof of one contention is not the proof of its opposite.

So we see that we may think freely about thought without first considering its ultimate character, insisting on its sources, or creating a pre-emptive inventory of its uses and productions.

When we speak of thought, we know on a fundamental level what activity we are speaking of. All we are seeking to do in thinking about thinking is deliver that fundamental awareness into the realm of thought itself. We do not have to raise linguistic questions, or questions as to whether computers are “thinking” in the same sense we are, or questions as to whether thought itself is merely an illusory by-product of the brain’s “software.”

All we really have to do is observe thought, and see what it does, and see how it moves, and see what it spends itself on. If we do that well enough, and long enough, I am sure that we will understand thought more deeply.

And here I see that I do entertain at least one prior assumption about thought. I assume that the purpose of thought is understanding. Understanding, not the display of expertise. Understanding, not statistics. Understanding, not the catalogue of a ramifying hierarchy of facts.

Why do humans seek to understand? Some might suppose it is to gain mastery. I suppose it is to create harmony. At the deepest level, perhaps harmony and mastery are the same. But if so, it is at a level that only harmony can reach, that mastery alone cannot touch. But what sort of harmony?

Many people appear to feel that the function of thought is to lift us out of the physical realm into the reaches of “higher” existence—the ideal, the abstract, the spiritual. I am beginning to feel that its function is actually to apprehend those “higher” modes of being, and return them to the body.

Why do we seek harmony? Well, what else is there to do?

You may well disagree with my assumption that the purpose of thought is understanding. I mention it here in an attempt to be as clear and honest as possible. I don’t intend to argue the assumption, because I do not wish it to function as one of those prior definitions. I wish this essay to be not a persuasion, but an invitation to observe. So you may safely trot along beside me for a while and see if you think any genuine observation is occurring.

 

The Mathematical Model of Thought

It is currently impossible to speak of thought itself without reference to mathematics. I love mathematics, and have had some degree of training in the subject. Mathematics is a valuable tool of thought. But I do not feel that it is identical to thought, or that it is the essence of thought.

Thought is prior to mathematics, and is the more fundamental behavior. Thought created mathematics. Mathematics did not create thought.

This would seem to be a perfectly evident statement. Yet in almost all contemporary fields of thought or research, the mathematical model predominates, even in those fields for which it is ill-suited.

We have the spectacle of physicists and mathematicians writing books presuming to explain, in more or less mathematical terms, such matters as the origin of the universe or the nature of consciousness, when we have no assurance, in either case, that these matters are mathematically treatable.

(A proviso: I am aware that when cosmologists speak of the “beginning” of the universe, in a first Planck interval and Planck distance, they are speaking of a description that seems to fit most of the facts and that is mathematically consistent. In fact, I accept their model (provisionally). Nevertheless, we have no assurance that the beginning they model is the “actual” beginning. How could we know, unless we were there to compare it to our model? Please note further that I do not maintain these questions will never be mathematically treatable—that would be as great a presumption. I am simply saying that we have as yet little indication.)

We have the further spectacle of thinkers in fields which are resolutely a-mathematical imitating the mathematical model—as when a French literary philosopher misconstrues the meaning of i, the square root of negative one, as some sort of universal phallic quality; as when the painter speaks of the uncertainty principle; as when the poet writes “experimental” work.

By “mathematics” I mean to include, for the time being, all of those deeply mathematized branches of science which give us so much conceptual trouble. Has no one ever observed that it is precisely in the mathematized branches of science that we do begin to have conceptual troubles?

Developments in biology in recent years have been as amazing and far-ranging in their effects as those in physics, and actually now promise (or threaten) to bring greater transformations into our lives. And yet, although developments in biology have certainly upset many of our earlier and long-held notions, they have never posed us the paradoxes regarding the fundamental nature of reality that developments in physics have posed.

I am sure that many would assume this is because physics is the more “fundamental” science, the science that pushes closer to the limits of the knowable. We typically create a neat little hierarchy running from physics at the extreme through chemistry then into biology, treating each successive discipline as “fuzzier” and therefore less fundamental. (The categories themselves get fuzzier after biology—which science comes next? Anthropology? Ecology? Where does psychology fit in, how far down the line?)

But is this hierarchy of fundamentalism in fact an accurate description? If it is, how do we know it is? Mightn’t we just as easily assume that the arrangement describes no hierarchy at all? That it describes nothing more than the relative ease with which mathematics may be applied to the given disciplines? That it is, in short, a mirror of our own assumptions?

Why do we assume that the paradoxes of physics are the paradoxes of nature? Is it not possible that they are the paradoxes of mathematics? Is it not possible that when we come up against Heisenberg’s uncertainty principle or against the four-slit photon experiment, we are simply meeting Goedel’s Unprovability Theorem in yet another guise?

Please observe that I am very carefully not saying two things: (1) I am not saying that the universe is ultimately knowable, and (2) I am not saying that only our stupid mathematics is getting in the way of knowing it.

Mathematics is a powerful tool for understanding, and we would be fools to abandon it. On the other hand, I at least would be deeply surprised if the universe (or multiverse) were fully knowable by any of its denizens.

What I am saying is that mathematics is only a method of thought, and has limits which thought itself does not have. (Thought has its own different and I think somehow larger limits—but I will discuss that in another place.)

In our age, because of its manifold successes, we have forgotten the limits of mathematics. That is, although we are all aware of Goedel’s Theorem, we treat it as a special result, of interest only to mathematicians, and almost universally fail to draw the implication that, in our study of reality, mathematics can never be a completely adequate tool.

There are many subjects worthy of genuine and rigorous thought which however may not be treated effectively by mathematics. In our culture these subjects have become marginalized, and assigned to the weaker thinkers. They are frequently treated as having no real existence whatsoever.

An example is the question of the existence of divinity. The fact that we cannot run an experiment on God or gods does not mean that we cannot think clearly and powerfully and usefully on the subject. It merely means that we cannot mathematize the subject. But because most strong thinkers today can only think mathematically, the question is usually dismissed out of hand.

Or worse, the question is assigned to theologians to chew over, so that it may become hopelessly confused, buried under a debris of ill-begotten concepts, and eventually completely useless to most humans.

Wait a minute, some might say: We have already had powerful thinkers going over the question of divinity. It happened in the late European middle ages, just prior to (and probably instrumental in) the various renaissances. And look at all the nonsense and folly that came out of that: angels dancing on the head of a pin, the perfection proof of God’s existence.

Not so, I would say. You’re looking at it the wrong way round. Your example in fact supports the point I am making.

What happened in the middle ages was that scholars discovered classical logic, and promptly went around applying it to every theological question they could come up with. Of course the result was absurdity, because those questions were not amenable to the application of logic in the first place.

Mathematics, as has been adequately demonstrated for many years now, may be understood as logic. The operations of mathematics are the operations of logic. They are the operations of logic on a selected and carefully defined set of concepts, but they are the operations of logic.

It is easy to see in retrospect that what the medieval theologians were doing was foolish and wrongheaded. Perhaps they were simply intoxicated with the beauty and potency of this new, to them, method of thought. It was such a powerful method, perhaps they thought it was powerful enough to do anything.

What is harder to see is that when we today dismiss such questions from all rigorous thought because they are not amenable to mathematics, we are committing a very similar, if reciprocal, error.

The question of the existence of divinity is a fundamental question, a very important question. It matters intrinsically to the life of every creature. We may never be able to resolve the question. But we are surrendering far too much territory to thoughtlessness if we do not consider it.

I believe it is possible to think productively and powerfully on such questions, even if we may not think entirely mathematically. I believe we may think with clarity, honesty, and precision, corrected but not ruled by logic. We may at least get rid of nonsense. And while we may not be able ever to settle whether there is a divinity, we could probably come to some reasonable concurrence about what the nature of that divinity might be if it did in fact exist.

We could create, to put it another way, a theoretical model of heaven.

“Not as a god,” Wallace Stevens says, “but as a god might be.”

The question of the existence of divinity is only one of many questions which are important but about which we no longer allow ourselves to do any serious thinking. We do not allow ourselves to do so because it is patent that we cannot yet think about these questions mathematically.

We have, then, a situation in which we either address extremely important questions in a spuriously mathematical manner, or do not address them at all. Surely this is not acceptable. Surely we can find a potency and efficacy in thought that does not depend entirely on mathematics.

 

The Limits of Mathematics

In order to discuss a sort of thinking which does not depend entirely on the mathematical model, it will be useful to first describe the limits of mathematical thought. Mathematicians and logicians who are familiar with Goedel’s Theorem, the Liar’s Paradox, or Russell’s set-of-all-sets paradox may already have a very good understanding of these limits.

I would like to present a more generally accessible description.

No system can fully comprehend the system of which it is a part—in chaos theory, this means sensitive dependence on initial conditions.

Descriptions of reality are systems. They may be thought of as emergent phenomena, and therefore by any approach they are parts of larger systems which themselves are parts of reality.

In human terms, no mind can understand itself fully. In cosmological terms, no model we create can approximate more than a portion of reality. In logical terms, no system can be both consistent and complete: paradox.

Quantum zero.

All logic and all mathematics, to put it another way, are discontinuous at the singularity of paradox. If we have no taste for paradox, we shall be unhappy creatures, for our existence is not only beyond mathematics, but beyond imagination.

 

Analysis and Synthesis

To think most powerfully and fruitfully, you must think more fundamentally. The most powerful tool of thought is not analysis, which after all is always only logic, but the ability to make connections.

But not all connections are equal. Some are indeed rich and deep, at once evocative and explanatory. Others are merely amusing, although there is frequently genius and passion in the merely amusing. Others are frivolous, a waste of time, like the artificial categories of information on Jeopardy. Still others are perverse, persisted in for the wrong reasons and against all harmony.

If connection is the most powerful tool of thought, how do we select, as individuals and then as tribes, and—one hopes—eventually as humans, those connections which are the most valuable?

We assume an identity between valuable connections and fundamental connections. Anyone may argue this identity does not exist, but we are free to posit it, use it as a working hypothesis, and see whether our results encourage us to continue with the hypothesis or abandon it.

The question then becomes: How do we identify the more fundamental connections? The answer is very simple: root and branch. How many roots (how many antecedents, and antecedents of antecedents) feed into the connection? And how many precedents and precedents of precedents radiate from it? And one further question: How fundamental are the rootlets and the branchlets themselves?

This last question of course brings the task of evaluating a connection into a sort of unprovable circularity. But we are not after proof, remember. We are after better and more useful thought. And we find ourselves in a world in which our attempts to estimate value must always be somewhat circular and self-referential. Only the self needs value. It needs it because it conceives of itself as distinct from everything else. Rankings arise with identities.

There is an interesting question regarding the way in which values arise in any thinking system. On the internet, there is something called a “driver” which powers search engines. The way this driver works, apparently, is to keep count of the most frequently used sites or clues and to strengthen those connections in proportion to the frequency with which they occur.

Such a system is probably a good model for how the human nervous system (or any other intelligence) has evolved. Such a system is also notably self-referential or fed back, and feedback, we have learned, is the hallmark of chaotic systems—in economics, for example, the theory of increasing returns.

 

Thought and Wellbeing

The essence of strong thought is not in tricks of memory, or in the ability to manipulate increasingly high-order abstractions. These in fact are what computers are so far capable of doing, and all they are capable of doing, and they are not the primary strengths of the human mind, though there are certainly sports out there with fantastic memories, as well as English professors who can deconstruct mathematics and mathematicians who can prove poetry doesn’t exist.

In the same way, though we admire LeBron James or Clayton Kershaw very greatly for their gifts, and for the perfection they have brought to those gifts, we do not mistake for overall good health the ability to jump vertically four feet in the air from a standing stop, or the ability to fling a baseball at a hundred miles an hour at a dime-sized target 60 feet 6 inches away.

Good health is the vigor of a body, any body, with reference to the whole of that body and not its exceptional abilities. So: a strong mind is not so much a mind with unusual abilities in certain arenas as a healthy mind, a mind that uses its own potential to the maximum. Any non-damaged human mind is capable of remarkable feats of thought, given its maximum health.

We may train our bodies to health, barring accidents and illness, and we may likewise train our minds to think powerfully. It’s a rigorous training, unless we happen to be one of those born with the ability. There are such physical sports, born with a constitution of such innate well-being that ordinary living can hardly damage it. There are such mental sports.

I consider health of thought to be the first requirement of thinking. It is a discipline little understood as a discipline—that is, as a path—and as a result, there are few masters to teach us, though there are all sorts of partial masters who may train us to produce prodigal results in a narrow arena.

And what is health of thought?

It is characterized by flexibility, connectivity, clarity, and the willingness to recognize preconceptions and surrender them. Thought is a gift of the being, as nimbleness, coordination, efficiency, and freshness are the gifts of the body.

Objectivity is an ideal, not a fact. Honesty is the only science.

 

IN SEARCH OF A GOLDEN EGG

It occurred to me that one ought to be able to fit a circle to any parabola, thus creating a smooth two-dimensional shape more or less like a plane section along the long axis of an egg. In fact, the resulting graph is not merely smooth, but continuous, which makes it interesting, since the continuity derives from the seamless fusion of two functions.

I decided to work with the basic parabolic equation, y = x2, since all of the others are transformations, and similar principles would apply. Besides, the curve I was most interested in was parabolic at its small end and circular at its large, and I wanted the bottom of that curve to pass through (0, 0). (It’s possible to construct a closed curve circular at each end and parabolic on the sides by calculating a smaller and a larger circle for each parabola and replacing the bottom of the parabola with the bottom of the smaller circle.)

In order to make the fusion seamless, it would be necessary to join the two curves precisely where they had the same slope, or tangent—in other words, where the derivatives of the two functions were equal.

Since the absolute value of Dx for the function y = x2 is always rising but never reaches infinity, and since the absolute value of Dx for the general equation of a circle can only reach infinity when x = the radius of the implied circle, the fusion of the two curves must always occur before that point. In other words, no matter how great the absolute value of x, you can always close the loop of the parabola by fusing it to a circle.

It develops that a circle fused to a parabola at x and –x will have its center at (0, x2 + ½ ) and a radius of (x2 + ¼ )1/2 (absolute value). From this it’s obvious that the radius of the circle can never be equal to or greater than the height of the center above (0,0), which verifies the assertion that one can close a parabola of any size with a circle. The single exception occurs when x = 0. When x = 0, the parabola disappears completely, leaving only a circle of radius ½ centered at (0, ½ )—the bottom of the curve still passing through (0,0).

The greater the absolute value of x, the larger the circle surmounting the parabola, obviously enough, and the more elongated the egg shape. I had been thinking all along of rotating the curve on the y-axis to generate a more or less ovoid three-dimensional surface. Now I wondered what the most pleasing proportions for such an object would be—in other words, what two-dimensional closed curve would produce the most pleasant proportions when rotated.

I appealed to the Golden Mean, and decided to find what value of x would produce a curve whose major axis (along the y-axis) was approximately 1.61803399 times the length of the diameter of its surmounting circle (twice the radius)—in short, a curve approximately 1/.61803399 times as tall as it is wide at the “shoulders.”

It turns out that to produce the desired curve, x must = approximately 2.05817103, the radius of the circle = approximately 1/.61803399 + ½ (or 2.11803399), and the major axis = approximately 6.85410197.

One could argue whether those are indeed the most pleasing proportions, but regardless, I got a good title out of it for this discourse.   

PARAFIBS

The so-called Fibonacci Sequence is only one of an infinite family of sequences in which the ratio of the penultimate term to the ultimate term converges to the Golden Mean as the terms approach infinity; further, there is an infinity of similar families of sequences, each family converging to a unique ratio of penultimate to ultimate term.

Families of families, so to speak. Infinitely.

Further, there is a continuous function such that this result may be generalized for any combination of real numbers whatsoever.

The most interesting topic in Dan Brown’s The Da Vinci Code was the Fibonacci Sequence. The rule of the sequence is that each term in the sequence is the sum of the previous two terms. There is a primary sequence, familiar to almost everyone in numbers, which begins with the seed terms 1, 1 and proceeds to 2, 3, 5, 8, 13, 21, and so on, producing terms inexorably and endlessly.

The ratio xn/xn+1 converges upon the golden mean as the terms grow larger, where we take xn as the nth term and xn+1 as the next term after the nth term. (The exact value of the golden mean is (1/2) ((5)1/2 -1), and the decimal approximation of this value to eight places is .61803399.) The golden mean is the only numerical value which, when added to 1, forms its own reciprocal. We know of applications in architecture and biology.

I would think there are applications to time.

Since the ratio of any two successive terms in the sequence converges to a distinct value, one suspects a feedback system is at work, and that the golden mean may be considered an attractor. It is not the sort of attractor we are accustomed to, however, since the convergence requires a two-stage feedback.

The familiar or primary Fibonacci sequence is not the only sequence which converges to the Golden Mean. In fact any two numbers taken as seed terms (including the terms of the Lucas set, of course) will produce a sequence in which the ratio of xn to xn+1 converges to the golden mean. It does not matter whether the seed numbers are integral, rational, irrational, positive, or negative. In all cases the ratio of two successive terms converges to the golden mean as the absolute value of the terms grows larger. The number of terms added to produce the next term is what controls this convergence, not the numbers used as seeds.

Since in these instances 2 successive terms are added to produce the next term, it’s the number 2 which specifies the Golden Mean.

If you have a spreadsheet or a mathematics program, you may wish to test various Fibonacci sequences to satisfy yourself that no matter which two numbers are used as seeds, the ratio of xn to xn+1 converges to the Golden Mean.

The fact that the number 2 specifies, via Fibonacci-type sequences, the golden mean, raises the question whether other numbers might specify other values in a similar way. 3, for example. Or 4. Let us designate as a sum number a number which indicates how many successive terms of a sequence must be summed to produce the next term, and represent sum numbers by s. If we say that a sequence has a sum number of 3 (s = 3), we mean that 3 successive terms are summed to produce the next term. A sum number of 4 implies that 4 successive terms are summed to produce the next term, and so on. For every sum number s there is what I will call the primary sequence for that number, in which s successive values of 1 are summed to initiate the sequence—1, 1, 1, 1, 1, would be the seed numbers for the primary sequence in which s = 5, for example. Let us refer to all sequences generated by a given sum number s, including of course the primary sequence, as a family of sequences, S.

It develops, as one might expect from the behavior of the family of Fibonacci sequences (s = 2), that one may begin by summing any s numbers, and the ratio xn/xn+1 will converge (as the terms of the sequence grow larger) to a distinct value determined only by s.

For convenience let us refer to the limit of the ratio xn/xn+1 for any family of sequences S produced by a sum number s as rs.

There exists an infinite group of families of sequences, each family classifiable by the number of successive terms summed to produce the next term. I refer to the members of this general group as parafibs (para for “beyond” and “fib” for Fibonacci). All parafibs behave alike, including the Fibonacci sequence and its family, in that each family S converges to a ratio rs unique to that family.

Since every value of s correlates with one and only one value of rs, every positive sum number s implies one and only one other positive number, rs. Therefore s <=> rs, and we have a function, which we will designate as f(s). Is every rs implied by s irrational? It seems so, but I cannot think how to prove this result.

When the value of s is zero, no sequence can occur. We may designate this as the null sequence. To say s = 1 is to say that the sequence consists of the same number repeated endlessly and therefore that the ratio r1 = 1.

Can f(s) be made continuous—that is to say, is there a way to create fractional and irrational values for s which produce results consistent with the results of integral values?

The conceptual problem is how to define a portion of an operation. In this situation, it develops that there is one definition which suits our requirements. If we define s as v.w, where v is understood as the integral portion of s and .w is understood as the decimal portion, and we define a term xn in the sequence s as the sum of the (v -1) previous terms plus (.w) times the vth term, then we have a continuous function whose domain is between 0 and infinity.

It is notable that in this definition if 1 > rs > the golden mean, then rs = (j – 1)/j => that s = 1 + j/(j – 1)2 for all real values of j.

In the successful definition, if s < 1, the successive terms rapidly approach zero and rs = 1/s, so that rs approaches infinity as s approaches 0.

I’ve investigated other potential definitions, in particular one which reverses the method above, treating xn as the sum of (.w)(xn-1) plus the (v – 1) terms previous (xn-2, xn-3, . . . , xn-v) and one in which rv.w is defined as rv + (.w)(rv+1 – rv), and the results, while formally interesting, do not produce a continuous function over the interval.

We discover that when s is greater than 1, rs is less than 1 but greater than ½. In fact rs approaches ½ as s approaches positive infinity. I haven’t found any algebraic representation of f(s). Perhaps the function is transcendental.

Again, set up a spreadsheet or mathematical program to produce sequences with s-values of your choosing. See if you do not find, for example, that r2 is approximately equal to .54368901, and r12 to .50006113.

It develops that in any sequence s, 1/rs approaches 1+[xn-1+ . . . +xn-(s-1)]/xn as n approaches positive infinity. In the Fibonacci family, this formula reduces to the familiar reciprocal effect. (Since s= 2, xn-(s-1) = xn-1, and therefore 1/ rs = 1+xn-1/xn = xn/xn-1).

It’s interesting that any two successive terms (using forward slant bars to indicate absolute value) of any sequence from the Fibonacci family (xn, xn+1) are related by the equation /(xn)2 – ((xn-1)(xn+1))/ = c, in which equation c is a constant specific to that particular sequence. In the traditional sequence, for example, c = 1. In a sequence from the Fibonacci family whose seeds are 4 and 11, c = 61. (If one does not use the absolute value of (xn)2 – (xn-1)(xn+1), then c alternates between positive and negative values.) Similar results do not appear to apply to sequences whose sum number is not 2.

(Addendum:  It seems clear to me, some time later, that what I have designated S ought to be designated S-sub-s, but I cannot find an editing tool on this site to generate subscripts, and while I am on this site, the formatting tool from the general overhead bar does not appear.)

SMALL MATTERS

How much reading can you possibly have done if you consistently get the “it’s/its” distinction not merely wrong, but exactly backwards? The people who screw it up always use “its” where “it’s” is the right word, and “it’s” where “its” is the right word.

How is that even possible unless you’re deliberately clinging to the wrong notion? I’m not talking making your B in your boring old grammar class. I’m talking your observation and your deductive reasoning.

“Its” is the possessive form of the pronoun “it.” “It’s” is the contraction of “it is.” That’s it. That’s the rule. No exceptions.

I see the error not merely in online comments, but in articles by people who consider themselves professionals. To me it’s like not having your head in the game, it’s like not being able to hit a hanging curve.

Surely a professional ought to pay better attention?

The subject, supposedly, is what counts. The disastrously incorrect meme is that if the subject is important we shouldn’t nitpick. What does it matter if the writer can’t spell?

One way it matters is that“small” errors can have big implications to the reader, implications concerning the quality of your intelligence and attention.

Just tell me how it’s possible to “tow a line”?

Or “hoe a road”?

Well, I guess you could hoe a road, if it was a dirt road. But I have no idea why you would want to.

How is the reader supposed to be impressed if you can’t even be bothered to get the cliches right?

It’s not just written grammar. Even pronunciation makes a difference. Two examples, one from a movie, one from Shakespeare :

The Big Lebowski—how do you say it? Almost everybody says it with the stress on the second syllable of Lebowski: The Big LeBOWski.

But it isn’t your movie about winning your bowling award, an award called The Big LeBOWski. It’s about two Lebowskis, one of whom is your larger. In other words, it’s about The BIG Lebowski.

Every sentence is a tune. If you get the music wrong, even by so much as one note, you’re messing with the message.

In Shakespeare’s Hamlet, Hamlet says to Horatio at one point, “There’s more to Heaven and Earth than is found in your philosophy, Horatio.”

The line is frequently quoted.

But when it’s quoted, most people put the stress on the word “your,” as if Hamlet were scolding Horatio for having the wrong “philosophy on life”: “There’s more to Heaven and Earth than is dreamt of in YOUR philosophy, Horatio.”

I think this is because nowadays everyone is assumed to have a “philosopy on life,” which usually turns out to be pretty much identical to their neighbor’s philosophy on life, especially if the neighbor is a successful pro football coach.

And most of what passes for the individual’s “philosophy on life” is not philosophy at all, but attitude. The difference is that genuine philosophy requires thought.

Considering that the Elizabethans (of whom Shakespeare was one, and humanly vain though they certainly were) did not pretend that just any Joe Blow could attain the lofty heights of philosophical reasoning, it seems to me the stress should be on the second syllable of the word “philosophy,” not on the “your”: “There’s more to Heaven and Earth than is dreamt in your phiLOSophy, Horatio.”

That was a common way of referring to an entire discipline. It didn’t mean the discipline belonged to you. It meant instead that whether or not you were individually capable, the discipline belonged to the species: You had your science, your mathematics, your courtly love, your politics, your blacksmithing, your finance, your schoolteaching, your archery. Your philosophy.